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

Abstract

Unequal allocations have been used in clinical trials motivated by ethical, efficiency, or feasibility concerns. Commonly used permuted block randomization faces a tradeoff between effective imbalance control with a small block size and accurate allocation target with a large block size. Few other unequal allocation randomization designs have been proposed in literature with applications in real trials hardly ever been reported, partly due to their complexity in implementation compared to the permuted block randomization. Proposed in this paper is the mass weighted urn design, in which the number of balls in the urn equals to the number of treatments, and remains unchanged during the study. The chance a ball being randomly selected is proportional to the mass of the ball. After each treatment assignment, a part of the mass of the selected ball is re-distributed to all balls based on the target allocation ratio. This design allows any desired optimal unequal allocations be accurately targeted without approximation, and provides a consistent imbalance control throughout the allocation sequence. The statistical properties of this new design is evaluated with the Euclidean distance between the observed treatment distribution and the desired treatment distribution as the treatment imbalance measure; and the Euclidean distance between the conditional allocation probability and the target allocation probability as the allocation predictability measure. Computer simulation results are presented comparing the mass weighted urn design with other randomization designs currently available for unequal allocations.

Guessing strategies for treatment prediction under restricted randomization with unequal allocation

“…In recent years, unequal allocation gains more applications in clinical trials [6], partially due to the emerging of Bayesian adaptive designs [7] and response adaptive randomization [8]. To evaluate the performance of different randomization designs under unequal allocations, the convergence guessing strategy has been extended from equal allocation to unequal allocation scenarios in recent publications [2][3][4][6][7][8][9][10][11][12]. In the book titled Selection bias and covariate imbalances in randomized clinical trials, Berger described two guessing strategies for unequal allocations under the names of convergent prediction and directional prediction with illustration examples [2].…”

“…In the book titled Selection bias and covariate imbalances in randomized clinical trials, Berger described two guessing strategies for unequal allocations under the names of convergent prediction and directional prediction with illustration examples [2]. Zhao et al used the so-called convergence guessing strategy for the treatment predictability assessment for unequal allocations when comparing different randomization schemes [4,[10][11][12]. However, explicit definitions for both the convergence guessing strategy and the directional guessing strategy for unequal allocation have not been given yet.…”

“…Therefore, the distinction between the two guessing strategies is not critical. For unequal allocations, the maximum probability guess has been used under the name of convergence guess [2][3][4][9][10][11][12], and the minimum imbalance guess is similar to the directional guess [2]. Without explicit definitions like (2) and (3), the name of convergence guess might be mistakenly considered as the minimal imbalance guess.…”

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

“…Allocation predictability is defined by the Euclidian distance between the conditional allocation probability and the target allocation probability for each treatment assignment [8], ${\phi }_i=\sqrt{{\sum }_{j=1}^m{\left(p_{\mathit{ij}}-r_j\right)}^2}$.…”

“…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.…”