Optimal Designs for Poisson Count Data with Gamma Block Effects

TL;DR

This paper investigates optimal experimental designs for Poisson count data modeled with Gamma block effects, establishing a connection between Poisson-Gamma and Poisson models and deriving optimal designs for various criteria.

Contribution

It derives D-optimal designs for the Poisson-Gamma model and links optimality criteria between models, extending to multiple covariates and linear criteria.

Findings

D-optimality for Poisson-Gamma equals a combined weighted criterion for Poisson.

Explicit D-optimal designs are obtained for multiple regression.

Optimal designs for linear criteria are the same in both models.

Abstract

The Poisson-Gamma model is a generalization of the Poisson model, which can be used for modelling count data. We show that the $D$-optimality criterion for the Poisson-Gamma model is equivalent to a combined weighted optimality criterion of $D$-optimality and $D_s$-optimality for the Poisson model. Moreover, we determine the $D$-optimal designs for the Poisson-Gamma model for multiple regression with an arbitrary number of covariates, obtaining the $D_s$-optimal designs for the Poisson and Poisson-Gamma model as a special case. For linear optimality criteria like $L$- and $c$-optimality it is shown that the optimal designs in the Poisson and Poisson-Gamma model coincide.