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Platform Trial with Arm Added Midway

2026-05-29

platform_study.Rmd

Introduction

This vignette demonstrates how appendMCP handles a platform-type group sequential design in which a new treatment arm is added midway through an ongoing trial. The design pattern illustrated here extends the standard two-arm group sequential framework described in the example_study_report vignette to accommodate:

  • A 3-arm structure (TrtA monotherapy, TrtAB combination, Control) where TrtAB is added after the trial has already started enrolling TrtA vs Control;
  • A biomarker stratification (B+ / B−) that is a genuine patient subgroup of clinical interest;
  • Two control-arm pooling strategies — pooled and concurrent — for the TrtAB vs Control comparison;
  • A large graphical multiple testing procedure spanning 12 hypotheses across 5 endpoints.

Five families of hypotheses are evaluated in parallel within a single run. Hence, 5×α was used (with alpha initially distributed among the families). Each family allows alpha reallocation within its own set of hypotheses, but not across families. The total one-sided type I error rate of 12.5% is distributed across 5 families (each starting with its primary hypothesis) with an initial weight of 0.2.

Study Design

Two-Part Enrollment

The trial enrolls in two parts:

  • Part 1 — TrtA vs Control (2:1 ratio), B+/B− stratified, over approximately 14 months.
  • Part 2 — TrtAB arm opens. All three arms enroll (TrtA : Control : TrtAB = 1:1:4 ratio). B+/B− stratification continues.

The total accrual period is 22 months.

The Four-Stratum Structure

appendMCP uses a stratum field to define subpopulations for enrollment and event modeling. This study uses 4 strata that encode two independent dimensions simultaneously:

Stratum Enrollment stage Biomarker
S1_Bpos Part 1 (before TrtAB opens) B+
S1_Bneg Part 1 (before TrtAB opens) B−
S2_Bpos Part 2 (after TrtAB opens) B+
S2_Bneg Part 2 (after TrtAB opens) B−

The S1/S2 dimension is introduced to track which patients were enrolled before vs after the TrtAB arm opened. The B+/B− dimension is a stratification factor that defines a subgroup of clinical interest (see H12).

Encoding “Arm Added Midway” via Zero Enrollment Rates

The platform design is encoded in the enroll_rate table using zero-rate placeholder rows. S2 strata have rate = 0 during Part 1 (months 0–14), signaling that no S2 patients exist yet. S1 strata have no rows after Part 1, so their enrollment stops when Part 1 ends.

Enrollment rate specification. S2 rows with rate = 0 are required placeholders during Part 1.
stratum treatments rate duration ratio
S1_Bpos TrtA , Control 13.2 6 2, 1
S1_Bneg TrtA , Control 19.8 6 2, 1
S1_Bpos TrtA , Control 23.2 8 2, 1
S1_Bneg TrtA , Control 34.8 8 2, 1
S2_Bpos TrtA , Control, TrtAB 0.0 6 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 0.0 6 1, 1, 4
S2_Bpos TrtA , Control, TrtAB 0.0 8 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 0.0 8 1, 1, 4
S2_Bpos TrtA , Control, TrtAB 15.0 8 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 22.5 8 1, 1, 4

The enrollment curves below show the resulting piecewise accrual by arm and stage.

Enrollment

Multiplicity Adjustment

The multiplicity strategy follows the graphical approach of Maurer and Bretz (2013), controlling the family-wise type I error rate across both multiple hypotheses and multiple analyses.

The 12 hypotheses span OS, PFS, EFS, TSP, TTE endpoints.

The following table provides a descriptive summary of each hypothesis for the reader’s convenience. (This table is specific to this vignette; in a formal report, abbreviated information could be captured in the Label column of Table 1.)

Descriptive summary of hypotheses H1–H12.
Hypothesis Endpoint Comparison Population Family
H1 OS TrtA vs Control S1 (Part 1 only) 1
H2 PFS TrtA vs Control S1 (Part 1 only) 1
H3 OS TrtAB vs Control S1+S2 (pooled) 2
H4 EFS TrtAB vs Control S1+S2 (pooled) 2
H5 PFS TrtAB vs Control S1+S2 (pooled) 2
H6 TSP TrtAB vs Control S1+S2 (pooled) 2
H7 TTE TrtAB vs Control S1+S2 (pooled) 2
H8 OS TrtAB vs Control S2 (concurrent) 3
H9 PFS TrtAB vs Control S2 (concurrent) 3
H10 OS TrtAB vs TrtA S1+S2 (pooled) 4
H11 PFS TrtAB vs TrtA S1+S2 (pooled) 4
H12 OS TrtAB vs Control B+ subgroup only 5

The following table summarizes the hypotheses with their alpha-spending functions and maximum statistical information.

Table 1. Summary of Hypotheses

Trial Design Summary
Label Endpoint Type Initial weight GSD spending fn Effect size* Maximum events / sample size
H1 OS Primary 0.2 KDM(2.5) HR = 0.72 444
H2 PFS Secondary 0 N/A HR = 0.57 475
H3 OS Primary 0.2 KDM(2.5) HR = 0.62 330
H4 EFS Secondary 0 N/A HR = 0.35 373
H5 PFS Secondary 0 N/A HR = 0.4 346
H6 TSP Secondary 0 KDM(2) HR = 0.55 377
H7 TTE Secondary 0 KDM(2) HR = 0.47 348
H8 OS Primary 0.2 KDM(2.5) HR = 0.62 148
H9 PFS Secondary 0 KDM(2.5) HR = 0.4 184
H10 OS Primary 0.2 KDM(2.5) HR = 0.861 465
H11 PFS Secondary 0 KDM(2.5) HR = 0.702 565
H12 OS Primary 0.2 N/A HR = 0.62 132
*For a hypothesis corresponding to a time-to-event variable subject to non-proportional hazards, the listed effect size is the average hazard ratio at the final analysis of this hypothesis.

Pooled vs Concurrent Control

For the TrtAB vs Control comparison, two separate sets of hypotheses are pre-specified:

  • Pooled (H3–H7): Control arm patients from both Part 1 and Part 2 are combined. This maximizes power but assumes no stage effect on the control arm.
  • Concurrent (H8–H9): Only control arm patients enrolled concurrently with TrtAB (Part 2, S2 strata) are used. This is conservative but robust to a potential stage effect.

Both strategies are included in the graphical testing procedure, with alpha propagating between them upon rejection.

Alpha Distribution

The overall family-wise type I error rate is controlled at 2.5% (one-sided) per primary hypothesis. The alpha = 0.025 × 5 setting in the config reflects the five families evaluated in parallel. Each family receives initial weight 0.2, giving it a significance level of 0.025. Alpha can be reallocated within a family upon rejection, but not across families.

Figure 1. Graphical Multiple Testing Procedure

Interim Analyses

Three event-driven analyses are planned:

  1. IA1 — 333 OS events in S1 strata (TrtA vs Control)
  2. IA2 — 444 OS events in S1 strata (TrtA vs Control, final for Part 1) and first look at TrtAB vs Control
  3. FA — 330 OS events in S1+S2 strata (TrtAB vs Control, final)

Table 2. Summary of Interim Analyses (by hypothesis)

Summary of interim analyses by hypothesis
Analysis Criteria for conduct Events / sample size Expected analysis time, mo Information fraction, %
H1: OS
1 333 OS events 333 20.5 75.0%
2 444 OS events 444 29.0 100.0%
H2: PFS
1 333 OS events 475 20.5 100.0%
H3: OS
2 444 OS events 269 29.0 81.5%
3 330 OS events 330 37.5 100.0%
H4: EFS
2 444 OS events 373 29.0 100.0%
H5: PFS
2 444 OS events 346 29.0 100.0%
H6: TSP
2 444 OS events 323 29.0 85.5%
3 330 OS events 377 37.5 100.0%
H7: TTE
2 444 OS events 292 29.0 83.9%
3 330 OS events 348 37.5 100.0%
H8: OS
2 444 OS events 105 29.0 70.6%
3 330 OS events 148 37.5 100.0%
H9: PFS
2 444 OS events 142 29.0 77.1%
3 330 OS events 184 37.5 100.0%
H10: OS
2 444 OS events 379 29.0 81.6%
3 330 OS events 465 37.5 100.0%
H11: PFS
2 444 OS events 493 29.0 87.2%
3 330 OS events 565 37.5 100.0%
H12: OS
3 330 OS events 132 37.5 100.0%

Table 3. Summary of Interim Analyses (by calendar analysis)

Summary of interim analyses by criteria for conduct
Hypothesis Analysis Events / sample size Information fraction, %
333 OS events (Expected analysis time: 20.5 mo)
H1: OS 1 333 75.0%
H2: PFS 1 475 100.0%
444 OS events (Expected analysis time: 29.0 mo)
H1: OS 2 444 100.0%
H3: OS 2 269 81.5%
H4: EFS 2 373 100.0%
H5: PFS 2 346 100.0%
H6: TSP 2 323 85.5%
H7: TTE 2 292 83.9%
H8: OS 2 105 70.6%
H9: PFS 2 142 77.1%
H10: OS 2 379 81.6%
H11: PFS 2 493 87.2%
330 OS events (Expected analysis time: 37.5 mo)
H3: OS 3 330 100.0%
H6: TSP 3 377 100.0%
H7: TTE 3 348 100.0%
H8: OS 3 148 100.0%
H9: PFS 3 184 100.0%
H10: OS 3 465 100.0%
H11: PFS 3 565 100.0%
H12: OS 3 132 100.0%

Hypothesis Testing

Table 4. Weight Allocation Scenarios

Summary of alpha allocation by hypothesis
Local alpha level Weight Testing scenario
H1: OS
0.02500 0.2 Initial allocation
H2: PFS
0.02500 0.2 Successful H1
H3: OS
0.02500 0.2 Initial allocation
H4: EFS
0.00250 0.02 Successful H3
0.00500 0.04 Successful H3, H5
0.00625 0.05 Successful H3, H7
0.00875 0.07 Successful H3, H5, H7
0.00917 0.0733 Successful H3, H5, H6
0.01083 0.0867 Successful H3, H6, H7
0.02500 0.2 Successful H3, H5, H6, H7
H5: PFS
0.00500 0.04 Successful H3
0.00625 0.05 Successful H3, H4
0.00917 0.0733 Successful H3, H4, H7
0.01000 0.08 Successful H3, H6
0.01125 0.09 Successful H3, H4, H6
0.01417 0.113 Successful H3, H6, H7
0.02500 0.2 Successful H3, H4, H6, H7
H6: TSP
0.01000 0.08 Successful H3
0.01250 0.1 Successful H3, H5
0.01375 0.11 Successful H3, H7
0.01417 0.113 Successful H3, H4, H5
0.01583 0.127 Successful H3, H4, H7
0.01625 0.13 Successful H3, H5, H7
0.02500 0.2 Successful H3, H4, H5, H7
H7: TTE
0.00750 0.06 Successful H3
0.00875 0.07 Successful H3, H4
0.01083 0.0867 Successful H3, H4, H5
0.01250 0.1 Successful H3, H6
0.01375 0.11 Successful H3, H4, H6
0.01583 0.127 Successful H3, H5, H6
0.02500 0.2 Successful H3, H4, H5, H6
H8: OS
0.02500 0.2 Initial allocation
H9: PFS
0.02500 0.2 Successful H8
H10: OS
0.02500 0.2 Initial allocation
H11: PFS
0.02500 0.2 Successful H10
H12: OS
0.02500 0.2 Initial allocation

Table 5. Boundary Specifications

Operating characteristics
Analysis Local alpha level Nominal p-value Exit hurdle Local power Information fraction, %
H1: OS
1 0.02500 0.01218 0.770 72.3% 75.0%
2 0.02500 0.02055 0.814 88.7% 100.0%
H2: PFS
1 0.02500 0.02500 0.826 100.0% 100.0%
H3: OS
1 0.02500 0.01378 0.701 79.5% 81.5%
2 0.02500 0.02015 0.746 89.2% 100.0%
H4: EFS
1 0.00250 0.00250 0.684 100.0% 100.0%
1 0.00500 0.00500 0.706 100.0% 100.0%
1 0.00625 0.00625 0.713 100.0% 100.0%
1 0.00875 0.00875 0.725 100.0% 100.0%
1 0.00917 0.00917 0.727 100.0% 100.0%
1 0.01083 0.01083 0.733 100.0% 100.0%
1 0.02500 0.02500 0.767 100.0% 100.0%
H5: PFS
1 0.00500 0.00500 0.693 100.0% 100.0%
1 0.00625 0.00625 0.701 100.0% 100.0%
1 0.00917 0.00917 0.715 100.0% 100.0%
1 0.01000 0.01000 0.718 100.0% 100.0%
1 0.01125 0.01125 0.723 100.0% 100.0%
1 0.01417 0.01417 0.732 100.0% 100.0%
1 0.02500 0.02500 0.756 100.0% 100.0%
H6: TSP
1 0.01000 0.00681 0.697 96.0% 85.5%
2 0.01000 0.00691 0.721 97.6% 100.0%
1 0.01250 0.00851 0.706 96.7% 85.5%
2 0.01250 0.00876 0.730 98.1% 100.0%
1 0.01375 0.00936 0.709 97.0% 85.5%
2 0.01375 0.00970 0.733 98.3% 100.0%
1 0.01417 0.00964 0.710 97.1% 85.5%
2 0.01417 0.01002 0.734 98.3% 100.0%
1 0.01583 0.01078 0.715 97.4% 85.5%
2 0.01583 0.01128 0.739 98.5% 100.0%
1 0.01625 0.01106 0.716 97.4% 85.5%
2 0.01625 0.01160 0.740 98.5% 100.0%
1 0.02500 0.01701 0.734 98.3% 85.5%
2 0.02500 0.01842 0.758 99.1% 100.0%
H7: TTE
1 0.00750 0.00477 0.663 99.2% 83.9%
2 0.00750 0.00517 0.696 99.5% 100.0%
1 0.00875 0.00557 0.669 99.4% 83.9%
2 0.00875 0.00609 0.702 99.6% 100.0%
1 0.01083 0.00690 0.677 99.5% 83.9%
2 0.01083 0.00765 0.710 99.7% 100.0%
1 0.01250 0.00796 0.683 99.6% 83.9%
2 0.01250 0.00891 0.715 99.7% 100.0%
1 0.01375 0.00875 0.686 99.6% 83.9%
2 0.01375 0.00986 0.719 99.8% 100.0%
1 0.01583 0.01008 0.692 99.7% 83.9%
2 0.01583 0.01146 0.725 99.8% 100.0%
1 0.02500 0.01591 0.712 99.8% 83.9%
2 0.02500 0.01867 0.745 99.9% 100.0%
H8: OS
1 0.02500 0.01046 0.569 35.0% 70.6%
2 0.02500 0.02103 0.659 60.9% 100.0%
H9: PFS
1 0.02500 0.01305 0.627 99.2% 77.1%
2 0.02500 0.02033 0.686 99.5% 100.0%
H10: OS
1 0.02500 0.01466 0.785 20.0% 81.6%
2 0.02500 0.01993 0.815 31.0% 100.0%
H11: PFS
1 0.02500 0.01688 0.812 93.5% 87.2%
2 0.02500 0.01944 0.829 96.8% 100.0%
H12: OS
1 0.02500 0.02500 0.642 56.5% 100.0%

Operating Characteristics

Table 6a. Operating Characteristics at Each Analysis

Operating characteristics by analysis
Analysis Metric Hypothesis subset Probability, %
1 Power H1 74.6%
H2 74.6%
H3 0.0%
H4 0.0%
H5 0.0%
H6 0.0%
H7 0.0%
H8 0.0%
H9 0.0%
H10 0.0%
H11 0.0%
H12 0.0%
2 H1 91.5%
H2 91.5%
H3 91.3%
H4 91.3%
H5 91.3%
H6 90.9%
H7 91.3%
H8 65.2%
H9 65.1%
H10 25.8%
H11 24.7%
H12 0.0%
3 H1 91.5%
H2 91.5%
H3 95.0%
H4 95.0%
H5 95.0%
H6 94.9%
H7 95.0%
H8 75.4%
H9 75.3%
H10 33.3%
H11 32.4%
H12 60.5%

Table 6b. Operating Characteristics Across Analyses

Operating characteristics across analyses
Metric Hypothesis subset Value
Expected Success Analysis H1 1.18
H2 1.18
H3 2.04
H4 2.04
H5 2.04
H6 2.04
H7 2.04
H8 2.14
H9 2.14
H10 2.22
H11 2.24
H12 3.00
Expected Success Time H1 22.1
H2 22.1
H3 29.4
H4 29.4
H5 29.4
H6 29.4
H7 29.4
H8 30.2
H9 30.2
H10 30.9
H11 31.0
H12 37.5

Distribution Assumptions

Survival Curves

Hazard Rates

Hazard Ratios

Average Hazard Ratios

Median Survival

Drop-out Rates

Input Data

Enrollment Rate Assumptions
stratum treatments rate duration ratio
S1_Bpos TrtA , Control 13.2 6 2, 1
S1_Bneg TrtA , Control 19.8 6 2, 1
S1_Bpos TrtA , Control 23.2 8 2, 1
S1_Bneg TrtA , Control 34.8 8 2, 1
S2_Bpos TrtA , Control, TrtAB 0.0 6 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 0.0 6 1, 1, 4
S2_Bpos TrtA , Control, TrtAB 0.0 8 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 0.0 8 1, 1, 4
S2_Bpos TrtA , Control, TrtAB 15.0 8 1, 1, 4
S2_Bneg TrtA , Control, TrtAB 22.5 8 1, 1, 4
Time-to-Event Distribution Parameters
endpoint stratum treatment duration fail_rate dropout_rate
OS S1_Bpos TrtA Inf 0.0555 0.0088
OS S1_Bpos Control Inf 0.0770 0.0088
OS S1_Bneg TrtA Inf 0.0555 0.0088
OS S1_Bneg Control Inf 0.0770 0.0088
OS S2_Bpos TrtA Inf 0.0555 0.0088
OS S2_Bpos Control Inf 0.0770 0.0088
OS S2_Bneg TrtA Inf 0.0555 0.0088
OS S2_Bneg Control Inf 0.0770 0.0088
OS S1_Bpos TrtAB Inf 0.0478 0.0088
OS S1_Bneg TrtAB Inf 0.0478 0.0088
OS S2_Bpos TrtAB Inf 0.0478 0.0088
OS S2_Bneg TrtAB Inf 0.0478 0.0088
PFS S1_Bpos TrtA Inf 0.0988 0.0088
PFS S1_Bpos Control Inf 0.1733 0.0088
PFS S1_Bneg TrtA Inf 0.0988 0.0088
PFS S1_Bneg Control Inf 0.1733 0.0088
PFS S2_Bpos TrtA Inf 0.0988 0.0088
PFS S2_Bpos Control Inf 0.1733 0.0088
PFS S2_Bneg TrtA Inf 0.0988 0.0088
PFS S2_Bneg Control Inf 0.1733 0.0088
PFS S1_Bpos TrtAB Inf 0.0693 0.0088
PFS S1_Bneg TrtAB Inf 0.0693 0.0088
PFS S2_Bpos TrtAB Inf 0.0693 0.0088
PFS S2_Bneg TrtAB Inf 0.0693 0.0088
EFS S1_Bpos TrtAB Inf 0.0866 0.0088
EFS S1_Bneg TrtAB Inf 0.0866 0.0088
EFS S2_Bpos TrtAB Inf 0.0866 0.0088
EFS S2_Bneg TrtAB Inf 0.0866 0.0088
EFS S1_Bpos Control Inf 0.2476 0.0088
EFS S1_Bneg Control Inf 0.2476 0.0088
EFS S2_Bpos Control Inf 0.2476 0.0088
EFS S2_Bneg Control Inf 0.2476 0.0088
TSP S1_Bpos TrtAB Inf 0.0657 0.0088
TSP S1_Bneg TrtAB Inf 0.0657 0.0088
TSP S2_Bpos TrtAB Inf 0.0657 0.0088
TSP S2_Bneg TrtAB Inf 0.0657 0.0088
TSP S1_Bpos Control Inf 0.1195 0.0088
TSP S1_Bneg Control Inf 0.1195 0.0088
TSP S2_Bpos Control Inf 0.1195 0.0088
TSP S2_Bneg Control Inf 0.1195 0.0088
TTE S1_Bpos TrtAB Inf 0.0479 0.0088
TTE S1_Bneg TrtAB Inf 0.0479 0.0088
TTE S2_Bpos TrtAB Inf 0.0479 0.0088
TTE S2_Bneg TrtAB Inf 0.0479 0.0088
TTE S1_Bpos Control Inf 0.1019 0.0088
TTE S1_Bneg Control Inf 0.1019 0.0088
TTE S2_Bpos Control Inf 0.1019 0.0088
TTE S2_Bneg Control Inf 0.1019 0.0088

Initial weights: 0.2, 0, 0.2, 0, 0, 0, 0, 0.2, 0, 0.2, 0, 0.2

Transition Matrix
0 1 0 0.0 0.0 0.0 0.0 0 0 0 0 0
0 0 0 0.0 0.0 0.0 0.0 0 0 0 0 0
0 0 0 0.1 0.2 0.4 0.3 0 0 0 0 0
0 0 0 0.0 0.5 0.0 0.5 0 0 0 0 0
0 0 0 0.5 0.0 0.5 0.0 0 0 0 0 0
0 0 0 0.0 0.5 0.0 0.5 0 0 0 0 0
0 0 0 0.5 0.0 0.5 0.0 0 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 0.0 0.0 0.0 0 0 0 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 0.0 0.0 0 0 0 0 0
0 0 0 0.0 0.0 0.0 0.0 0 0 0 0 0

References

Maurer W, Bretz F. Multiple testing in group sequential trials using graphical approaches. Statistics in Biopharmaceutical Research. 2013;5(4):311-320.