# MAS3913 : Linear & Generalised Linear Models

• Offered for Year: 2020/21
• Module Leader(s): Dr Pete Philipson
• Lecturer: Dr David Walshaw
• Owning School: Mathematics, Statistics and Physics
• Teaching Location: Newcastle City Campus
##### Semesters
 Semester 1 Credit Value: 10 Semester 2 Credit Value: 10 ECTS Credits: 10

#### 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

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

##### Teaching Activities
Category Activity Number Length Student Hours Comment
Guided Independent StudyAssessment preparation and completion301:0030:00Completion of in course assessments
Structured Guided LearningLecture materials361:0036:00Non-Synchronous Activities
Scheduled Learning And Teaching ActivitiesLecture91:009:00Synchronous On Line Material
Scheduled Learning And Teaching ActivitiesLecture91:009:00Present in Person
Structured Guided LearningStructured non-synchronous discussion181:0018:00Non-Synchronous Discussions of Lecture Material
Scheduled Learning And Teaching ActivitiesDrop-in/surgery41:004:00Office Hour or Discussion Board Activity
Guided Independent StudyIndependent study941:0094:00Lecture preparation, background reading, course review
Total200:00
##### Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer: a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

#### Assessment Methods

Please note that module leaders are reviewing the module teaching and assessment methods for Semester 2 modules, in light of the Covid-19 restrictions. There may also be a few further changes to Semester 1 modules. Final information will be available by the end of August 2020 in for Semester 1 modules and the end of October 2020 for Semester 2 modules.

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

##### Exams
Description Length Semester When Set Percentage Comment
Written Examination1202A80Alternative assessment - class test
##### Other Assessment
Description Semester When Set Percentage Comment
Written exercise1M10written exercises
Written exercise2M10written exercises
##### Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will 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.