Propensity Score

A propensity score is the estimated conditional probability that an individual receives a particular treatment given a set of observed baseline covariates. Introduced by Rosenbaum and Rubin in 1983, the propensity score condenses multiple confounding variables into a single summary score, enabling researchers to approximate the balance of observed covariates that randomization would achieve in an observational study.

In observational research, treatment assignment is not random. Patients who receive a particular treatment often differ systematically from those who do not, and these differences may confound the estimated treatment effect. The propensity score provides a principled method for reducing this confounding by summarizing the measured covariates included in its model into a single value that can be used for matching, stratification, weighting, or regression adjustment. Propensity score methods have become one of the most widely used tools in pharmacoepidemiology, health services research, and comparative effectiveness research, where randomized trials are often impractical, unethical, or insufficient to address the question.

The propensity score is typically estimated using logistic regression, where the dependent variable is treatment assignment (treated versus untreated) and the independent variables are the observed baseline covariates thought to influence both treatment selection and the outcome. The predicted probability from this model is the propensity score for each individual.

Once propensity scores are estimated, they can be used in several ways to reduce confounding. Propensity score matching pairs treated and untreated individuals with similar propensity scores, creating a pseudo randomized comparison. Stratification divides the sample into strata (typically five) based on propensity score quantiles and estimates the treatment effect within each stratum. Inverse probability of treatment weighting (IPTW) assigns each individual a weight based on the inverse of the probability of receiving their actual treatment, creating a pseudo population in which treatment assignment is independent of measured covariates. Covariate adjustment includes the propensity score as a covariate in the outcome regression model.

After applying any propensity score method, researchers must verify covariate balance between treated and untreated groups. Standardized mean differences below 0.10 for each covariate are generally considered evidence of adequate balance. If balance is not achieved, the propensity score model should be respecified by adding interaction terms, polynomial terms, or additional covariates.

A critical limitation of propensity score methods is that they only address confounding from measured covariates. Unmeasured confounders remain a threat, and sensitivity analyses (such as the E value or Rosenbaum bounds) should be conducted to assess how sensitive the results are to potential unmeasured confounding.

Use propensity scores when analyzing observational data where treatment assignment is not random and measured covariates differ between treatment groups. They are particularly useful when the number of confounders is large relative to the number of outcome events, as they reduce a multidimensional confounding problem to a single dimension. Propensity scores are not necessary in randomized trials where balance is achieved by design. They should not be used as a substitute for careful covariate selection based on subject matter knowledge, and they do not eliminate bias from unmeasured confounders.

A frequent error is including variables in the propensity score model that are affected by treatment (post treatment variables), which introduces bias rather than removing it. Another mistake is failing to check covariate balance after propensity score application, assuming that model fit statistics (such as c statistic) guarantee balance. A high c statistic for the propensity score model does not necessarily mean that treated and untreated groups are well balanced on individual covariates. Researchers also sometimes ignore the possibility of unmeasured confounding, presenting propensity score adjusted estimates as if they were causal without conducting sensitivity analyses.

A pharmacoepidemiologic study compares cardiovascular outcomes between 12,000 patients initiated on drug A and 8,000 patients initiated on drug B using a claims database. Patients receiving drug A are older, have more comorbidities, and have higher baseline medication use. A logistic regression model with 24 baseline covariates generates propensity scores for receiving drug A. After 1:1 nearest neighbor matching with a caliper of 0.2 standard deviations of the logit of the propensity score, 7,200 matched pairs are formed. All standardized mean differences fall below 0.05, confirming adequate balance on the 24 measured covariates; unmeasured confounders may still differ between the groups. The matched analysis estimates a hazard ratio of 0.88 (95% CI: 0.78 to 0.99) for major adverse cardiovascular events with drug A versus drug B.