INTRODUCTION

In this chapter we discuss a method for estimating causal effects given a regular assignment mechanism, based on subclassification on the estimated propensity score. We also refer to this method as blocking or stratification.

Given the assumptions of individualistic assigment and unconfoundedness, the definition of the propensity score in Chapter 3 implies that the super-population propensity score equals the conditional probability of receiving the treatment given the observed covariates. As shown in Chapter 12, the propensity score is a member of a class of functions of the covariates, collectively called balancing scores, that share an important property: within subpopulations with the same value of a balancing score, the super-population distribution of the covariates is identical in the treated and control subpopulations. This, in turn, was shown to imply that, under the assumption of super-population unconfoundedness, systematic biases in comparisons of outcomes for treated and control units associated with observed covariates can be eliminated entirely by adjusting solely for differences between treated and control units on a balancing score. The practical relevance of this result stems from the fact that a balancing score may be of lower dimension than the original covariates. (By definition, the covariates themselves form a balancing score, but one that has no dimension reduction.) When a balancing score is of lower dimension than the full set of covariates, adjustments for differences in this balancing score may be easier to implement than adjusting for differences in all covariates, because it avoids high-dimensional considerations. Within the class of balancing scores, the propensity score, as well as strictly monotonic transformations of it (such as the linearized propensity score or log odds ratio), have a special place. All balancing scores b(x) satisfy the property that if for two covariate values x and x′, b(x′) = b(x′), then it must be the case that e(x) = e(x′).