Preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation

Abstract

The demand for unequal allocation in clinical trials is growing. Most commonly, the unequal allocation is achieved through permuted block randomization. However, other allocation procedures might be required to better approximate the allocation ratio in small samples, reduce the selection bias in open-label studies, or balance on baseline covariates. When these allocation procedures are generalized to unequal allocation, special care is to be taken to preserve the allocation ratio at every allocation step. This paper offers a way to expand the biased coin randomization to unequal allocation that preserves the allocation ratio at every allocation. The suggested expansion works with biased coin randomization that balances only on treatment group totals and with covariate-adaptive procedures that use a random biased coin element at every allocation. Balancing properties of the allocation ratio preserving biased coin randomization and minimization are described through simulations. It is demonstrated that these procedures are asymptotically protected against the shift in the rerandomization distribution identified for some examples of minimization with 1:2 allocation. The asymptotic shift in the rerandomization distribution of the difference in treatment means for an arbitrary unequal allocation procedure is explicitly derived in the paper.

Mass weighted urn design — A new randomization algorithm for unequal allocations

“…A consistent unconditional allocation probability for each treatment assignment in a randomization sequence is a good feature for a randomization design [13,14,19]. To exam this feature for MWUD, consider a two-arm trial with target allocation w 1 : w 2 .…”

“…Equal probability sequence is important for designed-based randomization tests in which the type I error is assessed based on the permutation of all permissible sequences under the randomization design. Proposed by Kuznetsova and Tymofyeyev, BTR ensures that the unconditional allocation probability for each treatment assignment in the randomization sequence equals to the target allocation ratio [13,19]. Preserving the unconditional allocation ratio helps selection bias prevention.…”

Generalization of Wei's urn design to unequal allocations in sequential clinical trials

“…Let u ij be the unconditional allocation probability of assigning the i th subject to treatment j prior to the start of the trial. To prevent selection bias, it is desired that the unconditional allocation probability equals the target allocation probability for each treatment assignment [7, 10]. …”

“…Although it is desirable for a randomization design to have an unconditional allocation probability that equals the target allocation probability at each treatment assignment, not all randomization designs have this property [8, 10]. However, it is necessary for all randomization designs to have an unconditional allocation probability that converges to the target allocation asymptotically.…”

Considerations in the paper by Proschan, Brittain, and Kammerman are not an argument against minimization. In response to Vance W Berger ‘Minimization: not all it’s cracked up to be’, Clin Trials 2011; 8: 443

“…However, as shown in Ref. [3], the shift in unconditional randomization distribution is not peculiar to minimization. It is common to all unequal allocation procedures for which the allocation ratio varies from allocation to allocation -as was the case in the examples in Ref.…”