Aims

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
This module builds on 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 Linear Modelling 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

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 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. Practical classes are used to help the student’s practical ability to apply the methods to real data.

The teaching methods are appropriate to allow students to develop a wide range of skills, from understanding basic concepts and facts to higher-order thinking.

Assessment Rationale And Relationship

A substantial formal unseen examination is appropriate for the assessment of the material in this module. The format of the examination will enable students to reliably demonstrate their own knowledge, understanding and application of learning outcomes. The assurance of academic integrity forms a necessary part of programme accreditation.

Examination problems may require a synthesis of concepts and strategies from different sections, while they may have more than one ways for solution. The examination time allows the students to test different strategies, work out examples and gather evidence for deciding on an effective strategy, while carefully articulating their ideas and explicitly citing the theory they are using.

The problem solving exercises are expected to consist of two exercises of equal weight: the exact nature of assessment will be explained at the start of the module. The exercises and the group project 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.