Rasch used a Poisson model to analyze errors and speed in reading tests. An important property of the Poisson distribution is that the mean and variance are equal. However, in social science research, it is very common for the variance to be greater than the mean (i.e., the data are overdispersed). This study embeds the Rasch model within an overdispersion framework and proposes new estimation methods. The parameters in the proposed model can be estimated using the Markov chain Monte Carlo method implemented in WinBUGS and the marginal maximum likelihood method implemented in SAS. An empirical example based on models generated by the results of empirical data, which are fitted and discussed, is examined.

Longitudinal data describe developmental patterns and enable predictions of individual changes beyond sampled time points. Major methodological issues in longitudinal data include modeling random effects, subject effects, growth curve parameters, and autoregressive residuals. This study embedded the longitudinal model within a multigroup multilevel framework and allowed for autoregressive residuals. The parameter in the new model can be estimated using the computer program WinBUGS, which adopts Markov Chain Monte Carlo algorithms. Two simulation studies were conducted. An empirical example was raised and established based on models generated by the results of empirical data, which have been fitted and compared.

No abstract

The process-component approach has become quite popular for examining many psychological concepts. A typical example is the model with internal restrictions on item difficulty (MIRID) described by Butter (1994) and Butter, De Boeck, and Verhelst (1998). This study proposes a hierarchical generalized random-situation random-weight MIRID. The proposed model is more flexible for formulating endogenous latent variables within a multilevel framework, allowing the analysis of polytomous data with complex models (e.g., including item discriminations, random situations, random weights, and heteroskedasticity). The parameters in the proposed model can be estimated using the computer program WinBUGS, which adopts Markov Chain Monte Carlo algorithms. To illustrate the application of the proposed model, a real data set about guilt is analyzed and a comparison of MIRIDs for various conditions is conducted.