Causal Ordering Without Effect Estimation: A Framework for Using Proxies in Treatment Prioritization

TL;DR

The paper addresses prioritizing treatment when causal-effect estimation is impractical by formalizing causal ordering via a noncausal proxy signal with CAS $\theta(x)$ and comparing it to the true CATE $\beta(x)$. It identifies two principled conditions—Dominant Moderation and Signal Monotonicity—that ensure CAS-based rankings match the true effect ordering, and develops a practical alignment-SNR framework to assess proxies under imperfect alignment. The authors introduce a concrete diagnostic toolkit to quantify proxy usefulness, including SNR estimation, dilution factors, and an alignment threshold, and validate the approach empirically in online advertising, showing that simple baseline proxies can outperform CATE-based ordering for targeting decisions. The work reconciles practical targeting with causal reasoning by reframing the problem around moderators and signal quality, thereby enabling credible causal decisions even when direct effect estimation is infeasible. It provides a nuanced perspective on when predictive proxies should be preferred and offers actionable diagnostics to guide practice in real-world prioritization tasks.

Abstract

Who should we prioritize for treatment when causal effects cannot be estimated? In practice, organizations often rely on predictive proxies: ads are targeted using purchase probabilities, and retention incentives are allocated using churn-risk scores. These models are not causal, but they are often used with the aim of ranking individuals by treatment effects, a task we call causal ordering. We develop a decision-focused framework to reason about this practice. We identify conditions under which proxies recover the correct effect ordering, which hold when a proxy reflects a dominant moderator of treatment effects. We show how these conditions emerge as a useful approximation in discrete choice settings, where the propensity to act without an intervention moderates persuasion. Moreover, we extend beyond this case, demonstrating that proxies capturing a non-dominant moderator can still outperform CATE estimates when they target signals that are easier to estimate precisely. Building on these insights, we introduce diagnostic tools to assess proxy usefulness in practice. Finally, we illustrate the framework in advertising, where a simple predictive proxy outperforms heterogeneous-effect estimation methods.

Figures (7)

• Figure 1: Dominant moderation. All variation in the expected treatment effect $\beta$ (the CATE) of $T$ on $Y$ is governed by a dominant moderator $\theta$, which subsumes the moderators in $X$. The signal variable $S$ reflects $\theta$.
• Figure 2: Bias Can Reduce Ranking Errors. The figure shows the sampling distributions of scores, $\hat{\theta}(x_j)$ and $\hat{\theta}(x_i)$, under the true ordering $\beta(x_i)>\beta(x_j)$. Without bias (left), variance causes the distributions to overlap, raising the likelihood of misranking. When bias increases with effect size (right), the distributions shift apart, improving ranking accuracy.
• Figure 3: A larger expected baseline utility $\mu$ increases the probability of acting without intervention (the CAS, shown in blue). By contrast, the CATE (red, dashed) is bell-shaped, peaking when $\mu$ is near $-\delta/2$.
• Figure 4: A larger CAS does not always imply a larger CATE, so the causal ordering is biased. However, unbiased causal classification holds for all thresholds below $\tilde{\tau}_k^\star=28\%$ on the CAS and $\tau_k^\star=11\%$ on the CATE.
• Figure 5: Comparison of scoring models. While only the CATE-on-conversion model accurately estimates treatment effects (Figure \ref{['fig:subfig1']}), all models perform similarly in causal classification (Figure \ref{['fig:subfig2']}) and ordering (Figures \ref{['fig:subfig3']} and \ref{['fig:subfig4']}). Surprisingly, the conversion rate model performs best, despite not estimating causal effects directly.

Theorems & Definitions (4)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4