D-optimal designs for multinomial logistic models. (English) Zbl 1453.62606

Bu, Xianwei; Majumdar, Dibyen; Yang, Jie

Ann. Stat. 48, No. 2, 983-1000 (2020).

While there is a sizable and growing literature for optimal designs with binary response, the literature for experiments with more than two categories is limited. In the statistical literature, four kinds of multinomial logit models have been used: baseline-category, cumulative, adjacent-categories and continuation-ratio logit models. Combined with three odd assumptions (proportional, nonproportional, and partial proportional odds), 12 different models for multinomial responses can be generated.
The authors work in a general framework which covers all of the 12 models and they present results on (mainly \(D\)-) optimal designs. First they derive the Fisher information matrix and necessary and sufficient conditions for their positive definiteness. Furthermore, they develop efficient algorithms for searching local \(D\)-optimal designs.
The authors show that the optimal designs for multinomial responses with three or more categories are remarkably different from the ones for binary responses, in two major aspects:

(i)

the required minimum number of experimental settings is less than the number of parameters and

(ii)

even among minimally supported designs, unlike \(D\)-optimal designs for binary responses, uniform allocation often is not \(D\)-optimal (and can be quite inefficient).

The authors illustrate the results extensively by two real experiments.

Reviewer: Wolfgang Näther (Freiberg)

MSC:

62K05	Optimal statistical designs
62J12	Generalized linear models (logistic models)
62B10	Statistical aspects of information-theoretic topics

Keywords:

approximate design; exact design; Fisher information matrix; multinomial response; minimally supported design; lift-one algorithm

Software:

ordinal; VGAMdata; dobson; SAS

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