Minimisation and randomisation in clinical trials.


Since (Taves 1974; Pocock and Simon 1975) proposed mimisation the technique has been controversial. It appears to provide better balance as regards prognostic factors than does randomisation but it was not based on sound design principles and the procedure, using as it does marginal balance is extremely ad hoc. (Treasure and MacRae 1998) described minimisation as providing the platinum standard, that is to say even better than randomisation, which provided the gold standard. On the other hand, drug regulatory agencies have expressed their disapproval and suspicion of the procedure(Buyse and McEntegart 2004).


A logically superior approach was proposed by (Atkinson 1982) but seems to have been little used. This works directly with the design matrix and so ought to be more efficient than minimisation. However, neither minimisation nor Atkinsonís approach (AA) will be acceptable to regulators in strictly deterministic form. Hence, in practice they have to be used to guide a biased coin allocation(Efron 1971). In this context the literature is rather confusing. For example Atkinson himself seems to have found by simulation that AA is less efficient under some circumstances than minimisation(Atkinson 2002). This result seems almost inexplicable. The only plausible explanation in my view is that implementation of the biased coin element of the algorithm is such that stochastic convergence is weaker with AA than minimisation. This cannot, however, be an inherent feature of the method.


This project will attempt to elucidate some of these mysteries. In particular it seems that a careful analytic investigation of simple cases (in other words not using simulation) is needed. For example just how many patients are required for a given number of binary covariates before it is even theoretically possible that minimisation and AA could produce a different allocation. (Obviously with only one binary predictor it is not possible.) Related to the above, of course, is the question of for which configurations of the design matrix is marginal balance, which is what minimisation achieves, not equivalent to (reduced) D-optimality, which is what AA uses.


Extensions of the project will look at the relationship between predictability and efficiency in such allocation schemes (important work has already been done here by (Burman 1996)) and also the case of stochastic regressors, for which the Gauss-Markov theorem does not apply(Popper Shaffer 1991), as well as non-linear models for which optimal allocation cannot be determined using covariates alone. Finally, a much more practical element of the project could consider a survey of literature and statisticians to see to what extent such allocation methods are likely to gain acceptance in practice. Very much related to this is the extent to which statisticians are prepared to use more complicated methods of analysis.





Atkinson, A. C. (1982). "Optimum Biased Coin Designs for Sequential Clinical-Trials with Prognostic Factors." Biometrika 69(1): 61-67.

Atkinson, A. C. (2002). "The comparison of designs for sequential clinical trials with covariate information." Journal of the Royal Statistical Society Series a-Statistics in Society 165: 349-373.

Burman, C.-F. (1996). On Sequential Treatment Allocations in Clinical Trials. Department of Mathematics. Gothenburg, Chalmers University of Technology.

Buyse, M. and D. McEntegart (2004). "Achieving balance in clinical trials." Applied Clinical Trials 13(5): 36-40.

Efron, B. (1971). "Forcing a sequential experiment to be balanced." Biometrika 58(3): 403-417.

Pocock, S. J. and R. Simon (1975). "Sequential Treatment Assignment with Balancing for Prognostic Factors in Controlled Clinical Trial." Biometrics 31(1): 103-115.

Popper Shaffer, J. (1991). "The Gauss-Markov theorem and random regressors." The American Statistician 45: 269-272.

Taves, D. R. (1974). "Minimization: a new method of assigning patients to treatment and control groups." Clinical Pharmacology and Therapeutics 15(5): 443-53.

Treasure, T. and K. D. MacRae (1998). "Minimisation: the platinum standard for trials?. Randomisation doesn't guarantee similarity of groups; minimisation does." Bmj 317(7155): 362-3.