  Propensity score methods have become a part of the standard toolkit for
applied researchers who wish to ascertain causal effects from observational
data. While they were originally developed for binary treatments, several
researchers have proposed generalizations of the propensity score methodology
for non-binary treatment regimes. Such extensions have widened the
applicability of propensity score methods and are indeed becoming increasingly
popular themselves. In this article, we closely examine the two main
generalizations of propensity score methods, namely, the propensity function
(P-FUNCTION) of Imai and van Dyk (2004) and the generalized propensity score
(GPS) of Hirano and Imbens (2004), along with recent extensions of the GPS that
aim to improve its robustness. We compare the assumptions, theoretical
properties, and empirical performance of these alternative methodologies. On a
theoretical level, the GPS and its extensions are advantageous in that they can
be used to estimate the full dose response function rather than the simple
average treatment effect that is typically estimated with the P-FUNCTION.
Unfortunately, our analysis shows that in practice response models often used
with the original GPS are less flexible than those typically used with
propensity score methods and are prone to misspecification. We compare new and
existing methods that improve the robustness of the GPS and propose methods
that use the P-FUNCTION to estimate the dose response function. We illustrate
our findings and proposals through simulation studies, including one based on
an empirical application.
