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# BSc Actuarial Science and Mathematics / Course details

Year of entry: 2021

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## Course unit details:Generalised Linear Models

Unit code MATH38172 10 Level 3 Semester 2 Department of Mathematics No

### Overview

As an important modelling strategy Linear Models is concerned with investigating whether, and how, one or more so-called explanatory variables, such as age, sex, blood pressure, etc., influence a response variable, such as a patient's diagnosis, by taking random variations of data into account. In Linear Models, linear regression technique and Normal distribution are used to explore the possible linear relation between a continuous response and one or more explanatory variables. In this course unit we depart from linearity and normality, the very strict limitation in Linear Models. We study the extension of linearity to non-linearity and normality to a commonly encountered distribution family, called the exponential family of distributions. This extension forms Generalized Linear Models (GLM). The GLM, on the one hand, unifies linear and non-linear models in terms of statistical modelling. On the other hand, it can be used to analyze discrete data, including binary, binomial, counted and categorical data that arise very often in biomedical and industrial applications.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 2 MATH20701 Pre-Requisite Compulsory
Statistical Methods MATH20802 Pre-Requisite Compulsory
MATH38052 pre-requisites

Students are not permitted to take more than one of MATH38052 or MATH48052 for credit in the same or different undergraduate year.

Students are not permitted to take MATH48052 and MATH68052 for credit in an undergraduate programme and then a postgraduate programme.

Note that MATH68052 is an example of an enhanced level 3 module as it includes all the material from MATH38052

When a student has taken level 3 modules which are enhanced to produce level 6 modules on an MSc programme taken within the School of Mathematics, then they are limited to a maximum of two such modules (with no alternative arrangements available otherwise)

### Aims

To study an important aspect of modern statistical modelling in an integrated way, and to develop the properties and uses of GLM, focusing on those situations in which the response variable is discrete. To explore some of the wide range of real-life situations occurring in the fields of agriculture, biology, engineering, industrial experimentation, medicine and social science that can be investigated using GLM.

### Learning outcomes

On successful completion of this course unit students will be able to:

• use generalised linear models, including logistic regression and log linear models with a Poisson response, to analyse data with dependence on one or more explanatory variables;
• write down the fitted model, assess goodness-of-fit, test significance of parameters, compare models and use the chosen model to calculate various quantities of interest;
• write down a GLM with factors/covariates as appropriate, state the associated assumptions and constraints, derive the likelihood equation and algorithms for model fitting;
• prove that a given distribution belong to the exponential family, work out its mean, variance, variance function and derive the canonical link.

### Syllabus

1.Introduction: background, review of linear models in matrix notation, model assessment, some pre-required knowledge. 

2.The exponential family of distributions: Definition and examples. Mean and variance, variance function and scale parameter. 

3.Generalized linear models (GLM): linear predictor, link function, canonical link, maximum likelihood estimation, iterative reweighted least squares and Fisher scoring algorithms, significance of parameter estimates, deviance, Pearson and deviance residuals, Pearsonâ€'s chi-square test and the likelihood ratio test, model fitting using R. 

4.Normal linear regression models: least squares, analysis of variance, orthogonality of parameters, factors, interactions between factors. 

5.Binary and Binomial data analysis: distribution and models, logistic regression models, odds ratio, one- and two-way logistic regression analysis. 

6.Poisson count data analysis: Poisson regression models with offset, two-dimensional contingency tables, log-linear models. 

### Assessment methods

Method Weight
Other 20%
Written exam 80%
• Coursework: 20%
• End of semester examination: weighting 80%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• Dobson, A. J., An Introduction to Generalized Linear Models, Chapman & Hall 2002.
• Krzanowski, W., An Introduction to Statistical Modelling, Edward Arnold 1998.
• McCullagh, P. and Nelder, J. A., Generalized Linear Models, Chapman & Hall 1990.

### Study hours

Scheduled activity hours
Lectures 22
Tutorials 11
Independent study hours
Independent study 67

### Teaching staff

Staff member Role
Timothy Waite Unit coordinator