Imbalance in treatment assignments in stratified blocked randomization

Blocking and stratification are used in preparing randomization assignments to ensure that there will be nearly equal numbers of patients in each treatment group and that the groups will be similar with respect to important covariates. Stratified blocked randomization will create near balance within strata, but imbalance for the total trial may still occur. The variance for the total trial imbalance D is derived and examples from clinical trials are given. Under reasonable assumptions, if the blocking factor is size B in each of K strata, then max D = KB/2 and var D = K(B + 1)/6. These results may be used in planning a trial to estimate the overall imbalance expected for various choices of B and K. A conditional variance is given that allows the probability of an observed imbalance at the completion of a trial to be evaluated. Overall imbalance is about as likely with stratified blocked randomization as with simple randomization unless the total sample size N is appreciably larger than K X B. So long as the blinding is maintained, the block sizes should be chosen to be as small as possible.