Decomposing Treatment Effect Variation

TL;DR

This paper introduces a fully randomization-based framework for decomposing treatment effect variation into explained and unexplained parts, accounting for noncompliance, and provides tools for testing and measuring explained variation.

Contribution

It develops a novel, fully randomization-based approach for decomposing treatment effect variation, including methods for testing and quantifying explained variation with covariates and noncompliance.

Findings

Randomization-based estimates align with linear regression and two-stage least squares.

The proposed omnibus test effectively detects systematic treatment effect variation.

Application to the Head Start Impact Study demonstrates practical utility.

Abstract

Understanding and characterizing treatment effect variation in randomized experiments has become essential for going beyond the "black box" of the average treatment effect. Nonetheless, traditional statistical approaches often ignore or assume away such variation. In the context of randomized experiments, this paper proposes a framework for decomposing overall treatment effect variation into a systematic component explained by observed covariates and a remaining idiosyncratic component. Our framework is fully randomization-based, with estimates of treatment effect variation that are entirely justified by the randomization itself. Our framework can also account for noncompliance, which is an important practical complication. We make several contributions. First, we show that randomization-based estimates of systematic variation are very similar in form to estimates from fully-interacted linear regression and two stage least squares. Second, we use these estimators to develop an omnibus test for systematic treatment effect variation, both with and without noncompliance. Third, we propose an $R^2$-like measure of treatment effect variation explained by covariates and, when applicable, noncompliance. Finally, we assess these methods via simulation studies and apply them to the Head Start Impact Study, a large-scale randomized experiment.