Factor Analysis of Cross-Classified Data

Abstract

This thesis introduces a model hierarchy related to Principal Component Analysis and Factor Analysis, in which vector measurements are linearly decomposed into a relatively small set of hypothetical principal directions, for purposes of dimension reduction. The mathematical specification of unknown parameters in the models is unified. Identifiability of the suitably defined models is proved. The EM algorithm and the Newton-Raphson algorithm based on likelihoods and profile likelihoods are implemented to get computationally effective (maximum likelihood) estimators for the unknown parameters. A restricted model (with some error variances $0$) and a sufficient condition for a local maximum likelihood estimate are established. Score tests are constructed to check whether error variances are $0$, which is shown to be associated with non-identifiability of models. Statistical tests of goodness of fit of the models to data are established in a likelihood ratio testing framework, so that the most parsimoniously parameterized model consistent with the data can be chosen for purposes of description and classification of the experimental settings. The results are applied on a real data set involving coronal cross-sectional ultrasound pictures of the human tongue surface during speech. The likelihood ratio test is used to test the fit of the PARAFAC model on the real coronal tongue data, leading to a finding of inadequacy of the PARAFAC model.