Simultaneous causal inference for multiple treatments via sufficiency

TL;DR

This paper develops a framework for simultaneous causal inference across multiple treatments using the minimal sufficient statistic for covariate distributions, extending existing theory and ensuring unbiased treatment effect estimates.

Contribution

It introduces conditions for sufficiency and minimal sufficiency of statistics in multi-treatment causal inference, and establishes strong ignorability given the minimal sufficient statistic.

Findings

Minimal sufficient statistic is the coarsest balancing score.

Strong ignorability holds given the minimal sufficient statistic.

Unbiased estimation of treatment differences is achieved using this framework.

Abstract

Some units from a population receive the same treatment that is different from treatments available for other reservoir populations. The minimal sufficient statistic $s$ for the pre-treatment $x$-covariates's distributions in the populations is the coarsest balancing score. $s$ is used to select matching units for simultaneous causal comparisons of multiple treatments.Necessary and sufficient conditions on the posterior distribution of the treatment variable (given $x$) determine whether a statistic is either sufficient or minimal sufficient for the x-covariates' distributions. Results in the literature are thus extended. Strong ignorability of treatment assignment given $s(x)$ is also established. Consequently, the expected treatments' differences given $s(x)$ are shown to be simultaneously unbiased for the average causal effects of all treatments' differences. The existing statistical theory for $s$ and its estimates support their use in causal inference.