A flexible approach to modelling over-, under- and equidispersed count data in IRT: The Two-Parameter Conway-Maxwell-Poisson Model

Interactions between latent variables in count regression models

In psychology and the social sciences, researchers often model count outcome variables accounting for latent predictors and their interactions. Even though neglecting measurement error in such count regression models (e.g., Poisson or negative binomial regression) can have unfavorable consequences like attenuation bias, such analyses are often carried out in the generalized linear model (GLM) framework using fallible covariates such as sum scores. An alternative is count regression models based on structural equation modeling, which allow to specify latent covariates and thereby account for measurement error. However, the issue of how and when to include interactions between latent covariates or between latent and manifest covariates is rarely discussed for count regression models. In this paper, we present a latent variable count regression model (LV-CRM) allowing for latent covariates as well as interactions among both latent and manifest covariates. We conducted three simulation studies, investigating the estimation accuracy of the LV-CRM and comparing it to GLM-based count regression models. Interestingly, we found that even in scenarios with high reliabilities, the regression coefficients from a GLM-based model can be severely biased. In contrast, even for moderate sample sizes, the LV-CRM provided virtually unbiased regression coefficients. Additionally, statistical inferences yielded mixed results for the GLM-based models (i.e., low coverage rates, but acceptable empirical detection rates), but were generally acceptable using the LV-CRM. We provide an applied example from clinical psychology illustrating how the LV-CRM framework can be used to model count regressions with latent interactions.

Understanding Ability and Reliability Differences Measured with Count Items: The Distributional Regression Test Model and the Count Latent Regression Model

In psychology and education, tests (e.g., reading tests) and self-reports (e.g., clinical questionnaires) generate counts, but corresponding Item Response Theory (IRT) methods are underdeveloped compared to binary data. Recent advances include the Two-Parameter Conway-Maxwell-Poisson model (2PCMPM), generalizing Rasch's Poisson Counts Model, with item-specific difficulty, discrimination, and dispersion parameters. Explaining differences in model parameters informs item construction and selection but has received little attention. We introduce two 2PCMPM-based explanatory count IRT models: The Distributional Regression Test Model for item covariates, and the Count Latent Regression Model for (categorical) person covariates. Estimation methods are provided and satisfactory statistical properties are observed in simulations. Two examples illustrate how the models help understand tests and underlying constructs.