The Two-Parameter Quasi-Poisson Item Response Theory Model for Handling Heterogeneous Dispersion in Count Data

To correctly model heterogeneous dispersion in count data, several psychometric methods have been proposed. In this paper, we introduce a semiparametric two-parameter Quasi-Poisson model that extends a recently proposed one-parameter Quasi-Poisson model and relaxes its equal-discrimination assumption. The proposed model is computationally efficient for small to moderately sized instruments and yields dispersion parameters that are comparable to those of the Poisson and negative binomial models while also accommodating underdispersion. We adopt a generalized method of moments framework for fast model parameter estimation with accessible standard errors (SEs), together with an adaptive information-weighted scoring procedure for latent ability estimation. We conduct extensive simulation studies to examine the recovery of model parameters, their associated SEs, and latent abilities, and we present an empirical data analysis to illustrate the application of the proposed model in a real-world setting.