# R/fix_sample_pow.R In vanLunteren/plan_norm: R-tool for planning clinical studies on normally distributed data

#### Documented in fix_calc

```#' @title
#' Simulation of the sample size , the type I error rate and the power
#'
#' @description
#' This is an auxiliary function.
#' It calculates the sample size , simulated type I error rate and the simulated power of a fixed-sample
#' design.
#' The originally planned sample size is calculated on the basis of an assumed standard deviation.
#' A distinction is made between one-sided and two-sided tests. The test statistic is compared to the
#' quantiles of the t-distribution.
#'
#' @usage
#' fix_sample_pow(sd_ber, delta = 0, Delta, sd, test = 1, alpha = 0.05, beta = 0.2,
#'                simu = 10000)
#'
#' @param sd_ber
#' Vector of numbers. Interval of an area for the actual standard deviation in the data.
#'
#' @param delta
#' Number. Expectation difference of two samples.\cr
#' If you select a Test for superiority/ difference then select 'delta = 0'.\cr
#' Only if you select a Test for non-inferiority you can select 'delta != 0'.\cr
#' Attention: If you chose 'test = 1' and 'delta != 0', the test for non-inferiority will automatically \cr
#' be applied.\cr
#' If not specified, delta is set to 0.\cr
#'
#' @param Delta
#' Number. Relevant difference of expected values in the alternative hypothesis.
#'
#' @param sd
#' Number. Assumed standard deviation of the data. \cr
#' Used to calculate the originally planned sample size.
#'
#' @param test
#' Number. What type of hypothesis test should be performed, one-sided (Superiority/ \cr
#' Non-Inferiority test) or two-sided (Test for difference).\cr
#' One-sided (test = 1): Superiortity H0: mu_x - mu_y <= 0 vs. H1: mu_x - mu_y >0\cr
#'                       Non-Inferiority H0: mu_x - mu_y >= delta vs. H1: mu_x - mu_y < delta\cr
#' Two-sided (test = 2): Difference H0: |mu_x - mu_y| = 0 vs. H1: |mu_x -  mu_y| != 0\cr
#' Attention: Choice of delta. (see \code{delta})\cr
#' If not specified, the one-Sided Test (Superiority/ Non-Inferiority Test) is used.
#'
#' @param alpha
#' Number. Desired alpha-level of the test.\cr
#' If not specified, alpha is set to 0.05.
#'
#' @param beta
#' Number. Acceptable beta error of the test.\cr
#' If not specified, beta is set to 0.2.
#'
#' @param simu
#' Number. How many simulations should be performed?\cr
#' If not specified, simu is set to 10000.
#'
#' @return
#' This function only creates the sample size, type I error rate and Power values for multiple actual
#' standard deviation values.\cr
#' The output is used in the function sample_pow to visualize the sample size, the type I error rate and
#' the power.
#'
#' @author
#' Csilla van Lunteren
#'
#'  @seealso
#'
#' @import stats
#' @export
#'
fix_sample_pow <- function (sd_ber, delta = 0, Delta, sd, test = 1, alpha = 0.05, beta = 0.2,
simu = 10000){

if (delta != 0 & test == 2){
stop("When choosing a two-sided hypothesis test (test for differences), delta = 0 must be
selected.")
}

if (test == 1){
N0 <- ceiling(4 * (stats::qnorm(1 - alpha) + stats::qnorm(1 - beta))^2 * sd^2 /
(Delta - delta)^2)
} else {
N0 <- ceiling(4 * (stats::qnorm(1 - alpha / 2) + stats::qnorm(1 - beta))^2 * sd^2 / Delta^2)
}

mapply(sig <- function (sd_ber){
if (test == 1){
N <- ceiling(4 * (stats::qnorm(1 - alpha) + stats::qnorm(1 - beta))^2 * sd_ber^2 /
(Delta - delta)^2)
} else {
N <- ceiling(4 * (stats::qnorm(1 - alpha / 2) + stats::qnorm(1 - beta))^2 * sd_ber^2 /
Delta^2)
}

abg <- replicate(simu,
fix_calc(N0 = N0, sd_ber = sd_ber, delta = delta, Delta = Delta, test = test,
alpha = alpha, beta = beta))

return (c(N, mean(abg[1, ]), mean(abg[2, ])))
}, sd_ber)
}

#' @title
#' Conducted a study with a fixed sample size and normally distributed data
#'
#' @description
#' This is an auxiliary function of 'fix_sample_pow' and 'pow'.
#' It is conducting a study with fixed sample size and normally distributed data. The calculated value of
#' the test statistic is compared directly with the corresponding quantile of the normal distribution.
#'
#' @usage
#' fix_calc(N0, sd_ber, delta = 0, Delta, test = 1, alpha = 0.05, beta = 0.2)
#'
#' @param N0
#' Number. Size of the originally planned sample size.
#'
#' @param sd_ber
#' Number. Actual standard deviation in the data.
#'
#' @param delta
#' Number. Expectation difference of two samples.\cr
#' If you select a Test for superiority/ difference then select 'delta = 0'.\cr
#' Only if you select a Test for non-inferiority you can select 'delta != 0'.\cr
#' Attention: If you chose 'test = 1' and 'delta != 0', the test for non-inferiority will \cr
#' automatically be applied.\cr
#' If not specified, delta is set to 0.
#'
#' @param Delta
#' Number. Relevant difference of expected values in the alternative hypothesis.
#'
#' @param test
#' Number. What type of hypothesis test should be performed, one-sided (Superiority/ \cr
#' Non-Inferiority test) or two-sided (Test for difference).\cr
#' One-sided (test = 1): Superiortity H0: mu_x - mu_y <= 0 vs. H1: mu_x - mu_y > 0\cr
#'                       Non-Inferiority H0: mu_x - mu_y >= delta vs. H1: mu_x - mu_y < delta\cr
#' Two-sided (test = 2): Difference H0: |mu_x - mu_y| = 0 vs. H1: |mu_x -  mu_y| != 0\cr
#' Attention: Choice of delta. (see \code{delta})\cr
#' If not specified, the one-Sided Test (Superiority/ Non-Inferiority Test) is used.
#'
#' @param alpha
#' Number. Desired alpha-level of the test.\cr
#' If not specified, alpha is set to 0.05.
#'
#' @param beta
#' Number. Acceptable beta error of the test.\cr
#' If not specified, beta is set to 0.2.
#'
#' @return
#' This function returns a value if the simulation study rejects (1) or can not reject (0) the \cr
#' null hypothesis.
#'
#' @author
#' Csilla van Lunteren
#'
#'  @seealso
#'
#' @import stats
#' @export
#'
fix_calc <- function(N0, sd_ber, delta = 0, Delta, test = 1, alpha = 0.05, beta = 0.2){

X_h0 <- stats::rnorm(n = N0 / 2, mean = delta, sd = sd_ber)
Y_h0 <- stats::rnorm(n = N0 / 2, mean = 0, sd = sd_ber)

X_h1 <- stats::rnorm(n = N0 / 2, mean = Delta, sd = sd_ber)
Y_h1 <- stats::rnorm(n = N0 / 2, mean = 0, sd = sd_ber)

Sig_gepoolt_h0 <- sqrt(((length(X_h0) - 1) * stats::var(X_h0) +
(length(Y_h0) - 1) * stats::var(Y_h0) )/
(length(X_h0) + length(Y_h0) - 2))
if (length(X_h0) == 1 & length(Y_h0) == 1){
Sig_gepoolt_h0 <- sqrt(stats::var(c(X_h0, Y_h0)))
}

Sig_gepoolt_h1 <- sqrt(((length(X_h1) - 1) * stats::var(X_h1) +
(length(Y_h1) - 1) * stats::var(Y_h1) )/
(length(X_h1) + length(Y_h1) - 2))
if (length(X_h1) == 1 & length(Y_h1) == 1){
Sig_gepoolt_h1 <- sqrt(stats::var(c(X_h1, Y_h1)))
}

T_h0 <- sqrt((length(X_h0) * length(Y_h0)) / (length(X_h0) + length(Y_h0))) *
(mean(X_h0) - mean(Y_h0) - delta) / Sig_gepoolt_h0

T_h1 <- sqrt((length(X_h1) * length(Y_h1)) / (length(X_h1) + length(Y_h1))) *
(mean(X_h1) - mean(Y_h1) - delta) / Sig_gepoolt_h1
return (c(test_h0(test = test, T = T_h0, n1 = N0 / 2, n2 = N0 / 2, alpha = alpha, delta = delta),
test_h0(test = test, T = T_h1, n1 = N0 / 2, n2 = N0 / 2, alpha = alpha, delta = delta)))
}
```
vanLunteren/plan_norm documentation built on May 20, 2019, 12:32 p.m.