Propensity scores and M-structures

In a recent issue of Statistics in Medicine, Ian Shrier [Statist. Med. 2008; 27(14):2740-2741] posed a question regarding the use of propensity scores [Biometrika 1983; 70(1):41-55]. He considered an 'M-structure' illustrated by the directed acyclic graph (DAG) in Figure 1. In Figure 1, z is a binary exposure, r is a response of interest, x is a measured covariate, and u(1) and u(2) are two unmeasured covariates. Shrier stated that for the M-structure, '... it remains unclear if the propensity method described by Rubin would introduce selection bias or not'. In the same issue, Donald Rubin [Statist. Med. 2002; 27(14):2741-2742] replied by clarifying several key points in the use of propensity scores. He did not, however, discuss the original question posed by Shrier. Given the popularity of both propensity score methods and graphical models, I think any confusion regarding the appropriateness of these methods deserves serious attention and I would therefore like to answer Shrier's question here. The short answer is that for the M-structure, propensity score methods do indeed induce a bias. Below, I will clarify this statement. I will first briefly review the basic idea of propensity scores and then explain why the idea does not apply to the M-structure. I will use a notation which is consistent with Rosenbaum and Rubin [Biometrika 1983; 70(1):41-55].