  A general framework of latent trait item response models for continuous
responses is given. In contrast to classical test theory models, which
traditionally distinguish between true scores and error scores, the responses
are clearly linked to latent traits. It is shown that classical test theory
models can be derived as special cases but the model class is much wider. It
provides, in particular, appropriate modelling of responses that are restricted
in some way, for example, if responses are positive or are restricted to an
interval. Restrictions of this sort are easily incorporated in the modeling
framework. Restriction to an interval is typically ignored in common models
yielding inappropriate models, for example, when modeling Likert-type data. The
model also extends common response time models, which can be treated as special
cases. Properties of the model class are derived and the role of the total
score is investigated, which leads to a modified total score. Several
applications illustrate the use of the model including an example, in which
covariates that may modify the response are taken into account.
