Expansion of the modified Zelen's approach randomization and dynamic randomization with partial block supplies at the centers to unequal allocation

Modified Zelen's approach is a randomization technique useful in multi-center trials where balance in treatment assignments within a center is desired. It has great balancing properties in a study with equal allocation to several treatment arms. This technique can also be used in studies with unequal allocation, where it would provide an allocation ratio close to the targeted one within centers as well as across centers. However, the implementation of the modified Zelen's approach for unequal allocation involves more than just imposing a constraint on within-center imbalance in treatment assignments, as is the case with equal allocation. Failure to account for this might provide a potential for accidental bias and (even in a double-blind study) selection and evaluation bias. The naive expansion of the modified Zelen's approach to unequal allocation can also lead to a shift in the re-randomization distribution of the between-group difference in treatment means. This paper will explain why and will offer an easy way to expand the modified Zelen's approach to unequal allocation that allows one to avoid these issues. The expansion to unequal allocation of another very useful in multi-center trials technique where partial blocks of supplies are sent to the centers will be offered.