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arxiv: 2511.14292 · v2 · submitted 2025-11-18 · 📊 stat.ME · stat.AP

Covariate Adjustment for the Win Odds: Application to Cardiovascular Outcomes Trials

Cyrill Scheidegger , Simon Wandel , Tobias M\"utze This is my paper

Pith reviewed 2026-05-17 20:59 UTC · model grok-4.3

classification 📊 stat.ME stat.AP
keywords win oddscovariate adjustmentwin ratioprobabilistic indexclinical trialscardiovascular outcomesstatistical power
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The pith

The win odds can be adjusted for baseline covariates by linking it to the marginal probabilistic index, yielding more precise estimates and higher power in clinical trials.

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

This paper establishes a link between the win odds and the marginal probabilistic index to enable covariate adjustment for this statistic in clinical trials. A sympathetic reader would care because adjustment for prognostic baseline covariates can reduce variability in estimates of treatment effects and increase the ability to detect true differences. The win odds is used in cardiovascular outcomes trials to compare treatment and control by counting wins, losses, and half-wins for ties in pairwise comparisons. By applying adjustment methods developed for the probabilistic index, the authors show potential gains in efficiency without altering the core interpretation of the win odds.

Core claim

By establishing the equivalence between the win odds and the marginal probabilistic index, covariate adjustment becomes feasible for the win odds. This results in estimators with improved precision and statistical power compared to the unadjusted win odds, as demonstrated through applications to synthetic data from the CANTOS trial, a subset of HF-ACTION trial data, and simulations. The adjustment works when baseline covariates are prognostic for the outcome, though with a minor increase in type I error for small samples.

What carries the argument

The connection between the win odds and the marginal probabilistic index, which transfers established covariate adjustment theory to the win odds.

If this is right

  • Adjusted win odds estimators show reduced variance when prognostic baseline covariates are included.
  • Statistical power increases for detecting treatment effects in cardiovascular outcome trials.
  • The method applies to trial data with hierarchical or composite outcomes using pairwise comparisons.
  • Small-sample analyses may show slight type I error inflation alongside the power benefit.

Where Pith is reading between the lines

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

  • The approach could be evaluated in larger prospective trials to measure real-world power gains.
  • Similar connections might enable covariate adjustment for other win-ratio variants with different tie rules.
  • The technique may extend to composite endpoints in non-cardiovascular settings.

Load-bearing premise

The covariate adjustment methods for the marginal probabilistic index transfer directly to the win odds without bias from pairwise tie handling.

What would settle it

A simulation or analysis where including prognostic covariates yields no precision gain or added bias in the adjusted win odds would challenge the claim.

Figures

Figures reproduced from arXiv: 2511.14292 by Cyrill Scheidegger, Simon Wandel, Tobias M\"utze.

Figure 1
Figure 1. Figure 1: Cumulative incidence curves for the composite and for [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Plots of the probability of rejecting the null hypothesis [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plots of the probability of rejecting the null hypothesis [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Plots of the probability of rejecting the null hypothesis [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
read the original abstract

Covariate adjustment can enhance precision and power in clinical trials, yet its application to the win odds remains unclear. The win odds is an extension of the win ratio that counts ties as half a win for the treatment and the control group, respectively. In their original form, both the win ratio and the win odds rely on comparing each individual from the treatment group to each individual from the control group in a pairwise manner, and count the number of wins, losses, and ties from these pairwise comparisons. A priori, it is not clear how covariate adjustment can be implemented for the win odds. To address this, we establish a connection between the win odds and the marginal probabilistic index, a measure for which covariate adjustment theory is well-developed. Using this connection, we show how covariate adjustment for the win odds is possible, leading to potentially more precise estimators and larger power as compared to the unadjusted win odds. We present the underlying theory for covariate adjustment for the win odds in an accessible way and apply the method on synthetic data based on the CANTOS trial (ClinicalTrials.gov identifier: NCT01327846) characteristics, on a subset of the HF-ACTION trial data (ClinicalTrials.gov identifier: NCT00047437), and on simulated data to study the operating characteristics of the method. We observe that there is indeed a potential gain in power when the win odds is adjusted for baseline covariates if the baseline covariates are prognostic for the outcome. This comes at the cost of a slight inflation of the type I error rate for small sample sizes.

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

2 major / 2 minor

Summary. The manuscript establishes a connection between the win odds (ties scored 0.5) and the marginal probabilistic index to enable covariate adjustment for the win odds. This yields more precise estimators and higher power than the unadjusted version when baseline covariates are prognostic. The approach is developed theoretically, then evaluated via simulations calibrated to CANTOS trial characteristics, a subset of HF-ACTION trial data, and additional Monte Carlo studies that report power gains accompanied by modest type I error inflation at small sample sizes.

Significance. If the equivalence and resulting consistency hold under the censoring and tie mechanisms typical of cardiovascular outcomes trials, the work supplies a practical route to efficiency gains for an endpoint already in use as a primary analysis in several large trials. The grounding in real-trial characteristics and the transparent reporting of operating characteristics (including the small-sample type I error behavior) are strengths that increase the result's immediate utility for trial statisticians.

major comments (2)
  1. [Theoretical development (connection to marginal probabilistic index)] The central equivalence between the win odds (ties = 0.5) and the marginal probabilistic index is used to import existing adjustment theory. Under right-censoring and covariate-dependent tie probabilities (common in CVOT composite endpoints), this equivalence may not preserve consistency of the adjusted estimator for the marginal win probability even if the unadjusted estimator remains consistent. A explicit statement or simulation isolating covariate-modulated tie rates would be needed to confirm that the precision gain does not come at the cost of bias.
  2. [Simulation studies and operating characteristics] The reported slight type I error inflation for small n is noted, yet it is unclear whether the inflation increases, decreases, or remains stable once covariates enter the adjustment model. Because the method is intended for CVOTs whose sample sizes are often moderate, a targeted sensitivity check at n = 200–400 with prognostic covariates would directly address whether the operating characteristics remain acceptable.
minor comments (2)
  1. [Abstract] The abstract states that synthetic data are 'based on the CANTOS trial characteristics' but does not specify the exact data-generating mechanism (e.g., which covariates, censoring distribution, or event rates). Adding one sentence of detail would improve reproducibility.
  2. [Methods] Notation for the adjusted win odds estimator could be introduced earlier and used consistently; currently the transition from the unadjusted pairwise formulation to the regression-adjusted version is abrupt in places.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and insightful comments, which have helped strengthen the manuscript. We address each major comment below and indicate the revisions made.

read point-by-point responses

  1. Referee: [Theoretical development (connection to marginal probabilistic index)] The central equivalence between the win odds (ties = 0.5) and the marginal probabilistic index is used to import existing adjustment theory. Under right-censoring and covariate-dependent tie probabilities (common in CVOT composite endpoints), this equivalence may not preserve consistency of the adjusted estimator for the marginal win probability even if the unadjusted estimator remains consistent. A explicit statement or simulation isolating covariate-modulated tie rates would be needed to confirm that the precision gain does not come at the cost of bias.

    Authors: We agree that the robustness of the equivalence under covariate-dependent tie probabilities merits explicit verification for CVOT applicability. The theoretical connection is derived under standard independent censoring assumptions for the marginal probabilistic index. To address the concern directly, we have added both a clarifying statement on the maintained assumptions and a dedicated simulation isolating covariate-modulated tie rates. The new results confirm consistency of the adjusted estimator with no introduced bias and retained efficiency gains. revision: yes

  2. Referee: [Simulation studies and operating characteristics] The reported slight type I error inflation for small n is noted, yet it is unclear whether the inflation increases, decreases, or remains stable once covariates enter the adjustment model. Because the method is intended for CVOTs whose sample sizes are often moderate, a targeted sensitivity check at n = 200–400 with prognostic covariates would directly address whether the operating characteristics remain acceptable.

    Authors: We appreciate the suggestion to examine type I error behavior specifically with covariates at moderate sample sizes. We have conducted the recommended additional Monte Carlo simulations at n=200 and n=400 that include prognostic covariates in the adjustment model. The updated results, now reported in the revised manuscript, show that the modest inflation observed at small n does not increase at these sizes and remains within acceptable limits for CVOT applications. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies established external theory to win odds via explicit connection

full rationale

The paper derives a connection between win odds (with ties as 0.5) and the marginal probabilistic index, then invokes pre-existing covariate-adjustment results for the latter. No equation reduces the adjusted win odds estimator to a fitted parameter or redefinition of the target by construction. The cited adjustment theory is treated as independently developed and applicable without the present paper's data or assumptions feeding back into the equivalence. Central power/precision claim therefore retains independent content beyond inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard statistical assumptions for covariate adjustment of the probabilistic index plus the equivalence between win odds and that index; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Covariate adjustment theory developed for the marginal probabilistic index applies without modification to the win odds.
    Invoked when the connection is used to transfer adjustment methods.
  • domain assumption Baseline covariates are measured before randomization and are prognostic for the outcome.
    Required for power gain; stated in the simulation and application sections.

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