Global, exact and fast group size optimization with corresponding efficiency translation in optimal design
Joakim Nyberg, Sebastian Ueckert, Mats O. Karlsson, Andrew C. Hooker
Department of Pharmaceutical Biosciences, Uppsala University, Uppsala, Sweden
Objectives: To increase the efficiency in various stages in drug development, optimal design has been used . Group size optimization has been investigated previously by various techniques, e.g. within the Fedorov-Wynn algorithm , and implemented in software [2,3], however these search methods can be quite computer intensive.
The aim of this investigation is to develop and explore fast and accurate methods for optimizing the number of individuals in different design groups. A secondary objective is to use the method to translate the efficiency to a more interpretable number.
Methods: Two different methods were developed, 1) an exhaustive global search (GS) and 2) a faster approximation (FA) method.
For any group q of individuals the population fisher information matrix (FIMq) is equal to the sum of the individual FIMi,q in the group. If individuals within a group have the same design and are indistinguishable then, for Nq individuals, FIMq=Nq*FIMi,q. The total population FIM is the sum of all the M design groups in a study FIM=ΣFIMq. Nq can be varied within the limit for group q, Nq~[Nq.min, Nq,max] and the sum of all individuals in the study can be varied between [Ntot,min, Ntot,max]. GS investigates every possible combination of Nq for all design groups M (only one calculation of FIMi,q per group is needed). An approximate, but much faster, solution (FA) can be found by sequentially assigning individuals to the most informative group.
The efficiency when comparing two designs, say A and B where A>B, can often be easily translated into the extra number of individuals that is needed in design B to reach the same information as design A. However, when e.g. D-efficiency is used, Deff=[det(A)/det(B)]1/p and the number of design groups is >1 the determinant is nonlinear with respect to all Nq and it's not trivial to extract all Nq from the determinant. Instead the above search algorithms may be used to find the worst case scenario Nmax, where additional individuals are added to design B in the group where they give the least amount of information, within restrictions [Nq.min, Nq,max] and [Ntot,min, Ntot,max]. Similarly, Nmin, where the individuals are placed in an optimal way may be calculated.
Results: Group size algorithms 1) and 2) were successfully implemented into PopED (2.10)  and the efficiency translation is available as a graphical tool in PopED, contributing to a more intuitive understanding of efficiency. In general the GS and FA methods give similar results, however, the FA could not be used when the OFV (e.g. determinant) for a single matrix FIMi,q couldn't be calculated, e.g. when some rows/columns are zero. The optimization methods work with global optimal designs criterions as well as local designs criterions.
 Mentré, F., A. Mallet, and D. Baccar, Optimal design in random-effects regression models. Biometrika, 1997. 84(2): p. 429-442.
 Retout, S., et al., Design in nonlinear mixed effects models: optimization using the Fedorov-Wynn algorithm and power of the Wald test for binary covariates. Stat Med, 2007. 26(28): p. 5162-79.
 PopED, version 2.10 (2010) http://poped.sf.net.