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1.  A note on overadjustment in inverse probability weighted estimation 
Biometrika  2010;97(4):997-1001.
Summary
Standardized means, commonly used in observational studies in epidemiology to adjust for potential confounders, are equal to inverse probability weighted means with inverse weights equal to the empirical propensity scores. More refined standardization corresponds with empirical propensity scores computed under more flexible models. Unnecessary standardization induces efficiency loss. However, according to the theory of inverse probability weighted estimation, propensity scores estimated under more flexible models induce improvement in the precision of inverse probability weighted means. This apparent contradiction is clarified by explicitly stating the assumptions under which the improvement in precision is attained.
doi:10.1093/biomet/asq049
PMCID: PMC3371719  PMID: 22822256
Causal inference; Propensity score; Standardized mean

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