Randomization by minimization for unbalanced treatment allocation

Minimization is a dynamic randomization technique that has been widely used in clinical trials for achieving a balance of prognostic factors across treatment groups, but most often it has been used in the setting of equal treatment allocations. Although unequal treatment allocation is frequently encountered in clinical trials, an appropriate minimization procedure for such trials has not been published. The purpose of this paper is to present novel strategies for applying minimization methodology to such clinical trials. Two minimization techniques are proposed and compared by probability calculation and simulation studies. In the first method, called naïve minimization, probability assignment is based on a simple modification of the original minimization algorithm, which does not account for unequal allocation ratios. In the second method, called biased‐coin minimization (BCM), probability assignment is based on allocation ratios and optimized to achieve an ‘unbiased’ target allocation ratio. The performance of the two methods is investigated in various trial settings including different number of treatments, prognostic factors and sample sizes. The relative merits of the different distance metrics are also explored. On the basis of the results, we conclude that BCM is the preferable method for randomization in clinical trials involving unequal treatment allocations. The choice of different distance metrics slightly affects the performance of the minimization and may be optimized according to the specific feature of trials. Copyright © 2009 John Wiley & Sons, Ltd.     

Treatment by drawing lots in a therapeutic trial with a balance distribution of prognostic factors: the minimization method

In a controlled clinical trial, a more powerful design can be obtained by balancing the treatment groups for the main prognostic factors rather than ignoring these factors in the randomization. The method of random permuted blocks within strata becomes inadequate as the number of prognostic factors increases. Another method known as minimization is described, which balances the marginal distribution of prognostic factors; it was proposed by Taves, and generalized by Pocock and Simon. It avoids the problems arising from an imbalance in a major prognostic factor.     

Dynamic balancing randomization in controlled clinical trials

In the design of randomized clinical trials, balancing of treatment allocation across important prognostic factors (strata) improves the efficiency of the final comparisons. Whilst randomization methods exist which attempt to balance treatments across the strata (permuted blocks, minimization, biased coin), these approaches assign equal importance for all the strata. Dynamic balancing randomization (DBR) is a tree-based method proposed by Signorini et al. allowing different levels of imbalance in different strata which ensures a balance for each level of prognostic risk factors (conditional balance) whilst at the same time preserving randomness. We present a simple modification to the original approach to maintain a marginal balance over important strata and examine the properties of this modification. Two important measures of performance are used to provide comparisons between the approaches: a loss function, which can be interpreted as the squared norm of the imbalance vector, and a forcing index which conveys the degree of randomness. A comparison of DBR with minimization and a biased coin design is carried out by simulation on two simulated trials.     

Studies with group treatments required special power calculations, allocation methods, and statistical analyses

ObjectiveIn some trials, the intervention is delivered to individuals in groups, for example, groups that exercise together. The group structure of such trials has to be taken into consideration in the analysis and has an impact on the power of the trial. Our aim was to provide optimal methods for the design and analysis of such trials.Study Design and SettingWe described various treatment allocation methods and presented a new allocation algorithm: optimal batchwise minimization (OBM). We carried out a simulation study to evaluate the performance of unrestricted randomization, stratification, permuted block randomization, deterministic minimization, and OBM. Furthermore, we described appropriate analysis methods and derived a formula to calculate the study size.ResultsStratification, deterministic minimization, and OBM had considerably less risk of imbalance than unrestricted randomization and permuted block randomization. Furthermore, OBM led to unpredictable treatment allocation. The sample size calculation and the analysis of the study must be based on a multilevel model that takes the group structure of the trial into account.ConclusionTrials evaluating interventions that are carried out in subsequent groups require adapted treatment allocation, power calculation, and analysis methods. From the perspective of obtaining overall balance, we conclude that minimization is the method of choice. When the number of prognostic factors is low, stratification is an excellent alternative. OBM leads to better balance within the batches, but it is more complicated. It is probably most worthwhile in trials with many prognostic factors. From the perspective of predictability, a treatment allocation method, such as OBM, that allocates several subjects at the same time, is superior to other methods because it leads to the lowest possible predictability.     

Treatment allocation methods in clinical trials: a review

A comprehensive review of methods for allocation of treatments to patients in clinical trials is presented. Attention is restricted to controlled, prospective trials, as opposed to comparisons involving historical or other 'external' controls. We describe the features of each method and classify them according to whether their primary focus is randomization, efficiency, or balance with respect to prognostic factors. Methods which prevent bias, ensure an efficient treatment comparison and are simple to implement will contribute to the ability of clinical trials to provide precise and valid treatment comparisons. We assess critically the extent to which the methods achieve these goals, review the relationships of the allocation methods with subsequent analyses of the trial results, discuss current usage and provide guidelines for choice of method.     

Dynamic balanced randomization for clinical trials

Common methods of treatment allocation for multi-centre and/or stratified randomized clinical trials can result in substantial differences between the number of patients allocated to each treatment arm. This can occur in the overall trial for a permuted block design or within individual institutions/strata when using a minimization scheme. This may lead to a bias in the result. Also, these procedures can be predictable, with the possibility of an investigator-introduced selection bias. An easily implemented method of randomization is proposed which attempts to overcome these problems by balancing treatment allocations both within strata and across the trial as a whole. The method keeps a running tally on total treatment allocation numbers at all stratification levels. When a patient accrues a hierarchical decision rule is applied, and the allocation is deterministic if certain pre-defined limits are exceeded, and random otherwise. The method is an extension of the big stick design of Soares and Wu, and is related to both Zelen's key number randomization methods and the schemes of Nordle and Brantmark. Simulation studies are used to demonstrate that major imbalances possible with other schemes do not occur using this method, and that the potential for selection bias is much reduced.     

New algorithm for treatment allocation reduced selection bias and loss of power in small trials

OBJECTIVE: In clinical trials, patients become available for treatment sequentially. Especially in trials with a small number of patients, loss of power may become an important issue, if treatments are not allocated equally or if prognostic factors differ between the treatment groups. We present a new algorithm for sequential allocation of two treatments in small clinical trials, which is concerned with the reduction of both selection bias and imbalance. STUDY DESIGN AND SETTING: With the algorithm, an element of chance is added to the treatment as allocated by minimization. The amount of chance depends on the actual amount of imbalance of treatment allocations of the patients already enrolled. The sensitivity to imbalance may be tuned. We performed trial simulations with different numbers of patients and prognostic factors, in which we quantified loss of power and selection bias. RESULTS: With our method, selection bias is smaller than with minimization, and loss of power is lower than with pure randomization or treatment allocation according to a biased coin principle. CONCLUSION: Our method combines the conflicting aims of reduction of bias by predictability and reduction of loss of power, as a result of imbalance. The method may be of use in small trials.     

Comparison of balanced and random allocation in clinical trials: A simulation study

Background: Before analysing the results of a randomised controlled clinical trial in which 200 children were balanced over five prognostic factors, we wanted to know the efficiency of balanced allocation compared to simple randomisation as well as the efficiency of adjusted as compared to unadjusted statistical analysis in small and large sample sizes. Methods: A simulation study with 1000 replications of each assignment was performed for both relatively large trials (n = 100) and for small trials (n = 20). Four options for the design and analysis were studied: (1) simple randomisation with simple univariate analysis, (2) simple randomisation with multivariate modelling, (3) balanced allocation with simple univariate analysis and (4) balanced allocation with multivariate modelling. In addition, we also considered the effect of an unmeasured covariable, which was either uncorrelated or correlated with another covariate. Results/conclusion: The simulations show that a combination of balanced allocation and multivariate analysis as compared to simple randomisation and multivariate analysis leads to more valid and precise treatment effects as well as to smaller confidence intervals, especially in small trials (n = 20). Multivariate analysis with all known prognostic factors produces on average smaller Type I errors and Type II errors in balanced allocation compared to simple randomisation. If an unmeasured covariate is strongly correlated with another covariate the treatment effect is estimated more precisely as compared to an unmeasured covariate that is not correlate or less strongly correlated.     

The "best balance" allocation led to optimal balance in cluster-controlled trials

ObjectiveBalance of prognostic factors between treatment groups is desirable because it improves the accuracy, precision, and credibility of the results. In cluster-controlled trials, imbalance can easily occur by chance when the number of cluster is small. If all clusters are known at the start of the study, the “best balance” allocation method (BB) can be used to obtain optimal balance. This method will be compared with other allocation methods.Study Design and SettingWe carried out a simulation study to compare the balance obtained with BB, minimization, unrestricted randomization, and matching for four to 20 clusters and one to five categorical prognostic factors at cluster level.ResultsBB resulted in a better balance than randomization in 13–100% of the situations, in 0–61% for minimization, and in 0–88% for matching. The superior performance of BB increased as the number of clusters and/or the number of factors increased.ConclusionBB results in a better balance of prognostic factors than randomization, minimization, stratification, and matching in most situations. Furthermore, BB cannot result in a worse balance of prognostic factors than the other methods.     

An algorithm for prospective individual matching in a non-randomized clinical trial.

A method is described to achieve balance across prognostic factors in intervention trials for which randomized allocation to treatment group is not possible. The method involves prospective individual matching of patients that have already been assigned to treatment groups. Data can be analyzed using methods appropriate for prospective matched cohort studies. Successful implementation depends on the number and complexity of factors to be matched, and on the number of available control patients. Simulation studies suggest that, in order to yield satisfactory match rates and to reduce costs associated with screening unmatched controls, no more than three prognostic factors should generally be considered. Baseline prognostic indices, incorporating information from multiple variables, provide effective matching factors. The implementation of the method in a successful clinical trial, the Delirium Prevention Trial, is discussed. In that study, treatment group was determined by hospital admission to either an intervention floor or to one of two usual care hospital floors. The ratio of available control to intervention patients was 1.3, and 95% of the eligible intervention floor patients were successfully matched to control floor patients. Excellent balance was demonstrated for non-matching factors, due in part to the use of a composite baseline risk score as a matching factor. In addition, external validity is enhanced because most eligible intervention patients are enrolled as they present. The methods outlined in this report provide a methodologically rigorous alternative for achieving balance across treatment groups, with respect to important prognostic factors, in non-randomized clinical trials, and will have broad applicability in the numerous situations in which randomization is not possible.