185-2007: Local and Global Optimal Propensity Score Matching

Propensity score-matching methods are often used to control for bias in observational studies when randomization is not possible. This paper describes how to match samples using both local and global optimal matching algorithms. The paper includes macros to perform the nearest available neighbor, caliper, and radius matching methods with or without replacement and matching treated observations to one or many controls. The similarity between observations is evaluated using both the absolute value and the Mahalanobis distance that includes the propensity score along with other covariates. This paper also explains how to find a global optimal match with a variable number of controls using network flows. SAS® 9.1, SAS/STAT®, and SAS/OR® are required. INTRODUCTION The objective of randomization in statistics is to obtain groups that are comparable in terms of both observed and unobserved characteristics. When randomization is not possible, causal inference is complicated by the fact that a group that received a treatment or experienced an event maybe very different from another group that did not experience the event or receive the treatment. Thus, it is not clear whether a difference in certain outcome of interest is due to the treatment or is the product of prior differences among groups. One way of overcoming this problem is to adjust the estimates of treatment effect using the measured characteristics of each group as covariates in a model. Another way of establishing causality that has gained popularity in recent years is to select groups that are similar in terms of observed characteristics before making a comparison, which may still involve some type of model adjustment. Propensity score methods were developed to facilitate the creation of comparison groups that are similar. “Similar” in this sense refers to the distribution of observed characteristics. Propensity score methods were created to be used in the design stage of an observational study, sometimes before the outcome information has been collected or not using the outcome information when it is available (Rubin 2007). The first step involves estimating the likelihood (the propensity score) that a person would have received the treatment given certain characteristics. More formally, the propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates (D’Agostino 1998). If a treated person and a potential control unit have the same propensity score, then both units have the same distribution of the covariates that entered into the estimation of the propensity scores. Intuitively, comparing matched treated and untreated individuals with the same observable characteristics is like comparing the individuals in a randomized experiment. In prospective randomization, however, both observed and unobserved characteristics are guaranteed to be balanced. Two key assumptions of propensity scores are that both the outcome of interest and the treatment assignment do not depend on unobservable characteristics. For example, if the treatment effect depended on the presence of a gene, matching individuals on a multitude of demographic characteristics could fail to create treatment and control groups with a similar proportion of the relevant gene. In the same way, if the treatment was only given to persons who volunteered because of the severity of their symptoms, then matching without explicitly taking into account the severity of the symptoms may fail to produce similar groups (Coca-Perraillon 2006). After estimating the propensity scores, the scores are used to group observations that are close to each other. One way of accomplishing this is to classify treated and untreated observations into subgroups and then separately compare the outcome for each subgroup. This method is usually referred as subclassification on the propensity scores (Rosenbaum and Rubin 1984). The other way is to match one treated unit to one or more untreated controls, which is usually referred as matching on the propensity score (Rosenbaum and Rubin 1983). This paper focuses on matching on the propensity score. Key in the implementation of matching using propensity scores is to decide what metric to use when evaluating the distance between scores (usually the absolute value or the Mahalanobis metric) and what type of algorithm to implement (local or global optimal). The first part of this paper describes the most commonly used methods and introduces a SAS macro to implement them. The second part explains how to adapt the code to implement the Mahalanobis metric matching. Finally, the last part describes global optimum matching and provides code to implement it using SAS. This paper assumes that the propensity scores have been already estimated. See D’Agostino and Rubin (2000) and Rubin (2004) for an introduction on how to estimate the propensity scores. 1 See Gu and Rosenbaum (1993) and Smith and Todd (2005) for detailed descriptions of most of the matching methods currently available. SAS Global Forum 2007 Statistics and Data Analysis