The propensity score is the probability of treatment assignment conditional on observed baseline characteristics. The propensity score allows one to design and analyze an observational (nonrandomized) study so that it mimics some of the particular characteristics of a randomized controlled trial. In particular, the propensity score is a balancing score: conditional on the propensity score, the distribution of observed baseline covariates will be similar between treated and untreated subjects. I describe 4 different propensity score methods: matching on the propensity score, stratification on the propensity score, inverse probability of treatment weighting using the propensity score, and covariate adjustment using the propensity score. I describe balance diagnostics for examining whether the propensity score model has been adequately specified. Furthermore, I discuss differences between regression-based methods and propensity score-based methods for the analysis of observational data. I describe different causal average treatment effects and their relationship with propensity score analyses.

The propensity score is a subject's probability of treatment, conditional on observed baseline covariates. Conditional on the true propensity score, treated and untreated subjects have similar distributions of observed baseline covariates. Propensity-score matching is a popular method of using the propensity score in the medical literature. Using this approach, matched sets of treated and untreated subjects with similar values of the propensity score are formed. Inferences about treatment effect made using propensity-score matching are valid only if, in the matched sample, treated and untreated subjects have similar distributions of measured baseline covariates. In this paper we discuss the following methods for assessing whether the propensity score model has been correctly specified: comparing means and prevalences of baseline characteristics using standardized differences; ratios comparing the variance of continuous covariates between treated and untreated subjects; comparison of higher order moments and interactions; five-number summaries; and graphical methods such as quantile–quantile plots, side-by-side boxplots, and non-parametric density plots for comparing the distribution of baseline covariates between treatment groups. We describe methods to determine the sampling distribution of the standardized difference when the true standardized difference is equal to zero, thereby allowing one to determine the range of standardized differences that are plausible with the propensity score model having been correctly specified. We highlight the limitations of some previously used methods for assessing the adequacy of the specification of the propensity-score model. In particular, methods based on comparing the distribution of the estimated propensity score between treated and untreated subjects are uninformative. Copyright © 2009 John Wiley & Sons, Ltd.