Restricted Randomization Procedures Research Articles

Understanding an impact of patient enrollment pattern on predictability of central (unstratified) randomization in a multi-center clinical trial.

In a multi-center randomized controlled trial (RCT) with competitive recruitment, eligible patients are enrolled sequentially by different study centers and are randomized to treatment groups using the chosen randomization method. Given the stochastic nature of the recruitment process, some centers may enroll more patients than others, and in some instances, a center may enroll multiple patients in a row, for example, on a given day. If the study is open-label, the investigators might be able to make intelligent guesses on upcoming treatment assignments in the randomization sequence, even if the trial is centrally randomized and not stratified by center. In this paper, we use enrollment data inspired by a real multi-center RCT to quantify the susceptibility of two restricted randomization procedures, the permuted block design and the big stick design, to selection bias under the convergence strategy of Blackwell and Hodges (1957) applied at the center level. We provide simulation evidence that the expected proportion of correct guesses may be greater than 50% (i.e., an increased risk of selection bias) and depends on the chosen randomization method and the number of study patients recruited by a given center that takes consecutive positions on the central allocation schedule. We propose some strategies for ensuring stronger encryption of the randomization sequence to mitigate the risk of selection bias.

Balancing the Objectives of Statistical Efficiency and Allocation Randomness in Randomized Controlled Trials

Various restricted randomization procedures are available to achieve equal (1:1) allocation in a randomized clinical trial. However, for some procedures, there is a nonnegligible probability of imbalance in the final numbers which may result in an underpowered study. It is important to assess such probability at the study planning stage and make adjustments in the design if needed. In this article, we perform a quantitative assessment of the tradeoff between randomness, balance, and power of restricted randomization designs targeting equal allocation. First, we study the small-sample performance of biased coin designs with known asymptotic properties and identify a design with an excellent balance–randomness tradeoff. Second, we investigate the issue of randomization-induced treatment imbalance and the corresponding risk of an underpowered study. We propose two risk mitigation strategies: increasing the total sample size or fine-tuning the biased coin parameter to obtain the least restrictive randomization procedure that attains the target power with a high, user-defined probability for the given sample size. Additionally, we investigate an approach for finding the most balanced design that satisfies a constraint on the chosen measure of randomness. Our proposed methodology is simple and yet generalizable to more complex settings, such as trials with stratified randomization and multi-arm trials with possibly unequal randomization ratios.

A roadmap to using randomization in clinical trials

BackgroundRandomization is the foundation of any clinical trial involving treatment comparison. It helps mitigate selection bias, promotes similarity of treatment groups with respect to important known and unknown confounders, and contributes to the validity of statistical tests. Various restricted randomization procedures with different probabilistic structures and different statistical properties are available. The goal of this paper is to present a systematic roadmap for the choice and application of a restricted randomization procedure in a clinical trial.MethodsWe survey available restricted randomization procedures for sequential allocation of subjects in a randomized, comparative, parallel group clinical trial with equal (1:1) allocation. We explore statistical properties of these procedures, including balance/randomness tradeoff, type I error rate and power. We perform head-to-head comparisons of different procedures through simulation under various experimental scenarios, including cases when common model assumptions are violated. We also provide some real-life clinical trial examples to illustrate the thinking process for selecting a randomization procedure for implementation in practice.ResultsRestricted randomization procedures targeting 1:1 allocation vary in the degree of balance/randomness they induce, and more importantly, they vary in terms of validity and efficiency of statistical inference when common model assumptions are violated (e.g. when outcomes are affected by a linear time trend; measurement error distribution is misspecified; or selection bias is introduced in the experiment). Some procedures are more robust than others. Covariate-adjusted analysis may be essential to ensure validity of the results. Special considerations are required when selecting a randomization procedure for a clinical trial with very small sample size.ConclusionsThe choice of randomization design, data analytic technique (parametric or nonparametric), and analysis strategy (randomization-based or population model-based) are all very important considerations. Randomization-based tests are robust and valid alternatives to likelihood-based tests and should be considered more frequently by clinical investigators.

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.

Exact optimum coin bias in Efron's randomization procedure.

Efron's biased coin design is a restricted randomization procedure that has very favorable balancing properties, yet it is fully randomized, in that subjects are always randomized to one of two treatments with a probability less than 1. The parameter of interest is the bias p of the coin, which can range from 0.5 to 1. In this note, we propose a compound optimization strategy that selects p based on a subjected weighting of the relative importance of the two fundamental criteria of interest for restricted randomization mechanisms, namely balance between the treatment assignments and allocation randomness. We use exact and asymptotic distributional properties of Efron's coin to find the optimal p under compound criteria involving imbalance variability, expected imbalance, selection bias, and accidental bias, for both small/moderate trials and large samples.

Randomization, matching, and propensity scores in the design and analysis of experimental studies with measured baseline covariates.

In many experimental situations, researchers have information on a number of covariates prior to randomization. This information can be used to balance treatment assignment with respect to these covariates as well as in the analysis of the outcome data. In this paper, we investigate the use of propensity scores in both of these roles. We also introduce a randomization procedure in which the balance of all measured covariates is approximately indexed by the variance of the empirical propensity scores and randomization is restricted to those permutations with the least variable propensity scores. This procedure is compared with recently proposed methods in terms of resulting covariate balance and estimation efficiency. Properties of the estimators resulting from each procedure are compared with estimates which incorporate the propensity score in the analysis stage. Simulation results show that analytical adjustment for the propensity score yields results on par with those obtained through restricted randomization procedures and can be used in conjunction with such procedures to further improve inferential efficiency.

Sequential monitoring with conditional randomization tests

Sequential monitoring in clinical trials is often employed to allow for early stopping and other interim decisions, while maintaining the type I error rate. However, sequential monitoring is typically described only in the context of a population model. We describe a computational method to implement sequential monitoring in a randomization-based context. In particular, we discuss a new technique for the computation of approximate conditional tests following restricted randomization procedures and then apply this technique to approximate the joint distribution of sequentially computed conditional randomization tests. We also describe the computation of a randomization-based analog of the information fraction. We apply these techniques to a restricted randomization procedure, Efron's [Biometrika 58 (1971) 403--417] biased coin design. These techniques require derivation of certain conditional probabilities and conditional covariances of the randomization procedure. We employ combinatoric techniques to derive these for the biased coin design.

Exact properties of Efron’s biased coin randomization procedure

Efron [Biometrika 58 (1971) 403--417] developed a restricted randomization procedure to promote balance between two treatment groups in a sequential clinical trial. He called this the biased coin design. He also introduced the concept of accidental bias, and investigated properties of the procedure with respect to both accidental and selection bias, balance, and randomization-based inference using the steady-state properties of the induced Markov chain. In this paper we revisit this procedure, and derive closed-form expressions for the exact properties of the measures derived asymptotically in Efron's paper. In particular, we derive the exact distribution of the treatment imbalance and the variance-covariance matrix of the treatment assignments. These results have application in the design and analysis of clinical trials, by providing exact formulas to determine the role of the coin's bias probability in the context of selection and accidental bias, balancing properties and randomization-based inference.

Likelihood ratio tests for equality of shape under varying degrees of orientation invariance

We consider a problem from image cytometry where the objective is to describe possible changes in the shape and orientation of cellular nuclei after treatment with a toxin. The shapes of nuclei are represented by individual ellipses. It is argued that the shape comparison problem can be formulated as a generalization of a hypothesis test for the equality of covariance matrices. For many cell types, the test statistic should be invariant with respect to orientations of the cells. For other cell types, the test statistic should be equivariant with respect to orientations of the cells, but invariant with respect to orientations of the images. Likelihood ratio tests (LRTs) are derived under a Wishart model. The likelihood maximization uses a new result about the minimization of the determinant of a sum of matrices under individual rotations. The applicability and limitations of these LRTs are demonstrated by means of simulation experiments. The reference distributions of the test statistics under the null hypothesis are obtained using unrestricted and restricted randomization procedures. Justification for the Wishart model is provided using a residual diagnostic method. The scientific implications of the results are considered.

A New Proposal for Setting Parameter Values in Restricted Randomization Methods

Complete randomization could result in an undesirable imbalance in the number of patients assigned to each treatment, especially in small trials. Therefore, a variety of restricted randomization procedures has been developed. By varying parameters it is possible to appropriately modify the balancing characteristics of these designs. However, there is little information on what are sensible choices for the parameters. Therefore, we suggest a new method for suitable determination of parameter values of restricted randomization rules. For restriction to be effective, it need not yield exact equality. As the reliability of a test is not very sensitive to slight deviations from equal sample sizes we define that a given maximum tolerable imbalance d can be achieved or exceeded with a given probability p(*). By using this condition, parameter values of restricted procedures are determinable. For permuted-block, biased-coin, urn, and big-stick randomization we investigated the impact of parameters on balancing properties. For different extents of restriction and by using the submitted condition, the values of parameters to be chosen are determined. Up to now choice of parameter values has often occurred at random. Now it is possible to determine values of parameters by specifying the tolerable degree of imbalance and the risk to be worse. As a consequence restriction will, as much as possible, not be imposed and not imposed more than necessary in order to preserve the intrinsic quality of randomization.

An Alternative Proposal for “Mixed Randomization” by Schulz and Grimes

Randomization is an important part of clinical trials. Using permuted-block randomization for forcing equal group sizes potentially harms the unpredictability of treatment assignments. This can allow bias to creep into a trial. As an alternative, Schulz and Grimes suggest a "Mixed randomization" scheme which introduces more complexity to realize randomization. The objective of our research was to work out a model for randomization which is easier to handle than "Mixed randomization", with an equal level of performance in unpredictability and balance. We analyzed a "Mixed randomization" procedure regarding the degree of unpredictability and balancing power and compared performance using permuted-block randomization with very large block size in a worst case scenario. Our work was done by the application of Blackwell-Hodges model for evaluation of the unpredictability of treatment assignments. Regarding unpredictability, performance of permuted-block randomization with block size b = 36 was very similar to that of "Mixed randomization". Regarding balancing power it was more favourable than "Mixed randomization". Results of Schulz and Grimes are very important as they emphasized that mildly unequal sample sizes of therapy groups don't cause problems. But the suggested scheme of "Mixed randomization" to a large extent adds complexity and we do not believe that this proposal is very feasible. Basically, we rather recommend the use of only one restricted randomization procedure in the best way. This can be permuted-block randomization with optimum choice of a large block size.

Use of certain 5 × 5 Latin squares for increasing information from agroforestry experiments

A restricted randomization procedure for self-conjugate 5 × 5 Latin squares is shown to be useful for trials on intercropping, particularly agroforestry, as records from border plots need no longer be discarded.

Properties of the urn randomization in clinical trials

In this article we review the important statistical properties of the urn randomization (design) for assigning patients to treatment groups in a clinical trial. The urn design is the most widely studied member of the family of adaptive biased-coin designs. Such designs are a compromise between designs that yield perfect balance in treatment assignments and complete randomization which eliminates experimental bias. The urn design forces a small-sized trial to be balanced but approaches complete randomization as the size of the trial ( n) increases. Thus, the urn design is not as vulnerable to experimental bias as are other restricted randomization procedures. In a clinical trial it may be difficult to postulate that the study subjects constitute a random sample from a well-defined homogeneous population. In this case, a randomization model provides a preferred basis for statistical inference. We describe the large-sample permutational null distributions of linear rank statistics for testing the equality of treatment groups based on the urn design. In general, these permutation tests may be different from those based on the population model, which is equivalent to assuming complete randomization. Poststratified subgroup analyses can also be performed on the basis of the urn design permutational distribution. This provides a basis for analyzing the subset of patients with observed responses when some patients' responses can be assumed to be missing-at-random. For multiple mutually exclusive strata, these tests are correlated. For this case, a combined covariate-adjusted test of treatment effect is described. Finally, we show how to generalize the urn design to a prospectively stratified trial with a fairly large number of strata.

Spatial autocorrelation of ABO serotypes in mediaeval cemeteries as an indicator of ethnic and familial structure

Hungarian cemeteries in an effort to detect familial structure. Different ethnic groups buried in separate areas dominated the spatial pattern in two cemeteries, leaving insufficient power to test for familial patterns. In a third, ethnically homogeneous cemetery, no evidence of familial structure was found. A simulation showed that familial structure could readily be detected by the methods applied when it exists in an ethnically homogeneous population. The spatial autocorrelation methods employed would have detected the ethnic diversity in the two cemeteries containing graves from different populations, even in the absence of archaeological information to that effect. A restricted randomization procedure was developed to test two alternative hypotheses concerning the ethnic designations of the occupants of the cemetery at Szentendre. As a result of this test, the hypothesis that graves located in a rough circle (putative Lombards) differ serologically from those located at the periphery (putative nonLombards) is strongly preferred over a second hypothesis based on grave goods which would imply spatially random placement of the graves of the two ethnic units.

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.

Probabilistic reinforcement of reversal and dimensional shifts

Thirty subjects learned 120 two-dimensional, 2-valued concept-identification (CI) problems over 5 days. After each problem was learned, the solution was shifted by interchanging responses (reversal shift, RS) or by picking a new dimension (dimensional shift, DS). Two groups with complementary probabilities of the type of shift were run. The probabilities of an RS for one group on the 5 days were: 1, 0, .7, .7, .7. The shift was so introduced that the subject's first response indicated the type of shift expected. The subjects learned to predict the shift and showed a tendency to follow the probability of the shift but with a bias toward the RS prediction. Negative recency and alternation tendencies occurred, but seemed to adapt out. The subjects improved greatly in trials to solution for each CI problem. A generalization of the cue-selection model designed to handle initially correct concept selection did not adequately describe the statistic of the trial of the last error, but a restricted randomization procedure accounted for the major deviations. The error distributions were more accurately described. The parameter which measured the initial probability of selecting the correct concept on the first trial also indicated that subjects approached probability matching in their expectations of the type of shift.