Shift in re‐randomization distribution with conditional randomization test

Proschan, Brittain, and Kammerman made a very interesting observation that for some examples of the unequal allocation minimization, the mean of the unconditional randomization distribution is shifted away from 0. Kuznetsova and Tymofyeyev linked this phenomenon to the variations in the allocation ratio from allocation to allocation in the examples considered in the paper by Proschan et al. and advocated the use of unequal allocation procedures that preserve the allocation ratio at every step. In this paper, we show that the shift phenomenon extends to very common settings: using conditional randomization test in a study with equal allocation. This phenomenon has the same cause: variations in the allocation ratio among the allocation sequences in the conditional reference set, not previously noted. We consider two kinds of conditional randomization tests. The first kind is the often used randomization test that conditions on the treatment group totals; we describe the variations in the conditional allocation ratio with this test on examples of permuted block randomization and biased coin randomization. The second kind is the randomization test proposed by Zheng and Zelen for a multicenter trial with permuted block central allocation that conditions on the within‐center treatment totals. On the basis of the sequence of conditional allocation ratios, we derive the value of the shift in the conditional randomization distribution for specific vector of responses and the expected value of the shift when responses are independent identically distributed random variables. We discuss the asymptotic behavior of the shift for the two types of tests. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison of response adaptive randomization features in multiarm clinical trials with control

We investigate multiple features of response adaptive randomization (RAR) in the context of a multiple arm randomized trial with control, where the primary goal is the identification of the best arm for use in a broader patient population. We maintain constant control allocation and vary the length of time until RAR is started, interim frequency, the underlying quantity used to calculate the randomization probabilities, and a threshold resulting in temporary arm dropping. We evaluate the designs on five metrics measuring benefit to the internal trial population, the future external population, and statistical estimation. Our results indicate these features have minimal interaction within the space explored, with preference for earlier activation of RAR, more frequent interim analyses, randomizing in proportion to the probability each arm is the best, and aggressive thresholding for temporarily dropping arms. The results illustrate useful principles for maximizing the benefit of RAR in practice.     

A COMPARISON OF THE IMPRECISE BETA CLASS, THE RANDOMIZED PLAY-THE-WINNER RULE AND THE TRIANGULAR TEST FOR CLINICAL TRIALS WITH BINARY RESPONSES

This paper develops clinical trial designs that compare two treatments with a binary outcome. The imprecise beta class (IBC), a class of beta probability distributions, is used in a robust Bayesian framework to calculate posterior upper and lower expectations for treatment success rates using accumulating data. The posterior expectation for the difference in success rates can be used to decide when there is sufficient evidence for randomized treatment allocation to cease. This design is formally related to the randomized play‐the‐winner (RPW) design, an adaptive allocation scheme where randomization probabilities are updated sequentially to favour the treatment with the higher observed success rate. A connection is also made between the IBC and the sequential clinical trial design based on the triangular test. Theoretical and simulation results are presented to show that the expected sample sizes on the truly inferior arm are lower using the IBC compared with either the triangular test or the RPW design, and that the IBC performs well against established criteria involving error rates and the expected number of treatment failures.     

Balanced covariates with response adaptive randomization

Response adaptive randomization (RAR) methods for clinical trials are susceptible to imbalance in the distribution of influential covariates across treatment arms. This can make the interpretation of trial results difficult, because observed differences between treatment groups may be a function of the covariates and not necessarily because of the treatments themselves. We propose a method for balancing the distribution of covariate strata across treatment arms within RAR. The method uses odds ratios to modify global RAR probabilities to obtain stratum‐specific modified RAR probabilities. We provide illustrative examples and a simple simulation study to demonstrate the effectiveness of the strategy for maintaining covariate balance. The proposed method is straightforward to implement and applicable to any type of RAR method or outcome.     

Response‐adaptive designs for binary responses: How to offer patient benefit while being robust to time trends?

Response‐adaptive randomisation (RAR) can considerably improve the chances of a successful treatment outcome for patients in a clinical trial by skewing the allocation probability towards better performing treatments as data accumulates. There is considerable interest in using RAR designs in drug development for rare diseases, where traditional designs are not either feasible or ethically questionable. In this paper, we discuss and address a major criticism levelled at RAR: namely, type I error inflation due to an unknown time trend over the course of the trial. The most common cause of this phenomenon is changes in the characteristics of recruited patients—referred to as patient drift. This is a realistic concern for clinical trials in rare diseases due to their lengthly accrual rate. We compute the type I error inflation as a function of the time trend magnitude to determine in which contexts the problem is most exacerbated. We then assess the ability of different correction methods to preserve type I error in these contexts and their performance in terms of other operating characteristics, including patient benefit and power. We make recommendations as to which correction methods are most suitable in the rare disease context for several RAR rules, differentiating between the 2‐armed and the multi‐armed case. We further propose a RAR design for multi‐armed clinical trials, which is computationally efficient and robust to several time trends considered.     

Critical appraisal of Bayesian dynamic borrowing from an imperfectly commensurate historical control

Bayesian dynamic borrowing designs facilitate borrowing information from historical studies. Historical data, when perfectly commensurate with current data, have been shown to reduce the trial duration and the sample size, while inflation in the type I error and reduction in the power have been reported, when imperfectly commensurate. These results, however, were obtained without considering that Bayesian designs are calibrated to meet regulatory requirements in practice and even no‐borrowing designs may use information from historical data in the calibration. The implicit borrowing of historical data suggests that imperfectly commensurate historical data may similarly impact no‐borrowing designs negatively. We will provide a fair appraiser of Bayesian dynamic borrowing and no‐borrowing designs. We used a published selective adaptive randomization design and real clinical trial setting and conducted simulation studies under varying degrees of imperfectly commensurate historical control scenarios. The type I error was inflated under the null scenario of no intervention effect, while larger inflation was noted with borrowing. The larger inflation in type I error under the null setting can be offset by the greater probability to stop early correctly under the alternative. Response rates were estimated more precisely and the average sample size was smaller with borrowing. The expected increase in bias with borrowing was noted, but was negligible. Using Bayesian dynamic borrowing designs may improve trial efficiency by stopping trials early correctly and reducing trial length at the small cost of inflated type I error.     

Bayesian response adaptive randomization using longitudinal outcomes

The response adaptive randomization (RAR) method is used to increase the number of patients assigned to more efficacious treatment arms in clinical trials. In many trials evaluating longitudinal patient outcomes, RAR methods based only on the final measurement may not benefit significantly from RAR because of its delayed initiation. We propose a Bayesian RAR method to improve RAR performance by accounting for longitudinal patient outcomes (longitudinal RAR). We use a Bayesian linear mixed effects model to analyze longitudinal continuous patient outcomes for calculating a patient allocation probability. In addition, we aim to mitigate the loss of statistical power because of large patient allocation imbalances by embedding adjusters into the patient allocation probability calculation. Using extensive simulation we compared the operating characteristics of our proposed longitudinal RAR method with those of the RAR method based only on the final measurement and with an equal randomization method. Simulation results showed that our proposed longitudinal RAR method assigned more patients to the presumably superior treatment arm compared with the other two methods. In addition, the embedded adjuster effectively worked to prevent extreme patient allocation imbalances. However, our proposed method may not function adequately when the treatment effect difference is moderate or less, and still needs to be modified to deal with unexpectedly large departures from the presumed longitudinal data model. Copyright © 2015 John Wiley & Sons, Ltd.     

Quantitative comparison of randomization designs in sequential clinical trials based on treatment balance and allocation randomness

To evaluate the performance of randomization designs under various parameter settings and trial sample sizes, and identify optimal designs with respect to both treatment imbalance and allocation randomness, we evaluate 260 design scenarios from 14 randomization designs under 15 sample sizes range from 10 to 300, using three measures for imbalance and three measures for randomness. The maximum absolute imbalance and the correct guess (CG) probability are selected to assess the trade-off performance of each randomization design. As measured by the maximum absolute imbalance and the CG probability, we found that performances of the 14 randomization designs are located in a closed region with the upper boundary (worst case) given by Efron's biased coin design (BCD) and the lower boundary (best case) from the Soares and Wu's big stick design (BSD). Designs close to the lower boundary provide a smaller imbalance and a higher randomness than designs close to the upper boundary. Our research suggested that optimization of randomization design is possible based on quantified evaluation of imbalance and randomness. Based on the maximum imbalance and CG probability, the BSD, Chen's biased coin design with imbalance tolerance method, and Chen's Ehrenfest urn design perform better than popularly used permuted block design, EBCD, and Wei's urn design.     

Randomization of neighbour balanced generalized Youden designs

If a crossover design with more than two treatments is carryover balanced, then the usual randomization of experimental units and periods would destroy the neighbour structure of the design. As an alternative, Bailey [1985. Restricted randomization for neighbour-balanced designs. Statist. Decisions Suppl. 2, 237–248] considered randomization of experimental units and of treatment labels, which leaves the neighbour structure intact. She has shown that, if there are no carryover effects, this randomization validates the row–column model, provided the starting design is a generalized Latin square. We extend this result to generalized Youden designs where either the number of experimental units is a multiple of the number of treatments or the number of periods is equal to the number of treatments. For the situation when there are carryover effects we show for so-called totally balanced designs that the variance of the estimates of treatment differences does not change in the presence of carryover effects, while the estimated variance of this estimate becomes conservative.     

Effects of unstratified and centre‐stratified randomization in multi‐centre clinical trials

This paper deals with the analysis of randomization effects in multi‐centre clinical trials. The two randomization schemes most often used in clinical trials are considered: unstratified and centre‐stratified block‐permuted randomization. The prediction of the number of patients randomized to different treatment arms in different regions during the recruitment period accounting for the stochastic nature of the recruitment and effects of multiple centres is investigated. A new analytic approach using a Poisson‐gamma patient recruitment model (patients arrive at different centres according to Poisson processes with rates sampled from a gamma distributed population) and its further extensions is proposed. Closed‐form expressions for corresponding distributions of the predicted number of the patients randomized in different regions are derived. In the case of two treatments, the properties of the total imbalance in the number of patients on treatment arms caused by using centre‐stratified randomization are investigated and for a large number of centres a normal approximation of imbalance is proved. The impact of imbalance on the power of the study is considered. It is shown that the loss of statistical power is practically negligible and can be compensated by a minor increase in sample size. The influence of patient dropout is also investigated. The impact of randomization on predicted drug supply overage is discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Balanced and partially balanced incomplete block designs with autocorrelation errors

The paper aims to find variance balanced and variance partially balanced incomplete block designs when observations within blocks are autocorrelated and we call them BIBAC and PBIBAC designs. Orthogonal arrays of type I and type II when used as BIBAC designs have smaller average variance of elementary contrasts of treatment effects compared to the corresponding Balanced Incomplete Block (BIB) designs with homoscedastic, uncorrelated errors. The relative efficiency of BIB designs compared to BIBAC designs depends on the block size k and the autocorrelation ρ and is independent of the number of treatments. Further this relative efficiency increases with increasing k. Partially balanced incomplete block designs with autocorrelated errors are introduced using partially balanced incomplete block designs and orthogonal arrays of type I and type II.     

Efficient circular neighbour designs for spatial interference model

We consider circular block designs for field-trials when there are two-sided spatial interference between neighbouring plots of the same blocks. The parameter of interest is total effects that is the sum of direct effect of treatment and neighbour effects, which correspond to the use of a single treatment in the whole field. We determine universally optimal approximate designs. When the number of blocks may be large, we propose efficient exact designs generated by a single sequence of treatment. We also give efficiency factors of the usual binary block neighbour balanced designs which can be used when the number of blocks is small.     

A stochastically curtailed two-arm randomised phase II trial design for binary outcomes

Randomised controlled trials are considered the gold standard in trial design. However, phase II oncology trials with a binary outcome are often single-arm. Although a number of reasons exist for choosing a single-arm trial, the primary reason is that single-arm designs require fewer participants than their randomised equivalents. Therefore, the development of novel methodology that makes randomised designs more efficient is of value to the trials community. This article introduces a randomised two-arm binary outcome trial design that includes stochastic curtailment (SC), allowing for the possibility of stopping a trial before the final conclusions are known with certainty. In addition to SC, the proposed design involves the use of a randomised block design, which allows investigators to control the number of interim analyses. This approach is compared with existing designs that also use early stopping, through the use of a loss function comprised of a weighted sum of design characteristics. Comparisons are also made using an example from a real trial. The comparisons show that for many possible loss functions, the proposed design is superior to existing designs. Further, the proposed design may be more practical, by allowing a flexible number of interim analyses. One existing design produces superior design realisations when the anticipated response rate is low. However, when using this design, the probability of rejecting the null hypothesis is sensitive to misspecification of the null response rate. Therefore, when considering randomised designs in phase II, we recommend the proposed approach be preferred over other sequential designs.     

Response Adaptive Designs with Misclassified Responses

We consider response adaptive designs when the binary response may be misclassified and extend relevant results in the literature. We derive the optimal allocations under various objectives and examine the relationship between the power of statistical test and the variability of treatment allocation. Asymptotically best response adaptive randomization procedures and effects of misclassification on the optimal allocations are investigated. A real-life clinical trial is also discussed to illustrate our proposed approach.     

Subset selection in two-factor experiments using randomization restricted designs

This paper studies subset selection procedures for screening in two-factor treatment designs that employ either a split-plot or strip-plot randomization restricted experimental design laid out in blocks. The goal is to select a subset of treatment combinations associated with the largest mean. In the split-plot design, it is assumed that the block effects, the confounding effects (whole-plot error) and the measurement errors are normally distributed. None of the selection procedures developed depend on the block variances. Subset selection procedures are given for both the case of additive and non-additive factors and for a variety of circumstances concerning the confounding effect and measurement error variances. In particular, procedures are given for (1) known confounding effect and measurement error variances (2) unknown measurement error variance but known confounding effect (3) unknown confounding effect and measurement error variances. The constants required to implement the procedures are shown to be obtainable from available FORTRAN programs and tables. Generalization to the case of strip-plot randomization restriction is considered.     

Sample size determination for cluster randomization phase II trials of dichotomous data

One of the main goals for a phase II trial is to screen and select the best treatment to proceed onto further studies in a phase III trial. Under the flexible design proposed elsewhere, we discuss for cluster randomization trials sample size calculation with a given desired probability of correct selection to choose the best treatment when one treatment is better than all the others. We develop exact procedures for calculating the minimum required number of clusters with a given cluster size (or the minimum number of patients with a given number of repeated measurements) per treatment. An approximate sample size and the evaluation of its performance for two arms are also given. To help readers employ the results presented here, tables are provided to summarize the resulting minimum required sample sizes for cluster randomization trials with two arms and three arms in a variety of situations. Finally, to illustrate the sample size calculation procedures developed here, we use the data taken from a cluster randomization trial to study the association between the dietary sodium and the blood pressure.     

Use of simulation to compare the performance of minimization with stratified blocked randomization

Minimization is an alternative method to stratified permuted block randomization, which may be more effective at balancing treatments when there are many strata. However, its use in the regulatory setting for industry trials remains controversial, primarily due to the difficulty in interpreting conventional asymptotic statistical tests under restricted methods of treatment allocation. We argue that the use of minimization should be critically evaluated when designing the study for which it is proposed. We demonstrate by example how simulation can be used to investigate whether minimization improves treatment balance compared with stratified randomization, and how much randomness can be incorporated into the minimization before any balance advantage is no longer retained. We also illustrate by example how the performance of the traditional model-based analysis can be assessed, by comparing the nominal test size with the observed test size over a large number of simulations. We recommend that the assignment probability for the minimization be selected using such simulations.     

Weighted re‐randomization tests for minimization with unbalanced allocation

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.     

On the efficiency of some non-orthogonal split-plot×split-block designs with control treatments

Many split-plot×split-block (SPSB) type experiments used in agriculture, biochemistry or plant protection are designed to study new crop plant cultivars or chemical agents. In these experiments it is usually very important to compare test treatments with the so-called control treatments. It happens yet that experimental material is limited and it does not allow using a complete (orthogonal) SPSB design. In the paper we propose a non-orthogonal SPSB design for consideration. Two cases of the design are presented here, i.e. when its incompleteness is connected with a crossed treatment structure only or with a nested treatment structure only. It is assumed the factors' levels connected with the incompleteness of the design are split into two groups: a set of test treatments and a set of control treatments. The method of constructions involves applying augmented block designs for some factors' levels. In a modelling data obtained from such experiments the structure of experimental material and appropriate randomization scheme of the different kinds of units before they enter the experiment are taken into account. With respect to the analysis of the obtained randomization model the approach typical to the multistratum experiments with orthogonal block structure is adapted. The proposed statistical analysis of linear model obtained includes estimation of parameters, testing general and particular hypotheses defined by the (basic) treatment contrasts with special reference to the notion of general balance.