RPACT Multi-Arm Designs

Gernot Wassmer

November 13, 2025

RPACT Multi-Arm Designs

Theoretical Background of Adaptive Designs

Clue of Combination Testing Principle

  • Proposed by Peter Bauer (1989): Multistage testing with adaptive designs, in Biometrie und Informatik in Medizin und Biologie, Bauer and Köhne (1994) was the breakthrough.

  • Do not pool the data of the stages, combine the stage-wise \(p\,\)-values.

  • Then the distribution of the combination function under the null does not depend on design modifications, and the adaptive test is still a test at the level \(\alpha\) for the modified design.

  • In the two stages, even different hypotheses \(H_{01}\) and \(H_{02}\) can be considered, the considered global test is a test for \(H_0 = H_{01} \cap H_{02}\).

  • Or there are multiple hypotheses at the beginning of the trials and maybe some selected at interim.

  • Or there will be even hypotheses to be added at an interim stage (not of practical concern).

Clue of Combination Testing Principle

  • The rules for adapting the design need not be prespecified!

  • The combination test needs to be prespecified:

    • Bauer and Köhne (1994) proposed Fisher’s combination test (\(p_1 \cdot p_2\)) but also mentioned other combination tests, e.g., the inverse normal combination test (\(w_1 \Phi^{-1}(1 - p_1) + w_2 \Phi^{-1}(1 - p_2)\) ).

    • It was shown that the inverse normal combination test has the decisive advantage that then the adaptive confirmatory design simply generalizes the group sequential design (Lehmacher and Wassmer 1999).

    • So (initial) planning of an adaptive design is essentially the same as planning of a classical group sequential design and the same software can be used.

    • An equivalent procedure (not based on p-values but based on weighted \(Z\) score statistic) was proposed by Cui, Hung, and Wang (1999) .

Possible Data Dependent Changes of Design

Examples of data dependent changes of design are

  • Sample size recalculation

  • Change of allocation ratio

  • Change of test statistic

  • Flexible number of looks

  • Treatment arm selection (seamless phase II/III)

  • Population selection (population enrichment)

  • Selection of endpoints

For the latter three, in general, multiple hypotheses testing applies and a closed testing procedure can be used in order to control the experimentwise error rate in a strong sense.

Methods for Multi-Arm Multi-Stage (MAMS) Designs

  • Methods for predefined selection rules

    (Stallard and Todd (2003), Magirr, Stallard, and Jaki (2014), …)

  • Flexible Multi-Stage Closed Combination Tests

    (Bauer and Kieser (1999), Hommel (2001), …)

    • Do not require a predefined treatment and sample size selection rule.

    • Combine two methodology concepts:

      Combination Tests and Closed Testing Principle.

Closed Testing Principle, 3 Hypotheses, select one

Combination tests to be performed for the closed system of hypotheses (\(G = 3\)) for testing hypothesis \(H_0^3\) if treatment arm 3 is selected for the second stage

Closed Testing Principle, 3 Hypotheses, select two

Combination tests to be performed for the closed system of hypotheses (\(G = 3\)) for testing hypothesis \(H_0^3\) if treatment arms 2 and 3 are selected for the second stage

Adaptive Designs with Treatment Arm Selection

  • Can be applied to selection of one or more than one treatment arm. The number of selected arms and the way of how to select treatment arms needs not to be preplanned.

  • Choice of combination test is free.

  • Data-driven recalculation of sample size is possible.

  • Choice of intersection tests is free. You can choose between Dunnett, Bonferroni, Simes, Sidak, hierarchical testing, etc.

  • For two-stage designs, the CRP principle can be applied: adaptive Dunnett test (König et al. (2008), Wassmer and Brannath (2016), Section 11.1.5, Mehta and Kappler (2025)).

  • Confidence intervals based on stepwise testing are difficult to construct. This is a specific feature of multiple testing procedures and not of adaptive testing. Posch et al. (2005) proposed to construct repeated confidence intervals based on the single step adjusted overall p-values.

rpact Multi-Arm Analysis

Current Methods

  • Consider many-to-one comparisons comparing G active treatment arms to control.

  • Given a design and a dataset, at given stage the function

    getAnalysisResults(design, dataInput, ...)

    calculates the results of the closed test procedure, overall p-values and test statistics, conditional rejection probability (CRP), conditional power, repeated confidence intervals (RCIs), and repeated overall p-values.

  • design is either from getDesignInverseNormal() or getDesignFisher() (or NULL)

  • For two stages, design <- getDesignConditionalDunnett() can be selected.

Multi-Arm Analysis

  • The conditional power is calculated only if (at least) the sample size nPlanned for the subsequent stage(s) is specified.

  • dataInput is the summary data used for calculating the test results. This is either an element of DataSetMeans, of DataSetRates, or of DataSetSurvival.

  • dataInput is defined through getDataset(), rpact identifies the type of endpoint.

  • In rpact 3.0, getDataset() is generalized to an arbitrary number of treatment arms.

Multi-Arm Analysis

dataInput

  • An element of DataSetMeans for one sample is created by

    getDataset(means =, stDevs =, sampleSizes =)

    where means, stDevs, sampleSizes are vectors with stagewise means, standard deviations, and sample sizes of length given by the number of available stages.

  • An element of DataSetMeans for two samples is created by

    getDataset(means1 =, means2 =, stDevs1 =, stDevs2 =, sampleSizes1 =, sampleSizes2 =)

    where means1, means2, stDevs1, stDevs2, sampleSizes1, sampleSizes2 are vectors with stagewise means, standard deviations, and sample sizes for the two treatment groups of length given by the number of available stages.

Multi-Arm Analysis

dataInput

  • An element of DataSetMeans for G + 1 samples is created by

    getDataset(means1 =,..., means[G+1] =, stDevs1 =, ..., stDevs[G+1] =, sampleSizes1 =, ..., sampleSizes[G+1] =),

    where means1, ..., means[G+1], stDevs1, ..., stDevs[G+1], sampleSizes1, ..., sampleSizes[G+1] are vectors with stagewise means, standard deviations, and sample sizes for G+1 treatment groups of length given by the number of available stages.

  • Last treatment arm G + 1 always refers to the control group that cannot be deselected.

  • Only for the first stage all treatment arms needs to be specified, so treatment arm selection with an arbitrary number of treatment arms for subsequent stage can be considered.

  • Analogue definition of DataSetRates and DataSetSurvival.

Multi-Arm Analysis Example

exampleMeans <- getDataset(
    n1      = c( 23,  25),
    n2      = c( 25,  NA),
    n3      = c( 24,  27),
    n4      = c( 22,  29),
    means1  = c(2.41, 2.27),
    means2  = c(1.38,  NA),
    means3  = c(2.07, 2.01),
    means4  = c(0.92, 1.02),
    stDevs1 = c(2.24, 2.21),
    stDevs2 = c(2.12,  NA),
    stDevs3 = c(2.56, 2.32),
    stDevs4 = c(2.15, 2.21)
)

Multi-Arm Analysis Example

designIN <- getDesignInverseNormal(
  typeOfDesign = "WT",
  deltaWT = 0.25, 
  informationRates = c(0.25, 0.5, 1)
)
results <- designIN |> 
    getAnalysisResults(dataInput = exampleMeans)
results |> print()

Multi-Arm Analysis Example

Multi-arm analysis results (means of 4 groups, inverse normal combination test design)

Design parameters

  • Fixed weights: 0.500, 0.500, 0.707
  • Critical values: 2.904, 2.442, 2.053
  • Futility bounds (non-binding): -Inf, -Inf
  • Cumulative alpha spending: 0.001843, 0.008414, 0.025000
  • Local one-sided significance levels: 0.001843, 0.007307, 0.020021
  • Significance level: 0.0250
  • Test: one-sided

Default parameters

  • Normal approximation: FALSE
  • Direction upper: TRUE
  • Theta H0: 0
  • Intersection test: Dunnett
  • Variance option: overallPooled

Stage results

  • Cumulative effect sizes (1): 1.490, 1.360, NA
  • Cumulative effect sizes (2): 0.460, NA, NA
  • Cumulative effect sizes (3): 1.150, 1.061, NA
  • Cumulative (pooled) standard deviations (1): 2.197, 2.182, NA
  • Cumulative (pooled) standard deviations (2): 2.134, NA, NA
  • Cumulative (pooled) standard deviations (3): 2.373, 2.291, NA
  • Stage-wise test statistics (1): 2.196, 2.038, NA
  • Stage-wise test statistics (2): 0.691, NA, NA
  • Stage-wise test statistics (3): 1.712, 1.647, NA
  • Separate p-values (1): 0.01535, 0.02246, NA
  • Separate p-values (2): 0.24552, NA, NA
  • Separate p-values (3): 0.04516, 0.05176, NA

Adjusted stage-wise p-values

  • Treatments 1, 2, 3 vs. control: 0.03904, 0.04112, NA
  • Treatments 1, 2 vs. control: 0.02806, 0.02246, NA
  • Treatments 1, 3 vs. control: 0.02810, 0.04112, NA
  • Treatments 2, 3 vs. control: 0.07879, 0.05176, NA
  • Treatment 1 vs. control: 0.01535, 0.02246, NA
  • Treatment 2 vs. control: 0.24552, NA, NA
  • Treatment 3 vs. control: 0.04516, 0.05176, NA

Overall adjusted test statistics

  • Treatments 1, 2, 3 vs. control: 1.762, 2.475, NA
  • Treatments 1, 2 vs. control: 1.910, 2.769, NA
  • Treatments 1, 3 vs. control: 1.909, 2.579, NA
  • Treatments 2, 3 vs. control: 1.413, 2.151, NA
  • Treatment 1 vs. control: 2.161, 2.946, NA
  • Treatment 2 vs. control: 0.689, NA, NA
  • Treatment 3 vs. control: 1.694, 2.349, NA

Test actions

  • Rejected (1): FALSE, TRUE, NA
  • Rejected (2): FALSE, FALSE, NA
  • Rejected (3): FALSE, FALSE, NA

Further analysis results

  • Conditional rejection probability (1): 0.11367, 0.33392, NA
  • Conditional rejection probability (2): 0.02593, NA, NA
  • Conditional rejection probability (3): 0.07194, 0.22563, NA
  • Conditional power (1): NA, NA, NA
  • Conditional power (2): NA, NA, NA
  • Conditional power (3): NA, NA, NA
  • Repeated confidence intervals (lower) (1): -0.76273, 0.01443, NA
  • Repeated confidence intervals (lower) (2): -1.74824, NA, NA
  • Repeated confidence intervals (lower) (3): -1.07967, -0.26374, NA
  • Repeated confidence intervals (upper) (1): 3.743, 2.719, NA
  • Repeated confidence intervals (upper) (2): 2.668, NA, NA
  • Repeated confidence intervals (upper) (3): 3.380, 2.399, NA
  • Repeated p-values (1): 0.15176, 0.02326, NA
  • Repeated p-values (2): 0.46577, NA, NA
  • Repeated p-values (3): 0.23285, 0.04597, NA

Legend

  • (i): results of treatment arm i vs. control group 4

Multi-Arm Analysis Example

results |> summary()

Multi-Arm Analysis Example

Multi-arm analysis results for a continuous endpoint (3 active arms vs. control)

Sequential analysis with 3 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were calculated using a multi-arm t-test, Dunnett intersection test, overall pooled variances option. H0: mu(i) - mu(control) = 0 against H1: mu(i) - mu(control) > 0.

Stage 1 2 3
Fixed weight 0.5 0.5 0.707
Cumulative alpha spent 0.0018 0.0084 0.0250
Stage levels (one-sided) 0.0018 0.0073 0.0200
Efficacy boundary (z-value scale) 2.904 2.442 2.053
Cumulative effect size (1) 1.490 1.360
Cumulative effect size (2) 0.460
Cumulative effect size (3) 1.150 1.061
Cumulative (pooled) standard deviation 2.276 2.263
Stage-wise test statistic (1) 2.196 2.038
Stage-wise test statistic (2) 0.691
Stage-wise test statistic (3) 1.712 1.647
Stage-wise p-value (1) 0.0153 0.0225
Stage-wise p-value (2) 0.2455
Stage-wise p-value (3) 0.0452 0.0518
Adjusted stage-wise p-value (1, 2, 3) 0.0390 0.0411
Adjusted stage-wise p-value (1, 2) 0.0281 0.0225
Adjusted stage-wise p-value (1, 3) 0.0281 0.0411
Adjusted stage-wise p-value (2, 3) 0.0788 0.0518
Adjusted stage-wise p-value (1) 0.0153 0.0225
Adjusted stage-wise p-value (2) 0.2455
Adjusted stage-wise p-value (3) 0.0452 0.0518
Overall adjusted test statistic (1, 2, 3) 1.762 2.475
Overall adjusted test statistic (1, 2) 1.910 2.769
Overall adjusted test statistic (1, 3) 1.909 2.579
Overall adjusted test statistic (2, 3) 1.413 2.151
Overall adjusted test statistic (1) 2.161 2.946
Overall adjusted test statistic (2) 0.689
Overall adjusted test statistic (3) 1.694 2.349
Test action: reject (1) FALSE TRUE
Test action: reject (2) FALSE FALSE
Test action: reject (3) FALSE FALSE
Conditional rejection probability (1) 0.1137 0.3339
Conditional rejection probability (2) 0.0259
Conditional rejection probability (3) 0.0719 0.2256
95% repeated confidence interval (1) [-0.763; 3.743] [0.014; 2.719]
95% repeated confidence interval (2) [-1.748; 2.668]
95% repeated confidence interval (3) [-1.080; 3.380] [-0.264; 2.399]
Repeated p-value (1) 0.1518 0.0233
Repeated p-value (2) 0.4658
Repeated p-value (3) 0.2329 0.0460

Legend:

  • (i): results of treatment arm i vs. control arm
  • (i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Multi-Arm Analysis Example

results |> plot(, type = 2)

Multi-Arm Analysis Example

Conditional Power

result <- designIN |> 
    getAnalysisResults(
        dataInput = exampleMeans, 
        nPlanned = 80
    )
result |> summary()

Multi-Arm Analysis Example

Multi-arm analysis results for a continuous endpoint (3 active arms vs. control)

Sequential analysis with 3 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were calculated using a multi-arm t-test, Dunnett intersection test, overall pooled variances option. H0: mu(i) - mu(control) = 0 against H1: mu(i) - mu(control) > 0. The conditional power calculation with planned sample size is based on overall effect: thetaH1(1) = 1.36, thetaH1(2) = NA, thetaH1(3) = 1.06 and overall standard deviation: sd(1) = 2.18, sd(2) = NA, sd(3) = 2.29.

Stage 1 2 3
Fixed weight 0.5 0.5 0.707
Cumulative alpha spent 0.0018 0.0084 0.0250
Stage levels (one-sided) 0.0018 0.0073 0.0200
Efficacy boundary (z-value scale) 2.904 2.442 2.053
Cumulative effect size (1) 1.490 1.360
Cumulative effect size (2) 0.460
Cumulative effect size (3) 1.150 1.061
Cumulative (pooled) standard deviation 2.276 2.263
Stage-wise test statistic (1) 2.196 2.038
Stage-wise test statistic (2) 0.691
Stage-wise test statistic (3) 1.712 1.647
Stage-wise p-value (1) 0.0153 0.0225
Stage-wise p-value (2) 0.2455
Stage-wise p-value (3) 0.0452 0.0518
Adjusted stage-wise p-value (1, 2, 3) 0.0390 0.0411
Adjusted stage-wise p-value (1, 2) 0.0281 0.0225
Adjusted stage-wise p-value (1, 3) 0.0281 0.0411
Adjusted stage-wise p-value (2, 3) 0.0788 0.0518
Adjusted stage-wise p-value (1) 0.0153 0.0225
Adjusted stage-wise p-value (2) 0.2455
Adjusted stage-wise p-value (3) 0.0452 0.0518
Overall adjusted test statistic (1, 2, 3) 1.762 2.475
Overall adjusted test statistic (1, 2) 1.910 2.769
Overall adjusted test statistic (1, 3) 1.909 2.579
Overall adjusted test statistic (2, 3) 1.413 2.151
Overall adjusted test statistic (1) 2.161 2.946
Overall adjusted test statistic (2) 0.689
Overall adjusted test statistic (3) 1.694 2.349
Test action: reject (1) FALSE TRUE
Test action: reject (2) FALSE FALSE
Test action: reject (3) FALSE FALSE
Conditional rejection probability (1) 0.1137 0.3339
Conditional rejection probability (2) 0.0259
Conditional rejection probability (3) 0.0719 0.2256
Planned sample size 80
Conditional power (1) 0.9908
Conditional power (2)
Conditional power (3) 0.9064
95% repeated confidence interval (1) [-0.763; 3.743] [0.014; 2.719]
95% repeated confidence interval (2) [-1.748; 2.668]
95% repeated confidence interval (3) [-1.080; 3.380] [-0.264; 2.399]
Repeated p-value (1) 0.1518 0.0233
Repeated p-value (2) 0.4658
Repeated p-value (3) 0.2329 0.0460

Legend:

  • (i): results of treatment arm i vs. control arm
  • (i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

Multi-Arm Analysis Example

Conditional Power

result |> plot(type = 1)

Multi-Arm Analysis Example

Final stage

exampleMeans <- getDataset(
    n1      = c( 23,  25, NA),
    n2      = c( 25,  NA, NA),
    n3      = c( 24,  27, 42),
    n4      = c( 22,  29, 47),
    means1  = c(2.41, 2.27, NA),
    means2  = c(1.38,  NA, NA),
    means3  = c(2.07, 2.01, 2.05),
    means4  = c(0.92, 1.02, 1.05),
    stDevs1 = c(2.24, 2.21, NA),
    stDevs2 = c(2.12,  NA, NA),
    stDevs3 = c(2.56, 2.32, 2.15),
    stDevs4 = c(2.15, 2.21, 2.09)
)

designIN |> getAnalysisResults(
  dataInput = exampleMeans
) |> summary()

Multi-Arm Analysis Example

Multi-arm analysis results for a continuous endpoint (3 active arms vs. control)

Sequential analysis with 3 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were calculated using a multi-arm t-test, Dunnett intersection test, overall pooled variances option. H0: mu(i) - mu(control) = 0 against H1: mu(i) - mu(control) > 0.

Stage 1 2 3
Fixed weight 0.5 0.5 0.707
Cumulative alpha spent 0.0018 0.0084 0.0250
Stage levels (one-sided) 0.0018 0.0073 0.0200
Efficacy boundary (z-value scale) 2.904 2.442 2.053
Cumulative effect size (1) 1.490 1.360
Cumulative effect size (2) 0.460
Cumulative effect size (3) 1.150 1.061 1.032
Cumulative (pooled) standard deviation 2.276 2.263 2.201
Stage-wise test statistic (1) 2.196 2.038
Stage-wise test statistic (2) 0.691
Stage-wise test statistic (3) 1.712 1.647 2.223
Stage-wise p-value (1) 0.0153 0.0225
Stage-wise p-value (2) 0.2455
Stage-wise p-value (3) 0.0452 0.0518 0.0144
Adjusted stage-wise p-value (1, 2, 3) 0.0390 0.0411 0.0144
Adjusted stage-wise p-value (1, 2) 0.0281 0.0225
Adjusted stage-wise p-value (1, 3) 0.0281 0.0411 0.0144
Adjusted stage-wise p-value (2, 3) 0.0788 0.0518 0.0144
Adjusted stage-wise p-value (1) 0.0153 0.0225
Adjusted stage-wise p-value (2) 0.2455
Adjusted stage-wise p-value (3) 0.0452 0.0518 0.0144
Overall adjusted test statistic (1, 2, 3) 1.762 2.475 3.296
Overall adjusted test statistic (1, 2) 1.910 2.769
Overall adjusted test statistic (1, 3) 1.909 2.579 3.369
Overall adjusted test statistic (2, 3) 1.413 2.151 3.066
Overall adjusted test statistic (1) 2.161 2.946
Overall adjusted test statistic (2) 0.689
Overall adjusted test statistic (3) 1.694 2.349 3.207
Test action: reject (1) FALSE TRUE TRUE
Test action: reject (2) FALSE FALSE FALSE
Test action: reject (3) FALSE FALSE TRUE
Conditional rejection probability (1) 0.1137 0.3339
Conditional rejection probability (2) 0.0259
Conditional rejection probability (3) 0.0719 0.2256
95% repeated confidence interval (1) [-0.763; 3.743] [0.014; 2.719]
95% repeated confidence interval (2) [-1.748; 2.668]
95% repeated confidence interval (3) [-1.080; 3.380] [-0.264; 2.399] [0.244; 1.827]
Repeated p-value (1) 0.1518 0.0233
Repeated p-value (2) 0.4658
Repeated p-value (3) 0.2329 0.0460 0.0012

Legend:

  • (i): results of treatment arm i vs. control arm
  • (i, j, …): comparison of treatment arms ‘i, j, …’ vs. control arm

rpact Multi-Arm Simulation

Overview

getSimulationMultiArmMeans(design,...),

getSimulationMultiArmRates(design,...), and

getSimulationMultiArmSurvival(design,...)

  • perform simulations in multi-arm designs for testing means, rates, and hazard ratios, respectively.

  • You can assess different treatment arm selection strategies, sample size reassessment methods, general stopping, and stopping for futility rules.

  • Define selection strategy and effect size pattern appropriately (e.g., linear, sigmoidEmax, user defined, etc).

Time to Event example:

  • Time to disease progression event

  • 2 active arms, 1 control arm

  • Equal allocation between groups

  • Power 90%

  • \(\alpha\) = 0.025 one sided

  • 2 analyses (1 IA at 50% events) futility analysis at interim and select best dose based on highest HR

  • Assume median TTE in control arm: 25 months

  • Median TTE in active: 18 months so target HR 0.72

  • Accrual: Assume 10 for first 10 months, then 20 for next 10 then 30 per month thereafter for max 36 months (or feel free to use a constant accrual rate)

Sample Size Calculation

Around 390 events are needed to achieve 90% power for a two-sample comparison:

getSampleSizeSurvival(
    alpha = 0.025,
    beta = 0.1,
    median2 = 25,
    hazardRatio = 0.72,
    accrualTime = c(0, 10, 20, 36),
    accrualIntensity = c(10, 20, 30)
) |> summary()

Sample size calculation for a survival endpoint

Fixed sample analysis, one-sided significance level 2.5%, power 90%. The results were calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: hazard ratio = 0.72, control median(2) = 25, accrual time = c(10, 20, 36), accrual intensity = c(10, 20, 30).

Stage Fixed
Stage level (one-sided) 0.0250
Efficacy boundary (z-value scale) 1.960
Efficacy boundary (t) 0.820
Number of subjects 780.0
Number of events 389.5
Analysis time 52.02
Expected study duration under H1 52.02

Legend:

  • (t): treatment effect scale

Design with Interim Stage and Bonferroni

getDesignInverseNormal(
    kMax = 2,
    typeOfDesign = "noEarlyEfficacy",
    alpha = 0.0125,
    beta = 0.1,
    futilityBounds = 0
) |> getSampleSizeSurvival(
    median2 = 25,
    hazardRatio = 0.72,
    accrualTime = c(0, 10, 20, 36),
    accrualIntensity = c(10, 20, 30)
) |> summary()

Design with Interim Stage and Bonferroni

Sample size calculation for a survival endpoint

Sequential analysis with a maximum of 2 looks (inverse normal combination test design), one-sided overall significance level 1.25%, power 90%. The results were calculated for a two-sample logrank test, H0: hazard ratio = 1, H1: hazard ratio = 0.72, control median(2) = 25, accrual time = c(10, 20, 36), accrual intensity = c(10, 20, 30).

Stage 1 2
Fixed weight 0.707 0.707
Cumulative alpha spent 0 0.0125
Stage levels (one-sided) 0 0.0125
Efficacy boundary (z-value scale) Inf 2.241
Futility boundary (z-value scale) 0
Efficacy boundary (t) 0 0.812
Futility boundary (t) 1.000
Cumulative power 0 0.9000
Number of subjects 780.0 780.0
Expected number of subjects under H1 780.0
Cumulative number of events 230.8 461.6
Expected number of events under H1 460.2
Analysis time 37.53 60.77
Expected study duration under H1 60.62
Overall exit probability (under H0) 0.5000
Overall exit probability (under H1) 0.0063
Exit probability for efficacy (under H0) 0
Exit probability for efficacy (under H1) 0
Exit probability for futility (under H0) 0.5000
Exit probability for futility (under H1) 0.0063

Legend:

  • (t): treatment effect scale

A Multi-Arm Approach

  • Based on this result, we plan a multi-arm design with a maximum of 460 events in order to achieve power 90%

  • The procedure is designed in such a way that in case of selecting a treatment arm the specified events need to be observed for the remaining arms (i.e., no event number recalculation)

  • Note, for multi-armed designs, to simulate TTE on a patient level is not available yet. However, in rpact the approach of Deng et al. (2019) for simulating normally distributed log-rank statistics is implemented

  • The number of events for pairwise comparisons are estimated from the assumption about the hazard ratios

  • This provides a reasonable approximation for the assessment of test characteristics, i.e., for the estimation of power and selection probabilities.

Example Select All

design <- getDesignInverseNormal(
    kMax = 2,
    typeOfDesign = "noEarlyEfficacy",
    alpha = 0.025,
    futilityBounds = 0
)
effectMatrix = matrix(
    c(0.72, 0.72,
    0.72, 0.8,
    0.72, 0.9,
    0.8, 0.8,
    0.8, 0.9,
    1, 1), 
    ncol = 2, byrow = TRUE
)
design |> getSimulationMultiArmSurvival(
    activeArms = 2,
    directionUpper = FALSE,
    typeOfShape = "userDefined",
    effectMatrix = effectMatrix,
    typeOfSelection = "all",
    plannedEvents = c(230, 460),
    maxNumberOfIterations = 1000,
    seed = 123
) |> summary()

Example Select All

Simulation of a survival endpoint (multi-arm design)

Sequential analysis with a maximum of 2 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were simulated for a multi-arm logrank test (2 treatments vs. control), H0: hazard ratio(i) = 1, power directed towards smaller values, H1: omega_max as specified, planned cumulative events = c(230, 460), effect shape = user defined, intersection test = Dunnett, selection = all, effect measure based on effect estimate, success criterion: all, simulation runs = 1000, seed = 123.

Stage 1 2
Fixed weight 0.707 0.707
Cumulative alpha spent 0 0.0250
Stage levels (one-sided) 0 0.0250
Efficacy boundary (z-value scale) Inf 1.960
Futility boundary (z-value scale) 0
Reject at least one [1] 0.9080
Reject at least one [2] 0.8020
Reject at least one [3] 0.6990
Reject at least one [4] 0.5770
Reject at least one [5] 0.3670
Reject at least one [6] 0.0190
Rejected arms per stage [1]
Treatment arm 1 vs. control 0 0.8290
Treatment arm 2 vs. control 0 0.8240
Rejected arms per stage [2]
Treatment arm 1 vs. control 0 0.7630
Treatment arm 2 vs. control 0 0.4990
Rejected arms per stage [3]
Treatment arm 1 vs. control 0 0.6930
Treatment arm 2 vs. control 0 0.1530
Rejected arms per stage [4]
Treatment arm 1 vs. control 0 0.4490
Treatment arm 2 vs. control 0 0.4580
Rejected arms per stage [5]
Treatment arm 1 vs. control 0 0.3510
Treatment arm 2 vs. control 0 0.1170
Rejected arms per stage [6]
Treatment arm 1 vs. control 0 0.0150
Treatment arm 2 vs. control 0 0.0110
Success per stage [1] 0 0.7450
Success per stage [2] 0 0.4600
Success per stage [3] 0 0.1470
Success per stage [4] 0 0.3300
Success per stage [5] 0 0.1010
Success per stage [6] 0 0.0070
Cumulative number of events [1]
Treatment arm 1 vs. control 162.1 324.3
Treatment arm 2 vs. control 162.1 324.3
Cumulative number of events [2]
Treatment arm 1 vs. control 157.0 314.0
Treatment arm 2 vs. control 164.3 328.6
Cumulative number of events [3]
Treatment arm 1 vs. control 151.0 302.0
Treatment arm 2 vs. control 166.8 333.6
Cumulative number of events [4]
Treatment arm 1 vs. control 159.2 318.5
Treatment arm 2 vs. control 159.2 318.5
Cumulative number of events [5]
Treatment arm 1 vs. control 153.3 306.7
Treatment arm 2 vs. control 161.9 323.7
Cumulative number of events [6]
Treatment arm 1 vs. control 153.3 306.7
Treatment arm 2 vs. control 153.3 306.7
Expected number of events under H1 [1] 458.6
Expected number of events under H1 [2] 456.1
Expected number of events under H1 [3] 449.9
Expected number of events under H1 [4] 445.5
Expected number of events under H1 [5] 431.5
Expected number of events under H1 [6] 345.7
Overall exit probability [1] 0.0060
Overall exit probability [2] 0.0170
Overall exit probability [3] 0.0440
Overall exit probability [4] 0.0630
Overall exit probability [5] 0.1240
Overall exit probability [6] 0.4970
Selected arms [1]
Treatment arm 1 vs. control 1.0000 0.9940
Treatment arm 2 vs. control 1.0000 0.9940
Selected arms [2]
Treatment arm 1 vs. control 1.0000 0.9830
Treatment arm 2 vs. control 1.0000 0.9830
Selected arms [3]
Treatment arm 1 vs. control 1.0000 0.9560
Treatment arm 2 vs. control 1.0000 0.9560
Selected arms [4]
Treatment arm 1 vs. control 1.0000 0.9370
Treatment arm 2 vs. control 1.0000 0.9370
Selected arms [5]
Treatment arm 1 vs. control 1.0000 0.8760
Treatment arm 2 vs. control 1.0000 0.8760
Selected arms [6]
Treatment arm 1 vs. control 1.0000 0.5030
Treatment arm 2 vs. control 1.0000 0.5030
Number of active arms [1] 2.000 2.000
Number of active arms [2] 2.000 2.000
Number of active arms [3] 2.000 2.000
Number of active arms [4] 2.000 2.000
Number of active arms [5] 2.000 2.000
Number of active arms [6] 2.000 2.000
Conditional power (achieved) [1] 0.8718
Conditional power (achieved) [2] 0.8029
Conditional power (achieved) [3] 0.7665
Conditional power (achieved) [4] 0.6935
Conditional power (achieved) [5] 0.5708
Conditional power (achieved) [6] 0.2924
Exit probability for efficacy 0
Exit probability for futility [1] 0.0060
Exit probability for futility [2] 0.0170
Exit probability for futility [3] 0.0440
Exit probability for futility [4] 0.0630
Exit probability for futility [5] 0.1240
Exit probability for futility [6] 0.4970

Legend:

  • (i): results of treatment arm i vs. control arm
  • [j]: effect matrix row j (situation to consider)

Example Select the Best

design |> getSimulationMultiArmSurvival(
    activeArms = 2,
    directionUpper = FALSE,
    typeOfShape = "userDefined",
    effectMatrix = effectMatrix,
    typeOfSelection = "best",
    plannedEvents = c(230, 460),
    maxNumberOfIterations = 1000,
    seed = 123
) |> summary()

Example Select the Best

Simulation of a survival endpoint (multi-arm design)

Sequential analysis with a maximum of 2 looks (inverse normal combination test design), one-sided overall significance level 2.5%. The results were simulated for a multi-arm logrank test (2 treatments vs. control), H0: hazard ratio(i) = 1, power directed towards smaller values, H1: omega_max as specified, planned cumulative events = c(230, 460), effect shape = user defined, intersection test = Dunnett, selection = best, effect measure based on effect estimate, success criterion: all, simulation runs = 1000, seed = 123.

Stage 1 2
Fixed weight 0.707 0.707
Cumulative alpha spent 0 0.0250
Stage levels (one-sided) 0 0.0250
Efficacy boundary (z-value scale) Inf 1.960
Futility boundary (z-value scale) 0
Reject at least one [1] 0.9210
Reject at least one [2] 0.8220
Reject at least one [3] 0.7930
Reject at least one [4] 0.6350
Reject at least one [5] 0.4900
Reject at least one [6] 0.0260
Rejected arms per stage [1]
Treatment arm 1 vs. control 0 0.4310
Treatment arm 2 vs. control 0 0.4900
Rejected arms per stage [2]
Treatment arm 1 vs. control 0 0.6400
Treatment arm 2 vs. control 0 0.1820
Rejected arms per stage [3]
Treatment arm 1 vs. control 0 0.7710
Treatment arm 2 vs. control 0 0.0220
Rejected arms per stage [4]
Treatment arm 1 vs. control 0 0.3200
Treatment arm 2 vs. control 0 0.3150
Rejected arms per stage [5]
Treatment arm 1 vs. control 0 0.4410
Treatment arm 2 vs. control 0 0.0490
Rejected arms per stage [6]
Treatment arm 1 vs. control 0 0.0130
Treatment arm 2 vs. control 0 0.0130
Success per stage [1] 0 0.9210
Success per stage [2] 0 0.8220
Success per stage [3] 0 0.7930
Success per stage [4] 0 0.6350
Success per stage [5] 0 0.4900
Success per stage [6] 0 0.0260
Cumulative number of events [1]
Treatment arm 1 vs. control 162.1 340.9
Treatment arm 2 vs. control 162.1 347.1
Cumulative number of events [2]
Treatment arm 1 vs. control 157.0 358.4
Treatment arm 2 vs. control 164.3 325.0
Cumulative number of events [3]
Treatment arm 1 vs. control 151.0 372.5
Treatment arm 2 vs. control 166.8 308.1
Cumulative number of events [4]
Treatment arm 1 vs. control 159.2 337.8
Treatment arm 2 vs. control 159.2 338.4
Cumulative number of events [5]
Treatment arm 1 vs. control 153.3 360.9
Treatment arm 2 vs. control 161.9 310.7
Cumulative number of events [6]
Treatment arm 1 vs. control 153.3 324.6
Treatment arm 2 vs. control 153.3 327.1
Expected number of events under H1 [1] 457.7
Expected number of events under H1 [2] 457.5
Expected number of events under H1 [3] 447.1
Expected number of events under H1 [4] 447.1
Expected number of events under H1 [5] 436.3
Expected number of events under H1 [6] 337.2
Overall exit probability [1] 0.0100
Overall exit probability [2] 0.0110
Overall exit probability [3] 0.0560
Overall exit probability [4] 0.0560
Overall exit probability [5] 0.1030
Overall exit probability [6] 0.5340
Selected arms [1]
Treatment arm 1 vs. control 1.0000 0.4630
Treatment arm 2 vs. control 1.0000 0.5270
Selected arms [2]
Treatment arm 1 vs. control 1.0000 0.7120
Treatment arm 2 vs. control 1.0000 0.2770
Selected arms [3]
Treatment arm 1 vs. control 1.0000 0.8700
Treatment arm 2 vs. control 1.0000 0.0740
Selected arms [4]
Treatment arm 1 vs. control 1.0000 0.4690
Treatment arm 2 vs. control 1.0000 0.4750
Selected arms [5]
Treatment arm 1 vs. control 1.0000 0.7120
Treatment arm 2 vs. control 1.0000 0.1850
Selected arms [6]
Treatment arm 1 vs. control 1.0000 0.2280
Treatment arm 2 vs. control 1.0000 0.2380
Number of active arms [1] 2.000 1.000
Number of active arms [2] 2.000 1.000
Number of active arms [3] 2.000 1.000
Number of active arms [4] 2.000 1.000
Number of active arms [5] 2.000 1.000
Number of active arms [6] 2.000 1.000
Conditional power (achieved) [1] 0.8675
Conditional power (achieved) [2] 0.7846
Conditional power (achieved) [3] 0.7430
Conditional power (achieved) [4] 0.6803
Conditional power (achieved) [5] 0.5998
Conditional power (achieved) [6] 0.3370
Exit probability for efficacy 0
Exit probability for futility [1] 0.0100
Exit probability for futility [2] 0.0110
Exit probability for futility [3] 0.0560
Exit probability for futility [4] 0.0560
Exit probability for futility [5] 0.1030
Exit probability for futility [6] 0.5340

Legend:

  • (i): results of treatment arm i vs. control arm
  • [j]: effect matrix row j (situation to consider)

Summary

  • Introduction to design and analysis of multi-arm multi-stage designs with use of flexible closed combination testing principle which strongly controls the familywise Type I error rate

  • Fast simulation in getSimulationMultiArm...() for reasonable activeArms and maxNumberOfIterations

  • Through selection procedure there might be a gain in power

  • Consider different situations through specification of effectMatrix

  • Assessment of futility stops

  • Simulation for survival designs on the patient level will be possible in the next release.

Thank you

References

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Bauer, P., and M. Kieser. 1999. “Combining Different Phases in the Development of Medical Treatments Within a Single Trial.” Statistics in Medicine 34: 1833–48.
Bauer, P., and K. Köhne. 1994. “Evaluation of Experiments with Adaptive Interim Analyses.” Biometrics, 1029–41.
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Deng, Qiqi, Xiaofei Bai, Dacheng Liu, Dooti Roy, Zhiliang Ying, and Dan-Yu Lin. 2019. “Power and Sample Size for Dose-Finding Studies with Survival Endpoints Under Model Uncertainty.” Biometrics 75 (1): 308–14.
Hommel, G. 2001. “Adaptive Modifications of Hypotheses After an Interim Analysis.” Biometrical Journal 43: 581–89.
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Mehta, Cyrus, and Martin Kappler. 2025. “A Comparison of Two Methods for Adaptive Multi-Arm Two-Stage Design.” Statistics in Medicine 44 (15-17): e70162.
Posch, M., F. König, M. Branson, W. Brannath, C. Dunger–Baldauf, and P. Bauer. 2005. “Testing and Estimating in Flexible Group Sequential Designs with Adaptive Treatment Selection.” Statistics in Medicine 24: 3697–3714.
Stallard, N., and S. Todd. 2003. “Sequential Designs for Phase III Clinical Trials Incorporating Treatment Selection.” Statistics in Medicine 22 (5): 689–703.
Wassmer, Gernot, and Werner Brannath. 2016. Group Sequential and Confirmatory Adaptive Designs in Clinical Trials. Springer. https://link.springer.com/book/10.1007/978-3-319-32562-0.