Fitting a generalized linear model for the population size by year, region and gender

Abstract

Analysis of covariance is a very useful statistical technique based upon the general linear model, and as such can be presented as an extension of either analysis of variance or of regression analysis, or of both. The boundaries between these three kinds are not very sharp. Whereas in the analysis of covariance, some of the variables are factors and some are covariates, in the analysis of variance all the variables are factors and in regression analysis all variables are covariates. The emphasis of this study is on the practice of analysis of covariance from a regression point of view. It describes the general procedures of estimation and hypothesis testing for the analysis of covariance model and by means of an application onto a real-life data, clarifies the use of these techniques and demonstrates what conclusions could be made. The data analyzed in this dissertation consists of the total Maltese population classified by gender and region at each census year between 1861 and 1995. Even though this data (provided in Table 1.1) is balanced; emphasis is made onto unbalanced data. This is done since the analysis of covariance techniques for balanced data are merely special cases of those for unbalanced data. The dissertation proceeds by analyzing the data. This is done so as to achieve a suitable parsimonious analysis of covariance model with normal errors and identity link. This reveals some very important facts about the variables, such as which variables are significant and if there exists any dependence and/or interaction between them. This is done by means of the interactive statistical package GLIM. After this, a diagnostic analysis of the model, which includes analysis of the residuals, leverages and Cook's distances, is done. This diagnostic analysis shows that the fitted model is not applicable. Hence, an alternative generalized linear model with a gamma error distribution and log link function is proposed and fitted. This in turn provides an improved model.