Randomized Method for Dose Response Study Permuted Block by Block Randomization

Research Article

Austin Biom and Biostat. 2015;2(2): 1017.

Randomized Method for Dose Response Study Permuted Block by Block Randomization

Yoshiharu Horie1,2*, Fumiaki Takahashi¹ and Masahiro Takeuchi¹

¹Biostatistics Division, Graduate School of Kitasato University, Japan

²Medical Data Service Department, Nippon Boehringer Ingelheim Co., LTD., Japan

*Corresponding author: Yoshiharu Horie, 2-1-1 Osaki,Shinagawa, 141-6017, Tokyo, Japan.

Received: March 23, 2015; Accepted: May 25, 2015; Published: June 04, 2015


Randomization is necessary for reducing bias in clinical studies, especially confirmatory studies. Since the 1970s, researchers proposed various methods of randomization. This has resulted in randomization becoming one of the most effective methods for bias control in clinical studies. Zelen proposed the importance of randomization and applied the permuted block randomization to a clinical study. After his work, Efron proposed the biased coin design in order to balance treatment assignments. Using Efron’s biased coin design; the probability of assignment to the other group is constant, regardless of the degree of imbalance. Wei developed an adaptive biased coin design where the probabilities of assignment adapt according to the degree of imbalance. Despite the value of these randomization methods, the permuted block randomization is the preferred method used in clinical studies. This permuted block method is simple and easily controls the randomization of equal numbers of patients into each treatment group at each center. But the permuted randomization method has some shortcomings, one of which is predicting the allocation as it nears the end of block. This predictability of allocation induces some biases, especially selection bias. Increasing the block size leads to a lower predictability. However, increasing the block size is difficult in studies which includes several treatment groups because a large number of patients have to be randomized at each center. This causes incomplete randomization in several of the centers. Therefore, we propose the following method for improving permuted block randomization.

Keywords: Randomization; Permuted block randomization; Dose response study; Permuted block by block randomization; Relative efficiency


The randomized controlled multicenter study sets the highest standard for clinical research. Random allocation by blocks, with regards to permuted block randomization, is frequently used to balance the number of patients in each treatment group in randomized controlled studies [1]. Permuted Block (PB) randomization has several valuable properties [2]. First, investigators can easily control an equal number of randomized patients into each treatment group in their centers. Investigators can assess all treatment groups set in the study, leading to a reduced center effect on treatment assessment. In other words, if only one treatment group is randomized and evaluated in a center, the assessment by the investigator in this center is reflected in the one treatment group, which causes the center effect. Second, unrestricted randomizations can result in severe imbalances at some point during the study. This is particularly undesirable if there is a time-heterogonous covariate related to treatment outcome, because imbalances in treatment assignments can then lead to imbalances in those important covariates. To avoid this, PB randomization is often used to ensure balance throughout the course of the clinical study. Third, the randomized procedures are easier than some adaptive randomization proposed by Efron [3] or Wei [4-7]. Finally, the logistics (preparation of investigational drugs, generation of allocation codes, and supply of investigational drugs to each center) are simple.

In spite of these properties, the PB randomization has some shortcomings. One of the critical limitations of PB randomization is that it is possible to predict the treatment situated at the end of a block when the block length is known [8]. Lack of concealment of allocation can invite selection bias, which is the preferential enrollment of specific patients into one treatment group over another [9].

One of the solutions for this is to increase block size. When a block size is two in a study setting two treatments, the second randomized patient is sure to take the different drug from the first randomized patient. If a block size is four, then the second randomized patient has a possibility of being allocated the same drug (0.333) or the different drug (0.667). The probability of allocating the second randomized patient to either drug converges to 0.5 as the block size increases. This solution works in confirmatory studies, which basically compare a new drug with a standard drug or a placebo. These studies usually set two treatment groups, meaning the new drug, and the standard drug, or placebo.

However, several doses of an investigational drug are evaluated in most clinical studies during new drug development. The safety profile of a drug is evaluated by setting more than five doses in a phase I study. At least three doses and placebo are also included in a dose response study in order to investigate the dose response relationship. If a PB randomization is applied to a dose response study, which includes at least four treatment groups, the smallest block size are often four. In this case it is difficult to increase the block size since some investigators cannot enroll large numbers of patients in their centers. This causes the randomized number of patients to be imbalanced among treatment groups if enrollments cannot fulfill block sizes.

In general, the same number of patients is randomized into each group at each center participating in a clinical study in order to adjust the effect of the center. Additionally, the investigator at each center seeks to randomize patients into all treatment groups because the imbalance interferes with the relationship between patient and investigator, especially all randomized patients given placebo in a center.

In this paper, we propose the method for reducing the predictability in PB randomization. First, our proposed method is explained in method. Next, the imbalance produced by our proposed method was assessed by formula and simulations, and the results of those simulations are shown in results. In motivating data, an application of this method to an actual study is illustrated. Finally, the results will be discussed.


Our proposed method is illustrated in this Section. A study which includes several dose groups is assumed to be conducted for investigating the dose response relationship (dose response study). Suppose that the Low dose (L), Middle dose (M), High dose (H), and Placebo group (P) are set as treatment groups in a dose responsible study.

Generation of matrix

First, a matrix of each block is generated by the permuted block method [10]. L, M, and H are used as drug codes when the matrix is generated. The block size is then set at three. Therefore, there are the following six combinations. One of the six combinations is repeatedly sampled n times with a replacement. An n by three matrixes, called Matrix 1 (Mat1) is generated after replications.


Next, a matrix which determines the change from active to placebo is also generated by the permuted block method. For this matrix the block size is set at four. This matrix includes three active dose codes (L, M, and H) and a “Stay (S)” code. “Stay” stands for “No change”. Then, 24 combinations are generated (parts of these are shown below). One of the 24 combinations is repeatedly sampled m times with a replacement, where 4m equals n.

An one by n matrixes, called Matrix 2 (Mat2) is generated after replications.


This Mat2 is transposed. Mat2 is changed to a 4m by one matrix, or in other words, an n by one matrix. Then, Mat1 and Mat2 are matched for each block as shown in Table 1. If the same code exists in a block, for example, L exists in Mat1 and Mat2 in Block 1 of Table 1, and then L is changed to P. In other words, P, M, and H are allocated in order. If S exists in Mat2, Block 4 of Table 1, for example, then no code is changed. The new n by three matrixes is generated after this procedure is applied to all blocks (Table 2). Patients will be randomized into each treatment group according to this new matrix. Matching two matrices generated by the PB randomization twice provides the new matrix which is used for the randomization. Therefore, for the rest of this paper, our proposed method will be referred to as permuted block by block randomization.