Optimal response surface designs in the presence of model contamination

TL;DR

This paper develops a comprehensive framework for designing response surface experiments that balances model inference quality, lack-of-fit testing, and robustness to model contamination, especially in complex experimental settings.

Contribution

It introduces a novel compound optimality criterion that integrates multiple objectives and adapts to restricted randomization frameworks, with a new algorithm for design optimization.

Findings

Optimal designs balance inference quality and robustness to contamination.

The framework is adaptable to blocked and multistratum experiments.

Practical recommendations for design compromises are provided.

Abstract

Complete reliance on the fitted model in response surface experiments is risky and relaxing this assumption, whether out of necessity or intentionally, requires an experimenter to account for multiple conflicting objectives. This work provides a methodological framework of a compound optimality criterion comprising elementary criteria responsible for: (i) the quality of the confidence region-based inference to be done using the fitted model (DP-/LP-optimality); (ii) improving the ability to test for the lack-of-fit from specified potential model contamination in the form of extra polynomial terms; and (iii) simultaneous minimisation of the variance and bias of the fitted model parameters arising from this misspecification. The latter two components have been newly developed in accordance with the model-independent 'pure error' approach to the error estimation. The compound criteria and design construction were adapted to restricted randomisation frameworks: blocked and multistratum experiments, where the stratum-by-stratum approach was adopted. A point-exchange algorithm was employed for searching for nearly optimal designs. The theoretical work is accompanied by one real and two illustrative examples to explore the relationship patterns among the individual components and characteristics of the optimal designs, demonstrating the attainable compromises across the competing objectives and driving some general practical recommendations.