Separable Effects in Four-Arm and Two-Arm Designs

2026-05-07

AI Summary

This study addresses the limitations of existing decomposable treatment effect analyses, which are confined to two-arm designs with bundled components and cannot accommodate the more common four-arm designs featuring independently assigned components. The authors develop a unified framework encompassing both design types, establishing corresponding identification conditions and constructing efficient influence function–based estimators. By integrating machine learning with cross-fitting, the approach enables robust inference. Notably, the paper innovatively leverages four-arm data to conduct two falsification tests of key assumptions underlying two-arm designs, thereby strengthening causal interpretability. Simulation studies demonstrate the estimator’s favorable performance, and an application to National Assessment of Educational Progress data successfully isolates the independent causal effect of extended-time testing accommodations, offering clear guidance for policy decisions.

Abstract

Robins and Richardson (2010) reformulated mediation analysis by decomposing treatments into multiple components and examining separable effects of each component. While this approach is increasingly popular, existing work has analyzed ``two-arm'' data, where components are strictly bundled and manipulated simultaneously. However, in practice, four-arm data where components are assigned independently are often available. For example, testing accommodations might strictly bundle extra time with a separate session or allow them to be assigned separately. To address this distinction, we propose a general framework for analyzing separable effects in four-arm and two-arm designs. This framework provides distinct identification and estimation strategies for each design. For estimation, we utilize efficient influence function estimators coupled with machine learning and cross-fitting techniques. Additionally, we introduce two falsification tests for key identification assumptions required in the two-arm design by leveraging four-arm data. We investigate the performance of the proposed estimators via a simulation study and demonstrate their application by studying the effect of extended time accommodations using data from the National Assessment of Educational Progress. Ultimately, this separable effects analysis enables practitioners to clearly communicate underlying mechanisms and derive informative policy recommendations.

Problem

• separable effects
• mediation analysis
• four-arm design
• two-arm design
• identification assumptions

Innovation

• separable effects
• four-arm design
• efficient influence function
• falsification test
• cross-fitting