Justification for unequal allocation ratios in clinical trials: A scoping review.

OBJECTIVE: The objective of this review is to provide an overview of the justification reported for using unequal allocation ratios in randomized clinical trials (RCTs) testing a medical intervention. METHODS: Using the PICOS framework, we conducted a systematic search to find meta-studies within PubMed (a Medline database interface) that addressed the objective. RESULTS: The developed search strategy generated 525 results, of which, three studies met criteria for inclusion. These studies found that 22-43% of RCTs provided a justification for the use of unequal allocation based on publication alone, and between 38.7 and 66% after seeking input from trial authors. The most common reason given for this design was to gather increased safety data according to two reviews and to gain experience with an intervention according to the third review. CONCLUSION: Reporting of justification for RCTs designed with unequal allocation appears to occur less than half the time in the included studies. The reasons given for designing clinical trials with unequal participants encompass many domains, including ethical considerations. As such, this design feature should be implemented with intentionality to maximize the ethical features of clinical trials for participants. Coupling lack of justification with lack of adjusting for sample size estimations depicts an overall landscape in which there is significant room for improvement in methodological transparency within this area of RCTs.

Sample sizes required to estimate the protective efficacy of a vaccine when there is an unequal allocation of individuals across the vaccine and placebo groups.

The effectiveness of a vaccine is measured by means of protective vaccine efficacy, defined by VE=1-ARVARU, where ARV and ARU are, respectively, the disease attack rates in the vaccinated and the unvaccinated population. For each of the cohoret and case-control designs, methods have been presented in the literature for calculating the required sample size when the desired width of the confidence interval and the probability of coverage are pre-specified, where an equal number of individuals were assumed to be allocated to the vaccine and placebo group. In this article, we present a method for calculating the required sample size with a specified degree of precision when there is an unequal allocation of individuals across the two groups. The sample size required to achieve a desired power for the relevant level α test has also been explored, keeping the unequal allocation proportion in mind. The fraction of individuals allocated to the placebo group (ρ) can be so chosen that the total sample size or the expected number of people developing the disease or some other criteria of interest is minimized.

Optimal design of cluster randomized trials allowing unequal allocation of clusters and unequal cluster size between arms.

There are sometimes cost, scientific, or logistical reasons to allocate individuals unequally in an individually randomized trial. In cluster randomized trials we can allocate clusters unequally and/or allow different cluster size between trial arms. We consider parallel group designs with a continuous outcome, and optimal designs that require the smallest number of individuals to be measured given the number of clusters. Previous authors have derived the optimal allocation ratio for clusters under different variance and/or intracluster correlations (ICCs) between arms, allowing different but prespecified cluster sizes by arm. We derive closed-form expressions to identify the optimal proportions of clusters and of individuals measured for each arm, thereby defining optimal cluster sizes, when cluster size can be chosen freely. When ICCs differ between arms but the variance is equal, the optimal design allocates more than half the clusters to the arm with the higher ICC, but (typically only slightly) less than half the individuals and hence a smaller cluster size. We also describe optimal design under constraints on the number of clusters or cluster size in one or both arms. This methodology allows trialists to consider a range for the number of clusters in the trial and for each to identify the optimal design. Except if there is clear prior evidence for the ICC and variance by arm, a range of values will need to be considered. Researchers should choose a design with adequate power across the range, while also keeping enough clusters in each arm to permit the intended analysis method.

Implementing unequal randomization in clinical trials with heterogeneous treatment costs.

Equal randomization has been a popular choice in clinical trial practice. However, in trials with heterogeneous variances and/or variable treatment costs, as well as in settings where maximization of every trial participant's benefit is an important design consideration, optimal allocation proportions may be unequal across study treatment arms. In this paper, we investigate optimal allocation designs minimizing study cost under statistical efficiency constraints for parallel group clinical trials comparing several investigational treatments against the control. We show theoretically that equal allocation designs may be suboptimal, and unequal allocation designs can provide higher statistical power for the same budget or result in a smaller cost for the same level of power. We also show how optimal allocation can be implemented in practice by means of restricted randomization procedures and how to perform statistical inference following these procedures, using invoked population-based or randomization-based approaches. Our results provide further support to some previous findings in the literature that unequal randomization designs can be cost efficient and can be successfully implemented in practice. We conclude that the choice of the target allocation, the randomization procedure, and the statistical methodology for data analysis is an essential component in ensuring valid, powerful, and robust clinical trial results.

Implementing Optimal Designs for Dose-Response Studies Through Adaptive Randomization for a Small Population Group.

In dose-response studies with censored time-to-event outcomes, D-optimal designs depend on the true model and the amount of censored data. In practice, such designs can be implemented adaptively, by performing dose assignments according to updated knowledge of the dose-response curve at interim analysis. It is also essential that treatment allocation involves randomization-to mitigate various experimental biases and enable valid statistical inference at the end of the trial. In this work, we perform a comparison of several adaptive randomization procedures that can be used for implementing D-optimal designs for dose-response studies with time-to-event outcomes with small to moderate sample sizes. We consider single-stage, two-stage, and multi-stage adaptive designs. We also explore robustness of the designs to experimental (chronological and selection) biases. Simulation studies provide evidence that both the choice of an allocation design and a randomization procedure to implement the target allocation impact the quality of dose-response estimation, especially for small samples. For best performance, a multi-stage adaptive design with small cohort sizes should be implemented using a randomization procedure that closely attains the targeted D-optimal design at each stage. The results of the current work should help clinical investigators select an appropriate randomization procedure for their dose-response study.

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

Randomization designs for multiarm clinical trials are increasingly used in practice, especially in phase II dose-ranging studies. Many new methods have been proposed in the literature; however, there is lack of systematic, head-to-head comparison of the competing designs. In this paper, we systematically investigate statistical properties of various restricted randomization procedures for multiarm trials with fixed and possibly unequal allocation ratios. The design operating characteristics include measures of allocation balance, randomness of treatment assignments, variations in the allocation ratio, and statistical characteristics such as type I error rate and power. The results from the current paper should help clinical investigators select an appropriate randomization procedure for their clinical trial. We also provide a web-based R shiny application that can be used to reproduce all results in this paper and run simulations under additional user-defined experimental scenarios.

Group Sequential Survival Trial Design and Monitoring Using the Log-Rank Test.

For randomized group sequential survival trial designs with unbalanced treatment allocation, the widely used Schoenfeld formula is inaccurate, and the commonly used information time as the ratio of number of events at interim look to the number of events at the end of trial can be biased. In this paper, a sample size formula for the two-sample log-rank test under the proportional hazards model is proposed that provides more accurate sample size calculation for unbalanced survival trial designs. Furthermore, a new information time is introduced for the sequential survival trials such that the new information time is more accurate than the traditional information time when the allocation of enrollments is unbalanced in groups. Finally, we demonstrate the monitoring process using the sequential conditional probability ratio test and compare it with two other well known group sequential procedures. An example is given to illustrate unbalanced survival trial design using available software.

Sequence balance minimisation: minimising with unequal treatment allocations.

BACKGROUND: Minimisation ensures excellent balance between groups for several prognostic factors, even in small samples. However, its use with unequal allocation ratios has been problematic. This paper describes a new minimisation scheme named sequence balance minimisation for unequal treatment allocations. METHODS: Treatment- and factor-balancing properties were assessed in simulation studies for two- and three-arm trials with 1:2 and 1:2:3 allocation ratios. Sample sizes were set 30, 60 and 120. The number of prognostic factors on which to achieve balance was ranged from zero (treatment totals only) to ten with two levels occurring in equal probabilities. Random elements were set at 0.95, 0.9, 0.85, 0.80, 0.7, 0.6 and 0.5. Characteristics of the randomisation distributions and the impact of changing the block size while maintaining the allocation ratio were also examined. RESULTS: Sequence balance minimisation has good treatment- and factor-balancing capabilities, and the randomisation distribution was centred at zero for all scenarios. The mean and median number of allocations achieved were the same as the number expected in most scenarios, and including additional factors (up to ten) in the minimisation scheme had little impact on treatment balance. Treatment balance tended to depart from the target as the random element was lowered. The variability in allocations achieved increased slightly as the number of factors increased, as the random element was decreased and as the sample size increased. The mean and median factor imbalance remained tightly around zero even when the chosen factor was not included in the minimisation scheme, though the variability was greater. The variability in factor imbalance increased slightly as the random element decreased, as well as when the number of prognostic factors and sample size increased. Increasing block size while maintaining the allocation ratio improved treatment balance notably with little impact on factor imbalance. CONCLUSIONS: Sequence balance minimisation has good treatment- and factor-balancing properties and is particularly useful for small trials seeking to achieve balance across several prognostic factors.

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

The paper discusses three methods for expanding the biased coin randomization (BCR) to unequal allocation while preserving the unconditional allocation ratio at every step. The first method originally proposed in the contexts of BCR and minimization is based on mapping from an equal allocation multi-arm BCR. Despite the improvement proposed in this paper to ensure tighter adherence to the targeted unequal allocation, this method still distributes the probability mass at least as wide as the permuted block randomization (PBR). This works for smaller block sizes, but for larger block sizes, a tighter control of the imbalance in the treatment assignments is desired. The second method, which has two versions, allows to tighten the distribution of the imbalance compared with that achieved with the PBR. However, the distribution of the imbalance remains considerably wider than that of the brick tunnel randomization - the unequal allocation procedure with the tightest possible imbalance distribution among all allocation ratio preserving procedures with the same allocation ratio. Finally, the third method, the BCR with a preset proportion of maximal forcing, mimics the properties of the equal allocation BCR. With maximum forcing, it approaches the brick tunnel randomization, similar to how 1:1 BCR approaches 1:1 PBR with the permuted block size of 2 (the equal allocation procedure with the lowest possible imbalance) when the bias approaches 1. With minimum forcing, the BCR with a preset proportion of maximal forcing approaches complete randomization (similar to 1:1 BCR). Copyright © 2017 John Wiley & Sons, Ltd.

A surrogate-primary replacement algorithm for response-adaptive randomization in stroke clinical trials.

Response-adaptive randomization (RAR) offers clinical investigators benefit by modifying the treatment allocation probabilities to optimize the ethical, operational, or statistical performance of the trial. Delayed primary outcomes and their effect on RAR have been studied in the literature; however, the incorporation of surrogate outcomes has not been fully addressed. We explore the benefits and limitations of surrogate outcome utilization in RAR in the context of acute stroke clinical trials. We propose a novel surrogate-primary (S-P) replacement algorithm where a patient's surrogate outcome is used in the RAR algorithm only until their primary outcome becomes available to replace it. Computer simulations investigate the effect of both the delay in obtaining the primary outcome and the underlying surrogate and primary outcome distributional discrepancies on complete randomization, standard RAR and the S-P replacement algorithm methods. Results show that when the primary outcome is delayed, the S-P replacement algorithm reduces the variability of the treatment allocation probabilities and achieves stabilization sooner. Additionally, the S-P replacement algorithm benefit proved to be robust in that it preserved power and reduced the expected number of failures across a variety of scenarios.

Sample Size Calculations for Combination Drugs of 2 Monotherapies With a Single Approved Dose Level: Binary Endpoint Cases.

BACKGROUND: In this article, we study the sample size calculations for the combination drugs of 2 monotherapies with a single approved dose level when the primary endpoints are binary. METHODS: Two study cases are examined: In the first, each monotherapy has the same indication, while in the second, each monotherapy has a different indication. The sample sizes are calculated by using an asymptotic joint distribution of test statistics and employing unequal allocation for 3 popular measures of 2 proportions: the risk difference, the log relative risk, and the log odds ratio. RESULTS: Results show that our proposed method produces smaller total sample sizes compared with the heuristic method. CONCLUSIONS: The total sample sizes can be reduced by incorporating unequal allocation and dependency between 2 test statistics.

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

Wei's urn design was proposed in 1987 for subject randomization in trials comparing m ≥ 2 treatments with equal allocation. In this manuscript, two modified versions of Wei's urn design are presented to accommodate unequal allocations. First one uses a provisional allocation of [Formula: see text] to achieve the target allocation r1 : r2, and the second one uses equal allocation for r1 + r2 arms to achieve an unequal allocation r1 : r2 based on the concept Kaiser presented in his recent paper. The properties of these two designs are evaluated based on treatment imbalance and allocation predictability under different sample sizes and unequal allocation ratios. Simulations are performed to compare the two designs to other designs used for unequal allocations, include the complete randomization, permuted block randomization, block urn design, maximal procedure, and the mass weighted urn design.

Brick tunnel randomization and the momentum of the probability mass.

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.

Mass weighted urn design--A new randomization algorithm for unequal allocations.

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.

Hierarchical dynamic allocation procedures based on modified Zelen's approach in multiregional studies with unequal allocation.

Morrissey, McEntegart, and Lang (2010) showed that in multicenter studies with equal allocation to several treatment arms, the modified Zelen's approach provides excellent within-center and across-study balance in treatment assignments. In this article, hierarchical balancing procedures for equal allocation to more than two arms (with some elements different from earlier versions) and their unequal allocation expansions that incorporate modified Zelen's approach at the center level are described. The balancing properties of the described procedures for a case study of a multiregional clinical trial with 1:2 allocation where balance within regions as well as in other covariates is required are examined through simulations.

Wide brick tunnel randomization - an unequal allocation procedure that limits the imbalance in treatment totals.

In open-label studies, partial predictability of permuted block randomization provides potential for selection bias. To lessen the selection bias in two-arm studies with equal allocation, a number of allocation procedures that limit the imbalance in treatment totals at a pre-specified level but do not require the exact balance at the ends of the blocks were developed. In studies with unequal allocation, however, the task of designing a randomization procedure that sets a pre-specified limit on imbalance in group totals is not resolved. Existing allocation procedures either do not preserve the allocation ratio at every allocation or do not include all allocation sequences that comply with the pre-specified imbalance threshold. Kuznetsova and Tymofyeyev described the brick tunnel randomization for studies with unequal allocation that preserves the allocation ratio at every step and, in the two-arm case, includes all sequences that satisfy the smallest possible imbalance threshold. This article introduces wide brick tunnel randomization for studies with unequal allocation that allows all allocation sequences with imbalance not exceeding any pre-specified threshold while preserving the allocation ratio at every step. In open-label studies, allowing a larger imbalance in treatment totals lowers selection bias because of the predictability of treatment assignments. The applications of the technique in two-arm and multi-arm open-label studies with unequal allocation are described.