Quantifying the magnitude of baseline covariate imbalances resulting from selection bias in randomized clinical trials

Selection bias is most common in observational studies, when patients select their own treatments or treatments are assigned based on patient characteristics, such as disease severity. This first-order selection bias, as we call it, is eliminated by randomization, but there is residual selection bias that may occur even in randomized trials which occurs when, subconsciously or otherwise, an investigator uses advance knowledge of upcoming treatment allocations as the basis for deciding whom to enroll. For example, patients more likely to respond may be preferentially enrolled when the active treatment is due to be allocated, and patients less likely to respond may be enrolled when the control group is due to be allocated. If the upcoming allocations can be observed in their entirety, then we will call the resulting selection bias second-order selection bias. Allocation concealment minimizes the ability to observe upcoming allocations, yet upcoming allocations may still be predicted (imperfectly), or even determined with certainty, if at least some of the previous allocations are known, and if restrictions (such as randomized blocks) were placed on the randomization. This mechanism, based on prediction but not observation of upcoming allocations, is the third-order selection bias that is controlled by perfectly successful masking, but without perfect masking is not controlled even by the combination of advance randomization and allocation concealment. Our purpose is to quantify the magnitude of baseline imbalance that can result from third-order selection bias when the randomized block procedure is used. The smaller the block sizes, the more accurately one can predict future treatment assignments in the same block as known previous assignments, so this magnitude will depend on the block size, as well as on the level of certainty about upcoming allocations required to bias the patient selection. We find that a binary covariate can, on average, be up to 50% unbalanced by third-order selection bias.