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.
Keywords: 2PCMPM; Conway-Maxwell-Poisson distribution; EM algorithm; Item response theory; count data; item covariates; person covariates.