D-optimal Designs for Multinomial Logistic Models

Multinomial logistic models have been widely used for categorical responses. There exist four types of logit models, that is, baseline-category logit model for nominal responses, cumulative logit model for ordinal responses, adjacent-categories logit model for ordinal responses, and continuation-ratio logit model for hierarchical responses. There also exist three model assumptions, including proportional odds (PO), non-proportional odds (NPO) and partial proportional odds (PPO). In order to construct a general framework towards D-optimal designs for these models, we unified all the models into a common form and derived three types of Fisher information matrix expressions. Based on them, we discussed positive definiteness of Fisher information matrix and obtained the number of minimally supported points for different models. Furthermore, we derived the general formula for Fisher information matrix's determinant, which applies to all types of multinomial logistic models. In some special cases, the determinant can be expressed explicitly (in a closed form) so D-optimal designs can be obtained directly. Two real experiments are used to illustrate locally and non-locally D-optimal design's efficiency improvement, which provides us a benchmark and criterion. The D-optimal designs recommended in these examples are minimally supported. Generally the uniform design is not D-optimal design for multinomial logistic models.