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

Abstract

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 overand 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.

A Comparison of Different Item Response Theory Models for Scaling Speeded C-Tests

“…In particular, reliability estimates were the same (when rounded to three decimals) for both ordinal models (Rel(θ) = .961), and the CMPCM models (Rel(θ) = .969), clearly performed on par with these estimates. The slightly lower reliability of the RPCM as compared with the CMPCM models was due to the underdispersion in the data that is not taken into account by the RPCM (Forthmann et al, 2019).…”

“…This strong model assumption of equidispersion makes the model less useful than other members of the Rasch model family. Although subsequent methodological advances somewhat widened the applicability of the RPCM (Hung, 2012), only recently Forthmann et al (2019) showed how the original RPCM can be generalized into the Conway–Maxwell–Poisson Counts Model (CMPCM). The CMPCM is of similar flexibility as many linear latent trait models, as the mean is not directly related to the error variance.…”

“…The person parameter θ j is assumed to be a Gaussian latent variable, with variance normalσnormalθ2 estimated freely. Moreover, the CMPCM, as compared with RSM and PCM, is a model with fewer parameters (two parameters per item and a latent ability variance) and can be estimated with small sample sizes (Forthmann et al, 2019).…”

Residual Analysis in Rasch Poisson Counts Models

“…We show that violin plots are interesting to show the fit of the test items. Considering the proposed method of analysis of residuals in the analyzed data, our results show that the model is not the best model for the data, so other models such as those of Hung (2012) and Forthmann, Gühne and Doebler (2019) can be studied in the future with this dataset.…”

A Discrete Gamma Model Approach to Flexible Count Regression Analysis: Maximum Likelihood Inference

“…Indeed, real count data are often overdispersed (variance larger than mean) or underdispersed (variance smaller than mean). In either case, the inability of a model to account for underdispersion or overdispersion can cause standard errors to be biased downward or upward, thus under or over estimating the statistical significance of associated explanatory variables [8,9].…”