Dichotomous and polytomous item response model estimation using the marginal maximum likelihood and the EM algorithm

Abstract

Item Response Theory is a set oflatent variable techniques specifically designed to model the interaction between a respondent's latent trait/ability and the test items' characteristics such as difficulties, discrimination powers and guessing liabilities. The Item Response Theory framework emphasizes on how responses can be modelled in probabilistic terms, with the focus being on the response patterns rather than on the total test scores. fu Item Response Theory, the item responses are considered as the dependent variables, while the respondents' abilities and the items' characteristics are the independent, latent predictor variables. For dichotomously scored items, the probability of a correct response can be described by one of the various dichotomous Item Response Models, namely the Rasch Model (Rasch, 1961), the Two-Parameter Logistic Model (Birnbaum, 1968) and the Three-Parameter Logistic Model (Birnbaum, 1968). The assumptions underlying these models, as well as the concepts of Item and Test fuformation Functions are discussed in detail. The Bock and Lieberman Marginal Maximum T .ikelihood solution, the Bock and Aitkin Marginal Maximum Likelihood solution, and the Maximum Likelihood ability parameter estimation technique are also presented. fu addition to dichotomous models, several polytomous Item Response Models have been proposed, among which are the Partial Credit Model (Masters, 1982) and the Rating Scale Model (Andrich, 1978), belonging to the polytomous family of Rasch Models, and the Graded Response Model (Samejima, 1969), belonging to the San1ejima family of models. These polytomous models generalize dichotomous models, and are appropriate for rating scales characterized by ordered options. Parameter estimation methods for these models are also presented. Using the statistical package STATA, the Rasch Model and the Two-Parameter Logistic Model were applied to the dataset obtained through a questionnaire assessing perceptions about abortion. This questionnaire was distributed to around 200 individuals and included items describing abortion-related statements which required to be rated on a 5-point Likert scale by the respondents. Moreover, the EM Algorithm for Finite Mixtures was applied to this dataset using GLIM. The Expectation-Maximization algorithm is an iterative algorithm for Marginal Likelihood Estimation in the presence of unobserved random variables, in this case represented by the respondents' latent traits.