Randomization is employed in trials to ensure internal validity. One of the key determinants of trial quality is the specific randomization method used to allocate study participants to treatment groups.

Unrestricted Randomization: Many investigators are aware of unrestricted randomization. In concept, unrestricted randomization is similar to tossing a fair coin for each participant assignment, with no dependence or restrictions based on prior allocations. There are two major concerns with unrestricted randomization. First it allows for the possibility of unequal group sizes. And second, it enables chronological bias [1] by allowing not only unbalanced group sizes but also imbalances over time. That is, even if the final group sizes are nearly or exactly the same size, it still may happen that significantly more early allocations are to one treatment group and significantly more late ones are to the other treatment group. This becomes a problem when there are time trends, because then, through a common association with time, treatment assignment and some predictor of success will become confounded. Moreover, it is also impractical to use unrestricted randomization in the way intended [8]. These issues render unrestricted randomization less than ideal.

Permuted Block Randomization: Given the limitations of unrestricted randomization and concerns about chronological bias, many investigators choose to employ a method called permuted block randomization. However, blocked randomization can be subject to selection bias if participant allocation is not concealed appropriately. Allocations may be observed directly, but, even if allocations are not observed directly, they may be predicted based on known patterns imposed by the allocation scheme. For example, if the block size is four, then the first four participants enrolled form the first block, and the next four form the next block, and so on. Randomization occurs within the blocks, so with blocks of size four there are six possible patterns: AABB, ABAB, ABBA, BAAB, BABA, and BBAA. Each block will randomize according to one of these six patterns, with the patterns chosen randomly and, generally, independently from block to block. Knowledge of prior allocations and block sizes makes it possible to easily predict the last allocation in each block, and the possible selection bias can interfere with the internal validity of a trial. Though any restricted randomization procedure will allow some prediction of future allocations, some allow more and others allow less [2]. By this measure, permuted block randomization is a suboptimal method.

Alternative Randomization Procedures: Despite the ubiquity of permuted block randomization, there are alternative randomization procedures available that simultaneously lower allocation predictability and protect against chronological bias. These procedures are based not on blocks, but rather a maximally tolerated imbalance (MTI). As the name suggests, the MTI is a quantity the investigator specifies to indicate how unbalanced the group sizes can ever become during the course of randomization. For example, if the MTI is chosen to be three, then at no time can one treatment group have more than three patients more than the other one. The larger the MTI, the less restricted is the randomization.

There are four primary MTI randomization procedures: the big stick procedure [3], Chen's procedure [4], the maximal procedure [5], and the block urn design [6]. Each of these four procedures allows the user to specify the MTI value.

The big stick procedure continues to use equal allocation probabilities for any level of imbalance short of the MTI, and Chen's procedure uses a common biasing probability (which the user selects), independent of the level of imbalance, for any imbalance short of the MTI. The maximal procedure and the block urn design both use strictly monotonic allocation probabilities. In other words, as the level of imbalance increases, the allocation probability also shifts incrementally to more forcefully encourage a return to balance. It is important to note that all four of these procedures consistently and substantially reduce predictability, and, consequentially, the risk of selection bias, as compared to computed permuted blocks. See Table II of [7] for a graphical illustration of how these MTI procedures compare to one another.

This website provides tools to generate an allocation scheme using the asymptotic maximal procedure.

Technically, the asymptotic maximal procedure is not the same as the standard maximal procedure, but it does retain the primary characteristics, and differs only quantitatively, as it uses stable Markovian allocation probabilities that do not vary over time. It can be considered a modified maximal procedure so as to be easier to automate and more transparent in terms of understanding the allocation probabilities. This procedure may be used in the common case of two treatment groups with equal allocation to each.

1. Matts, JP, McHugh, RB. Conditional Markov Chain Design for Accrual Clinical Trials. Biometrical Journal 1983; 25 }: 563-577.
2. Berger, VW. Is Allocation Concealment A Binary Phenomenon?. 2005; 183 (3)}: 165.
3. Soares, JF, We, CFL. Some Restricted Randomization Rules in Sequential Designs. Communications in Statistics Theory and Methods 1983; 12 }: 2017-2034.
4. Chen, YP. Biased Coin Design With Imbalance Intolerance. Communications in Statistics Stochastic Models 1999; 15 }: 953-975.
5. Berger, VW, Ivanova, A, Deloria-Knoll, M. Minimizing Predictability while Retaining Balance through the Use of Less Restrictive Randomization Procedures. Statistics in Medicine 2003; 22 (19)}: 3017-3028.
6. Zhao, W, Weng, Y. Block Urn Design -- A New Randomization Algorithm for Sequential Trials with Two or More Treatments and Balanced or Unbalanced Allocation. Contemporary Clinical Trials 2011; 32 (6)}: 953-961.
7. Berger, VW, Bejleri, K, Agnor, R. Comparing MTI Randomization Procedures To Blocked Randomization. Statistics In Medicine 2016; 35 }: 685-694.
8. Berger, VW. The Alleged Benefits of Unrestricted Randomization in "Randomization, Masking, and Allocation Concealment". Boca Raton: Vance W. Berger, Editor, CRC Press, Chapman and Hall; 2018.