Sufficient Covariate, Propensity Variable and Doubly Robust Estimation

TL;DR

This paper explores the role of propensity variables in causal inference within a decision-theoretic framework, demonstrating their equivalence to multivariate adjustment in linear models and analyzing the properties of doubly robust estimators.

Contribution

It provides a theoretical analysis of propensity variables, their equivalence to multivariate adjustment, and the properties of augmented inverse probability weighted estimators in linear models.

Findings

Propensity variables can yield the same causal effect estimates as multivariate adjustment in linear models.

Estimated propensity variables may improve precision over true ones under certain conditions.

The augmented inverse probability weighted estimator is doubly robust and can enhance precision if correctly specified.

Abstract

Statistical causal inference from observational studies often requires adjustment for a possibly multi-dimensional variable, where dimension reduction is crucial. The propensity score, first introduced by Rosenbaum and Rubin, is a popular approach to such reduction. We address causal inference within Dawid's decision-theoretic framework, where it is essential to pay attention to sufficient covariates and their properties. We examine the role of a propensity variable in a normal linear model. We investigate both population-based and sample-based linear regressions, with adjustments for a multivariate covariate and for a propensity variable. In addition, we study the augmented inverse probability weighted estimator, involving a combination of a response model and a propensity model. In a linear regression with homoscedasticity, a propensity variable is proved to provide the same estimated causal effect as multivariate adjustment. An estimated propensity variable may, but need not, yield better precision than the true propensity variable. The augmented inverse probability weighted estimator is doubly robust and can improve precision if the propensity model is correctly specified.