Propensity Scores. Propensity scores are balancing scores that are most often used to control for various types of bias in observational studies, including but not limited to selection bias and confounding. They also can be used to test for ignorable treatment assignment, an important assumption of Xxxxx’x Causal Model. The first methods for propensity scores were developed by Xxxxxxxxx and Xxxxx (1983), but the generation of propensity scores and their application to various types of data have been a key area of interest for Causal Inference research (Xxxxxxxxx & Xxxxx 1983). The general idea of any balancing score is conditionally remove any inherent differences between groups. The balancing score b(X) is a function of the observed covariates X such that X is independent of treatment Z conditional on b(X). Xxxxxxxxx and Xxxxx present five theorems to support the use of propensity scores and other balancing scores, summarized below.
1. The propensity score e(X) is a balancing score.
2. Any score finer than the propensity score, such as b(X) = X, is also a balancing score.
3. If treatment assignment is strongly ignorable given X, then it is also strongly ignorable given a balancing score b(X).
4. At any value of a balancing score, comparison of means of an outcome in treated and untreated groups is an average treatment effect, if strongly ignorable treatment is met. This also indicates that use of balancing scores for matching, subclassification, and covariate adjustment produces unbiased treatment effect estimates, so long as treatment assignment is strongly ignorable.
5. Sample estimates of balancing scores produces sample balance on X. While there are many advantages to using propensity scores, the major disad- vantage to these methods is that bias is that one must assume that there are no unmeasured confounders of the treatment effect. In other words, propen- sity score methods can only account for confounding by covariates that are observed. This assumption is quite strong, especially in data that is poor in covariate measurements.
1. In addition to being used in the manners that Xxxxxxxxx and Xxxxx described (matching, subclassification or stratification, and covariate adjustment), the traditional propensity scores may also be used for inverse probability of treatment weighting in the likelihood for the outcome variable Y . eXβ
Propensity Scores. The theoretical groundwork for propensity scores was laid by Xxxxxxxxx and Xxxxx in a series of papers [1983a, 1983b, 1984, 1984]. The method has been used in a variety of fields over the past two decades [Imai and xxx Xxx, 0000, Xxxxxxxxxxx et al., 2002], with a growing body of literature expanding on these initial applications and analyzing the performance of propensity score analyses under a variety of circumstances. However, there currently still lacks a consensus regarding whether and how the estimated treatment effect size differs between propensity score and traditional adjustment methods, particularly when the confounders of interest are dichotomous. Robins et al [1992b] generalized propensity scores from the case of two groups (treatment and control or exposed and unexposed) to continuous, ordinal, or discrete treatments or exposures. Drake [1993] conducted simulations to compare different model specifications for the propensity model to traditional linear regression adjustment (with two normally dis- tributed covariates), Dehejia and Xxxxx [1999] conducted a sensitivity analysis of propen- sity score performance under varying model specifications and variable selections, Xxxxxx et al [2003] and Austin et al [2007] compared propensity score analyses to traditional logistic regression, and Kang and Xxxxxxx compared the performance of traditional and propensity score adjustment methods to doubly robust methods [2007]. Recent papers outside the statistics literature have compared traditional and propensity score methods in specific case studies [Austin and Mamdani, 2006, Posner et al., 2001], examined potentially biased results when estimating hazard and odds ratios using propen- sity score methods [Austin et al., 2007], compared different propensity score methods to each other [Xxxxx et al., 2006], and literature reviews have summarized recent usage of traditional versus propensity score methods [Shah et al., 2005, Sturmer et al., 2006] and summarized the use of propensity score methods in specific fields, providing some basic guidelines for their implementation [Xxxxx et al., 2006]. The literature reviews by Shah and Sturmer focused on publications including both propen- sity score and traditional methods, and compared whether or not a significant effect was detected with each method. Between these two reviews, more than 200 publications were summarized. Both reviews found few differences between traditional and propensity score methods and claim t...
