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REVIEW 4 major objections 5 minor 51 references

Robust Design and Analysis of Clinical Trials With Non-proportional Hazards: A Straw Man Guidance from a Cross-pharma Working Group

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proposes the MaxCombo test—the maximum of four Fleming-Harrington weighted log-rank statistics—as a primary analysis for confirmatory trials with non-proportional hazards, with design and sample-size guidance to match.

desk verdict MaxCombo is a genuinely useful straw-man proposal, but as written the test is internally inconsistent: the p-value is an upper-tail maximum while the design boundary is a lower-tail cutoff. read the letter →

arxiv 1908.07112 v4 pith:5ILS2AKC submitted 2019-08-20 stat.AP

Satrajit Roychoudhury , Keaven M Anderson , Jiabu Ye , Pralay Mukhopadhyay This is my paper · ORCID
classification stat.AP
keywords non-proportionalhazardsMaxCombotestFleming-Harringtonweightedlog-rankcombinationconfirmatorytrialdesignsamplesizecalculationinterimanalysistypeIerrorcontrol
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a single log-rank test and a hazard ratio can fail badly when treatment effects emerge late, cross over time, or fade, and that in such settings no one number adequately describes the treatment effect. It proposes MaxCombo, the maximum of four Fleming-Harrington weighted log-rank statistics, as a primary analysis test with robust power across proportional and non-proportional hazards. The authors give a full design-and-analysis package: how to compute the p-value from the joint normal distribution, how to adjust the significance level for the four correlated tests, how to size a trial by simulation, and how to set interim and final boundaries. Three reconstructed oncology trials illustrate that MaxCombo can catch a crossing-survival benefit that the log-rank test misses, while losing only a little power when proportional hazards hold.

What carries the argument

The central object is the MaxCombo statistic $Z_{max} = \max\{G^{0,0}, G^{0,1}, G^{1,1}, G^{1,0}\}$, where each $G^{\rho,\gamma}$ is a Fleming-Harrington weighted log-rank statistic; the weights emphasize early events ($\rho$), late events ($\gamma$), or both. The load-bearing identity is the covariance formula $\eta_{ij} = V(G_{(\rho_i+\rho_j)/2,(\gamma_i+\gamma_j)/2}) / \sqrt{V(G_{\rho_i,\gamma_i}) V(G_{\rho_j,\gamma_j})}$, which turns the joint null distribution of the four statistics into an estimable multivariate normal and lets the design compute an adjusted boundary without per-trial simulation. Around this identity the paper builds an iterative design procedure: simulate one large null trial to estimate the correlation matrix, solve for the per-component significance level, size each of the four tests with a published weighted-log-rank sample-size formula, and confirm the operating characteristics by simulation. For trials with an interim analysis, the same framework uses the independent-increments property to correlate the interim log-rank statistic with the final MaxCombo statistic.

What would settle it

Simulate a null trial with no treatment effect but with a survival or censoring pattern that departs from the assumed piecewise-exponential setup (for example, a cure fraction or heavy late censoring), apply the paper's Appendix D boundary and MaxCombo p-value calculation, and check whether the one-sided rejection rate stays at 2.5%; a material excess would show the calibration depends on the assumed null correlation matrix.

Watch

Extended reading notes

Core claim

The paper's central claim is that the MaxCombo test—the maximum of four correlated Fleming-Harrington weighted log-rank statistics $G^{0,0}$, $G^{0,1}$, $G^{1,1}$, and $G^{1,0}$—is suitable as the primary analysis test in a confirmatory trial when non-proportional hazards are a real possibility. Under the null hypothesis the four statistics are asymptotically multivariate normal, so a one-sided p-value can be obtained by integrating a four-dimensional normal density above the observed maximum, and the significance boundary can be adjusted using their correlation matrix rather than a conservative Bonferroni correction. Simulation and three reconstructed trial examples are used to argue that the test controls type I error at 2.5%, has strong power for delayed, crossing, early-separation, and mixed non-proportional-hazard patterns, and loses only modest power relative to the log-rank test under proportional hazards. The paper pairs the test with a three-step analysis strategy (test the null, assess proportional hazards, then report either the hazard ratio or a set of time-dependent summaries) and a simulation-based design procedure for sample size and interim boundaries.

Load-bearing premise

The design treats the correlation matrix of the four test statistics, estimated from one large simulated null trial under assumed piecewise-exponential survival, enrollment, and follow-up, as the true correlation matrix of the actual trial; if the real null survival or censoring pattern differs materially, the significance boundary and final p-value can be miscalibrated.

Editorial extensions

If this is right

  • A trial facing uncertain non-proportional hazards can pre-specify MaxCombo as the primary test and still control one-sided type I error at 2.5%.
  • In the delayed-effect design example, MaxCombo needs 472 patients and 372 events versus 690 patients and 544 events for the log-rank test, a substantial saving.
  • Designs should specify minimum follow-up (roughly twice the control median) alongside event count, because MaxCombo power depends on follow-up, not just events.
  • For a trial with an interim log-rank analysis, the final MaxCombo boundary can be adjusted using the independent-increments correlation between the interim statistic and the four final statistics.
  • Under proportional hazards with a modest benefit, MaxCombo can lose power relative to the log-rank test, so the paper recommends reserving it for settings where non-proportional hazards are plausible.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • One implication not developed in the paper is that the practical bottleneck for MaxCombo will be regulator agreement on the simulation plan and the null correlation matrix in advance, not the test statistic itself.
  • A natural extension would be to let the data at the final analysis determine the correlation matrix from the observed event and censoring pattern instead of fixing it at the design stage, and to study how sensitive the reported p-value is to that choice.
  • The same maximum-of-correlated-weighted-log-rank construction could be applied to a larger or differently chosen set of weights; everything needed for the p-value would follow from the same covariance identity, and the operating characteristics would need rerunning.
  • A testable refinement motivated by the paper's examples is to treat the MaxCombo test as a gatekeeper and then quantify how much the treatment-effect estimate changes across the four component weights, exposing whether the smallest p-value also corresponds to a clinically meaningful effect.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper proposes the MaxCombo test, defined as the maximum of four correlated Fleming-Harrington weighted log-rank statistics (G0,0, G0,1, G1,1, G1,0), as a robust primary analysis for confirmatory clinical trials with potential non-proportional hazards (NPH). It presents an analysis workflow (test of the null, assessment of proportional hazards, and adaptive treatment-effect summaries), a design approach with sample-size calculation based on an adjusted significance boundary and an iterative simulation step, an interim-analysis strategy, and three illustrative applications from published oncology trials. The authors claim that MaxCombo controls type I error at 2.5% and provides robust power across PH, delayed effect, crossing survival, early separation, and mixtures of NPH patterns, and they frame the proposal as a 'straw man' guidance for discussion.

Significance. If the proposal were internally consistent and its operating characteristics were fully supported, the paper would provide valuable, practical guidance for an important methodological need: a pre-specified primary test with robust power under various non-proportional-hazards patterns while controlling type I error at the conventional level. The paper draws on established asymptotic results (Karrison 2016), provides worked examples and a concrete design algorithm, and explicitly acknowledges the limitations of single summary measures under NPH. However, the central test definition is marred by a sign/direction inconsistency between the formal definition and the numerical cutoffs and examples, and the main robust-power evidence is delegated to a companion paper by the same working group. These issues currently prevent the manuscript from fulfilling its stated goal of a directly implementable guidance.

major comments (4)
  1. [Appendix A vs. Appendix D] The definition of the MaxCombo test is internally inconsistent. Appendix A defines Zmax = max{G0,0, G0,1, G1,0, G1,1} and gives the one-sided p-value as P(Zmax > zmax|H0) = 1 - Φ_4(zmax), which is an upper-tail procedure (large positive values are significant). Appendix D, however, solves Φ_4(Zcutoff, 0, ΣH0) ≤ 0.025 and obtains Zcutoff = -2.286, a lower-tail critical value for the maximum: rejection occurs when the maximum is below -2.286, i.e., when all four statistics are simultaneously strongly negative. For a maximum of four positively correlated normals, the one-sided 2.5% upper-tail cutoff is approximately +2.0, not -2.286. If negative values of the Fleming-Harrington statistics are meant to favor treatment, the appropriate combination is the minimum (or the negative of the maximum), not the maximum as written. The worked examples in Table 2 (e.g., IM211 row: selected G0,1 with p=0.002 and MaxCombo p=0.005) are consistent with a minimum-p/lower-tail procedure, contradicting the stated 'max' definition. Because Section 3.1 uses the -2.286 boundary for sample-size calculation and the final p-value is tied to the same statistic, a practitioner following the equations literally cannot implement a test with the claimed operating characteristics. This is a load-bearing inconsistency, independent of the correlation-matrix calibration issue.
  2. [Section 3.1, Step 2 and Appendix D] The correlation matrix of the four Fleming-Harrington statistics is estimated from a single large simulated null trial with an assumed piecewise-exponential control survival, enrollment rate, and follow-up pattern, and is then treated as the true correlation matrix when computing the adjusted significance boundary and the final MaxCombo p-value. If the actual trial's null survival, censoring, or dropout pattern differs materially from the simulation assumptions, the boundary and p-value can be miscalibrated. The paper does not provide sensitivity analyses, an analytic characterization of the correlation as a function of the underlying censoring pattern, or a conservative fallback. Given that the proposal is for confirmatory regulatory use, the claim of type I error control at 2.5% needs stronger justification than a single simulation scenario.
  3. [Section 2.1.1 and Section 3.3] The core evidence for robust power and type I error control is delegated to Lin, Lin, Roychoudhury et al. (2020), a companion paper by the same working group. This manuscript itself reports only a limited set of strong-null simulations (Section 2.1.2) and the worked design example (Section 3.3). For a self-contained guidance paper that states the MaxCombo test 'fulfills the necessary regulatory standards,' the reader needs either a summary of the companion's operating characteristics, or at least a verification of the design example's power and type I error under the stated assumptions, rather than a citation. The single simulation check in Section 3.3 ('we confirm type I error and power using simulation') is not described in enough detail to be independently reproducible.
  4. [Appendix C and Appendix D] The same direction inconsistency appears in the interim-analysis boundary equations. Appendix C states the final boundary condition as P(ZI > zI|H0) + P(ZI ≤ zI, ZF_max > zF_max|H0) ≤ 0.025, which is an upper-tail formulation with zI positive. Appendix D instead writes P(ZI < -2.34|H0) + P(ZI > -2.34, MF < zF|H0) ≤ 0.025, using a negative interim boundary and a lower-tail condition on MF. The sign of the interim efficacy boundary and the direction of the final condition are thus inconsistent between the two appendices, which further complicates implementation.
minor comments (5)
  1. [Section 1.1] The phrase 'non-proportional hazard is a possibility' in the abstract and elsewhere should be 'non-proportional hazards' for consistency.
  2. [Section 2.3] The examples use reconstructed and unstratified data, while the published results are stratified; the paper acknowledges this, but it would be helpful to state explicitly that the reported p-values are not directly comparable to the original trial analyses.
  3. [Section 3.3] The sentence 'Further details of the sample size calculation including correlation matrix for null distribution are provided in Appendix D' is followed by another sentence about interim analysis; reordering for clarity would help.
  4. [Section 2.1.2] In the strong-null 1 description, 'The curves meet at the 36 month.' has a grammatical error and should probably read 'at 36 months.'
  5. [Appendix B] The simultaneous confidence interval formula 'HRMaxCombo ± C*×SE(HRMaxCombo)' is written before defining C*; the notation should be introduced before the formula.

Circularity Check

2 steps flagged · score 5.0 of 10

Robustness claim rests on a companion self-citation, and the MaxCombo definition is internally inconsistent between an upper-tail max p-value and a lower-tail design boundary; the central operating-characteristics claim is therefore not fully derived within the paper.

  1. self citation load bearing [Section 2.1.1, paragraph on simulation study; echoed in Section 4.]
    "An extensive simulation study under the null hypothesis and different treatment effect scenarios was performed by the cross-pharma working group (Lin, Lin, Roychoudhury et al. (2020)) to understand the statistical operating characteristics (type I error and power) of MaxCombo test. ... In summary, the MaxCombo test fulfills the necessary regulatory standards and is suitable for a confirmatory trial with potential NPH."

    The paper's headline conclusion that MaxCombo 'fulfills the necessary regulatory standards' is justified by citing a companion simulation study by the same cross-pharma working group, with overlapping authorship (Roychoudhury and Anderson are authors of both this paper and Lin et al. 2020). The present paper's own simulations only assess strong-null and severe-late-crossing rejection rates; they do not independently produce the PH, delayed-effect, and crossing-survival power comparisons that are used to conclude robustness. The cited work is not reproduced or machine-checked in this paper, and its assumptions include exactly the MaxCombo test whose operating characteristics are claimed.

  2. self definitional [Appendix A p-value formula vs Appendix D boundary and Section 3.3, Step 2.]
    "P (Zmax>z max|H0) = P (max(G0,0,G 0,1,G 1,0,G 1,1)>z max|H0) = 1 − ∫ zmax −∞ ∫ zmax −∞ ∫ zmax −∞ ∫ zmax −∞ φ4(ω, 0, Γ)dω ... Therefore, the boundary for MaxCombo (Zcutoff ) is obtained by solving the equation below; Φ4(Zcutoff, 0, ΣH0)≤ 0.025 ... This yields Zcutoff = -2.286."

    Appendix A defines the MaxCombo statistic as the maximum of four FH statistics and evaluates an upper-tail probability, rejecting for large positive zmax. Appendix D and Section 3.3 instead compute the boundary by solving the lower-tail CDF at -2.286, rejecting for a sufficiently negative maximum. These two definitions cannot define the same test. The sample-size boundary, the final p-value, and the claimed type I error and power operating characteristics are computed under the second convention, while the paper's stated statistic and p-value formula are the first.

full rationale

The main derivation chain--null multivariate normality of the four FH statistics, the correlation-matrix formula, the boundary search, and the Hasegawa sample-size calculation--is anchored in an external asymptotic result (Karrison 2016) and a standard design formula, so it is not circular by construction. The simulation-based estimate of the correlation matrix is an acknowledged design calibration rather than a prediction, and confirming type I error by simulating the same null model is normal numerical design practice. The principal circularity risk is the load-bearing self-citation: the paper's robustness and 'regulatory standards' conclusion is assigned to a companion paper by the same working group rather than demonstrated inside this paper. Separately, there is a serious self-definitional break between the upper-tail max p-value in Appendix A and the lower-tail cutoff -2.286 in Appendix D; this is a contradiction between two stated definitions and prevents the claimed operating characteristics from being a consequence of the paper's own equations. These issues make the paper only partially circular: the external Karrison anchor and the paper's own strong-null simulations give it independent content, but the central claim is not fully self-contained.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The method introduces hand-picked weight sets and heuristic design rules rather than fitted parameters. The central statistical machinery, multivariate normality and independent increments, is imported from the cited literature. The main non-mathematical assumption is that extreme strong-null scenarios are rare enough to ignore for the primary analysis, which is asserted rather than demonstrated.

free parameters (4)
  • MaxCombo weight set (rho,gamma) = (0,0),(0,1),(1,0),(1,1) = four exponent pairs
    Hand-selected to cover PH, delayed, crossing, and early separation scenarios; no optimality criterion or data fitting, but the test's power profile is sensitive to this choice.
  • Modified MaxCombo weight set (0,0),(0,0.5),(0.5,0.5),(0.5,0) = four exponent pairs
    Ad hoc alternative introduced to reduce rejection under strong null 2 and severe late crossing; chosen without a formal rule or external benchmark.
  • Minimum follow-up recommendation: twice the control median = 2 times control median
    General rule stated in Section 3.1 step 1; affects sample size and power but is a heuristic, not derived from a formal optimization.
  • Futility threshold HR > 1.5 = 1.5
    Recommended early stopping only if treatment appears harmful; arbitrary operational threshold with no formal derivation.
assumptions (5)
  • standard math The four Fleming-Harrington weighted log-rank statistics follow an asymptotic multivariate normal distribution under the null with the correlation structure given by Karrison (2016).
    Invoked throughout Sections 2.1.1 and Appendix A for p-value computation and boundary calculation.
  • domain assumption Independent increments (Tsiatis 1982) hold for the interim log-rank statistic and the final MaxCombo statistics.
    Used in Appendix C to compute the correlation between the interim and final test statistics for group sequential boundaries.
  • domain assumption The empirical correlation matrix from one large simulated null trial with piecewise-exponential survival and planned enrollment and follow-up equals the correlation matrix of the actual trial.
    The entire sample-size and boundary procedure in Section 3.1 steps 2 and 3 relies on this simulated null correlation matrix.
  • ad hoc to paper Strong null and severe late crossing scenarios are unlikely in confirmatory trials and would be stopped early by a data monitoring committee.
    Used in Section 2.1.2 and the Conclusion to argue that the 48.9% rejection probability under strong null 2 does not disqualify the unmodified MaxCombo test.
  • domain assumption Patient-level data reconstructed from published Kaplan-Meier curves using the Guyot et al. (2012) method faithfully represent the original trials for the worked examples.
    The re-analyses of IM211 and PA3 in Section 2.3 are based on digitized KM data rather than original patient-level data with stratification factors.

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Cite this review

Pith. "Pith review of Robust Design and Analysis of Clinical Trials With Non-proportional Hazards: A Straw Man Guidance from a Cross-pharma Working Group." pith.science (2026). https://pith.science/paper/5ILS2AKC

@misc{pith2026190807112,
  author       = {Pith},
  title        = {Pith review of: Robust Design and Analysis of Clinical Trials With Non-proportional Hazards: A Straw Man Guidance from a Cross-pharma Working Group},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ILS2AKC}},
  note         = {Machine review of arXiv:1908.07112}
}
read the original abstract

Loss of power and clear description of treatment differences are key issues in designing and analyzing a clinical trial where non-proportional hazard is a possibility. A log-rank test may be very inefficient and interpretation of the hazard ratio estimated using Cox regression is potentially problematic. In this case, the current ICH E9 (R1) addendum would suggest designing a trial with a clinically relevant estimand, e.g., expected life gain. This approach considers appropriate analysis methods for supporting the chosen estimand. However, such an approach is case specific and may suffer lack of power for important choices of the underlying alternate hypothesis distribution. On the other hand, there may be a desire to have robust power under different deviations from proportional hazards. Also, we would contend that no single number adequately describes treatment effect under non-proportional hazards scenarios. The cross-pharma working group has proposed a combination test to provide robust power under a variety of alternative hypotheses. These can be specified for primary analysis at the design stage and methods appropriately accounting for combination test correlations are efficient for a variety of scenarios. We have provided design and analysis considerations based on a combination test under different non-proportional hazard types and present a straw man proposal for practitioners. The proposals are illustrated with real life example and simulation.

Figures

Figures reproduced from arXiv: 1908.07112 by the authors.

Figure 1
Figure 1. Simulation scenarios for evaluation of MaxCombo under strong null and severe late [PITH_FULL_IMAGE:figures/full_fig_p037_1.png] view at source ↗
Figure 2
Figure 2. Schoenfeld Residual Plots for Three Examples: a) IM211: Digitized (top left), b) [PITH_FULL_IMAGE:figures/full_fig_p038_2.png] view at source ↗
Figure 3
Figure 3. Milestone Survival (95% CI) and Piecewise Hazard Ratio (95% CI) at Clinically Relevant [PITH_FULL_IMAGE:figures/full_fig_p039_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Overview of Sample Size Calculation 40 [PITH_FULL_IMAGE:figures/full_fig_p040_4.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.