## Minimization

In minimization, the treatment allocated to a patient depends on the allocations given to all previous patients with the same characteristics, and the allocation is done in a manner which minimizes the treatment imbalance in each of those characteristics - hence the name.

For example, suppose in a trial comparing treatment A and treatment B, we wanted to ensure balance across three age groups, <50, 50-60, >60. Suppose also that currently fourteen patients aged <50 have been allocated A and twelve have been allocated B. If the next patient is aged <50, then we allocate them to the treatment which minimizes the treatment imbalance for this age group - in this case, we would allocate B. An element of pure randomization must be introduced whenever the current allocations for a given group are exactly balanced, including the first allocations within an age group, but can be introduced at any stage (see later in this section).

More often, just as with stratified randomization, you wish to balance a number of factors. The method here is to sum the imbalances across all the factors, and allocate the treatment which minimizes the sum of the imbalances. This is best shown by an example.

Example 1: Suppose sixteen patients have been randomized into a trial, and their characteristics are distributed as in Table 4.1.

Suppose the next patient is from hospital X, aged 38 and has stage II disease. Currently:

for hospital X there is no treatment imbalance, A — B = 0, forage <50 A — B = —2, for stage, A — B = +1.

The overall imbalance, A — B, = —1, therefore we allocate treatment A. This is equivalent to simply summing the number of previous patients with the same characteristics on each treatment and allocating the patient to the group with the lowest total:

allocated treatment A: 4 + 3 + 3 = 10, allocated treatment B: 4 + 5 + 2 = 11.

Treatment A B | ||

Hospital |
X |
44 |

Y |
3 2 | |

Z |
1 2 | |

Age |
<50 |
3 5 |

>50 |
5 3 | |

III/IV |
32 56 |

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