Package 'lrstat'

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, [email protected]

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)

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, [email protected]

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)

Description

Obtains the conditional power for specified incremental information given the interim results, parameter value, and data-dependent changes in the error spending function, and the number and spacing of interim looks. Conversely, obtains the incremental information needed to attain a specified conditional power given the interim results, parameter value, and 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, [email protected]

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, 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, 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)

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.

Description

Performs simulation for two-endpoint two-arm group sequential trials. The first endpoint is a binary endpoint and the Mantel-Haenszel test is used to test risk difference. The second endpoint is a time-to-event endpoint and the log-rank test is used to test the treatment difference. The analysis times of the first endpoint are determined by the specified calendar times, while the analysis times for the second endpoint is based on the planned number of events at each look. The binary endpoint is assessed at the first post-treatment follow-up visit.

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. Therefor 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, [email protected]

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)

Description

Obtains the decision table for the Bayesian optimal interval (BOIN) design.

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: The decision data frame for the BOIN design. It includes the following variables:

  • n: The sample size.

  • y: The number of toxicities.

  • decision: The dosing decision.

• decisionMatrix: The decision matrix corresponding to the decision data frame.

Author(s)

Kaifeng Lu, [email protected]

Examples

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

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, [email protected]

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)

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, [email protected]

Examples

ClopperPearsonCI(20, 3)

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, [email protected]

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)

Description

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

Usage

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

Arguments

Value

A vector of errors spent up to the interim look.

Author(s)

Kaifeng Lu, [email protected]

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)

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, [email protected]

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))

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, [email protected]

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)

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, [email protected]

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)

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, [email protected]

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)

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, [email protected]

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)

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, [email protected]

Examples

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

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, [email protected]

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))

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, [email protected]

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)

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, [email protected]

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)

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, [email protected]

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")

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, [email protected]

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))

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, [email protected]

Examples

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

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, [email protected]

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)

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, [email protected]

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)

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, [email protected]

Examples

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

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, [email protected]

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)

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, [email protected]

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")

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, [email protected]

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"))

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, [email protected]

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))

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, [email protected]

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))

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, [email protected]

Examples

(design1 <- getDesignANOVAContrast(
  beta = 0.1, ngroups = 4, means = c(1.5, 2.5, 2, 0),
  stDev = 3.5, contrast = c(1, 1, 1, -3)/3,
  allocationRatioPlanned = c(2, 2, 2, 1),
  alpha = 0.025))

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_,
  varianceRatioH10 = 1,
  varianceRatioH20 = 1,
  varianceRatioH12 = 1,
  varianceRatioH21 = 1
)

Arguments

Details

Consider the equivalence design with two one-sided hypotheses:

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

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