The Construction of a Model-Robust IV-Optimal Mixture Designs Using a Genetic Algorithm

Abstract

Among the numerous alphabetical optimality criteria is the IV-criterion that is focused on prediction variance. We propose a new criterion, called the weighted IV-optimality. It is similar to IV-optimality, because the researcher must first specify a model. However, unlike IV-optimality, a suite of "reduced" models is also proposed if the original model is misspecified via over-parameterization. In this research, weighted IV-optimality is applied to mixture experiments with a set of prior weights assigned to the potential mixture models of interest. To address the issue of implementation, a genetic algorithm was developed to generate weighted IV-optimal mixture designs that are robust across multiple models. In our examples, we assign models with p parameters to have equal weights, but weights will vary based on varying p. Fraction-of-design-space (FDS) plots are used to compare the performance of an experimental design in terms of the prediction variance properties. An illustrating example is presented. The result shows that the GA-generated designs studied are robust across a set of potential mixture models.

Using geometric mean to compute robust mixture designs

“…In this research, we zero in on the problem of robustness of the design against model change, the so‐called model robustness. We extend the work by Limmun et al 14–17 . The key difference is that in Limmun et al., 14 a single model was used to generate an optimal design, while in this research, the possible reduced models are taken parts in the generation of a robust optimal design.…”

“…Stallings and Morgan 25 proposed a weighted information matrix for any positive definite weight matrix. Limmun et al 15–17 . 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.…”

“…The key difference is that in Limmun et al., 14 a single model was used to generate an optimal design, while in this research, the possible reduced models are taken parts in the generation of a robust optimal design. Another distinction is that in Limmun et al., 15–17 the weighted arithmetic mean was employed to construct the objective function of the GA, whereas in this research, the geometric mean is proposed and employed to formulate the objective function of the GA because of desirable properties in comparison to the arithmetic mean.…”

“…Recent applications of GAs provided an alternative approach to classic exchange‐point algorithms for constructing optimal designs. Uses of GAs to find optimal designs can be found in Heredia‐Langner et al., 8,9 Borkowski, 10 Park et al., 11 Drain et al., 12 Rodriguez et al., 13 and Limmun et al 14–17 …”

“…Recent applications of GAs provided an alternative approach to classic exchange-point algorithms for constructing optimal designs. Uses of GAs to find optimal designs can be found in Heredia-Langner et al, 8,9 Borkowski, 10 Park et al, 11 Drain et al, 12 Rodriguez et al, 13 and Limmun et al [14][15][16][17] In general, the actual relationship between the response of interest and the number of associated control variables is unknown, but it can be estimated by using a good approximating model. The final model in the analysis phase frequently differs from the initial model when the experimenter found that some effect terms in the initial model are meaningless and are subsequently removed from the model.…”