Robust Blocked Response-Adaptive Randomization Designs
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In most clinical trials, patients are randomized with equal probability among treatments to obtain an unbiased estimate of the treatment effect. Response-adaptive randomization (RAR) has been proposed for ethical reasons, where the randomization ratio is tilted successively to favor the better performing treatment. However, the substantial disagreement regarding bias due to time-trends in adaptive randomization is not fully recognized. The type-I error is inflated in the traditional Bayesian RAR approaches when a time-trend is present. In our approach, patients are assigned in blocks and the randomization ratio is recomputed for blocks rather than traditional adaptive randomization where it is done per patient. We further investigate the design with a range of scenarios for both frequentist and Bayesian designs. We compare our method with equal randomization and with different numbers of blocks including the traditional RAR design where randomization ratio is altered patient by patient basis. The analysis is stratified if there are two or more patients in each block. Small blocks should be avoided due to the possibility of not acquiring any information from the $\mu_i$. On the other hand, RAR with large blocks has a good balance between efficiency and treating more subjects to the better-performing treatment, while retaining blocked RAR's unique unbiasedness.Keywords:
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Stratified permuted block randomization has been the dominant covariate‐adaptive randomization procedure in clinical trials for several decades. Its high probability of deterministic assignment and low capacity of covariate balancing have been well recognized. The popularity of this sub‐optimal method is largely due to its simplicity in implementation and the lack of better alternatives. Proposed in this paper is a two‐stage covariate‐adaptive randomization procedure that uses the block urn design or the big stick design in stage one to restrict the treatment imbalance within each covariate stratum, and uses the biased‐coin minimization method in stage two to control imbalances in the distribution of additional covariates that are not included in the stratification algorithm. Analytical and simulation results show that the new randomization procedure significantly reduces the probability of deterministic assignments, and improve the covariate balancing capacity when compared to the traditional stratified permuted block randomization. Copyright © 2014 John Wiley & Sons, Ltd.
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In a randomized clinical trial, response-adaptive randomization procedures use the information gathered, including the previous patients' responses, to allocate the next patient. In this setting, we consider randomization-based inference. We provide an algorithm to obtain exact p-values for statistical tests that compare two treatments with dichotomous responses. This algorithm can be applied to a family of response adaptive randomization procedures which share the following property: the distribution of the allocation rule depends only on the imbalance between treatments and on the imbalance between successes for treatments 1 and 2 in the previous step. This family includes some outstanding response adaptive randomization procedures. We study a randomization test to contrast the null hypothesis of equivalence of treatments and we show that this test has a similar performance to that of its parametric counterpart. Besides, we study the effect of a covariate in the inferential process. First, we obtain a parametric test, constructed assuming a logit model which relates responses to treatments and covariate levels, and we give conditions that guarantee its asymptotic normality. Finally, we show that the randomization test, which is free of model specification, performs as well as the parametric test that takes the covariate into account.
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The theocratical properties of the power of the conventional testing hypotheses and the selection bias are usually unknown under covariate-adaptive randomized clinical trials. In the literature, most studies are based on simulations. In this article, we provide theoretical foundation of the power of the hypothesis testing and the selection bias under covariate-adaptive randomization based on linear models. We study the asymptotic relative loss of power of hypothesis testing to compare the treatment effects and the asymptotic selection bias. Under the covariate-adaptive randomization, (i) the hypothesis testing usually losses power, the more covariates in testing model are not incorporated in the randomization procedure, the more the power is lost; (ii) the hypothesis testing is usually more powerful than the one under complete randomization; and (iii) comparing to complete randomization, most of the popular covariate-adaptive randomization procedures in the literature, for example, Pocock and Simon's marginal procedure, stratified permuted block design, etc, produce nontrivial selection bias. A new family of covariate-adaptive randomization procedures are proposed for considering the power and selection bias simultaneously, under which, the covariate imbalances are small enough so that the power of testing the treatment effects would be asymptotically the largest and at the same time, the selection bias is asymptotically the optimal. The theocratical properties give a full picture how the power of the hypothesis testing, the selection bias of the randomization procedure, and the randomization method affect each other.
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Abstract Response-adaptive designs allow the randomization probabilities to change during the course of a trial based on cumulated response data so that a greater proportion of patients can be allocated to the better performing treatments. A major concern over the use of response-adaptive designs in practice, particularly from a regulatory viewpoint, is controlling the type I error rate. In particular, we show that the naïve z-test can have an inflated type I error rate even after applying a Bonferroni correction. Simulation studies have often been used to demonstrate error control but do not provide a guarantee. In this article, we present adaptive testing procedures for normally distributed outcomes that ensure strong familywise error control by iteratively applying the conditional invariance principle. Our approach can be used for fully sequential and block randomized trials and for a large class of adaptive randomization rules found in the literature. We show there is a high price to pay in terms of power to guarantee familywise error control for randomization schemes with extreme allocation probabilities. However, for proposed Bayesian adaptive randomization schemes in the literature, our adaptive tests maintain or increase the power of the trial compared to the z-test. We illustrate our method using a three-armed trial in primary hypercholesterolemia.
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Summary Despite the widespread use of equal randomization in clinical trials, response-adaptive randomization has attracted considerable interest. There is typically a prerun of equal randomization before the implementation of response-adaptive randomization, although it is often not clear how many subjects are needed in this prephase, and in practice the number of patients in the equal randomization stage is often arbitrary. Another concern that is associated with realtime response-adaptive randomization is that trial conduct often requires patients' responses to be immediately available after the treatment, whereas clinical responses may take a relatively long period of time to exhibit. To resolve these two issues, we propose a two-stage procedure to achieve a balance between power and response, which is equipped with a likelihood ratio test before skewing the allocation probability towards a better treatment. Furthermore, we develop a non-parametric fractional model and a parametric survival design with an optimal allocation scheme to tackle the common problem caused by delayed response. We evaluate the operating characteristics of the two-stage designs through extensive simulation studies and illustrate them with a human immunodeficiency virus clinical trial. Numerical results show that the methods proposed satisfactorily resolve the arbitrary size of the equal randomization phase and the delayed response problem in response-adaptive randomization.
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Re-randomization test has been considered as a robust alternative to the traditional population model-based methods for analyzing randomized clinical trials. This is especially so when the clinical trials are randomized according to minimization, which is a popular covariate-adaptive randomization method for ensuring balance among prognostic factors. Among various re-randomization tests, fixed-entry-order re-randomization is advocated as an effective strategy when a temporal trend is suspected. Yet when the minimization is applied to trials with unequal allocation, fixed-entry-order re-randomization test is biased and thus compromised in power. We find that the bias is due to non-uniform re-allocation probabilities incurred by the re-randomization in this case. We therefore propose a weighted fixed-entry-order re-randomization test to overcome the bias. The performance of the new test was investigated in simulation studies that mimic the settings of a real clinical trial. The weighted re-randomization test was found to work well in the scenarios investigated including the presence of a strong temporal trend. Copyright © 2013 John Wiley & Sons, Ltd.
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The simplest Bayesian adaptive randomization scheme is to randomize patients to a treatment with probability equal to the probability p that the treatment is better. We examine three variations on adaptive randomization which are used to compromise between this scheme and equal randomization. The first variation is to apply a power transformation to p to obtain randomization probabilities. The second is to clip p to live within specified lower and upper bounds. The third is to begin the trial with a burn-in period of equal randomization. We illustrate how each approach effects statistical power and the number of patients assigned to each treatment. We conclude with recommendations for designing adaptively randomized clinical trials. Comparing methods of tuning adaptively randomized trials John D. Cook∗ January 9, 2007 Abstract The simplest Bayesian adaptive randomization scheme is to randomize patients to a treatment with probability equal to the probability p that the treatment is better. We examine three variations on adaptive randomization which are used to compromise between this scheme and equal randomization. The first variation is to apply a power transformation to p to obtain randomization probabilities. The second is to clip p to live within specified lower and upper bounds. The third is to begin the trial with a burn-in period of equal randomization. We illustrate how each approach effects statistical power and the number of patients assigned to each treatment. We conclude with recommendations for designing adaptively randomized clinical trials. ∗University of Texas M. D. Anderson Cancer Center, cook@mdanderson.orgThe simplest Bayesian adaptive randomization scheme is to randomize patients to a treatment with probability equal to the probability p that the treatment is better. We examine three variations on adaptive randomization which are used to compromise between this scheme and equal randomization. The first variation is to apply a power transformation to p to obtain randomization probabilities. The second is to clip p to live within specified lower and upper bounds. The third is to begin the trial with a burn-in period of equal randomization. We illustrate how each approach effects statistical power and the number of patients assigned to each treatment. We conclude with recommendations for designing adaptively randomized clinical trials. ∗University of Texas M. D. Anderson Cancer Center, cook@mdanderson.org
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As randomization methods use more information in more complex ways to assign patients to treatments, analysis of the resulting data becomes challenging. The treatment assignment vector and outcome vector become correlated whenever randomization probabilities depend on data correlated with outcomes. One straightforward analysis method is a re‐randomization test that fixes outcome data and creates a reference distribution for the test statistic by repeatedly re‐randomizing according to the same randomization method used in the trial. This article reviews re‐randomization tests, especially in nonstandard settings like covariate‐adaptive and response‐adaptive randomization. We show that re‐randomization tests provide valid inference in a wide range of settings. Nonetheless, there are simple examples demonstrating limitations.
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Summary Covariate-adaptive randomization is popular in clinical trials with sequentially arrived patients for balancing treatment assignments across prognostic factors that may have influence on the response. However, existing theory on tests for the treatment effect under covariate-adaptive randomization is limited to tests under linear or generalized linear models, although the covariate-adaptive randomization method has been used in survival analysis for a long time. Often, practitioners will simply adopt a conventional test to compare two treatments, which is controversial since tests derived under simple randomization may not be valid in terms of type I error under other randomization schemes. We derive the asymptotic distribution of the partial likelihood score function under covariate-adaptive randomization and a working model that is subject to possible model misspecification. Using this general result, we prove that the partial likelihood score test that is robust against model misspecification under simple randomization is no longer robust but conservative under covariate-adaptive randomization. We also show that the unstratified log-rank test is conservative and the stratified log-rank test remains valid under covariate-adaptive randomization. We propose a modification to variance estimation in the partial likelihood score test, which leads to a score test that is valid and robust against arbitrary model misspecification under a large family of covariate-adaptive randomization schemes including simple randomization. Furthermore, we show that the modified partial likelihood score test derived under a correctly specified model is more powerful than log-rank-type tests in terms of Pitman’s asymptotic relative efficiency. Simulation studies about the type I error and power of various tests are presented under several popular randomization schemes.
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Response adaptive randomization (RAR) is appealing from methodological, ethical, and pragmatic perspectives in the sense that subjects are more likely to be randomized to better performing treatment groups based on accumulating data. However, applications of RAR in confirmatory drug clinical trials with multiple active arms are limited largely due to its complexity, and lack of control of randomization ratios to different treatment groups. To address the aforementioned issues, we propose a Response Adaptive Block Randomization (RABR) design allowing arbitrarily prespecified randomization ratios for the control and high‐performing groups to meet clinical trial objectives. We show the validity of the conventional unweighted test in RABR with a controlled type I error rate based on the weighted combination test for sample size adaptive design invoking no large sample approximation. The advantages of the proposed RABR in terms of robustly reaching target final sample size to meet regulatory requirements and increasing statistical power as compared with the popular Doubly Adaptive Biased Coin Design are demonstrated by statistical simulations and a practical clinical trial design example.
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