Estimation for components of variance models

The main purpose of this thesis is to investigate the fitting, using maximum likelihood methods, of certain variance components models with a covariate. Random effects models and split plot models in particular have been employed in a variety of different fields over the years. It is the inclusion of a covariate term in such models that makes the investigation of this thesis differ from much of the literature. This covariate term models a linear dependence relationship between covariate and response variables, while at the same time allowing the estimation of the variance components.

In this thesis we begin by investigating a one-way random effects model with a covariate. We estimate parameters of the model using a sequential approach to maximum likelihood. Several key contributions are made to the understanding of the model. Firstly we show that the level of global and local linearity evident in a given data set determines the form that the loglikelihood function will take. For data sets with moderate to strong levels of both global and local linearity we find that the loglikelihood function has two local maxima and thus there are two candidates for the maximum likelihood estimate. Secondly we introduce an interval which we know must contain the slope parameter and which is found to be a significant improvement on the investigation of a previous author. This interval most often contains only one of the candidate maximum likelihood estimates. Finally we develop an heuristic algorithm which details the parameter estimation routine.

This thesis also looks at a number of generalisations of the one-way random effects model with a covariate. These include allowing the slope to differ between treatments, consideration of multiple covariates and the addition of a blocking term. We develop parameter estimation routines for each of these generalisations and thus allow the user to choose a model which best suits a given data set. A real data set is used to illustrate the parameter estimation outine for each model.

The motivating question for this thesis concerns the inclusion of a covariate term in a split plot model. Several formulations of the split plot model are examined in order to investigate conditions under which the two variance components can be separately estimated. A parameter estimation routine is developed and illustrated using the motivating forestry data set.

The final contribution of this thesis is a thorough investigation of the use of the EM algorithm as an alternative route to maximum likelihood estimation. The success of this algorithm is confirmed when the resulting parameter estimates are found to be equivalent to those found using the earlier analytically based maximum likelihood approach.