**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 Atkinson’s 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.