Optimal experiment design for cardinal values estimation: guidelines for data collection

Abstract

Optimal experiment design for parameter estimation (OED/PE) is an interesting technique for modelling practices when aiming for maximum parameter estimation accuracy. Nowadays, experimental designs for secondary modelling within the field of predictive microbiology are mostly arbitrary or based on factorial design. The latter type of design is common practice in response surface modelling approaches. A number of levels of the factor(s) under study are selected and all possible treatment combinations are performed. It is however not always clear which levels and treatment combinations are most relevant. An answer to this question can be obtained from optimal experiment design for—in this particular case—parameter estimation. This technique is based on the extremisation of a scalar function of the Fisher information matrix. The type of scalar function determines the final focus of the optimised design. In this paper, optimal experiment designs are computed for the cardinal temperature model with inflection point (CTMI) and the cardinal pH model (CPM). A model output sensitivity analysis (depicting the sensitivity of the model output to a small change in the model parameters) yields a first indication of relevant temperature or pH treatments. Performed designs are: D-optimal design aiming for a maximum global parameter estimation accuracy (by minimising the determinant of the Fisher information matrix), and E-optimal design improving the confidence in the most uncertain model parameter (by maximising the smallest eigenvalue of the Fisher information matrix). Although lowering the information content of a set of experiments, boundary values on the design region need to be imposed during optimisation to exclude unworkable experiments and partly account for incorrect nominal parameter values. Opposed to the frequently applied equidistant or arbitrary treatment placement, optimal design results show that typically four informative temperature or pH levels are selected and replicate experiments are to be performed at these points. Informative experiments are typically placed at points with an extreme model output sensitivity.