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Author: dr. J.M. Binnekade
Co-author: dr. ir. N. van Geloven
Authorship

This page was translated from the Dutch-language original using AI and has not been fully reviewed. Where the two differ, the Dutch version takes precedence.

• What is randomization?
• When should you randomize?
• Why randomize?
• What does randomization not do?
• Which methods are unsuitable for randomization?
• Which software is available for randomization?
• What types of randomization are there?
  • Simple randomization
  • Block randomization
  • Stratified randomization
  • Minimization
• Questions about randomization
  • Can I randomize by order of registration?
  • With stratified randomization, can it happen that certain patients can no longer be included?
  • One of our trial patients has dropped out; is it wise to free up the randomization slot and assign it to a new patient?
• References

Randomization is a technique whereby patients, subjects or laboratory animals are assigned to one of the groups of an experimental study on the basis of chance (a draw).

A randomized controlled trial is the gold standard for evaluating interventions in healthcare. The idea behind the study design is that subjects are assigned to one of the study arms on the basis of chance (at random). The intended result is an equal distribution of the characteristics of known and unknown prognostic factors across the study arms. After randomization, you expect that the effect of natural course, placebo effects, confounders and measurement errors will cancel out between the two groups. As a result, you ultimately only measure the "net" effect of the intervention as the difference between the intervention and control group.

In experimental research where two or more groups are compared and where the researcher can manipulate the intervention, thereby potentially influencing the assignment of the intervention or the treatment of the subjects.

There are three reasons to randomize:

1. Ideally, the difference between the arms of a study consists only of the effect of the therapy, the intervention. By randomizing, all other factors that could influence the treatment effect are distributed by chance across both groups. This ensures that the effect of known and unknown (prognostic) factors is distributed at random across both study arms.
2. Randomization helps blind the intervention to caregivers and subjects. It prevents patterns in patient responses from being recognized, which could reveal the assignment to one of the study arms to the caregiver.
3. Randomization yields groups that are a random sample of the study population, thereby satisfying a condition for applying the standard statistical tests that are based on probability theory.

Randomization is no guarantee that prognostic factors will actually be equally distributed between study groups; chance can turn out unfavorably. The probability of an unequal distribution increases as the sample size decreases. With an arbitrarily large sample, prognostic factors will become equally distributed.

There are several methods for assigning patients to a study arm that do not count as random. The objection to these methods is the predictability of the assignment. Because physicians strive to treat every patient optimally, a predictable assignment can create a conflict between what the physician considers desirable, at that moment, under those specific circumstances, and what the care associated with the assigned study arm entails.
Examples of assignment methods associated with a high degree of predictability are: according to a fixed order (patient 1 gets A, patient 2 gets B, patient 3 gets A etc.); based on the date or day of the week (patients included on even days get A, patients included on odd days get B); based on the patient's initials (A-K → treatment A and M-Z → treatment B). These methods are strongly discouraged because they can easily be manipulated.

In general, randomization at Amsterdam UMC is done in Castor. If the desired method or settings are not available in Castor, researchers at Amsterdam UMC can request support.

There are several ways to randomize, including simple, block, stratified randomization and minimization.

With two study arms, this method is comparable to flipping a coin. In practice, it works with random numbers. A treatment is assigned according to a rule set in advance. For example, a computer generates a random number between 0 and 1. If this number is less than or equal to 0.5, treatment A is applied; if the number is greater than 0.5, treatment B is applied. A disadvantage of this method is that it does not guarantee that the sample will be divided exactly equally across the study arms. By chance, a long sequence of one-sided assignment can occur, for example AAAABBA. The smaller the sample, the greater the risk of an unequal distribution.
To avoid an unequal distribution between the study arms, you can generate a list of random numbers with a computer before the start of the study and assess it for usability. If the list has an unequal distribution of more than 10 between the study arms, you can generate a new list of random numbers. Experience shows that the simple randomization method is only applicable for an intended sample size of roughly more than 200 [1].

The disadvantage of simple randomization is the risk of an unequal distribution of the number of patients across the study arms. By randomizing in blocks you can avoid this. A block has a certain size, for example 4, 6 or 8, in which both study arms are proportionally distributed (example ABAB or BAAB etc.). After each block you therefore have an equal distribution across both study arms. The block size is a multiple of the number of study arms in the study.

If blocks are not used in full, an unequal distribution between study arms can still occur. For example, if a study with a block size of 8 stops halfway through, the maximum imbalance is 4 patients.

Criticism of the use of small blocks is that the randomization can become partly predictable. Therefore, it is preferable to use large blocks or blocks of randomly varying size.

Example of randomization with a block size of 4 (therapy = A, control = B) R → ABAB BABA AABB ABBA ABBA ABAB BABA AABB ABBA etc.

With simple or block randomization, there is no guarantee that known prognostic factors will be equally distributed across the study arms. The probability that prognostic factors will be unequally distributed is greater with a small sample than with a large sample. One method to prevent an unequal distribution of known prognostic factors is stratified randomization. Stratification means that a separate randomization list is used for each combination of prognostic factors (for each stratum). Stratified randomization is often used in combination with the block design to distribute the intervention proportionally across both study arms.

Example of a stratified block randomization stratified by sex and age group (therapy = A, control = B)

	Patient characteristics		Randomization order
List1	male	median age-	ABAB AABB ABBA BABA
List2	male	median age+	BAAB BBAA ABAB AABB
List3	female	median age-	AABB ABBA BAAB BBAA
List4	female	median age+	ABBA BABA ABAB AABB

Based on the patient's characteristics, the correct list is chosen; a male who is older than the median age is randomized from list 2. Stratification becomes more important as the sample size decreases, because the risk of a disproportionate distribution of prognostic factors is then greater. However, the number of randomization lists you have to maintain grows quickly as you want to take more prognostic factors into account. Before you know it, you have as many strata as patients in your sample. With this method, a disproportionate distribution of patient numbers between the study arms is equally possible (see block randomization). Here, block size and the number of strata play a role. If, in the example above, the last blocks in all strata happened to be AABB and the study stops after only the AA's have been used, the maximum (highly theoretical) imbalance is 8 patients. If you want to take multiple prognostic factors into account, you can combine several factors into an index and then use the index to define the strata. Note: when stratified randomization is used, the analysis must also correct for the stratification factors. See for example the articles by Kahan and Morris [2,3] and the EMA guideline on adjustment for baseline covariates. An accessible explanation of this can be found in this post by Jonathan Bartlett.

Minimization also aims for an equal distribution of patients and their prognostic factors among the study arms. The difference with stratified block randomization is that with minimization, no strata (subgroups) are defined in advance from which randomization takes place. Before a new patient is randomized, the minimization method assesses how his or her distribution of prognostic factors compares with the patients randomized so far. If this distribution is in balance, the patient is simply randomized (heads or tails). If there is an imbalance, the draw will be deliberately influenced in favor of the group that reduces the imbalance.

Example of randomization using the minimization method.

Overview of previously randomized patients
Prognostic factor	Intervention	Control
Men	3	5
Women	5	3
Age 21-30	4	4
Age 31-40	2	3
Age 41-50	2	1
High risk	4	5
Low risk	4	3

Suppose the next patient is a man, between 31 and 40 years old, with high risk.
So far, 3 men have been randomized to the intervention group, 2 patients are aged between 31-40, and 4 patients have high risk.
In the control group, 5 men have been randomized, 3 patients are aged between 31-40, and 5 patients have high risk.
The imbalance that would now arise if randomized to the intervention arm is, for men, 5 - 4 = 1, for the age category 3 - 3 = 0, and for high risk 5 - 5 = 0, a total of 1 (1+0+0). If randomized to the control arm, an imbalance arises of 6 - 3 = 3 (men), 4 - 2 = 2 (age 31-40) and 6 - 4 = 2 (high risk), a total imbalance of 7 (3+2+2). There are now several options for assigning the patient. With strict minimization, the patient is assigned to the intervention group because 1 < 7. The disadvantage of this method is predictability. Another option is to adjust the probability ratio for assignment, for example 80:20. The probability ratio can be determined by the degree of imbalance (common practice). If groups are in balance, assignment takes place on a fifty-fifty basis.

We are going to conduct a double-blind crossover study. A group of adult patients will receive an additional supplement over 2 periods of 4 weeks, containing a possibly active substance in one period and a placebo in the other period. The goal is to include 20 patients. We would like your advice on how to randomize. There will be a group 1, which first receives the blue supplement and the green supplement in the second period, and a group 2, which first receives the green supplement and the blue supplement in the second period. My proposal would be to randomize in order of registration, with odd numbers (1st, 3rd, 5th registration etc.) going into group 1 and even numbers (2nd, 4th, 6th etc.) registration going into group 2. Is this a correct method, or are there pitfalls?

Alternating randomization is dangerous because blinding can then easily be broken (once you know that number 5 received green, you also immediately know all subsequent assignments!). In meta-analyses, studies are even excluded if they were randomized based on, for example, day of the week. It is advisable to use a randomization program. Whoever administers the agent must not know whether it is green or blue. In the second (crossover) period, the 'randomizer' must switch all green to blue and vice versa. The caregiver again remains blinded.

I am about to start an RCT in which I want 100 patients in the intervention group and 100 in the control group. Stratification will be for sex (male/female), age (50-70/70-90) and operating surgeon. My question now is: can it happen (especially later in the randomization process) that patients are NOT included because a randomization stratum is already full and the patient does not meet the requirements set by the stratification procedure (e.g. not the right age, sex, surgeon already full, etc.)? I ask this because the patients I want to include are hard to find and every patient really counts.

Generally, when using stratified randomization, no limit is set on, for example, the 'absolute' number of men or women to be included. Suppose only 10% of the enrolled patients are women; then the stratification aims to achieve that 10% women in both treatment groups. It is not the case that at some point 'no more women may be included' or the like. What does need to be taken into account is that when using (multiple) stratification factors, it becomes more difficult to achieve that balance, and it can also become more difficult to ensure the overall balance (100 vs 100). It might then, for example, become 95-105, or 5% women in one arm and 15% in the other. You need to consider in advance how many stratification factors can be included to ensure a reasonable balance. But it is therefore not the case that patients 'cannot participate'.

At the moment, a number of patients have been randomized, but now a patient drops out midway through the list. This patient was included incorrectly. We would prefer to reassign this slot to a new patient. Is that allowed, or are there methodological objections? In principle, the randomization list is long enough to leave the slot empty.

The approach to 'dropout' of randomized patients depends on a number of factors:

1. intention-to-treat principle: According to the 'intention-to-treat' principle, everyone who was ever included in a study should continue to be followed until the end according to the randomized allocation. Even if there are protocol violations such as switching treatment, not adhering to the prescribed treatment, or later turning out not to fully meet the inclusion criteria. The idea behind this is that when the treatment is recommended (for example in guidelines) for patients with certain criteria, similar 'misjudgments'/protocol violations may occur, and a comparison of effectiveness within a study population including these violators gives the best estimate of the effect in reality/practice. If you want to adhere to the intention-to-treat principle in the trial, this patient should not be excluded.

2. sample size calculation: The sample size calculation can take dropout into account in various ways. The sample size calculation often already accounts for a certain degree of dropout. For example, 100 patients are needed for the analysis, and 110 are included because some are expected to drop out/not be usable in the analysis. In this case, you would randomize 110 patients (a randomization list of 110). Ultimately, 10 of these will not be included in the analysis; that is already accounted for, and a vacated slot therefore does not need to be filled. Very occasionally, the sample size calculation is set up such that, for example, 10 men and 10 women are needed who must have strictly followed the entire protocol (power calculation based on the per-protocol analysis). This is often the case in very early-phase, experimental studies. In such a case, a slot can be filled by a new patient upon dropout. In some situations, it is advisable to fill the vacated randomization slots (see point 3 below).

3. balance in the randomization list: When there is a small sample size, and possibly also stratification in the randomization (for example 10 men in one stratum and 10 women in another stratum), dropout of one of the patients and adding a new patient with a new randomization outcome can produce an undesired skewed allocation (for example men 4:6 instead of the desired 5:5). In such cases, one may choose to have a vacated randomization slot filled by the new 'replacement' patient. A disadvantage of this, however, is that (in unblinded studies) the randomization outcome of the replacement is already known before inclusion. When there is a larger sample size, a small imbalance in allocation is generally not a problem. For example, a trial with 47:53 allocation is fine and has just as much power as a 50:50 allocation. Assuming that patient dropout does not occur in only one of the two arms (which would indicate a completely different problem), there is generally no expected problem with allocation balance, and it is recommended to simply use the next position in the randomization list after a patient drops out.

In summary, it is generally inadvisable to 'free up and refill' randomization positions. Exceptions to this are early-phase experimental studies with a strong desire for balanced allocations. Another exception is when randomization was done incorrectly due to, for example, a technical error such as 'pressing the randomization button twice'.

1. Pocock SJ. Allocation of patients to treatment in clinical trials. Biometrics 1979; 35: 183-197 DOI 10.2307/2529944
2. Kahan, B. C. and Morris, T. P. (2012), Improper analysis of trials randomised using stratified blocks or minimisation. Statist. Med., 31: 328–340 DOI: 10.1002/sim.4431
3. Kahan BC, Morris TP. Reporting and analysis of trials using stratified randomisation in leading medical journals: review and reanalysis. BMJ 2012; 345 :e5840 DOI 10.1136/bmj.e5840
4. Kernan WN, Viscoli CM, Makuch RW, Brass LM, Horwitz RI. Stratified randomization for clinical trials. J Clin Epidemiol. 1999 Jan;52(1):19-26. DOI 10.1016/S0895-4356(98)00138-3
5. DG Altman, JM Bland. Treatment allocation by minimization. BMJ. 2005 April 9; 330(7495): 843. DOI 10.1136/bmj.330.7495.843
6. Scott NW, McPherson GC, Ramsay CR, Campbell MK. The method of minimization for allocation to clinical trials. a review. Control Clin Trials. 2002 Dec;23(6):662-74. DOI 10.1016/S0197-2456(02)00242-8
7. Schulz KF, Grimes DA. Generation of allocation sequences in randomised trials: chance, not choice. Lancet. 2002 Feb 9;359(9305):515-9. DOI 10.1016/S0140-6736(02)07683-3
8. Kang M, Ragan BG, Park JH. Issues in outcomes research: an overview of randomization techniques for clinical trials. J Athl Train. 2008 Apr-Jun;43(2):215-21. DOI 10.4085/1062-6050-43.2.215.

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