Loglinear Models as Item Response Models

For analyzing item response data, item response theory (IRT) models treat the discrete responses to the items as driven by underlying continuous latent traits, and consider the form of conditional probability of the response to each item given the latent traits. In a similar fashion, log-linear models directly consider the form of the manifest probability of response patterns. Researchers have been connecting the two paradigms by establishing equivalence relationships between IRT models and log-linear models. This has lead to the notion of obtaining IRT solutions by fitting their equivalent log-linear models. In this research, I have established a family of log-linear models, log linear-by-linear association (LLLA) models, that incorporate a variety of IRT models, particularly, a family of generalized Rasch models. I have derived an extension of the Dutch Identity theorem to polytomous items and utilized it to develop the models that incorporate item covariates and person covariates. Noteworthy features of the models include both polytomous responses and multiple latent traits. Along with developing this new family of models, I have conducted extensive research on the development of an accompanying estimation method. Historically, a significant barrier to the application of log-linear models in analyzing item responses has been the high computational cost of maximum likelihood estimation (MLE), due to the fact that the number of response patterns grows exponentially as the number of items increases. To solve this computational problem, a pseudo-likelihood estimation (PLE) method is proposed and it dramatically decreases the computational cost. To demonstrate the effectiveness of the developed models and the pseudolikelihood estimation method, I will present results of a series of simulation studies. To demonstrate the practical advantages of the methods, I will give a detailed description of an application to a real data set from a study on verbally aggressive behavior