Nonparametric inference for sublevel-set probabilities of conditional average treatment effect functions

TL;DR

This paper introduces a novel nonparametric approach to estimate and visualize heterogeneity in treatment effects using sublevel-set probabilities of the CATE function, addressing non-differentiability issues with advanced estimators.

Contribution

It develops a Grenander-type estimator and a debiased machine learning estimator for sublevel-set probabilities of the CATE, enabling better heterogeneity analysis.

Findings

The proposed estimators perform well in finite samples.

The methods effectively visualize treatment effect heterogeneity.

Application to diabetes trial data demonstrates practical utility.

Abstract

The average treatment effect can obscure important heterogeneity when individuals respond differently to a treatment. While the conditional average treatment effect (CATE) function captures such heterogeneity, it is difficult to communicate when it depends on many covariates. Sublevels sets of a multivariate CATE function are equally complicated objects, but the probability of a sublevel set of a CATE function is a single number with a simple interpretation as the proportion of individuals whose expected treatment effect does not exceed a prespecified threshold. By varying the threshold, a univariate monotone curve appears which can be used to visualize the overall type and degree of heterogeneity in a population. We formalize this curve as a target parameter and show that it is not pathwise differentiable under a nonparametric model. To address this nonstandard estimation problem, we leverage recent advances in monotone function estimation and develop a Grenander-type estimator that incorporates machine learning. We also show that the best piecewise linear approximation to the curve of interest is a pathwise differentiable parameter, and we develop a debiased machine learning estimator of this approximation. We investigate our proposed estimators' finite sample performance in a sequence of numerical studies based on data synthesized from a randomized trial. The methods are illustrated in data from a randomized trial on diabetes medication.