The propensity score was defined by Rosenbaum and Rubin (1983a) to be the probability of treatment assignment conditional on observed baseline covariates: ei = Pr(Zi = 1|Xi). The propensity score is a balancing score: conditional on the propensity score, the distribution of measured baseline covariates is similar between treated and untreated subjects. Thus, in a set of subjects all of whom have the same propensity score, the distribution of observed baseline covariates will be the same between the treated and untreated subjects.
The propensity score exists in both randomized experiments and in observational studies. In randomized experiments the true propensity score is known and is defined by the study design. In observational studies, the true propensity score is not, in general, known. However, it can be estimated using the study data. In practice, the propensity score is most often estimated using a logistic regression model, in which treatment status is regressed on observed baseline characteristics. The estimated propensity score is the predicted probability of treatment derived from the fitted regression model. Although logistic regression appears to be the most commonly used method for estimating the propensity score, the use of bagging or boosting (Lee, Lessler, & Stuart, 2010; (link) McCaffrey, Ridgeway, & Morral, 2004 (link)), recursive partitioning or tree-based methods (Lee et al., 2010 (link); Setoguchi, Schneeweiss, Brookhart, Glynn, & Cook, 2008 (link)), random forests (Lee et al., 2010 (link)), and neural networks (Setoguchi et al., 2008 (link)) for estimating the propensity score have been examined.
Four different propensity score methods are used for removing the effects of confounding when estimating the effects of treatment on outcomes: propensity score matching, stratification (or subclassification) on the propensity score, inverse probability of treatment weighting (IPTW) using the propensity score, and covariate adjustment using the propensity score (Austin & Mamdani, 2006 (link); Rosenbaum, 1987a ; Rosenbaum & Rubin, 1983a ). I describe each of these methods separately in the following subsections.
Rosenbaum and Rubin (1983a) defined treatment assignment to be strongly ignorable if the following two conditions hold: (a) (Y (1), Y(0)) ╩ Z|X and (b) 0 < P(Z = 1|X) < 1. The first condition says that treatment assignment is independent of the potential outcomes conditional on the observed baseline covariates. The second condition says that every subject has a nonzero probability to receive either treatment. They demonstrated that if treatment assignment is strongly ignorable, conditioning on the propensity score allows one to obtain unbiased estimates of average treatment effects. The aforementioned first condition is also referred to as the “no unmeasured confounders” assumption: the assumption that all variables that affect treatment assignment and outcome have been measured. Because this is the crucial assumption that underlies propensity score analyses, Rosenbaum and Rubin (1983b) proposed analyses to assess the sensitivity of study conclusions to the assumption that there were no unmeasured confounders that influenced treatment assignment. Furthermore, Rosenbaum (1987b) proposed the use of a second control group to examine the plausibility that adjustment for measured covariates has eliminated bias in estimating treatment effects. It should be noted that although the assumption of strongly ignorable treatment assignment/no unmeasured confounding is explicitly stated in the context of propensity score analyses, this assumption also underlies regression-based approaches for estimating treatment effects in observational studies.
The propensity score exists in both randomized experiments and in observational studies. In randomized experiments the true propensity score is known and is defined by the study design. In observational studies, the true propensity score is not, in general, known. However, it can be estimated using the study data. In practice, the propensity score is most often estimated using a logistic regression model, in which treatment status is regressed on observed baseline characteristics. The estimated propensity score is the predicted probability of treatment derived from the fitted regression model. Although logistic regression appears to be the most commonly used method for estimating the propensity score, the use of bagging or boosting (Lee, Lessler, & Stuart, 2010; (link) McCaffrey, Ridgeway, & Morral, 2004 (link)), recursive partitioning or tree-based methods (Lee et al., 2010 (link); Setoguchi, Schneeweiss, Brookhart, Glynn, & Cook, 2008 (link)), random forests (Lee et al., 2010 (link)), and neural networks (Setoguchi et al., 2008 (link)) for estimating the propensity score have been examined.
Four different propensity score methods are used for removing the effects of confounding when estimating the effects of treatment on outcomes: propensity score matching, stratification (or subclassification) on the propensity score, inverse probability of treatment weighting (IPTW) using the propensity score, and covariate adjustment using the propensity score (Austin & Mamdani, 2006 (link); Rosenbaum, 1987a ; Rosenbaum & Rubin, 1983a ). I describe each of these methods separately in the following subsections.
Rosenbaum and Rubin (1983a) defined treatment assignment to be strongly ignorable if the following two conditions hold: (a) (Y (1), Y(0)) ╩ Z|X and (b) 0 < P(Z = 1|X) < 1. The first condition says that treatment assignment is independent of the potential outcomes conditional on the observed baseline covariates. The second condition says that every subject has a nonzero probability to receive either treatment. They demonstrated that if treatment assignment is strongly ignorable, conditioning on the propensity score allows one to obtain unbiased estimates of average treatment effects. The aforementioned first condition is also referred to as the “no unmeasured confounders” assumption: the assumption that all variables that affect treatment assignment and outcome have been measured. Because this is the crucial assumption that underlies propensity score analyses, Rosenbaum and Rubin (1983b) proposed analyses to assess the sensitivity of study conclusions to the assumption that there were no unmeasured confounders that influenced treatment assignment. Furthermore, Rosenbaum (1987b) proposed the use of a second control group to examine the plausibility that adjustment for measured covariates has eliminated bias in estimating treatment effects. It should be noted that although the assumption of strongly ignorable treatment assignment/no unmeasured confounding is explicitly stated in the context of propensity score analyses, this assumption also underlies regression-based approaches for estimating treatment effects in observational studies.