propensity score matching

When treated and untreated groups differ in many background variables at once, comparing their raw outcomes is unfair. Propensity score matching collapses all of that imbalance into a single number, the estimated probability that a unit received treatment given its covariates, and then pairs each treated unit with an untreated unit of nearly identical propensity. Within such pairs the two groups look, on average, as if treatment had been assigned at random, so their outcome difference estimates a causal effect.

The justification is Rosenbaum and Rubin's result that the propensity score is a balancing score: conditioning on the scalar e(x) = P(T=1 given X=x) achieves the same covariate balance as conditioning on the full vector X, provided unconfoundedness and overlap hold. In practice one estimates e(x) (logistic regression or a flexible learner), forms matches (nearest-neighbor, caliper, or one-to-many), and then checks that covariate distributions are actually balanced after matching before reading off the treated-control outcome gap.

It is intuitive and widely used, but fragile in known ways: it only adjusts for measured confounders, it discards unmatched units, and matching can paradoxically increase imbalance if the model is misspecified. Modern practice often prefers inverse-propensity weighting or doubly robust estimators that pair the propensity model with an outcome model so that a mistake in either one need not bias the answer.