# Minimisation

Minimisation is a method of randomisation that allocates participants to the treatment group that best maintains balance in prognostic factors (Taves 1974). 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 participant 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 participant age, so that younger participants, 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:

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 |

To decide which treatment to allocate to the participant, the balance of treatments in the trial is compared for participants with the same characteristics as the participant 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 participant to be randomised is a man age 23, so before randomisation we have the following treatment counts for the strata.

Prognostic factor |
Placebo |
New drug |
---|---|---|

Male | 3 | 1 |

<30 | 2 | 1 |

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

The method (Pocock 1975) 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.

Prognostic factor |
Placebo |
New drug |
---|---|---|

Male | 4 | 1 |

<30 | 3 | 1 |

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.

Prognostic factor |
Placebo |
New drug |
---|---|---|

Male | 3 | 2 |

<30 | 2 | 2 |

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 participant 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 participants already randomised in the trial and the participant 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:

*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 method (Kuznetsova 2012) 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 F _{1}, F_{2}, and F_{3}, where minimisation to F_{1} would result in an allocation of the placebo, and minimisation to either F_{2} or F_{3} 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.

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

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

Kuznetsova OM, Tymofyeyev Y. Preserving the allocation ratio at every allocation with biased coin randomization and minimization in studies with unequal allocation. *Statist. Med.* 2012;31:701–723.