Flexible Item Response Models for Count Data: The Count Thresholds Model

A new item response theory model for count data is introduced. In contrast to models in common use, it does not assume a fixed distribution for the responses as, for example, the Poisson count model and extensions do. The distribution of responses is determined by difficulty functions which reflect the characteristics of items in a flexible way. Sparse parameterizations are obtained by choosing fixed parametric difficulty functions, more general versions use an approximation by basis functions. The model can be seen as constructed from binary response models as the Rasch model or the normal-ogive model to which it reduces if responses are dichotomized. It is demonstrated that the model competes well with advanced count data models. Simulations demonstrate that parameters and response distributions are recovered well. An application shows the flexibility of the model to account for strongly varying distributions of responses.