Conditional Win Ratio in Causal Inference

• Conditional win ratio is a statistical measure comparing treatment outcomes on an ordinal scale while adjusting for individual covariates.
• It generalizes the standard win ratio by incorporating subject-level heterogeneity, stratification, and detailed covariate structures.
• Robust estimation methods like NN pairing, IPW, and AIPW enable reliable inference for composite and ordinal outcomes in diverse applications.

The conditional win ratio is a causal and distributional parameter that quantifies, conditional on covariates, the probability that an outcome under one condition (“treatment”) is lexicographically superior to that under another (“control”), aggregated according to a clinically or application-defined ordering. This metric generalizes the standard “win ratio” used in hierarchical or composite outcome analyses by incorporating adjustment for subject-level heterogeneity, stratification, and covariate structure. It plays a central role in causal inference with composite or ordinal outcomes, as well as in precision medicine, policy learning, and game-theoretic/statistical evaluation of actions.

1. Formal Definition and Causal Foundations

Consider a potential outcomes framework with units indexed by $i$, each possessing baseline covariates $X_i \in \mathcal{X}$ and two potential outcomes $Y_i(1)$, $Y_i(0)$. A “win function” $w:Y \times Y \to [0,1]$ encodes, for any outcome pair, the degree to which $y$ is preferred to $y'$. For a total or hierarchical (often lexicographic) order $\succ$ on $Y$, the canonical choice is: $$w(y|y') = \begin{cases} 1 & \text{if } y \succ y' \ \tfrac12 & \text{if } y \sim y' \ 0 & \text{if } y \prec y' \end{cases}$$ The conditional win-proportion at $x$ is: $$\tau_\star(x) = \mathbb{E}[w(Y^{(x)}(1)\mid Y_i(0)) \mid X_i = x]$$ where $Y^{(x)}(t)$ denotes an independent draw from the (potential) outcome distribution under treatment $t$ given $X = x$. The marginalization yields an identifiable, individual-level causal effect: $$\tau_\star = \mathbb{E}[w(Y^{(X_i)}(1)\mid Y_i(0))]$$ This contrasts with the standard population win proportion

$\tau_{\rm pop} = \mathbb{E}[w(Y_i(1)|Y_j(0))]$

where $i$, $j$ are independent draws; the population estimand is subject to paradoxical reversals in the presence of effect heterogeneity (Even et al., 28 Jan 2025, Parnas et al., 3 Feb 2026).

Given any win-proportion $\tau$, the associated conditional win ratio is

$R_{\rm WR} = \frac{\tau}{1 - \tau}$

and the net-benefit is $2\tau - 1$.

2. Model-Based and Nonparametric Estimation

Several estimators for the conditional win ratio have been introduced, each with specific assumptions and application domains:

• Nearest Neighbor (NN) Pairing: For each control subject $i$, find the closest treated unit $\sigma^\star(i)$ in covariate space. The empirical estimator

$$\hat{p}_W^{\rm NN} = \frac{1}{n_0} \sum_{i \in N_0} w(Y_{\sigma^\star(i)} | Y_i)$$

is consistent for $\tau_\star$ under randomization and mild continuity assumptions (Even et al., 28 Jan 2025).

• Propensity Score Weighted (IPW–NN): For observational data, inverse propensity weighting corrects for confounding,

$$\hat{\tau}_{\rm IPW} = \frac{1}{n} \sum_{i=1}^n \frac{1-T_i}{1-\hat{e}(X_i)} w(Y_{\sigma^\star(i)} | Y_i)$$

where $\hat{e}(x)$ estimates the treatment probability at $x$ (Even et al., 28 Jan 2025, Cao et al., 28 Aug 2025).

• Doubly Robust (AIPW): Regression adjustment using distributional regression estimators for $\mathbb{E}[w(Y_i(t)|y)|X_i=x]$ results in a robust estimator $\hat{\tau}_{\rm AIPW}$. Consistency is achieved if either the PS or outcome model is correctly specified (Even et al., 28 Jan 2025, Scheidegger et al., 18 Nov 2025, Cao et al., 28 Aug 2025).
• Matching and Distributional Regression: The k-NN and quantile/distributional regression approaches provide flexible nonparametric or semiparametric estimators for $q_W(x)$, the conditional win-proportion at $x$, facilitating high-dimensional or non-linear scenarios (Parnas et al., 3 Feb 2026).
• Probabilistic Index Models (PIMs): Regression models for pairwise win probabilities can yield covariate-adjusted win ratios and benefit from established asymptotic and inference properties (Scheidegger et al., 18 Nov 2025, Gasparyan et al., 2019).
• Stratified and Covariate-Adjusted Estimators: Win ratio estimation can be further refined by stratifying or linearly adjusting for numeric covariates, thereby reducing variance and aligning target estimands with causal parameters of interest (Gasparyan et al., 2019).

3. Identifiability, Causal and Statistical Guarantees

Identifiability of the conditional win ratio relies on

• SUTVA (Stable Unit Treatment Value Assumption),
• Unconfoundedness ($\{Y_i(0), Y_i(1)\} \perp T_i | X_i$), and
• Positivity ($e(x) \in (0,1)$ for all $x$).

Under these assumptions, $q_W(x)$, and hence the conditional win ratio, are identifiable as functionals of observed data distributions: $$q_W(x) = \mathbb{E}[w(Y_i|Y_j)|X_i = X_j = x, T_i = 1, T_j = 0]$$ Estimation is further justified by consistency, asymptotic normality (via U-statistic theory), and weak double robustness for AIPW variants (Even et al., 28 Jan 2025, Parnas et al., 3 Feb 2026, Cao et al., 28 Aug 2025, Scheidegger et al., 18 Nov 2025).

4. Applications and Interpretational Guidelines

The conditional win ratio has become a preferred effect measure in hierarchical composite endpoint analysis in cardiovascular, neurological, and critical care trials—particularly in settings with strong baseline effect heterogeneity, complex competing risks, and precision medicine objectives (Even et al., 28 Jan 2025, Cao et al., 28 Aug 2025).

It also underpins optimal policy learning frameworks in conditional treatment effect estimation. Policy value functionals of the form

$$V(\pi) = \mathbb{E} \left[\pi(X) q_W(X) + (1-\pi(X)) q_L(X)\right]$$

give rise to optimal treatment rules $\pi^*(x) = 1_{\{q_W(x) > q_L(x)\}}$ (Parnas et al., 3 Feb 2026).

In gaming and reinforcement learning, conditional win ratios must be corrected for selection effects (“skill bias”), as naive conditioning can produce misleading conclusions about action value. Explicit bias-correction formulas based on cross-group skill-matching address this limitation (Yang, 2018).

5. Algorithmic and Computational Implementations

All conditional win ratio estimators admit precise, algorithmic definitions:

• Distributional Regression: (1) Fit models for the conditional quantiles of $Y|T=t,X=x$; (2) Monte Carlo sampling or analytic computation of $w(Q^{(1)}_u(x), Q^{(0)}_v(x))$ over $u,v$; (3) Aggregate over the sample. This is robust in high-dimensional or ordinal outcome spaces (Even et al., 28 Jan 2025, Parnas et al., 3 Feb 2026, Cao et al., 28 Aug 2025).
• Probabilistic Index Model Workflow:

1. Construct all pairwise contrasts with indicators $I_{win}, I_{loss}, I_{tie}$.
2. Fit logistic regression for $m(A_i, A_j, X_i, X_j) = P(Y_i \leq Y_j| \cdot)$.
3. Marginalize predictions for covariate-adjusted ν, then compute win odds or ratios (Scheidegger et al., 18 Nov 2025, Gasparyan et al., 2019).

• Variance Estimation & Inference: All estimators have closed-form or influence-function based variance estimation procedures, supporting confidence intervals and hypothesis tests via normal-theory approximations or delta methods (Scheidegger et al., 18 Nov 2025, Cao et al., 28 Aug 2025, Gasparyan et al., 2019).
• Software: Flexible R implementations exist for most methods, e.g., the winPSW package for propensity-based adjustment (Cao et al., 28 Aug 2025).

6. Comparison to Alternative Metrics and Recommendations

The table below summarizes key estimands and estimators relevant to conditional win ratio analysis (Even et al., 28 Jan 2025, Parnas et al., 3 Feb 2026, Cao et al., 28 Aug 2025, Gasparyan et al., 2019):

Estimand/Estimator	Target	Covariate Handling
Population win proportion ($\tau_{pop}$)	$E[w(Y_i(1)|Y_j(0))]$	Marginal only
Conditional win proportion ($\tau_\star$)	$E[w(Y^{(X_i)}(1)|Y_i(0)]$	Adjusted (individual)
NN Pairing	$\tau_\star$	Fully nonparametric
IPW / OW	$\tau_\star$	PS weighting
AIPW / AOW	$\tau_\star$	Doubly robust
PIM/Logit	$m(A_i,A_j,X_i,X_j)$	Parametric/regression

When to use the conditional win ratio:

• In the presence of individual-level heterogeneity, noncollapsible effects, or complex covariates.
• When targeting precision medicine, stratified policy learning, or individualized effect measures.
• In observational studies requiring robustness to confounding and modeling biases.
• For correcting selection-induced biases in reinforcement learning or game outcome statistics.

Practical guidance suggests combining distributional regression or IPW/AIPW estimators for scalability, robustness, and inference, with careful attention to model calibration and the regularity of covariate spaces (Even et al., 28 Jan 2025, Scheidegger et al., 18 Nov 2025, Cao et al., 28 Aug 2025).

7. Limitations, Extensions, and Controversies

While the conditional win ratio resolves fundamental ambiguities of the population win ratio in heterogeneous or stratified data, it is not immune to identifiability failure when key effect modifiers are unmeasured. Further, plug-in and nonparametric estimators remain sensitive to kernel bandwidth, curse of dimensionality, and covariate overlap; double robustness mitigates some, but not all, of these concerns (Even et al., 28 Jan 2025, Parnas et al., 3 Feb 2026).

Naive conditional win statistics in gaming applications not accounting for skill bias risk systematic misestimation of action value, which can be both theoretically and empirically corrected using group-matching and variance renormalization (Yang, 2018).

The conditional win ratio framework is extendable to multivariate, preference-based, or generalized pairwise comparison settings, connecting to contemporary developments in causal inference, empirical welfare maximization, and subpopulation policy optimization (Parnas et al., 3 Feb 2026).

In summary, the conditional win ratio and its associated estimation methodologies are central and evolving tools in comparative, stratified, and individualized effect evaluation for ordered, composite, or multivariate outcomes across clinical, statistical, and applied domains.