## Stratified randomization

In multi-centre trials, it is good practice to ensure that within a hospital, or perhaps an individual clinician, a roughly equal number of patients are allocated each treatment. This might be because variability between centres in some aspect of patient care might be anticipated - for example surgical technique - or perhaps because of the cost or difficulty of one of the treatments. One may also want to ensure that important known prognostic factors are balanced between treatments (randomization does not guarantee this - particularly in small trials). In many cancer trials, it may well be the case that differences in outcome according to factors such as stage or age are much larger than the anticipated differences due to treatment. In small trials, therefore, imbalances in prognostic factors may occur across treatment groups which could obscure, or lead to over-estimation of, differences due to treatment itself if a simple unadjusted analysis is performed.

To avoid such imbalances, a widely used method is 'stratified randomization'. With this method, blocked randomization is used, but instead of a single randomization 'list', a number will be prepared such that there is a separate list for each combination of the important stratification factors. For example, if you wish to balance treatment allocation by one factor only, say hospital, simply prepare a block randomization list for each hospital using the methods described previously.

If you wish to balance treatment allocation across a number of factors, for example hospital (A versus B) stage (I/II versus III/IV) and age (<50 versus >50), then you must prepare blocked lists for each combination of factors - here there are eight combinations:

hospital A, stage I/II aged <50; hospital A, stage I/II aged >50; hospital A, stage III/IV aged <50; hospital A, stage III/IV aged >50;

hospital B, stage I/II aged <50; hospital B, stage I/II aged >50; hospital B, stage III/IV aged <50; hospital B, stage III/IV aged >50.

Note that using stratified randomization without blocking serves little purpose -without blocking you would simply have random treatment allocations for each of the eight lists, possibly with fairly small numbers of patients on each list, and balance across treatments cannot be guaranteed.

Clearly, if there are many stratification factors, or some factors with many categories, this method may become impractical, even if computerized. In the previous example, if the number of hospitals was twenty rather than two, there would be eighty combinations of factors, meaning eighty 'lists.' For those randomizing without the aid of a computer this is time-consuming, but there are potential problems even with computerized stratified randomization. The larger the number of lists, the smaller the number of patients likely on each. With a moderately large block size and trials which stop when a specified total number of patients is reached, rather than when specified numbers are achieved on each list, each list can be imbalanced at the time of stopping, and potentially large overall imbalances in treatment allocation could occur. They would have to be substantial to affect the power of the study considerably, but even moderate imbalances can give the impression of a poorly conducted study which may even affect its publication.

Also, it has been pointed out [28] that in general we are less interested in whether the number (proportion) of patients aged <50, with stage II disease, from hospital X are balanced between treatments, and more interested in whether the individual factors are balanced - i.e. the number (proportion) aged <50 is balanced and the number with stage II disease is balanced across treatments. Minimization is a method of treatment allocation that ensures this in a more direct manner than stratification.

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