A-Optimal Designs for Rasch Poisson Gamma Counts Models.

Abstract

In this article, we develop A-optimal designs for a longitudinal Rasch Poisson Gamma counts model (RPGCM). The model is particularly well suited for repeated testing designs in educational and psychological research when the outcomes are count data. Within the RPGCM framework, test scores depend on two types of parameters: respondents' abilities, which are modeled as gamma-distributed random effects, and item difficulties, which are treated as fixed parameters. To account for dependencies among repeated measurements, we incorporate a compound symmetry structure into the RPGCM. Parameter estimation is performed using maximum quasi-likelihood methods. Based on this framework, we derive A-optimal designs that allow for efficient estimation of mean abilities and commonly used functions thereof. In analogy to classical ANOVA models, no constraints are imposed on the mean structure. The resulting A-optimal designs are substantially less restrictive than commonly used D-optimal designs. Moreover, the derived designs are also valid for a more general class of longitudinal Poisson Gamma models with repeated measurements. These models can be applied to analyze growth curves for count data in other disciplines, including epidemiology, medicine, and criminology.