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Propensity score methods for bias reduction in the comparison of a treatment & 8230; Med . 17,2265 — 2281 (1998) TUTORIAL IN BIOSTATISTICS PROPENSITY SCORE METHODS FOR BIAS REDUCTION IN THE & 8230; Propensity score methods for bias reduction in the comparison of a treatment to a non-randomized control group* Correspondence to: Ralph B. D& 8217;Agostino, Jr, Department of Public Health Sciences, Section on Biostatistics, Wake ForestUniversity School of Medicine, Medical left Boulevard, Winston-Salem, NC 27157-1063, U.S.A. E-mail: rdagosti@ & 8230;

In observational studies, investigators have no control over the treatment assignment. The treated and non-treated (that is, control) groups may have largedi?erencesontheir observedcovariates, and these di?erencescanleadto biased estimates of treatmente?ects. Even traditional covariance analysis adjust- mentsmaybe inadequate to eliminate this bias. The propensity score, defined as the conditional probability of being treated given thecovariates, can be used to balance the covariatesin the two groups, and therefore reduce this bias. In order to estimate the propensity score, one must model the distribution of the treatment indicator variable given the observedcovariates. Once estimated the propensity score can be used to reduce bias through matching, stratification (subclassification), regression adjustment, or some combination of all three. In this tutorial we discuss the uses of propensity score methods for bias reduction, give references to the literature and illustrate the uses through applied examples. (1998 John Wiley&Sons, Ltd. INTRODUCTION Observational studies occur frequently in medical research. In these studies, investigators have no control over the treatment assignment. Therefore, largedi?erenceson observedcovariates in the two groups may exist, and thesedi?erencescouldlead to biased estimates of treatmente?ects. The propensity score for an individual, defined as the conditional probability of being treated given the individual& 8217;s covariates, can be used to balance the covariatesin 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 journals 1—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& 8230;.. DEFINITION With complete data, Rosenbaumand Rubin 22 introduced the propensity score for subject i ( i & 8220;1, 2 , N ) as the conditional probability of assignment to a particular treatment ( Z i & 8220;1) versus control ( Z i & 8220;0) given a vector of observedcovariates, x i : e ( x i ) & 8220;pr( Z i & 8220;1D X i & 8221; x i ) where it is assumed that, given the X & 8217;s, the Z i are independent: pr( Z 1 & 8221; z 1 , 2 , Z & 8221; z N D X 1 & 8221; x 1 , 2 , X N & 8221; x N ) & 8221; N < i /1 e ( x i ) z i M1! e ( x i ) N1~ z i . The propensity score is the& 8217;coarsest function& 8217;of thecovariates that is a balancing score, where a balancing score, b ( X ), is defined as & 8216;afunctionofthe observedcovariates X such that the conditional distribution of X given b ( X ) is the same for treated ( Z & 8220;1) and control ( Z & 8220;0) units& 8217;. 22 Fora specific value of the propensity score, thedi?erencebetweenthe treatment and control means for all units with that value of the propensity score is an unbiased estimate of the average treatmente?ectatthat propensity score, if the treatment assignment is strongly ignorable, given thecovariates. Thus, matching, stratification, or regression (covariance) adjustment on the propensity score tends to produce unbiased estimates of the treatmente?ectswhen treatment assignment is strongly ignorable. Treatment assignment is considered strongly ignorable if the treatment assignment, Z , and the response, ‰, are known to be conditionally independent given thecovariates, X (that is, when Yo ZDX).

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