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Evaluation of imbalance in stratified blocked randomization: some remarks on the range of validity of the model by Hallstrom and Davis.

OBJECTIVES
If in a clinical trial prognostic factors are known in advance, it is often recommended that randomization of patients should be stratified. The best-known method is permuted-block randomization within strata. But it suffers from the disadvantage that imbalance still occurs in the trial as a whole if there are a large number of strata, or/and the block sizes are too large for the number of patients. The results of Hallstrom and Davis are appropriate for evaluating the risk of such a troubled situation by using two special cases of their general variance formula. But it is merely generally argued for whichever practical situations these special cases are valid. Consequently, additional investigations are required to reveal the conditions for correct application.


METHODS
We investigated the range of validity of special cases by performing computer simulations, varying a number of trial characteristics, and discuss the application of results for practical situations.


RESULTS
The validity of special cases is not given in each situation. Depending on block size, a binomial distribution model is valid for a permitted average maximum number of patients per stratum between 36% and 57% of considered block sizes, whereas the uniform distribution model works adequately from at least 70%. In an intermediate range of invalidity, implementation of a simulation study is necessary to compute the probability distribution of differences.


CONCLUSIONS
Our results are important if choosing the stratified permuted-block randomization to estimate the risk for an intolerable overall imbalance when planning a trial.