Robust optimal experimental design for integer-valued design spaces: A case study in functional response experiments

Functional response models are important in understanding predator-prey interactions. The development of functional response methodology has progressed from mechanistic models to more statistically motivated models that can account for variance and the over-dispersion commonly seen in the datasets collected from functional response experiments. However, little information seems to be available to those wishing to prepare optimal parameter estimation designs for functional response experiments. In this paper, we develop a so-called exchange design optimisation algorithm suitable for integer-valued design spaces, which for the motivating functional response experiment involves selecting the number of prey used for each observation. Further, we develop and compare new utility functions for performing robust optimal design in the presence of parameter uncertainty, which are generally applicable. The methods are illustrated using a published beta-binomial functional response model for an experiment involving the freshwater predator Notonecta glauca (an aquatic insect) preying on Asellus aquaticus (a small crustacean) as a case study. As a by-product of the case study, we also derive necessary quantities to perform optimal design for beta-binomial regression models, which may be useful in other applications.