Optimal design: A computer program to study the best possible spacing of design points for model discrimination

Abstract

Optimal design is largely concerned with selecting design points so as to maximise the precision of parameter estimates once the correct model has been identified. Often, however, a priority is that the correct model has first to be identified from a set of candidate models, and this procedure requires a different set of design criteria. We suppose that a decision has to be made on statistical grounds between accepting either a correct model g 2( x, Θ) or a deficient model g 1( x, Φ), using goodness of fit criteria, where weighting is dictated by a weighting function w( x) and spacing is specified by a spacing function ∝( x). We describe, for the first time, the choice of spacing function required to generate points which are uniformly spaced with respect to the y axis (Uniform- Y design). Then we introduce appropriate norms S n ( Θ), Q( Θ) and R( n), to quantify the effects of alternative choices of spacing, density and number of design points on the probability of correct model identification. Typical problems of this general type in biochemistry, for example, would be determining the correct number of classes of receptors in a ligand binding experiment, or fixing the number of exponential components in a pharmacokinetic experiment. A computer program which performs the necessary calculations for these and similar problems is described, and illustrated by analysing some typical test cases. Results from our extended investigations are also summarised, leading to more general conclusions as to the best spacing and density of design points for optimal model discrimination. We also prove the remarkable and previously unsuspected fact that, in differentiating two site cooperative ligand binding from one site, the two best designs give identical parameter estimates for the deficient model but are not actually equivalent from the point of view of model discrimination.