Variable selection for propensity score estimation via balancing covariates.

We first define some notation. Let Y denote the response of interest and X denote a p-dimensional vector of covariates. Let T denote a binary indicator of treatment exposure: T = 1 if treated, T = 0 if control. (Yi,Xi, Ti), i = 1, . . . , n, is a random sample from (Y,X, T ). We further define Y (1) as the potential outcome if the subject were treated and Y (0) as the potential outcome if the subject were assigned to the control group. Depending on the type of the outcome variable, the causal estimands can be defined as the difference between Y (1) and Y (0) if Y is continuous or the odds ratio if Y is binary. We further denote the propensity score as π(X) = Pr(T = 1|X). In observational studies, π(X) is unknown and needs to be estimated from the data. Next, we briefly review the generalized boosted model (GBM) and covariate balancing propensity scores (CBPS). Both approaches estimate propensity scores by achieving balance in the covariates.