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arxiv: 1907.10976 · v1 · pith:SUNGXHGYnew · submitted 2019-07-25 · 📊 stat.ME

Non-constant hazard ratios in randomized controlled trials with composite endpoints

Jordi Cort\'es Mart\'inez , Mois\`es G\'omez Mateu , KyungMann Kim , Guadalupe G\'omez Melis This is my paper

Pith reviewed 2026-05-24 16:08 UTC · model grok-4.3

classification 📊 stat.ME
keywords composite endpointshazard rationon-proportional hazardssurvival analysisclinical trialstime-to-event datasample size
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The pith

The hazard ratio for a composite endpoint is often non-constant over time even when each component has a constant hazard ratio.

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

This paper examines the behavior of the hazard ratio for a two-component composite time-to-event endpoint when each component is modeled with its own constant hazard ratio. The composite hazard ratio varies with time depending on the baseline hazards of the components and their association, with larger variation when the component baseline hazards differ markedly even if the treatment effects on each component are similar. This matters because trial analyses, power calculations, and effect summaries routinely assume a single constant hazard ratio for the composite, which can produce misleading averages or incorrect sample sizes when the assumption fails. The authors quantify the degree of non-constancy with the range D between the maximum and minimum composite hazard ratios and with a relative sample-size measure R, then illustrate the patterns through simulation across hazard ratios, event rates, and correlations plus a re-analysis of the ZODIAC trial.

Core claim

Under the modeling assumption that the hazard ratio for each component of the composite endpoint is constant over time, the hazard ratio for the composite endpoint itself varies as a function of time, the component-specific baseline hazards, and the degree of association between components. The variation, measured by the difference D between maximum and minimum composite hazard ratios and by the relative sample size measure R, is pronounced when the component hazard ratios are near 1 and when the component baseline hazards differ markedly.

What carries the argument

The time-dependent composite hazard ratio obtained by combining two components each having constant individual hazard ratios, quantified via the range D and the sample-size ratio R.

If this is right

  • The average hazard ratio may not serve as a valid summary measure of treatment effect when D is large.
  • Common sample-size formulas that assume a constant hazard ratio can be inappropriate for composite endpoints.
  • Non-constant composite hazard ratios arise even when treatment effects on the components are similar, provided the baseline hazards differ.
  • Interpretation of results from trials using composite endpoints such as progression-free survival may need to account for time variation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Trial designers could simulate the expected path of the composite hazard ratio before finalizing the primary endpoint.
  • Alternative measures such as restricted mean survival time that avoid proportionality assumptions might be preferable for composites in some settings.
  • The patterns could be checked in settings with more than two components or with mild time variation in the component hazard ratios themselves.

Load-bearing premise

The hazard ratio for each individual component endpoint remains constant over time.

What would settle it

A trial dataset in which the two component hazard ratios are constant, the component baseline hazards differ substantially, yet the observed composite hazard ratio shows no variation over time would falsify the central claim.

Figures

Figures reproduced from arXiv: 1907.10976 by Guadalupe G\'omez Melis, Jordi Cort\'es Mart\'inez, KyungMann Kim, Mois\`es G\'omez Mateu.

Figure 1
Figure 1. Figure 1: HRs over time in ZODIAC trial under two different scenarios. Dark and light [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Behaviour of the HR∗(t) (orange) for PFS in three different scenarios. HROS and HROT P are the cause-specific hazard ratios of each component: E1 = death (dark blue) and E2 = OTP (light blue). The probabilities of observing the events during the study in the control group are p (0) OS = 0.15 and p (0) OT P = 0.5, respectively, with a correlation coefficient ρ = 0.3. The three scenarios are characterized by… view at source ↗
Figure 3
Figure 3. Figure 3: Quantification of the impact of various factors on the sample size. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

The hazard ratio is routinely used as a summary measure to assess the treatment effect in clinical trials with time-to-event endpoints. It is frequently assumed as constant over time although this assumption often does not hold. When the hazard ratio deviates considerably from being constant, the average of its plausible values is not a valid measure of the treatment effect, can be clinically misleading and common sample size formulas are not appropriate. In this paper, we study the hazard ratio along time of a two-component composite endpoint under the assumption that the hazard ratio for each component is constant. This work considers two measures for quantifying the non-proportionality of the hazard ratio: the difference $D$ between the maximum and minimum values of hazard ratio over time and the relative measure $R$ representing the ratio between the sample sizes for the minimum detectable and the average effects. We illustrate $D$ and $R$ by means of the ZODIAC trial where the primary endpoint was progression-free survival. We have run a simulation study deriving scenarios for different values of the hazard ratios, different event rates and different degrees of association between the components. We illustrate situations that yield non-constant hazard ratios for the composite endpoints and consider the likely impact on sample size. Results show that the distance between the two component hazard ratios plays an important role, especially when they are close to 1. Furthermore, even when the treatment effects for each component are similar, if the two-component hazards are markedly different, hazard ratio of the composite is often non-constant.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that, under the modeling assumption of constant component-specific hazard ratios, the hazard ratio of a two-component composite endpoint is generally time-varying (non-constant) whenever the baseline hazards of the components differ substantially. The authors quantify the degree of non-proportionality via two measures, D (difference between maximum and minimum composite HR over time) and R (ratio of sample sizes needed to detect the minimum versus average effect), map the regions where D and R become large through simulation (varying component HRs, event rates, and association strength), and illustrate the phenomenon with the ZODIAC trial's progression-free survival endpoint.

Significance. If the central derivation holds, the result is significant for the analysis of randomized trials with composite time-to-event endpoints: it shows that a single summary HR can be misleading and that standard sample-size formulas may be inappropriate even when each component obeys proportional hazards. Credit is due for the explicit forward simulation design that isolates the effect of differing baselines and for the concrete ZODIAC illustration that demonstrates practical impact.

minor comments (3)
  1. [Simulation study] The abstract states that 'the distance between the two component hazard ratios plays an important role, especially when they are close to 1,' yet the simulation section should explicitly tabulate the grid of component HR values (e.g., 0.6–1.4) and baseline hazard ratios used to generate the reported D and R surfaces.
  2. [Methods] The definition and computation of the relative measure R (sample-size ratio) is central to the practical message; the manuscript should supply the exact formula or algorithm used to obtain the 'minimum detectable' and 'average' effects so that readers can replicate the reported values.
  3. [ZODIAC illustration] In the ZODIAC illustration, the estimated component-specific baseline hazards and HRs should be reported numerically (with confidence intervals) so that the resulting D and R can be verified directly from the published Kaplan–Meier or Cox output.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the supportive summary of our work and for recommending minor revision. The assessment correctly captures the core finding that composite HRs can be time-varying even under constant component HRs, and we appreciate the credit given to the simulation design and ZODIAC illustration.

Circularity Check

0 steps flagged

Derivation is self-contained; no circularity detected

full rationale

The paper assumes constant component-specific HRs (explicit modeling premise) and shows via direct definition that the composite HR equals a time-dependent weighted average of the component hazards. This follows immediately from the first-event hazard being the sum of component hazards; the weights evolve unless baselines are proportional in a specific way. Simulations vary HR values, event rates, and association to map regions where D and R are large, without fitting parameters to observed data or reducing any result to a prior fitted quantity. No self-citation is load-bearing, no uniqueness theorem is invoked, and no ansatz is smuggled. The central claim is therefore a straightforward mathematical consequence of the stated assumptions rather than a re-expression of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the modeling premise that each component hazard ratio is constant; all other elements (event rates, association structure, simulation design) are treated as inputs chosen by the authors. No free parameters are fitted inside the reported results; no new physical entities are postulated.

axioms (1)
  • domain assumption Hazard ratio for each component endpoint is constant over time
    Explicitly stated as the assumption under which the composite hazard ratio is studied

pith-pipeline@v0.9.0 · 5820 in / 1255 out tokens · 22588 ms · 2026-05-24T16:08:34.685022+00:00 · methodology

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Reference graph

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17 extracted references · 17 canonical work pages

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