A Flexible Approach to Modeling Over-, Under- and Equidispersed Count Data in IRT: The Two-Parameter Conway-Maxwell-Poisson Model

Abstract

Several psychometric tests generate count data, e.g. the number of ideas in divergent thinkingtasks. The most prominent count data IRT model, the Rasch Poisson Counts Model (RPCM)assumes constant discriminations across items as well as the equidispersion assumption of thePoisson distribution (i.e., E(X) = Var(X)), considerably limiting modeling flexibility. Violationsof these assumptions are associated with impaired ability, reliability, and standard error estimates.Models have been proposed to loose the one or the other assumption. The Two-Parameter PoissonCounts Model (2PPCM) allows varying discriminations but retains the equidispersion assumption.The Conway-Maxwell-Poisson Counts Model (CMPCM) that allows for modeling equi- but alsoover- and underdispersion (more or less variance than implied by the mean under the Poisson distribution)but assumes constant discriminations. The present work introduces the Two-ParameterConway-Maxwell-Poisson (2PCMP) model which generalizes the RPCM, the 2PPCM, and the CMPCM(all contained as special cases) to allow for varying discriminations and dispersions withinone model. A marginal maximum likelihood method based on a fixed quadrature Expectation-Maximization (EM) algorithm is derived. Standard errors as well as two methods for latent abilityestimation are provided. An implementation of the 2PCMP model in R and C++ is provided. Twosimulation studies examine the model’s statistical properties and compare the 2PCMP model toestablished methods. Data from divergent thinking tasks are re-analyzed with the 2PCMP modelto illustrate the model’s flexibility and ability to test assumptions of special cases.