INTRODUCTION

In Part III of this text we leave the conceptually straightforward world of perfect randomized experiments and move toward the more common setting of observational studies. Although in simple situations we can still directly apply the tools from randomized experiments and exploit the exact results that accompany them, quickly we will be forced to make approximations in our inferences. No longer will estimators be exactly unbiased as in Chapter 6, nor will we be able to calculate exact p-values of the type considered in Chapter 5.

The first step toward addressing observational studies is to relax the classical randomized experiment assumption that the probability of treatment assignment is a known function. We do maintain, however, in this part of the text, the unconfoundedness assumption that states that assignment is free from dependence on the potential outcomes. Moreover, we continue to assume that the assignment mechanism is individualistic, so that the probability for unit i is essentially a function of the pre-treatment variables for unit i only, free of dependence on the values of pre-treatment variables for other units. We also maintain the assumption that the assignment mechanism is probabilistic, so that the probability of receiving any level of the treatment is strictly between zero and one for all units.

The implication of these assumptions is that the assignment mechanism can be interpreted as if, within subpopulations of units with the same value for the covariates, a completely randomized experiment of the type discussed in Chapters 5–8 was conducted, although an experiment with unknown assignment probabilities for the units. Thus, under these assumptions, we can analyze data from a subsample with the same value of the covariates as if it came from such an experiment. Although we do not know a priori the assignment probabilities for each of these units, we know these probabilities are identical because their covariate values are identical, and hence, conditional on the number of treated and control units composing such a subpopulation, the probability of receiving the treatment, the propensity score, is equal to e(x) = Nt(x) / (Nc(x) + Nt(x)) for all units with Xi = x; here Nt(x) and Nc(x) are the number of units in the control and treatment groups respectively with pre-treatment value Xi = x.