CPTE: Conditional Preference Treatment Effect

• Conditional Preference-based Treatment Effect (CPTE) is a framework that applies user-specified preference functions to quantify and compare heterogeneous treatment effects in complex outcome spaces.
• It provides robust estimation methods—such as distributional regression, quantile regression, and influence-function correction—that address non-identifiability in causal settings.
• CPTE enables personalized policy learning and optimization by capturing multi-attribute, ordinal, or non-linear treatment effects that extend beyond conventional mean comparisons.

The Conditional Preference-based Treatment Effect (CPTE) is a central concept in modern causal inference, enabling flexible characterization, estimation, and policy optimization for heterogeneous treatment effects under arbitrary preference relations over outcomes. Unlike conventional metrics such as the Conditional Average Treatment Effect (CATE), which summarize effects via means, the CPTE framework utilizes a user-specified preference function to encapsulate complex, potentially multi-attribute, ordinal, or non-linear outcome analysis. This paradigm addresses non-identifiable comparison-based estimands, delivers new identifiability conditions, and supplies robust estimation and inference methods for personalized policy learning (Parnas et al., 3 Feb 2026, Kallus et al., 2022).

1. Definition and Formalization

The CPTE is defined with respect to a user-specified preference function $w:Y\times Y\to[0,1]$, encoding the degree to which outcome $y$ is preferred over $y'$. This generalizes several important causal estimands, including conditional probability of necessity and sufficiency (PNS), conditional Win Ratio, and Generalized Pairwise Comparisons.

Let $Y(1)$, $Y(0)$ be the potential outcomes, $X$ the covariates. The individual preference-based treatment effect is

$\Delta_i = w(Y_i(1)\mid Y_i(0))$

Given the fundamental problem of causal inference (unobservability of both potential outcomes), $E[ \Delta_i \mid X_i=x ]$ is generally non-identifiable. The CPTE circumvents this by defining

$$q_W(x) = \mathbb{E}[ w(Y(1,x)\mid Y_j(0)) \mid X_j = x ]$$

where $Y(1,x)$ and $Y_j(0)$ are drawn independently, conditional on $X = x$. The anti-preference is similarly

$$q_L(x) = \mathbb{E}[ w(Y_i(0)\mid Y_j(1)) \mid X_i = X_j = x ]$$

The net preference-based effect is then

$\Delta(x) = q_W(x) - q_L(x)$

This framework unifies special cases: for $w(y|y')=1\{y>y'\}$, $q_W(x)$ quantifies the probability that treatment yields a better outcome than control for individuals with covariates $x$ (conditional PNS). For lexicographic or ordinal $w$, CPTE specializes to conditional win-probabilities and related statistics (Parnas et al., 3 Feb 2026).

2. Identification and Theoretical Guarantees

CPTE estimands are identifiable under standard causal conditions:

• SUTVA: No interference between units.
• Unconfoundedness: $\{Y(0),Y(1)\} \perp T | X$.
• Positivity: $0 < P(T=1|X=x) < 1$ for all $x$.

Under these, $q_W(x)$ and $q_L(x)$ are identified as functionals of the observed data via observed treated and control pairs within covariate strata. Importantly, when all "structural treatment-effect modifiers" (variables determining the treatment effect under $w$) lie in $X$, CPTE recovers the true conditional individual-level estimand: $$q_W(x) = \mathbb{E}[ w(Y_i(1)\mid Y_i(0)) | X_i=x ]$$ In the general case, CPTE serves as a proxy estimand encoding the cross-sectional average preference-based comparison, which remains interpretable even when the structural modifiers are not fully observed. Lemma 2.1 and Theorem 2.6 in (Parnas et al., 3 Feb 2026) clarify the boundaries of identifiability and recovery of individual effects.

3. Estimation Strategies

Estimation of CPTE proceeds in two key stages: learning the conditional outcome distributions and evaluating the average preference under these distributions.

Methods for Estimating $q_W(x)$ and $q_L(x)$

Approach	Key Steps	Notes
Distributional k-NN	Find $k$-nearest neighbors in each arm, compute pairwise $w$ over neighbor pairs	Consistent if $k\to\infty$, $k \log n / n \to 0$
Quantile Regression	Fit conditional quantile functions, generate pseudo-samples via inverse CDF sampling	Accommodates arbitrary $w$ via Monte Carlo averaging
Distributional Regression	Fit full conditional CDFs (e.g. via DRF), simulate samples for evaluation	Generalizes to multivariate, ordinal, or complex outcome spaces

The estimator

$$\widehat q_W(x) = \frac{1}{m} \sum_{k=1}^m w(\hat Y_1^{(k)} \mid \hat Y_0^{(k)})$$

converges as $m\to\infty$, provided $\hat Y_t^{(k)} \sim \widehat P(Y\mid T=t, X=x)$ (Parnas et al., 3 Feb 2026).

Efficient Influence Function Estimation

Policy value estimation incorporates influence-function-based debiasing: $$\phi_\pi(P)(o) = -\psi_\pi(P) + \left[\pi(X)q_W(X) + (1-\pi(X))q_L(X) \right] + \ldots$$ as detailed in Proposition 5.1 (Parnas et al., 3 Feb 2026). Cross-fitting and one-step correction enable root-$n$-consistent, semiparametrically efficient inference.

4. Relationship to Distributional and Risk-based Effects

The CPTE generalizes and is unified with conditional distributional treatment effects (CDTEs), encompassing quantile, super-quantile, and general risk measure contrasts (Kallus et al., 2022). The pseudo-outcome regression framework allows estimation of any preference-based functional by designing an orthogonal pseudo-outcome $\psi$ such that $\mathbb{E}[\psi|X]=\text{CPTE}(x)$:

• CQTE (quantile): $w(y|y') = \mathbb{I}\{y>y'\}$, functional is quantile difference.
• CSQTE (superquantile): uses tail expectations; pseudo-outcomes reflect excess losses.
• $C_f$RTE (coherent risk): preference expressed via dual $f$-divergence risk functionals; plug-in minus orthogonalization achieves robustness.

This connect highlights the non-parametric, model-agnostic and robust nature of CPTE estimation: convergence rates are unaffected by slow or misspecified nuisance estimates given the orthogonality property (Theorem 1 in (Kallus et al., 2022)). The approach allows valid inference on linear projections, matching oracle rates (Theorems 2–4).

5. Policy Learning Applications

Individualized policies $\pi:X\to\{0,1\}$ maximize population-average preference:

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

The optimal unconstrained rule recommends treatment when $q_W(x) > q_L(x)$.

Estimation for constrained intervention—but practically relevant—settings (where resource limits restrict intervention to a fraction $\delta$ of patients) utilizes the CPTE curve as a prioritization score. Targeting the top-$\delta$ stratum of $\beta(x)$ maximizes the net gain (Levis et al., 2024). The constrained optimization

$$\max_{E[\Delta(X)]\leq\delta} E[\Delta(X) \cdot \beta(X)]$$

is solved by thresholding the CPTE at the $1-\delta$ quantile.

Influence-function-based plug-in estimators allow for Wald confidence intervals for policy value. Empirical illustrations demonstrate the effectiveness of CPTE-based policies, e.g., in ICU transfer and 401(k) eligibility studies, showing superiority or complementarity to mean-based CATE policies, especially under heterogeneity or heavy-tailed outcomes (Parnas et al., 3 Feb 2026, Kallus et al., 2022, Levis et al., 2024).

6. Practical Implementation and Empirical Evaluation

Implementing CPTE-based analysis involves:

1. Cross-fitting and estimation of nuisance functions (propensity, conditional distribution).
2. Monte Carlo or pseudo-outcome regression procedures to estimate $q_W(x)$, $q_L(x)$.
3. Plug-in or one-step influence-function-corrected policy evaluation.
4. Thresholding for constrained policy design.

Empirical applications confirm rapid convergence of distributional and preference-based policies (e.g., PNS-policy, win-ratio policy) to the optimal rule, even when CATE-based policies fail due to non-mean outcome structure or heavy-tailed distributions. One-step correction improves value estimation stability in moderate sample regimes. For complex hierarchical outcomes, lexicographic CPTE policies yield superior performance compared to those optimizing on a single outcome or mean differences only (Parnas et al., 3 Feb 2026).

A concrete example: in ICU transfer, the CPB/CPTE curve as a function of age and SOFA score identified gain strata (e.g., $>5\%$ survival lift for older or sicker patients) (Levis et al., 2024). In 401(k) analysis, tail-based CPTEs revealed hidden heterogeneity that mean-based estimates did not.

7. Connections, Extensions, and Limitations

CPTE unifies multiple strands in causal inference—spanning risk-sensitive, ordinal, and preference-driven effects—via its general preference rule $w$. It delivers estimation and policy learning tools with proven efficiency, minimax properties, and robust identification under minimal assumptions.

Limitations stem from intrinsic non-identifiability when structural effect modifiers are unobserved, as true individual-level effects are not generally functionals of the data-generating law (Parnas et al., 3 Feb 2026). Nevertheless, CPTE targets remain interpretable and relevant for policy, subsuming foundational statistics such as PNS and win-ratio.

Ongoing research continues to refine estimation procedures, explore copula-induced dependencies, and generalize to dynamic, longitudinal, or multi-armed treatment settings. The CPTE framework thus constitutes a central organizing principle in modern preference-based and distributional causal inference.