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Learn About Randomization

Clinical trials are research studies "in which one or more human subjects are prospectively assigned to one or more interventions (which may include placebo or other control) to evaluate the effects of those interventions on health-related biomedical or behavioral outcomes," as defined by the NIH.

A key determinant of randomized clinical trial quality is the process by which subjects (or "participants") are assigned an intervention (or "treatment group").

Reducing predictability with randomization

Treatment groups must comprise similar participants so that the comparison of treatments is unbiased. It is imperative to avoid selection bias by ensuring that participants are not assigned to a group based on identifiable health characteristics. To this end, participants are assigned to groups randomly. All randomization procedures are intended to reduce selection bias, but they differ in how effectively they achieve this goal. Therefore, the method of randomization chosen must aim to reduce the predictability of assignment to each treatment group.

Maintaining balance over time

According to a trial's design requirements, each treatment group ideally will have been assigned a certain number of participants by each time point. If treatments are assigned randomly, without restriction, it could happen by chance that one treatment group is assigned too frequently, while another group is assigned too infrequently. Even if the treatment groups end in balance, it may still happen that at some point during the trial the imbalance reaches unacceptable levels. The method of randomization chosen must ensure that, as the randomization proceeds, the expected number of participants are assigned to each treatment group and that balance is maintained over time.

A maximally tolerated imbalance (MTI) procedure is an ideal way to randomly assign participants to treatment groups in a clinical trial. As the name suggests, an MTI is a quantity which specifies how unbalanced the group sizes can ever become during the randomization sequence. In other words, as each participant is randomized, the difference between any two treatment groups never exceeds the MTI.

MTI procedures have an important advantage compared to other methods. While randomization procedures in general maintain balance across treatment groups, MTI procedures do so with reduced predictability of group assignment. This is a crucial consideration, as the ability to predict treatment group must be minimized. MTI procedures are designed for that purpose.

Choosing an MTI

MTI procedures don't have a defined value for the MTI, so a value must be chosen by the investigator when planning the randomization process. A small MTI will maintain a tight group balance but suffer from increased group predictability. Conversely, a larger MTI decreases group predictability but is susceptible to greater group imbalance at any given time. For large trials, a larger MTI may be appropriate, and small trials might employ a smaller MTI. The choice of the MTI is up to the investigator. – If you are using the randomization tool on this site, a default MTI value is provided.

As to the computational calculation of the randomization sequence, a variety of MTI procedures are available: the maximal procedure [1], the asymptotic maximal procedure [2], the big stick procedure [3], Chen's procedure [4], and the block urn design [5]. See Table II of [6] for a graphical illustration of how these MTI procedures compare. The choice of procedure is up to the investigator. – If you are using the randomization tool on this site, the default procedure is the asymptotic maximal procedure.

Ultimately, the choice of MTI value and MTI procedure are left to the investigator.

Example of MTI randomization

Suppose participants are randomized into two groups, one treatment and one control. The investigator chooses an MTI of 3. At no point in time can the imbalance between treatment groups exceed 3.

If at any particular time during the trial we see, for example, 20 subjects allocated to treatment A and 23 subjects allocated to treatment B, and if the MTI value is 3, then we have reached the MTI boundary. Therefore, any MTI procedure will ensure that the next allocation (44th subject) is to treatment A, so as to avoid violating the MTI condition.

Maximal procedure

Among MTI procedures, the maximal procedure is often considered best at reducing the overall predictability of treatment assignment. It does this by constructing all possible ways participants might be randomized into groups, and constructing a course of randomization that is just as likely as any other course.

In a maximal procedure the probability of assignment to a treatment depends on the current imbalance between groups and on the number of participants remaining to be randomized (i.e., how many allocations remain in the randomization sequence).

Asymptotic maximal procedure

The asymptotic maximal procedure is based upon the same principles as the maximal procedure but is simpler in design and implementation. In the asymptotic maximal procedure, the probability of group assignment depends only on the imbalance between the group sizes. The asymptotic maximal procedure uses fixed allocation probabilities, so that it is more easily described to clinical colleagues than the maximal procedure.

As trial size increases, the asymptotic maximal procedure becomes more like a maximal procedure.

Chen's procedure

Like the maximal and asymptotic maximal procedures, Chen's procedure applies a balance-forcing probability when treatment group sizes are unbalanced. Unlike the maximal and asymptotic maximal procedures, Chen's procedure ignores the extent of imbalance and considers only whether groups are or are not balanced.

Big stick procedure

The big stick procedure is in essence a special case of Chen's procedure. In the big stick procedure, the MTI constrains imbalance between treatment groups, but the probability of treatment assignment is otherwise independent of whether the treatment has more or fewer participants than expected by that point in time.

Block urn design

Another method is the block urn design, which in some ways is analogous to permuted blocks (see Suboptimal Methods below), but much improved in that the blocks "reset" under certain conditions, reducing the overall predictability of treatment group assignments.

Unrestricted randomization

In concept, unrestricted randomization is similar to tossing a fair coin for each participant assignment, with no consideration given to the number of participants currently assigned to each treatment group. An obvious concern with unrestricted randomization is the risk of large group imbalances and, relatedly, chronological bias [7]. Chronological bias means that, even if final group sizes are their intended size, it may happen that a treatment is assigned too frequently during one period of time and too infrequently at another time. 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 the need to maintain group balance, many investigators employ a method called permuted block randomization. As the name suggests, treatment group assignment order is arranged within small blocks and participants are assigned accordingly within each block of treatments. By the end of each block, an exact and known number of participants have been assigned to each treatment group, and a new block begins. A major disadvantage to blocked randomization is its high group assignment predictability and susceptibility to selection bias, and so it is not preferred when better methods are available [9].

As an example, consider 2 treatment groups, A and B, where each group will have the same number of subjects. If the block size is 4, treatments can be arranged six different ways: AABB, ABAB, ABBA, BAAB, BABA, and BBAA. At the start of randomization, one of these blocks is chosen randomly. If after the first 3 participants are randomized the pattern ABB is observed, then it is known for certain that the next participant will be assigned to treatment A. In other words, the 4th allocation is deterministic for treatment A. In this case, even the 2nd allocation is predictable, though not deterministic.

Even if block procedures employ varied block size (such as randomly alternating between blocks of 4 and 6) in an attempt (generally unsuccessful) to eliminate deterministic assignments at the end of each block, permuted blocks force balance between treatment groups so frequently that the overall predictability of group assignment is much higher than the preferred MTI procedures.

  1. 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. doi:10.1002/sim.1538
  2. Zhao, W, Berger, VW, Yu, Z. The Asymptotic Maximal Procedure for Subject Randomization in Clinical Trials. Statistical Methods in Medical Research. 2018;27(7):2142-2153. doi:10.1177/0962280216677107
  3. Soares, JF, We, CFL. Some Restricted Randomization Rules in Sequential Designs. Communications in Statistics Theory and Methods. 1983;12:2017-2034. doi:10.1080/03610928308828586
  4. Chen, YP. Biased Coin Design With Imbalance Intolerance. Communications in Statistics Stochastic Models. 1999;15:953-975. doi:10.1080/15326349908807570
  5. 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. doi:10.1016/j.cct.2011.08.004
  6. Berger, VW, Bejleri, K, Agnor, R. Comparing MTI Randomization Procedures To Blocked Randomization. Statistics In Medicine. 2016;35:685-694. doi:10.1002/sim.6637
  7. Matts, JP, McHugh, RB. Conditional Markov Chain Design for Accrual Clinical Trials. Biometrical Journal. 1983;25:563-577. doi:10.1002/bimj.19830250608
  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.
  9. Berger, VW. Allocation Concealment and Blinding: When Ignorance is Bliss. Medical Journal of Australia. 2005;183(3):165. doi:10.5694/j.1326-5377.2005.tb06974.x