The Analysis of Multivariate, Longitudinal Categorical Data by Log-Multilinear Models

362 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1993.The models developed in this thesis are designed to analyze the nature of the interactions present in 3-mode, multivariate, longitudinal categorical data. They are extensions of loglinear models that provide graphical representations of interactions or associations among discrete variables. The models are generalizations of association models for 2-way tables and are similar to canonical correlation models and their generalizations.Basic concepts regarding the modeling and analysis of categorical data are reviewed, as well as association and correlation models for 2-way tables. The association and correlation models are inherently limited because they rely on 2-way singular value decompositions. Various strategies for using these models to analyze 3- or higher-way tables and existing generalizations of these models are reviewed. To demonstrate both the usefulness and the limitations of these approaches, they are applied to a multivariate, longitudinal data set from a study in developmental psychology (Kramer & Gottman, 1992).To overcome the limitations and problems associated with the existing models, a new class of models is developed. This class consists of extensions of loglinear models for 3-way tables in which particular combinations of the 2- and higher-way interactions are decomposed by Tucker's 3-mode principal components model (Tucker, 1963, 1964, 1966; Kroonenberg, 1983). Tucker's 3-mode model, which is a generalization of 2-way singular value decomposition to 3-way matrices, is quite useful for analyzing longitudinal data.Typical of longitudinal data, the data analyzed in this study contain missing observations, and the standard sampling assumption of independent, Poisson variables is not valid. Three different strategies for dealing with missing data are employed. The violations of the sampling assumptions invalidate statistical tests and affect estimates of standard errors. Residual analyses are performed to ascertain the degree to which violations of the sampling assumptions affect the results of analyses.All of the models and techniques reviewed and used to analyze the data in this study are compared theoretically and empirically. Equivalences among them are pointed out. There is a remarkable similarity of results from fitting the various models to the data, but the new models yield the most parsimonious representations.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD