Nothing
R/getDesignMeans.R
In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond
Defines functions
hedgesg
getDesignSlopeDiffMMRM
getDesignSlopeDiff
getDesignOneSlope
getDesignRepeatedANOVAContrast
getDesignRepeatedANOVA
getDesignANOVAContrast
getDesignTwoWayANOVA
getDesignANOVA
getDesignMeanDiffCarryover
getDesignMeanDiffMMRM
getDesignWilcoxon
getDesignMeanRatioXOEquiv
getDesignMeanDiffXOEquiv
getDesignMeanRatioEquiv
getDesignMeanDiffEquiv
getDesignPairedMeanRatioEquiv
getDesignPairedMeanDiffEquiv
getDesignMeanRatioXO
getDesignMeanDiffXO
getDesignMeanRatio
getDesignMeanDiff
getDesignPairedMeanRatio
getDesignPairedMeanDiff
getDesignOneMean
Documented in
getDesignANOVA
getDesignANOVAContrast
getDesignMeanDiff
getDesignMeanDiffCarryover
getDesignMeanDiffEquiv
getDesignMeanDiffMMRM
getDesignMeanDiffXO
getDesignMeanDiffXOEquiv
getDesignMeanRatio
getDesignMeanRatioEquiv
getDesignMeanRatioXO
getDesignMeanRatioXOEquiv
getDesignOneMean
getDesignOneSlope
getDesignPairedMeanDiff
getDesignPairedMeanDiffEquiv
getDesignPairedMeanRatio
getDesignPairedMeanRatioEquiv
getDesignRepeatedANOVA
getDesignRepeatedANOVAContrast
getDesignSlopeDiff
getDesignSlopeDiffMMRM
getDesignTwoWayANOVA
getDesignWilcoxon
hedgesg
#' @title Group Sequential Design for One-Sample Mean
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for one-sample mean.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanH0 The mean under the null hypothesis.
#' Defaults to 0.
#' @param mean The mean under the alternative hypothesis.
#' @param stDev The standard deviation.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designOneMean} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanH0}: The mean under the null hypothesis.
#'
#' - \code{mean}: The mean under the alternative hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMean}: The efficacy boundaries on the mean scale.
#'
#' - \code{futilityMean}: The futility boundaries on the mean scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignOneMean(
#' beta = 0.1, n = NA, meanH0 = 7, mean = 6, stDev = 2.5,
#' kMax = 5, alpha = 0.025, typeAlphaSpending = "sfOF",
#' typeBetaSpending = "sfP"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignOneMean(
#' beta = 0.1, n = NA, meanH0 = 7, mean = 6, stDev = 2.5,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignOneMean <- function(
beta = NA_real_,
n = NA_real_,
meanH0 = 0,
mean = 0.5,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
directionUpper = mean > meanH0
theta = ifelse(directionUpper, mean - meanH0, meanH0 - mean)
# variance for one sampling unit
v1 = stDev^2
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
power = pt(b, n-1, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMean = delta + meanH0
des$byStageResults$futilityMean = delta + meanH0
} else {
des$byStageResults$efficacyMean = -delta + meanH0
des$byStageResults$futilityMean = -delta + meanH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMean =
des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
des$byStageResults$futilityTheta + meanH0
} else {
des$byStageResults$efficacyMean =
-des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
-des$byStageResults$futilityTheta + meanH0
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
pt(b, n-1, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
power = pt(b, n-1, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMean = delta + meanH0
des$byStageResults$futilityMean = delta + meanH0
} else {
des$byStageResults$efficacyMean = -delta + meanH0
des$byStageResults$futilityMean = -delta + meanH0
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMean =
des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
des$byStageResults$futilityTheta + meanH0
} else {
des$byStageResults$efficacyMean =
-des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
-des$byStageResults$futilityTheta + meanH0
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanH0 = meanH0
des$overallResults$mean = mean
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designOneMean"
des
}
#' @title Group Sequential Design for Paired Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for paired mean
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedDiffH0 The paired difference under the null hypothesis.
#' Defaults to 0.
#' @param pairedDiff The paired difference under the alternative hypothesis.
#' @param stDev The standard deviation for paired difference.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanDiff} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedDiffH0}: The paired difference under the null
#' hypothesis.
#'
#' - \code{pairedDiff}: The paired difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for paired difference.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyPairedDiff}: The efficacy boundaries on the paired
#' difference scale.
#'
#' - \code{futilityPairedDiff}: The futility boundaries on the paired
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanDiff(
#' beta = 0.1, n = NA, pairedDiffH0 = 0, pairedDiff = -2, stDev = 5,
#' kMax = 5, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignPairedMeanDiff(
#' beta = 0.1, n = NA, pairedDiffH0 = 0, pairedDiff = -2, stDev = 5,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignPairedMeanDiff <- function(
beta = NA_real_,
n = NA_real_,
pairedDiffH0 = 0,
pairedDiff = 0.5,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
des = getDesignOneMean(
beta, n, pairedDiffH0, pairedDiff, stDev,
normalApproximation,
rounding, kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
nov = names(des$overallResults)
names(des$overallResults)[nov == "meanH0"] <- "pairedDiffH0"
names(des$overallResults)[nov == "mean"] <- "pairedDiff"
nby = names(des$byStageResults)
names(des$byStageResults)[nby == "efficacyMean"] <- "efficacyPairedDiff"
names(des$byStageResults)[nby == "futilityMean"] <- "futilityPairedDiff"
attr(des, "class") = "designPairedMeanDiff"
des
}
#' @title Group Sequential Design for Paired Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for paired mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedRatioH0 The paired ratio under the null hypothesis.
#' @param pairedRatio The paired ratio under the alternative
#' hypothesis.
#' @param CV The coefficient of variation for paired ratio.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanRatio} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedRatioH0}: The paired ratio under the null hypothesis.
#'
#' - \code{pairedRatio}: The paired ratio under the alternative
#' hypothesis.
#'
#' - \code{CV}: The coefficient of variation for paired ratio.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyPairedRatio}: The efficacy boundaries on the paired
#' ratio scale.
#'
#' - \code{futilityPairedRatio}: The futility boundaries on the paired
#' ratio scale.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanRatio(
#' beta = 0.1, n = NA, pairedRatio = 1.2, CV = 0.35,
#' kMax = 5, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignPairedMeanRatio(
#' beta = 0.1, n = NA, pairedRatio = 1.2, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanRatio <- function(
beta = NA_real_,
n = NA_real_,
pairedRatioH0 = 1,
pairedRatio = 1.2,
CV = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (pairedRatioH0 <= 0) {
stop("pairedRatioH0 must be positive")
}
if (pairedRatio <= 0) {
stop("pairedRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignPairedMeanDiff(
beta, n, pairedDiffH0 = log(pairedRatioH0),
pairedDiff = log(pairedRatio),
stDev = sqrt(log(1 + CV^2)),
normalApproximation,
rounding, kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$pairedRatioH0 = pairedRatioH0
des$overallResults$pairedRatio = pairedRatio
des$overallResults$CV = CV
des$overallResults$pairedDiffH0 = NULL
des$overallResults$pairedDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyPairedRatio =
exp(des$byStageResults$efficacyPairedDiff)
des$byStageResults$futilityPairedRatio =
exp(des$byStageResults$futilityPairedDiff)
des$byStageResults$efficacyPairedDiff = NULL
des$byStageResults$futilityPairedDiff = NULL
attr(des, "class") = "designPairedMeanRatio"
des
}
#' @title Group Sequential Design for Two-Sample Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference under the null hypothesis.
#' Defaults to 0.
#' @param meanDiff The mean difference under the alternative hypothesis.
#' @param stDev The standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiff} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiff(
#' beta = NA, n = 456, meanDiff = 9, stDev = 32,
#' kMax = 5, alpha = 0.025, typeAlphaSpending = "sfOF",
#' typeBetaSpending = "sfP"))
#'
#' # Example 2: sample size calculation for two-sample t-test
#' (design2 <- getDesignMeanDiff(
#' beta = 0.1, n = NA, meanDiff = 0.3, stDev = 1,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignMeanDiff <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# variance for one sampling unit
v1 = stDev^2/(r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiff"
des
}
#' @title Group Sequential Design for Two-Sample Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioH0 The mean ratio under the null hypothesis.
#' Defaults to 1.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation. The standard deviation on the
#' log scale is equal to \code{sqrt(log(1 + CV^2))}.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatio} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioH0}: The mean ratio under the null hypothesis.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyMeanRatio}: The efficacy boundaries on the mean
#' ratio scale.
#'
#' - \code{futilityMeanRatio}: The futility boundaries on the mean
#' ratio scale.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanRatio(
#' beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
#' alpha = 0.05, normalApproximation = FALSE))
#'
#' @export
getDesignMeanRatio <- function(
beta = NA_real_,
n = NA_real_,
meanRatioH0 = 1,
meanRatio = 1.25,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (meanRatioH0 <= 0) {
stop("meanRatioH0 must be positive")
}
if (meanRatio <= 0) {
stop("meanRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiff(
beta, n, meanDiffH0 = log(meanRatioH0),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$meanRatioH0 = meanRatioH0
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffH0 = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatio =
exp(des$byStageResults$efficacyMeanDiff)
des$byStageResults$futilityMeanRatio =
exp(des$byStageResults$futilityMeanDiff)
des$byStageResults$efficacyMeanDiff = NULL
des$byStageResults$futilityMeanDiff = NULL
attr(des, "class") = "designMeanRatio"
des
}
#' @title Group Sequential Design for Mean Difference in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' difference in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference under the null hypothesis.
#' Defaults to 0.
#' @param meanDiff The mean difference under the alternative hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffXO} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for within-subject random
#' error.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanDiffXO(
#' beta = 0.2, n = NA, meanDiff = 75, stDev = 150,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffXO <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# variance for one sampling unit
v1 = stDev^2/(2*r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiffXO"
des
}
#' @title Group Sequential Design for Mean Ratio in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' ratio in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioH0 The mean ratio under the null hypothesis.
#' Defaults to 1.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation. The standard deviation on the
#' log scale is equal to \code{sqrt(log(1 + CV^2))}.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioXO} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioH0}: The mean ratio under the null hypothesis.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyMeanRatio}: The efficacy boundaries on the mean
#' ratio scale.
#'
#' - \code{futilityMeanRatio}: The futility boundaries on the mean
#' ratio scale.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanRatioXO(
#' beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
#' alpha = 0.05, normalApproximation = FALSE))
#'
#' @export
getDesignMeanRatioXO <- function(
beta = NA_real_,
n = NA_real_,
meanRatioH0 = 1,
meanRatio = 1.25,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (meanRatioH0 <= 0) {
stop("meanRatioH0 must be positive")
}
if (meanRatio <= 0) {
stop("meanRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffXO(
beta, n, meanDiffH0 = log(meanRatioH0),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$meanRatioH0 = meanRatioH0
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffH0 = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatio =
exp(des$byStageResults$efficacyMeanDiff)
des$byStageResults$futilityMeanRatio =
exp(des$byStageResults$futilityMeanDiff)
des$byStageResults$efficacyMeanDiff = NULL
des$byStageResults$futilityMeanDiff = NULL
attr(des, "class") = "designMeanRatioXO"
des
}
#' @title Group Sequential Design for Equivalence in Paired Mean
#' Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' paired mean difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedDiffLower The lower equivalence limit of paired difference.
#' @param pairedDiffUpper The upper equivalence limit of paired difference.
#' @param pairedDiff The paired difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation for paired difference.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanDiffEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level under H0.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedDiffLower}: The lower equivalence limit of paired
#' difference.
#'
#' - \code{pairedDiffUpper}: The upper equivalence limit of paired
#' difference.
#'
#' - \code{pairedDiff}: The paired difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for paired difference.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyPairedDiffLower}: The efficacy boundaries on the
#' paired difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyPairedDiffUpper}: The efficacy boundaries on the
#' paired difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanDiffEquiv(
#' beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
#' pairedDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignPairedMeanDiffEquiv(
#' beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
#' pairedDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanDiffEquiv <- function(
beta = NA_real_,
n = NA_real_,
pairedDiffLower = NA_real_,
pairedDiffUpper = NA_real_,
pairedDiff = 0,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(pairedDiffLower)) {
stop("pairedDiffLower must be provided")
}
if (is.na(pairedDiffUpper)) {
stop("pairedDiffUpper must be provided")
}
if (pairedDiffLower >= pairedDiff) {
stop("pairedDiffLower must be less than pairedDiff")
}
if (pairedDiffUpper <= pairedDiff) {
stop("pairedDiffUpper must be greater than pairedDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
v1 = stDev^2
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-1)
ncpLower = (pairedDiff - pairedDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-1, ncpLower, lower.tail = FALSE)
ncpUpper = (pairedDiffUpper - pairedDiff)*sqrt(n/v1)
powerUpper = pt(b, n-1, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = pairedDiffLower,
thetaUpper = pairedDiffUpper, theta = pairedDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = pairedDiffLower,
thetaUpper = pairedDiffUpper, theta = pairedDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-1)
ncp = (pairedDiffUpper - pairedDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-1)*x*x, (n-1)/2, 1/2)*2*(n-1)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyPairedDiffLower = b*sqrt(v1/n) +
pairedDiffLower
des$byStageResults$efficacyPairedDiffUpper = -b*sqrt(v1/n) +
pairedDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyPairedDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyPairedDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$pairedDiffLower = pairedDiffLower
des$overallResults$pairedDiffUpper = pairedDiffUpper
des$overallResults$pairedDiff = pairedDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "pairedDiffLower",
"pairedDiffUpper", "pairedDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyPairedDiffLower",
"efficacyPairedDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"normalApproximation", "rounding")]
attr(des, "class") = "designPairedMeanDiffEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Paired Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' paired mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedRatioLower The lower equivalence limit of paired ratio.
#' @param pairedRatioUpper The upper equivalence limit of paired ratio.
#' @param pairedRatio The paired ratio under the alternative
#' hypothesis.
#' @param CV The coefficient of variation for paired ratio.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level under H0.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedRatioLower}: The lower equivalence limit of paired
#' ratio.
#'
#' - \code{pairedRatioUpper}: The upper equivalence limit of paired
#' ratio.
#'
#' - \code{pairedRatio}: The paired ratio under the alternative
#' hypothesis.
#'
#' - \code{CV}: The coefficient of variation for paired ratios.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative alpha attained under
#' H0.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyPairedRatioLower}: The efficacy boundaries on the
#' paired ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyPairedRatioUpper}: The efficacy boundaries on the
#' paired ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanRatioEquiv(
#' beta = 0.1, n = NA, pairedRatioLower = 0.8, pairedRatioUpper = 1.25,
#' pairedRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignPairedMeanRatioEquiv(
#' beta = 0.1, n = NA, pairedRatioLower = 0.8, pairedRatioUpper = 1.25,
#' pairedRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanRatioEquiv <- function(
beta = NA_real_,
n = NA_real_,
pairedRatioLower = NA_real_,
pairedRatioUpper = NA_real_,
pairedRatio = 1,
CV = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(pairedRatioLower)) {
stop("pairedRatioLower must be provided")
}
if (is.na(pairedRatioUpper)) {
stop("pairedRatioUpper must be provided")
}
if (pairedRatioLower <= 0) {
stop("pairedRatioLower must be positive")
}
if (pairedRatioLower >= pairedRatio) {
stop("pairedRatioLower must be less than pairedRatio")
}
if (pairedRatioUpper <= pairedRatio) {
stop("pairedRatioUpper must be greater than pairedRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignPairedMeanDiffEquiv(
beta, n, pairedDiffLower = log(pairedRatioLower),
pairedDiffUpper = log(pairedRatioUpper),
pairedDiff = log(pairedRatio),
stDev = sqrt(log(1 + CV^2)),
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$pairedRatioLower = pairedRatioLower
des$overallResults$pairedRatioUpper = pairedRatioUpper
des$overallResults$pairedRatio = pairedRatio
des$overallResults$CV = CV
des$overallResults$pairedDiffLower = NULL
des$overallResults$pairedDiffUpper = NULL
des$overallResults$pairedDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyPairedRatioLower =
exp(des$byStageResults$efficacyPairedDiffLower)
des$byStageResults$efficacyPairedRatioUpper =
exp(des$byStageResults$efficacyPairedDiffUpper)
des$byStageResults$efficacyPairedDiffLower = NULL
des$byStageResults$efficacyPairedDiffUpper = NULL
attr(des, "class") = "designPairedMeanRatioEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Two-Sample
#' Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' two-sample mean difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffLower The lower equivalence limit of mean difference.
#' @param meanDiffUpper The upper equivalence limit of mean difference.
#' @param meanDiff The mean difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffLower}: The lower equivalence limit of mean
#' difference.
#'
#' - \code{meanDiffUpper}: The upper equivalence limit of mean
#' difference.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanDiffLower}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanDiffUpper}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiffEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanDiffEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanDiffLower = NA_real_,
meanDiffUpper = NA_real_,
meanDiff = 0,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(meanDiffLower)) {
stop("meanDiffLower must be provided")
}
if (is.na(meanDiffUpper)) {
stop("meanDiffUpper must be provided")
}
if (meanDiffLower >= meanDiff) {
stop("meanDiffLower must be less than meanDiff")
}
if (meanDiffUpper <= meanDiff) {
stop("meanDiffUpper must be greater than meanDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
v1 = stDev^2/(r*(1-r))
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-2)
ncpLower = (meanDiff - meanDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-2, ncpLower, lower.tail = FALSE)
ncpUpper = (meanDiffUpper - meanDiff)*sqrt(n/v1)
powerUpper = pt(b, n-2, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-2)
ncp = (meanDiffUpper - meanDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-2)*x*x, (n-2)/2, 1/2)*2*(n-2)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyMeanDiffLower = b*sqrt(v1/n) +
meanDiffLower
des$byStageResults$efficacyMeanDiffUpper = -b*sqrt(v1/n) +
meanDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyMeanDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyMeanDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffLower = meanDiffLower
des$overallResults$meanDiffUpper = meanDiffUpper
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "meanDiffLower",
"meanDiffUpper", "meanDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyMeanDiffLower",
"efficacyMeanDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"allocationRatioPlanned",
"normalApproximation", "rounding")]
attr(des, "class") = "designMeanDiffEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Two-Sample
#' Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' two-sample mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioLower The lower equivalence limit of mean ratio.
#' @param meanRatioUpper The upper equivalence limit of mean ratio.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioLower}: The lower equivalence limit of mean ratio.
#'
#' - \code{meanRatioUpper}: The upper equivalence limit of mean ratio.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyMeanRatioLower}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanRatioUpper}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanRatioEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanRatioEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanRatioEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanRatioLower = NA_real_,
meanRatioUpper = NA_real_,
meanRatio = 1,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(meanRatioLower)) {
stop("meanRatioLower must be provided")
}
if (is.na(meanRatioUpper)) {
stop("meanRatioUpper must be provided")
}
if (meanRatioLower <= 0) {
stop("meanRatioLower must be positive")
}
if (meanRatioLower >= meanRatio) {
stop("meanRatioLower must be less than meanRatio")
}
if (meanRatioUpper <= meanRatio) {
stop("meanRatioUpper must be greater than meanRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffEquiv(
beta, n, meanDiffLower = log(meanRatioLower),
meanDiffUpper = log(meanRatioUpper),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$meanRatioLower = meanRatioLower
des$overallResults$meanRatioUpper = meanRatioUpper
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffLower = NULL
des$overallResults$meanDiffUpper = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatioLower =
exp(des$byStageResults$efficacyMeanDiffLower)
des$byStageResults$efficacyMeanRatioUpper =
exp(des$byStageResults$efficacyMeanDiffUpper)
des$byStageResults$efficacyMeanDiffLower = NULL
des$byStageResults$efficacyMeanDiffUpper = NULL
attr(des, "class") = "designMeanRatioEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Mean Difference
#' in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' mean difference in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffLower The lower equivalence limit of mean difference.
#' @param meanDiffUpper The upper equivalence limit of mean difference.
#' @param meanDiff The mean difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffXOEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffLower}: The lower equivalence limit of mean
#' difference.
#'
#' - \code{meanDiffUpper}: The upper equivalence limit of mean
#' difference.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for within-subject random
#' error.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanDiffLower}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanDiffUpper}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiffXOEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanDiffXOEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffXOEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanDiffLower = NA_real_,
meanDiffUpper = NA_real_,
meanDiff = 0,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(meanDiffLower)) {
stop("meanDiffLower must be provided")
}
if (is.na(meanDiffUpper)) {
stop("meanDiffUpper must be provided")
}
if (meanDiffLower >= meanDiff) {
stop("meanDiffLower must be less than meanDiff")
}
if (meanDiffUpper <= meanDiff) {
stop("meanDiffUpper must be greater than meanDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
v1 = stDev^2/(2*r*(1-r))
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-2)
ncpLower = (meanDiff - meanDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-2, ncpLower, lower.tail = FALSE)
ncpUpper = (meanDiffUpper - meanDiff)*sqrt(n/v1)
powerUpper = pt(b, n-2, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-2)
ncp = (meanDiffUpper - meanDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-2)*x*x, (n-2)/2, 1/2)*2*(n-2)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyMeanDiffLower = b*sqrt(v1/n) +
meanDiffLower
des$byStageResults$efficacyMeanDiffUpper = -b*sqrt(v1/n) +
meanDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyMeanDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyMeanDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffLower = meanDiffLower
des$overallResults$meanDiffUpper = meanDiffUpper
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "meanDiffLower",
"meanDiffUpper", "meanDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyMeanDiffLower",
"efficacyMeanDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"allocationRatioPlanned",
"normalApproximation", "rounding")]
attr(des, "class") = "designMeanDiffXOEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Mean Ratio
#' in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence
#' mean ratio in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioLower The lower equivalence limit of mean ratio.
#' @param meanRatioUpper The upper equivalence limit of mean ratio.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioLower}: The lower equivalence limit of mean ratio.
#'
#' - \code{meanRatioUpper}: The upper equivalence limit of mean ratio.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanRatioLower}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanRatioUpper}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanRatioXOEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanRatioXOEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanRatioXOEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanRatioLower = NA_real_,
meanRatioUpper = NA_real_,
meanRatio = 1,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(meanRatioLower)) {
stop("meanRatioLower must be provided")
}
if (is.na(meanRatioUpper)) {
stop("meanRatioUpper must be provided")
}
if (meanRatioLower <= 0) {
stop("meanRatioLower must be positive")
}
if (meanRatioLower >= meanRatio) {
stop("meanRatioLower must be less than meanRatio")
}
if (meanRatioUpper <= meanRatio) {
stop("meanRatioUpper must be greater than meanRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffXOEquiv(
beta, n, meanDiffLower = log(meanRatioLower),
meanDiffUpper = log(meanRatioUpper),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$meanRatioLower = meanRatioLower
des$overallResults$meanRatioUpper = meanRatioUpper
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffLower = NULL
des$overallResults$meanDiffUpper = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatioLower =
exp(des$byStageResults$efficacyMeanDiffLower)
des$byStageResults$efficacyMeanRatioUpper =
exp(des$byStageResults$efficacyMeanDiffUpper)
des$byStageResults$efficacyMeanDiffLower = NULL
des$byStageResults$efficacyMeanDiffUpper = NULL
attr(des, "class") = "designMeanRatioXOEquiv"
des
}
#' @title Group Sequential Design for Two-Sample Wilcoxon Test
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample
#' Wilcoxon test.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pLarger The probability that a randomly chosen sample from
#' the treatment group is larger than a randomly chosen sample from the
#' control group under the alternative hypothesis.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designWilcoxon} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping..
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pLarger}: The probability that a randomly chosen sample from
#' the treatment group is larger than a randomly chosen sample from the
#' control group under the alternative hypothesis.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyPLarger}: The efficacy boundaries on the proportion
#' of pairs of samples from the two treatment groups with the sample
#' from the treatment group greater than that from the control group.
#'
#' - \code{futilityPLarger}: The futility boundaries on the proportion
#' of pairs of samples from the two treatment groups with the sample
#' from the treatment group greater than that from the control group.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: fixed design
#' (design1 <- getDesignWilcoxon(
#' beta = 0.1, n = NA,
#' pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025))
#'
#' # Example 2: group sequential design
#' (design2 <- getDesignWilcoxon(
#' beta = 0.1, n = NA,
#' pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025,
#' kMax = 3, typeAlphaSpending = "sfOF"))
#'
#' @export
getDesignWilcoxon <- function(
beta = NA_real_,
n = NA_real_,
pLarger = 0.6,
allocationRatioPlanned = 1,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (pLarger <= 0 || pLarger >= 1) {
stop("pLarger must lie between 0 and 1")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = pLarger > 0.5
theta = ifelse(directionUpper, pLarger - 0.5, 0.5 - pLarger)
# variance for one sampling unit
v1 = 1/(12*r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
} else { # sample size calculation
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$pLarger = pLarger
if (directionUpper) {
des$byStageResults$efficacyPLarger =
des$byStageResults$efficacyTheta + 0.5
des$byStageResults$futilityPLarger =
des$byStageResults$futilityTheta + 0.5
} else {
des$byStageResults$efficacyPLarger =
-des$byStageResults$efficacyTheta + 0.5
des$byStageResults$futilityPLarger =
-des$byStageResults$futilityTheta + 0.5
}
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designWilcoxon"
des
}
#' @title Group Sequential Design for Two-Sample Mean Difference From the
#' MMRM Model
#' @description Obtains the power and sample size for two-sample
#' mean difference at the last time point from the mixed-model
#' for repeated measures (MMRM) model.
#'
#' @param beta The type II error.
#' @param meanDiffH0 The mean difference at the last time point
#' under the null hypothesis. Defaults to 0.
#' @param meanDiff The mean difference at the last time point
#' under the alternative hypothesis.
#' @param k The number of postbaseline time points.
#' @param t The postbaseline time points.
#' @param covar1 The covariance matrix for the repeated measures
#' given baseline for the active treatment group.
#' @param covar2 The covariance matrix for the repeated measures
#' given baseline for the control group. If missing, it will be
#' set equal to the covariance matrix for the active treatment group.
#' @inheritParams param_accrualTime
#' @inheritParams param_accrualIntensity
#' @inheritParams param_piecewiseSurvivalTime
#' @inheritParams param_gamma1
#' @inheritParams param_gamma2
#' @inheritParams param_accrualDuration
#' @inheritParams param_allocationRatioPlanned
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The
#' degrees of freedom for the t-distribution is the total effective
#' sample size minus 2. The exact calculation using the t distribution
#' is only implemented for the fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Defaults to
#' \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffMMRM} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{studyDuration}: The maximum study duration.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{expectedStudyDurationH1}: The expected study duration
#' under H1.
#'
#' - \code{expectedStudyDurationH0}: The expected study duration
#' under H0.
#'
#' - \code{accrualDuration}: The accrual duration.
#'
#' - \code{followupTime}: The follow-up time.
#'
#' - \code{fixedFollowup}: Whether a fixed follow-up design is used.
#'
#' - \code{meanDiffH0}: The mean difference under H0.
#'
#' - \code{meanDiff}: The mean difference under H1.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{analysisTime}: The average time since trial start.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: The allocation ratio for the active
#' treatment versus control.
#'
#' - \code{accrualTime}: A vector that specifies the starting time of
#' piecewise Poisson enrollment time intervals.
#'
#' - \code{accrualIntensity}: A vector of accrual intensities.
#' One for each accrual time interval.
#'
#' - \code{piecewiseSurvivalTime}: A vector that specifies the
#' starting time of piecewise exponential survival time intervals.
#'
#' - \code{gamma1}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the active treatment group.
#'
#' - \code{gamma2}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the control group.
#'
#' - \code{k}: The number of postbaseline time points.
#'
#' - \code{t}: The postbaseline time points.
#'
#' - \code{covar1}: The covariance matrix for the repeated measures
#' given baseline for the active treatment group.
#'
#' - \code{covar2}: The covariance matrix for the repeated measures
#' given baseline for the control group.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # function to generate the AR(1) correlation matrix
#' ar1_cor <- function(n, corr) {
#' exponent <- abs(matrix((1:n) - 1, n, n, byrow = TRUE) - ((1:n) - 1))
#' corr^exponent
#' }
#'
#' (design1 = getDesignMeanDiffMMRM(
#' beta = 0.2,
#' meanDiffH0 = 0,
#' meanDiff = 0.5,
#' k = 4,
#' t = c(1,2,3,4),
#' covar1 = ar1_cor(4, 0.7),
#' accrualIntensity = 10,
#' gamma1 = 0.02634013,
#' gamma2 = 0.02634013,
#' accrualDuration = NA,
#' allocationRatioPlanned = 1,
#' kMax = 3,
#' alpha = 0.025,
#' typeAlphaSpending = "sfOF"))
#'
#' @export
#'
getDesignMeanDiffMMRM <- function(
beta = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
k = 1,
t = NA_real_,
covar1 = diag(k),
covar2 = NA_real_,
accrualTime = 0,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0,
gamma1 = 0,
gamma2 = 0,
accrualDuration = NA_real_,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
nintervals = length(piecewiseSurvivalTime)
if (is.na(beta) && is.na(accrualDuration)) {
stop("beta and accrualDuration cannot be both missing")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (k <= 0 || k != round(k)) {
stop("k must be a positive integer")
}
if (any(is.na(t))) {
stop("t must be provided")
} else if (length(t) != k) {
stop("Invalid length for t")
} else if (any(t <= 0)) {
stop("Elements of t must be positive")
} else if (k > 1 && any(diff(t) <= 0)) {
stop("Elements of t must be increasing")
}
if (!all(eigen(covar1)$values > 0)) {
stop("covar1 must be positive definite")
}
if (any(is.na(covar2))) {
covar2 = covar1
} else if (!all(eigen(covar2)$values > 0)) {
stop("covar2 must be positive definite")
}
if (accrualTime[1] != 0) {
stop("accrualTime must start with 0")
}
if (length(accrualTime) > 1 && any(diff(accrualTime) <= 0)) {
stop("accrualTime should be increasing")
}
if (any(is.na(accrualIntensity))) {
stop("accrualIntensity must be provided")
}
if (length(accrualTime) != length(accrualIntensity)) {
stop("accrualTime and accrualIntensity must have the same length")
}
if (any(accrualIntensity < 0)) {
stop("accrualIntensity must be non-negative")
}
if (piecewiseSurvivalTime[1] != 0) {
stop("piecewiseSurvivalTime must start with 0")
}
if (nintervals > 1 && any(diff(piecewiseSurvivalTime) <= 0)) {
stop("piecewiseSurvivalTime should be increasing")
}
if (any(gamma1 < 0)) {
stop("gamma1 must be non-negative")
}
if (any(gamma2 < 0)) {
stop("gamma2 must be non-negative")
}
if (length(gamma1) != 1 && length(gamma1) != nintervals) {
stop("Invalid length for gamma1")
}
if (length(gamma2) != 1 && length(gamma2) != nintervals) {
stop("Invalid length for gamma2")
}
if (length(gamma1) == 1) {
gamma1 = rep(gamma1, nintervals)
}
if (length(gamma2) == 1) {
gamma2 = rep(gamma2, nintervals)
}
if (!is.na(accrualDuration) && accrualDuration <= 0) {
stop("accrualDuration must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# function to obtain the info for mean difference
f_info <- function(tau, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r) {
# total number of enrolled subjects at interim analysis
n = accrual(tau, accrualTime, accrualIntensity, accrualDuration)
# total number of enrolled subjects at each time point
ns = accrual(tau - t, accrualTime, accrualIntensity, accrualDuration)
# probability of not dropping out at each time point by treatment
q1 = ptpwexp(t, piecewiseSurvivalTime, gamma1, lower.tail = FALSE)
q2 = ptpwexp(t, piecewiseSurvivalTime, gamma2, lower.tail = FALSE)
# number of subjects remaining at each time point by treatment
m1 = r*ns*q1
m2 = (1-r)*ns*q2
# number of subjects dropping out at each time point by treatment
n1 = c(-diff(m1), m1[k])
n2 = c(-diff(m2), m2[k])
# information matrix in each treatment group
I1 = 0
I2 = 0
I = matrix(0, k, k)
for (j in 1:k) {
I[1:j, 1:j] = solve(covar1[1:j, 1:j])
I1 = I1 + n1[j]*I
I[1:j, 1:j] = solve(covar2[1:j, 1:j])
I2 = I2 + n2[j]*I
}
# variance for the treatment mean at the last time point by treatment
V1 = solve(I1)
V2 = solve(I2)
# information for treatment difference
IMax = 1/(V1[k,k] + V2[k,k])
# variance for treatment difference for one sampling unit
v1 = (V1[k,k] + V2[k,k])*n
# inflation factors for each treatment
phi1 = V1[k,k]/(covar1[k,k]/(r*n))
phi2 = V2[k,k]/(covar2[k,k]/((1-r)*n))
# information for treatment mean difference at the last time point
list(IMax = IMax, v1 = v1, phi1 = phi1, phi2 = phi2)
}
# power calculation
if (is.na(beta)) {
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
}
studyDuration = accrualDuration + t[k]
out = f_info(studyDuration, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r)
IMax = out$IMax
phi1 = out$phi1
phi2 = out$phi2
des = getDesign(
beta, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta + meanDiffH0
des$byStageResults$futilityMeanDiff = delta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff = -delta + meanDiffH0
des$byStageResults$futilityMeanDiff = -delta + meanDiffH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta + meanDiffH0
}
}
} else { # sample size calculation
des = getDesign(
beta, NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
IMax = des$overallResults$information
out = f_info(1 + t[k], k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, 1, r)
v1 = out$v1
phi1 = out$phi1
phi2 = out$phi2
n = IMax*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
# power for t-test
f = function(n, r, phi1, phi2, alpha, theta, v1) {
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
pt(b, nu, ncp, lower.tail = FALSE)
}
n = uniroot(function(n) {
f(n, r, phi1, phi2, alpha, theta, v1) - (1-beta)
}, c(0.5*n, 1.5*n))$root
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (is.na(accrualDuration)) {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else { # calculate accrualIntensity
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.001, 240))$root*accrualIntensity
}
# final maximum information
IMax = n/v1
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta + meanDiffH0
des$byStageResults$futilityMeanDiff = delta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff = -delta + meanDiffH0
des$byStageResults$futilityMeanDiff = -delta + meanDiffH0
}
} else {
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (is.na(accrualDuration)) {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else { # calculate accrualIntensity
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.001, 240))$root*accrualIntensity
}
# final maximum information
IMax = n/v1
des = getDesign(
NA, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta + meanDiffH0
}
}
}
# timing of interim analysis
studyDuration = accrualDuration + t[k]
information = IMax*informationRates
analysisTime = rep(0, kMax)
if (kMax > 1) {
for (i in 1:(kMax-1)) {
analysisTime[i] = uniroot(function(tau) {
out = f_info(tau, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r)
out$IMax - information[i]
}, c(t[k]+0.001, studyDuration))$root
}
}
analysisTime[kMax] = studyDuration
numberOfSubjects = accrual(analysisTime, accrualTime, accrualIntensity,
accrualDuration)
p = des$byStageResults$rejectPerStage +
des$byStageResults$futilityPerStage
pH0 = des$byStageResults$rejectPerStageH0 +
des$byStageResults$futilityPerStageH0
des$overallResults$numberOfSubjects = n
des$overallResults$studyDuration = studyDuration
des$overallResults$expectedNumberOfSubjectsH1 = sum(p*numberOfSubjects)
des$overallResults$expectedNumberOfSubjectsH0 = sum(pH0*numberOfSubjects)
des$overallResults$expectedStudyDurationH1 = sum(p*analysisTime)
des$overallResults$expectedStudyDurationH0 = sum(pH0*analysisTime)
des$overallResults$accrualDuration = accrualDuration
des$overallResults$followupTime = t[k]
des$overallResults$fixedFollowup = TRUE
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$byStageResults$numberOfSubjects = numberOfSubjects
des$byStageResults$analysisTime = analysisTime
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$accrualTime = accrualTime
des$settings$accrualIntensity = accrualIntensity
des$settings$piecewiseSurvivalTime = piecewiseSurvivalTime
des$settings$gamma1 = gamma1
des$settings$gamma2 = gamma2
des$settings$k = k
des$settings$t = t
des$settings$covar1 = covar1
des$settings$covar2 = covar2
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiffMMRM"
des
}
#' @title Power and Sample Size for Direct Treatment Effects in Crossover
#' Trials Accounting for Carryover Effects
#' @description Obtains the power and sample size for direct treatment
#' effects in crossover trials accounting for carryover effects.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference at the last time point
#' under the null hypothesis. Defaults to 0.
#' @param meanDiff The mean difference at the last time point
#' under the alternative hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param corr The intra-subject correlation due to subject random effect.
#' @param design The crossover design represented by a matrix with
#' rows indexing the sequences, columns indexing the periods, and
#' matrix entries indicating the treatments.
#' @param cumdrop The cumulative dropout rate over periods.
#' @param allocationRatioPlanned Allocation ratio for the sequences.
#' Defaults to equal randomization if not provided.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @details
#' The linear mixed-effects model to assess the direct treatment effect
#' in the presence of carryover treatment effect is given by
#' \deqn{y_{ijk} = \mu + \alpha_i + b_{ij} + \gamma_k + \tau_{d(i,k)}
#' + \lambda_{c(i,k-1)} + e_{ijk},}
#' \deqn{i=1,\ldots,n; j=1,\ldots,r_i; k = 1,\ldots,p; d,c = 1,\ldots,t,}
#' where \eqn{\mu} is the general mean, \eqn{\alpha_i} is the effect of
#' the \eqn{i}th treatment sequence, \eqn{b_{ij}} is the random effect
#' with variance \eqn{\sigma_b^2} for the \eqn{j}the subject of the
#' \eqn{i}th treatment sequence, \eqn{\gamma_k} is the period effect,
#' and \eqn{e_{ijk}} is the random error with variance \eqn{\sigma^2}
#' for the subject in period \eqn{k}. The direct effect of the treatment
#' administered in period \eqn{k} of sequence \eqn{i} is
#' \eqn{\tau_{d(i,k)}}, and \eqn{\lambda_{c(i,k-1)}} is the carryover
#' effect of the treatment administered in period \eqn{k-1} of sequence
#' \eqn{i}. The value of the carryover effect for the observed
#' response in the first period is \eqn{\lambda_{c(i,0)} = 0} since
#' there is no carryover effect in the first period. The intra-subject
#' correlation due to the subject random effect is
#' \deqn{\rho = \frac{\sigma_b^2}{\sigma_b^2 + sigma^2}.}
#' By constructing the design matrix \eqn{X} for the linear model with
#' a compound symmetry covariance matrix for the response vector of
#' a subject, we can obtain \deqn{Var(\hat{\beta}) = (X'V^{-1}X)^{-1}.}
#'
#' The covariance matrix for the direct treatment effects and the
#' carryover treatment effects can be extracted from the appropriate
#' sub-matrices. The covariance matrix for the direct treatment effects
#' without accounting for the carryover treatment effects can be obtained
#' by omitting the carryover effect terms from the model.
#'
#' The power and relative efficiency are for the direct treatment
#' effect comparing the first treatment to the last treatment
#' accounting for carryover effects.
#'
#' The degrees of freedom for the t-test can be calculated as the
#' total number of observations minus the number of subjects minus
#' \eqn{p-1} minus \eqn{2(t-1)} to account for the subject effect,
#' period effect, and direct and carryover treatment effects.
#'
#' @return An S3 class \code{designMeanDiffCarryover} object with the
#' following components:
#'
#' * \code{power}: The power to reject the null hypothesis.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{numberOfSubjects}: The maximum number of subjects.
#'
#' * \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' * \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' * \code{stDev}: The standard deviation for within-subject random error.
#'
#' * \code{corr}: The intra-subject correlation due to subject random effect.
#'
#' * \code{design}: The crossover design represented by a matrix with
#' rows indexing the sequences, columns indexing the periods, and
#' matrix entries indicating the treatments.
#'
#' * \code{nseq}: The number of sequences.
#'
#' * \code{nprd}: The number of periods.
#'
#' * \code{ntrt}: The number of treatments.
#'
#' * \code{cumdrop}: The cumulative dropout rate over periods.
#'
#' * \code{V_direct_only}: The covariance matrix for direct treatment
#' effects without accounting for carryover effects.
#'
#' * \code{V_direct_carry}: The covariance matrix for direct and
#' carryover treatment effects.
#'
#' * \code{v_direct_only}: The variance of direct treatment effects
#' without accounting for carryover effects.
#'
#' * \code{v_direct}: The variance of direct treatment effects
#' accounting for carryover effects.
#'
#' * \code{v_carry}: The variance of carryover treatment effects.
#'
#' * \code{releff_direct}: The relative efficiency of the design
#' for estimating direct treatment effects after accounting
#' for carryover effects with respect to that without
#' accounting for carryover effects. This is equal to
#' \code{v_direct_only/v_direct}.
#'
#' * \code{releff_carry}: The relative efficiency of the design
#' for estimating carryover effects. This is equal to
#' \code{v_direct_only/v_carry}.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the sequences.
#'
#' * \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Robert O. Kuehl. Design of Experiments: Statistical Principles of
#' Research Design and Analysis. Brooks/Cole: Pacific Grove, CA. 2000.
#'
#' @examples
#'
#' # Williams design for 4 treatments
#'
#' (design1 = getDesignMeanDiffCarryover(
#' beta = 0.2, n = NA,
#' meanDiff = 0.5, stDev = 1,
#' design = matrix(c(1, 4, 2, 3,
#' 2, 1, 3, 4,
#' 3, 2, 4, 1,
#' 4, 3, 1, 2),
#' 4, 4, byrow = TRUE),
#' alpha = 0.025))
#'
#' @export
#'
getDesignMeanDiffCarryover <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
corr = 0.5,
design = NA_real_,
cumdrop = NA_real_,
allocationRatioPlanned = NA_real_,
normalApproximation = FALSE,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between -1 and 1")
}
if (any(is.na(design))) {
stop("design must be provided")
}
nseq = nrow(design)
nprd = ncol(design)
ntrt = length(unique(c(design)))
if (any(design <= 0 | design != round(design)) ||
!all.equal(unique(c(design)), 1:ntrt)) {
stop(paste("Elements of design must be positive integers",
"ranging from 1 to", ntrt))
}
# number of model parameters consisting of the intercept, sequence effects,
# period effects, direct treatment effects, and carryover treatment effects
q = 1 + (nseq-1) + (nprd-1) + (ntrt-1) + (ntrt-1)
# start of direct treatment effect
l = 1 + (nseq-1) + (nprd-1) + 1
# end of direct treatment effect
m = 1 + (nseq-1) + (nprd-1) + (ntrt-1)
if (q > nseq*nprd) {
stop("The crossover design is overparameterized")
}
if (any(is.na(cumdrop))) {
cumdrop = rep(0, nseq)
}
if (any(cumdrop < 0)) {
stop("Elements of cumdrop must be nonnegative")
} else if (nprd > 1 && any(diff(cumdrop) < 0)) {
stop("Elements of cumdrop must be nondecreasing")
} else if (any(cumdrop >= 1)) {
stop("Elements of cumdrop must be less than 1")
}
# observed data pattern probabilities
p = diff(c(cumdrop, 1))
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, nrow(design))
}
if (length(allocationRatioPlanned) != nseq-1 &&
length(allocationRatioPlanned) != nseq) {
stop(paste("allocationRatioPlanned should have", nseq-1,
"or", nseq, "elements"))
}
if (length(allocationRatioPlanned) == nseq-1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
# treatment sequence randomization probabilities
r = allocationRatioPlanned/sum(allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# model design matrix with carryover effect
X = matrix(0, nseq*nprd, q)
X[,1] = 1
for (i in 1:nseq) {
for (j in 1:nprd) {
k = (i-1)*nprd + j
# sequence effects
if (i < nseq) {
X[k, 1 + i] = 1
}
# period effects
if (j < nprd) {
X[k, 1 + (nseq-1) + j] = 1
}
# direct treatment effects
if (design[i,j] < ntrt) {
X[k, 1 + (nseq-1) + (nprd-1) + design[i,j]] = 1
}
# carryover treatment effects
if (j > 1 && design[i,j-1] < ntrt) {
X[k, 1 + (nseq-1) + (nprd-1) + (ntrt-1) + design[i,j-1]] = 1
}
}
}
# compound symmetry covariance matrix for repeated measures
Sigma = stDev^2/(1-corr) * ((1-corr)*diag(nprd) + corr)
# information matrix for model parameters with carryover effects
I = 0
for (i in 1:nseq) {
offset = (i-1)*nprd
J = 0
for (j in 1:nprd) {
idx = offset + (1:j)
if (j==1) {
x = t(as.matrix(X[idx,]))
} else {
x = X[idx,]
}
J = J + p[j]*t(x) %*% solve(Sigma[1:j,1:j]) %*% x
}
I = I + r[i]*J
}
# covariance matrix for model parameters with carryover effects
V = solve(I)
# variance for direct treatment effect for one sampling unit
v1 = V[l,l]
# information matrix for model parameters without carryover effects
X0 = X[, 1:m]
I0 = 0
for (i in 1:nseq) {
offset = (i-1)*nprd
J = 0
for (j in 1:nprd) {
idx = offset+(1:j)
if (j==1) {
x = t(as.matrix(X0[idx,]))
} else {
x = X0[idx,]
}
J = J + p[j]*t(x) %*% solve(Sigma[1:j,1:j]) %*% x
}
I0 = I0 + r[i]*J
}
# covariance matrix for model parameters without carryover effects
V0 = solve(I0)
# power for t test
f = function(n) {
# residual degrees of freedom after accounting for the subject effects
mean_nprd = sum(p*(1:nprd))
nu = n*mean_nprd - 1 - (n-1) - (nprd-1) - (ntrt-1) - (ntrt-1)
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
pt(b, nu, ncp, lower.tail = FALSE)
}
if (is.na(n)) { # calculate the sample size
if (normalApproximation) {
n = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
} else {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, 1.5*n0))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (normalApproximation) {
power = pnorm(theta*sqrt(n/v1) - qnorm(1-alpha))
} else {
power = f(n)
}
rownames(design) = paste0("Seq", 1:nseq)
colnames(design) = paste0("Prd", 1:nprd)
des = list(
power = power, alpha = alpha, numberOfSubjects = n,
meanDiffH0 = meanDiffH0, meanDiff = meanDiff,
stDev = stDev, corr = corr, design = design,
nseq = nseq, nprd = nprd, ntrt = ntrt,
cumdrop = cumdrop,
V_direct_only = V0[l:m,l:m]/n,
V_direct_carry = V[l:q, l:q]/n,
v_direct_only = V0[l,l]/n,
v_direct = V[l,l]/n,
v_carry = V[m+1,m+1]/n,
releff_direct = V0[l,l]/V[l,l],
releff_carry = V0[l,l]/V[m+1,m+1],
allocationRatioPlanned = allocationRatioPlanned,
normalApproximation = normalApproximation,
rounding = rounding)
attr(des, "class") = "designMeanDiffCarryover"
des
}
#' @title Power and Sample Size for One-Way ANOVA
#' @description Obtains the power and sample size for one-way
#' analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The common standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the treatment
#' groups. It has length \code{ngroups - 1} or \code{ngroups}. If it is
#' of length \code{ngroups - 1}, then the last treatment group will
#' assume value 1 for allocation ratio.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @details
#'
#' Let \eqn{\{\mu_i: i=1,\ldots,k\}} denote the group means, and
#' \eqn{\{r_i: i=1,\ldots,k\}} denote the randomization probabilities
#' to the \eqn{k} treatment groups. Let \eqn{\sigma} denote the
#' common standard deviation, and \eqn{n} denote the total sample
#' size. Then the \eqn{F}-statistic
#' \deqn{F = \frac{SSR/(k-1)}{SSE/(n-k)}
#' \sim F_{k-1, n-k, \lambda},} where
#' \deqn{\lambda = n \sum_{i=1}^k r_i (\mu_i - \bar{\mu})^2/\sigma^2}
#' is the noncentrality parameter, and
#' \eqn{\bar{\mu} = \sum_{i=1}^k r_i \mu_i}.
#'
#' @return An S3 class \code{designANOVA} object with the following
#' components:
#'
#' * \code{power}: The power to reject the null hypothesis that
#' there is no difference among the treatment groups.
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the treatment
#' groups.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignANOVA(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 3.5, allocationRatioPlanned = c(2, 2, 2, 1),
#' alpha = 0.05))
#'
#' @export
#'
getDesignANOVA <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
allocationRatioPlanned = NA_real_,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (!length(means) == ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, ngroups)
}
if (length(allocationRatioPlanned) != ngroups - 1 &&
length(allocationRatioPlanned) != ngroups) {
stop(paste("allocationRatioPlanned should have", ngroups - 1,
"or", ngroups, "elements"))
}
if (length(allocationRatioPlanned) == ngroups - 1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
r = allocationRatioPlanned/sum(allocationRatioPlanned)
mubar = sum(r*means)
vmu = sum(r*(means - mubar)^2)
# power for F-test
f <- function(n) {
lambda = n*vmu/stDev^2
b = qf(1 - alpha, ngroups - 1, n - ngroups)
pf(b, ngroups - 1, n - ngroups, lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = ngroups - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/(vmu/stDev^2)
while (f(n0) < 1-beta) n0 <- 2*n0
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
effectsize = vmu/stDev^2,
allocationRatioPlanned = allocationRatioPlanned,
rounding = rounding)
attr(des, "class") = "designANOVA"
des
}
#' @title Power and Sample Size for Two-Way ANOVA
#' @description Obtains the power and sample size for two-way
#' analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param nlevelsA The number of groups for Factor A.
#' @param nlevelsB The number of levels for Factor B.
#' @param means The matrix of treatment means for Factors A and B
#' combination.
#' @param stDev The common standard deviation.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @return An S3 class \code{designTwoWayANOVA} object with the following
#' components:
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{nlevelsA}: The number of levels for Factor A.
#'
#' * \code{nlevelsB}: The number of levels for Factor B.
#'
#' * \code{means}: The matrix of treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{effectsizeA}: The effect size for Factor A.
#'
#' * \code{effectsizeB}: The effect size for Factor B.
#'
#' * \code{effectsizeAB}: The effect size for Factor A and Factor B
#' interaction.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' * \code{powerdf}: The data frame containing the power and sample size
#' results. It has the following variables:
#'
#' - \code{n}: The sample size.
#'
#' - \code{powerA}: The power to reject the null hypothesis that
#' there is no difference among Factor A levels.
#'
#' - \code{powerB}: The power to reject the null hypothesis that
#' there is no difference among Factor B levels.
#'
#' - \code{powerAB}: The power to reject the null hypothesis that
#' there is no interaction between Factor A and Factor B.
#'
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignTwoWayANOVA(
#' beta = 0.1, nlevelsA = 2, nlevelsB = 2,
#' means = matrix(c(0.5, 4.7, 0.4, 6.9), 2, 2, byrow = TRUE),
#' stDev = 2, alpha = 0.05))
#'
#' @export
#'
getDesignTwoWayANOVA <- function(
beta = NA_real_,
n = NA_real_,
nlevelsA = 2,
nlevelsB = 2,
means = NA_real_,
stDev = 1,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (nlevelsA < 2 || nlevelsA != round(nlevelsA)) {
stop("nlevelsA must be a positive integer >= 2")
}
if (nlevelsB < 2 || nlevelsB != round(nlevelsB)) {
stop("nlevelsB must be a positive integer >= 2")
}
if (any(is.na(means))) {
stop("means must be provided")
}
if (!is.matrix(means) || nrow(means) != nlevelsA ||
ncol(means) != nlevelsB) {
stop(paste("means must be a matrix with", nlevelsA, "rows and",
nlevelsB, "columns"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
muA = as.numeric(rowMeans(means))
muB = as.numeric(colMeans(means))
mu = mean(means)
vmuA = var(muA)*(nlevelsA - 1)/nlevelsA
vmuB = var(muB)*(nlevelsB - 1)/nlevelsB
mmuA = matrix(muA, nlevelsA, nlevelsB)
mmuB = matrix(muB, nlevelsA, nlevelsB, byrow = TRUE)
vmuAB = sum((means - mmuA - mmuB + mu)^2)/(nlevelsA*nlevelsB)
effectsizeA = vmuA/stDev^2
effectsizeB = vmuB/stDev^2
effectsizeAB = vmuAB/stDev^2
# power for F-test for Factor A
fA <- function(n) {
lambda = n*effectsizeA
b = qf(1-alpha, nlevelsA - 1, n - nlevelsA*nlevelsB)
pf(b, nlevelsA - 1, n - nlevelsA*nlevelsB, lambda, lower.tail = FALSE)
}
# power for F-test for Factor A
fB <- function(n) {
lambda = n*effectsizeB
b = qf(1-alpha, nlevelsB - 1, n - nlevelsA*nlevelsB)
pf(b, nlevelsB - 1, n - nlevelsA*nlevelsB, lambda, lower.tail = FALSE)
}
# power for F-test for Factor A and Factor B interaction
fAB <- function(n) {
lambda = n*effectsizeAB
b = qf(1-alpha, (nlevelsA - 1)*(nlevelsB - 1), n - nlevelsA*nlevelsB)
pf(b, (nlevelsA - 1)*(nlevelsB - 1), n - nlevelsA*nlevelsB,
lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = nlevelsA - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeA
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fA(n0) < 1-beta) n0 <- 2*n0
nA = uniroot(function(n) fA(n) - (1-beta), c(0.5*n0, n0))$root
nu = nlevelsB - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeB
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fB(n0) < 1-beta) n0 <- 2*n0
nB = uniroot(function(n) fB(n) - (1-beta), c(0.5*n0, n0))$root
nu = (nlevelsA - 1)*(nlevelsB - 1)
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeAB
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fAB(n0) < 1-beta) n0 <- 2*n0
nAB = uniroot(function(n) fAB(n) - (1-beta), c(0.5*n0, n0))$root
} else {
nA = n
nB = n
nAB = n
}
if (rounding) {
nA = ceiling(nA - 1.0e-12)
nB = ceiling(nB - 1.0e-12)
nAB = ceiling(nAB - 1.0e-12)
}
m = unique(c(nA, nB, nAB))
powerdf = data.frame(n = m, powerA = sapply(m, fA),
powerB = sapply(m, fB), powerAB = sapply(m, fAB))
des = list(
alpha = alpha, nlevelsA = nlevelsA,
nlevelsB = nlevelsB, means = means, stDev = stDev,
effectsizeA = effectsizeA, effectsizeB = effectsizeB,
effectsizeAB = effectsizeAB,
rounding = rounding,
powerdf = powerdf)
attr(des, "class") = "designTwoWayANOVA"
des
}
#' @title Power and Sample Size for One-Way ANOVA Contrast
#' @description Obtains the power and sample size for a single contrast
#' in one-way analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The common standard deviation.
#' @param contrast The coefficients for the single contrast.
#' @param meanContrastH0 The mean of the contrast under the
#' null hypothesis.
#' @param allocationRatioPlanned Allocation ratio for the treatment
#' groups. It has length \code{ngroups - 1} or \code{ngroups}. If it is
#' of length \code{ngroups - 1}, then the last treatment group will
#' assume value 1 for allocation ratio.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @return An S3 class \code{designANOVAContrast} object with the following
#' components:
#'
#' * \code{power}: The power to reject the null hypothesis for the
#' treatment contrast.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{contrast}: The coefficients for the single contrast.
#'
#' * \code{meanContrastH0}: The mean of the contrast under the null
#' hypothesis.
#'
#' * \code{meanContrast}: The mean of the contrast under the alternative
#' hypothesis.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the treatment
#' groups.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignANOVAContrast(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 3.5, contrast = c(1, 1, 1, -3)/3,
#' allocationRatioPlanned = c(2, 2, 2, 1),
#' alpha = 0.025))
#'
#' @export
#'
getDesignANOVAContrast <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
contrast = NA_real_,
meanContrastH0 = 0,
allocationRatioPlanned = NA_real_,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (length(means) != ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (any(is.na(contrast))) {
stop("contrast must be provided")
}
if (length(contrast) != ngroups) {
stop(paste("contrast must have", ngroups, "elements"))
}
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, ngroups)
}
if (length(allocationRatioPlanned) != ngroups - 1 &&
length(allocationRatioPlanned) != ngroups) {
stop(paste("allocationRatioPlanned should have", ngroups - 1,
"or", ngroups, "elements"))
}
if (length(allocationRatioPlanned) == ngroups - 1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
r = allocationRatioPlanned/sum(allocationRatioPlanned)
meanContrast = sum(contrast*means)
v1 = sum(contrast^2/r)*stDev^2
directionUpper = meanContrast > meanContrastH0
theta = ifelse(directionUpper, meanContrast - meanContrastH0,
meanContrastH0 - meanContrast)
# power for t-test
f <- function(n) {
b = qt(1-alpha, n-ngroups)
ncp = theta*sqrt(n/v1)
power = pt(b, n-ngroups, ncp, lower.tail = FALSE)
}
if (is.na(n)) {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(n0, 2*n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
contrast = contrast, meanContrastH0 = meanContrastH0,
meanContrast = meanContrast, effectsize = theta^2/v1,
allocationRatioPlanned = allocationRatioPlanned,
rounding = rounding)
attr(des, "class") = "designANOVAContrast"
des
}
#' @title Power and Sample Size for Repeated-Measures ANOVA
#' @description Obtains the power and sample size for one-way repeated
#' measures analysis of variance. Each subject takes all treatments
#' in the longitudinal study.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The total standard deviation.
#' @param corr The correlation among the repeated measures.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @details
#' Let \eqn{y_{ij}} denote the measurement under treatment condition
#' \eqn{j (j=1,\ldots,k)} for subject \eqn{i (i=1,\ldots,n)}. Then
#' \deqn{y_{ij} = \alpha + \beta_j + b_i + e_{ij},} where \eqn{b_i}
#' denotes the subject random effect, \eqn{b_i \sim N(0, \sigma_b^2),}
#' and \eqn{e_{ij} \sim N(0, \sigma_e^2)} denotes the within-subject
#' residual. If we set \eqn{\beta_k = 0}, then \eqn{\alpha} is the
#' mean of the last treatment (control), and \eqn{\beta_j} is the
#' difference in means between the \eqn{j}th treatment and the control
#' for \eqn{j=1,\ldots,k-1}.
#'
#' The repeated measures have a compound symmetry covariance structure.
#' Let \eqn{\sigma^2 = \sigma_b^2 + \sigma_e^2}, and
#' \eqn{\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma_e^2}}. Then
#' \eqn{Var(y_i) = \sigma^2 \{(1-\rho) I_k + \rho 1_k 1_k^T\}}.
#' Let \eqn{X_i} denote the design matrix for subject \eqn{i}.
#' Let \eqn{\theta = (\alpha, \beta_1, \ldots, \beta_{k-1})^T}.
#' It follows that
#' \deqn{Var(\hat{\theta}) = \left(\sum_{i=1}^{n} X_i^T V_i^{-1}
#' X_i\right)^{-1}.} It can be shown that
#' \deqn{Var(\hat{\beta}) = \frac{\sigma^2 (1-\rho)}{n} (I_{k-1} +
#' 1_{k-1} 1_{k-1}^T).} It follows that
#' \eqn{\hat{\beta}^T \hat{V}_{\hat{\beta}}^{-1} \hat{\beta} \sim
#' F_{k-1,(n-1)(k-1), \lambda},} where the noncentrality parameter for
#' the \eqn{F} distribution is \deqn{\lambda =
#' \beta^T V_{\hat{\beta}}^{-1} \beta = \frac{n \sum_{j=1}^{k}
#' (\mu_j - \bar{\mu})^2}{\sigma^2(1-\rho)}.}
#'
#' @return An S3 class \code{designRepeatedANOVA} object with the
#' following components:
#'
#' * \code{power}: The power to reject the null hypothesis that
#' there is no difference among the treatment groups.
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The total standard deviation.
#'
#' * \code{corr}: The correlation among the repeated measures.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignRepeatedANOVA(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 5, corr = 0.2, alpha = 0.05))
#'
#' @export
#'
getDesignRepeatedANOVA <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
corr = 0,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (!length(means) == ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between 0 and 1")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
vmu = var(means)*(ngroups-1)/ngroups
# power for F-test
f <- function(n) {
lambda = n*ngroups*vmu/(stDev^2*(1-corr))
b = qf(1 - alpha, ngroups - 1, (n-1)*(ngroups-1))
pf(b, ngroups - 1, (n-1)*(ngroups-1), lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = ngroups - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/
(ngroups*vmu/(stDev^2*(1-corr)))
while (f(n0) < 1-beta) n0 <- 2*n0
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
corr = corr, effectsize = ngroups*vmu/(stDev^2*(1-corr)),
rounding = rounding)
attr(des, "class") = "designRepeatedANOVA"
des
}
#' @title Power and Sample Size for One-Way Repeated Measures ANOVA Contrast
#' @description Obtains the power and sample size for a single contrast
#' in one-way repeated measures analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The total standard deviation.
#' @param corr The correlation among the repeated measures.
#' @param contrast The coefficients for the single contrast.
#' @param meanContrastH0 The mean of the contrast under the
#' null hypothesis.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @return An S3 class \code{designRepeatedANOVAContrast} object with
#' the following components:
#'
#' * \code{power}: The power to reject the null hypothesis for the
#' treatment contrast.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The total standard deviation.
#'
#' * \code{corr}: The correlation among the repeated measures.
#'
#' * \code{contrast}: The coefficients for the single contrast.
#'
#' * \code{meanContrastH0}: The mean of the contrast under the null
#' hypothesis.
#'
#' * \code{meanContrast}: The mean of the contrast under the alternative
#' hypothesis.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignRepeatedANOVAContrast(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 5, corr = 0.2, contrast = c(1, 1, 1, -3)/3,
#' alpha = 0.025))
#'
#' @export
#'
getDesignRepeatedANOVAContrast <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
corr = 0,
contrast = NA_real_,
meanContrastH0 = 0,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (length(means) != ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between -1 and 1")
}
if (any(is.na(contrast))) {
stop("contrast must be provided")
}
if (length(contrast) != ngroups) {
stop(paste("contrast must have", ngroups, "elements"))
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
meanContrast = sum(contrast*means)
v1 = sum(contrast^2)*stDev^2*(1-corr)
directionUpper = meanContrast > meanContrastH0
theta = ifelse(directionUpper, meanContrast - meanContrastH0,
meanContrastH0 - meanContrast)
# power for t-test
f <- function(n) {
b = qt(1-alpha, (n-1)*(ngroups-1))
ncp = theta*sqrt(n/v1)
power = pt(b, (n-1)*(ngroups-1), ncp, lower.tail = FALSE)
}
if (is.na(n)) {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(n0, 2*n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev, corr = corr,
contrast = contrast, meanContrastH0 = meanContrastH0,
meanContrast = meanContrast, effectsize = theta^2/v1,
rounding = rounding)
attr(des, "class") = "designRepeatedANOVAContrast"
des
}
#' @title Group Sequential Design for One-Sample Slope
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for one-sample slope.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param slopeH0 The slope under the null hypothesis.
#' Defaults to 0.
#' @param slope The slope under the alternative hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevCovariate The standard deviation of the covariate.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designOneSlope} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{slopeH0}: The slope under the null hypothesis.
#'
#' - \code{slope}: The slope under the alternative hypothesis.
#'
#' - \code{stDev}: The standard deviation of the residual.
#'
#' - \code{stDevCovariate}: The standard deviation of the covariate.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlope}: The efficacy boundaries on the slope scale.
#'
#' - \code{futilitySlope}: The futility boundaries on the slope scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignOneSlope(
#' beta = 0.1, n = NA, slope = 0.5,
#' stDev = 15, stDevCovariate = 9,
#' normalApproximation = FALSE,
#' alpha = 0.025))
#'
#' @export
getDesignOneSlope <- function(
beta = NA_real_,
n = NA_real_,
slopeH0 = 0,
slope = 0.5,
stDev = 1,
stDevCovariate = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (stDevCovariate <= 0) {
stop("stDevCovariate must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
directionUpper = slope > slopeH0
theta = ifelse(directionUpper, slope - slopeH0, slopeH0 - slope)
# variance for one sampling unit
v1 = stDev^2/stDevCovariate^2
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlope = delta + slopeH0
des$byStageResults$futilitySlope = delta + slopeH0
} else {
des$byStageResults$efficacySlope = -delta + slopeH0
des$byStageResults$futilitySlope = -delta + slopeH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlope =
des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
des$byStageResults$futilityTheta + slopeH0
} else {
des$byStageResults$efficacySlope =
-des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
-des$byStageResults$futilityTheta + slopeH0
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlope = delta + slopeH0
des$byStageResults$futilitySlope = delta + slopeH0
} else {
des$byStageResults$efficacySlope = -delta + slopeH0
des$byStageResults$futilitySlope = -delta + slopeH0
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlope =
des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
des$byStageResults$futilityTheta + slopeH0
} else {
des$byStageResults$efficacySlope =
-des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
-des$byStageResults$futilityTheta + slopeH0
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$slopeH0 = slopeH0
des$overallResults$slope = slope
des$overallResults$stDev = stDev
des$overallResults$stDevCovariate = stDevCovariate
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designOneSlope"
des
}
#' @title Group Sequential Design for Two-Sample Slope Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample slope
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param slopeDiffH0 The slope difference under the null hypothesis.
#' Defaults to 0.
#' @param slopeDiff The slope difference under the alternative hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevCovariate The standard deviation of the covariate.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designSlopeDiff} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{slopeDiffH0}: The slope difference under the null hypothesis.
#'
#' - \code{slopeDiff}: The slope difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation of the residual.
#'
#' - \code{stDevCovariate}: The standard deviation of the covariate.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlopeDiff}: The efficacy boundaries on the slope
#' difference scale.
#'
#' - \code{futilitySlopeDiff}: The futility boundaries on the slope
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignSlopeDiff(
#' beta = 0.1, n = NA, slopeDiff = -0.5,
#' stDev = 10, stDevCovariate = 6,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignSlopeDiff <- function(
beta = NA_real_,
n = NA_real_,
slopeDiffH0 = 0,
slopeDiff = 0.5,
stDev = 1,
stDevCovariate = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = slopeDiff > slopeDiffH0
theta = ifelse(directionUpper, slopeDiff - slopeDiffH0,
slopeDiffH0 - slopeDiff)
# variance for one sampling unit
v1 = stDev^2/stDevCovariate^2/(r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
power = pt(b, n-4, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta
des$byStageResults$futilitySlopeDiff = delta
} else {
des$byStageResults$efficacySlopeDiff = -delta
des$byStageResults$futilitySlopeDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
pt(b, n-4, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
power = pt(b, n-4, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta
des$byStageResults$futilitySlopeDiff = delta
} else {
des$byStageResults$efficacySlopeDiff = -delta
des$byStageResults$futilitySlopeDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$slopeDiffH0 = slopeDiffH0
des$overallResults$slopeDiff = slopeDiff
des$overallResults$stDev = stDev
des$overallResults$stDevCovariate = stDevCovariate
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designSlopeDiff"
des
}
#' @title Group Sequential Design for Two-Sample Slope Difference
#' From the MMRM Model
#' @description Obtains the power given sample size or obtains the sample
#' size given power for two-sample slope difference from the growth curve
#' MMRM model.
#'
#' @param beta The type II error.
#' @param slopeDiffH0 The slope difference under the null hypothesis.
#' Defaults to 0.
#' @param slopeDiff The slope difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevIntercept The standard deviation of the random intercept.
#' @param stDevSlope The standard deviation of the random slope.
#' @param corrInterceptSlope The correlation between the random
#' intercept and random slope.
#' @param w The number of time units per measurement visit in a period.
#' @param N The number of measurement visits in a period.
#' @inheritParams param_accrualTime
#' @inheritParams param_accrualIntensity
#' @inheritParams param_piecewiseSurvivalTime
#' @inheritParams param_gamma1
#' @inheritParams param_gamma2
#' @inheritParams param_accrualDuration
#' @inheritParams param_followupTime
#' @inheritParams param_allocationRatioPlanned
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The
#' degrees of freedom for the t-distribution for testing the slope
#' difference is calculated using the containment method, and
#' is equal to the total number of observations minus two times
#' the total number of subjects. The exact calculation using the
#' t distribution is only implemented for the fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Defaults to
#' \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @details
#'
#' We use the following random-effects model to compare two slopes:
#' \deqn{y_{ij} = \alpha + (\beta + \gamma x_i) t_j + a_i + b_i t_j
#' + e_{ij},} where
#'
#' * \eqn{\alpha}: overall intercept common across treatment groups
#' due to randomization
#'
#' * \eqn{\beta}: slope for the control group
#'
#' * \eqn{\gamma}: difference in slopes between the active treatment and
#' control groups
#'
#' * \eqn{x_i}: treatment indicator for subject \eqn{i},
#' 1 for the active treatment and 0 for the control
#'
#' * \eqn{t_j}: time point \eqn{j} for repeated measurements,
#' \eqn{t_1 = 0 < t_2 < \ldots < t_k}
#'
#' * \eqn{(a_i, b_i)}: random intercept and random slope
#' for subject \eqn{i}, \eqn{Var(a_i) = \sigma_a^2},
#' \eqn{Var(b_i) = \sigma_b^2}, \eqn{Corr(a_i, b_i) = \rho}
#'
#' * \eqn{e_{ij}}: within-subject residual with variance \eqn{\sigma_e^2}
#'
#' By accounting for randomization, we improve the efficiency for
#' estimating the difference in slopes. We also allow for non-equal
#' spacing of the time points and missing data due to dropouts.
#'
#' @return An S3 class \code{designSlopeDiffMMRM} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{studyDuration}: The maximum study duration.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{expectedStudyDurationH1}: The expected study duration
#' under H1.
#'
#' - \code{expectedStudyDurationH0}: The expected study duration
#' under H0.
#'
#' - \code{accrualDuration}: The accrual duration.
#'
#' - \code{followupTime}: The follow-up time.
#'
#' - \code{fixedFollowup}: Whether a fixed follow-up design is used.
#'
#' - \code{slopeDiffH0}: The slope difference under H0.
#'
#' - \code{slopeDiff}: The slope difference under H1.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlopeDiff}: The efficacy boundaries on the slope
#' difference scale.
#'
#' - \code{futilitySlopeDiff}: The futility boundaries on the slope
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{analysisTime}: The average time since trial start.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: The allocation ratio for the active
#' treatment versus control.
#'
#' - \code{accrualTime}: A vector that specifies the starting time of
#' piecewise Poisson enrollment time intervals.
#'
#' - \code{accrualIntensity}: A vector of accrual intensities.
#' One for each accrual time interval.
#'
#' - \code{piecewiseSurvivalTime}: A vector that specifies the
#' starting time of piecewise exponential survival time intervals.
#'
#' - \code{gamma1}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the active treatment group.
#'
#' - \code{gamma2}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the control group.
#'
#' - \code{w}: The number of time units per measurement visit in a
#' period.
#'
#' - \code{N}: The number of measurement visits in a period.
#'
#' - \code{stdDev}: The standard deviation of the residual.
#'
#' - \code{G}: The covariance matrix for the random intercept and
#' random slope.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Daniel O. Scharfstein, Anastasios A. Tsiatis, and James M. Robins.
#' Semiparametric efficiency and its implication on the
#' design and analysis of group-sequential studies.
#' Journal of the American Statistical Association 1997; 92:1342-1350.
#'
#' @examples
#'
#' (design1 <- getDesignSlopeDiffMMRM(
#' beta = 0.2, slopeDiff = log(1.15)/52,
#' stDev = sqrt(.182),
#' stDevIntercept = sqrt(.238960),
#' stDevSlope = sqrt(.000057),
#' corrInterceptSlope = .003688/sqrt(.238960*.000057),
#' w = 8,
#' N = 10000,
#' accrualIntensity = 15,
#' gamma1 = 1/(4.48*52),
#' gamma2 = 1/(4.48*52),
#' accrualDuration = NA,
#' followupTime = 8,
#' alpha = 0.025))
#'
#' @export
getDesignSlopeDiffMMRM <- function(
beta = NA_real_,
slopeDiffH0 = 0,
slopeDiff = 0.5,
stDev = 1,
stDevIntercept = 1,
stDevSlope = 1,
corrInterceptSlope = 0.5,
w = NA_real_,
N = NA_real_,
accrualTime = 0,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0,
gamma1 = 0,
gamma2 = 0,
accrualDuration = NA_real_,
followupTime = NA_real_,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
m = length(w)
cumN = c(0, cumsum(N))
cumwN = c(0, cumsum(w*N))
nintervals = length(piecewiseSurvivalTime)
if (is.na(beta) + is.na(accrualDuration) + is.na(followupTime) > 1) {
stop(paste("At most one can be missing among beta, accrualDuration,",
"and followupTime"))
}
if (is.na(beta)) {
unknown = "beta"
} else if (is.na(accrualDuration)) {
unknown = "accrualDuration"
} else if (is.na(followupTime)) {
unknown = "followupTime"
} else {
unknown = "accrualIntensity"
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (any(w <= 0)) {
stop("Elements of w must be positive")
}
if (length(N) != m) {
stop("w and N must have the same length")
}
if (any(N <= 0)) {
stop("Elements of N must be positive")
}
if (sum(N) < 10000) {
stop("The max # of measurements must be >=10000 to enable root finding")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (stDevIntercept <= 0) {
stop("stDevIntercept must be positive")
}
if (stDevSlope <= 0) {
stop("stDevSlope must be positive")
}
if (corrInterceptSlope <= -1 || corrInterceptSlope >= 1) {
stop("corrInterceptSlope must lie between -1 and 1")
}
if (accrualTime[1] != 0) {
stop("accrualTime must start with 0")
}
if (length(accrualTime) > 1 && any(diff(accrualTime) <= 0)) {
stop("accrualTime should be increasing")
}
if (any(is.na(accrualIntensity))) {
stop("accrualIntensity must be provided")
}
if (length(accrualTime) != length(accrualIntensity)) {
stop("accrualTime and accrualIntensity must have the same length")
}
if (any(accrualIntensity < 0)) {
stop("accrualIntensity must be non-negative")
}
if (piecewiseSurvivalTime[1] != 0) {
stop("piecewiseSurvivalTime must start with 0")
}
if (nintervals > 1 && any(diff(piecewiseSurvivalTime) <= 0)) {
stop("piecewiseSurvivalTime should be increasing")
}
if (any(gamma1 < 0)) {
stop("gamma1 must be non-negative")
}
if (any(gamma2 < 0)) {
stop("gamma2 must be non-negative")
}
if (length(gamma1) != 1 && length(gamma1) != nintervals) {
stop("Invalid length for gamma1")
}
if (length(gamma2) != 1 && length(gamma2) != nintervals) {
stop("Invalid length for gamma2")
}
if (length(gamma1) == 1) {
gamma1 = rep(gamma1, nintervals)
}
if (length(gamma2) == 1) {
gamma2 = rep(gamma2, nintervals)
}
if (!is.na(accrualDuration) && accrualDuration <= 0) {
stop("accrualDuration must be positive")
}
if (!is.na(followupTime) && followupTime < 0) {
stop("followupTime must be nonnegative")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = slopeDiff > slopeDiffH0
theta = ifelse(directionUpper, slopeDiff - slopeDiffH0,
slopeDiffH0 - slopeDiff)
v11 = stDevIntercept^2
v12 = corrInterceptSlope*stDevIntercept*stDevSlope
v22 = stDevSlope^2
G = matrix(c(v11, v12, v12, v22), 2, 2)
# function to obtain the info for slope difference
f_info <- function(tau, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r) {
# total number of enrolled subjects at interim analysis
n = accrual(tau, accrualTime, accrualIntensity, accrualDuration)
i = findInterval(tau, cumwN)
k = floor((tau - cumwN[i])/w[i]) + cumN[i] + 1
j = 1:k
i = pmax(findInterval(j-2, cumN), 1) # period indicators
t = cumwN[i] + (j - cumN[i] - 1)*w[i] # time points
# total number of enrolled subjects at each time point
ns = accrual(tau - t, accrualTime, accrualIntensity, accrualDuration)
# probability of not dropping out at each time point by treatment
q1 = ptpwexp(t, piecewiseSurvivalTime, gamma1, lower.tail = FALSE)
q2 = ptpwexp(t, piecewiseSurvivalTime, gamma2, lower.tail = FALSE)
# number of subjects remaining at each time point by treatment
m1 = r*ns*q1
m2 = (1-r)*ns*q2
# number of subjects dropping out at each time point by treatment
n1 = c(-diff(m1), m1[k])
n2 = c(-diff(m2), m2[k])
Z = matrix(c(rep(1,k), t), k, 2)
covar = Z %*% G %*% t(Z) + stDev^2*diag(k)
X1 = matrix(c(rep(1,k), t, t), k, 3)
X2 = matrix(c(rep(1,k), t, rep(0,k)), k, 3)
# information matrix per subject
I1 = 0
I2 = 0
for (j in 1:k) {
x1 = X1[1:j, , drop=FALSE]
x2 = X2[1:j, , drop=FALSE]
I = solve(covar[1:j, 1:j])
I1 = I1 + n1[j]*t(x1) %*% I %*% x1
I2 = I2 + n2[j]*t(x2) %*% I %*% x2
}
# variance for common intercept, control slope, and slope difference
V = solve(I1 + I2)
# information for slope difference
IMax = 1/V[3,3]
# residual degrees of freedom
nu = sum((n1 + n2)*(1:k)) - 2*n
list(IMax = IMax, nu = nu)
}
# power calculation
if (is.na(beta)) {
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
}
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
nu = out$nu
des = getDesign(
beta, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = delta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff = -delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = -delta + slopeDiffH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta + slopeDiffH0
}
}
} else { # sample size calculation
des = getDesign(
beta, NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
IMax = des$overallResults$information
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root*accrualIntensity
}
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
nu = out$nu # this serves as the lower bound for degrees of freedom
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
# power for t-test
b = qt(1-alpha, nu)
info = uniroot(function(info) {
ncp = theta*sqrt(info)
pt(b, nu, ncp, lower.tail = FALSE) - (1-beta)
}, c(0.5*IMax, 1.5*IMax))$root
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - info
}, c(0.5*accrualDuration, 1.5*accrualDuration))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - info
}, c(0.5*followupTime, 1.5*followupTime))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - info
}, c(0.5, 1.5))$root*accrualIntensity
}
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
if (unknown == "accrualDuration" || unknown == "followupTime") {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else {
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.5, 1.5))$root*accrualIntensity
}
}
# update maximum information and degrees of freedom
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
nu = out$nu
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = delta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff = -delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = -delta + slopeDiffH0
}
} else {
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root*accrualIntensity
}
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
if (unknown == "accrualDuration" || unknown == "followupTime") {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else {
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.5, 1.5))$root*accrualIntensity
}
# final maximum information
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
des = getDesign(
NA, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta + slopeDiffH0
}
}
}
# timing of interim analysis
studyDuration = accrualDuration + followupTime
information = IMax*informationRates
analysisTime = rep(0, kMax)
if (kMax > 1) {
for (i in 1:(kMax-1)) {
analysisTime[i] = uniroot(function(tau) {
out = f_info(tau, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - information[i]
}, c(w[1]+0.001, studyDuration))$root
}
}
analysisTime[kMax] = studyDuration
numberOfSubjects = accrual(analysisTime, accrualTime, accrualIntensity,
accrualDuration)
p = des$byStageResults$rejectPerStage +
des$byStageResults$futilityPerStage
pH0 = des$byStageResults$rejectPerStageH0 +
des$byStageResults$futilityPerStageH0
des$overallResults$numberOfSubjects = n
des$overallResults$studyDuration = studyDuration
des$overallResults$expectedNumberOfSubjectsH1 = sum(p*numberOfSubjects)
des$overallResults$expectedNumberOfSubjectsH0 = sum(pH0*numberOfSubjects)
des$overallResults$expectedStudyDurationH1 = sum(p*analysisTime)
des$overallResults$expectedStudyDurationH0 = sum(pH0*analysisTime)
des$overallResults$accrualDuration = accrualDuration
des$overallResults$followupTime = followupTime
des$overallResults$fixedFollowup = FALSE
des$overallResults$slopeDiffH0 = slopeDiffH0
des$overallResults$slopeDiff = slopeDiff
des$byStageResults$numberOfSubjects = numberOfSubjects
des$byStageResults$analysisTime = analysisTime
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$accrualTime = accrualTime
des$settings$accrualIntensity = accrualIntensity
des$settings$piecewiseSurvivalTime = piecewiseSurvivalTime
des$settings$gamma1 = gamma1
des$settings$gamma2 = gamma2
des$settings$w = w
des$settings$N = N
des$settings$stDev = stDev
des$settings$G = G
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designSlopeDiffMMRM"
des
}
#' @title Hedges' g Effect Size
#' @description Obtains Hedges' g estimate and confidence interval of
#' effect size.
#'
#' @param tstat The value of the t-test statistic for comparing two
#' treatment conditions.
#' @param m The degrees of freedom for the t-test.
#' @param ntilde The normalizing sample size to convert the
#' standardized treatment difference to the t-test statistic, i.e.,
#' \code{tstat = sqrt(ntilde)*meanDiff/stDev}.
#' @param cilevel The confidence interval level. Defaults to 0.95.
#'
#' @details
#'
#' Hedges' \eqn{g} is an effect size measure commonly used in meta-analysis
#' to quantify the difference between two groups. It's an improvement
#' over Cohen's \eqn{d}, particularly when dealing with small sample sizes.
#'
#' The formula for Hedges' \eqn{g} is \deqn{g = c(m) d,} where \eqn{d}
#' is Cohen's \eqn{d} effect size estimate, and \eqn{c(m)} is the bias
#' correction factor, \deqn{d = (\hat{\mu}_1 - \hat{\mu}_2)/\hat{\sigma},}
#' \deqn{c(m) = 1 - \frac{3}{4m-1}.}
#' Since \eqn{c(m) < 1}, Cohen's \eqn{d} overestimates the true effect size.
#' \eqn{\delta = (\mu_1 - \mu_2)/\sigma.}
#' Since \deqn{t = \sqrt{\tilde{n}} d,} we have
#' \deqn{g = \frac{c(m)}{\sqrt{\tilde{n}}} t,} where \eqn{t}
#' has a noncentral \eqn{t} distribution with \eqn{m} degrees of freedom
#' and noncentrality parameter \eqn{\sqrt{\tilde{n}} \delta}.
#'
#' The asymptotic variance of \eqn{g} can be approximated by
#' \deqn{Var(g) = \frac{1}{\tilde{n}} + \frac{g^2}{2m}.}
#' The confidence interval for \eqn{\delta}
#' can be constructed using normal approximation.
#'
#' For two-sample mean difference with sample size \eqn{n_1} for the
#' treatment group and \eqn{n_2} for the control group, we have
#' \eqn{\tilde{n} = \frac{n_1n_2}{n_1+n_2}} and \eqn{m=n_1+n_2-2}
#' for pooled variance estimate.
#'
#' @return A data frame with the following variables:
#'
#' * \code{tstat}: The value of the \code{t} test statistic.
#'
#' * \code{m}: The degrees of freedom for the t-test.
#'
#' * \code{ntilde}: The normalizing sample size to convert the
#' standardized treatment difference to the t-test statistic.
#'
#' * \code{g}: Hedges' \code{g} effect size estimate.
#'
#' * \code{varg}: Variance of \code{g}.
#'
#' * \code{lower}: The lower confidence limit for effect size.
#'
#' * \code{upper}: The upper confidence limit for effect size.
#'
#' * \code{cilevel}: The confidence interval level.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Larry V. Hedges. Distribution theory for Glass's estimator of
#' effect size and related estimators.
#' Journal of Educational Statistics 1981; 6:107-128.
#'
#' @examples
#'
#' n1 = 7
#' n2 = 8
#' meanDiff = 0.444
#' stDev = 1.201
#' m = n1+n2-2
#' ntilde = n1*n2/(n1+n2)
#' tstat = sqrt(ntilde)*meanDiff/stDev
#'
#' hedgesg(tstat, m, ntilde)
#'
#' @export
hedgesg <- function(tstat, m, ntilde, cilevel = 0.95) {
d = 1/sqrt(ntilde)*tstat
g = (1 - 3/(4*m-1))*d
varg = 1/ntilde + g^2/(2*m)
lower = g - qnorm((1 + cilevel)/2)*sqrt(varg)
upper = g + qnorm((1 + cilevel)/2)*sqrt(varg)
data.frame(tstat, m, ntilde, g, varg, lower, upper, cilevel)
}
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Defines functions hedgesg getDesignSlopeDiffMMRM getDesignSlopeDiff getDesignOneSlope getDesignRepeatedANOVAContrast getDesignRepeatedANOVA getDesignANOVAContrast getDesignTwoWayANOVA getDesignANOVA getDesignMeanDiffCarryover getDesignMeanDiffMMRM getDesignWilcoxon getDesignMeanRatioXOEquiv getDesignMeanDiffXOEquiv getDesignMeanRatioEquiv getDesignMeanDiffEquiv getDesignPairedMeanRatioEquiv getDesignPairedMeanDiffEquiv getDesignMeanRatioXO getDesignMeanDiffXO getDesignMeanRatio getDesignMeanDiff getDesignPairedMeanRatio getDesignPairedMeanDiff getDesignOneMean
Documented in getDesignANOVA getDesignANOVAContrast getDesignMeanDiff getDesignMeanDiffCarryover getDesignMeanDiffEquiv getDesignMeanDiffMMRM getDesignMeanDiffXO getDesignMeanDiffXOEquiv getDesignMeanRatio getDesignMeanRatioEquiv getDesignMeanRatioXO getDesignMeanRatioXOEquiv getDesignOneMean getDesignOneSlope getDesignPairedMeanDiff getDesignPairedMeanDiffEquiv getDesignPairedMeanRatio getDesignPairedMeanRatioEquiv getDesignRepeatedANOVA getDesignRepeatedANOVAContrast getDesignSlopeDiff getDesignSlopeDiffMMRM getDesignTwoWayANOVA getDesignWilcoxon hedgesg
#' @title Group Sequential Design for One-Sample Mean
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for one-sample mean.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanH0 The mean under the null hypothesis.
#' Defaults to 0.
#' @param mean The mean under the alternative hypothesis.
#' @param stDev The standard deviation.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designOneMean} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanH0}: The mean under the null hypothesis.
#'
#' - \code{mean}: The mean under the alternative hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMean}: The efficacy boundaries on the mean scale.
#'
#' - \code{futilityMean}: The futility boundaries on the mean scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignOneMean(
#' beta = 0.1, n = NA, meanH0 = 7, mean = 6, stDev = 2.5,
#' kMax = 5, alpha = 0.025, typeAlphaSpending = "sfOF",
#' typeBetaSpending = "sfP"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignOneMean(
#' beta = 0.1, n = NA, meanH0 = 7, mean = 6, stDev = 2.5,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignOneMean <- function(
beta = NA_real_,
n = NA_real_,
meanH0 = 0,
mean = 0.5,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
directionUpper = mean > meanH0
theta = ifelse(directionUpper, mean - meanH0, meanH0 - mean)
# variance for one sampling unit
v1 = stDev^2
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
power = pt(b, n-1, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMean = delta + meanH0
des$byStageResults$futilityMean = delta + meanH0
} else {
des$byStageResults$efficacyMean = -delta + meanH0
des$byStageResults$futilityMean = -delta + meanH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMean =
des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
des$byStageResults$futilityTheta + meanH0
} else {
des$byStageResults$efficacyMean =
-des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
-des$byStageResults$futilityTheta + meanH0
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
pt(b, n-1, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-1)
ncp = theta*sqrt(n/v1)
power = pt(b, n-1, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMean = delta + meanH0
des$byStageResults$futilityMean = delta + meanH0
} else {
des$byStageResults$efficacyMean = -delta + meanH0
des$byStageResults$futilityMean = -delta + meanH0
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMean =
des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
des$byStageResults$futilityTheta + meanH0
} else {
des$byStageResults$efficacyMean =
-des$byStageResults$efficacyTheta + meanH0
des$byStageResults$futilityMean =
-des$byStageResults$futilityTheta + meanH0
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanH0 = meanH0
des$overallResults$mean = mean
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designOneMean"
des
}
#' @title Group Sequential Design for Paired Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for paired mean
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedDiffH0 The paired difference under the null hypothesis.
#' Defaults to 0.
#' @param pairedDiff The paired difference under the alternative hypothesis.
#' @param stDev The standard deviation for paired difference.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanDiff} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedDiffH0}: The paired difference under the null
#' hypothesis.
#'
#' - \code{pairedDiff}: The paired difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for paired difference.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyPairedDiff}: The efficacy boundaries on the paired
#' difference scale.
#'
#' - \code{futilityPairedDiff}: The futility boundaries on the paired
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanDiff(
#' beta = 0.1, n = NA, pairedDiffH0 = 0, pairedDiff = -2, stDev = 5,
#' kMax = 5, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignPairedMeanDiff(
#' beta = 0.1, n = NA, pairedDiffH0 = 0, pairedDiff = -2, stDev = 5,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignPairedMeanDiff <- function(
beta = NA_real_,
n = NA_real_,
pairedDiffH0 = 0,
pairedDiff = 0.5,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
des = getDesignOneMean(
beta, n, pairedDiffH0, pairedDiff, stDev,
normalApproximation,
rounding, kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
nov = names(des$overallResults)
names(des$overallResults)[nov == "meanH0"] <- "pairedDiffH0"
names(des$overallResults)[nov == "mean"] <- "pairedDiff"
nby = names(des$byStageResults)
names(des$byStageResults)[nby == "efficacyMean"] <- "efficacyPairedDiff"
names(des$byStageResults)[nby == "futilityMean"] <- "futilityPairedDiff"
attr(des, "class") = "designPairedMeanDiff"
des
}
#' @title Group Sequential Design for Paired Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for paired mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedRatioH0 The paired ratio under the null hypothesis.
#' @param pairedRatio The paired ratio under the alternative
#' hypothesis.
#' @param CV The coefficient of variation for paired ratio.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanRatio} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedRatioH0}: The paired ratio under the null hypothesis.
#'
#' - \code{pairedRatio}: The paired ratio under the alternative
#' hypothesis.
#'
#' - \code{CV}: The coefficient of variation for paired ratio.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyPairedRatio}: The efficacy boundaries on the paired
#' ratio scale.
#'
#' - \code{futilityPairedRatio}: The futility boundaries on the paired
#' ratio scale.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanRatio(
#' beta = 0.1, n = NA, pairedRatio = 1.2, CV = 0.35,
#' kMax = 5, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for one-sample t-test
#' (design2 <- getDesignPairedMeanRatio(
#' beta = 0.1, n = NA, pairedRatio = 1.2, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanRatio <- function(
beta = NA_real_,
n = NA_real_,
pairedRatioH0 = 1,
pairedRatio = 1.2,
CV = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (pairedRatioH0 <= 0) {
stop("pairedRatioH0 must be positive")
}
if (pairedRatio <= 0) {
stop("pairedRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignPairedMeanDiff(
beta, n, pairedDiffH0 = log(pairedRatioH0),
pairedDiff = log(pairedRatio),
stDev = sqrt(log(1 + CV^2)),
normalApproximation,
rounding, kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$pairedRatioH0 = pairedRatioH0
des$overallResults$pairedRatio = pairedRatio
des$overallResults$CV = CV
des$overallResults$pairedDiffH0 = NULL
des$overallResults$pairedDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyPairedRatio =
exp(des$byStageResults$efficacyPairedDiff)
des$byStageResults$futilityPairedRatio =
exp(des$byStageResults$futilityPairedDiff)
des$byStageResults$efficacyPairedDiff = NULL
des$byStageResults$futilityPairedDiff = NULL
attr(des, "class") = "designPairedMeanRatio"
des
}
#' @title Group Sequential Design for Two-Sample Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference under the null hypothesis.
#' Defaults to 0.
#' @param meanDiff The mean difference under the alternative hypothesis.
#' @param stDev The standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiff} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiff(
#' beta = NA, n = 456, meanDiff = 9, stDev = 32,
#' kMax = 5, alpha = 0.025, typeAlphaSpending = "sfOF",
#' typeBetaSpending = "sfP"))
#'
#' # Example 2: sample size calculation for two-sample t-test
#' (design2 <- getDesignMeanDiff(
#' beta = 0.1, n = NA, meanDiff = 0.3, stDev = 1,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignMeanDiff <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# variance for one sampling unit
v1 = stDev^2/(r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiff"
des
}
#' @title Group Sequential Design for Two-Sample Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioH0 The mean ratio under the null hypothesis.
#' Defaults to 1.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation. The standard deviation on the
#' log scale is equal to \code{sqrt(log(1 + CV^2))}.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatio} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioH0}: The mean ratio under the null hypothesis.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyMeanRatio}: The efficacy boundaries on the mean
#' ratio scale.
#'
#' - \code{futilityMeanRatio}: The futility boundaries on the mean
#' ratio scale.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanRatio(
#' beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
#' alpha = 0.05, normalApproximation = FALSE))
#'
#' @export
getDesignMeanRatio <- function(
beta = NA_real_,
n = NA_real_,
meanRatioH0 = 1,
meanRatio = 1.25,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (meanRatioH0 <= 0) {
stop("meanRatioH0 must be positive")
}
if (meanRatio <= 0) {
stop("meanRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiff(
beta, n, meanDiffH0 = log(meanRatioH0),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$meanRatioH0 = meanRatioH0
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffH0 = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatio =
exp(des$byStageResults$efficacyMeanDiff)
des$byStageResults$futilityMeanRatio =
exp(des$byStageResults$futilityMeanDiff)
des$byStageResults$efficacyMeanDiff = NULL
des$byStageResults$futilityMeanDiff = NULL
attr(des, "class") = "designMeanRatio"
des
}
#' @title Group Sequential Design for Mean Difference in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' difference in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference under the null hypothesis.
#' Defaults to 0.
#' @param meanDiff The mean difference under the alternative hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffXO} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for within-subject random
#' error.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanDiffXO(
#' beta = 0.2, n = NA, meanDiff = 75, stDev = 150,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffXO <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# variance for one sampling unit
v1 = stDev^2/(2*r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta
des$byStageResults$futilityMeanDiff = delta
} else {
des$byStageResults$efficacyMeanDiff = -delta
des$byStageResults$futilityMeanDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiffXO"
des
}
#' @title Group Sequential Design for Mean Ratio in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample mean
#' ratio in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioH0 The mean ratio under the null hypothesis.
#' Defaults to 1.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation. The standard deviation on the
#' log scale is equal to \code{sqrt(log(1 + CV^2))}.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioXO} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioH0}: The mean ratio under the null hypothesis.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyMeanRatio}: The efficacy boundaries on the mean
#' ratio scale.
#'
#' - \code{futilityMeanRatio}: The futility boundaries on the mean
#' ratio scale.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignMeanRatioXO(
#' beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
#' alpha = 0.05, normalApproximation = FALSE))
#'
#' @export
getDesignMeanRatioXO <- function(
beta = NA_real_,
n = NA_real_,
meanRatioH0 = 1,
meanRatio = 1.25,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (meanRatioH0 <= 0) {
stop("meanRatioH0 must be positive")
}
if (meanRatio <= 0) {
stop("meanRatio must be positive")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffXO(
beta, n, meanDiffH0 = log(meanRatioH0),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
des$overallResults$meanRatioH0 = meanRatioH0
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffH0 = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatio =
exp(des$byStageResults$efficacyMeanDiff)
des$byStageResults$futilityMeanRatio =
exp(des$byStageResults$futilityMeanDiff)
des$byStageResults$efficacyMeanDiff = NULL
des$byStageResults$futilityMeanDiff = NULL
attr(des, "class") = "designMeanRatioXO"
des
}
#' @title Group Sequential Design for Equivalence in Paired Mean
#' Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' paired mean difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedDiffLower The lower equivalence limit of paired difference.
#' @param pairedDiffUpper The upper equivalence limit of paired difference.
#' @param pairedDiff The paired difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation for paired difference.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanDiffEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level under H0.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedDiffLower}: The lower equivalence limit of paired
#' difference.
#'
#' - \code{pairedDiffUpper}: The upper equivalence limit of paired
#' difference.
#'
#' - \code{pairedDiff}: The paired difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for paired difference.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyPairedDiffLower}: The efficacy boundaries on the
#' paired difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyPairedDiffUpper}: The efficacy boundaries on the
#' paired difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanDiffEquiv(
#' beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
#' pairedDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignPairedMeanDiffEquiv(
#' beta = 0.1, n = NA, pairedDiffLower = -1.3, pairedDiffUpper = 1.3,
#' pairedDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanDiffEquiv <- function(
beta = NA_real_,
n = NA_real_,
pairedDiffLower = NA_real_,
pairedDiffUpper = NA_real_,
pairedDiff = 0,
stDev = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(pairedDiffLower)) {
stop("pairedDiffLower must be provided")
}
if (is.na(pairedDiffUpper)) {
stop("pairedDiffUpper must be provided")
}
if (pairedDiffLower >= pairedDiff) {
stop("pairedDiffLower must be less than pairedDiff")
}
if (pairedDiffUpper <= pairedDiff) {
stop("pairedDiffUpper must be greater than pairedDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
v1 = stDev^2
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-1)
ncpLower = (pairedDiff - pairedDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-1, ncpLower, lower.tail = FALSE)
ncpUpper = (pairedDiffUpper - pairedDiff)*sqrt(n/v1)
powerUpper = pt(b, n-1, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = pairedDiffLower,
thetaUpper = pairedDiffUpper, theta = pairedDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = pairedDiffLower,
thetaUpper = pairedDiffUpper, theta = pairedDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-1)
ncp = (pairedDiffUpper - pairedDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-1)*x*x, (n-1)/2, 1/2)*2*(n-1)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyPairedDiffLower = b*sqrt(v1/n) +
pairedDiffLower
des$byStageResults$efficacyPairedDiffUpper = -b*sqrt(v1/n) +
pairedDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyPairedDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyPairedDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$pairedDiffLower = pairedDiffLower
des$overallResults$pairedDiffUpper = pairedDiffUpper
des$overallResults$pairedDiff = pairedDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "pairedDiffLower",
"pairedDiffUpper", "pairedDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyPairedDiffLower",
"efficacyPairedDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"normalApproximation", "rounding")]
attr(des, "class") = "designPairedMeanDiffEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Paired Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' paired mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pairedRatioLower The lower equivalence limit of paired ratio.
#' @param pairedRatioUpper The upper equivalence limit of paired ratio.
#' @param pairedRatio The paired ratio under the alternative
#' hypothesis.
#' @param CV The coefficient of variation for paired ratio.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designPairedMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level under H0.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pairedRatioLower}: The lower equivalence limit of paired
#' ratio.
#'
#' - \code{pairedRatioUpper}: The upper equivalence limit of paired
#' ratio.
#'
#' - \code{pairedRatio}: The paired ratio under the alternative
#' hypothesis.
#'
#' - \code{CV}: The coefficient of variation for paired ratios.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative alpha attained under
#' H0.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyPairedRatioLower}: The efficacy boundaries on the
#' paired ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyPairedRatioUpper}: The efficacy boundaries on the
#' paired ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignPairedMeanRatioEquiv(
#' beta = 0.1, n = NA, pairedRatioLower = 0.8, pairedRatioUpper = 1.25,
#' pairedRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignPairedMeanRatioEquiv(
#' beta = 0.1, n = NA, pairedRatioLower = 0.8, pairedRatioUpper = 1.25,
#' pairedRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignPairedMeanRatioEquiv <- function(
beta = NA_real_,
n = NA_real_,
pairedRatioLower = NA_real_,
pairedRatioUpper = NA_real_,
pairedRatio = 1,
CV = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(pairedRatioLower)) {
stop("pairedRatioLower must be provided")
}
if (is.na(pairedRatioUpper)) {
stop("pairedRatioUpper must be provided")
}
if (pairedRatioLower <= 0) {
stop("pairedRatioLower must be positive")
}
if (pairedRatioLower >= pairedRatio) {
stop("pairedRatioLower must be less than pairedRatio")
}
if (pairedRatioUpper <= pairedRatio) {
stop("pairedRatioUpper must be greater than pairedRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignPairedMeanDiffEquiv(
beta, n, pairedDiffLower = log(pairedRatioLower),
pairedDiffUpper = log(pairedRatioUpper),
pairedDiff = log(pairedRatio),
stDev = sqrt(log(1 + CV^2)),
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$pairedRatioLower = pairedRatioLower
des$overallResults$pairedRatioUpper = pairedRatioUpper
des$overallResults$pairedRatio = pairedRatio
des$overallResults$CV = CV
des$overallResults$pairedDiffLower = NULL
des$overallResults$pairedDiffUpper = NULL
des$overallResults$pairedDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyPairedRatioLower =
exp(des$byStageResults$efficacyPairedDiffLower)
des$byStageResults$efficacyPairedRatioUpper =
exp(des$byStageResults$efficacyPairedDiffUpper)
des$byStageResults$efficacyPairedDiffLower = NULL
des$byStageResults$efficacyPairedDiffUpper = NULL
attr(des, "class") = "designPairedMeanRatioEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Two-Sample
#' Mean Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' two-sample mean difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffLower The lower equivalence limit of mean difference.
#' @param meanDiffUpper The upper equivalence limit of mean difference.
#' @param meanDiff The mean difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffLower}: The lower equivalence limit of mean
#' difference.
#'
#' - \code{meanDiffUpper}: The upper equivalence limit of mean
#' difference.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanDiffLower}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanDiffUpper}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiffEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanDiffEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanDiffLower = NA_real_,
meanDiffUpper = NA_real_,
meanDiff = 0,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(meanDiffLower)) {
stop("meanDiffLower must be provided")
}
if (is.na(meanDiffUpper)) {
stop("meanDiffUpper must be provided")
}
if (meanDiffLower >= meanDiff) {
stop("meanDiffLower must be less than meanDiff")
}
if (meanDiffUpper <= meanDiff) {
stop("meanDiffUpper must be greater than meanDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
v1 = stDev^2/(r*(1-r))
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-2)
ncpLower = (meanDiff - meanDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-2, ncpLower, lower.tail = FALSE)
ncpUpper = (meanDiffUpper - meanDiff)*sqrt(n/v1)
powerUpper = pt(b, n-2, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-2)
ncp = (meanDiffUpper - meanDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-2)*x*x, (n-2)/2, 1/2)*2*(n-2)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyMeanDiffLower = b*sqrt(v1/n) +
meanDiffLower
des$byStageResults$efficacyMeanDiffUpper = -b*sqrt(v1/n) +
meanDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyMeanDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyMeanDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffLower = meanDiffLower
des$overallResults$meanDiffUpper = meanDiffUpper
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "meanDiffLower",
"meanDiffUpper", "meanDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyMeanDiffLower",
"efficacyMeanDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"allocationRatioPlanned",
"normalApproximation", "rounding")]
attr(des, "class") = "designMeanDiffEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Two-Sample
#' Mean Ratio
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' two-sample mean ratio.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioLower The lower equivalence limit of mean ratio.
#' @param meanRatioUpper The upper equivalence limit of mean ratio.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioLower}: The lower equivalence limit of mean ratio.
#'
#' - \code{meanRatioUpper}: The upper equivalence limit of mean ratio.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{efficacyMeanRatioLower}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanRatioUpper}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanRatioEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanRatioEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanRatioEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanRatioLower = NA_real_,
meanRatioUpper = NA_real_,
meanRatio = 1,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(meanRatioLower)) {
stop("meanRatioLower must be provided")
}
if (is.na(meanRatioUpper)) {
stop("meanRatioUpper must be provided")
}
if (meanRatioLower <= 0) {
stop("meanRatioLower must be positive")
}
if (meanRatioLower >= meanRatio) {
stop("meanRatioLower must be less than meanRatio")
}
if (meanRatioUpper <= meanRatio) {
stop("meanRatioUpper must be greater than meanRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffEquiv(
beta, n, meanDiffLower = log(meanRatioLower),
meanDiffUpper = log(meanRatioUpper),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$meanRatioLower = meanRatioLower
des$overallResults$meanRatioUpper = meanRatioUpper
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffLower = NULL
des$overallResults$meanDiffUpper = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatioLower =
exp(des$byStageResults$efficacyMeanDiffLower)
des$byStageResults$efficacyMeanRatioUpper =
exp(des$byStageResults$efficacyMeanDiffUpper)
des$byStageResults$efficacyMeanDiffLower = NULL
des$byStageResults$efficacyMeanDiffUpper = NULL
attr(des, "class") = "designMeanRatioEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Mean Difference
#' in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence in
#' mean difference in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffLower The lower equivalence limit of mean difference.
#' @param meanDiffUpper The upper equivalence limit of mean difference.
#' @param meanDiff The mean difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffXOEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanDiffLower}: The lower equivalence limit of mean
#' difference.
#'
#' - \code{meanDiffUpper}: The upper equivalence limit of mean
#' difference.
#'
#' - \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation for within-subject random
#' error.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanDiffLower}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanDiffUpper}: The efficacy boundaries on the
#' mean difference scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanDiffXOEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanDiffXOEquiv(
#' beta = 0.1, n = NA, meanDiffLower = -1.3, meanDiffUpper = 1.3,
#' meanDiff = 0, stDev = 2.2,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanDiffXOEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanDiffLower = NA_real_,
meanDiffUpper = NA_real_,
meanDiff = 0,
stDev = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (is.na(meanDiffLower)) {
stop("meanDiffLower must be provided")
}
if (is.na(meanDiffUpper)) {
stop("meanDiffUpper must be provided")
}
if (meanDiffLower >= meanDiff) {
stop("meanDiffLower must be less than meanDiff")
}
if (meanDiffUpper <= meanDiff) {
stop("meanDiffUpper must be greater than meanDiff")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
# variance for one sampling unit
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
v1 = stDev^2/(2*r*(1-r))
f <- function(n) { # power for two one-sided t-tests
b = qt(1-alpha, n-2)
ncpLower = (meanDiff - meanDiffLower)*sqrt(n/v1)
powerLower = pt(b, n-2, ncpLower, lower.tail = FALSE)
ncpUpper = (meanDiffUpper - meanDiff)*sqrt(n/v1)
powerUpper = pt(b, n-2, ncpUpper, lower.tail = FALSE)
power = powerLower + powerUpper - 1
power
}
if (is.na(n)) { # calculate sample size
des = getDesignEquiv(
beta = beta, IMax = NA, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
n = des$overallResults$information*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n, 1.5*n))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
if (is.na(beta) || rounding) { # calculate power
des = getDesignEquiv(
beta = NA, IMax = n/v1, thetaLower = meanDiffLower,
thetaUpper = meanDiffUpper, theta = meanDiff,
kMax = kMax, informationRates = informationRates,
alpha = alpha, typeAlphaSpending = typeAlphaSpending,
parameterAlphaSpending = parameterAlphaSpending,
userAlphaSpending = userAlphaSpending,
spendingTime = spendingTime)
}
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
power = f(n)
b = qt(1-alpha, n-2)
ncp = (meanDiffUpper - meanDiffLower)*sqrt(n/v1)
attainedAlpha = integrate(function(x) {
t1 = pnorm(-b*x + ncp) - pnorm(b*x)
t2 = dgamma((n-2)*x*x, (n-2)/2, 1/2)*2*(n-2)*x
t1*t2
}, 0, ncp/(2*b))$value
des$overallResults$overallReject = power
des$overallResults$attainedAlpha = attainedAlpha
des$overallResults$information = n/v1
des$overallResults$expectedInformationH1 = n/v1
des$overallResults$expectedInformationH0 = n/v1
des$byStageResults$efficacyBounds = b
des$byStageResults$rejectPerStage = power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeAttainedAlpha = attainedAlpha
des$byStageResults$efficacyMeanDiffLower = b*sqrt(v1/n) +
meanDiffLower
des$byStageResults$efficacyMeanDiffUpper = -b*sqrt(v1/n) +
meanDiffUpper
des$byStageResults$information = n/v1
} else {
des$overallResults$attainedAlpha =
des$overallResults$attainedAlphaH10
des$overallResults$expectedInformationH0 =
des$overallResults$expectedInformationH10
des$byStageResults$cumulativeAttainedAlpha =
des$byStageResults$cumulativeAttainedAlphaH10
des$byStageResults$efficacyMeanDiffLower =
des$byStageResults$efficacyThetaLower
des$byStageResults$efficacyMeanDiffUpper =
des$byStageResults$efficacyThetaUpper
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$meanDiffLower = meanDiffLower
des$overallResults$meanDiffUpper = meanDiffUpper
des$overallResults$meanDiff = meanDiff
des$overallResults$stDev = stDev
des$overallResults <-
des$overallResults[, c("overallReject", "alpha", "attainedAlpha",
"kMax", "information", "expectedInformationH1",
"expectedInformationH0", "numberOfSubjects",
"expectedNumberOfSubjectsH1",
"expectedNumberOfSubjectsH0", "meanDiffLower",
"meanDiffUpper", "meanDiff", "stDev")]
des$byStageResults$numberOfSubjects = n*informationRates
des$byStageResults <-
des$byStageResults[, c("informationRates", "efficacyBounds",
"rejectPerStage", "cumulativeRejection",
"cumulativeAlphaSpent", "cumulativeAttainedAlpha",
"efficacyMeanDiffLower",
"efficacyMeanDiffUpper", "efficacyP",
"information", "numberOfSubjects")]
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings <-
des$settings[c("typeAlphaSpending", "parameterAlphaSpending",
"userAlphaSpending", "spendingTime",
"allocationRatioPlanned",
"normalApproximation", "rounding")]
attr(des, "class") = "designMeanDiffXOEquiv"
des
}
#' @title Group Sequential Design for Equivalence in Mean Ratio
#' in 2x2 Crossover
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for equivalence
#' mean ratio in 2x2 crossover.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanRatioLower The lower equivalence limit of mean ratio.
#' @param meanRatioUpper The upper equivalence limit of mean ratio.
#' @param meanRatio The mean ratio under the alternative hypothesis.
#' @param CV The coefficient of variation.
#' @param allocationRatioPlanned Allocation ratio for sequence A/B
#' versus sequence B/A. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_criticalValues
#' @param alpha The significance level for each of the two one-sided
#' tests. Defaults to 0.05.
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanRatioEquiv} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{meanRatioLower}: The lower equivalence limit of mean ratio.
#'
#' - \code{meanRatioUpper}: The upper equivalence limit of mean ratio.
#'
#' - \code{meanRatio}: The mean ratio under the alternative hypothesis.
#'
#' - \code{CV}: The coefficient of variation.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale for
#' each of the two one-sided tests.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha for each of
#' the two one-sided tests.
#'
#' - \code{cumulativeAttainedAlpha}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{efficacyMeanRatioLower}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' lower equivalence limit.
#'
#' - \code{efficacyMeanRatioUpper}: The efficacy boundaries on the
#' mean ratio scale for the one-sided null hypothesis on the
#' upper equivalence limit.
#'
#' - \code{efficacyP}: The efficacy bounds on the p-value scale for
#' each of the two one-sided tests.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for sequence A/B
#' versus sequence B/A.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: group sequential trial power calculation
#' (design1 <- getDesignMeanRatioXOEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' kMax = 4, alpha = 0.05, typeAlphaSpending = "sfOF"))
#'
#' # Example 2: sample size calculation for t-test
#' (design2 <- getDesignMeanRatioXOEquiv(
#' beta = 0.1, n = NA, meanRatioLower = 0.8, meanRatioUpper = 1.25,
#' meanRatio = 1, CV = 0.35,
#' normalApproximation = FALSE, alpha = 0.05))
#'
#' @export
getDesignMeanRatioXOEquiv <- function(
beta = NA_real_,
n = NA_real_,
meanRatioLower = NA_real_,
meanRatioUpper = NA_real_,
meanRatio = 1,
CV = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
alpha = 0.05,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(meanRatioLower)) {
stop("meanRatioLower must be provided")
}
if (is.na(meanRatioUpper)) {
stop("meanRatioUpper must be provided")
}
if (meanRatioLower <= 0) {
stop("meanRatioLower must be positive")
}
if (meanRatioLower >= meanRatio) {
stop("meanRatioLower must be less than meanRatio")
}
if (meanRatioUpper <= meanRatio) {
stop("meanRatioUpper must be greater than meanRatio")
}
if (CV <= 0) {
stop("CV must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
des = getDesignMeanDiffXOEquiv(
beta, n, meanDiffLower = log(meanRatioLower),
meanDiffUpper = log(meanRatioUpper),
meanDiff = log(meanRatio),
stDev = sqrt(log(1 + CV^2)),
allocationRatioPlanned,
normalApproximation, rounding,
kMax, informationRates,
alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
spendingTime)
des$overallResults$meanRatioLower = meanRatioLower
des$overallResults$meanRatioUpper = meanRatioUpper
des$overallResults$meanRatio = meanRatio
des$overallResults$CV = CV
des$overallResults$meanDiffLower = NULL
des$overallResults$meanDiffUpper = NULL
des$overallResults$meanDiff = NULL
des$overallResults$stDev = NULL
des$byStageResults$efficacyMeanRatioLower =
exp(des$byStageResults$efficacyMeanDiffLower)
des$byStageResults$efficacyMeanRatioUpper =
exp(des$byStageResults$efficacyMeanDiffUpper)
des$byStageResults$efficacyMeanDiffLower = NULL
des$byStageResults$efficacyMeanDiffUpper = NULL
attr(des, "class") = "designMeanRatioXOEquiv"
des
}
#' @title Group Sequential Design for Two-Sample Wilcoxon Test
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample
#' Wilcoxon test.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param pLarger The probability that a randomly chosen sample from
#' the treatment group is larger than a randomly chosen sample from the
#' control group under the alternative hypothesis.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designWilcoxon} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping..
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{pLarger}: The probability that a randomly chosen sample from
#' the treatment group is larger than a randomly chosen sample from the
#' control group under the alternative hypothesis.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyPLarger}: The efficacy boundaries on the proportion
#' of pairs of samples from the two treatment groups with the sample
#' from the treatment group greater than that from the control group.
#'
#' - \code{futilityPLarger}: The futility boundaries on the proportion
#' of pairs of samples from the two treatment groups with the sample
#' from the treatment group greater than that from the control group.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # Example 1: fixed design
#' (design1 <- getDesignWilcoxon(
#' beta = 0.1, n = NA,
#' pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025))
#'
#' # Example 2: group sequential design
#' (design2 <- getDesignWilcoxon(
#' beta = 0.1, n = NA,
#' pLarger = pnorm((8 - 2)/sqrt(2*25^2)), alpha = 0.025,
#' kMax = 3, typeAlphaSpending = "sfOF"))
#'
#' @export
getDesignWilcoxon <- function(
beta = NA_real_,
n = NA_real_,
pLarger = 0.6,
allocationRatioPlanned = 1,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (pLarger <= 0 || pLarger >= 1) {
stop("pLarger must lie between 0 and 1")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = pLarger > 0.5
theta = ifelse(directionUpper, pLarger - 0.5, 0.5 - pLarger)
# variance for one sampling unit
v1 = 1/(12*r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
} else { # sample size calculation
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
}
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$pLarger = pLarger
if (directionUpper) {
des$byStageResults$efficacyPLarger =
des$byStageResults$efficacyTheta + 0.5
des$byStageResults$futilityPLarger =
des$byStageResults$futilityTheta + 0.5
} else {
des$byStageResults$efficacyPLarger =
-des$byStageResults$efficacyTheta + 0.5
des$byStageResults$futilityPLarger =
-des$byStageResults$futilityTheta + 0.5
}
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designWilcoxon"
des
}
#' @title Group Sequential Design for Two-Sample Mean Difference From the
#' MMRM Model
#' @description Obtains the power and sample size for two-sample
#' mean difference at the last time point from the mixed-model
#' for repeated measures (MMRM) model.
#'
#' @param beta The type II error.
#' @param meanDiffH0 The mean difference at the last time point
#' under the null hypothesis. Defaults to 0.
#' @param meanDiff The mean difference at the last time point
#' under the alternative hypothesis.
#' @param k The number of postbaseline time points.
#' @param t The postbaseline time points.
#' @param covar1 The covariance matrix for the repeated measures
#' given baseline for the active treatment group.
#' @param covar2 The covariance matrix for the repeated measures
#' given baseline for the control group. If missing, it will be
#' set equal to the covariance matrix for the active treatment group.
#' @inheritParams param_accrualTime
#' @inheritParams param_accrualIntensity
#' @inheritParams param_piecewiseSurvivalTime
#' @inheritParams param_gamma1
#' @inheritParams param_gamma2
#' @inheritParams param_accrualDuration
#' @inheritParams param_allocationRatioPlanned
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The
#' degrees of freedom for the t-distribution is the total effective
#' sample size minus 2. The exact calculation using the t distribution
#' is only implemented for the fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Defaults to
#' \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designMeanDiffMMRM} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{studyDuration}: The maximum study duration.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{expectedStudyDurationH1}: The expected study duration
#' under H1.
#'
#' - \code{expectedStudyDurationH0}: The expected study duration
#' under H0.
#'
#' - \code{accrualDuration}: The accrual duration.
#'
#' - \code{followupTime}: The follow-up time.
#'
#' - \code{fixedFollowup}: Whether a fixed follow-up design is used.
#'
#' - \code{meanDiffH0}: The mean difference under H0.
#'
#' - \code{meanDiff}: The mean difference under H1.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacyMeanDiff}: The efficacy boundaries on the mean
#' difference scale.
#'
#' - \code{futilityMeanDiff}: The futility boundaries on the mean
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{analysisTime}: The average time since trial start.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: The allocation ratio for the active
#' treatment versus control.
#'
#' - \code{accrualTime}: A vector that specifies the starting time of
#' piecewise Poisson enrollment time intervals.
#'
#' - \code{accrualIntensity}: A vector of accrual intensities.
#' One for each accrual time interval.
#'
#' - \code{piecewiseSurvivalTime}: A vector that specifies the
#' starting time of piecewise exponential survival time intervals.
#'
#' - \code{gamma1}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the active treatment group.
#'
#' - \code{gamma2}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the control group.
#'
#' - \code{k}: The number of postbaseline time points.
#'
#' - \code{t}: The postbaseline time points.
#'
#' - \code{covar1}: The covariance matrix for the repeated measures
#' given baseline for the active treatment group.
#'
#' - \code{covar2}: The covariance matrix for the repeated measures
#' given baseline for the control group.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' # function to generate the AR(1) correlation matrix
#' ar1_cor <- function(n, corr) {
#' exponent <- abs(matrix((1:n) - 1, n, n, byrow = TRUE) - ((1:n) - 1))
#' corr^exponent
#' }
#'
#' (design1 = getDesignMeanDiffMMRM(
#' beta = 0.2,
#' meanDiffH0 = 0,
#' meanDiff = 0.5,
#' k = 4,
#' t = c(1,2,3,4),
#' covar1 = ar1_cor(4, 0.7),
#' accrualIntensity = 10,
#' gamma1 = 0.02634013,
#' gamma2 = 0.02634013,
#' accrualDuration = NA,
#' allocationRatioPlanned = 1,
#' kMax = 3,
#' alpha = 0.025,
#' typeAlphaSpending = "sfOF"))
#'
#' @export
#'
getDesignMeanDiffMMRM <- function(
beta = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
k = 1,
t = NA_real_,
covar1 = diag(k),
covar2 = NA_real_,
accrualTime = 0,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0,
gamma1 = 0,
gamma2 = 0,
accrualDuration = NA_real_,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
nintervals = length(piecewiseSurvivalTime)
if (is.na(beta) && is.na(accrualDuration)) {
stop("beta and accrualDuration cannot be both missing")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (k <= 0 || k != round(k)) {
stop("k must be a positive integer")
}
if (any(is.na(t))) {
stop("t must be provided")
} else if (length(t) != k) {
stop("Invalid length for t")
} else if (any(t <= 0)) {
stop("Elements of t must be positive")
} else if (k > 1 && any(diff(t) <= 0)) {
stop("Elements of t must be increasing")
}
if (!all(eigen(covar1)$values > 0)) {
stop("covar1 must be positive definite")
}
if (any(is.na(covar2))) {
covar2 = covar1
} else if (!all(eigen(covar2)$values > 0)) {
stop("covar2 must be positive definite")
}
if (accrualTime[1] != 0) {
stop("accrualTime must start with 0")
}
if (length(accrualTime) > 1 && any(diff(accrualTime) <= 0)) {
stop("accrualTime should be increasing")
}
if (any(is.na(accrualIntensity))) {
stop("accrualIntensity must be provided")
}
if (length(accrualTime) != length(accrualIntensity)) {
stop("accrualTime and accrualIntensity must have the same length")
}
if (any(accrualIntensity < 0)) {
stop("accrualIntensity must be non-negative")
}
if (piecewiseSurvivalTime[1] != 0) {
stop("piecewiseSurvivalTime must start with 0")
}
if (nintervals > 1 && any(diff(piecewiseSurvivalTime) <= 0)) {
stop("piecewiseSurvivalTime should be increasing")
}
if (any(gamma1 < 0)) {
stop("gamma1 must be non-negative")
}
if (any(gamma2 < 0)) {
stop("gamma2 must be non-negative")
}
if (length(gamma1) != 1 && length(gamma1) != nintervals) {
stop("Invalid length for gamma1")
}
if (length(gamma2) != 1 && length(gamma2) != nintervals) {
stop("Invalid length for gamma2")
}
if (length(gamma1) == 1) {
gamma1 = rep(gamma1, nintervals)
}
if (length(gamma2) == 1) {
gamma2 = rep(gamma2, nintervals)
}
if (!is.na(accrualDuration) && accrualDuration <= 0) {
stop("accrualDuration must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# function to obtain the info for mean difference
f_info <- function(tau, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r) {
# total number of enrolled subjects at interim analysis
n = accrual(tau, accrualTime, accrualIntensity, accrualDuration)
# total number of enrolled subjects at each time point
ns = accrual(tau - t, accrualTime, accrualIntensity, accrualDuration)
# probability of not dropping out at each time point by treatment
q1 = ptpwexp(t, piecewiseSurvivalTime, gamma1, lower.tail = FALSE)
q2 = ptpwexp(t, piecewiseSurvivalTime, gamma2, lower.tail = FALSE)
# number of subjects remaining at each time point by treatment
m1 = r*ns*q1
m2 = (1-r)*ns*q2
# number of subjects dropping out at each time point by treatment
n1 = c(-diff(m1), m1[k])
n2 = c(-diff(m2), m2[k])
# information matrix in each treatment group
I1 = 0
I2 = 0
I = matrix(0, k, k)
for (j in 1:k) {
I[1:j, 1:j] = solve(covar1[1:j, 1:j])
I1 = I1 + n1[j]*I
I[1:j, 1:j] = solve(covar2[1:j, 1:j])
I2 = I2 + n2[j]*I
}
# variance for the treatment mean at the last time point by treatment
V1 = solve(I1)
V2 = solve(I2)
# information for treatment difference
IMax = 1/(V1[k,k] + V2[k,k])
# variance for treatment difference for one sampling unit
v1 = (V1[k,k] + V2[k,k])*n
# inflation factors for each treatment
phi1 = V1[k,k]/(covar1[k,k]/(r*n))
phi2 = V2[k,k]/(covar2[k,k]/((1-r)*n))
# information for treatment mean difference at the last time point
list(IMax = IMax, v1 = v1, phi1 = phi1, phi2 = phi2)
}
# power calculation
if (is.na(beta)) {
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
}
studyDuration = accrualDuration + t[k]
out = f_info(studyDuration, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r)
IMax = out$IMax
phi1 = out$phi1
phi2 = out$phi2
des = getDesign(
beta, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta + meanDiffH0
des$byStageResults$futilityMeanDiff = delta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff = -delta + meanDiffH0
des$byStageResults$futilityMeanDiff = -delta + meanDiffH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta + meanDiffH0
}
}
} else { # sample size calculation
des = getDesign(
beta, NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
IMax = des$overallResults$information
out = f_info(1 + t[k], k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, 1, r)
v1 = out$v1
phi1 = out$phi1
phi2 = out$phi2
n = IMax*v1
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
# power for t-test
f = function(n, r, phi1, phi2, alpha, theta, v1) {
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
pt(b, nu, ncp, lower.tail = FALSE)
}
n = uniroot(function(n) {
f(n, r, phi1, phi2, alpha, theta, v1) - (1-beta)
}, c(0.5*n, 1.5*n))$root
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (is.na(accrualDuration)) {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else { # calculate accrualIntensity
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.001, 240))$root*accrualIntensity
}
# final maximum information
IMax = n/v1
nu = n*r/phi1 + n*(1-r)/phi2 - 2
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff = delta + meanDiffH0
des$byStageResults$futilityMeanDiff = delta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff = -delta + meanDiffH0
des$byStageResults$futilityMeanDiff = -delta + meanDiffH0
}
} else {
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (is.na(accrualDuration)) {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else { # calculate accrualIntensity
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.001, 240))$root*accrualIntensity
}
# final maximum information
IMax = n/v1
des = getDesign(
NA, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (directionUpper) {
des$byStageResults$efficacyMeanDiff =
des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
des$byStageResults$futilityTheta + meanDiffH0
} else {
des$byStageResults$efficacyMeanDiff =
-des$byStageResults$efficacyTheta + meanDiffH0
des$byStageResults$futilityMeanDiff =
-des$byStageResults$futilityTheta + meanDiffH0
}
}
}
# timing of interim analysis
studyDuration = accrualDuration + t[k]
information = IMax*informationRates
analysisTime = rep(0, kMax)
if (kMax > 1) {
for (i in 1:(kMax-1)) {
analysisTime[i] = uniroot(function(tau) {
out = f_info(tau, k, t, covar1, covar2, accrualTime,
accrualIntensity, piecewiseSurvivalTime,
gamma1, gamma2, accrualDuration, r)
out$IMax - information[i]
}, c(t[k]+0.001, studyDuration))$root
}
}
analysisTime[kMax] = studyDuration
numberOfSubjects = accrual(analysisTime, accrualTime, accrualIntensity,
accrualDuration)
p = des$byStageResults$rejectPerStage +
des$byStageResults$futilityPerStage
pH0 = des$byStageResults$rejectPerStageH0 +
des$byStageResults$futilityPerStageH0
des$overallResults$numberOfSubjects = n
des$overallResults$studyDuration = studyDuration
des$overallResults$expectedNumberOfSubjectsH1 = sum(p*numberOfSubjects)
des$overallResults$expectedNumberOfSubjectsH0 = sum(pH0*numberOfSubjects)
des$overallResults$expectedStudyDurationH1 = sum(p*analysisTime)
des$overallResults$expectedStudyDurationH0 = sum(pH0*analysisTime)
des$overallResults$accrualDuration = accrualDuration
des$overallResults$followupTime = t[k]
des$overallResults$fixedFollowup = TRUE
des$overallResults$meanDiffH0 = meanDiffH0
des$overallResults$meanDiff = meanDiff
des$byStageResults$numberOfSubjects = numberOfSubjects
des$byStageResults$analysisTime = analysisTime
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$accrualTime = accrualTime
des$settings$accrualIntensity = accrualIntensity
des$settings$piecewiseSurvivalTime = piecewiseSurvivalTime
des$settings$gamma1 = gamma1
des$settings$gamma2 = gamma2
des$settings$k = k
des$settings$t = t
des$settings$covar1 = covar1
des$settings$covar2 = covar2
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designMeanDiffMMRM"
des
}
#' @title Power and Sample Size for Direct Treatment Effects in Crossover
#' Trials Accounting for Carryover Effects
#' @description Obtains the power and sample size for direct treatment
#' effects in crossover trials accounting for carryover effects.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param meanDiffH0 The mean difference at the last time point
#' under the null hypothesis. Defaults to 0.
#' @param meanDiff The mean difference at the last time point
#' under the alternative hypothesis.
#' @param stDev The standard deviation for within-subject random error.
#' @param corr The intra-subject correlation due to subject random effect.
#' @param design The crossover design represented by a matrix with
#' rows indexing the sequences, columns indexing the periods, and
#' matrix entries indicating the treatments.
#' @param cumdrop The cumulative dropout rate over periods.
#' @param allocationRatioPlanned Allocation ratio for the sequences.
#' Defaults to equal randomization if not provided.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @details
#' The linear mixed-effects model to assess the direct treatment effect
#' in the presence of carryover treatment effect is given by
#' \deqn{y_{ijk} = \mu + \alpha_i + b_{ij} + \gamma_k + \tau_{d(i,k)}
#' + \lambda_{c(i,k-1)} + e_{ijk},}
#' \deqn{i=1,\ldots,n; j=1,\ldots,r_i; k = 1,\ldots,p; d,c = 1,\ldots,t,}
#' where \eqn{\mu} is the general mean, \eqn{\alpha_i} is the effect of
#' the \eqn{i}th treatment sequence, \eqn{b_{ij}} is the random effect
#' with variance \eqn{\sigma_b^2} for the \eqn{j}the subject of the
#' \eqn{i}th treatment sequence, \eqn{\gamma_k} is the period effect,
#' and \eqn{e_{ijk}} is the random error with variance \eqn{\sigma^2}
#' for the subject in period \eqn{k}. The direct effect of the treatment
#' administered in period \eqn{k} of sequence \eqn{i} is
#' \eqn{\tau_{d(i,k)}}, and \eqn{\lambda_{c(i,k-1)}} is the carryover
#' effect of the treatment administered in period \eqn{k-1} of sequence
#' \eqn{i}. The value of the carryover effect for the observed
#' response in the first period is \eqn{\lambda_{c(i,0)} = 0} since
#' there is no carryover effect in the first period. The intra-subject
#' correlation due to the subject random effect is
#' \deqn{\rho = \frac{\sigma_b^2}{\sigma_b^2 + sigma^2}.}
#' By constructing the design matrix \eqn{X} for the linear model with
#' a compound symmetry covariance matrix for the response vector of
#' a subject, we can obtain \deqn{Var(\hat{\beta}) = (X'V^{-1}X)^{-1}.}
#'
#' The covariance matrix for the direct treatment effects and the
#' carryover treatment effects can be extracted from the appropriate
#' sub-matrices. The covariance matrix for the direct treatment effects
#' without accounting for the carryover treatment effects can be obtained
#' by omitting the carryover effect terms from the model.
#'
#' The power and relative efficiency are for the direct treatment
#' effect comparing the first treatment to the last treatment
#' accounting for carryover effects.
#'
#' The degrees of freedom for the t-test can be calculated as the
#' total number of observations minus the number of subjects minus
#' \eqn{p-1} minus \eqn{2(t-1)} to account for the subject effect,
#' period effect, and direct and carryover treatment effects.
#'
#' @return An S3 class \code{designMeanDiffCarryover} object with the
#' following components:
#'
#' * \code{power}: The power to reject the null hypothesis.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{numberOfSubjects}: The maximum number of subjects.
#'
#' * \code{meanDiffH0}: The mean difference under the null hypothesis.
#'
#' * \code{meanDiff}: The mean difference under the alternative
#' hypothesis.
#'
#' * \code{stDev}: The standard deviation for within-subject random error.
#'
#' * \code{corr}: The intra-subject correlation due to subject random effect.
#'
#' * \code{design}: The crossover design represented by a matrix with
#' rows indexing the sequences, columns indexing the periods, and
#' matrix entries indicating the treatments.
#'
#' * \code{nseq}: The number of sequences.
#'
#' * \code{nprd}: The number of periods.
#'
#' * \code{ntrt}: The number of treatments.
#'
#' * \code{cumdrop}: The cumulative dropout rate over periods.
#'
#' * \code{V_direct_only}: The covariance matrix for direct treatment
#' effects without accounting for carryover effects.
#'
#' * \code{V_direct_carry}: The covariance matrix for direct and
#' carryover treatment effects.
#'
#' * \code{v_direct_only}: The variance of direct treatment effects
#' without accounting for carryover effects.
#'
#' * \code{v_direct}: The variance of direct treatment effects
#' accounting for carryover effects.
#'
#' * \code{v_carry}: The variance of carryover treatment effects.
#'
#' * \code{releff_direct}: The relative efficiency of the design
#' for estimating direct treatment effects after accounting
#' for carryover effects with respect to that without
#' accounting for carryover effects. This is equal to
#' \code{v_direct_only/v_direct}.
#'
#' * \code{releff_carry}: The relative efficiency of the design
#' for estimating carryover effects. This is equal to
#' \code{v_direct_only/v_carry}.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the sequences.
#'
#' * \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Robert O. Kuehl. Design of Experiments: Statistical Principles of
#' Research Design and Analysis. Brooks/Cole: Pacific Grove, CA. 2000.
#'
#' @examples
#'
#' # Williams design for 4 treatments
#'
#' (design1 = getDesignMeanDiffCarryover(
#' beta = 0.2, n = NA,
#' meanDiff = 0.5, stDev = 1,
#' design = matrix(c(1, 4, 2, 3,
#' 2, 1, 3, 4,
#' 3, 2, 4, 1,
#' 4, 3, 1, 2),
#' 4, 4, byrow = TRUE),
#' alpha = 0.025))
#'
#' @export
#'
getDesignMeanDiffCarryover <- function(
beta = NA_real_,
n = NA_real_,
meanDiffH0 = 0,
meanDiff = 0.5,
stDev = 1,
corr = 0.5,
design = NA_real_,
cumdrop = NA_real_,
allocationRatioPlanned = NA_real_,
normalApproximation = FALSE,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between -1 and 1")
}
if (any(is.na(design))) {
stop("design must be provided")
}
nseq = nrow(design)
nprd = ncol(design)
ntrt = length(unique(c(design)))
if (any(design <= 0 | design != round(design)) ||
!all.equal(unique(c(design)), 1:ntrt)) {
stop(paste("Elements of design must be positive integers",
"ranging from 1 to", ntrt))
}
# number of model parameters consisting of the intercept, sequence effects,
# period effects, direct treatment effects, and carryover treatment effects
q = 1 + (nseq-1) + (nprd-1) + (ntrt-1) + (ntrt-1)
# start of direct treatment effect
l = 1 + (nseq-1) + (nprd-1) + 1
# end of direct treatment effect
m = 1 + (nseq-1) + (nprd-1) + (ntrt-1)
if (q > nseq*nprd) {
stop("The crossover design is overparameterized")
}
if (any(is.na(cumdrop))) {
cumdrop = rep(0, nseq)
}
if (any(cumdrop < 0)) {
stop("Elements of cumdrop must be nonnegative")
} else if (nprd > 1 && any(diff(cumdrop) < 0)) {
stop("Elements of cumdrop must be nondecreasing")
} else if (any(cumdrop >= 1)) {
stop("Elements of cumdrop must be less than 1")
}
# observed data pattern probabilities
p = diff(c(cumdrop, 1))
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, nrow(design))
}
if (length(allocationRatioPlanned) != nseq-1 &&
length(allocationRatioPlanned) != nseq) {
stop(paste("allocationRatioPlanned should have", nseq-1,
"or", nseq, "elements"))
}
if (length(allocationRatioPlanned) == nseq-1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
# treatment sequence randomization probabilities
r = allocationRatioPlanned/sum(allocationRatioPlanned)
directionUpper = meanDiff > meanDiffH0
theta = ifelse(directionUpper, meanDiff - meanDiffH0,
meanDiffH0 - meanDiff)
# model design matrix with carryover effect
X = matrix(0, nseq*nprd, q)
X[,1] = 1
for (i in 1:nseq) {
for (j in 1:nprd) {
k = (i-1)*nprd + j
# sequence effects
if (i < nseq) {
X[k, 1 + i] = 1
}
# period effects
if (j < nprd) {
X[k, 1 + (nseq-1) + j] = 1
}
# direct treatment effects
if (design[i,j] < ntrt) {
X[k, 1 + (nseq-1) + (nprd-1) + design[i,j]] = 1
}
# carryover treatment effects
if (j > 1 && design[i,j-1] < ntrt) {
X[k, 1 + (nseq-1) + (nprd-1) + (ntrt-1) + design[i,j-1]] = 1
}
}
}
# compound symmetry covariance matrix for repeated measures
Sigma = stDev^2/(1-corr) * ((1-corr)*diag(nprd) + corr)
# information matrix for model parameters with carryover effects
I = 0
for (i in 1:nseq) {
offset = (i-1)*nprd
J = 0
for (j in 1:nprd) {
idx = offset + (1:j)
if (j==1) {
x = t(as.matrix(X[idx,]))
} else {
x = X[idx,]
}
J = J + p[j]*t(x) %*% solve(Sigma[1:j,1:j]) %*% x
}
I = I + r[i]*J
}
# covariance matrix for model parameters with carryover effects
V = solve(I)
# variance for direct treatment effect for one sampling unit
v1 = V[l,l]
# information matrix for model parameters without carryover effects
X0 = X[, 1:m]
I0 = 0
for (i in 1:nseq) {
offset = (i-1)*nprd
J = 0
for (j in 1:nprd) {
idx = offset+(1:j)
if (j==1) {
x = t(as.matrix(X0[idx,]))
} else {
x = X0[idx,]
}
J = J + p[j]*t(x) %*% solve(Sigma[1:j,1:j]) %*% x
}
I0 = I0 + r[i]*J
}
# covariance matrix for model parameters without carryover effects
V0 = solve(I0)
# power for t test
f = function(n) {
# residual degrees of freedom after accounting for the subject effects
mean_nprd = sum(p*(1:nprd))
nu = n*mean_nprd - 1 - (n-1) - (nprd-1) - (ntrt-1) - (ntrt-1)
b = qt(1-alpha, nu)
ncp = theta*sqrt(n/v1)
pt(b, nu, ncp, lower.tail = FALSE)
}
if (is.na(n)) { # calculate the sample size
if (normalApproximation) {
n = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
} else {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, 1.5*n0))$root
}
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
if (normalApproximation) {
power = pnorm(theta*sqrt(n/v1) - qnorm(1-alpha))
} else {
power = f(n)
}
rownames(design) = paste0("Seq", 1:nseq)
colnames(design) = paste0("Prd", 1:nprd)
des = list(
power = power, alpha = alpha, numberOfSubjects = n,
meanDiffH0 = meanDiffH0, meanDiff = meanDiff,
stDev = stDev, corr = corr, design = design,
nseq = nseq, nprd = nprd, ntrt = ntrt,
cumdrop = cumdrop,
V_direct_only = V0[l:m,l:m]/n,
V_direct_carry = V[l:q, l:q]/n,
v_direct_only = V0[l,l]/n,
v_direct = V[l,l]/n,
v_carry = V[m+1,m+1]/n,
releff_direct = V0[l,l]/V[l,l],
releff_carry = V0[l,l]/V[m+1,m+1],
allocationRatioPlanned = allocationRatioPlanned,
normalApproximation = normalApproximation,
rounding = rounding)
attr(des, "class") = "designMeanDiffCarryover"
des
}
#' @title Power and Sample Size for One-Way ANOVA
#' @description Obtains the power and sample size for one-way
#' analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The common standard deviation.
#' @param allocationRatioPlanned Allocation ratio for the treatment
#' groups. It has length \code{ngroups - 1} or \code{ngroups}. If it is
#' of length \code{ngroups - 1}, then the last treatment group will
#' assume value 1 for allocation ratio.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @details
#'
#' Let \eqn{\{\mu_i: i=1,\ldots,k\}} denote the group means, and
#' \eqn{\{r_i: i=1,\ldots,k\}} denote the randomization probabilities
#' to the \eqn{k} treatment groups. Let \eqn{\sigma} denote the
#' common standard deviation, and \eqn{n} denote the total sample
#' size. Then the \eqn{F}-statistic
#' \deqn{F = \frac{SSR/(k-1)}{SSE/(n-k)}
#' \sim F_{k-1, n-k, \lambda},} where
#' \deqn{\lambda = n \sum_{i=1}^k r_i (\mu_i - \bar{\mu})^2/\sigma^2}
#' is the noncentrality parameter, and
#' \eqn{\bar{\mu} = \sum_{i=1}^k r_i \mu_i}.
#'
#' @return An S3 class \code{designANOVA} object with the following
#' components:
#'
#' * \code{power}: The power to reject the null hypothesis that
#' there is no difference among the treatment groups.
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the treatment
#' groups.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignANOVA(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 3.5, allocationRatioPlanned = c(2, 2, 2, 1),
#' alpha = 0.05))
#'
#' @export
#'
getDesignANOVA <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
allocationRatioPlanned = NA_real_,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (!length(means) == ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, ngroups)
}
if (length(allocationRatioPlanned) != ngroups - 1 &&
length(allocationRatioPlanned) != ngroups) {
stop(paste("allocationRatioPlanned should have", ngroups - 1,
"or", ngroups, "elements"))
}
if (length(allocationRatioPlanned) == ngroups - 1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
r = allocationRatioPlanned/sum(allocationRatioPlanned)
mubar = sum(r*means)
vmu = sum(r*(means - mubar)^2)
# power for F-test
f <- function(n) {
lambda = n*vmu/stDev^2
b = qf(1 - alpha, ngroups - 1, n - ngroups)
pf(b, ngroups - 1, n - ngroups, lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = ngroups - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/(vmu/stDev^2)
while (f(n0) < 1-beta) n0 <- 2*n0
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
effectsize = vmu/stDev^2,
allocationRatioPlanned = allocationRatioPlanned,
rounding = rounding)
attr(des, "class") = "designANOVA"
des
}
#' @title Power and Sample Size for Two-Way ANOVA
#' @description Obtains the power and sample size for two-way
#' analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param nlevelsA The number of groups for Factor A.
#' @param nlevelsB The number of levels for Factor B.
#' @param means The matrix of treatment means for Factors A and B
#' combination.
#' @param stDev The common standard deviation.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @return An S3 class \code{designTwoWayANOVA} object with the following
#' components:
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{nlevelsA}: The number of levels for Factor A.
#'
#' * \code{nlevelsB}: The number of levels for Factor B.
#'
#' * \code{means}: The matrix of treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{effectsizeA}: The effect size for Factor A.
#'
#' * \code{effectsizeB}: The effect size for Factor B.
#'
#' * \code{effectsizeAB}: The effect size for Factor A and Factor B
#' interaction.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' * \code{powerdf}: The data frame containing the power and sample size
#' results. It has the following variables:
#'
#' - \code{n}: The sample size.
#'
#' - \code{powerA}: The power to reject the null hypothesis that
#' there is no difference among Factor A levels.
#'
#' - \code{powerB}: The power to reject the null hypothesis that
#' there is no difference among Factor B levels.
#'
#' - \code{powerAB}: The power to reject the null hypothesis that
#' there is no interaction between Factor A and Factor B.
#'
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignTwoWayANOVA(
#' beta = 0.1, nlevelsA = 2, nlevelsB = 2,
#' means = matrix(c(0.5, 4.7, 0.4, 6.9), 2, 2, byrow = TRUE),
#' stDev = 2, alpha = 0.05))
#'
#' @export
#'
getDesignTwoWayANOVA <- function(
beta = NA_real_,
n = NA_real_,
nlevelsA = 2,
nlevelsB = 2,
means = NA_real_,
stDev = 1,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (nlevelsA < 2 || nlevelsA != round(nlevelsA)) {
stop("nlevelsA must be a positive integer >= 2")
}
if (nlevelsB < 2 || nlevelsB != round(nlevelsB)) {
stop("nlevelsB must be a positive integer >= 2")
}
if (any(is.na(means))) {
stop("means must be provided")
}
if (!is.matrix(means) || nrow(means) != nlevelsA ||
ncol(means) != nlevelsB) {
stop(paste("means must be a matrix with", nlevelsA, "rows and",
nlevelsB, "columns"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
muA = as.numeric(rowMeans(means))
muB = as.numeric(colMeans(means))
mu = mean(means)
vmuA = var(muA)*(nlevelsA - 1)/nlevelsA
vmuB = var(muB)*(nlevelsB - 1)/nlevelsB
mmuA = matrix(muA, nlevelsA, nlevelsB)
mmuB = matrix(muB, nlevelsA, nlevelsB, byrow = TRUE)
vmuAB = sum((means - mmuA - mmuB + mu)^2)/(nlevelsA*nlevelsB)
effectsizeA = vmuA/stDev^2
effectsizeB = vmuB/stDev^2
effectsizeAB = vmuAB/stDev^2
# power for F-test for Factor A
fA <- function(n) {
lambda = n*effectsizeA
b = qf(1-alpha, nlevelsA - 1, n - nlevelsA*nlevelsB)
pf(b, nlevelsA - 1, n - nlevelsA*nlevelsB, lambda, lower.tail = FALSE)
}
# power for F-test for Factor A
fB <- function(n) {
lambda = n*effectsizeB
b = qf(1-alpha, nlevelsB - 1, n - nlevelsA*nlevelsB)
pf(b, nlevelsB - 1, n - nlevelsA*nlevelsB, lambda, lower.tail = FALSE)
}
# power for F-test for Factor A and Factor B interaction
fAB <- function(n) {
lambda = n*effectsizeAB
b = qf(1-alpha, (nlevelsA - 1)*(nlevelsB - 1), n - nlevelsA*nlevelsB)
pf(b, (nlevelsA - 1)*(nlevelsB - 1), n - nlevelsA*nlevelsB,
lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = nlevelsA - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeA
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fA(n0) < 1-beta) n0 <- 2*n0
nA = uniroot(function(n) fA(n) - (1-beta), c(0.5*n0, n0))$root
nu = nlevelsB - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeB
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fB(n0) < 1-beta) n0 <- 2*n0
nB = uniroot(function(n) fB(n) - (1-beta), c(0.5*n0, n0))$root
nu = (nlevelsA - 1)*(nlevelsB - 1)
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/effectsizeAB
n0 = max(n0, nlevelsA*nlevelsB + 1)
while (fAB(n0) < 1-beta) n0 <- 2*n0
nAB = uniroot(function(n) fAB(n) - (1-beta), c(0.5*n0, n0))$root
} else {
nA = n
nB = n
nAB = n
}
if (rounding) {
nA = ceiling(nA - 1.0e-12)
nB = ceiling(nB - 1.0e-12)
nAB = ceiling(nAB - 1.0e-12)
}
m = unique(c(nA, nB, nAB))
powerdf = data.frame(n = m, powerA = sapply(m, fA),
powerB = sapply(m, fB), powerAB = sapply(m, fAB))
des = list(
alpha = alpha, nlevelsA = nlevelsA,
nlevelsB = nlevelsB, means = means, stDev = stDev,
effectsizeA = effectsizeA, effectsizeB = effectsizeB,
effectsizeAB = effectsizeAB,
rounding = rounding,
powerdf = powerdf)
attr(des, "class") = "designTwoWayANOVA"
des
}
#' @title Power and Sample Size for One-Way ANOVA Contrast
#' @description Obtains the power and sample size for a single contrast
#' in one-way analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The common standard deviation.
#' @param contrast The coefficients for the single contrast.
#' @param meanContrastH0 The mean of the contrast under the
#' null hypothesis.
#' @param allocationRatioPlanned Allocation ratio for the treatment
#' groups. It has length \code{ngroups - 1} or \code{ngroups}. If it is
#' of length \code{ngroups - 1}, then the last treatment group will
#' assume value 1 for allocation ratio.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @return An S3 class \code{designANOVAContrast} object with the following
#' components:
#'
#' * \code{power}: The power to reject the null hypothesis for the
#' treatment contrast.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The common standard deviation.
#'
#' * \code{contrast}: The coefficients for the single contrast.
#'
#' * \code{meanContrastH0}: The mean of the contrast under the null
#' hypothesis.
#'
#' * \code{meanContrast}: The mean of the contrast under the alternative
#' hypothesis.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{allocationRatioPlanned}: Allocation ratio for the treatment
#' groups.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignANOVAContrast(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 3.5, contrast = c(1, 1, 1, -3)/3,
#' allocationRatioPlanned = c(2, 2, 2, 1),
#' alpha = 0.025))
#'
#' @export
#'
getDesignANOVAContrast <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
contrast = NA_real_,
meanContrastH0 = 0,
allocationRatioPlanned = NA_real_,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (length(means) != ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (any(is.na(contrast))) {
stop("contrast must be provided")
}
if (length(contrast) != ngroups) {
stop(paste("contrast must have", ngroups, "elements"))
}
if (any(is.na(allocationRatioPlanned))) {
allocationRatioPlanned = rep(1, ngroups)
}
if (length(allocationRatioPlanned) != ngroups - 1 &&
length(allocationRatioPlanned) != ngroups) {
stop(paste("allocationRatioPlanned should have", ngroups - 1,
"or", ngroups, "elements"))
}
if (length(allocationRatioPlanned) == ngroups - 1) {
allocationRatioPlanned = c(allocationRatioPlanned, 1)
}
if (any(allocationRatioPlanned <= 0)) {
stop("Elements of allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
r = allocationRatioPlanned/sum(allocationRatioPlanned)
meanContrast = sum(contrast*means)
v1 = sum(contrast^2/r)*stDev^2
directionUpper = meanContrast > meanContrastH0
theta = ifelse(directionUpper, meanContrast - meanContrastH0,
meanContrastH0 - meanContrast)
# power for t-test
f <- function(n) {
b = qt(1-alpha, n-ngroups)
ncp = theta*sqrt(n/v1)
power = pt(b, n-ngroups, ncp, lower.tail = FALSE)
}
if (is.na(n)) {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(n0, 2*n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
contrast = contrast, meanContrastH0 = meanContrastH0,
meanContrast = meanContrast, effectsize = theta^2/v1,
allocationRatioPlanned = allocationRatioPlanned,
rounding = rounding)
attr(des, "class") = "designANOVAContrast"
des
}
#' @title Power and Sample Size for Repeated-Measures ANOVA
#' @description Obtains the power and sample size for one-way repeated
#' measures analysis of variance. Each subject takes all treatments
#' in the longitudinal study.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The total standard deviation.
#' @param corr The correlation among the repeated measures.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The two-sided significance level. Defaults to 0.05.
#'
#' @details
#' Let \eqn{y_{ij}} denote the measurement under treatment condition
#' \eqn{j (j=1,\ldots,k)} for subject \eqn{i (i=1,\ldots,n)}. Then
#' \deqn{y_{ij} = \alpha + \beta_j + b_i + e_{ij},} where \eqn{b_i}
#' denotes the subject random effect, \eqn{b_i \sim N(0, \sigma_b^2),}
#' and \eqn{e_{ij} \sim N(0, \sigma_e^2)} denotes the within-subject
#' residual. If we set \eqn{\beta_k = 0}, then \eqn{\alpha} is the
#' mean of the last treatment (control), and \eqn{\beta_j} is the
#' difference in means between the \eqn{j}th treatment and the control
#' for \eqn{j=1,\ldots,k-1}.
#'
#' The repeated measures have a compound symmetry covariance structure.
#' Let \eqn{\sigma^2 = \sigma_b^2 + \sigma_e^2}, and
#' \eqn{\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma_e^2}}. Then
#' \eqn{Var(y_i) = \sigma^2 \{(1-\rho) I_k + \rho 1_k 1_k^T\}}.
#' Let \eqn{X_i} denote the design matrix for subject \eqn{i}.
#' Let \eqn{\theta = (\alpha, \beta_1, \ldots, \beta_{k-1})^T}.
#' It follows that
#' \deqn{Var(\hat{\theta}) = \left(\sum_{i=1}^{n} X_i^T V_i^{-1}
#' X_i\right)^{-1}.} It can be shown that
#' \deqn{Var(\hat{\beta}) = \frac{\sigma^2 (1-\rho)}{n} (I_{k-1} +
#' 1_{k-1} 1_{k-1}^T).} It follows that
#' \eqn{\hat{\beta}^T \hat{V}_{\hat{\beta}}^{-1} \hat{\beta} \sim
#' F_{k-1,(n-1)(k-1), \lambda},} where the noncentrality parameter for
#' the \eqn{F} distribution is \deqn{\lambda =
#' \beta^T V_{\hat{\beta}}^{-1} \beta = \frac{n \sum_{j=1}^{k}
#' (\mu_j - \bar{\mu})^2}{\sigma^2(1-\rho)}.}
#'
#' @return An S3 class \code{designRepeatedANOVA} object with the
#' following components:
#'
#' * \code{power}: The power to reject the null hypothesis that
#' there is no difference among the treatment groups.
#'
#' * \code{alpha}: The two-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The total standard deviation.
#'
#' * \code{corr}: The correlation among the repeated measures.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignRepeatedANOVA(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 5, corr = 0.2, alpha = 0.05))
#'
#' @export
#'
getDesignRepeatedANOVA <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
corr = 0,
rounding = TRUE,
alpha = 0.05) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (!length(means) == ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between 0 and 1")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
vmu = var(means)*(ngroups-1)/ngroups
# power for F-test
f <- function(n) {
lambda = n*ngroups*vmu/(stDev^2*(1-corr))
b = qf(1 - alpha, ngroups - 1, (n-1)*(ngroups-1))
pf(b, ngroups - 1, (n-1)*(ngroups-1), lambda, lower.tail = FALSE)
}
if (is.na(n)) {
nu = ngroups - 1
n0 = (qchisq(1-alpha, nu) - nu + qnorm(1-beta)*sqrt(2*nu))/
(ngroups*vmu/(stDev^2*(1-corr)))
while (f(n0) < 1-beta) n0 <- 2*n0
n = uniroot(function(n) f(n) - (1-beta), c(0.5*n0, n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev,
corr = corr, effectsize = ngroups*vmu/(stDev^2*(1-corr)),
rounding = rounding)
attr(des, "class") = "designRepeatedANOVA"
des
}
#' @title Power and Sample Size for One-Way Repeated Measures ANOVA Contrast
#' @description Obtains the power and sample size for a single contrast
#' in one-way repeated measures analysis of variance.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param ngroups The number of treatment groups.
#' @param means The treatment group means.
#' @param stDev The total standard deviation.
#' @param corr The correlation among the repeated measures.
#' @param contrast The coefficients for the single contrast.
#' @param meanContrastH0 The mean of the contrast under the
#' null hypothesis.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @param alpha The one-sided significance level. Defaults to 0.025.
#'
#' @return An S3 class \code{designRepeatedANOVAContrast} object with
#' the following components:
#'
#' * \code{power}: The power to reject the null hypothesis for the
#' treatment contrast.
#'
#' * \code{alpha}: The one-sided significance level.
#'
#' * \code{n}: The number of subjects.
#'
#' * \code{ngroups}: The number of treatment groups.
#'
#' * \code{means}: The treatment group means.
#'
#' * \code{stDev}: The total standard deviation.
#'
#' * \code{corr}: The correlation among the repeated measures.
#'
#' * \code{contrast}: The coefficients for the single contrast.
#'
#' * \code{meanContrastH0}: The mean of the contrast under the null
#' hypothesis.
#'
#' * \code{meanContrast}: The mean of the contrast under the alternative
#' hypothesis.
#'
#' * \code{effectsize}: The effect size.
#'
#' * \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignRepeatedANOVAContrast(
#' beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
#' stDev = 5, corr = 0.2, contrast = c(1, 1, 1, -3)/3,
#' alpha = 0.025))
#'
#' @export
#'
getDesignRepeatedANOVAContrast <- function(
beta = NA_real_,
n = NA_real_,
ngroups = 2,
means = NA_real_,
stDev = 1,
corr = 0,
contrast = NA_real_,
meanContrastH0 = 0,
rounding = TRUE,
alpha = 0.025) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (length(means) != ngroups) {
stop(paste("means must have", ngroups, "elements"))
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (corr <= -1 || corr >= 1) {
stop("corr must lie between -1 and 1")
}
if (any(is.na(contrast))) {
stop("contrast must be provided")
}
if (length(contrast) != ngroups) {
stop(paste("contrast must have", ngroups, "elements"))
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
meanContrast = sum(contrast*means)
v1 = sum(contrast^2)*stDev^2*(1-corr)
directionUpper = meanContrast > meanContrastH0
theta = ifelse(directionUpper, meanContrast - meanContrastH0,
meanContrastH0 - meanContrast)
# power for t-test
f <- function(n) {
b = qt(1-alpha, (n-1)*(ngroups-1))
ncp = theta*sqrt(n/v1)
power = pt(b, (n-1)*(ngroups-1), ncp, lower.tail = FALSE)
}
if (is.na(n)) {
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) f(n) - (1-beta), c(n0, 2*n0))$root
}
if (rounding) {
n = ceiling(n - 1.0e-12)
}
power = f(n)
des = list(
power = power, alpha = alpha, n = n,
ngroups = ngroups, means = means, stDev = stDev, corr = corr,
contrast = contrast, meanContrastH0 = meanContrastH0,
meanContrast = meanContrast, effectsize = theta^2/v1,
rounding = rounding)
attr(des, "class") = "designRepeatedANOVAContrast"
des
}
#' @title Group Sequential Design for One-Sample Slope
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for one-sample slope.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param slopeH0 The slope under the null hypothesis.
#' Defaults to 0.
#' @param slope The slope under the alternative hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevCovariate The standard deviation of the covariate.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designOneSlope} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{slopeH0}: The slope under the null hypothesis.
#'
#' - \code{slope}: The slope under the alternative hypothesis.
#'
#' - \code{stDev}: The standard deviation of the residual.
#'
#' - \code{stDevCovariate}: The standard deviation of the covariate.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlope}: The efficacy boundaries on the slope scale.
#'
#' - \code{futilitySlope}: The futility boundaries on the slope scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignOneSlope(
#' beta = 0.1, n = NA, slope = 0.5,
#' stDev = 15, stDevCovariate = 9,
#' normalApproximation = FALSE,
#' alpha = 0.025))
#'
#' @export
getDesignOneSlope <- function(
beta = NA_real_,
n = NA_real_,
slopeH0 = 0,
slope = 0.5,
stDev = 1,
stDevCovariate = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (stDevCovariate <= 0) {
stop("stDevCovariate must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
directionUpper = slope > slopeH0
theta = ifelse(directionUpper, slope - slopeH0, slopeH0 - slope)
# variance for one sampling unit
v1 = stDev^2/stDevCovariate^2
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlope = delta + slopeH0
des$byStageResults$futilitySlope = delta + slopeH0
} else {
des$byStageResults$efficacySlope = -delta + slopeH0
des$byStageResults$futilitySlope = -delta + slopeH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlope =
des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
des$byStageResults$futilityTheta + slopeH0
} else {
des$byStageResults$efficacySlope =
-des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
-des$byStageResults$futilityTheta + slopeH0
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
pt(b, n-2, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-2)
ncp = theta*sqrt(n/v1)
power = pt(b, n-2, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlope = delta + slopeH0
des$byStageResults$futilitySlope = delta + slopeH0
} else {
des$byStageResults$efficacySlope = -delta + slopeH0
des$byStageResults$futilitySlope = -delta + slopeH0
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlope =
des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
des$byStageResults$futilityTheta + slopeH0
} else {
des$byStageResults$efficacySlope =
-des$byStageResults$efficacyTheta + slopeH0
des$byStageResults$futilitySlope =
-des$byStageResults$futilityTheta + slopeH0
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$slopeH0 = slopeH0
des$overallResults$slope = slope
des$overallResults$stDev = stDev
des$overallResults$stDevCovariate = stDevCovariate
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designOneSlope"
des
}
#' @title Group Sequential Design for Two-Sample Slope Difference
#' @description Obtains the power given sample size or obtains the sample
#' size given power for a group sequential design for two-sample slope
#' difference.
#'
#' @param beta The type II error.
#' @param n The total sample size.
#' @param slopeDiffH0 The slope difference under the null hypothesis.
#' Defaults to 0.
#' @param slopeDiff The slope difference under the alternative hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevCovariate The standard deviation of the covariate.
#' @param allocationRatioPlanned Allocation ratio for the active treatment
#' versus control. Defaults to 1 for equal randomization.
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The exact
#' calculation using the t distribution is only implemented for the
#' fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Fixed prior to the trial.
#' Defaults to \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @return An S3 class \code{designSlopeDiff} object with three components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{slopeDiffH0}: The slope difference under the null hypothesis.
#'
#' - \code{slopeDiff}: The slope difference under the alternative
#' hypothesis.
#'
#' - \code{stDev}: The standard deviation of the residual.
#'
#' - \code{stDevCovariate}: The standard deviation of the covariate.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlopeDiff}: The efficacy boundaries on the slope
#' difference scale.
#'
#' - \code{futilitySlopeDiff}: The futility boundaries on the slope
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: Allocation ratio for the active
#' treatment versus control.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @examples
#'
#' (design1 <- getDesignSlopeDiff(
#' beta = 0.1, n = NA, slopeDiff = -0.5,
#' stDev = 10, stDevCovariate = 6,
#' normalApproximation = FALSE, alpha = 0.025))
#'
#' @export
getDesignSlopeDiff <- function(
beta = NA_real_,
n = NA_real_,
slopeDiffH0 = 0,
slopeDiff = 0.5,
stDev = 1,
stDevCovariate = 1,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
if (is.na(beta) && is.na(n)) {
stop("beta and n cannot be both missing")
}
if (!is.na(beta) && !is.na(n)) {
stop("Only one of beta and n should be provided")
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (!is.na(n) && n <= 0) {
stop("n must be positive")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = slopeDiff > slopeDiffH0
theta = ifelse(directionUpper, slopeDiff - slopeDiffH0,
slopeDiffH0 - slopeDiff)
# variance for one sampling unit
v1 = stDev^2/stDevCovariate^2/(r*(1-r))
if (is.na(beta)) { # power calculation
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = round(n*informationRates)/n
}
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
power = pt(b, n-4, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta
des$byStageResults$futilitySlopeDiff = delta
} else {
des$byStageResults$efficacySlopeDiff = -delta
des$byStageResults$futilitySlopeDiff = -delta
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta
}
}
} else { # sample size calculation
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
n0 = (qnorm(1-alpha) + qnorm(1-beta))^2*v1/theta^2
n = uniroot(function(n) {
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
pt(b, n-4, ncp, lower.tail = FALSE) - (1 - beta)
}, c(n0, 2*n0))$root
if (rounding) n = ceiling(n - 1.0e-12)
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
b = qt(1-alpha, n-4)
ncp = theta*sqrt(n/v1)
power = pt(b, n-4, ncp, lower.tail = FALSE)
delta = b/sqrt(n/v1)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta
des$byStageResults$futilitySlopeDiff = delta
} else {
des$byStageResults$efficacySlopeDiff = -delta
des$byStageResults$futilitySlopeDiff = -delta
}
} else {
des = getDesign(
beta, IMax = NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
n = des$overallResults$information*v1
if (rounding) {
n = ceiling(n - 1.0e-12)
informationRates = des$byStageResults$informationRates
informationRates = round(n*informationRates)/n
des = getDesign(
beta = NA, IMax = n/v1, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta
}
}
}
des$overallResults$theta = theta
des$overallResults$numberOfSubjects = n
des$overallResults$expectedNumberOfSubjectsH1 =
des$overallResults$expectedInformationH1*v1
des$overallResults$expectedNumberOfSubjectsH0 =
des$overallResults$expectedInformationH0*v1
des$overallResults$slopeDiffH0 = slopeDiffH0
des$overallResults$slopeDiff = slopeDiff
des$overallResults$stDev = stDev
des$overallResults$stDevCovariate = stDevCovariate
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$byStageResults$numberOfSubjects =
des$byStageResults$informationRates*n
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designSlopeDiff"
des
}
#' @title Group Sequential Design for Two-Sample Slope Difference
#' From the MMRM Model
#' @description Obtains the power given sample size or obtains the sample
#' size given power for two-sample slope difference from the growth curve
#' MMRM model.
#'
#' @param beta The type II error.
#' @param slopeDiffH0 The slope difference under the null hypothesis.
#' Defaults to 0.
#' @param slopeDiff The slope difference under the alternative
#' hypothesis.
#' @param stDev The standard deviation of the residual.
#' @param stDevIntercept The standard deviation of the random intercept.
#' @param stDevSlope The standard deviation of the random slope.
#' @param corrInterceptSlope The correlation between the random
#' intercept and random slope.
#' @param w The number of time units per measurement visit in a period.
#' @param N The number of measurement visits in a period.
#' @inheritParams param_accrualTime
#' @inheritParams param_accrualIntensity
#' @inheritParams param_piecewiseSurvivalTime
#' @inheritParams param_gamma1
#' @inheritParams param_gamma2
#' @inheritParams param_accrualDuration
#' @inheritParams param_followupTime
#' @inheritParams param_allocationRatioPlanned
#' @param normalApproximation The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution. The
#' degrees of freedom for the t-distribution for testing the slope
#' difference is calculated using the containment method, and
#' is equal to the total number of observations minus two times
#' the total number of subjects. The exact calculation using the
#' t distribution is only implemented for the fixed design.
#' @param rounding Whether to round up sample size. Defaults to 1 for
#' sample size rounding.
#' @inheritParams param_kMax
#' @param informationRates The information rates. Defaults to
#' \code{(1:kMax) / kMax} if left unspecified.
#' @inheritParams param_efficacyStopping
#' @inheritParams param_futilityStopping
#' @inheritParams param_criticalValues
#' @inheritParams param_alpha
#' @inheritParams param_typeAlphaSpending
#' @inheritParams param_parameterAlphaSpending
#' @inheritParams param_userAlphaSpending
#' @inheritParams param_futilityBounds
#' @inheritParams param_typeBetaSpending
#' @inheritParams param_parameterBetaSpending
#' @inheritParams param_userBetaSpending
#' @param spendingTime A vector of length \code{kMax} for the error spending
#' time at each analysis. Defaults to missing, in which case, it is the
#' same as \code{informationRates}.
#'
#' @details
#'
#' We use the following random-effects model to compare two slopes:
#' \deqn{y_{ij} = \alpha + (\beta + \gamma x_i) t_j + a_i + b_i t_j
#' + e_{ij},} where
#'
#' * \eqn{\alpha}: overall intercept common across treatment groups
#' due to randomization
#'
#' * \eqn{\beta}: slope for the control group
#'
#' * \eqn{\gamma}: difference in slopes between the active treatment and
#' control groups
#'
#' * \eqn{x_i}: treatment indicator for subject \eqn{i},
#' 1 for the active treatment and 0 for the control
#'
#' * \eqn{t_j}: time point \eqn{j} for repeated measurements,
#' \eqn{t_1 = 0 < t_2 < \ldots < t_k}
#'
#' * \eqn{(a_i, b_i)}: random intercept and random slope
#' for subject \eqn{i}, \eqn{Var(a_i) = \sigma_a^2},
#' \eqn{Var(b_i) = \sigma_b^2}, \eqn{Corr(a_i, b_i) = \rho}
#'
#' * \eqn{e_{ij}}: within-subject residual with variance \eqn{\sigma_e^2}
#'
#' By accounting for randomization, we improve the efficiency for
#' estimating the difference in slopes. We also allow for non-equal
#' spacing of the time points and missing data due to dropouts.
#'
#' @return An S3 class \code{designSlopeDiffMMRM} object with three
#' components:
#'
#' * \code{overallResults}: A data frame containing the following variables:
#'
#' - \code{overallReject}: The overall rejection probability.
#'
#' - \code{alpha}: The overall significance level.
#'
#' - \code{attainedAlpha}: The attained significance level, which is
#' different from the overall significance level in the presence of
#' futility stopping.
#'
#' - \code{kMax}: The number of stages.
#'
#' - \code{theta}: The parameter value.
#'
#' - \code{information}: The maximum information.
#'
#' - \code{expectedInformationH1}: The expected information under H1.
#'
#' - \code{expectedInformationH0}: The expected information under H0.
#'
#' - \code{drift}: The drift parameter, equal to
#' \code{theta*sqrt(information)}.
#'
#' - \code{inflationFactor}: The inflation factor (relative to the
#' fixed design).
#'
#' - \code{numberOfSubjects}: The maximum number of subjects.
#'
#' - \code{studyDuration}: The maximum study duration.
#'
#' - \code{expectedNumberOfSubjectsH1}: The expected number of subjects
#' under H1.
#'
#' - \code{expectedNumberOfSubjectsH0}: The expected number of subjects
#' under H0.
#'
#' - \code{expectedStudyDurationH1}: The expected study duration
#' under H1.
#'
#' - \code{expectedStudyDurationH0}: The expected study duration
#' under H0.
#'
#' - \code{accrualDuration}: The accrual duration.
#'
#' - \code{followupTime}: The follow-up time.
#'
#' - \code{fixedFollowup}: Whether a fixed follow-up design is used.
#'
#' - \code{slopeDiffH0}: The slope difference under H0.
#'
#' - \code{slopeDiff}: The slope difference under H1.
#'
#' * \code{byStageResults}: A data frame containing the following variables:
#'
#' - \code{informationRates}: The information rates.
#'
#' - \code{efficacyBounds}: The efficacy boundaries on the Z-scale.
#'
#' - \code{futilityBounds}: The futility boundaries on the Z-scale.
#'
#' - \code{rejectPerStage}: The probability for efficacy stopping.
#'
#' - \code{futilityPerStage}: The probability for futility stopping.
#'
#' - \code{cumulativeRejection}: The cumulative probability for efficacy
#' stopping.
#'
#' - \code{cumulativeFutility}: The cumulative probability for futility
#' stopping.
#'
#' - \code{cumulativeAlphaSpent}: The cumulative alpha spent.
#'
#' - \code{efficacyP}: The efficacy boundaries on the p-value scale.
#'
#' - \code{futilityP}: The futility boundaries on the p-value scale.
#'
#' - \code{information}: The cumulative information.
#'
#' - \code{efficacyStopping}: Whether to allow efficacy stopping.
#'
#' - \code{futilityStopping}: Whether to allow futility stopping.
#'
#' - \code{rejectPerStageH0}: The probability for efficacy stopping
#' under H0.
#'
#' - \code{futilityPerStageH0}: The probability for futility stopping
#' under H0.
#'
#' - \code{cumulativeRejectionH0}: The cumulative probability for
#' efficacy stopping under H0.
#'
#' - \code{cumulativeFutilityH0}: The cumulative probability for futility
#' stopping under H0.
#'
#' - \code{efficacySlopeDiff}: The efficacy boundaries on the slope
#' difference scale.
#'
#' - \code{futilitySlopeDiff}: The futility boundaries on the slope
#' difference scale.
#'
#' - \code{numberOfSubjects}: The number of subjects.
#'
#' - \code{analysisTime}: The average time since trial start.
#'
#' * \code{settings}: A list containing the following input parameters:
#'
#' - \code{typeAlphaSpending}: The type of alpha spending.
#'
#' - \code{parameterAlphaSpending}: The parameter value for alpha
#' spending.
#'
#' - \code{userAlphaSpending}: The user defined alpha spending.
#'
#' - \code{typeBetaSpending}: The type of beta spending.
#'
#' - \code{parameterBetaSpending}: The parameter value for beta spending.
#'
#' - \code{userBetaSpending}: The user defined beta spending.
#'
#' - \code{spendingTime}: The error spending time at each analysis.
#'
#' - \code{allocationRatioPlanned}: The allocation ratio for the active
#' treatment versus control.
#'
#' - \code{accrualTime}: A vector that specifies the starting time of
#' piecewise Poisson enrollment time intervals.
#'
#' - \code{accrualIntensity}: A vector of accrual intensities.
#' One for each accrual time interval.
#'
#' - \code{piecewiseSurvivalTime}: A vector that specifies the
#' starting time of piecewise exponential survival time intervals.
#'
#' - \code{gamma1}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the active treatment group.
#'
#' - \code{gamma2}: The hazard rate for exponential dropout or
#' a vector of hazard rates for piecewise exponential dropout
#' for the control group.
#'
#' - \code{w}: The number of time units per measurement visit in a
#' period.
#'
#' - \code{N}: The number of measurement visits in a period.
#'
#' - \code{stdDev}: The standard deviation of the residual.
#'
#' - \code{G}: The covariance matrix for the random intercept and
#' random slope.
#'
#' - \code{normalApproximation}: The type of computation of the p-values.
#' If \code{TRUE}, the variance is assumed to be known, otherwise
#' the calculations are performed with the t distribution.
#'
#' - \code{rounding}: Whether to round up sample size.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Daniel O. Scharfstein, Anastasios A. Tsiatis, and James M. Robins.
#' Semiparametric efficiency and its implication on the
#' design and analysis of group-sequential studies.
#' Journal of the American Statistical Association 1997; 92:1342-1350.
#'
#' @examples
#'
#' (design1 <- getDesignSlopeDiffMMRM(
#' beta = 0.2, slopeDiff = log(1.15)/52,
#' stDev = sqrt(.182),
#' stDevIntercept = sqrt(.238960),
#' stDevSlope = sqrt(.000057),
#' corrInterceptSlope = .003688/sqrt(.238960*.000057),
#' w = 8,
#' N = 10000,
#' accrualIntensity = 15,
#' gamma1 = 1/(4.48*52),
#' gamma2 = 1/(4.48*52),
#' accrualDuration = NA,
#' followupTime = 8,
#' alpha = 0.025))
#'
#' @export
getDesignSlopeDiffMMRM <- function(
beta = NA_real_,
slopeDiffH0 = 0,
slopeDiff = 0.5,
stDev = 1,
stDevIntercept = 1,
stDevSlope = 1,
corrInterceptSlope = 0.5,
w = NA_real_,
N = NA_real_,
accrualTime = 0,
accrualIntensity = NA_real_,
piecewiseSurvivalTime = 0,
gamma1 = 0,
gamma2 = 0,
accrualDuration = NA_real_,
followupTime = NA_real_,
allocationRatioPlanned = 1,
normalApproximation = TRUE,
rounding = TRUE,
kMax = 1L,
informationRates = NA_real_,
efficacyStopping = NA_integer_,
futilityStopping = NA_integer_,
criticalValues = NA_real_,
alpha = 0.025,
typeAlphaSpending = "sfOF",
parameterAlphaSpending = NA_real_,
userAlphaSpending = NA_real_,
futilityBounds = NA_real_,
typeBetaSpending = "none",
parameterBetaSpending = NA_real_,
userBetaSpending = NA_real_,
spendingTime = NA_real_) {
m = length(w)
cumN = c(0, cumsum(N))
cumwN = c(0, cumsum(w*N))
nintervals = length(piecewiseSurvivalTime)
if (is.na(beta) + is.na(accrualDuration) + is.na(followupTime) > 1) {
stop(paste("At most one can be missing among beta, accrualDuration,",
"and followupTime"))
}
if (is.na(beta)) {
unknown = "beta"
} else if (is.na(accrualDuration)) {
unknown = "accrualDuration"
} else if (is.na(followupTime)) {
unknown = "followupTime"
} else {
unknown = "accrualIntensity"
}
if (!is.na(beta) && (beta >= 1-alpha || beta < 0.0001)) {
stop("beta must lie in [0.0001, 1-alpha)")
}
if (any(w <= 0)) {
stop("Elements of w must be positive")
}
if (length(N) != m) {
stop("w and N must have the same length")
}
if (any(N <= 0)) {
stop("Elements of N must be positive")
}
if (sum(N) < 10000) {
stop("The max # of measurements must be >=10000 to enable root finding")
}
if (stDev <= 0) {
stop("stDev must be positive")
}
if (stDevIntercept <= 0) {
stop("stDevIntercept must be positive")
}
if (stDevSlope <= 0) {
stop("stDevSlope must be positive")
}
if (corrInterceptSlope <= -1 || corrInterceptSlope >= 1) {
stop("corrInterceptSlope must lie between -1 and 1")
}
if (accrualTime[1] != 0) {
stop("accrualTime must start with 0")
}
if (length(accrualTime) > 1 && any(diff(accrualTime) <= 0)) {
stop("accrualTime should be increasing")
}
if (any(is.na(accrualIntensity))) {
stop("accrualIntensity must be provided")
}
if (length(accrualTime) != length(accrualIntensity)) {
stop("accrualTime and accrualIntensity must have the same length")
}
if (any(accrualIntensity < 0)) {
stop("accrualIntensity must be non-negative")
}
if (piecewiseSurvivalTime[1] != 0) {
stop("piecewiseSurvivalTime must start with 0")
}
if (nintervals > 1 && any(diff(piecewiseSurvivalTime) <= 0)) {
stop("piecewiseSurvivalTime should be increasing")
}
if (any(gamma1 < 0)) {
stop("gamma1 must be non-negative")
}
if (any(gamma2 < 0)) {
stop("gamma2 must be non-negative")
}
if (length(gamma1) != 1 && length(gamma1) != nintervals) {
stop("Invalid length for gamma1")
}
if (length(gamma2) != 1 && length(gamma2) != nintervals) {
stop("Invalid length for gamma2")
}
if (length(gamma1) == 1) {
gamma1 = rep(gamma1, nintervals)
}
if (length(gamma2) == 1) {
gamma2 = rep(gamma2, nintervals)
}
if (!is.na(accrualDuration) && accrualDuration <= 0) {
stop("accrualDuration must be positive")
}
if (!is.na(followupTime) && followupTime < 0) {
stop("followupTime must be nonnegative")
}
if (allocationRatioPlanned <= 0) {
stop("allocationRatioPlanned must be positive")
}
if (alpha < 0.00001 || alpha >= 1) {
stop("alpha must lie in [0.00001, 1)")
}
if (any(is.na(informationRates))) {
informationRates = (1:kMax)/kMax
}
r = allocationRatioPlanned/(1 + allocationRatioPlanned)
directionUpper = slopeDiff > slopeDiffH0
theta = ifelse(directionUpper, slopeDiff - slopeDiffH0,
slopeDiffH0 - slopeDiff)
v11 = stDevIntercept^2
v12 = corrInterceptSlope*stDevIntercept*stDevSlope
v22 = stDevSlope^2
G = matrix(c(v11, v12, v12, v22), 2, 2)
# function to obtain the info for slope difference
f_info <- function(tau, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r) {
# total number of enrolled subjects at interim analysis
n = accrual(tau, accrualTime, accrualIntensity, accrualDuration)
i = findInterval(tau, cumwN)
k = floor((tau - cumwN[i])/w[i]) + cumN[i] + 1
j = 1:k
i = pmax(findInterval(j-2, cumN), 1) # period indicators
t = cumwN[i] + (j - cumN[i] - 1)*w[i] # time points
# total number of enrolled subjects at each time point
ns = accrual(tau - t, accrualTime, accrualIntensity, accrualDuration)
# probability of not dropping out at each time point by treatment
q1 = ptpwexp(t, piecewiseSurvivalTime, gamma1, lower.tail = FALSE)
q2 = ptpwexp(t, piecewiseSurvivalTime, gamma2, lower.tail = FALSE)
# number of subjects remaining at each time point by treatment
m1 = r*ns*q1
m2 = (1-r)*ns*q2
# number of subjects dropping out at each time point by treatment
n1 = c(-diff(m1), m1[k])
n2 = c(-diff(m2), m2[k])
Z = matrix(c(rep(1,k), t), k, 2)
covar = Z %*% G %*% t(Z) + stDev^2*diag(k)
X1 = matrix(c(rep(1,k), t, t), k, 3)
X2 = matrix(c(rep(1,k), t, rep(0,k)), k, 3)
# information matrix per subject
I1 = 0
I2 = 0
for (j in 1:k) {
x1 = X1[1:j, , drop=FALSE]
x2 = X2[1:j, , drop=FALSE]
I = solve(covar[1:j, 1:j])
I1 = I1 + n1[j]*t(x1) %*% I %*% x1
I2 = I2 + n2[j]*t(x2) %*% I %*% x2
}
# variance for common intercept, control slope, and slope difference
V = solve(I1 + I2)
# information for slope difference
IMax = 1/V[3,3]
# residual degrees of freedom
nu = sum((n1 + n2)*(1:k)) - 2*n
list(IMax = IMax, nu = nu)
}
# power calculation
if (is.na(beta)) {
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
}
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
nu = out$nu
des = getDesign(
beta, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = delta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff = -delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = -delta + slopeDiffH0
}
} else {
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta + slopeDiffH0
}
}
} else { # sample size calculation
des = getDesign(
beta, NA, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
IMax = des$overallResults$information
if (kMax == 1 && !normalApproximation) { # t-test for fixed design
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root*accrualIntensity
}
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
nu = out$nu # this serves as the lower bound for degrees of freedom
if (!any(is.na(criticalValues))) {
alpha = 1 - pnorm(criticalValues)
}
# power for t-test
b = qt(1-alpha, nu)
info = uniroot(function(info) {
ncp = theta*sqrt(info)
pt(b, nu, ncp, lower.tail = FALSE) - (1-beta)
}, c(0.5*IMax, 1.5*IMax))$root
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - info
}, c(0.5*accrualDuration, 1.5*accrualDuration))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - info
}, c(0.5*followupTime, 1.5*followupTime))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - info
}, c(0.5, 1.5))$root*accrualIntensity
}
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
if (unknown == "accrualDuration" || unknown == "followupTime") {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else {
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.5, 1.5))$root*accrualIntensity
}
}
# update maximum information and degrees of freedom
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
nu = out$nu
b = qt(1-alpha, nu)
ncp = theta*sqrt(IMax)
power = pt(b, nu, ncp, lower.tail = FALSE)
des$overallResults$overallReject = power
des$byStageResults$rejectPerStage = power
des$byStageResults$futilityPerStage = 1 - power
des$byStageResults$cumulativeRejection = power
des$byStageResults$cumulativeFutility = 1 - power
des$byStageResults$efficacyBounds = b
des$byStageResults$futilityBounds = b
delta = b/sqrt(IMax)
if (directionUpper) {
des$byStageResults$efficacySlopeDiff = delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = delta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff = -delta + slopeDiffH0
des$byStageResults$futilitySlopeDiff = -delta + slopeDiffH0
}
} else {
if (unknown == "accrualDuration") {
accrualDuration = uniroot(function(x) {
studyDuration = x + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
x, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else if (unknown == "followupTime") {
followupTime = uniroot(function(x) {
studyDuration = accrualDuration + x
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root
} else {
accrualIntensity = uniroot(function(x) {
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, x*accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - IMax
}, c(0.001, 240))$root*accrualIntensity
}
n = accrual(accrualDuration, accrualTime, accrualIntensity,
accrualDuration)
if (rounding) {
n = ceiling(n - 1.0e-12)
if (unknown == "accrualDuration" || unknown == "followupTime") {
accrualDuration = getAccrualDurationFromN(n, accrualTime,
accrualIntensity)
} else {
accrualIntensity = uniroot(function(x) {
accrual(accrualDuration, accrualTime, x*accrualIntensity,
accrualDuration) - n
}, c(0.5, 1.5))$root*accrualIntensity
}
# final maximum information
studyDuration = accrualDuration + followupTime
out = f_info(studyDuration, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
IMax = out$IMax
des = getDesign(
NA, IMax, theta,
kMax, informationRates,
efficacyStopping, futilityStopping,
criticalValues, alpha, typeAlphaSpending,
parameterAlphaSpending, userAlphaSpending,
futilityBounds, typeBetaSpending,
parameterBetaSpending, userBetaSpending,
spendingTime)
}
if (directionUpper) {
des$byStageResults$efficacySlopeDiff =
des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
des$byStageResults$futilityTheta + slopeDiffH0
} else {
des$byStageResults$efficacySlopeDiff =
-des$byStageResults$efficacyTheta + slopeDiffH0
des$byStageResults$futilitySlopeDiff =
-des$byStageResults$futilityTheta + slopeDiffH0
}
}
}
# timing of interim analysis
studyDuration = accrualDuration + followupTime
information = IMax*informationRates
analysisTime = rep(0, kMax)
if (kMax > 1) {
for (i in 1:(kMax-1)) {
analysisTime[i] = uniroot(function(tau) {
out = f_info(tau, w, m, cumN, cumwN, stDev, G,
accrualTime, accrualIntensity,
piecewiseSurvivalTime, gamma1, gamma2,
accrualDuration, r)
out$IMax - information[i]
}, c(w[1]+0.001, studyDuration))$root
}
}
analysisTime[kMax] = studyDuration
numberOfSubjects = accrual(analysisTime, accrualTime, accrualIntensity,
accrualDuration)
p = des$byStageResults$rejectPerStage +
des$byStageResults$futilityPerStage
pH0 = des$byStageResults$rejectPerStageH0 +
des$byStageResults$futilityPerStageH0
des$overallResults$numberOfSubjects = n
des$overallResults$studyDuration = studyDuration
des$overallResults$expectedNumberOfSubjectsH1 = sum(p*numberOfSubjects)
des$overallResults$expectedNumberOfSubjectsH0 = sum(pH0*numberOfSubjects)
des$overallResults$expectedStudyDurationH1 = sum(p*analysisTime)
des$overallResults$expectedStudyDurationH0 = sum(pH0*analysisTime)
des$overallResults$accrualDuration = accrualDuration
des$overallResults$followupTime = followupTime
des$overallResults$fixedFollowup = FALSE
des$overallResults$slopeDiffH0 = slopeDiffH0
des$overallResults$slopeDiff = slopeDiff
des$byStageResults$numberOfSubjects = numberOfSubjects
des$byStageResults$analysisTime = analysisTime
des$byStageResults$efficacyTheta = NULL
des$byStageResults$futilityTheta = NULL
des$settings$allocationRatioPlanned = allocationRatioPlanned
des$settings$accrualTime = accrualTime
des$settings$accrualIntensity = accrualIntensity
des$settings$piecewiseSurvivalTime = piecewiseSurvivalTime
des$settings$gamma1 = gamma1
des$settings$gamma2 = gamma2
des$settings$w = w
des$settings$N = N
des$settings$stDev = stDev
des$settings$G = G
des$settings$normalApproximation = normalApproximation
des$settings$rounding = rounding
des$settings$varianceRatio = NULL
attr(des, "class") = "designSlopeDiffMMRM"
des
}
#' @title Hedges' g Effect Size
#' @description Obtains Hedges' g estimate and confidence interval of
#' effect size.
#'
#' @param tstat The value of the t-test statistic for comparing two
#' treatment conditions.
#' @param m The degrees of freedom for the t-test.
#' @param ntilde The normalizing sample size to convert the
#' standardized treatment difference to the t-test statistic, i.e.,
#' \code{tstat = sqrt(ntilde)*meanDiff/stDev}.
#' @param cilevel The confidence interval level. Defaults to 0.95.
#'
#' @details
#'
#' Hedges' \eqn{g} is an effect size measure commonly used in meta-analysis
#' to quantify the difference between two groups. It's an improvement
#' over Cohen's \eqn{d}, particularly when dealing with small sample sizes.
#'
#' The formula for Hedges' \eqn{g} is \deqn{g = c(m) d,} where \eqn{d}
#' is Cohen's \eqn{d} effect size estimate, and \eqn{c(m)} is the bias
#' correction factor, \deqn{d = (\hat{\mu}_1 - \hat{\mu}_2)/\hat{\sigma},}
#' \deqn{c(m) = 1 - \frac{3}{4m-1}.}
#' Since \eqn{c(m) < 1}, Cohen's \eqn{d} overestimates the true effect size.
#' \eqn{\delta = (\mu_1 - \mu_2)/\sigma.}
#' Since \deqn{t = \sqrt{\tilde{n}} d,} we have
#' \deqn{g = \frac{c(m)}{\sqrt{\tilde{n}}} t,} where \eqn{t}
#' has a noncentral \eqn{t} distribution with \eqn{m} degrees of freedom
#' and noncentrality parameter \eqn{\sqrt{\tilde{n}} \delta}.
#'
#' The asymptotic variance of \eqn{g} can be approximated by
#' \deqn{Var(g) = \frac{1}{\tilde{n}} + \frac{g^2}{2m}.}
#' The confidence interval for \eqn{\delta}
#' can be constructed using normal approximation.
#'
#' For two-sample mean difference with sample size \eqn{n_1} for the
#' treatment group and \eqn{n_2} for the control group, we have
#' \eqn{\tilde{n} = \frac{n_1n_2}{n_1+n_2}} and \eqn{m=n_1+n_2-2}
#' for pooled variance estimate.
#'
#' @return A data frame with the following variables:
#'
#' * \code{tstat}: The value of the \code{t} test statistic.
#'
#' * \code{m}: The degrees of freedom for the t-test.
#'
#' * \code{ntilde}: The normalizing sample size to convert the
#' standardized treatment difference to the t-test statistic.
#'
#' * \code{g}: Hedges' \code{g} effect size estimate.
#'
#' * \code{varg}: Variance of \code{g}.
#'
#' * \code{lower}: The lower confidence limit for effect size.
#'
#' * \code{upper}: The upper confidence limit for effect size.
#'
#' * \code{cilevel}: The confidence interval level.
#'
#' @author Kaifeng Lu, \email{kaifenglu@@gmail.com}
#'
#' @references
#'
#' Larry V. Hedges. Distribution theory for Glass's estimator of
#' effect size and related estimators.
#' Journal of Educational Statistics 1981; 6:107-128.
#'
#' @examples
#'
#' n1 = 7
#' n2 = 8
#' meanDiff = 0.444
#' stDev = 1.201
#' m = n1+n2-2
#' ntilde = n1*n2/(n1+n2)
#' tstat = sqrt(ntilde)*meanDiff/stDev
#'
#' hedgesg(tstat, m, ntilde)
#'
#' @export
hedgesg <- function(tstat, m, ntilde, cilevel = 0.95) {
d = 1/sqrt(ntilde)*tstat
g = (1 - 3/(4*m-1))*d
varg = 1/ntilde + g^2/(2*m)
lower = g - qnorm((1 + cilevel)/2)*sqrt(varg)
upper = g + qnorm((1 + cilevel)/2)*sqrt(varg)
data.frame(tstat, m, ntilde, g, varg, lower, upper, cilevel)
}
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