Valid adjustment of randomized comparisons for binary covariates

In randomized trials, the treatment influences not only endpoints but also other variables measured after randomization which, when used as covariates to adjust for the observed imbalance, become pseudo-covariates. There is a logical circularity in adjusting for a pseudo-covariate because the variability in the endpoint that is attributed not to the treatment but rather to the pseudo-covariate may actually represent an effect of the treatment modulated by the pseudo-covariate. This potential bias is well known, but we offer new insight into how it can lead to reversals in the direction of the apparent treatment effect by way of stage migration. We then discuss a related problem that is not generally appreciated, specifically how the absence of allocation concealment can lead to this reversal of the direction of the apparent treatment effect even when adjustment is for a true covariate measured prior to randomization.