Abstract: The propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates. Both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates. Applications include: (i) matched sampling on the univariate propensity score, which is a generalization of discriminant matching, (ii) multivariate adjustment by subclassification on the propensity score where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations, and (iii) visual representation of multivariate covariance adjustment by a two-dimensional plot.

DEFINITIONS

The Structure of Studies for Causal Effects

Inferences about the effects of treatments involve speculations about the effect one treatment would have had on a unit which, in fact, received some other treatment. We consider the case of two treatments, numbered 1 and 0. In principle, the ith of the N units under study has both a response r1i that would have resulted if it had received treatment 1, and a response r0i that would have resulted if it had received treatment 0. In this formulation, causal effects are comparisons of r1i and r0i, for example r1i – r0i or r1i/r0i. Since each unit receives only one treatment, either r1i or r0i is observed, but not both, so comparisons of r1i and r0i imply some degree of speculation.