Causal Inference with General Treatment Regimes: Generalizing the Propensity Score
Imai, Kosuke, van Dyk, David A., Journal of the American Statistical Association
Establishing the effect of a treatment that is not randomly assigned is a common goal in empirical research. But the lack of random assignment means that groups with different levels of the treatment variable can systematically differ in important ways other than the observed treatment. Because these differences may exhibit complex correlations with the outcome variable, ascertaining the causal effect of the treatment may be difficult. It is in this setting that the propensity score of Rosenbaum and Rubin (1983b) has found wide applicability in empirical research; in particular, the method has rapidly become popular in the social sciences (e.g., Heckman, Ichimura, and Todd 1998; Lechner 1999; Imai 2004).
The propensity score aims to control for differences between the treatment groups when the treatment is binary; it is defined as the conditional probability of assignment to the treatment group given a set of observed pretreatment variables. Under the assumption of strongly ignorable treatment assignment, multivariate adjustment methods based on the propensity score have the desirable property of effectively reducing the bias that frequently arises in observational studies. In fact, there exists empirical evidence that in certain situations the propensity score method produces more reliable estimates of causal effects than other estimation methods (e.g., Dehejia and Wahba 1999; Imai 2004).
The propensity score is called a balancing score because, conditional on the propensity score, the binary treatment assignment and the observed covariates are independent (Rosenbaum and Rubin 1983b). If we further assume the conditional independence between treatment assignment and potential outcomes given the observed covariates, then it is possible to obtain unbiased estimates of treatment effects. In practice, matching or subclassification is used to adjust for the estimated propensity score, which is ordinarily generated by logistic regression (Rosenbaum and Rubin 1984, 1985). The advantage of using estimated propensity scores in place of true propensity scores has been discussed at length in the literature (e.g., Rosenbaum 1987; Robins, Rotnitzky, and Zhao 1995; Rubin and Thomas 1996; Heckmen et al. 1998; Hirano, Imbens, and Ridder 2003); see also Section 5.3. Indeed, even in randomized experiments where the randomization scheme specifies the true propensity score, adjusting for the estimated propensity score can reduce the variance of the estimated treatment effect. One of the principle advantages of this method is that adjusting for the propensity score amounts to matching or subclassifying on a scalar, which is significantly easier than matching or subclassifying on many covariates.
In this article we extend and generalize the propensity score method so that it can be applied to arbitrary treatment regimes. The original propensity score was developed to estimate the causal effects of a binary treatment; however, in many observational studies, the treatment may not be binary or even categorical. For example, in clinical trials, one may be interested in estimating the dose-response function where the drug dose may take on a continuum of values (e.g., Efron and Feldman 1991). Alternatively, the treatment may be ordinal. In economics, an important quantity of interest is the effect of schooling on wages, where schooling is measured as years of education in school (e.g., Card 1995). The treatment can also consist of multiple factors and their interactions. In political science, one may be interested in the combined effects of different voter mobilization strategies, such as phone calls and door-to-door visits (e.g., Gerber and Green 2000). Treatment can also be measured in terms of frequency and duration, for example, the health effects of smoking. These examples illustrate the need to extend the propensity score, a prominent methodology of causal inference, for application to general treatment regimes. …