189-2007: Introducing a SAS® Macro for Doubly Robust Estimation

Estimation of the effect of a treatment or exposure with a causal interpretation from studies where exposure is not randomized may be biased if confounding and selection bias are not taken into appropriate account. Such adjustment for confounding is often carried out through regression modeling of the relationships among treatment, confounders, and outcome. Correct specification of the regression model is one of the most fundamental assumptions in statistical analysis. Even when all relevant confounders have been measured, an unbiased estimate of the treatment effect will be obtained only if the model itself reflects the true relationship among treatment, confounders, and the outcome. Outside of simulation studies, we can never know whether or not the model we have constructed includes all relevant confounders and accurately depicts those relationships. Doubly robust estimation of the effect of exposure on outcome combines inverse probability weighting by a propensity score with regression modeling in such a way that as long as either the propensity score model is correctly specified or the regression model is correctly specified the effect of the exposure on the outcome will be correctly estimated, assuming that there are no unmeasured confounders. While several authors have shown doubly-robust estimators to be powerful tools for modeling, they are not in common usage yet in part because they are difficult to implement. We have developed a simple SAS® macro for obtaining doubly robust estimates. We will present sample code and results from analyses of simulated data. INTRODUCTION Correct specification of the regression model is one of the most fundamental assumptions in statistical analysis. Even when all relevant confounders have been measured, an unbiased estimate will be obtained only if the model itself reflects the true relationship among treatment, confounders, and the outcome. Outside of simulation studies, we can never know whether or not the model we have constructed accurately depicts those relationships. So the correct specification of the regression model is typically an unverifiable assumption. Doubly robust (DR) estimation builds on the propensity score approach of Rosenbaum & Rubin (1983) and the inverse probability of weighting (IPW) approach of Robins and colleagues (Robins, 1998; Robins, 1998a; Robins, 1999; Robins, 1999a; Robins, Hernan, and Brumback, 2000). DR estimation combines inverse probability weighting by a propensity score with regression modeling of the relationship between covariates and outcome in such a way that as long as either the propensity score model or the regression model is correctly specified, the effect of the exposure on the outcome will be correctly estimated, assuming that there are no unmeasured confounders (Robins, Rotnitzky, and Zhao, 1994; Robins, 2000; van der Laan and Robins, 2003; Bang and Robins 2005). Specifically, one estimates the probability that a particular patient receives a given treatment as a function of that individual’s covariates (the propensity score). Each individual observation is then given a weight equal to the inverse of this propensity score to create two pseudopopulations of exposed and unexposed subjects that now represent what would have happened to the entire population under those two treatment conditions. Maximum likelihood regression is conducted within these pseudopopulations with adjustment for confounders and risk factors. Results from extensive simulations by Lunceford and Davidian (2004) as well as Bang and Robins (2005) confirm the theoretical properties of this estimator. MATHEMATICS OF DOUBLY ROBUST ESTIMATION We use the following notation: Y is the observed response or outcome, Z is a binary treatment (exposure) variable, and X represents a vector of baseline covariates. Y1 and Y0 are the counterfactual responses under treatment and no treatment, respectively. All of these variables are further subscripted by i for subjects i=1, . . ., n. In this example, the causal effect of interest is the difference in means if everyone in the population received treatment versus SAS Global Forum 2007 Statistics and Data Analysis