Multilevel, Two-Parameter, and Random-Weights Generalizations of a Model With Internal Restrictions on Item Difficulty

In this study, all the advantages of slope parameters, random weights, and latent regression are acknowledged when dealing with component and composite items by adding slope parameters and random weights into the standard item response model with internal restrictions on item difficulty and formulating this new model within a multilevel framework in which Level 2 predictors are added to account for variation in the latent trait. The resulting model is a nonlinear mixed model (NLMM) so that existing parameter estimation procedures and computer packages for NLMMs can be directly adopted to estimate the parameters. Through simulations, it was found that the SAS NLMIXED procedure could recover the parameters in the new model fairly well and produce appropriate standard errors. To illustrate applications of the new model, a real data set pertaining to guilt was analyzed with gender as a Level 2 predictor. Further model generalization is discussed.