Principal Score Methods: Assumptions and Extensions
  Researchers addressing post-treatment complications in randomized trials often turn to principal stratification to define relevant assumptions and quantities of interest. One approach for estimating causal effects in this framework is to use methods based on the "principal score," typically assuming that stratum membership is as-good-as-randomly assigned given a set of covariates. In this paper, we clarify the key assumption in this context, known as Principal Ignorability, and argue that versions of this assumption are quite strong in practice. We describe different estimation approaches and demonstrate that weighting-based methods are generally preferable to subgroup-based approaches that discretize the principal score. We then extend these ideas to the case of two-sided noncompliance and propose a natural framework for combining Principal Ignorability with exclusion restrictions and other assumptions. Finally, we apply these ideas to the Head Start Impact Study, a large-scale randomized evaluation of the Head Start program. Overall, we argue that, while principal score methods are useful tools, applied researchers should fully understand the relevant assumptions when using them in practice. 
Avi Feller and Fabrizia Mealli and Luke Miratrix
stat.ME