Causal inference with general treatment regimes: Generalizing the propensity score

In this article we develop the theoretical properties of the propensity function, which is a generalization of the propensity score of Rosenbaum and Rubin. Methods based on the propensity score have long been used for causal inference in observational studies; they are easy to use and can effectively reduce the bias caused by nonrandom treatment assignment. Although treatment regimes need not be binary in practice, the propensity score methods are generally confined to binary treatment scenarios. Two possible exceptions have been suggested for ordinal and categorical treatments. In this article we develop theory and methods that encompass all of these techniques and widen their applicability by allowing for arbitrary treatment regimes. We illustrate our propensity function methods by applying them to two datasets; we estimate the effect of smoking on medical expenditure and the effect of schooling on wages. We also conduct simulation studies to investigate the performance of our methods.