On Associative Confounder Bias

Conditioning on some set of confounders that causally affect both treatmentand outcome variables can be sufficient for eliminating bias introduced by allsuch confounders when estimating causal effect of the treatment on the outcomefrom observational data. It is done by including them in propensity score modelin so-called potential outcome framework for causal inference whereas in causalgraphical modeling framework usual conditioning on them is done. However inthe former framework, it is confusing when modeler finds a variable that is noncausallyassociated with both the treatment and the outcome. Some argue that suchvariables should also be included in the analysis for removing bias. But others arguethat they introduce no bias so they should be excluded and conditioning onthem introduces spurious dependence between the treatment and the outcome, thusresulting extra bias in the estimation. We show that there may be errors in boththe arguments in different contexts. When such a variable is found neither of theactions may give the correct causal effect estimate. Selecting one action over theother is needed in order to be less wrong.We discuss how to select the better action