Revisiting dispersion in count data item response theory models: The Conway-Maxwell-Poisson counts model.

Count data naturally arise in several areas of cognitive ability testing, such as processing speed, memory, verbal fluency, and divergent thinking. Contemporary count data item response theory models, however, are not flexible enough, especially to account for over- and underdispersion at the same time. For example, the Rasch Poisson counts model (RPCM) assumes equidispersion (conditional mean and variance coincide) which is often violated in empirical data. This work introduces the Conway-Maxwell-Poisson counts model (CMPCM) that can handle underdispersion (variance lower than the mean), equidispersion, and overdispersion (variance larger than the mean) in general and specifically at the item level. A simulation study revealed satisfactory parameter recovery at moderate sample sizes and mostly unbiased standard errors for the proposed estimation approach. In addition, plausible empirical reliability estimates resulted, while those based on the RPCM were biased downwards (underdispersion) and biased upwards (overdispersion) when the simulation model deviated from equidispersion. Finally, verbal fluency data were analysed and the CMPCM with item-specific dispersion parameters fitted the data best. Dispersion parameter estimates indicated underdispersion for three out of four items. Overall, these findings indicate the feasibility and importance of the suggested flexible count data modelling approach.