Introduction

Observational studies occur frequently in medical research. In these studies, investigators have no control over the treatment assignment. Therefore, large differences on observed covariates in the two groups may exist, and these differences could lead to biased estimates of treatment effects. The propensity score for an individual, defined as the conditional probability of being treated given the individual's covariates, can be used to balance the covariates in the two groups, and thus reduce this bias. The propensity score has been used to reduce bias in observational studies in many fields. In particular, there are good recent examples in the literature where propensity scores were discussed in either applied statistical journals1 7 or in medical journals.8"21 Topics discussed in these articles come from a variety of fields including epidemiology, health services research, economics and social sciences.

* Correspondence to: Ralph B. D'Agostino, Jr, Department of Public Health Sciences, Section on Biostatistics, Wake Forest University School of Medicine, Medical Center Boulevard, Winston-Salem, NC 27157-1063, U.S.A. E-mail: [email protected]

Tutorials in Biostatistics Volume 1: Statistical Methods in Clinical Studies Edited by R. B. D'Agostino © 2004 John Wiley & Sons, Ltd. ISBN: 0-470-02365-1

In a randomized experiment, the randomization of units (that is, subjects) to different treatments guarantees that on average there should be no systematic differences in observed or unobserved covariates (that is, bias) between units assigned to the different treatments. However, in a non-randomized observational study, investigators have no control over the treatment assignment, and therefore direct comparisons of outcomes from the treatment groups may be misleading. This difficulty may be partially avoided if information on measured covariates is incorporated into the study design (for example, through matched sampling) or into estimation of the treatment effect (for example, through stratification or covariance adjustment). Traditional methods of adjustment (matching, stratification and covariance adjustment) are often limited since they can only use a limited number of covariates for adjustment. However, propensity scores, which provide a scalar summary of the covariate information, do not have this limitation.

Formally, the propensity score22 for an individual is the probability of being treated conditional on (or based only on) the individual's covariate values. Intuitively, the propensity score is a measure of the likelihood that a person would have been treated using only their covariate scores. Rosenbaum and Rubin22 showed that the propensity score is a balancing score and can be used in observational studies to reduce bias through the adjustment methods mentioned above.

The three goals of this tutorial are: to present the formal definition of propensity scores with some theoretical findings; to illustrate common uses of the propensity score; and to present applied examples that illustrate applications of the propensity score. The Appendix includes SAS code used to perform some of the analyses presented. The tutorial will conclude with a discussion about areas of current and future research.

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