class: center, middle, inverse, title-slide # IDS 702: Module 6.6 ## Propensity scores ### Dr. Olanrewaju Michael Akande --- ## Propensity scores - The .hlight[propensity score] (ps) is defined as the conditional probability of receiving a treatment given pre-treatment covariates X. -- - That is, .block[ .small[ $$ e(X) = \mathbb{Pr}[W = 1 | X] = \mathbb{E}[W | X], $$ ] ] where `\(X = (X_1, \ldots, X_p)\)` is the vector of `\(p\)` covariates/predictors. -- - Propensity score is a probability, analogous to a summary statistic. -- - Propensity score has really nice properties which makes it desirable to use within our causal inference framework. --- ## Balancing property of propensity score - .hlight[Property 1]. The propensity score e(X) balances the distribution of all `\(X\)` between the treatment groups: .block[ .small[ $$ W \perp X | e(X) $$ ] ] -- - Equivalently, .block[ .small[ $$ \mathbb{Pr}[W_i = 1 | X_i, e(X_i)] = \mathbb{Pr}[W_i = 1 | e(X_i)]. $$ ] ] -- - The propensity score is NOT the only .hlight[balancing score]. Generally, a balancing score `\(b(x)\)` is a function of the covariates such that: .block[ .small[ $$ W \perp X | b(X) $$ ] ] --- ## Remarks on the balancing property - Rosenbaum and Rubin (1983) show that all balancing scores are a function of `\(e(X)\)`. -- - If a subclass of units or a matched treatment-control pair are homogeneous in `\(e(X)\)`, then the treatment and control units have the same distribution of `\(X\)`. -- - The balancing property is a statement on the distribution of `\(X\)`, NOT on assignment mechanism or potential outcomes. --- ## Propensity score: unconfoundedness - .hlight[Property 2]. If `\(W\)` is unconfounded given `\(X\)`, then `\(W\)` is unconfounded given `\(e(X)\)`, i.e., -- - That is, if .block[ .small[ $$ \{Y_i(0), Y_i(1)\} \perp W_i | X_i $$ ] ] holds, then .block[ .small[ $$ \{Y_i(0), Y_i(1)\} \perp W_i | e(X_i), $$ ] ] also holds. -- - Given a vector of covariates that ensure unconfoundedness, adjustment for differences in propensity scores removes all biases associated with differences in the covariates. --- ## Propensity score: unconfoundedness - `\(e(X)\)` can be viewed as a summary score of the observed covariates. -- - This is great because causal inference can then be drawn through stratification, matching, regression, etc. using the scalar `\(e(X)\)` instead of the high dimensional covariates. -- - The propensity score balances the **observed covariates**, but does not generally balance **unobserved covariates**. -- - In most observational studies, the propensity score e(X) is unknown and thus needs to be estimated. -- - However, since we always observe `\(X\)` and `\(W\)`, estimation can be done using models for binary outcomes. --- class: center, middle # What's next? ### Move on to the readings for the next module!