Model averaged double robust estimation

Abstract

Researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high-dimensional set of confounders. Typically, a single propensity score model is chosen and used to adjust for confounding, while the uncertainty surrounding which covariates to include into the propensity score model is often ignored, and failure to include even one important confounder will results in bias. We propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging. We develop and evaluate this approach in the context of double robust estimation. More specifically, we introduce the model averaged double robust (MA-DR) estimators, which account for model uncertainty in both the propensity score and outcome model through the use of model averaging. The MA-DR estimators are defined as weighted averages of double robust estimators, where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model. The MA-DR estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified, possibly large, class of models. Using simulation studies, we also assessed small sample properties, and found that MA-DR estimators can reduce mean squared error substantially, particularly when the set of potential confounders is large relative to the sample size. We apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one-year survival in a cohort of 1887 Medicare enrollees who were diagnosed with glioblastoma between June 2005 and December 2009.

Keywords: causal inference; confounding; double robustness; model averaging; propensity score; variable selection.

Figure 1

Model specific double robust estimator for the glioblastoma example and corresponding 95% confidence intervals for 1000 randomly chosen outcome models and 1000 randomly chosen propensity score models, sorted by point estimates. The unadjusted estimator (X), the model averaged double robust estimator (square), the double robust estimator that includes all covariates into both models (circle), and the model selected double robust estimator (triangle) are included. Model selection is performed using BIC.