  A comprehensive class of models is proposed that can be used for continuous,
binary, ordered categorical and count type responses. The difficulty of items
is described by difficulty functions, which replace the item difficulty
parameters that are typically used in item response models. They crucially
determine the response distribution and make the models very flexible with
regard to the range of distributions that are covered. The model class contains
several widely used models as the binary Rasch model and the graded response
model as special cases, allows for simplifications, and offers a distribution
free alternative to count type items. A major strength of the models is that
they can be used for mixed item formats, when different types of items are
combined to measure abilities or attitudes. It is an immediate consequence of
the comprehensive modeling approach that allows that difficulty functions
automatically adapt to the response distribution. Basic properties of the model
class are shown. Several real data sets are used to illustrate the flexibility
of the models.
