PropensityScoreMatchings

Propensity score is a statistical concept used in observational studies to address confounding when estimating treatment effects. It is defined as the probability of receiving the treatment given a set of observed covariates, denoted e(x) = P(T = 1 | X = x). The key idea is that, conditional on the propensity score, the distribution of observed covariates should be similar between treated and untreated units, making the groups more comparable.

Estimation of the propensity score is typically done with a regression model, most commonly logistic regression, but alternative models such as probit, boosted trees, or machine learning algorithms can be used. Once scores are estimated, several methods can be employed to influence causal estimates. Matching pairs treated and control units with similar scores; stratification or subclassification into groups (for example, quintiles) based on score percentiles; weighting through inverse probability of treatment (IPTW) to create a synthetic population; and including the score as a covariate in the outcome model.

The approach relies on key assumptions. No unmeasured confounding, meaning all relevant covariates that influence both treatment and outcome are observed and properly modeled; the overlap or common support assumption, which requires that treated and untreated units have similar propensity scores in the data. It is also important to assess balance after applying propensity score methods and to consider positivity constraints.

Limitations include dependence on observed covariates only, making residual confounding possible if important confounders are unmeasured. Model misspecification, insufficient overlap, and insufficient balance can bias results. Propensity scores are a tool for causal inference in observational data, complementing careful study design and sensitivity analyses.