A multidimensional Rasch-type item response model, the multidimensional random coefficients multinomial logit model, is presented as an extension to the Adams & Wilson (1996) random coefficients multinomial logit model. The model is developed in a form that permits generalization to the multidimensional case of a wide class of Rasch models, including the simple logistic model, Masters' partial credit model, Wilson's ordered partition model, and Fischer's linear logistic model. Moreover, the model includes several existing multidimensional models as special cases, including Whitely's multicomponent latent trait model, Andersen's multidimensional Rasch model for repeated testing, and Embretson's multidimensional Rasch model for learning and change. Marginal maximum likelihood estimators for the model are derived and the estimation is examined using a simulation study. Implications and applications of the model are discussed and an example is given.

In this article we show how certain analytic problems that arise when one attempts to use latent variables as outcomes in regression analyses can be addressed by taking a multilevel perspective on item response modeling. Under a multilevel, or hierarchical, perspective we cast the item response model as a within-student model and the student population distribution as a between-student model. Taking this perspective leads naturally to an extension of the student population model to include a range of student-level variables, and it invites the possibility of further extending the models to additional levels so that multilevel models can be applied with latent outcome variables. In the two-level case, the model that we employ is formally equivalent to the plausible value procedures that are used as part of the National Assessment of Educational Progress (NAEP), but we present the method for a different class of measurement models, and we use a simultaneous estimation method rather than two-step estimation. In our application of the models to the appropriate treatment of measurement error in the dependent variable of a between-student regression, we also illustrate the adequacy of some approximate procedures that are used in NAEP.

The Rasch rating (or partial credit) model is a widely applied item response model that is used to model ordinal observed variables that are assumed to collectively reflect a common latent variable. In the application of the model there is considerable controversy surrounding the assessment of fit. This controversy is most notable when the set of parameters that are associated with the categories of an item have estimates that are not ordered in value in the same order as the categories. Some consider this disordering to be inconsistent with the intended order of the response categories in a variable and often term it reversed deltas. This article examines a variety of derivations of the model to illuminate the controversy. The examination of the derivations shows that the so-called parameter disorder and order of the response categories are separate phenomena. When the data fit the Rasch rating model the response categories are ordered regardless of the (order of the) values of the parameter estimates. In summary, reversed deltas are not necessarily evidence of a problem. In fact the reversed deltas phenomenon is indicative of specific patterns in the relative numbers of respondents in each category. When there are preferences about such relative numbers in categories, the patterns of deltas may be a useful diagnostic.

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