Minimisation (Trial Allocation)

Minimisation (Trial Allocation) in Randomised Trials

Minimisation here means allocation decided participant by participant from the running totals in each arm, rather than from a list drawn up before recruitment, so that the arms stay alike on a short list of prognostic factors such as age or disease severity. It is unrelated to cost-minimisation analysis, the form of economic evaluation that compares only costs once outcomes have been shown to be equivalent; only the word is shared. Taves described the method in 1974, and Pocock and Simon set out a general family of such procedures in 1975. Health economists meet it because some of the trials that supply treatment effects, costs and QALYs to health technology assessment are allocated this way, and the allocation method shapes how the trial data, economic data included, should be analysed and appraised.

Why chance imbalance matters and why stratification runs out

Simple randomisation gives arms that are alike on average, but in a single small trial one arm can by chance hold more older or sicker participants. ICH E9 notes that randomisation tends to produce groups in which the distributions of prognostic factors, known and unknown, are similar, and that stratifying on important baseline factors has greater potential benefit in small trials. Such an imbalance does not bias the method across repeated trials, but it can distort the estimate from the one trial a decision maker actually has. The EMA guideline on adjustment for baseline covariates describes this as conditional bias from chance covariate imbalance.

Stratified randomisation, usually with block randomisation inside each stratum, is the traditional remedy. Its weakness is the number of cells. Two levels of each of five factors already make 32 strata, and Pocock and Simon observed that permuted blocks within strata prove inadequate once the number of strata approaches the sample size. ICH E9 adds that using more than two or three stratification factors is rarely necessary, less successful at achieving balance and logistically troublesome. Minimisation was proposed for exactly this situation, where several factors matter and the trial is too small to stratify on all of them.

How the minimisation algorithm assigns each participant

Minimisation balances each factor on its own across the arms, rather than balancing every combination of factors. The CONSORT 2025 explanation and elaboration describes the usual procedure. The first participant is allocated at random. For each later participant, the allocation that would minimise imbalance on the selected factors at that moment is identified, and it is either used outright or chosen at random with a heavy weighting, for example a probability of 0.8. Taves's original method is the simplest member of the family that Pocock and Simon later described.

For two arms, A and B, the imbalance that would follow each possible allocation can be written as a weighted sum over the factors.

$$D_k = \sum_{j=1}^{J} w_j \left| a_{j}^{(k)} - b_{j}^{(k)} \right|$$

where $D_k$ is the total imbalance if the new participant joined arm $k$, $J$ is the number of minimisation factors, $a_{j}^{(k)}$ and $b_{j}^{(k)}$ are the numbers of participants in arms A and B at the new participant's level of factor $j$ after that allocation, and $w_j$ is a weight for factor $j$, equal to 1 for every factor when all factors count equally. The arm with the smaller $D_k$ is favoured, and a tie is broken at random. The EMA guideline describes the same idea in general terms, measuring imbalance marginally over each prognostic factor and minimising a weighted sum of imbalances over all of them.

Each allocation depends on the factor levels of everyone already allocated, so the sequence cannot be prepared in advance as a fixed list. ICH E9 gives the example of a central trial office from which dynamic allocation is controlled, where knowledge of the treatment code can be restricted and eligibility checked before entry.

Worked example of one allocation decision

All figures in this example are illustrative. A two-arm trial minimises on age (under 65 or 65 and over) and disease severity (mild or severe), with equal weights. Twenty participants have been allocated, ten to each arm, as shown below. The next participant is 65 or over with severe disease.

Step 1. Matching levels. Only the rows for the new participant's own levels count, which are age 65 and over (6 in arm A, 4 in arm B) and severe disease (5 in arm A, 3 in arm B). The current imbalance on these two levels is 2 + 2 = 4.

Step 2. Imbalance after each possible allocation. Allocation to arm A would give 7 against 4 for older participants and 6 against 3 for severe disease, so $D_A$ is 3 + 3 = 6. Allocation to arm B would give 6 against 5 and 5 against 4, so $D_B$ is 1 + 1 = 2. Arm B has the smaller total and is favoured.

Step 3. Random element. With a deterministic rule the participant goes to arm B. With a probability of 0.8 for the favoured arm, the participant goes to arm B with probability 0.8 and to arm A with probability 0.2, so a recruiter who has tracked every past allocation still cannot be certain of the next one.

The random element, predictability and regulatory views

A purely deterministic version makes the next allocation knowable to anyone who can see the running totals, which opens the door to selection bias at recruitment. Scott and colleagues, reviewing the method, noted that concerns have centred on assignments being predictable with certainty in some situations. Regulators therefore expect a random component. ICH E9 states that deterministic dynamic allocation procedures should be avoided and that an appropriate element of randomisation should be incorporated for each treatment allocation. The 2015 EMA guideline on baseline covariates, which calls the approach dynamic allocation, repeats that deterministic schemes should be avoided.

The trial reporting and evidence review literature has accepted the method with some reservations. The CONSORT 2025 explanation and elaboration calls minimisation the only acceptable alternative to randomisation, says that the use of a random component is generally preferable, and states that trials using minimisation are in general considered methodologically equivalent to randomised trials even without a random element. It also notes that minimisation lacks the theoretical basis for eliminating bias on all known and unknown factors. CONSORT asks reports to name minimisation explicitly, list the variables used and state whether a random element was included. The Cochrane Handbook chapter on risk of bias says that minimisation generally includes a random element, should be paired with clear strategies for allocation concealment, and is viewed cautiously by some methodologists and favourably by others.

Analysing a minimised trial

Because minimisation forces balance on the chosen factors, the analysis should take those factors into account. Kahan and Morris showed by simulation that ignoring stratification or minimisation variables makes the standard error of the treatment effect too large. The result is confidence intervals that are too wide, a type I error rate below the nominal level and lost statistical power. An adjusted analysis gives valid inference. The EMA guideline asks that stratification factors normally enter the primary analysis as covariates, and for dynamic allocation it warns that conventional methods do not always control the type I error in some designs, suggesting re-randomisation methods in the analysis.

Practice has lagged behind this advice. Kahan and Morris reviewed 258 trials published in 2010 in four general medical journals. Of the 163 trials that balanced on centre or on a prognostic factor, only 42 (26%) adjusted for all balancing factors in the primary analysis, and 27 of the 29 trials that used minimisation did not say whether it was deterministic.

What minimisation means for trial-based economic evaluation

An economic evaluation run alongside a trial inherits its allocation method. Minimisation factors are picked for their prognostic value for the trial's primary outcome, so variables that drive costs or quality of life may be left off the list and remain imbalanced. Manca, Hawkins and Sculpher noted that baseline utility values are invariably imbalanced between arms in trial-based cost-effectiveness analysis and that baseline utility is likely to be highly correlated with QALYs over follow-up, so failing to control for the imbalance can produce a misleading ICER. They showed that regression analysis adjusting for baseline utility gives appropriate estimates.

Regression frameworks for costs and effects allow such adjustment. Willan, Briggs and Hoch set out a system of seemingly unrelated regressions for prognostic factor adjustment that works with either the incremental cost-effectiveness ratio or incremental net benefit, and that allows the covariates for costs and for effects to differ. The guidance on adjusting for minimisation factors was written for the primary clinical outcome. Including the same factors as covariates in the cost and QALY regressions is a reasoned extension that keeps the economic analysis consistent with the trial design. Kahan and Morris recommend that the protocol or statistical analysis plan prespecify adjustment for all balancing factors, and an economic analysis plan can state the same choice.

Since CONSORT regards minimised trials as methodologically equivalent to randomised trials, there is no methodological reason to exclude them from a synthesis of randomised evidence for health technology assessment. Reviewers judging risk of bias still look at the reported method, since an unreported or deterministic scheme raises the question of whether allocation could have been predicted.

Minimisation compared with neighbouring allocation ideas

Several terms sit close to minimisation, and one of them is a false friend. The comparison turns on what each method balances and how much chance it keeps.

The EMA guideline points out that dynamic allocation does not guarantee balance within combinations of prognostic factors, which is the price paid for handling many factors at once. Cost-minimisation analysis belongs to economic evaluation, not trial design.

Limitations and common misunderstandings of minimisation

Minimisation is a design tool with a narrow purpose. Several errors recur when trials that use it are reported, analysed or used in models.

• Balance on the listed factors says nothing about factors left off the list. CONSORT notes that minimisation lacks the theoretical basis for eliminating bias on all known and unknown factors, and any protection against chance confounding by unlisted factors rests on the random element.
• Marginal balance is weaker than joint balance. Arms can match on age and on severity separately yet differ on the share of older participants with severe disease.
• A balanced baseline table does not make adjustment unnecessary. The minimisation factors still belong in the analysis model, and variables that matter for costs or utility may need adjustment even when they look balanced.
• Minimisation does not replace allocation concealment or blinding.
• The method adds organisational demands, since the allocation service must know every participant's factor levels in real time, a point raised in the review by Scott and colleagues.

Sources

• Committee for Medicinal Products for Human Use. Guideline on adjustment for baseline covariates in clinical trials. EMA/CHMP/295050/2013. London: European Medicines Agency; 2015. https://www.ema.europa.eu/en/documents/scientific-guideline/guideline-adjustment-baseline-covariates-clinical-trials_en.pdf
• Higgins JPT, Savović J, Page MJ, Elbers RG, Sterne JAC. Chapter 8: Assessing risk of bias in a randomized trial. In: Higgins JPT, Thomas J, Chandler J, et al, editors. Cochrane Handbook for Systematic Reviews of Interventions version 6.5. Cochrane; 2024. https://training.cochrane.org/handbook/current/chapter-08
• Hopewell S, Chan AW, Collins GS, Hróbjartsson A, Moher D, Schulz KF, et al. CONSORT 2025 explanation and elaboration: updated guideline for reporting randomised trials. BMJ. 2025;389:e081124. doi:10.1136/bmj-2024-081124
• International Conference on Harmonisation. ICH Topic E9. Statistical principles for clinical trials. CPMP/ICH/363/96. London: European Medicines Agency; 1998. https://www.ema.europa.eu/en/documents/scientific-guideline/ich-e-9-statistical-principles-clinical-trials-step-5_en.pdf
• Kahan BC, Morris TP. Improper analysis of trials randomised using stratified blocks or minimisation. Statistics in Medicine. 2012;31(4):328-340. doi:10.1002/sim.4431
• 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
• Manca A, Hawkins N, Sculpher MJ. Estimating mean QALYs in trial-based cost-effectiveness analysis: the importance of controlling for baseline utility. Health Economics. 2005;14(5):487-496. doi:10.1002/hec.944
• Pocock SJ, Simon R. Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trial. Biometrics. 1975;31(1):103-115. doi:10.2307/2529712
• Scott NW, McPherson GC, Ramsay CR, Campbell MK. The method of minimization for allocation to clinical trials: a review. Controlled Clinical Trials. 2002;23(6):662-674. doi:10.1016/s0197-2456(02)00242-8
• Taves DR. Minimization: a new method of assigning patients to treatment and control groups. Clinical Pharmacology and Therapeutics. 1974;15(5):443-453. doi:10.1002/cpt1974155443
• Willan AR, Briggs AH, Hoch JS. Regression methods for covariate adjustment and subgroup analysis for non-censored cost-effectiveness data. Health Economics. 2004;13(5):461-475. doi:10.1002/hec.843