	Department of Mathematics & Computer Science

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MA2251 Linear Regression Models

MA2251(=MC262) Linear Regression Models

Credits: 10

Convenor: Dr. M.J. Phillips

Semester: 2 (weeks 1 to 6)

Prerequisites:	essential: MA1061(=MC160), MA2201(=MC265)
Assessment:	Coursework: 20%	One and a half hour examination: 80%

Lectures:	18	Problem Classes:	3
Tutorials:	none	Private Study:	51
Labs:	3	Seminars:	none
Project:	none	Other:	none
Surgeries:	none	Total:	75

Explanation of Pre-requisites

The module MA1061 provides the basic ideas of probability which are required to understand statistical methods. The module MA2201 presents the basic concept of a statistic and its use in estimation and hypothesis testing. Then there is a brief introduction to the basic methods of data analysis based on the classical normal theory results of mathematical statistics.

Course Description

The single most important method in statistical analysis is the natural extension of the simple linear (regression) model to include several explanatory variables to give the general linear model. With a judicious choice of variables it is possible to cover, within a general framework, a number of techniques for analysing data including the analysis of variance (ANOVA) techniques. A prime objective of an analysis using this model is 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 method of least squares estimation, where the `residual sum of squares' not only provides an estimate of the error variance but, perhaps more importantly, also offers a method for assessing the acceptability of any proposed model.

Note: this module cannot be taken in conjunction with MA2261.

Aims

This module covers the general linear model. A brief introduction to the least squares estimation of regression lines aims to show how linear relationships can be established between two variables. This module aims to present the method of least squares 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 general linear regression model is to enable the extension of the theory to cover multiple linear regression, polynomial regression, analysis of variance and covariance. This module aims to provide a necessary foundation for the study of generalized linear models in MA3201.

Objectives

To know the distributions of the least squares estimators for parameters of the simple linear regression model.

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

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

To perform hypothesis tests using an analysis of variance (ANOVA) table and to understand how it is used to explain variation.

To assess the lack of fit of a linear model using repeated observations.

Transferable Skills

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

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

Knowledge of using an analysis of variance (ANOVA) table to explain variation.

The ability to assess the lack of fit of a linear model using repeated observations.

Knowledge of some of the theory of the general linear model.

Syllabus

Relationships between variables are considered. Inferences about the linear association of two variables are studied using the coefficient of correlation. The simple linear regression model is briefly explained. Least squares estimators of slope and intercept parameters are obtained. Sampling distributions are derived for these estimators as well as for `error' variance parameter. Inference for the model parameters is covered using confidence intervals, prediction intervals and hypothesis tests. Calibration. Least squares estimators of the parameters of some other simple models are obtained. Use of residuals for checking model assumptions. Analysis of variance. Distributions of sums of squares of standard normal variables. Lack of fit using repeated observations. The theory of the general linear model.

Reading list

Essential:

W. Mendenhall, R. L. Scheaffer and D. D. Wackerly, Mathematical Statistics with Applications, 4th edition, Duxbury Press, 1990.

Details of Assessment

The final assessment of this module will consist of 20% coursework and 80% from a one and a half hour examination during the Summer exam period. The 20% coursework contribution will be determined by students' solutions to coursework problems. The examination paper, which is OPEN BOOK, will contain 3 questions, with full marks on the paper obtainable from 2 complete answers.