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MC361 Generalized Linear Models

MC361 Generalized Linear Models

Credits: 20 Convenor: Dr. M.J. Phillips Semester: 2
Prerequisites: essential: MC160, MC260, MC264, MC265
Assessment: Coursework: 20% Three-hour examination: 80%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Explanation of Pre-requisites

The module MC160 provides the basic ideas of probability which are required to understand statistical methods. The module MC265 presents the basic concept of a statistic and its use in estimation and hypothesis testing. The module MC260 provides the techniques for study of the joint distributions of random variables which enable the classical normal theory results of mathematical statistics to be derived. Then there is a brief introduction to the basic methods of data analysis based on the classical normal theory results of mathematical statistics. The module MC264 presents the basic methods of data analysis based on the general linear model and the theoretical results of this model.

Course Description

This module extends the ideas used in Linear Modelling to a more general framework, which allows the possibility of including a number of analyses in one general approach. This occurs in the case when the response variable is dependent through some link function on a predictor of an unknown linear combination of the explanatory variables as well as an error random variable. With a suitable choice of link function and error structure it is possible to cover, within a general framework, a number of techniques for analysing data: linear modelling of continuous variables, log-linear modelling for the analysis of counts and proportions and linear logistic regression modelling for binary data. Two prime objectives of an analysis using these models include a determination of which explanatory variables are important, and exactly how these variables are related to the response variable. It is possible to associate confidence intervals to estimates or predictions obtained from the model and assign p-values to hypotheses to be tested. The analysis of the model is based on the deviance which offers a method for assessing the acceptability of any proposed model.

Aims

This module aims to introduce the generalized linear model. It presents the methods of estimation of the linear model parameters and the methods of making inferences about these parameters through the calculation of confidence intervals and the use of hypothesis tests. In particular it is an aim of this course to impart understanding of how statistical models explain variation. The aim of covering the theory for the generalized linear model is to understand the extension of the theory to cover log-linear models for the analysis of counts and proportions and linear logistic regression models for binary data. Illustrations of how to use statistical software package GLIM to analyse data using the generalized linear model are given.

Objectives

To present the assumptions made in using the generalized linear regression model.

To enable the calculation of confidence intervals and the use of hypothesis tests for model parameters.

To assess the fit of a log-linear model using a nested hierarchy of log-linear models.

To understand the theory of the generalized linear model.

To understand how to use GLIM to analyse data with the generalized linear model.

Transferable Skills

The ability to calculate estimates and make inferences about the parameters of the generalized linear regression model.

The ability to construct confidence intervals and to perform hypothesis tests for model parameters.

The ability to assess the fit of a log-linear model using change in deviance.

Knowledge of the theory of the general linear model.

The ability to use GLIM to analyse data with the generalized linear model.

Syllabus

Examples of two dimensional contingency tables. Pearson goodness-of-fit test for 2 by 2 table. Three sampling models. Maximum likelihood estimates (MLEs) for the expected cell frequencies for two dimensional contingency tables. Properties of the odds ratio. Likelihood ratio goodness-of-fit test statistic. Partitioning two dimensional tables. Three dimensional tables. Hierarchical log-linear models for count data. Maximum likelihood estimates (MLEs) for the expected cell frequencies for two dimensional contingency tables. Bartlett's test. Conditional likelihood ratio goodness-of-fit test statistic. Theorem for collapsing tables. Birch's results. A nested hierarchy of log-linear models. Model choice. Higher dimensional tables. Stepwise selection procedures. The theory of the generalized linear model. Deviance. Residuals. Models for binary data - linear logistic regression models. Incomplete two dimensional contingency tables. Fitting models in GLIM.

Reading list

Essential:

P. McCullagh and J. A. Nelder, Generalized Linear Models, 2nd edition, Chapman and Hall.


Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a three hour examination, which is OPEN BOOK, during the Summer exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper will contain 6 questions with full marks on the paper obtainable from 4 complete answers.

Next: MC380 Ordinary Differential Equations Up: Year 3 Previous: MC356 Topology

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Last updated: 2001-09-20
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