Omitted Variables, Countervailing Effects, and the Possibility of Overadjustment

The effect of conditioning on an additional covariate on confounding bias depends, in part, on covariates that are unobserved. We characterize the conditions under which the interaction between a covariate that is available for conditioning and one that is not can affect bias. When the confounding effects of two covariates, one of which is observed, are countervailing (in opposite directions), conditioning on the observed covariate can increase bias. We demonstrate this possibility analytically, and then show that these conditions are not rare in actual data. We also consider whether balance tests or sensitivity analysis can be used to justify the inclusion of an additional covariate. Our results indicate that neither provide protection against overadjustment.