# Minimisation

Minimisation is a method of randomisation that allocates subjects to the treatment group that best maintains balance in stratifying factors. It is effective even at small sample sizes and with multiple stratification variables.

## Example

The method is best illustrated by example. Suppose it is important to balance subject sex in a trial of a new drug, because women are expected to respond more strongly to the drug. It would be unfortunate if, by chance, more women received the new drug rather than placebo and more men were allocated to placebo rather than the new drug. For similar reasons we would also like to balance subject age, so that younger subjects, who are expected to have a better outcome, are evenly distributed to the placebo and drug groups.

The randomisations to the trial so far look like this:

Number Sex Age Treatment group
1 Male <30 Placebo
2 Male 30+ Placebo
3 Female 30+ New drug
4 Male <30 Placebo
5 Female <30 New drug
6 Male 30+ New drug

The next subject to be randomised is a man age 23.

To decide which treatment to allocate the subject to, the balance of treatments in the trial is compared for subjects with the same characteristics as the subject to be randomised. There are various ways of calculating the imbalance, but the most popular method1 (and the one Sealed Envelope uses) is to simply sum the frequencies across the strata for each treatment. In this example the frequencies are:

Stratifying factor Placebo New drug
Male 3 1
<30 2 1
Total 5 2

Clearly in males and those under 30 there is an imbalance in favour of placebo so far. The next treatment allocation is the one with the lowest total score - in this case the next subject will be allocated to the new drug. Note that if the scores were tied, the treatment allocation would be chosen purely at random.

## Incorporating a random element

Minimisation as described above is a largely deterministic procedure - given the characteristics of subjects in the trial and the subject to be randomised, the new treatment allocation is almost entirely predictable.

It is desirable to inject a random element into the procedure and, in fact, ICH E9 guidelines require it:

Deterministic dynamic allocation procedures should be avoided and an appropriate element of randomisation should be incorporated for each treatment allocation.

ICH Topic E9 Statistical Principles for Clinical Trials

The Sealed Envelope randomisation system defines a probability that a purely random allocation will be made, instead of using minimisation. So for each randomisation there is a chance (usually around 30%) that the treatment will be chosen at random. This is equivalent to using a biased coin to determine the next treatment, with the bias in favour of the treatment that would make the treatment groups more balanced2. If there are two treatments allocated in a 1:1 ratio, and a 30% chance of choosing the treatment at random, then the probability that the under-represented treatment will be chosen is 0.85 (0.3 × 0.5 + 0.7). This probability can be viewed for your trial on the specification page .

## Factorial trials

In factorial trials, 2 or more treatments comparisons are evaluated in the same subjects. The most common design is the 2 × 2 factorial trial:

Placebo Aspirin
Placebo x x
β-carotene x x

where subjects are allocated to one of four treatment groups. In the above example these are:

• Placebo
• Aspirin alone
• β-carotene alone
• Aspirin and β-carotene

Suppose we want to make sure subject age is balanced between the four groups and the next subject to be randomised is aged under 30. The frequency table for allocations to each treatment group in subjects <30 years old is:

Age <30 Placebo Aspirin Total
Placebo 3 2 5
β-carotene 2 2 4
Total 5 4 9

To calculate the minimisation scores for each treatment group, the frequency in the relevant cell plus the marginal totals are used:

• Placebo: 3 + 5 + 5 = 13
• Aspirin alone: 2 + 5 + 4 = 11
• β-carotene alone: 2 + 4 + 5 = 11
• Aspirin and β-carotene: 2 + 4 + 4 = 10

So in this case the next allocation will be to the aspirin and β-carotene group. As before, if scores are tied the treatment is chosen at random from the tied groups.

1. Taves DR. Minimization: a new method of assigning subjects to treatment and control groups. Clin Pharmacol Therapeut. 1974;15:443-453.

2. Pocock SJ, Simon R. Sequential treatment assignment with balancing for prognostic factors in the controlled clinical trials. Biometrics 1975;31:103-115.

Page updated 4 Oct 2017