Minimisation

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2. Red Pill and Randomisation
3. Version 25
4. Minimisation

Minimisation¹ is a method of randomisation that allocates subjects to the treatment group that best maintains balance in prognostic factors. It is effective even at small sample sizes and with multiple prognostic 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. Here sex and age are prognostic factors for the trial.

The randomisations to the trial so far look like this:

To decide which treatment to allocate to the subject the balance of treatments in the trial is compared for subjects with the same characteristics as the subject to be randomised. The treatment choice that would result in the smallest treatment imbalance for that combination of characteristics is then the preferred treatment for that subject.

The next subject to be randomised is a man age 23, so before randomisation we have the following treatment counts for the strata.

Clearly in males and those under 30 there is an imbalance in favour of placebo so far.

The method² that Sealed Envelope uses proceeds by first calculating for each treatment the resulting counts for each prognostic factor assuming that that treatment was allocated next. Then we calculate the absolute difference of the treatment counts for each factor, and sum those differences to give the imbalance for that treatment.

If the next treatment allocation is to the placebo we would have the following counts.

So for sex with level Male we have absolute difference |4 - 1| = 3, and for age with level <30 we have difference |3 - 1| = 2. Summing the differences gives a treatment imbalance for the placebo of 3 + 2 = 5.

If the next allocation is to the new drug we would instead have the following counts.

Here for sex with level Male we have absolute difference |3 - 2| = 1, and for age with level <30 we have difference |2 - 2| = 0. Hence the sum of differences gives a treatment imbalance for the new drug of 1 + 0 = 1.

Now we rank the treatment imbalances in order of increasing treatment imbalance and choose the treatment with the lowest score. Since 1 < 5, we see that allocating the new drug treatment to the subject would best decrease the total imbalance. So the new drug is the preferred treatment.

Note that if there is a tie in the lowest treatment imbalance scores the preferred treatment is chosen at random from those with the tied score.

Incorporating a random element

The procedure above is deterministic unless there is a tie in the lowest treatment imbalance scores. Given the characteristics of subjects already randomised in the trial and the subject to be randomised, the preferred treatment 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

Instead of immediately allocating the preferred treatment we specify a probability for choosing the preferred treatment. The remaining probability is split equally between the other treatments, and the treatment to be allocated is then chosen randomly based on those probabilities. So for each randomisation there is a chance that the preferred treatment will not be chosen. 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 balanced.

The probability of choosing the preferred treatment is specified when we set up a trial and can be viewed on the specification page.

Each step of the calculation for every minimisation is recorded in the trial database.

Preserving the allocation ratio in trials with unequal allocation

Research published in the last decade has highlighted some issues when using this minimisation method in trials with unbalanced allocation ratios.

To address these issues we use a modified minimisation method³ that preserves the allocation ratio at every step.

This method works similarly to our standard minimisation method with the important difference that it is carried out on a set of fake treatments which are then mapped back to the real treatments for allocation.

Say that for our example above we had an allocation ratio of 1:2 for placebo to new drug. Then the minimisation would be carried out on fake treatments F₁, F₂, and F₃, where minimisation to F₁ would result in an allocation of the placebo and minimisation to either F₂ or F₃ would result in an allocation to the new drug.

The allocations to the fake treatments are stored separately from the real allocations in the trial database so that we can minimise their imbalance at randomisation.