Log-linear models, parameter symmetry, and exchangeable discrete random variables.

Abstract

Log-linear models for the analysis of classes of potentially exchangeable variables are the focus of this dissertation. The previous work of other authors concerning exchangeable categorical models is extended to a more general case, and a theoretical basis for this work is established. Moreover, a hierarchy of nested log-linear models assuming various forms of exchangeability and parameter symmetry is made evident by a new reparameterization of the log-linear model. This hierarchy encompasses the special cases of exchangeable and quasisymmetric models presented in the literature. The relationships among exchangeability, parameter symmetry and invariance, and table collapsibility are examined. Specifically, the necessary and sufficient conditions for exchangeability, which involve parameter symmetry and invariance, are proven to depend on the orders of interactions included in a log-linear model. It is also demonstrated that imposing exchangeability upon the saturated model of a full cross-classification yields the saturated model of the corresponding collapsed table. Moreover, an interesting relationship among likelihood ratio statistics is derived for the saturated model and those nested underneath it when they are fitted to full and collapsed tables. A reparameterization of the unconstrained log-linear model is developed such that the resulting parameters characterize departures from exchangeability or parameter symmetry. Subsets of parameters in this new reparameterization resemble effects models for one- and two-dimensional factorial designs. A hierarchy of parameter symmetric models and models for exchangeable random variables is formed by setting various sets of parameters in the new reparameterization equal to 0. The resulting hierarchy suggests a set of conditional likelihood ratio tests of symmetry and exchangeability. Relatively awkward approaches that have been proposed in the literature for comparing certain models in this hierarchy are no longer necessary. The log-linear models in the hierarchy are fitted to trichotomous measures of alveolar bone loss for analogous teeth in the upper and lower arches. The clinical interest in the variants of exchangeability and parameter symmetry arises because these properties constrain the associations among teeth in scientifically revealing ways. Although pairwise associations appear to be symmetric, the data do not support exchangeability. Instead those estimated associations involving adjacent teeth are more significant than those corresponding to nonadjacent teeth. Furthermore, a well-fitting model that is more parsimonious than the parameter symmetric model but less constrained than those models assuming exchangeability suggests that inter-arch first order associations involving certain upper arch teeth are stronger than those involving other upper arch teeth regardless of which lower tooth is involved.