Brick tunnel randomization and the momentum of the probability mass

Abstract

The allocation space of an unequal-allocation permuted block randomization can be quite wide. The development of unequal-allocation procedures with a narrower allocation space, however, is complicated by the need to preserve the unconditional allocation ratio at every step (the allocation ratio preserving (ARP) property). When the allocation paths are depicted on the K-dimensional unitary grid, where allocation to the l-th treatment is represented by a step along the l-th axis, l = 1 to K, the ARP property can be expressed in terms of the center of the probability mass after i allocations. Specifically, for an ARP allocation procedure that randomizes subjects to K treatment groups in w1 :⋯:wK ratio, w1 +⋯+wK =1, the coordinates of the center of the mass are (w1 i,…,wK i). In this paper, the momentum with respect to the center of the probability mass (expected imbalance in treatment assignments) is used to compare ARP procedures in how closely they approximate the target allocation ratio. It is shown that the two-arm and three-arm brick tunnel randomizations (BTR) are the ARP allocation procedures with the tightest allocation space among all allocation procedures with the same allocation ratio; the two-arm BTR is the minimum-momentum two-arm ARP allocation procedure. Resident probabilities of two-arm and three-arm BTR are analytically derived from the coordinates of the center of the probability mass; the existence of the respective transition probabilities is proven. Probability of deterministic assignments with BTR is found generally acceptable. Copyright © 2015 John Wiley & Sons, Ltd.

A comparative study of restricted randomization procedures for multiarm trials with equal or unequal treatment allocation ratios

“…Since the achieved allocation vector ( N 1 ( n ), …, N K ( n )) is generally random, so is the value of imbalance. Kuznetsova suggested using the unconditional expected value of imbalance as a measure of closeness of the probability mass at a given allocation step to the point of perfect balance for respective n . For ARP procedures (cf Section 3.5) that have the center of the probability mass at the point of perfect balance for every n , this measure can be interpreted as the momentum of probability mass (MPM).…”

Approaches to expanding the two‐arm biased coin randomization to unequal allocation while preserving the unconditional allocation ratio

“…The center of the probability mass of the i ‐th generation is the point Cent i = ( cent i 1 , … , cent iK ), where . An allocation procedure is an ARP procedure if Cent i belongs to the AR for all i ≥ 1 , that is, c ent il = w l i . Thus, all ARP procedures with the same allocation ratio have the same sequence of the centers of the probability mass Cent i .…”

“…Luckily, the permuted block randomization (PBR) used in the overwhelming majority of clinical trials with equal or unequal allocation is an ARP procedure. Other examples of ARP unequal allocation procedures are the brick tunnel randomization (BTR) and the wide brick tunnel randomization by Kuznetsova and Tymofyeyev, the drop‐the‐loser urn design by Ivanova applied to fixed unequal allocation, and the urn block design by Zhao and Weng .…”

“…Variations in the unconditional allocation ratio present a serious problem. First, while perfect blinding eliminates the opportunity for selection bias in a study with an ARP randomization, variations in the unconditional allocation ratio provide potential for selection and evaluation bias even in double‐blind studies . Indeed, if the investigator knows that the probability of the active allocation is much higher than its target allocation ratio at the 2nd, 5th, 8th, and so on allocations, they can assign subjects with better prognosis at these places in the allocation sequence.…”

“…For two‐arm studies with unequal allocation, the BTR is the allocation procedure with the narrowest allocation space among all allocation procedures with the same allocation ratio . However, in open‐label single center studies, where selection bias might be a problem, allocation procedures with a wider allocation space might be preferred to reduce the selection bias.…”