The development and application of a fit test for use with marginal maximum likelihood estimation and generalised item response models

Thesis (M.Ed.) -- University of Melbourne (Faculty of Education), 1997The construction of fit tests for item response models that are estimated with marginal maximum likelihood methods has proven problematic, largely because the theoretical reference distribution for the test statistics cannot be easily determined or approximated. The problems faced are further confounded if generalised IRT models for polytomous (Masters, 1982) and multi-faceted (Linacre, 1994) data, such as the generalised Rasch type item response models described by Adams and Wilson (1996), are considered. This dissertation proposes a number of residual based fit statistics, at the parameter level, for the generalised Rasch model when marginal maximum likelihood estimation is used. The theoretical asymptotic distributions of the fit statistics are also derived. Simulation results show that the sample distributions of the fit statistics are close to the predicted theoretical distributions, thus these statistics are able to provide a frame of reference as indices of fit. Various model violations are introduced to simulated data and the extent to which the fit statistics are able to detect such violations is examined. The results show that the fit statistic is useful for detecting a number of types of misfit of the model. Examples of the analyses of two real datasets are given to illustrate how the fit statistics can be used to detect problems in the data, in particular, problems associated with the violation of the assumptions of unidimensionality and independence between item responses