Type:	Package
Title:	Power and Sample Size Calculation for Non-Proportional Hazards and Beyond
Version:	0.2.15
Date:	2025-07-14
Description:	Performs power and sample size calculation for non-proportional hazards model using the Fleming-Harrington family of weighted log-rank tests. The sequentially calculated log-rank test score statistics are assumed to have independent increments as characterized in Anastasios A. Tsiatis (1982) <doi:10.1080/01621459.1982.10477898>. The mean and variance of log-rank test score statistics are calculated based on Kaifeng Lu (2021) <doi:10.1002/pst.2069>. The boundary crossing probabilities are calculated using the recursive integration algorithm described in Christopher Jennison and Bruce W. Turnbull (2000, ISBN:0849303168). The package can also be used for continuous, binary, and count data. For continuous data, it can handle missing data through mixed-model for repeated measures (MMRM). In crossover designs, it can estimate direct treatment effects while accounting for carryover effects. For binary data, it can design Simon's 2-stage, modified toxicity probability-2 (mTPI-2), and Bayesian optimal interval (BOIN) trials. For count data, it can design group sequential trials for negative binomial endpoints with censoring. Additionally, it facilitates group sequential equivalence trials for all supported data types. Moreover, it can design adaptive group sequential trials for changes in sample size, error spending function, number and spacing or future looks. Finally, it offers various options for adjusted p-values, including graphical and gatekeeping procedures.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Imports:	Rcpp (≥ 1.0.9), mvtnorm (≥ 1.1-3), lpSolve (≥ 5.6.1), shiny (≥ 1.7.1)
LinkingTo:	Rcpp
Suggests:	testthat (≥ 3.0.0), dplyr (≥ 1.1.4), tidyr (≥ 1.3.0), knitr (≥ 1.45), rmarkdown (≥ 2.25), survival (≥ 3.5-7)
VignetteBuilder:	knitr
RoxygenNote:	7.3.2
Encoding:	UTF-8
NeedsCompilation:	yes
Depends:	R (≥ 2.10)
LazyData:	true
URL:	https://github.com/kaifenglu/lrstat
BugReports:	https://github.com/kaifenglu/lrstat/issues
Config/testthat/edition:	3
Packaged:	2025-07-14 05:21:27 UTC; kaife
Author:	Kaifeng Lu [aut, cre]
Maintainer:	Kaifeng Lu <kaifenglu@gmail.com>
Repository:	CRAN
Date/Publication:	2025-07-14 06:20:02 UTC

Power and Sample Size Calculation for Non-Proportional Hazards and Beyond

Description

Performs power and sample size calculation for non-proportional hazards model using the Fleming-Harrington family of weighted log-rank tests.

Details

For proportional hazards, the power is determined by the total number of events and the constant hazard ratio along with information rates and spending functions. For non-proportional hazards, the hazard ratio varies over time and the calendar time plays a key role in determining the mean and variance of the log-rank test score statistic. It requires an iterative algorithm to find the calendar time at which the targeted number of events will be reached for each interim analysis. The lrstat package uses the analytic method in Lu (2021) to find the mean and variance of the weighted log-rank test score statistic at each interim analysis. In addition, the package approximates the variance and covariance matrix of the sequentially calculated log-rank test statistics under the alternative hypothesis with that under the null hypothesis to take advantage of the independent increments structure in Tsiatis (1982) applicable for the Fleming-Harrington family of weighted log-rank tests.

The most useful functions in the package are lrstat, lrpower, lrsamplesize, and lrsim, which calculate the mean and variance of log-rank test score statistic at a sequence of given calendar times, the power of the log-rank test, the sample size in terms of accrual duration and follow-up duration, and the log-rank test simulation, respectively. The accrual function calculates the number of patients accrued at given calendar times. The caltime function finds the calendar times to reach the targeted number of events. The exitprob function calculates the stagewise exit probabilities for specified boundaries with a varying mean parameter over time based on an adaptation of the recursive integration algorithm described in Chapter 19 of Jennison and Turnbull (2000).

The development of the lrstat package is heavily influenced by the rpact package. We find their function arguments to be self-explanatory. We have used the same names whenever appropriate so that users familiar with the rpact package can learn the lrstat package quickly. However, there are notable differences:

• lrstat uses direct approximation, while rpact uses the Schoenfeld method for log-rank test power and sample size calculation.

• lrstat uses accrualDuration to explicitly set the end of accrual period, while rpact incorporates the end of accrual period in accrualTime.

• lrstat considers the trial a failure at the last stage if the log-rank test cannot reject the null hypothesis up to this stage and cannot stop for futility at an earlier stage.

• the lrsim function uses the variance of the log-rank test score statistic as the information.

In addition to the log-rank test power and sample size calculations, the lrstat package can also be used for the following tasks:

• design generic group sequential trials.

• design generic group sequential equivalence trials.

• design adaptive group sequential trials for changes in sample size, error spending function, number and spacing or future looks.

• calculate the terminating and repeated confidence intervals for standard and adaptive group sequential trials.

• calculate the conditional power for non-proportional hazards with or without design changes.

• perform multiplicity adjustment based on graphical approaches using weighted Bonferroni tests, Bonferroni mixture of weighted Simes test, and Bonferroni mixture of Dunnett test as well as group sequential trials with multiple hypotheses.

• perform multiplicity adjustment using stepwise gatekeeping procedures for two sequences of hypotheses and the standard or modified mixture gatekeeping procedures in the general case.

• design parallel-group trials with the primary endpoint analyzed using mixed-model for repeated measures (MMRM).

• design crossover trials to estimate direct treatment effects while accounting for carryover effects.

• design one-way repeated measures ANOVA trials.

• design two-way ANOVA trials.

• design Simon's 2-stage trials.

• design modified toxicity probability-2 (mTPI-2) trials.

• design Bayesian optimal interval (BOIN) trials.

• design group sequential trials for negative binomial endpoints with censoring.

• design trials using Wilcoxon, Fisher's exact, and McNemar's test.

• calculate Clopper-Pearson confidence interval for single proportions.

• calculate Brookmeyer-Crowley confidence interval for quantiles of censored survival data.

• calculate Miettinen & Nurminen confidence interval for stratified risk difference, risk ratio, odds ratio, rate difference, and rate ratio.

• perform power and sample size calculation for logistic regression.

• perform power and sample size calculation for Cohen's kappa.

• calculate Hedges' g effect size.

• generate random numbers from truncated piecewise exponential distribution.

• perform power and sample size calculations for negative binomial data.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Anastasios A. Tsiatis. Repeated significance testing for a general class of statistics used in censored survival analysis. J Am Stat Assoc. 1982;77:855-861.

Christopher Jennison, Bruce W. Turnbull. Group Sequential Methods with Applications to Clinical Trials. Chapman & Hall/CRC: Boca Raton, 2000, ISBN:0849303168

Kaifeng Lu. Sample size calculation for logrank test and prediction of number of events over time. Pharm Stat. 2021;20:229-244.

See Also

rpact, gsDesign

Examples

lrpower(kMax = 2, informationRates = c(0.8, 1),
        criticalValues = c(2.250, 2.025), accrualIntensity = 20,
        piecewiseSurvivalTime = c(0, 6),
        lambda1 = c(0.0533, 0.0309), lambda2 = c(0.0533, 0.0533),
        gamma1 = 0.00427, gamma2 = 0.00427,
        accrualDuration = 22, followupTime = 18)

BOIN Decision Table for Dose-Finding Trials

Description

Generates the decision table for the Bayesian Optimal Interval (BOIN) design, a widely used approach for dose-escalation trials that guides dose-finding decisions based on observed toxicity rates.

Usage

BOINTable(
  nMax = NA_integer_,
  pT = 0.3,
  phi1 = 0.6 * pT,
  phi2 = 1.4 * pT,
  a = 1,
  b = 1,
  pExcessTox = 0.95
)

Arguments

Value

An S3 class BOINTable object with the following components:

• settings: The input settings data frame with the following variables:

  • nMax: The maximum number of subjects in a dose cohort.

  • pT: The target toxicity probability.

  • phi1: The lower equivalence limit for target toxicity probability.

  • phi2: The upper equivalence limit for target toxicity probability.

  • lambda1: The lower decision boundary for observed toxicity probability.

  • lambda2: The upper decision boundary for observed toxicity probability.

  • a: The prior toxicity parameter for the beta prior.

  • b: The prior non-toxicity parameter for the beta prior.

  • pExcessTox: The threshold for excessive toxicity.

• decisionDataFrame: A data frame listing dose-finding decisions for each combination of sample size (n) and number of observed toxicities (y):

  • n: Cohort size.

  • y: Number of observed toxicities.

  • decision: Recommended action: escalate, de-escalate, or stay at the current dose.

• decisionMatrix: A matrix version of the decision table showing the recommended action based on the number of toxicities for each possible cohort size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Liu, S., & Yuan, Y. (2015). Bayesian optimal interval designs for phase I clinical trials. Journal of the Royal Statistical Society: Series C (Applied Statistics), 64(3), 507-523.

Examples

BOINTable(nMax = 18, pT = 0.3, phi = 0.6*0.3, phi2 = 1.4*0.3)

Clopper-Pearson Confidence Interval for One-Sample Proportion

Description

Obtains the Clopper-Pearson exact confidence interval for a one-sample proportion.

Usage

ClopperPearsonCI(n, y, cilevel = 0.95)

Arguments

Value

A data frame with the following variables:

• n: The sample size.

• y: The number of responses.

• phat: The observed proportion of responses.

• lower: The lower limit of the confidence interval.

• upper: The upper limit of the confidence interval.

• cilevel: The confidence interval level.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Clopper, C. J., & Pearson, E. S. (1934). The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika, 26(4), 404-413.

Examples

ClopperPearsonCI(20, 3)

Number of Enrolled Subjects

Description

Obtains the number of subjects enrolled by given calendar times.

Usage

accrual(
  time = NA_real_,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  accrualDuration = NA_real_
)

Arguments

Value

A vector of total number of subjects enrolled by the specified calendar times.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Example 1: Uniform enrollment with 20 patients per month for 12 months.

accrual(time = 3, accrualTime = 0, accrualIntensity = 20,
        accrualDuration = 12)


# Example 2: Piecewise accrual, 10 patients per month for the first
# 3 months, and 20 patients per month thereafter. Patient recruitment
# ends at 12 months for the study.

accrual(time = c(2, 9), accrualTime = c(0, 3),
        accrualIntensity = c(10, 20), accrualDuration = 12)

Number of Patients Enrolled During an Interval and Having an Event by Specified Calendar Times

Description

Obtains the number of patients who are enrolled during a specified enrollment time interval and have an event by the specified calendar times.

Usage

ad(
  time = NA_real_,
  u1 = NA_real_,
  u2 = NA_real_,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  lambda = NA_real_,
  gamma = 0L
)

Arguments

Value

A vector of number of patients who are enrolled during a specified enrollment time interval and have an event by the specified calendar times for a given treatment group had the enrollment being restricted to the treatment group. By definition, we must have time >= u2.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Piecewise accrual, 10 patients per month for the first 3 months, and
# 20 patients per month thereafter. Piecewise exponential survival with
# hazard 0.0533 in the first 6 months, and hazard 0.0309 thereafter,
# and 5% dropout by the end of 1 year.

ad(time = c(9, 15), u1 = 1, u2 = 8, accrualTime = c(0, 3),
   accrualIntensity = c(10, 20), piecewiseSurvivalTime=c(0, 6),
   lambda = c(0.0533, 0.0309), gamma = -log(1-0.05)/12)

Adaptive Design at an Interim Look

Description

Calculates the conditional power for specified incremental information, given the interim results, parameter value, data-dependent changes in the error spending function, and the number and spacing of interim looks. Conversely, calculates the incremental information required to attain a specified conditional power, given the interim results, parameter value, data-dependent changes in the error spending function, and the number and spacing of interim looks.

Usage

adaptDesign(
  betaNew = NA_real_,
  INew = NA_real_,
  L = NA_integer_,
  zL = NA_real_,
  theta = NA_real_,
  IMax = NA_real_,
  kMax = NA_integer_,
  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_,
  spendingTime = NA_real_,
  MullerSchafer = 0L,
  kNew = NA_integer_,
  informationRatesNew = NA_real_,
  efficacyStoppingNew = NA_integer_,
  futilityStoppingNew = NA_integer_,
  typeAlphaSpendingNew = "sfOF",
  parameterAlphaSpendingNew = NA_real_,
  typeBetaSpendingNew = "none",
  parameterBetaSpendingNew = NA_real_,
  userBetaSpendingNew = NA_real_,
  spendingTimeNew = NA_real_,
  varianceRatio = 1
)

Arguments

Value

An adaptDesign object with two list components:

• primaryTrial: A list of selected information for the primary trial, including L, zL, theta, kMax, informationRates, efficacyBounds, futilityBounds, and MullerSchafer.

• secondaryTrial: A design object for the secondary trial.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Lu Chi, H. M. James Hung, and Sue-Jane Wang. Modification of sample size in group sequential clinical trials. Biometrics 1999;55:853-857.

Hans-Helge Muller and Helmut Schafer. Adaptive group sequential designs for clinical trials: Combining the advantages of adaptive and of classical group sequential approaches. Biometrics 2001;57:886-891.

See Also

getDesign

Examples

# original group sequential design with 90% power to detect delta = 6
delta = 6
sigma = 17
n = 282
(des1 = getDesign(IMax = n/(4*sigma^2), theta = delta, kMax = 3,
                  alpha = 0.05, typeAlphaSpending = "sfHSD",
                  parameterAlphaSpending = -4))

# interim look results
L = 1
n1 = n/3
delta1 = 4.5
sigma1 = 20
zL = delta1/sqrt(4/n1*sigma1^2)

t = des1$byStageResults$informationRates

# conditional power with sample size increase
(des2 = adaptDesign(
  betaNew = NA, INew = 420/(4*sigma1^2),
  L = L, zL = zL, theta = delta1,
  IMax = n/(4*sigma1^2), kMax = 3, informationRates = t,
  alpha = 0.05, typeAlphaSpending = "sfHSD",
  parameterAlphaSpending = -4))

# Muller & Schafer (2001) method to design the secondary trial:
# 3-look gamma(-2) spending with 84% power at delta = 4.5 and sigma = 20
(des2 = adaptDesign(
  betaNew = 0.16, INew = NA,
  L = L, zL = zL, theta = delta1,
  IMax = n/(4*sigma1^2), kMax = 3, informationRates = t,
  alpha = 0.05, typeAlphaSpending = "sfHSD",
  parameterAlphaSpending = -4,
  MullerSchafer = TRUE,
  kNew = 3, typeAlphaSpendingNew = "sfHSD",
  parameterAlphaSpendingNew = -2))

# incremental sample size for sigma = 20
(nNew = 4*sigma1^2*des2$secondaryTrial$overallResults$information)

Acute myelogenous leukemia survival data from the survival package

Description

Survival in patients with acute myelogenous leukemia.

time

Survival or censoring time

status

censoring status

x

maintenance chemotherapy given or not

Usage

aml

Format

An object of class data.frame with 23 rows and 3 columns.

Simulation for a Binary and a Time-to-Event Endpoint in Group Sequential Trials

Description

Performs simulation for two-endpoint two-arm group sequential trials.

• Endpoint 1: Binary endpoint, analyzed using the Mantel-Haenszel test for risk difference.

• Endpoint 2: Time-to-event endpoint, analyzed using the log-rank test for treatment effect.

The analysis times for the binary endpoint are based on calendar times, while the time-to-event analyses are triggered by reaching the pre-specified number of events. The binary endpoint is assessed at the first post-treatment follow-up visit (PTFU1).

Usage

binary_tte_sim(
  kMax1 = 1L,
  kMax2 = 1L,
  riskDiffH0 = 0,
  hazardRatioH0 = 1,
  allocation1 = 1L,
  allocation2 = 1L,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  stratumFraction = 1L,
  globalOddsRatio = 1,
  pi1 = NA_real_,
  pi2 = NA_real_,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  delta1 = 0L,
  delta2 = 0L,
  upper1 = NA_real_,
  upper2 = NA_real_,
  accrualDuration = NA_real_,
  plannedTime = NA_real_,
  plannedEvents = NA_integer_,
  maxNumberOfIterations = 1000L,
  maxNumberOfRawDatasetsPerStage = 0L,
  seed = NA_integer_
)

Arguments

Details

We consider dual primary endpoints with endpoint 1 being a binary endpoint and endpoint 2 being a time-to-event endpoint. The analyses of endpoint 1 will be based on calendar times, while the analyses of endpoint 2 will be based on the number of events. Therefore, the analyses of the two endpoints are not at the same time points. The correlation between the two endpoints is characterized by the global odds ratio of the Plackett copula. In addition, the time-to-event endpoint will render the binary endpoint as a non-responder, and so does the dropout. In addition, the treatment discontinuation will impact the number of available subjects for analysis. The administrative censoring will exclude subjects from the analysis of the binary endpoint.

Value

A list with 4 components:

• sumdataBIN: A data frame of summary data by iteration and stage for the binary endpoint:

  • iterationNumber: The iteration number.

  • stageNumber: The stage number, covering all stages even if the trial stops at an interim look.

  • analysisTime: The time for the stage since trial start.

  • accruals1: The number of subjects enrolled at the stage for the treatment group.

  • accruals2: The number of subjects enrolled at the stage for the control group.

  • totalAccruals: The total number of subjects enrolled at the stage.

  • source1: The total number of subjects with response status determined by the underlying latent response variable.

  • source2: The total number of subjects with response status (non-responder) determined by experiencing the event for the time-to-event endpoint.

  • source3: The total number of subjects with response status (non-responder) determined by dropping out prior to the PTFU1 visit.

  • n1: The number of subjects included in the analysis of the binary endpoint for the treatment group.

  • n2: The number of subjects included in the analysis of the binary endpoint for the control group.

  • n: The total number of subjects included in the analysis of the binary endpoint at the stage.

  • y1: The number of responders for the binary endpoint in the treatment group.

  • y2: The number of responders for the binary endpoint in the control group.

  • y: The total number of responders for the binary endpoint at the stage.

  • riskDiff: The estimated risk difference for the binary endpoint.

  • seRiskDiff: The standard error for risk difference based on the Sato approximation.

  • mnStatistic: The Mantel-Haenszel test Z-statistic for the binary endpoint.

• sumdataTTE: A data frame of summary data by iteration and stage for the time-to-event endpoint:

  • iterationNumber: The iteration number.

  • eventsNotAchieved: Whether the target number of events is not achieved for the iteration.

  • stageNumber: The stage number, covering all stages even if the trial stops at an interim look.

  • analysisTime: The time for the stage since trial start.

  • accruals1: The number of subjects enrolled at the stage for the treatment group.

  • accruals2: The number of subjects enrolled at the stage for the control group.

  • totalAccruals: The total number of subjects enrolled at the stage.

  • events1: The number of events at the stage for the treatment group.

  • events2: The number of events at the stage for the control group.

  • totalEvents: The total number of events at the stage.

  • dropouts1: The number of dropouts at the stage for the treatment group.

  • dropouts2: The number of dropouts at the stage for the control group.

  • totalDropouts: The total number of dropouts at the stage.

  • logRankStatistic: The log-rank test Z-statistic for the time-to-event endpoint.

• rawdataBIN (exists if maxNumberOfRawDatasetsPerStage is a positive integer): A data frame for subject-level data for the binary endpoint for selected replications, containing the following variables:

  • iterationNumber: The iteration number.

  • stageNumber: The stage under consideration.

  • analysisTime: The time for the stage since trial start.

  • subjectId: The subject ID.

  • arrivalTime: The enrollment time for the subject.

  • stratum: The stratum for the subject.

  • treatmentGroup: The treatment group (1 or 2) for the subject.

  • survivalTime: The underlying survival time for the time-to-event endpoint for the subject.

  • dropoutTime: The underlying dropout time for the time-to-event endpoint for the subject.

  • ptfu1Time:The underlying assessment time for the binary endpoint for the subject.

  • timeUnderObservation: The time under observation since randomization for the binary endpoint for the subject.

  • responder: Whether the subject is a responder for the binary endpoint.

  • source: The source of the determination of responder status for the binary endpoint: = 1 based on the underlying latent response variable, = 2 based on the occurrence of the time-to-event endpoint before the assessment time of the binary endpoint (imputed as a non-responder), = 3 based on the dropout before the assessment time of the binary endpoint (imputed as a non-responder), = 4 excluded from analysis due to administrative censoring.

• rawdataTTE (exists if maxNumberOfRawDatasetsPerStage is a positive integer): A data frame for subject-level data for the time-to-event endpoint for selected replications, containing the following variables:

  • iterationNumber: The iteration number.

  • stageNumber: The stage under consideration.

  • analysisTime: The time for the stage since trial start.

  • subjectId: The subject ID.

  • arrivalTime: The enrollment time for the subject.

  • stratum: The stratum for the subject.

  • treatmentGroup: The treatment group (1 or 2) for the subject.

  • survivalTime: The underlying survival time for the time-to-event endpoint for the subject.

  • dropoutTime: The underlying dropout time for the time-to-event endpoint for the subject.

  • timeUnderObservation: The time under observation since randomization for the time-to-event endpoint for the subject.

  • event: Whether the subject experienced the event for the time-to-event endpoint.

  • dropoutEvent: Whether the subject dropped out for the time-to-event endpoint.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

tcut = c(0, 12, 36, 48)
surv = c(1, 0.95, 0.82, 0.74)
lambda2 = (log(surv[1:3]) - log(surv[2:4]))/(tcut[2:4] - tcut[1:3])

sim1 = binary_tte_sim(
  kMax1 = 1,
  kMax2 = 2,
  accrualTime = 0:8,
  accrualIntensity = c(((1:8) - 0.5)/8, 1)*40,
  piecewiseSurvivalTime = c(0,12,36),
  globalOddsRatio = 1,
  pi1 = 0.80,
  pi2 = 0.65,
  lambda1 = 0.65*lambda2,
  lambda2 = lambda2,
  gamma1 = -log(1-0.04)/12,
  gamma2 = -log(1-0.04)/12,
  delta1 = -log(1-0.02)/12,
  delta2 = -log(1-0.02)/12,
  upper1 = 15*28/30.4,
  upper2 = 12*28/30.4,
  accrualDuration = 20,
  plannedTime = 20 + 15*28/30.4,
  plannedEvents = c(130, 173),
  maxNumberOfIterations = 1000,
  maxNumberOfRawDatasetsPerStage = 1,
  seed = 314159)

Calendar Times for Target Number of Events

Description

Obtains the calendar times needed to reach the target number of subjects experiencing an event.

Usage

caltime(
  nevents = NA_real_,
  allocationRatioPlanned = 1,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  stratumFraction = 1L,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  accrualDuration = NA_real_,
  followupTime = NA_real_,
  fixedFollowup = 0L
)

Arguments

Value

A vector of calendar times expected to yield the target number of events.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Piecewise accrual, piecewise exponential survivals, and 5% dropout by
# the end of 1 year.

caltime(nevents = c(24, 80), allocationRatioPlanned = 1,
        accrualTime = seq(0, 8),
        accrualIntensity = 26/9*seq(1, 9),
        piecewiseSurvivalTime = c(0, 6),
        lambda1 = c(0.0533, 0.0309),
        lambda2 = c(0.0533, 0.0533),
        gamma1 = -log(1-0.05)/12,
        gamma2 = -log(1-0.05)/12,
        accrualDuration = 22,
        followupTime = 18, fixedFollowup = FALSE)

Covariance Between Restricted Mean Survival Times

Description

Obtains the covariance between restricted mean survival times at two different time points.

Usage

covrmst(
  t2 = NA_real_,
  tau1 = NA_real_,
  tau2 = NA_real_,
  allocationRatioPlanned = 1,
  accrualTime = 0L,
  accrualIntensity = NA_real_,
  piecewiseSurvivalTime = 0L,
  lambda1 = NA_real_,
  lambda2 = NA_real_,
  gamma1 = 0L,
  gamma2 = 0L,
  accrualDuration = NA_real_,
  maxFollowupTime = NA_real_
)

Arguments

Value

The covariance between the restricted mean survival times for each treatment group.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

covrmst(t2 = 25, tau1 = 16, tau2 = 18, allocationRatioPlanned = 1,
        accrualTime = c(0, 3), accrualIntensity = c(10, 20),
        piecewiseSurvivalTime = c(0, 6),
        lambda1 = c(0.0533, 0.0309), lambda2 = c(0.0533, 0.0533),
        gamma1 = -log(1-0.05)/12, gamma2 = -log(1-0.05)/12,
        accrualDuration = 12, maxFollowupTime = 30)

Error Spending

Description

Obtains the error spent at given spending times for the specified error spending function.

Usage

errorSpent(t, error, sf = "sfOF", sfpar = NA)

Arguments

Details

This function implements a variety of error spending functions commonly used in group sequential designs, assuming one-sided hypothesis testing.

O'Brien-Fleming-Type Spending Function

This spending function allocates very little alpha early on and more alpha later in the trial. It is defined as:

\alpha(t) = 2 - 2\Phi\left(\frac{z_{\alpha/2}}{\sqrt{t}}\right),

where \Phi is the standard normal cumulative distribution function, z_{\alpha/2} is the critical value from the standard normal distribution, and t \in [0, 1] denotes the information fraction.

Pocock-Type Spending Function

This function spends alpha more evenly throughout the study:

\alpha(t) = \alpha \log(1 + (e - 1)t),

where e is Euler's number (approximately 2.718).

Kim and DeMets Power-Type Spending Function

This family of spending functions is defined as:

\alpha(t) = \alpha t^{\rho}, \quad \rho > 0.

• When \rho = 1, the function mimics Pocock-type boundaries.

• When \rho = 3, it approximates O’Brien-Fleming-type boundaries.

Hwang, Shih, and DeCani Spending Function

This flexible family of functions is given by:

\alpha(t) = \begin{cases} \alpha \frac{1 - e^{-\gamma t}}{1 - e^{-\gamma}}, & \text{if } \gamma \ne 0 \\ \alpha t, & \text{if } \gamma = 0. \end{cases}

• When \gamma = -4, the spending function resembles O’Brien-Fleming boundaries.

• When \gamma = 1, it resembles Pocock boundaries.

Value

A vector of errors spent up to the interim look.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

errorSpent(t = 0.5, error = 0.025, sf = "sfOF")

errorSpent(t = c(0.5, 0.75, 1), error = 0.025, sf = "sfHSD", sfpar = -4)

Stagewise Exit Probabilities

Description

Obtains the stagewise exit probabilities for both efficacy and futility stopping.

Usage

exitprob(b, a = NA, theta = 0, I = NA)

Arguments

Value

A list of stagewise exit probabilities:

• exitProbUpper: The vector of efficacy stopping probabilities

• exitProbLower: The vector of futility stopping probabilities.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

exitprob(b = c(3.471, 2.454, 2.004), theta = -log(0.6),
         I = c(50, 100, 150)/4)

exitprob(b = c(2.963, 2.359, 2.014),
         a = c(-0.264, 0.599, 2.014),
         theta = c(0.141, 0.204, 0.289),
         I = c(81, 121, 160))

Adjusted p-Values for Bonferroni-Based Graphical Approaches

Description

Obtains the adjusted p-values for graphical approaches using weighted Bonferroni tests.

Usage

fadjpbon(w, G, p)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Frank Bretz, Willi Maurer, Werner Brannath and Martin Posch. A graphical approach to sequentially rejective multiple test procedures. Statistics in Medicine. 2009; 28:586-604.

Examples

pvalues <- matrix(c(0.01,0.005,0.015,0.022, 0.02,0.015,0.010,0.023),
                  nrow=2, ncol=4, byrow=TRUE)
w <- c(0.5,0.5,0,0)
g <- matrix(c(0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0),
            nrow=4, ncol=4, byrow=TRUE)
fadjpbon(w, g, pvalues)

Adjusted p-Values for Dunnett-Based Graphical Approaches

Description

Obtains the adjusted p-values for graphical approaches using weighted Dunnett tests.

Usage

fadjpdun(wgtmat, p, family = NULL, corr = NULL)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi Maurer, and Kornelius Rohmeyer. Graphical approach for multiple comparison procedures using weighted Bonferroni, Simes, or parameter tests. Biometrical Journal. 2011; 53:894-913.

Examples

pvalues <- matrix(c(0.01,0.005,0.015,0.022, 0.02,0.015,0.010,0.023),
                  nrow=2, ncol=4, byrow=TRUE)
w <- c(0.5,0.5,0,0)
g <- matrix(c(0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0),
            nrow=4, ncol=4, byrow=TRUE)
wgtmat = fwgtmat(w,g)

family = matrix(c(1,1,0,0,0,0,1,1), nrow=2, ncol=4, byrow=TRUE)
corr = matrix(c(1,0.5,NA,NA, 0.5,1,NA,NA,
                NA,NA,1,0.5, NA,NA,0.5,1),
              nrow = 4, byrow = TRUE)
fadjpdun(wgtmat, pvalues, family, corr)

Adjusted p-Values for Simes-Based Graphical Approaches

Description

Obtains the adjusted p-values for graphical approaches using weighted Simes tests.

Usage

fadjpsim(wgtmat, p, family = NULL)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Frank Bretz, Martin Posch, Ekkehard Glimm, Florian Klinglmueller, Willi Maurer, and Kornelius Rohmeyer. Graphical approach for multiple comparison procedures using weighted Bonferroni, Simes, or parameter tests. Biometrical Journal. 2011; 53:894-913.

Kaifeng Lu. Graphical approaches using a Bonferroni mixture of weighted Simes tests. Statistics in Medicine. 2016; 35:4041-4055.

Examples

pvalues <- matrix(c(0.01,0.005,0.015,0.022, 0.02,0.015,0.010,0.023),
                  nrow=2, ncol=4, byrow=TRUE)
w <- c(0.5,0.5,0,0)
g <- matrix(c(0,0,1,0,0,0,0,1,0,1,0,0,1,0,0,0),
            nrow=4, ncol=4, byrow=TRUE)
wgtmat = fwgtmat(w,g)

family = matrix(c(1,1,0,0,0,0,1,1), nrow=2, ncol=4, byrow=TRUE)
fadjpsim(wgtmat, pvalues, family)

Find Interval Numbers of Indices

Description

The implementation of findInterval() in R from Advanced R by Hadley Wickham. Given a vector of non-decreasing breakpoints in v, find the interval containing each element of x; i.e., if i <- findInterval3(x,v), for each index j in x, v[i[j]] <= x[j] < v[i[j] + 1], where v[0] := -Inf, v[N+1] := +Inf, and N = length(v).

Usage

findInterval3(x, v)

Arguments

Value

A vector of length(x) with values in 0:N where N = length(v).

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

x <- 2:18
v <- c(5, 10, 15) # create two bins [5,10) and [10,15)
cbind(x, findInterval3(x, v))

Converting a decimal to a fraction

Description

Converts a decimal to a fraction based on the algorithm from http://stackoverflow.com/a/5128558/221955.

Usage

float_to_fraction(x, tol = 1e-06)

Arguments

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

float_to_fraction(5/3)

Adjusted p-Values for Modified Mixture Gatekeeping Procedures

Description

Obtains the adjusted p-values for the modified gatekeeping procedures for multiplicity problems involving serial and parallel logical restrictions.

Usage

fmodmix(
  p,
  family = NULL,
  serial,
  parallel,
  gamma,
  test = "hommel",
  exhaust = TRUE
)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Alex Dmitrienko, George Kordzakhia, and Thomas Brechenmacher. Mixture-based gatekeeping procedures for multiplicity problems with multiple sequences of hypotheses. Journal of Biopharmaceutical Statistics. 2016; 26(4):758–780.

George Kordzakhia, Thomas Brechenmacher, Eiji Ishida, Alex Dmitrienko, Winston Wenxiang Zheng, and David Fuyuan Li. An enhanced mixture method for constructing gatekeeping procedures in clinical trials. Journal of Biopharmaceutical Statistics. 2018; 28(1):113–128.

Examples

p = c(0.0194, 0.0068, 0.0271, 0.0088, 0.0370, 0.0018, 0.0814, 0.0066)
family = matrix(c(1, 1, 0, 0, 0, 0, 0, 0,
                  0, 0, 1, 1, 0, 0, 0, 0,
                  0, 0, 0, 0, 1, 1, 0, 0,
                  0, 0, 0, 0, 0, 0, 1, 1),
                nrow=4, byrow=TRUE)

serial = matrix(c(0, 0, 0, 0, 0, 0, 0, 0,
                  0, 0, 0, 0, 0, 0, 0, 0,
                  1, 0, 0, 0, 0, 0, 0, 0,
                  0, 1, 0, 0, 0, 0, 0, 0,
                  0, 0, 1, 0, 0, 0, 0, 0,
                  0, 0, 0, 1, 0, 0, 0, 0,
                  0, 0, 0, 0, 1, 0, 0, 0,
                  0, 0, 0, 0, 0, 1, 0, 0),
                nrow=8, byrow=TRUE)

parallel = matrix(0, 8, 8)
gamma = c(0.6, 0.6, 0.6, 1)
fmodmix(p, family, serial, parallel, gamma, test = "hommel", exhaust = TRUE)

The Quantiles of a Survival Distribution

Description

Obtains the quantiles of a survival distribution.

Usage

fquantile(S, probs, ...)

Arguments

Value

A vector of length(probs) for the quantiles.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

fquantile(pweibull, probs = c(0.25, 0.5, 0.75),
          shape = 1.37, scale = 1/0.818, lower.tail = FALSE)

Group Sequential Trials Using Bonferroni-Based Graphical Approaches

Description

Obtains the test results for group sequential trials using graphical approaches based on weighted Bonferroni tests.

Usage

fseqbon(
  w,
  G,
  alpha = 0.025,
  kMax,
  typeAlphaSpending = NULL,
  parameterAlphaSpending = NULL,
  incidenceMatrix = NULL,
  maxInformation = NULL,
  p,
  information,
  spendingTime = NULL
)

Arguments

Value

A vector to indicate the first look the specific hypothesis is rejected (0 if the hypothesis is not rejected).

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Willi Maurer and Frank Bretz. Multiple testing in group sequential trials using graphical approaches. Statistics in Biopharmaceutical Research. 2013; 5:311-320.

Examples

# Case study from Maurer & Bretz (2013)

fseqbon(
  w = c(0.5, 0.5, 0, 0),
  G = matrix(c(0, 0.5, 0.5, 0,  0.5, 0, 0, 0.5,
               0, 1, 0, 0,  1, 0, 0, 0),
             nrow=4, ncol=4, byrow=TRUE),
  alpha = 0.025,
  kMax = 3,
  typeAlphaSpending = rep("sfOF", 4),
  maxInformation = rep(1, 4),
  p = matrix(c(0.0062, 0.017, 0.009, 0.13,
               0.0002, 0.0035, 0.002, 0.06),
             nrow=4, ncol=2),
  information = matrix(c(rep(1/3, 4), rep(2/3, 4)),
                       nrow=4, ncol=2))

Adjusted p-Values for Standard Mixture Gatekeeping Procedures

Description

Obtains the adjusted p-values for the standard gatekeeping procedures for multiplicity problems involving serial and parallel logical restrictions.

Usage

fstdmix(
  p,
  family = NULL,
  serial,
  parallel,
  gamma,
  test = "hommel",
  exhaust = TRUE
)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Alex Dmitrienko and Ajit C Tamhane. Mixtures of multiple testing procedures for gatekeeping applications in clinical trials. Statistics in Medicine. 2011; 30(13):1473–1488.

Examples

p = c(0.0194, 0.0068, 0.0271, 0.0088, 0.0370, 0.0018, 0.0814, 0.0066)
family = matrix(c(1, 1, 0, 0, 0, 0, 0, 0,
                  0, 0, 1, 1, 0, 0, 0, 0,
                  0, 0, 0, 0, 1, 1, 0, 0,
                  0, 0, 0, 0, 0, 0, 1, 1),
                nrow=4, byrow=TRUE)

serial = matrix(c(0, 0, 0, 0, 0, 0, 0, 0,
                  0, 0, 0, 0, 0, 0, 0, 0,
                  1, 0, 0, 0, 0, 0, 0, 0,
                  0, 1, 0, 0, 0, 0, 0, 0,
                  0, 0, 1, 0, 0, 0, 0, 0,
                  0, 0, 0, 1, 0, 0, 0, 0,
                  0, 0, 0, 0, 1, 0, 0, 0,
                  0, 0, 0, 0, 0, 1, 0, 0),
                nrow=8, byrow=TRUE)

parallel = matrix(0, 8, 8)
gamma = c(0.6, 0.6, 0.6, 1)
fstdmix(p, family, serial, parallel, gamma, test = "hommel",
        exhaust = FALSE)

Adjusted p-Values for Stepwise Testing Procedures for Two Sequences

Description

Obtains the adjusted p-values for the stepwise gatekeeping procedures for multiplicity problems involving two sequences of hypotheses.

Usage

fstp2seq(p, gamma, test = "hochberg", retest = TRUE)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

p = c(0.0194, 0.0068, 0.0271, 0.0088, 0.0370, 0.0018, 0.0814, 0.0066)
gamma = c(0.6, 0.6, 0.6, 1)
fstp2seq(p, gamma, test="hochberg", retest=1)

Adjusted p-Values for Holm, Hochberg, and Hommel Procedures

Description

Obtains the adjusted p-values for possibly truncated Holm, Hochberg, and Hommel procedures.

Usage

ftrunc(p, test = "hommel", gamma = 1)

Arguments

Value

A matrix of adjusted p-values.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Alex Dmitrienko, Ajit C. Tamhane, and Brian L. Wiens. General multistage gatekeeping procedures. Biometrical Journal. 2008; 5:667-677.

Examples

pvalues <- matrix(c(0.01,0.005,0.015,0.022, 0.02,0.015,0.010,0.023),
                  nrow=2, ncol=4, byrow=TRUE)
ftrunc(pvalues, "hochberg")

Weight Matrix for All Intersection Hypotheses

Description

Obtains the weight matrix for all intersection hypotheses.

Usage

fwgtmat(w, G)

Arguments

Value

The weight matrix starting with the global null hypothesis.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

w = c(0.5,0.5,0,0)
g = matrix(c(0,0,1,0, 0,0,0,1, 0,1,0,0, 1,0,0,0),
           nrow=4, ncol=4, byrow=TRUE)
(wgtmat = fwgtmat(w,g))

Confidence Interval After Adaptation

Description

Obtains the p-value, median unbiased point estimate, and confidence interval after the end of an adaptive trial.

Usage

getADCI(
  L = NA_integer_,
  zL = NA_real_,
  IMax = NA_real_,
  kMax = NA_integer_,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.25,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  spendingTime = NA_real_,
  L2 = NA_integer_,
  zL2 = NA_real_,
  INew = NA_real_,
  MullerSchafer = 0L,
  informationRatesNew = NA_real_,
  efficacyStoppingNew = NA_integer_,
  typeAlphaSpendingNew = "sfOF",
  parameterAlphaSpendingNew = NA_real_,
  spendingTimeNew = NA_real_
)

Arguments

Value

A data frame with the following variables:

• pvalue: p-value for rejecting the null hypothesis.

• thetahat: Median unbiased point estimate of the parameter.

• cilevel: Confidence interval level.

• lower: Lower bound of confidence interval.

• upper: Upper bound of confidence interval.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Ping Gao, Lingyun Liu and Cyrus Mehta. Exact inference for adaptive group sequential designs. Stat Med. 2013;32(23):3991-4005.

See Also

adaptDesign

Examples

# original group sequential design with 90% power to detect delta = 6
delta = 6
sigma = 17
n = 282
(des1 = getDesign(IMax = n/(4*sigma^2), theta = delta, kMax = 3,
                  alpha = 0.05, typeAlphaSpending = "sfHSD",
                  parameterAlphaSpending = -4))

# interim look results
L = 1
n1 = n/3
delta1 = 4.5
sigma1 = 20
zL = delta1/sqrt(4/n1*sigma1^2)

t = des1$byStageResults$informationRates

# Muller & Schafer (2001) method to design the secondary trial:
des2 = adaptDesign(
  betaNew = 0.2, L = L, zL = zL, theta = 5,
  kMax = 3, informationRates = t,
  alpha = 0.05, typeAlphaSpending = "sfHSD",
  parameterAlphaSpending = -4,
  MullerSchafer = TRUE,
  kNew = 3, typeAlphaSpendingNew = "sfHSD",
  parameterAlphaSpendingNew = -2)

n2 = ceiling(des2$secondaryTrial$overallResults$information*4*20^2)
ns = round(n2*(1:3)/3)
 (des2 = adaptDesign(
   INew = n2/(4*20^2), L = L, zL = zL, theta = 5,
   kMax = 3, informationRates = t,
   alpha = 0.05, typeAlphaSpending = "sfHSD",
   parameterAlphaSpending = -4,
   MullerSchafer = TRUE,
   kNew = 3, informationRatesNew = ns/n2,
   typeAlphaSpendingNew = "sfHSD",
   parameterAlphaSpendingNew = -2))

# termination at the second look of the secondary trial
L2 = 2
delta2 = 6.86
sigma2 = 21.77
zL2 = delta2/sqrt(4/197*sigma2^2)

t2 = des2$secondaryTrial$byStageResults$informationRates[1:L2]

# confidence interval
getADCI(L = L, zL = zL,
        IMax = n/(4*sigma1^2), kMax = 3,
        informationRates = t,
        alpha = 0.05, typeAlphaSpending = "sfHSD",
        parameterAlphaSpending = -4,
        L2 = L2, zL2 = zL2,
        INew = n2/(4*sigma2^2),
        MullerSchafer = TRUE,
        informationRatesNew = t2,
        typeAlphaSpendingNew = "sfHSD",
        parameterAlphaSpendingNew = -2)

Repeated Confidence Interval After Adaptation

Description

Obtains the repeated p-value, conservative point estimate, and repeated confidence interval for an adaptive group sequential trial.

Usage

getADRCI(
  L = NA_integer_,
  zL = NA_real_,
  IMax = NA_real_,
  kMax = NA_integer_,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  spendingTime = NA_real_,
  L2 = NA_integer_,
  zL2 = NA_real_,
  INew = NA_real_,
  MullerSchafer = 0L,
  informationRatesNew = NA_real_,
  efficacyStoppingNew = NA_integer_,
  typeAlphaSpendingNew = "sfOF",
  parameterAlphaSpendingNew = NA_real_,
  spendingTimeNew = NA_real_
)

Arguments

Value

A data frame with the following variables:

• pvalue: Repeated p-value for rejecting the null hypothesis.

• thetahat: Point estimate of the parameter.

• cilevel: Confidence interval level.

• lower: Lower bound of repeated confidence interval.

• upper: Upper bound of repeated confidence interval.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Cyrus R. Mehta, Peter Bauer, Martin Posch and Werner Brannath. Repeated confidence intervals for adaptive group sequential trials. Stat Med. 2007;26:5422–5433.

See Also

adaptDesign

Examples

# original group sequential design with 90% power to detect delta = 6
delta = 6
sigma = 17
n = 282
(des1 = getDesign(IMax = n/(4*sigma^2), theta = delta, kMax = 3,
                  alpha = 0.05, typeAlphaSpending = "sfHSD",
                  parameterAlphaSpending = -4))

# interim look results
L = 1
n1 = n/3
delta1 = 4.5
sigma1 = 20
zL = delta1/sqrt(4/n1*sigma1^2)

t = des1$byStageResults$informationRates

# Muller & Schafer (2001) method to design the secondary trial:
des2 = adaptDesign(
  betaNew = 0.2, L = L, zL = zL, theta = 5,
  kMax = 3, informationRates = t,
  alpha = 0.05, typeAlphaSpending = "sfHSD",
  parameterAlphaSpending = -4,
  MullerSchafer = TRUE,
  kNew = 3, typeAlphaSpendingNew = "sfHSD",
  parameterAlphaSpendingNew = -2)

n2 = ceiling(des2$secondaryTrial$overallResults$information*4*20^2)
ns = round(n2*(1:3)/3)
(des2 = adaptDesign(
  INew = n2/(4*20^2), L = L, zL = zL, theta = 5,
  kMax = 3, informationRates = t,
  alpha = 0.05, typeAlphaSpending = "sfHSD",
  parameterAlphaSpending = -4,
  MullerSchafer = TRUE,
  kNew = 3, informationRatesNew = ns/n2,
  typeAlphaSpendingNew = "sfHSD",
  parameterAlphaSpendingNew = -2))

# termination at the second look of the secondary trial
L2 = 2
delta2 = 6.86
sigma2 = 21.77
zL2 = delta2/sqrt(4/197*sigma2^2)

t2 = des2$secondaryTrial$byStageResults$informationRates[1:L2]

# repeated confidence interval
getADRCI(L = L, zL = zL,
         IMax = n/(4*sigma1^2), kMax = 3,
         informationRates = t,
         alpha = 0.05, typeAlphaSpending = "sfHSD",
         parameterAlphaSpending = -4,
         L2 = L2, zL2 = zL2,
         INew = n2/(4*sigma2^2),
         MullerSchafer = TRUE,
         informationRatesNew = t2,
         typeAlphaSpendingNew = "sfHSD",
         parameterAlphaSpendingNew = -2)

Accrual Duration to Enroll Target Number of Subjects

Description

Obtains the accrual duration to enroll the target number of subjects.

Usage

getAccrualDurationFromN(
  nsubjects = NA_real_,
  accrualTime = 0L,
  accrualIntensity = NA_real_
)

Arguments

Value

A vector of accrual durations.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

getAccrualDurationFromN(nsubjects = c(20, 150), accrualTime = c(0, 3),
                        accrualIntensity = c(10, 20))

Efficacy Boundaries for Group Sequential Design

Description

Obtains the efficacy stopping boundaries for a group sequential design.

Usage

getBound(
  k = NA,
  informationRates = NA,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA,
  userAlphaSpending = NA,
  spendingTime = NA,
  efficacyStopping = NA
)

Arguments

Details

If typeAlphaSpending is "OF", "P", or "WT", then the boundaries will be based on equally spaced looks.

Value

A numeric vector of critical values up to the current look.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

getBound(k = 2, informationRates = c(0.5,1),
         alpha = 0.025, typeAlphaSpending = "sfOF")

Confidence Interval After Trial Termination

Description

Obtains the p-value, median unbiased point estimate, and confidence interval after the end of a group sequential trial.

Usage

getCI(
  L = NA_integer_,
  zL = NA_real_,
  IMax = NA_real_,
  informationRates = NA_real_,
  efficacyStopping = NA_integer_,
  criticalValues = NA_real_,
  alpha = 0.025,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  spendingTime = NA_real_
)

Arguments

Value

A data frame with the following components:

• pvalue: p-value for rejecting the null hypothesis.

• thetahat: Median unbiased point estimate of the parameter.

• cilevel: Confidence interval level.

• lower: Lower bound of confidence interval.

• upper: Upper bound of confidence interval.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Anastasios A. Tsiatis, Gary L. Rosner and Cyrus R. Mehta. Exact confidence intervals following a group sequential test. Biometrics 1984;40:797-803.

Examples

# group sequential design with 90% power to detect delta = 6
delta = 6
sigma = 17
n = 282
(des1 = getDesign(IMax = n/(4*sigma^2), theta = delta, kMax = 3,
                  alpha = 0.05, typeAlphaSpending = "sfHSD",
                  parameterAlphaSpending = -4))

# crossed the boundary at the second look
L = 2
n1 = n*2/3
delta1 = 7
sigma1 = 20
zL = delta1/sqrt(4/n1*sigma1^2)

# confidence interval
getCI(L = L, zL = zL, IMax = n/(4*sigma1^2),
      informationRates = c(1/3, 2/3), alpha = 0.05,
      typeAlphaSpending = "sfHSD", parameterAlphaSpending = -4)

Conditional Power Allowing for Varying Parameter Values

Description

Obtains the conditional power for specified incremental information given the interim results, parameter values, and data-dependent changes in the error spending function, as well as the number and spacing of interim looks.

Usage

getCP(
  INew = NA_real_,
  L = NA_integer_,
  zL = NA_real_,
  theta = NA_real_,
  IMax = NA_real_,
  kMax = NA_integer_,
  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_,
  spendingTime = NA_real_,
  MullerSchafer = 0L,
  kNew = NA_integer_,
  informationRatesNew = NA_real_,
  efficacyStoppingNew = NA_integer_,
  futilityStoppingNew = NA_integer_,
  typeAlphaSpendingNew = "sfOF",
  parameterAlphaSpendingNew = NA_real_,
  typeBetaSpendingNew = "none",
  parameterBetaSpendingNew = NA_real_,
  spendingTimeNew = NA_real_,
  varianceRatio = 1
)

Arguments

Value

The conditional power given the interim results, parameter values, and data-dependent design changes.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Cyrus R. Mehta and Stuart J. Pocock. Adaptive increase in sample size when interim results are promising: A practical guide with examples. Stat Med. 2011;30:3267–3284.

See Also

getDesign

Examples

# Conditional power calculation with delayed treatment effect

# Two interim analyses have occurred with 179 and 266 events,
# respectively. The observed hazard ratio at the second interim
# look is 0.81.

trialsdt = as.Date("2020-03-04")                       # trial start date
iadt = c(as.Date("2022-02-01"), as.Date("2022-11-01")) # interim dates
mo1 = as.numeric(iadt - trialsdt + 1)/30.4375          # interim months

# Assume a piecewise Poisson enrollment process with a 8-month ramp-up
# and 521 patients were enrolled after 17.94 months
N = 521                   # total number of patients
Ta = 17.94                # enrollment duration
Ta1 = 8                   # assumed end of enrollment ramp-up
enrate = N / (Ta - Ta1/2) # enrollment rate after ramp-up

# Assume a median survival of 16.7 months for the control group, a
# 5-month delay in treatment effect, and a hazard ratio of 0.7 after
# the delay
lam1 = log(2)/16.7  # control group hazard of exponential distribution
t1 = 5              # months of delay in treatment effect
hr = 0.7            # hazard ratio after delay
lam2 = hr*lam1      # treatment group hazard after delay

# Assume an annual dropout rate of 5%
gam = -log(1-0.05)/12  # hazard for dropout

# The original target number of events was 298 and the new target is 335
mo2 <- caltime(
  nevents = c(298, 335),
  allocationRatioPlanned = 1,
  accrualTime = seq(0, Ta1),
  accrualIntensity = enrate*seq(1, Ta1+1)/(Ta1+1),
  piecewiseSurvivalTime = c(0, t1),
  lambda1 = c(lam1, lam2),
  lambda2 = c(lam1, lam1),
  gamma1 = gam,
  gamma2 = gam,
  accrualDuration = Ta,
  followupTime = 1000)

# expected number of events and average hazard ratios
(lr1 <- lrstat(
  time = c(mo1, mo2),
  accrualTime = seq(0, Ta1),
  accrualIntensity = enrate*seq(1, Ta1+1)/(Ta1+1),
  piecewiseSurvivalTime = c(0, t1),
  lambda1 = c(lam1, lam2),
  lambda2 = c(lam1, lam1),
  gamma1 = gam,
  gamma2 = gam,
  accrualDuration = Ta,
  followupTime = 1000,
  predictTarget = 3))


hr2 = 0.81                    # observed hazard ratio at interim 2
z2 = (-log(hr2))*sqrt(266/4)  # corresponding z-test statistic value

# expected mean of -log(HR) at the original looks and the new final look
theta = -log(lr1$HR[c(1,2,3,4)])

# conditional power with sample size increase
getCP(INew = (335 - 266)/4,
      L = 2, zL = z2, theta = theta,
      IMax = 298/4, kMax = 3,
      informationRates = c(179, 266, 298)/298,
      alpha = 0.025, typeAlphaSpending = "sfOF")

Power and Sample Size for a Generic Group Sequential Design

Description

Obtains the maximum information and stopping boundaries for a generic group sequential design assuming a constant treatment effect, or obtains the power given the maximum information and stopping boundaries.

Usage

getDesign(
  beta = NA_real_,
  IMax = NA_real_,
  theta = NA_real_,
  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_,
  varianceRatio = 1
)

Arguments

Value

An S3 class design 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).

• 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.

  • efficacyTheta: The efficacy boundaries on the parameter scale.

  • futilityTheta: The futility boundaries on the parameter scale.

  • 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.

• 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.

  • varianceRatio: The ratio of the variance under H0 to the variance under H1.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Christopher Jennison, Bruce W. Turnbull. Group Sequential Methods with Applications to Clinical Trials. Chapman & Hall/CRC: Boca Raton, 2000, ISBN:0849303168

Examples

# Example 1: obtain the maximum information given power
(design1 <- getDesign(
  beta = 0.2, theta = -log(0.7),
  kMax = 2, informationRates = c(0.5,1),
  alpha = 0.025, typeAlphaSpending = "sfOF",
  typeBetaSpending = "sfP"))

# Example 2: obtain power given the maximum information
(design2 <- getDesign(
  IMax = 72.5, theta = -log(0.7),
  kMax = 3, informationRates = c(0.5, 0.75, 1),
  alpha = 0.025, typeAlphaSpending = "sfOF",
  typeBetaSpending = "sfP"))

Power and Sample Size for One-Way ANOVA

Description

Obtains the power and sample size for one-way analysis of variance.

Usage

getDesignANOVA(
  beta = NA_real_,
  n = NA_real_,
  ngroups = 2,
  means = NA_real_,
  stDev = 1,
  allocationRatioPlanned = NA_real_,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

Details

Let \{\mu_i: i=1,\ldots,k\} denote the group means, and \{r_i: i=1,\ldots,k\} denote the randomization probabilities to the k treatment groups. Let \sigma denote the common standard deviation, and n denote the total sample size. Then the F-statistic

F = \frac{SSR/(k-1)}{SSE/(n-k)} \sim F_{k-1, n-k, \lambda}

where

\lambda = n \sum_{i=1}^k r_i (\mu_i - \bar{\mu})^2/\sigma^2

is the noncentrality parameter, and \bar{\mu} = \sum_{i=1}^k r_i \mu_i.

Value

An S3 class designANOVA object with the following components:

• power: The power to reject the null hypothesis that there is no difference among the treatment groups.

• alpha: The two-sided significance level.

• n: The number of subjects.

• ngroups: The number of treatment groups.

• means: The treatment group means.

• stDev: The common standard deviation.

• effectsize: The effect size.

• allocationRatioPlanned: Allocation ratio for the treatment groups.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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))

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.

Usage

getDesignANOVAContrast(
  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
)

Arguments

Value

An S3 class designANOVAContrast object with the following components:

• power: The power to reject the null hypothesis for the treatment contrast.

• alpha: The one-sided significance level.

• n: The number of subjects.

• ngroups: The number of treatment groups.

• means: The treatment group means.

• stDev: The common standard deviation.

• contrast: The coefficients for the single contrast.

• meanContrastH0: The mean of the contrast under the null hypothesis.

• meanContrast: The mean of the contrast under the alternative hypothesis.

• effectsize: The effect size.

• allocationRatioPlanned: Allocation ratio for the treatment groups.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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),
  allocationRatioPlanned = c(2, 2, 2, 1),
  alpha = 0.025))

Power and Sample Size for Cohen's kappa

Description

Obtains the power given sample size or obtains the sample size given power for Cohen's kappa.

Usage

getDesignAgreement(
  beta = NA_real_,
  n = NA_real_,
  ncats = NA_integer_,
  kappaH0 = NA_real_,
  kappa = NA_real_,
  p1 = NA_real_,
  p2 = NA_real_,
  rounding = TRUE,
  alpha = 0.025
)

Arguments

Details

The kappa coefficient is defined as

\kappa = \frac{\pi_o - \pi_e}{1 - \pi_e},

where \pi_o = \sum_i \pi_{ii} is the observed agreement, and \pi_e = \sum_i \pi_{i.} \pi_{.i} is the expected agreement by chance.

By Fleiss et al. (1969), the variance of \hat{\kappa} is given by

Var(\hat{\kappa}) = \frac{v_1}{n},

where

v_1 = \frac{Q_1 + Q_2 - Q3 - Q4}{(1-\pi_e)^4},

Q_1 = \pi_o(1-\pi_e)^2,

Q_2 = (1-\pi_o)^2 \sum_i \sum_j \pi_{ij}(\pi_{i.} + \pi_{.j})^2,

Q_3 = 2(1-\pi_o)(1-\pi_e) \sum_i \pi_{ii}(\pi_{i.} + \pi_{.i}),

Q_4 = (\pi_o \pi_e - 2\pi_e + \pi_o)^2.

Given \kappa and marginals \{(\pi_{i.}, \pi_{.i}): i=1,\ldots,k\}, we obtain \pi_o. The only unknowns are the double summation in Q_2 and the single summation in Q_3.

We find the optimal configuration of cell probabilities that yield the maximum variance of \hat{\kappa} by treating the problem as a linear programming problem with constraints to match the given marginal probabilities and the observed agreement and ensure that the cell probabilities are nonnegative. This is an extension of Flack et al. (1988) by allowing unequal marginal probabilities of the two raters.

We perform the optimization under both the null and alternative hypotheses to obtain \max Var(\hat{\kappa} | \kappa = \kappa_0) and \max Var(\hat{\kappa} | \kappa = \kappa_1) for a single subject, and then calculate the sample size or power according to the following equation:

\sqrt{n} |\kappa - \kappa_0| = z_{1-\alpha} \sqrt{\max Var(\hat{\kappa} | \kappa = \kappa_0)} + z_{1-\beta} \sqrt{\max Var(\hat{\kappa} | \kappa = \kappa_1)}.

Value

An S3 class designAgreement object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The one-sided significance level.

• n: The total sample size.

• ncats: The number of categories.

• kappaH0: The kappa coefficient under the null hypothesis.

• kappa: The kappa coefficient under the alternative hypothesis.

• p1: The marginal probabilities for the first rater.

• p2: The marginal probabilities for the second rater.

• piH0: The cell probabilities that maximize the variance of estimated kappa under H0.

• pi: The cell probabilities that maximize the variance of estimated kappa under H1.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

V. F. Flack, A. A. Afifi, and P. A. Lachenbruch. Sample size determinations for the two rater kappa statistic. Psychometrika 1988; 53:321-325.

Examples

(design1 <- getDesignAgreement(
  beta = 0.2, n = NA, ncats = 4, kappaH0 = 0.4, kappa = 0.6,
  p1 = c(0.1, 0.2, 0.3, 0.4), p2 = c(0.15, 0.2, 0.24, 0.41),
  rounding = TRUE, alpha = 0.05))

Power and Sample Size for a Generic Group Sequential Equivalence Design

Description

Obtains the maximum information and stopping boundaries for a generic group sequential equivalence design assuming a constant treatment effect, or obtains the power given the maximum information and stopping boundaries.

Usage

getDesignEquiv(
  beta = NA_real_,
  IMax = NA_real_,
  thetaLower = NA_real_,
  thetaUpper = NA_real_,
  theta = 0,
  kMax = 1L,
  informationRates = NA_real_,
  criticalValues = NA_real_,
  alpha = 0.05,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  spendingTime = NA_real_
)

Arguments

Details

Consider the equivalence design with two one-sided hypotheses:

H_{10}: \theta \leq \theta_{10},

H_{20}: \theta \geq \theta_{20}.

We reject H_{10} at or before look k if

Z_{1j} = (\hat{\theta}_j - \theta_{10})\sqrt{I_j} \geq b_j

for some j=1,\ldots,k, where \{b_j:j=1,\ldots,K\} are the critical values associated with the specified alpha-spending function, and I_j is the information for \theta (inverse variance of \hat{\theta}) at the jth look. For example, for estimating the risk difference \theta = \pi_1 - \pi_2,

I_j = \left\{\frac{\pi_1 (1-\pi_1)}{n_{1j}} + \frac{\pi_2(1-\pi_2)}{n_{2j}}\right\}^{-1}.

It follows that

(Z_{1j} \geq b_j) = (Z_j \geq b_j + \theta_{10}\sqrt{I_j}),

where Z_j = \hat{\theta}_j \sqrt{I_j}.

Similarly, we reject H_{20} at or before look k if

Z_{2j} = (\hat{\theta}_j - \theta_{20})\sqrt{I_j} \leq -b_j

for some j=1,\ldots,k. We have

(Z_{2j} \leq -b_j) = (Z_j \leq - b_j + \theta_{20}\sqrt{I_j}).

Let l_j = b_j + \theta_{10}\sqrt{I_j}, and u_j = -b_j + \theta_{20}\sqrt{I_j}. The cumulative probability to reject H_0 = H_{10} \cup H_{20} at or before look k under the alternative hypothesis H_1 is given by

P_\theta\left(\cup_{j=1}^{k} (Z_{1j} \geq b_j) \cap \cup_{j=1}^{k} (Z_{2j} \leq -b_j)\right) = p_1 + p_2 - p_{12},

where

p_1 = P_\theta\left(\cup_{j=1}^{k} (Z_{1j} \geq b_j)\right) = P_\theta\left(\cup_{j=1}^{k} (Z_j \geq l_j)\right),

p_2 = P_\theta\left(\cup_{j=1}^{k} (Z_{2j} \leq -b_j)\right) = P_\theta\left(\cup_{j=1}^{k} (Z_j \leq u_j)\right),

and

p_{12} = P_\theta\left(\cup_{j=1}^{k} (Z_j \geq l_j) \cup (Z_j \leq u_j)\right).

Of note, both p_1 and p_2 can be evaluated using one-sided exit probabilities for group sequential designs. If there exists j\leq k such that l_j \leq u_j, then p_{12} = 1. Otherwise, p_{12} can be evaluated using two-sided exit probabilities for group sequential designs.

Since the equivalent hypothesis is tested using two one-sided tests, the type I error is controlled. To evaluate the attained type I error of the equivalence trial under H_{10} (or H_{20}), we simply fix the control group parameters, update the active treatment group parameters according to the null hypothesis, and use the parameters in the power calculation outlined above.

Value

An S3 class designEquiv object with three components:

• overallResults: A data frame containing the following variables:

  • overallReject: The overall rejection probability.

  • alpha: The overall significance level.

  • attainedAlphaH10: The attained significance level under H10.

  • attainedAlphaH20: The attained significance level under H20.

  • kMax: The number of stages.

  • thetaLower: The parameter value at the lower equivalence limit.

  • thetaUpper: The parameter value at the upper equivalence limit.

  • theta: The parameter value under the alternative hypothesis.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH10: The expected information under H10.

  • expectedInformationH20: The expected information under H20.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

  • cumulativeAttainedAlphaH10: The cumulative probability for efficacy stopping under H10.

  • cumulativeAttainedAlphaH20: The cumulative probability for efficacy stopping under H20.

  • efficacyThetaLower: The efficacy boundaries on the parameter scale for the one-sided null hypothesis at the lower equivalence limit.

  • efficacyThetaUpper: The efficacy boundaries on the parameter scale for the one-sided null hypothesis at the upper equivalence limit.

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

• settings: A list containing the following components:

  • typeAlphaSpending: The type of alpha spending.

  • parameterAlphaSpending: The parameter value for alpha spending.

  • userAlphaSpending: The user defined alpha spending.

  • spendingTime: The error spending time at each analysis.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Example 1: obtain the maximum information given power
(design1 <- getDesignEquiv(
  beta = 0.2, thetaLower = log(0.8), thetaUpper = log(1.25),
  kMax = 2, informationRates = c(0.5, 1),
  alpha = 0.05, typeAlphaSpending = "sfOF"))


# Example 2: obtain power given the maximum information
(design2 <- getDesignEquiv(
  IMax = 72.5, thetaLower = log(0.7), thetaUpper = -log(0.7),
  kMax = 3, informationRates = c(0.5, 0.75, 1),
  alpha = 0.05, typeAlphaSpending = "sfOF"))

Power and Sample Size for Fisher's Exact Test for Two Proportions

Description

Obtains the power given sample size or obtains the sample size given power for Fisher's exact test for two proportions.

Usage

getDesignFisherExact(
  beta = NA_real_,
  n = NA_real_,
  pi1 = NA_real_,
  pi2 = NA_real_,
  allocationRatioPlanned = 1,
  alpha = 0.05
)

Arguments

Value

A data frame with the following variables:

• alpha: The two-sided significance level.

• power: The power.

• n: The sample size.

• pi1: The assumed probability for the active treatment group.

• pi2: The assumed probability for the control group.

• allocationRatioPlanned: Allocation ratio for the active treatment versus control.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

(design1 <- getDesignFisherExact(
  beta = 0.2, pi1 = 0.5, pi2 = 0.2, alpha = 0.05))

Power and Sample Size for Logistic Regression

Description

Obtains the power given sample size or obtains the sample size given power for logistic regression of a binary response given the covariate of interest and other covariates.

Usage

getDesignLogistic(
  beta = NA_real_,
  n = NA_real_,
  ncovariates = NA_integer_,
  nconfigs = NA_integer_,
  x = NA_real_,
  pconfigs = NA_real_,
  corr = 0,
  oddsratios = NA_real_,
  responseprob = NA_real_,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

Details

We consider the logistic regression of a binary response variable Y on a set of predictor variables x = (x_1,\ldots,x_K)^T with x_1 being the covariate of interest: \log \frac{P(Y_i=1)}{1 - P(Y_i = 1)} = \psi_0 + x_i^T \psi, where \psi = (\psi_1,\ldots,\psi_K)^T. Similar to Self et al (1992), we assume that all covariates are either inherently discrete or discretized from continuous distributions (e.g. using the quantiles). Let m denote the total number of configurations of the covariate values. Let

\pi_i = P(x = x_i), i = 1,\ldots, m

denote the probabilities for the configurations of the covariates under independence. The likelihood ratio test statistic for testing H_0: \psi_1 = 0 can be approximated by a noncentral chi-square distribution with one degree of freedom and noncentrality parameter

\Delta = 2 \sum_{i=1}^m \pi_i [b'(\theta_i)(\theta_i - \theta_i^*) - \{b(\theta_i) - b(\theta_i^*)\}],

where

\theta_i = \psi_0 + \sum_{j=1}^{k} \psi_j x_{ij},

\theta_i^* = \psi_0^* + \sum_{j=2}^{k} \psi_j^* x_{ij},

for \psi_0^* = \psi_0 + \psi_1 \mu_1, and \psi_j^* = \psi_j for j=2,\ldots,K. Here \mu_1 is the mean of x_1, e.g., \mu_1 = \sum_i \pi_i x_{i1}. In addition, by formulating the logistic regression in the framework of generalized linear models,

b(\theta) = \log(1 + \exp(\theta)),

and

b'(\theta) = \frac{\exp(\theta)}{1 + \exp(\theta)}.

The regression coefficients \psi can be obtained by taking the log of the odds ratios for the covariates. The intercept \psi_0 can be derived as

\psi_0 = \log(\bar{\mu}/(1- \bar{\mu})) - \sum_{j=1}^{K} \psi_j \mu_j,

where \bar{\mu} denotes the response probability when all predictor variables are equal to their means.

Finally, let \rho denote the multiple correlation between the predictor and other covariates. The noncentrality parameter of the chi-square test is adjusted downward by multiplying by 1-\rho^2.

Value

An S3 class designLogistic object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The two-sided significance level.

• n: The total sample size.

• ncovariates: The number of covariates.

• nconfigs: The number of configurations of discretized covariate values.

• x: The matrix of covariate values.

• pconfigs: The vector of probabilities for the configurations.

• corr: The multiple correlation between the predictor and other covariates.

• oddsratios: The odds ratios for one unit increase in the covariates.

• responseprob: The response probability in the full model when all predictor variables are equal to their means.

• effectsize: The effect size for the chi-square test.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

References

Steven G. Self, Robert H. Mauritsen and Jill Ohara. Power calculations for likelihood ratio tests in generalized linear models. Biometrics 1992; 48:31-39.

Examples

# two ordinal covariates
x1 = c(5, 10, 15, 20)
px1 = c(0.2, 0.3, 0.3, 0.2)

x2 = c(2, 4, 6)
px2 = c(0.4, 0.4, 0.2)

# discretizing a normal distribution with mean 4 and standard deviation 2
nbins = 10
x3 = qnorm(((1:nbins) - 0.5)/nbins)*2 + 4
px3 = rep(1/nbins, nbins)

# combination of covariate values
nconfigs = length(x1)*length(x2)*length(x3)
x = expand.grid(x3 = x3, x2 = x2, x1 = x1)
x = as.matrix(x[, ncol(x):1])

# probabilities for the covariate configurations under independence
pconfigs = as.numeric(px1 %x% px2 %x% px3)

# convert the odds ratio for the predictor variable in 5-unit change
# to the odds ratio in 1-unit change
(design1 <- getDesignLogistic(
  beta = 0.1, ncovariates = 3,
  nconfigs = nconfigs,
  x = x,
  pconfigs = pconfigs,
  oddsratios = c(1.2^(1/5), 1.4, 1.3),
  responseprob = 0.25,
  alpha = 0.1))

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.

Usage

getDesignMeanDiff(
  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_
)

Arguments

Value

An S3 class designMeanDiff 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanDiffH0: The mean difference under the null hypothesis.

  • meanDiff: The mean difference under the alternative hypothesis.

  • stDev: The standard deviation.

• 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.

  • efficacyMeanDiff: The efficacy boundaries on the mean difference scale.

  • futilityMeanDiff: The futility boundaries on the mean difference scale.

  • numberOfSubjects: The number of subjects.

• 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: Allocation ratio for the active treatment versus control.

  • 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, 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))

Power and Sample Size for Direct Treatment Effects in Crossover Trials

Description

Obtains the power and sample size for direct treatment effects in crossover trials accounting or without accounting for carryover effects.

Usage

getDesignMeanDiffCarryover(
  beta = NA_real_,
  n = NA_real_,
  trtpair = NA_real_,
  carryover = TRUE,
  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
)

Arguments

Details

The linear mixed-effects model to assess the direct treatment effects in the presence of carryover treatment effects is given by

y_{ijk} = \mu + \alpha_i + b_{ij} + \gamma_k + \tau_{d(i,k)} + \lambda_{c(i,k-1)} + e_{ijk}

i=1,\ldots,n; j=1,\ldots,r_i; k = 1,\ldots,p; d,c = 1,\ldots,t

where \mu is the general mean, \alpha_i is the effect of the ith treatment sequence, b_{ij} is the random effect with variance \sigma_b^2 for the jth subject of the ith treatment sequence, \gamma_k is the period effect, and e_{ijk} is the random error with variance \sigma^2 for the subject in period k. The direct effect of the treatment administered in period k of sequence i is \tau_{d(i,k)}, and \lambda_{c(i,k-1)} is the carryover effect of the treatment administered in period k-1 of sequence i. The value of the carryover effect for the observed response in the first period is \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

\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma^2}.

Therefore, stDev = \sigma^2 and corr = \rho. By constructing the design matrix X for the linear model with a compound symmetry covariance matrix for the response vector of a subject, we can obtain

Var(\hat{\beta}) = (X'V^{-1}X)^{-1}.

The covariance matrix for the direct treatment effects and 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 is for the direct treatment effect for the treatment pair of interest with or without accounting for carryover effects as determined by the input parameter carryover. The relative efficiency is for the direct treatment effect for the treatment pair of interest accounting for carryover effects relative to that without accounting for carryover effects.

The degrees of freedom for the t-test accounting for carryover effects can be calculated as the total number of observations minus the number of subjects minus p-1 minus 2(t-1) to account for the subject effect, period effect, and direct and carryover treatment effects. The degrees of freedom for the t-test without accounting for carryover effects is equal to the total number of observations minus the number of subjects minus p-1 minus t-1 to account for the subject effect, period effect, and direct treatment effects.

Value

An S3 class designMeanDiffCarryover object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The one-sided significance level.

• numberOfSubjects: The maximum number of subjects.

• trtpair: The treatment pair of interest to power the study.

• carryover: Whether to account for carryover effects in the power calculation.

• meanDiffH0: The mean difference for the treatment pair of interest under the null hypothesis.

• meanDiff: The mean difference for the treatment pair of interest under the alternative hypothesis.

• stDev: The standard deviation for within-subject random error.

• corr: The intra-subject correlation due to subject random effect.

• design: The crossover design represented by a matrix with rows indexing the sequences, columns indexing the periods, and matrix entries indicating the treatments.

• designMatrix: The design matrix accounting for intercept, sequence, period, direct treatment effects and carryover treatment effects when carryover = TRUE, or the design matrix accounting for intercept, sequence, period, and direct treatment effects when carryover = FALSE.

• nseq: The number of sequences.

• nprd: The number of periods.

• ntrt: The number of treatments.

• cumdrop: The cumulative dropout rate over periods.

• V_direct_only: The covariance matrix for direct treatment effects without accounting for carryover effects. The treatment comparisons for the covariance matrix are for the first t-1 treatments relative to the last treatment.

• V_direct_carry: The covariance matrix for direct and carryover treatment effects.

• v_direct_only: The variance of the direct treatment effect for the treatment pair of interest without accounting for carryover effects.

• v_direct: The variance of the direct treatment effect for the treatment pair of interest accounting for carryover effects.

• v_carry: The variance of the carryover treatment effect for the treatment pair of interest.

• releff_direct: The relative efficiency of the design for estimating the direct treatment effect for the treatment pair of interest after accounting for carryover effects with respect to that without accounting for carryover effects. This is equal to v_direct_only/v_direct.

• releff_carry: The relative efficiency of the design for estimating the carryover effect for the treatment pair of interest. This is equal to v_direct_only/v_carry.

• half_width: The half-width of the confidence interval for the direct treatment effect for the treatment pair of interest.

• nu: Degrees of freedom for the t-test.

• allocationRatioPlanned: Allocation ratio for the sequences.

• 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 the sample size.

Author(s)

Kaifeng Lu, 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))

Power and Sample Size for Equivalence in Direct Treatment Effects in Crossover Trials

Description

Obtains the power and sample size for equivalence in direct treatment effects in crossover trials accounting or without accounting for carryover effects.

Usage

getDesignMeanDiffCarryoverEquiv(
  beta = NA_real_,
  n = NA_real_,
  trtpair = NA_real_,
  carryover = TRUE,
  meanDiffLower = NA_real_,
  meanDiffUpper = NA_real_,
  meanDiff = 0,
  stDev = 1,
  corr = 0.5,
  design = NA_real_,
  cumdrop = NA_real_,
  allocationRatioPlanned = NA_real_,
  normalApproximation = FALSE,
  rounding = TRUE,
  alpha = 0.025
)

Arguments

Details

The linear mixed-effects model to assess the direct treatment effects in the presence of carryover treatment effects is given by

y_{ijk} = \mu + \alpha_i + b_{ij} + \gamma_k + \tau_{d(i,k)} + \lambda_{c(i,k-1)} + e_{ijk}

i=1,\ldots,n; j=1,\ldots,r_i; k = 1,\ldots,p; d,c = 1,\ldots,t

where \mu is the general mean, \alpha_i is the effect of the ith treatment sequence, b_{ij} is the random effect with variance \sigma_b^2 for the jth subject of the ith treatment sequence, \gamma_k is the period effect, and e_{ijk} is the random error with variance \sigma^2 for the subject in period k. The direct effect of the treatment administered in period k of sequence i is \tau_{d(i,k)}, and \lambda_{c(i,k-1)} is the carryover effect of the treatment administered in period k-1 of sequence i. The value of the carryover effect for the observed response in the first period is \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

\rho = \frac{\sigma_b^2}{\sigma_b^2 + \sigma^2}.

Therefore, stDev = \sigma^2 and corr = \rho. By constructing the design matrix X for the linear model with a compound symmetry covariance matrix for the response vector of a subject, we can obtain

Var(\hat{\beta}) = (X'V^{-1}X)^{-1}.

The covariance matrix for the direct treatment effects and 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 is for the direct treatment effect for the treatment pair of interest with or without accounting for carryover effects as determined by the input parameter carryover. The relative efficiency is for the direct treatment effect for the treatment pair of interest accounting for carryover effects relative to that without accounting for carryover effects.

The degrees of freedom for the t-test accounting for carryover effects can be calculated as the total number of observations minus the number of subjects minus p-1 minus 2(t-1) to account for the subject effect, period effect, and direct and carryover treatment effects. The degrees of freedom for the t-test without accounting for carryover effects is equal to the total number of observations minus the number of subjects minus p-1 minus t-1 to account for the subject effect, period effect, and direct treatment effects.

Value

An S3 class designMeanDiffCarryover object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The one-sided significance level.

• numberOfSubjects: The maximum number of subjects.

• trtpair: The treatment pair of interest to power the study.

• carryover: Whether to account for carryover effects in the power calculation.

• meanDiffLower: The lower equivalence limit of mean difference for the treatment pair of interest.

• meanDiffUpper: The upper equivalence limit of mean difference for the treatment pair of interest.

• meanDiff: The mean difference for the treatment pair of interest under the alternative hypothesis.

• stDev: The standard deviation for within-subject random error.

• corr: The intra-subject correlation due to subject random effect.

• design: The crossover design represented by a matrix with rows indexing the sequences, columns indexing the periods, and matrix entries indicating the treatments.

• designMatrix: The design matrix accounting for intercept, sequence, period, direct treatment effects and carryover treatment effects when carryover = TRUE, or the design matrix accounting for intercept, sequence, period, and direct treatment effects when carryover = FALSE.

• nseq: The number of sequences.

• nprd: The number of periods.

• ntrt: The number of treatments.

• cumdrop: The cumulative dropout rate over periods.

• V_direct_only: The covariance matrix for direct treatment effects without accounting for carryover effects. The treatment comparisons for the covariance matrix are for the first t-1 treatments relative to the last treatment.

• V_direct_carry: The covariance matrix for direct and carryover treatment effects.

• v_direct_only: The variance of the direct treatment effect for the treatment pair of interest without accounting for carryover effects.

• v_direct: The variance of the direct treatment effect for the treatment pair of interest accounting for carryover effects.

• v_carry: The variance of the carryover treatment effect for the treatment pair of interest.

• releff_direct: The relative efficiency of the design for estimating the direct treatment effect for the treatment pair of interest after accounting for carryover effects with respect to that without accounting for carryover effects. This is equal to v_direct_only/v_direct.

• releff_carry: The relative efficiency of the design for estimating the carryover effect for the treatment pair of interest. This is equal to v_direct_only/v_carry.

• half_width: The half-width of the confidence interval for the direct treatment effect for the treatment pair of interest.

• nu: Degrees of freedom for the t-test.

• allocationRatioPlanned: Allocation ratio for the sequences.

• 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 the sample size.

Author(s)

Kaifeng Lu, 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 = getDesignMeanDiffCarryoverEquiv(
  beta = 0.2, n = NA,
  meanDiffLower = -1.3, meanDiffUpper = 1.3,
  meanDiff = 0, stDev = 2.2,
  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))

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.

Usage

getDesignMeanDiffEquiv(
  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_
)

Arguments

Value

An S3 class designMeanDiffEquiv object with three components:

• overallResults: A data frame containing the following variables:

  • overallReject: The overall rejection probability.

  • alpha: The significance level for each of the two one-sided tests. Defaults to 0.05.

  • attainedAlpha: The attained significance level.

  • kMax: The number of stages.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH0: The expected information under H0.

  • numberOfSubjects: The maximum number of subjects.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanDiffLower: The lower equivalence limit of mean difference.

  • meanDiffUpper: The upper equivalence limit of mean difference.

  • meanDiff: The mean difference under the alternative hypothesis.

  • stDev: The standard deviation.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

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

  • efficacyMeanDiffLower: The efficacy boundaries on the mean difference scale for the one-sided null hypothesis on the lower equivalence limit.

  • efficacyMeanDiffUpper: The efficacy boundaries on the mean difference scale for the one-sided null hypothesis on the upper equivalence limit.

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

  • numberOfSubjects: The number of subjects.

• 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.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: Allocation ratio for the active treatment versus control.

  • 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 exact calculation using the t distribution is only implemented for the fixed design.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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))

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.

Usage

getDesignMeanDiffMMRM(
  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_
)

Arguments

Details

Consider a longitudinal study with two treatment groups. The outcome is measured at baseline and at k postbaseline time points. For each treatment group, the outcomes are assumed to follow a multivariate normal distribution. Conditional on baseline, the covariance matrix of the post-baseline outcomes is denoted by \Sigma_1 for the active treatment group and \Sigma_2 for the control group. Let \mu_1 and \mu_2 denote the mean vectors of post-baseline outcomes for the active and control groups, respectively. We are interested in testing the null hypothesis H_0: \mu_{1,k} - \mu_{2,k} = \delta_0 against the alternative H_1: \mu_{1,k} - \mu_{2,k} = \delta.

The study design is based on the information for treatment difference at the last postbaseline time point. This information is given by

I = 1/\text{Var}(\hat{\mu}_{1,k} - \hat{\mu}_{2,k})

In the presence of monotone missing data, let p_{g,1},\ldots,p_{g,k} denote the proportions of subjects in observed data patterns 1 through k for treatment group g=1 (active) or 2 (control). A subject in pattern j has complete data up to time t_j, i.e., the observed outcomes are y_{i,1},\ldots,y_{i,j}, with missing values for y_{i,j+1},\ldots,y_{i,k}.

According to Lu et al. (2008), the information matrix for the post-baseline mean vector in group g is

I_g = n \pi_g J_g

where \pi_g is the proportion of subjects in group g, and

J_g = \sum_{j=1}^k p_{g,j} \left( \begin{array}{cc} \Sigma_{g,j}^{-1} & 0 \\ 0 & 0 \end{array}\right)

Here, \Sigma_{g,j} is the leading j\times j principal submatrix of \Sigma_g. It follows that

\text{Var}(\hat{\mu}_{1,k} - \hat{\mu}_{2,k}) = \frac{1}{n}\left(\frac{1}{\pi_1} J_1^{-1}[k,k] + \frac{1}{\pi_2} J_2^{-1}[k,k]\right)

The observed data pattern probabilities depend on the accrual and dropout distributions. Let H(u) denote the cumulative distribution function of enrollment time u, G_g(t) denote the survival function of dropout time t for treatment group g, and \tau denote the calendar time at interim or final analysis. Then, for j=1,\ldots,k-1, the probability that a subject in group g falls into observed data pattern j is

p_{g,j} = H(\tau - t_j)G_g(t_j) - H(\tau - t_{j+1})G_g(t_{j+1})

For the last pattern (j=k, i.e., completers),

p_{g,k} = H(\tau - t_k)G_g(t_k)

For the final analysis, \tau is the study duration, so H(\tau - t_j) = 1 for all j. Therefore, the pattern probabilities depend only on the dropout distribution:

p_{g,j} = G_g(t_j) - G_g(t_{j+1}), \quad j=1,\ldots,k-1

and

p_{g,k} = G_g(t_k)

Cumulative dropout probabilities at post-baseline time points can be used to define a piecewise exponential dropout distribution. Let F_g(t_j) denote the cumulative probability of dropout by time t_j for treatment group g. The left endpoints of the piecewise survival time intervals are given by t_0=0,t_1,\ldots,t_{k-1}. The hazard rate in the interval (t_{j-1},t_j] is given by

\gamma_{g,j} = -\log\left(\frac{1 - F_g(t_j)}{1 - F_g(t_{j-1})} \right) / (t_j - t_{j-1}), \quad j=1,\ldots,k

Value

An S3 class designMeanDiffMMRM 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.

  • meanDiffH0: The mean difference under H0.

  • meanDiff: The mean 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.

  • efficacyMeanDiff: The efficacy boundaries on the mean difference scale.

  • futilityMeanDiff: The futility boundaries on the mean difference scale.

  • numberOfSubjects: The number of subjects.

  • numberOfCompleters: The number of completers.

  • 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.

  • k: The number of postbaseline time points.

  • t: The postbaseline time points.

  • covar1: The covariance matrix for the repeated measures given baseline for the active treatment group.

  • covar2: The covariance matrix for the repeated measures given baseline for the control group.

  • 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, kaifenglu@gmail.com

References

Kaifeng Lu, Xiaohui Luo, and Pei-Yun Chen. Sample size estimation for repeated measures analysis in randomized clinical trials with missing data. The International Journal of Biostatistics 2008; 14(1), Article 9.

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"))

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.

Usage

getDesignMeanDiffXO(
  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_
)

Arguments

Value

An S3 class designMeanDiffXO 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanDiffH0: The mean difference under the null hypothesis.

  • meanDiff: The mean difference under the alternative hypothesis.

  • stDev: The standard deviation for within-subject random error.

• 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.

  • efficacyMeanDiff: The efficacy boundaries on the mean difference scale.

  • futilityMeanDiff: The futility boundaries on the mean difference scale.

  • numberOfSubjects: The number of subjects.

• 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: Allocation ratio for sequence A/B versus sequence B/A.

  • 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, kaifenglu@gmail.com

Examples

(design1 <- getDesignMeanDiffXO(
  beta = 0.2, n = NA, meanDiff = 75, stDev = 150,
  normalApproximation = FALSE, alpha = 0.05))

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.

Usage

getDesignMeanDiffXOEquiv(
  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_
)

Arguments

Value

An S3 class designMeanDiffXOEquiv 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.

  • kMax: The number of stages.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH0: The expected information under H0.

  • numberOfSubjects: The maximum number of subjects.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanDiffLower: The lower equivalence limit of mean difference.

  • meanDiffUpper: The upper equivalence limit of mean difference.

  • meanDiff: The mean difference under the alternative hypothesis.

  • stDev: The standard deviation for within-subject random error.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

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

  • efficacyMeanDiffLower: The efficacy boundaries on the mean difference scale for the one-sided null hypothesis on the lower equivalence limit.

  • efficacyMeanDiffUpper: The efficacy boundaries on the mean difference scale for the one-sided null hypothesis on the upper equivalence limit.

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

  • numberOfSubjects: The number of subjects.

• 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.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: Allocation ratio for sequence A/B versus sequence B/A.

  • 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 exact calculation using the t distribution is only implemented for the fixed design.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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))

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.

Usage

getDesignMeanRatio(
  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_
)

Arguments

Value

An S3 class designMeanRatio 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanRatioH0: The mean ratio under the null hypothesis.

  • meanRatio: The mean ratio under the alternative hypothesis.

  • CV: The coefficient of variation.

• 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.

  • numberOfSubjects: The number of subjects.

  • efficacyMeanRatio: The efficacy boundaries on the mean ratio scale.

  • futilityMeanRatio: The futility boundaries on the mean ratio scale.

• 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: Allocation ratio for the active treatment versus control.

  • 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, kaifenglu@gmail.com

Examples

(design1 <- getDesignMeanRatio(
  beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
  alpha = 0.05, normalApproximation = FALSE))

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.

Usage

getDesignMeanRatioEquiv(
  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_
)

Arguments

Value

An S3 class designMeanRatioEquiv object with three components:

• overallResults: A data frame containing the following variables:

  • overallReject: The overall rejection probability.

  • alpha: The significance level for each of the two one-sided tests. Defaults to 0.05.

  • attainedAlpha: The attained significance level.

  • kMax: The number of stages.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH0: The expected information under H0.

  • numberOfSubjects: The maximum number of subjects.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanRatioLower: The lower equivalence limit of mean ratio.

  • meanRatioUpper: The upper equivalence limit of mean ratio.

  • meanRatio: The mean ratio under the alternative hypothesis.

  • CV: The coefficient of variation.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

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

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

  • numberOfSubjects: The number of subjects.

  • efficacyMeanRatioLower: The efficacy boundaries on the mean ratio scale for the one-sided null hypothesis on the lower equivalence limit.

  • efficacyMeanRatioUpper: The efficacy boundaries on the mean ratio scale for the one-sided null hypothesis on the upper equivalence limit.

• 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.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: Allocation ratio for the active treatment versus control.

  • 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 exact calculation using the t distribution is only implemented for the fixed design.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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))

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.

Usage

getDesignMeanRatioXO(
  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_
)

Arguments

Value

An S3 class designMeanRatioXO 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanRatioH0: The mean ratio under the null hypothesis.

  • meanRatio: The mean ratio under the alternative hypothesis.

  • CV: The coefficient of variation.

• 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.

  • efficacyMeanRatio: The efficacy boundaries on the mean ratio scale.

  • futilityMeanRatio: The futility boundaries on the mean ratio scale.

  • 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.

  • numberOfSubjects: The number of subjects.

• 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: Allocation ratio for sequence A/B versus sequence B/A.

  • 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, kaifenglu@gmail.com

Examples

(design1 <- getDesignMeanRatioXO(
  beta = 0.1, n = NA, meanRatio = 1.25, CV = 0.25,
  alpha = 0.05, normalApproximation = FALSE))

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.

Usage

getDesignMeanRatioXOEquiv(
  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_
)

Arguments

Value

An S3 class designMeanRatioEquiv 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.

  • kMax: The number of stages.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH0: The expected information under H0.

  • numberOfSubjects: The maximum number of subjects.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanRatioLower: The lower equivalence limit of mean ratio.

  • meanRatioUpper: The upper equivalence limit of mean ratio.

  • meanRatio: The mean ratio under the alternative hypothesis.

  • CV: The coefficient of variation.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

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

  • efficacyMeanRatioLower: The efficacy boundaries on the mean ratio scale for the one-sided null hypothesis on the lower equivalence limit.

  • efficacyMeanRatioUpper: The efficacy boundaries on the mean ratio scale for the one-sided null hypothesis on the upper equivalence limit.

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

  • numberOfSubjects: The number of subjects.

• 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.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: Allocation ratio for sequence A/B versus sequence B/A.

  • 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 exact calculation using the t distribution is only implemented for the fixed design.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, 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))

Group Sequential Design for Two-Sample Odds Ratio

Description

Obtains the power given sample size or obtains the sample size given power for a group sequential design for two-sample odds ratio.

Usage

getDesignOddsRatio(
  beta = NA_real_,
  n = NA_real_,
  oddsRatioH0 = 1,
  pi1 = NA_real_,
  pi2 = NA_real_,
  nullVariance = FALSE,
  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_
)

Arguments

Details

Consider a group sequential design for two-sample odds ratio. The parameter of interest is

\psi = \frac{\pi_1(1-\pi_2)}{(1-\pi_1)\pi_2}

where \pi_1 is the response probability for the active treatment group and \pi_2 is the response probability for the control group. For statistical inference, the parameter is often transformed to the log scale:

\theta = \log(\psi) = \log(\pi_1/(1-\pi_1)) - \log(\pi_2/(1-\pi_2))

The variance of the estimator \hat{\theta} can be derived from the binomial distributions as follows:

Var(\hat{\theta}) = \frac{1}{n} \{ \frac{1}{\pi_1(1-\pi_1) r} + \frac{1}{\pi_2(1-\pi_2)(1-r)} \}

where n is the total number of subjects and r is the randomization probability for the active treatment group. When nullVariance = TRUE, the variance is computed under the null hypothesis. In this case, the values of \pi_1 and \pi_2 in the variance formula are replaced with their restricted maximum likelihood counterparts, subject to the constraint

\frac{\pi_1(1-\pi_2)}{(1-\pi_1)\pi_2}= \psi_0

Value

An S3 class designOddsRatio 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • oddsRatioH0: The odds ratio under the null hypothesis.

  • pi1: The assumed probability for the active treatment group.

  • pi2: The assumed probability for the control group.

• 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.

  • efficacyOdddsRatio: The efficacy boundaries on the odds ratio scale.

  • futilityOddsRatio: The futility boundaries on the odds ratio scale.

  • numberOfSubjects: The number of subjects.

• 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.

  • varianceRatio: The ratio of the variance under H0 to the variance under H1.

  • nullVariance: Whether to use the variance under the null or the empirical variance under the alternative.

  • allocationRatioPlanned: Allocation ratio for the active treatment versus control.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

(design1 <- getDesignOddsRatio(
  beta = 0.1, n = NA, pi1 = 0.5, pi2 = 0.3,
  alpha = 0.05))

Group Sequential Design for Equivalence in Two-Sample Odds 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 odds ratio.

Usage

getDesignOddsRatioEquiv(
  beta = NA_real_,
  n = NA_real_,
  oddsRatioLower = NA_real_,
  oddsRatioUpper = NA_real_,
  pi1 = NA_real_,
  pi2 = NA_real_,
  allocationRatioPlanned = 1,
  rounding = TRUE,
  kMax = 1L,
  informationRates = NA_real_,
  criticalValues = NA_real_,
  alpha = 0.05,
  typeAlphaSpending = "sfOF",
  parameterAlphaSpending = NA_real_,
  userAlphaSpending = NA_real_,
  spendingTime = NA_real_
)

Arguments

Value

An S3 class designOddsRatioEquiv object with three components:

• overallResults: A data frame containing the following variables:

  • overallReject: The overall rejection probability.

  • alpha: The significance level for each of the two one-sided tests. Defaults to 0.05.

  • attainedAlphaH10: The attained significance level under H10.

  • attainedAlphaH20: The attained significance level under H20.

  • kMax: The number of stages.

  • information: The maximum information.

  • expectedInformationH1: The expected information under H1.

  • expectedInformationH10: The expected information under H10.

  • expectedInformationH20: The expected information under H20.

  • numberOfSubjects: The maximum number of subjects.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH10: The expected number of subjects under H10.

  • expectedNumberOfSubjectsH20: The expected number of subjects under H20.

  • oddsRatioLower: The lower equivalence limit of odds ratio.

  • oddsRatioUpper: The upper equivalence limit of odds ratio.

  • pi1: The assumed probability for the active treatment group.

  • pi2: The assumed probability for the control group.

  • oddsRatio: The odds ratio.

• byStageResults: A data frame containing the following variables:

  • informationRates: The information rates.

  • efficacyBounds: The efficacy boundaries on the Z-scale for each of the two one-sided tests.

  • rejectPerStage: The probability for efficacy stopping.

  • cumulativeRejection: The cumulative probability for efficacy stopping.

  • cumulativeAlphaSpent: The cumulative alpha for each of the two one-sided tests.

  • cumulativeAttainedAlphaH10: The cumulative alpha attained under H10.

  • cumulativeAttainedAlphaH20: The cumulative alpha attained under H20.

  • efficacyOddsRatioLower: The efficacy boundaries on the odds ratio scale for the one-sided null hypothesis on the lower equivalence limit.

  • efficacyOddsRatioUpper: The efficacy boundaries on the odds ratio scale for the one-sided null hypothesis on the upper equivalence limit.

  • efficacyP: The efficacy bounds on the p-value scale for each of the two one-sided tests.

  • information: The cumulative information.

  • numberOfSubjects: The number of subjects.

• 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.

  • spendingTime: The error spending time at each analysis.

  • allocationRatioPlanned: Allocation ratio for the active treatment versus control.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

(design1 <- getDesignOddsRatioEquiv(
  beta = 0.2, n = NA, oddsRatioLower = 0.8,
  oddsRatioUpper = 1.25, pi1 = 0.12, pi2 = 0.12,
  kMax = 3, alpha = 0.05, typeAlphaSpending = "sfOF"))

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.

Usage

getDesignOneMean(
  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_
)

Arguments

Value

An S3 class designOneMean 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • meanH0: The mean under the null hypothesis.

  • mean: The mean under the alternative hypothesis.

  • stDev: The standard deviation.

• 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.

  • efficacyMean: The efficacy boundaries on the mean scale.

  • futilityMean: The futility boundaries on the mean scale.

  • numberOfSubjects: The number of subjects.

• 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.

  • 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, 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))

Power and Sample Size for One-Sample Multinomial Response

Description

Obtains the power given sample size or obtains the sample size given power for one-sample multinomial response.

Usage

getDesignOneMultinom(
  beta = NA_real_,
  n = NA_real_,
  ncats = NA_integer_,
  piH0 = NA_real_,
  pi = NA_real_,
  rounding = TRUE,
  alpha = 0.05
)

Arguments

Details

A single-arm multinomial response design is used to test whether the prevalence of each category is different from the null hypothesis prevalence. The null hypothesis is that the prevalence of each category is equal to \pi_{0i}, while the alternative hypothesis is that the prevalence of each category is equal to \pi_i, for i=1,\ldots,C, where C is the number of categories.

The sample size is calculated based on the chi-square test for multinomial response. The test statistic is given by

X^2 = \sum_{i=1}^{C} \frac{(n_i - n\pi_{0i})^2}{n\pi_{0i}}

where n_i is the number of subjects in category i, and n is the total sample size.

• Under the null hypothesis, X^2 follows a chi-square distribution with C-1 degrees of freedom.

• Under the alternative hypothesis, X^2 follows a non-central chi-square distribution with non-centrality parameter

  \lambda = n \sum_{i=1}^{C} \frac{(\pi_i - \pi_{0i})^2}{\pi_{0i}}

The sample size is chosen such that the power to reject the null hypothesis is at least 1-\beta for a given significance level \alpha.

Value

An S3 class designOneMultinom object with the following components:

• power: The power to reject the null hypothesis.

• alpha: The two-sided significance level.

• n: The maximum number of subjects.

• ncats: The number of categories of the multinomial response.

• piH0: The prevalence of each category under the null hypothesis.

• pi: The prevalence of each category.

• effectsize: The effect size for the chi-square test.

• rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

(design1 <- getDesignOneMultinom(
  beta = 0.1, ncats = 3, piH0 = c(0.25, 0.25),
  pi = c(0.3, 0.4), alpha = 0.05))

Group Sequential Design for One-Sample Proportion

Description

Obtains the power given sample size or obtains the sample size given power for a group sequential design for one-sample proportion.

Usage

getDesignOneProportion(
  beta = NA_real_,
  n = NA_real_,
  piH0 = 0.1,
  pi = 0.2,
  nullVariance = TRUE,
  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

Value

An S3 class designOneProportion 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 as well as for the binomial exact test in a fixed design.

  • 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • piH0: The response probability under the null hypothesis.

  • pi: The response probability under the alternative hypothesis.

• 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.

  • efficacyResponses: The efficacy boundaries on the number of responses scale.

  • futilityResponses: The futility boundaries on the number of responses scale.

  • numberOfSubjects: The number of subjects.

• 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.

  • varianceRatio: The ratio of the variance under H0 to the variance under H1.

  • nullVariance: Whether to use the variance under the null or the empirical variance under the alternative.

  • 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 binomial distribution.

  • rounding: Whether to round up sample size.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Example 1: group sequential trial power calculation
(design1 <- getDesignOneProportion(
  beta = 0.2, n = NA, piH0 = 0.15, pi = 0.25,
  kMax = 3, alpha = 0.05, typeAlphaSpending = "sfOF"))

# Example 2: sample size calculation for one-sample binomial exact test
(design2 <- getDesignOneProportion(
  beta = 0.2, n = NA, piH0 = 0.15, pi = 0.25,
  normalApproximation = FALSE, alpha = 0.05))

Power and Sample Size for One-Sample Poisson Rate Exact Test

Description

Obtains the power given sample size or obtains the sample size given power for one-sample Poisson rate.

Usage

getDesignOneRateExact(
  beta = NA_real_,
  n = NA_real_,
  lambdaH0 = NA_real_,
  lambda = NA_real_,
  D = 1,
  alpha = 0.025
)

Arguments

Value

A data frame containing the following variables:

• alpha: The specified significance level.

• attainedAlpha: The attained type I error of the exact test.

• power: The actual power of the exact test.

• n: The sample size.

• lambdaH0: The Poisson rate under the null hypothesis.

• lambda: The Poisson rate under the alternative hypothesis.

• D: The average exposure per subject.

• r: The critical value of the number of events for rejecting the null hypothesis. Reject H0 if Y >= r for upper-tailed test, and reject H0 if Y <= r for lower-tailed test.

Author(s)

Kaifeng Lu, kaifenglu@gmail.com

Examples

# Example 1: power calculation
(design1 <- getDesignOneRateExact(
  n = 525, lambdaH0 = 0.049, lambda = 0.012,
  D = 0.5, alpha = 0.025))

# Example 2: sample size calculation
(design2 <- getDesignOneRateExact(
  beta = 0.2, lambdaH0 = 0.2, lambda = 0.3,
  D = 1, alpha = 0.05))

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.

Usage

getDesignOneSlope(
  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_
)

Arguments

Details

We assume a simple linear regression of the form

y_i = \alpha + \beta x_i + \epsilon_i

where \epsilon_i is the residual error, which is assumed to be normally distributed with mean 0 and standard deviation \sigma_\epsilon. The covariate x_i is assumed to be normally distributed with mean 0 and standard deviation \sigma_x. The slope under the null hypothesis is \beta_0, and the slope under the alternative hypothesis is \beta. Since

\hat{\beta} = \frac{\sum_{i=1}^{n} (x_i-\bar{x}) y_i} {\sum_{i=1}^{n}(x_i-\bar{x})^2}

it follows that

\hat{\beta} \sim N(\beta, \frac{\sigma_\epsilon^2}{\sum_{i=1}^{n}(x_i-\bar{x})^2}).

Since the variance of \hat{\beta} is

\frac{\sigma_\epsilon^2}{n\sigma_x^2}

we can use it to calculate the power and sample size for the group sequential design.

Value

An S3 class designOneSlope 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.

  • expectedNumberOfSubjectsH1: The expected number of subjects under H1.

  • expectedNumberOfSubjectsH0: The expected number of subjects under H0.

  • slopeH0: The slope under the null hypothesis.

  • slope: The slope under the alternative hypothesis.

  • stDev: The standard deviation of the residual.

  • stDevCovariate: The standard deviation of the covariate.

• 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.

  • efficacySlope: The efficacy boundaries on the slope scale.

  • futilitySlope: The futility boundaries on the slope scale.

  • numberOfSubjects: The number of subjects.

• 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.

  • 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.