Some problems in the theory & application of graphical models
A graphical model is simply a representation of the results of an analysis of relationships between sets of variables. It can include the study of the dependence of one variable, or a set of variables on another variable or sets of variables, and can be extended to include variables which could be considered as intermediate to the others. This leads to the concept of representing these chains of relationships by means of a graph; where variables are represented by vertices, and relationships between the variables are represented by edges. These edges can be either directed or undirected, depending upon the type of relationship being represented. The thesis investigates a number of outstanding problems in the area of statistical modelling, with particular emphasis on representing the results in terms of a graph. The thesis will study models for multivariate discrete data and in the case of binary responses, some theoretical results are given on the relationship between two common models. In the more general setting of multivariate discrete responses, a general class of models is studied and an approximation to the maximum likelihood estimates in these models is proposed. This thesis also addresses the problem of measurement errors. An investigation into the effect that measurement error has on sample size calculations is given with respect to a general measurement error specification in both linear and binary regression models. Finally, the thesis presents, in terms of a graphical model, a re-analysis of a set of childhood growth data, collected in South Wales during the 1970s. Within this analysis, a new technique is proposed that allows the calculation of derived variables under the assumption that the joint relationships between the variables are constant at each of the time points.