A loglinear item response theory (IRT) model is proposed that relates polytomously scored item responses to a multidimensional latent space. Each item may have a different response function where each item response may be explained by one or more latent traits. Item response functions may follow a partial credit model (D. Andrich, 1978; and G. N. Masters, 1982), a multidimensional Rasch model (G. Rasch, 1961; and E. B. Andersen, 1973, 1983), or other forms of response functions to be defined by the user. Conditional maximum likelihood estimates are derived, and the models may be tested generally or against alternative loglinear models. The latter tests are sensitive to deviations from local independence subgroup invariance or assumptions about the form of the operating characteristic curves. The model was illustrated through application to data from a test to identify learning problems in Dutch children from 4 to 6.5 years of age. Fifteen items were administered to 66 children aged 4 to 5 years, 132 children aged 5 to 5.5 years, and 65 children aged 5.5 to 6 years. Three appendices illustrate the dichotomous Rasch model, the partial credit model, and Rasch's multidimensional model. Two tables contain data from the illustrative study and three figures give examples to support the text. (SLD)