appendix C Technical lemma for the propensity scores

In chapter 5, section 5.1.4, we learned that to find the ATE, we can group together all the patients who have the same propensity score. This works because with the same propensity score, the characteristics of the population are evenly distributed between the treated and control groups. In this appendix, we’ll show the math that proves this is true. We will use conditional probabilities, so if necessary, review the definitions and concepts about conditioning described in chapter 1, section 1.6.2.

Let’s start with an example. In figure C.1, we grouped together children and elderly people because their ratio (1 to 2) is consistent in both the treatment and control groups. Generally, when we group patients who share the same propensity score, the distribution of characteristics like age is the same between those receiving the treatment and those who do not.