Brick tunnel randomization for unequal allocation to two or more treatment groups

Abstract

Studies with unequal allocation to two or more treatment groups often require a large block size for permuted block allocation. This could present a problem in small studies, multi-center studies, or adaptive design dose-finding studies. In this paper, an allocation procedure, which generalizes the maximal procedure by Berger, Ivanova, and Knoll to the case of K≥2 treatment groups and any allocation ratio, is offered. Brick tunnel (BT) randomization requires the allocation path drawn in the k-dimensional space to stay close to the allocation ray that corresponds to the targeted allocation ratio. Specifically, it requires the allocation path to be confined to the set of the k-dimensional unitary cubes that are pierced by the allocation ray (the 'brick tunnel'). The important property of the BT randomization is that the transition probabilities at each node within the tunnel are defined in such a way that the unconditional allocation ratio is the same for every allocation step. This property is not necessarily met by other allocation procedures that implement unequal allocation.

Adaptive Optimal Designs for Dose-Finding Studies with Time-to-Event Outcomes

“…Treatment allocation ratio for a given cohort of subjects is frequently different from equal allocation. To implement an unequal allocation in practice, one can use randomization procedures with established statistical properties such as brick tunnel randomization (36) or wide brick tunnel randomization (37). These procedures preserve the allocation ratio at each step and lead to valid statistical inference at the end of the trial.…”

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

“…Currently available unequal allocation randomization designs include the complete randomization (CP), the permuted block randomization (PBR), the modified Wei's urn design (mUD) described by Rosenberger and Lachin [1], the maximal procedure (MP) proposed by Berger et al [11,12], the brick tunnel randomization (BTR) and its modified version, the wide brick tunnel randomization (WBT) propose by Kuznetsova and Tymofyeyev [13,14], and the block urn design (BUD) proposed by Zhao and Weng [15]. Consider a target allocation r 1 : r 2 :···: r m , where r j ( j =1,2,···, m ) are integers without common divisor greater than 1.…”

“…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.…”

“…Consider a trial with a target allocation 2:3 between arms A and B, using MP with MTI = 2, there are 8 possible sequences for the first 5 assignments: ABABB, ABBAB, ABBBA, BAABB, BABAB, BABBA, BBAAB, and BBABA. Giving each sequence the same probability of 1/8, the unconditional allocation ratio is 1:1 for the third subject and 3:5 for the other 4 subjects; none of them equals to the target allocation ratio 2:3 [13]. On the other hand, if BTR is applied to preserve the unconditional allocation ratio, sequences 1, 4, 7 and 8 have 1/10 chance being used, and the other 4 sequences have 3/ 20 chance being used [13], none of them equals to 1/ 8.…”

“…Giving each sequence the same probability of 1/8, the unconditional allocation ratio is 1:1 for the third subject and 3:5 for the other 4 subjects; none of them equals to the target allocation ratio 2:3 [13]. On the other hand, if BTR is applied to preserve the unconditional allocation ratio, sequences 1, 4, 7 and 8 have 1/10 chance being used, and the other 4 sequences have 3/ 20 chance being used [13], none of them equals to 1/ 8. What is common to MP and BTR is that both do not offer an explicit formula for the calculation of conditional allocation probabilities.…”