Can Nonrandomized Experiments Yield Accurate Answers? 

 Here is some latest progress (at least to me) on causal inference. William R. Shadish, M. H. Clark, and Peter M. Steiner published a paper on  JASA (December 1, 2008, 103(484): 1334-1344.)  based on "a randomized experiment comparing random and nonrandom assignments".  Basically "In the randomized experiment, participants were randomly assigned to mathematics or vocabulary training; in the nonrandomized experiment, participants chose their training." As the authors acknowledged, unsurprisingly, the randomized and nonrandomized experiments provided different estimates of the training effects, very likely through the selection bias caused by math phobia. The key finding is that statistical adjustment including propensity score stratification, weighting, and covariance adjustment can reduce estimation bias by about 58-96%. 

  Here is a link  to the PPT of the paper.  The comments  on the paper are also very insightful.