Global Opportunities

MAS3913 : Linear & Generalised Linear Models

Semesters
Semester 1 Credit Value: 10
Semester 2 Credit Value: 10
ECTS Credits: 10.0

Aims

To achieve an understanding of linear models, and how regression, Analysis of Variance (ANOVA) and Analysis of Covariance (ANCOVA) models arise as special cases. To understand the problem of identifiability in ANOVA, and the role played by parameter constraints and dummy variables in solving it.

To achieve an understanding of Generalized Linear Models and achieve familiarity with the most common families, understanding how logistic regression and log linear models arise as special cases. To understand asymptotic maximum likelihood theory for more than one parameter and its application to Generalized Linear Models. To understand the Exponential family, and demonstrate that certain distributions belong to this.

Module summary

The first part of this module is concerned with building and applying statistical models for data. How does a mixture of quantitative and qualitative variables affect the body mass index of an individual? Suppose we find an association between age and body mass index, how can we study if this association varies between men and women, or between those with different educational backgrounds? In this course we consider the issues involved when we wish to construct realistic and useful statistical models for problems which can arise in a range of fields: medicine, finance, social research and environmental issues being some of the main areas.
We revise multiple linear regression models, and see how they are special cases of a General Linear Model. We move on to consider Analysis of Variance (ANOVA) as another special case of a general linear model – this is the problem of investigating contrasts between different levels of a factor in affecting a response and then we generalize to the case of several factors. We consider Analysis of Covariance (ANCOVA) which involves mixing linear regression and factor effects, and the idea of interaction between explanatory variables in the way they affect a response. The module provides a comprehensive introduction to the issues involved in using statistics to model real data, and to draw relevant conclusions. There is an emphasis on hands-on application of the theory and methods throughout, with extensive use of R.

The second part of the module builds on the first part (Linear Modelling) by introducing a generalized framework of models which allow us to generalize away from Normally distributed errors to different kinds of random outcomes, building in an appropriate transformation of the linear function of explanatory variables to match. We note that the general linear models studied in the first part of the module exist as a special case.

We generalize linear models to study the topic of Generalized Linear Models, allowing us to build non-linear relationships into our models, and to study many different types of outcome measure which could not have been handled using general linear models. We consider asymptotic maximum likelihood estimation for the multi-parameter case, including the use of information matrices in parameter estimation and likelihood ratio tests for comparing nested models. These ideas are applied to Generalized Linear Models. We study in depth the special cases involved with Binomial outcomes, logistic regression, where we are interested in how explanatory variables affect the success rate, and then log-linear models, which enable us to study among other things, contingency tables involving more than two factors.

This module opens up the possibility to study many kinds of real life situations which were inaccessible to linear models, allowing us to study many realistic and important problems. There is an emphasis on hands-on application of the theory and methods throughout, with extensive use of R.

Outline Of Syllabus

The general linear model: maximum likelihood in the multi-parameter case; estimation of parameters; prediction; model adequacy; regression, ANOVA and ANCOVA as special cases. Model choice. Analysis of designs with 1, 2 or 3 factors. Model identifiability, parameter constraints and dummy variables. Use of transformations. Various extended examples of statistical modelling using R.

Generalized linear models: overall construction as generalization of linear models; binomial regression with various links; Poisson regression; log-linear models and their use for contingency tables. Asymptotic distribution of the maximum likelihood estimator in the multi-parameter case. The Exponential family of distributions. Various extended examples of statistical modelling using R.

Teaching Methods

Teaching Activities
Category Activity Number Length Student Hours Comment
Scheduled Learning And Teaching ActivitiesLecture401:0040:00Formal Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture41:004:00Revision Lectures – Present in Person
Scheduled Learning And Teaching ActivitiesLecture101:0010:00Problem Classes – Synchronous On-Line
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Guided Independent StudyIndependent study1161:00116:00Preparation time for lectures, background reading, coursework review
Total200:00
Teaching Rationale And Relationship

Lectures are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on marked work. Problem Classes are used to help develop the students’ abilities at applying the theory to solving problems.

Assessment Methods

The format of resits will be determined by the Board of Examiners

Exams
Description Length Semester When Set Percentage Comment
Written Examination1502A80N/A
Other Assessment
Description Semester When Set Percentage Comment
Prob solv exercises1M10Coursework assignment
Prob solv exercises2M10Coursework assignment
Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The coursework assignments allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.

In the event of on-campus examinations not being possible, an on-line alternative assessment will be used for written examination 1.

Reading Lists

Timetable