Is listwise deletion / complete case analysis biased if data are not missing completely at random? In the comments to the answer to my question I stated "Many rows have only 1 missing variable, so to exclude the row think leads to bias (they are not MCAR)" and in reply I was told "You're wrong, see Rubin's Statistical Analysis with Missing Data 2nd ed. CC is unbiased with MAR data,"
I don't have Rubin and Little's book but I had been able to borrow it a few months ago and I am sure I learned that complete case analysis is biased unless the data are missing completely at random. 
Now I am terribly confused.
Can anyone explain / clarify my misunderstanding ?
 A: You are not wrong.
From: 
Statistical Analysis with Missing Data,
Second Edition,
Roderick J.A. Little & Donald B Rubin,
John Wiley and Sons, 2002. p41:

Complete Case analysis confines attention to cases where all  variables are present. Advantages of this approach are .... . Disadvantages stem from the potential loss of information in discarding incomplete cases. This loss of information has two aspects: loss of precision, and bias when the missing-data mechanism is not MCAR, and the complete cases are not a random sample of all the cases".

From: Bias and efficiency of multiple imputation compared with complete-case analysis for
missing covariate values. Ian R.White and John B. Carlin. Statistics in Medicine, Volume 29, Issue 28, 2010

In particular, while MI has negligible bias and CC is biased under MAR mechanisms, there are other mechanisms under which CC has negligible bias and MI is biased. This point is widely misunderstood, but it has important implications.
  http://onlinelibrary.wiley.com/doi/10.1002/sim.3944/pdf

A: In general, complete case analysis is biased when data are not MCAR. However, when the analysis consists of fitting a regression model, complete case analysis is unbiased under the weaker condition that missingness is independent of the outcome variable, conditional on the covariates. Depending on which variable(s) contain missing values, this condition sometimes corresponds to MAR mechanisms, and sometimes to MNAR mechanisms.
For example, suppose the outcome Y is the variable with missing values. Then missingness being independent of outcome conditional on covariates corresponds to the MAR assumption, which says the probability of missingness is independent of the partially observed variable conditional on the fully observed variables. Alternatively, suppose that missingness in a covariate X depends on the value of that covariate, so that data are MNAR. Provided missingness in X is independent of Y, conditional on X and other covariates in the analysis model, complete case analysis is unbiased.
For more on this, see:
1) the paper cited in a previous answer by White and Carlin: Bias and efficiency of multiple imputation compared with complete-case analysis for missing covariate values. Ian R.White and John B. Carlin. Statistics in Medicine, Volume 29, Issue 28, 2010
2) a paper by myself and colleagues published in Biostatistics: http://doi.org/10.1093/biostatistics/kxu023
3) a blog post I previously wrote about this here: http://thestatsgeek.com/2013/07/06/when-is-complete-case-analysis-unbiased/
