Inference on Local Variable Importance Measures for Heterogeneous Treatment Effects

TL;DR

This paper develops a model-agnostic, RKHS-embedded framework to perform global-inference on local variable importance measures for heterogeneous treatment effects. It introduces a general weighted parameter $\gamma_\omega(P)$, constructs a one-step estimator with an efficient influence function, and proves asymptotic Gaussianity, enabling Wald-type tests and confidence bands via bootstrap. The approach yields valid inference even when nuisance components are estimated with flexible machine learning methods and is demonstrated through simulations in 5- and 10-dimensional settings and an infectious disease vaccination study, where country and baseline antibody titers emerge as non-zero modifiers of vaccine effect. The work advances interpretable causal inference by enabling principled, global conclusions about which variables modify treatment effects at the individual level, with practical implications for high-stakes domains like medicine and epidemiology.

Abstract

We provide an inferential framework to assess variable importance for heterogeneous treatment effects. This assessment is especially useful in high-risk domains such as medicine, where decision makers hesitate to rely on black-box treatment recommendation algorithms. The variable importance measures we consider are local in that they may differ across individuals, while the inference is global in that it tests whether a given variable is important for any individual. Our approach builds on recent developments in semiparametric theory for function-valued parameters, and is valid even when statistical machine learning algorithms are employed to quantify treatment effect heterogeneity. We demonstrate the applicability of our method to infectious disease prevention strategies.

Figures (7)

• Figure 1: Illustration of functions and their RKHS embeddings. Smooth functions remain almost unchanged (left), whereas rough ones become noticeably smoother (right).
• Figure 2: Simulation results of different variable importance measures (rows) under different data-generating processes (columns).
• Figure 3: Simulation results of different variable importance measures (rows) under different data-generating processes (columns).
• Figure 4: Variable importance, defined as the norm of the RKHS embedding of the local variable importance measures: KOI, LOO, and Shapley values, along with corresponding confidence intervals. Intervals are truncated at zero because variable importance measures are non-negative.
• Figure S1: Illustration of the basic notions of semiparametric theory.

Theorems & Definitions (16)

• Theorem 1: Pathwise differentiability
• Theorem 2: Asymptotic linearity and weak convergence
• Theorem 3: Asymptotically valid confidence set
• Lemma 1: Delta method
• Theorem 4: Confidence interval for the variable importance measure
• Lemma S1
• Theorem S1: Consistency against fixed alternatives
• Theorem S2: Local power of the hypothesis test of no importance
• proof : Proof of Lemma \ref{['lem:pd']}
• proof : Proof of Theorem \ref{['Theorem1']}