Propensity Score Matching

The propensity score is the probability of receiving treatment given observed covariates, estimated via logistic regression: P(T=1|Covariates). Two individuals with the same propensity score have similar baseline characteristics (in expectation) even if treated differently. Matching on propensity scores creates comparable groups: in the matched sample, treatment status is independent of covariates (conditional independence achieved). This is equivalent to having made randomization at baseline, enabling causal inference. The propensity score is a scalar (one number per person), reducing matching from many dimensions to one (curse of dimensionality resolved).

Propensity score matching: pairs treated individuals with untreated individuals having similar propensity scores; analyzes only matched pairs, discarding unmatched individuals. IPTW (inverse probability weighting): weights all individuals by inverse propensity scores (weight treated by 1/p, controls by 1/(1−p)), creating a pseudo-population where treatment is independent of covariates. IPTW retains all data, improving efficiency and power. Both achieve covariate balance; IPTW is statistically more efficient. Trade-off: IPTW can produce extreme weights if some individuals have very high or very low propensity scores, increasing variance; matching avoids this but loses data.

Calculate standardized mean difference (SMD) for each covariate: SMD = (mean_treated − mean_control) / pooled SD. SMD < 0.10 is generally accepted as 'balanced'; < 0.05 is excellent. Plot SMDs pre- and post-matching (love plot): before/after balance visible at a glance. Calculate SMD for all confounders; if any remain imbalanced (SMD > 0.10) after matching, consider: widening caliper (match more loosely), including interactions in propensity model, or using alternative methods (stratification, regression adjustment). Never judge balance by p-values (p-values are sample size-dependent); always examine SMD numerically and graphically.

No. PSM reduces bias from measured confounders (those you've collected data on). Unmeasured confounders (e.g., patient motivation, unmeasured disease severity, hidden selection mechanisms) remain uncontrolled. Conduct sensitivity analysis to assess how sensitive conclusions are to unmeasured confounding (Rosenbaum bounds: how strong an unmeasured confounder must be to change conclusions). Always discuss unmeasured confounding limitations. Consider instrumental variables or natural experiments if unmeasured confounding is a major concern.