getDesignSlopeDiffMMRM: Group Sequential Design for Two-Sample Slope Difference From...

In lrstat: Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

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.

Usage

getDesignSlopeDiffMMRM(
  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_,
  fixedFollowup = FALSE,
  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_
)

Arguments

beta	The type II error.
slopeDiffH0	The slope difference under the null hypothesis. Defaults to 0.
slopeDiff	The slope difference under the alternative hypothesis.
stDev	The standard deviation of the residual.
stDevIntercept	The standard deviation of the random intercept.
stDevSlope	The standard deviation of the random slope.
corrInterceptSlope	The correlation between the random intercept and random slope.
w	The number of time units (e.g. weeks) per measurement visit in a period. In general, visits are more frequent in the beginning of the study and less frequent towards the end.
N	The number of measurement visits in a period. For example, w = c(8, 16) and N = c(2, Inf) means that the response variable will be collected at baseline, week 8, week 16, and every 16 weeks thereafter.
accrualTime	A vector that specifies the starting time of piecewise Poisson enrollment time intervals. Must start with 0, e.g., c(0, 3) breaks the time axis into 2 accrual intervals: [0, 3) and [3, Inf).
accrualIntensity	A vector of accrual intensities. One for each accrual time interval.
piecewiseSurvivalTime	A vector that specifies the starting time of piecewise exponential survival time intervals. Must start with 0, e.g., c(0, 6) breaks the time axis into 2 event intervals: [0, 6) and [6, Inf). Defaults to 0 for exponential distribution.
gamma1	The hazard rate for exponential dropout, or a vector of hazard rates for piecewise exponential dropout for the active treatment group.
gamma2	The hazard rate for exponential dropout, or a vector of hazard rates for piecewise exponential dropout for the control group.
accrualDuration	Duration of the enrollment period.
followupTime	Follow-up time for the last enrolled subject.
fixedFollowup	Whether a fixed follow-up design is used. Defaults to 0 for variable follow-up.
allocationRatioPlanned	Allocation ratio for the active treatment versus control. Defaults to 1 for equal randomization.
normalApproximation	The type of computation of the p-values. If 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.
rounding	Whether to round up sample size. Defaults to 1 for sample size rounding.
kMax	The maximum number of stages.
informationRates	The information rates. Defaults to (1:kMax) / kMax if left unspecified.
efficacyStopping	Indicators of whether efficacy stopping is allowed at each stage. Defaults to true if left unspecified.
futilityStopping	Indicators of whether futility stopping is allowed at each stage. Defaults to true if left unspecified.
criticalValues	Upper boundaries on the z-test statistic scale for stopping for efficacy.
alpha	The significance level. Defaults to 0.025.
typeAlphaSpending	The type of alpha spending. One of the following: "OF" for O'Brien-Fleming boundaries, "P" for Pocock boundaries, "WT" for Wang & Tsiatis boundaries, "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early efficacy stopping. Defaults to "sfOF".
parameterAlphaSpending	The parameter value for the alpha spending. Corresponds to Delta for "WT", rho for "sfKD", and gamma for "sfHSD".
userAlphaSpending	The user defined alpha spending. Cumulative alpha spent up to each stage.
futilityBounds	Lower boundaries on the z-test statistic scale for stopping for futility at stages 1, ..., kMax-1. Defaults to rep(-6, kMax-1) if left unspecified. The futility bounds are non-binding for the calculation of critical values.
typeBetaSpending	The type of beta spending. One of the following: "sfOF" for O'Brien-Fleming type spending function, "sfP" for Pocock type spending function, "sfKD" for Kim & DeMets spending function, "sfHSD" for Hwang, Shi & DeCani spending function, "user" for user defined spending, and "none" for no early futility stopping. Defaults to "none".
parameterBetaSpending	The parameter value for the beta spending. Corresponds to rho for "sfKD", and gamma for "sfHSD".
userBetaSpending	The user defined beta spending. Cumulative beta spent up to each stage.
spendingTime	A vector of length kMax for the error spending time at each analysis. Defaults to missing, in which case, it is the same as informationRates.

Details

We use the following random-effects model to compare two slopes:

y_{ij} = \alpha + (\beta + \gamma x_i) t_j + a_i + b_i t_j + e_{ij}

where

• \alpha: overall intercept common across treatment groups due to randomization

• \beta: slope for the control group

• \gamma: difference in slopes between the active treatment and control groups

• x_i: treatment indicator for subject i, 1 for the active treatment and 0 for the control

• t_j: time point j for repeated measurements, t_1 = 0 < t_2 < \ldots < t_k

• (a_i, b_i): random intercept and random slope for subject i, Var(a_i) = \sigma_a^2, Var(b_i) = \sigma_b^2, Corr(a_i, b_i) = \rho

• e_{ij}: within-subject residual with variance \sigma_e^2

By accounting for randomization, we improve the efficiency for estimating the difference in slopes. The model also accommodates unequally spaced time points and missing data. Specifically, given a calendar time \tau for an interim or final analysis, let k be the number of scheduled time points up to and including \tau, subject to the follow-up duration for fixed follow-up designs. Let the observed time points be t_1, t_2, \ldots, t_k, where t_1 = 0 denotes baseline.

For a subject in treatment group g with observed data pattern j, the design matrix for the fixed effects (\alpha, \beta, \gamma)' is given by

X_{g,j} = (\bm{1}_j, \vec{t}_j, I(g=1)\vec{t}_j)

where \bm{1}_j is a j-vector of ones, and \vec{t}_j = (t_1,\ldots,t_j)' is the column vector of observed time points. The design matrix for the random effects (a_i, b_i)' is

Z_j = (\bm{1}_j, \vec{t}_j)

The variance-covariance matrix of the random effects is

D = \left(\begin{array}{cc} \sigma_a^2 & \rho \sigma_a \sigma_b \\ \rho \sigma_a \sigma_b & \sigma_b^2 \end{array}\right)

Therefore, the variance-covariance matrix for the observed data for the subject is

V_{j} = Z_j D Z_j' + \sigma_e^2 I_j

where I_j is the j\times j identity matrix. Let \pi_g denote the proportion of subjects in group g. The information matrix for the fixed effects is

I = nJ

where

J = \sum_{g=1}^{2} \pi_g \sum_{j=1}^{k} p_{g,j} X_{g,j}' V_j^{-1} X_{g,j}

and p_{g,j} is the proportion of subjects in group g with observed data pattern j.

The variance of the estimator for the slope difference \hat{\gamma} is given by

\text{Var}(\hat{\gamma}) = \frac{1}{n} J^{-1}[3,3]

which can be used to calculate the power and sample size for the group sequential design to detect a slope difference.

Value

An S3 class designSlopeDiffMMRM object with three components:

• overallResults: A data frame containing the following variables:

  • overallReject: The overall rejection probability.

  • alpha: The overall significance level.

  • attainedAlpha: The attained significance level, which is different from the overall significance level in the presence of futility stopping.

  • kMax: The number of stages.

  • theta: The parameter value.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH0: The expected information under H0.

  • drift: The drift parameter, equal to theta*sqrt(information).

  • inflationFactor: The inflation factor (relative to the fixed design).

  • numberOfSubjects: The maximum number of subjects.

  • studyDuration: The maximum study duration.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • expectedStudyDurationH1: The expected study duration under H1.

  • expectedStudyDurationH0: The expected study duration under H0.

  • accrualDuration: The accrual duration.

  • followupTime: The follow-up time.

  • fixedFollowup: Whether a fixed follow-up design is used.

  • slopeDiffH0: The slope difference under H0.

  • slopeDiff: The slope difference under H1.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale.

  • futilityBounds: The futility boundaries on the Z-scale.

  • rejectPerStage: The probability for efficacy stopping.

  • futilityPerStage: The probability for futility stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeFutility: The cumulative probability for futility stopping.

  • cumulativeAlphaSpent: The cumulative alpha spent.

  • efficacyP: The efficacy boundaries on the p-value scale.

  • futilityP: The futility boundaries on the p-value scale.

  • information: The cumulative information.

  • efficacyStopping: Whether to allow efficacy stopping.

  • futilityStopping: Whether to allow futility stopping.

  • rejectPerStageH0: The probability for efficacy stopping under H0.

  • futilityPerStageH0: The probability for futility stopping under H0.

  • cumulativeRejectionH0: The cumulative probability for efficacy stopping under H0.

  • cumulativeFutilityH0: The cumulative probability for futility stopping under H0.

  • efficacySlopeDiff: The efficacy boundaries on the slope difference scale.

  • futilitySlopeDiff: The futility boundaries on the slope difference scale.

  • numberOfSubjects: The number of subjects.

  • analysisTime: The average time since trial start.

• settings: A list containing the following input parameters:

  • typeAlphaSpending: The type of alpha spending.

  • parameterAlphaSpending: The parameter value for alpha spending.

  • userAlphaSpending: The user defined alpha spending.

  • typeBetaSpending: The type of beta spending.

  • parameterBetaSpending: The parameter value for beta spending.

  • userBetaSpending: The user defined beta spending.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: The allocation ratio for the active treatment versus control.

  • accrualTime: A vector that specifies the starting time of piecewise Poisson enrollment time intervals.

  • accrualIntensity: A vector of accrual intensities. One for each accrual time interval.

  • piecewiseSurvivalTime: A vector that specifies the starting time of piecewise exponential survival time intervals.

  • gamma1: The hazard rate for exponential dropout or a vector of hazard rates for piecewise exponential dropout for the active treatment group.

  • gamma2: The hazard rate for exponential dropout or a vector of hazard rates for piecewise exponential dropout for the control group.

  • w: The number of time units per measurement visit in a period.

  • N: The number of measurement visits in a period.

  • stdDev: The standard deviation of the residual.

  • G: The covariance matrix for the random intercept and random slope.

  • normalApproximation: The type of computation of the p-values. If TRUE, the variance is assumed to be known, otherwise the calculations are performed with the t distribution.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, [e-mail]

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))

lrstat documentation built on Aug. 8, 2025, 7:36 p.m.