Efron's Biased Coin Research Articles

Saddlepoint approximation p-values of weighted log-rank tests based on censored clustered data under block Efron's biased-coin design.

Clustered survival data frequently occurs in biomedical research fields and clinical trials. The log-rank tests are used for two independent samples of clustered data tests. We use the block Efron's biased-coin randomization (design) to assign patients to treatment groups in a clinical trial by forcing a sequential experiment to be balanced. In this article, the -values of the null permutation distribution of log-rank tests for clustered data are approximated via the double saddlepoint approximation method. Comprehensive numerical studies are carried out to assess the accuracy of the saddlepoint approximation. This approximation demonstrates great accuracy over the asymptotic normal approximation.

Generalized Efron's biased coin design and its theoretical properties

AbstractIn clinical trials with two treatment arms, Efron's biased coin design, Efron (1971), sequentially assigns a patient to the underrepresented arm with probabilityp> ½. Under this design the proportion of patients in any arm converges to ½, and the convergence rate isn-1, as opposed ton-½under some other popular designs. The generalization of Efron's design toK≥ 2 arms and an unequal target allocation ratio (q1, . . .,qK) can be found in some papers, most of which determine the allocation probabilitiesps in a heuristic way. Nonetheless, it has been noted that by using inappropriateps, the proportion of patients in theKarms never converges to the target ratio. We develop a general theory to answer the question of what allocation probabilities ensure that the realized proportions under a generalized design still converge to the target ratio (q1, . . .,qK) with raten-1.

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.

Balance and randomness in sequential clinical trials: the dominant biased coin design

Efron's biased coin design (BCD) is a well-known randomization technique that helps neutralize selection bias, while keeping the experiment fairly balanced for every sample size. Several extensions of this rule have been proposed, and their properties were analyzed from an asymptotic viewpoint and compared via simulations in a finite setup. The aim of this paper is to push forward these comparisons by taking also into account the adjustable BCD, which is never considered up to now. Firstly, we show that the adjustable BCD performs better than Efron's coin with respect to both loss of precision and randomness. Moreover, the adjustable BCD is always more balanced than the other coins and, only for some sample sizes, slightly more predictable. Therefore, we suggest the dominant BCD, namely a new and flexible class of procedures that can change the allocation rule step by step in order to ensure very good performance in terms of both balance and selection bias for any sample size. Our simulations demonstrate that the dominant BCD is more balanced and, at the same time, less or equally predictable than Atkinson's optimum BCD.

Algorithm for balancing both continuous and categorical covariates in randomized controlled trials

Minimization as proposed by Pocock and Simon for balancing categorical covariates in clinical trials with individual-level, consecutive randomization has been increasingly used. An extension of the method exists that uses the symmetric Kullback–Leibler divergence index to balance both categorical and continuous covariates, albeit for two-arm randomized controlled trials only. To date, the algorithm has not been made widely available to researchers via publicly accessible software.We adapted Endo et al.’s algorithm to randomized trials with two or more arms. In addition, our algorithm incorporates Efron's biased coin method to decrease the predictability of assignment even when a predefined threshold of difference in the numbers of subjects between treatment arms is reached, whereas Endo et al.’s algorithm assigns the next subject to the treatment of smaller size with certainty. We developed code in the free statistical software R to make the algorithm readily available to trialists. While there are no definitive answers regarding the optimal choices for certain statistical parameters that must be defined prior to algorithm application (Pk, Dn, and p_Dn), we provide guidance on these.We conducted simulations with actual data from a three-arm randomized trial to compare the modified algorithm and R code to another published biased coin minimization method that can accommodate continuous and categorical covariates in multi-arm trials. The three-arm trial used three categorical covariates (sex, race/ethnicity, and online personal health record access) and four continuous covariates (age, fasting blood glucose, body mass index, and waist circumference). All of the continuous and categorical covariates were well balanced, and the results were comparable to the comparison method.

Block urn design — A new randomization algorithm for sequential trials with two or more treatments and balanced or unbalanced allocation

Permuted block design is the most popular randomization method used in clinical trials, especially for trials with more than two treatments and unbalanced allocation, because of its consistent imbalance control and simplicity in implementation. However, the risk of selection biases caused by high proportion of deterministic assignments is a cause of concern. Efron's biased coin design and Wei's urn design provide better allocation randomness without deterministic assignments, but they do not consistently control treatment imbalances. Alternative randomization designs with improved performances have been proposed over the past few decades, including Soares and Wu's big stick design, which has high allocation randomness, but is limited to two-treatment balanced allocation scenarios only, and Berger's maximal procedure design which has a high allocation randomness and a potential for more general trial scenarios, but lacks the explicit function for the conditional allocation probability and is more complex to implement than most other designs. The block urn design proposed in this paper combines the advantages of existing randomization designs while overcoming their limitations. Statistical properties of the new algorithm are assessed and compared to currently available designs via analytical and computer simulation approaches. The results suggest that the block urn design simultaneously provides consistent imbalance control and high allocation randomness. It can be easily implemented for sequential clinical trials with two or more treatments and balanced or unbalanced allocation.

Efficient randomized-adaptive designs

Response-adaptive randomization has recently attracted a lot of attention in the literature. In this paper, we propose a new and simple family of response-adaptive randomization procedures that attain the Cramer--Rao lower bounds on the allocation variances for any allocation proportions, including optimal allocation proportions. The allocation probability functions of proposed procedures are discontinuous. The existing large sample theory for adaptive designs relies on Taylor expansions of the allocation probability functions, which do not apply to nondifferentiable cases. In the present paper, we study stopping times of stochastic processes to establish the asymptotic efficiency results. Furthermore, we demonstrate our proposal through examples, simulations and a discussion on the relationship with earlier works, including Efron's biased coin design.

A theoretical analysis of the power of biased coin designs

Outlining some recently obtained results of Hu and Rosenberger [2003. Optimality, variability, power: evaluating response-adaptive randomization procedures for treatment comparisons. J. Amer. Statist. Assoc. 98, 671–678] and Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835] on the relationship between sequential randomized designs and the power of the usual statistical procedures for testing the equivalence of two competing treatments, the aim of this paper is to provide theoretical proofs of the numerical results of Chen [2006. The power of Efron's biased coin design. J. Statist. Plann. Inference 136, 1824–1835]. Furthermore, we prove that the Adjustable Biased Coin Design [Baldi Antognini A., Giovagnoli, A., 2004. A new “biased coin design” for the sequential allocation of two treatments. J. Roy. Statist. Soc. Ser. C 53, 651–664] is uniformly more powerful than the other “coin” designs proposed in the literature for any sample size.

The power of Efron's biased coin design

The power of a statistical test depends on the sample size. Moreover, in a randomized trial where two treatments are compared, the power also depends on the number of assignments of each treatment. We can treat the power as the conditional probability of correctly detecting a treatment effect given a particular treatment allocation status. This paper uses a simple z-test and a t-test to demonstrate and analyze the power function under the biased coin design proposed by Efron in 1971. We numerically show that Efron's biased coin design is uniformly more powerful than the perfect simple randomization.

SEQUENTIALLY ADJUSTED RANDOMIZATION TO FORCE BALANCE IN CONTROLLED TRIALS WITH UNKNOWN PREVALENCE OF COVARIATES: APPLICATION TO ALCOHOLISM RESEARCH

In treatment outcome studies with small to medium sample sizes (n < 200), the balance of groups with regard to important factors, which sometimes occur at low prevalence, is indispensable for adequate interpretation. This study tested a method for use in clinical alcoholism research, an uncomplicated procedure for satisfactory randomization of patients to different treatments, taking into account relevant background variables. An easily applicable modification of Efron's biased coin method for the randomization of treatments within strata of unknown but low prevalence was compared with the original approach and alternative methods by computer simulation (10,000 runs). An application example for a clinical trial in alcoholism research is given. The sequentially adjusted randomization procedure revealed results similar to Efron's approach without the necessity for monitoring the assignment history throughout the trial. The new method was slightly superior to Efron's approach in randomizing subjects in strata with n < or = 20, whereas strata with n > 20 favoured randomization with Efron's approach. Taking into account all results from simulation, the new approach reached a proportion of acceptable balanced randomization > 95% in all stratum sizes. The approach, a special case of the standard urn design, provides three major advantages in clinical trials: (i) it can be easily implemented in any trial without technical equipment; (ii) it works with high accuracy in trials with a priori unknown but low numbers of subjects (4 < or = n < or = 20) in prognostic relevant strata; and (iii) a deterministic assignment tendency is completely avoided, as a random process takes place throughout the assignment procedure. The modified biased coin method can be recommended as one possible strategy for special purposes, particularly in alcoholism research.

Asymptotic properties of doubly adaptive biased coin designs for multitreatment clinical trials

A general doubly adaptive biased coin design is proposed for the allocation of subjects to K treatments in a clinical trial. This design follows the same spirit as Efron's biased coin design and applies to the cases where the desired allocation proportions are unknown, but estimated sequentially. Strong consistency, a law of the iterated logarithm and asymptotic normality of this design are obtained under some widely satisfied conditions. For two treatments, a new family of designs is proposed and shown to be less variable than both the randomized play-the-winner rule and the adaptive randomized design. Also the proposed design tends toward a randomization scheme (with a fixed target proportion) as the size of the experiment increases.