Equivalence of weighted and partial optimality of experimental designs

Abstract

When the experimental objective is expressed by a set of estimable functions, and any eigenvalue-based optimality criterion is selected, we prove the equivalence of the recently introduced weighted optimality and the 'standard' optimality criteria for estimating this set of functions of interest. Also, given a weighted eigenvalue-based criterion, we construct a system of estimable functions, so that the optimality for estimating this system of functions is equivalent to the weighted optimality. This allows one to use the large body of existing theoretical and computational results for the standard optimality criteria for estimating a system of interest to derive theorems and numerical algorithms for the weighted optimality of experimental designs. Moreover, we extend the theory of weighted optimality so that it captures the experimental objective consisting of any system of estimable functions, which was not the case in the literature on weighted optimality so far.For any set of estimable functions, we propose a corresponding weight matrix of a simple form, and with a straightforward interpretation. Given a set of estimable functions with their corresponding weights, we show that it is useful to distinguish between the primary weights selected by the experimenters and the secondary weights implied by the weight matrix.

Using geometric mean to compute robust mixture designs

“…Stallings and Morgan 25 proposed a weighted information matrix for any positive definite weight matrix. Limmun et al [15][16][17] developed the weighted A-optimality criterion, weighted IV-optimality criterion, and the weighted G-efficiency based on an arithmetic mean of efficiencies as the objective function of a GA. Rosa 26 extended the work of Stallings and Morgan 25 and proposed a weighted eigenvalue-based criterion. Kristoffersen and Smucker 27 proposed a computationally tractable algorithm for generating model-robust mixture designs based on the weighted D-and IV-optimality criteria using a set of models defined by a user-specified number of higher-order terms.…”