A broad framework for examining the class of
unidimensional and multidimensional models for
item responses is provided by nonlinear factor
analysis, with a classification of models as strictly
linear, linear in their coefficients, or strictly nonlinear.
These groups of models are compared and
contrasted with respect to the associated problems
of estimation, testing fit, and scoring an examinee.
The invariance of item parameters is related to the
congruence of common factors in linear theory.