D-optimal Designs for Multinomial Logistic Models

TL;DR

This paper develops methods to find optimal experimental designs for complex multinomial logistic models, showing that traditional uniform designs are often suboptimal and providing algorithms to improve experimental efficiency.

Contribution

It introduces new Fisher information matrix formulations and algorithms for D-optimal designs tailored to various multinomial logistic models, highlighting their practical advantages.

Findings

Feasible designs may have fewer settings than parameters.

Uniform allocation is generally not optimal for multinomial models.

Optimized designs significantly improve experimental efficiency.

Abstract

We consider optimal designs for general multinomial logistic models, which cover baseline-category, cumulative, adjacent-categories, and continuation-ratio logit models, with proportional odds, non-proportional odds, or partial proportional odds assumption. We derive the corresponding Fisher information matrices in three different forms to facilitate their calculations, determine the conditions for their positive definiteness, and search for optimal designs. We conclude that, unlike the designs for binary responses, a feasible design for a multinomial logistic model may contain less experimental settings than parameters, which is of practical significance. We also conclude that even for a minimally supported design, a uniform allocation, which is typically used in practice, is not optimal in general for a multinomial logistic model. We develop efficient algorithms for searching D-optimal designs. Using examples based on real experiments, we show that the efficiency of an experiment can be significantly improved if our designs are adopted.