## MAS6003 Linear Modelling

 Both semesters, 2017/18 20 Credits Lecturer: Prof Richard Wilkinson uses MOLE Timetable Reading List Aims Outcomes Teaching Methods Assessment

The unit develops students' understanding of the general theory of linear models for regression modelling and analysing experiments, and introduces extensions to these models. Many important applications are considered, including the modelling of binary and count data, and the analysis of contingency tables and structured data. Discussion in the unit covers regression model building and model checking, multiple regression, generalised linear models, and the analysis of complete factorial experiments. It then considers mixed effects models, which are useful when the data are structured, with different levels of variation. Finally, data structures with missing parts (known as missing data) are considered in detail and relevant methods are studied.

There are no prerequisites for this module.
No other modules have this module as a prerequisite.

## Outline syllabus

• Semester 1: Linear and Generalised Linear Models
• Linear Regression, LS estimators and fitted model
Brief introductory examples on regression and the analysis of variance.
• Hypothesis testing
General hypothesis testing, confidence intervals and tests of individual parameters.
• Deviations from assumptions - transformations
Model irregularities, variance stabilizing transformations, Box-Cox transformations.
• Variable selection
Variable selection methods, F-tests, penalized likelihood (AIC/BIC). Automated methods, subsets, stepwise, forward selection, sparse linear regression, LASSO, big data.
• Introduction to generalise linear models (GLMs)
Motivating GLMs, assumptions relating to GLMs. Fitting GLMs, common GLM distributions.
• Estimation and model building
Parameter estimation, use of deviance in GLMs to test model fit. Model building (analysis of deviance), types of residuals, quasi likelihood.
• Binary responses
Binary response: likelihood, links, examples, odds, odds ratios and logistic regression.
• Count data
Poisson regression for count data, using offsets to adjust for exposure.
• Contignecy tables
Two-way contingency tables, response & controlled variables, association and homogeneity, probability distributions for two-way tables. Using log-linear models when analysing two-way tables, MLEs, examples.
• Semester 2: Extended Linear Models
• Review of Linear Models
• Mixed Effects Models
Mixed Effects models and REML estimation. Motivating example for mixed effects models, illustrating the classical approach for estimating variance components. Fitting a mixed effects model in R. Further examples: multilevel models (split plots and nested arrangements).
• Repeated measures and the bootstrap
Further examples: repeated measures. Checking model assumptions. Comparing random effects structures with the GLRT. Bootstrapping for comparing fixed effects structures.
• Missing data
Mechanisms for missing data (missing at random, missing completely at random etc.). Naive methods (i.e. analysing complete cases only). Exact missing data methods for linear models.
• The EM algorithm
Introduction, structure and implementation of the EM algorithm for missing data.
• Imputation method
Single imputation methods. Estimation of single imputation uncertainty. Multiple imputation methods.

## Office hours

Monday 10-11 Friday 10-11

## Aims

• To review and extend the student's knowledge of the standard linear model.
• To introduce the more general ideas of Mixed Effects Models and Generalized Linear Models (GLM) by building on the familiar concepts of the linear model.
• To develop enough of the theory to allow a proper understanding of what these methods can achieve.
• To show how these methods are applied to data, and what kinds of conclusion are possible.

## Learning outcomes

• understand the basic concepts.
• carry out straightforward regression analysis.
• derive minor extensions and applications of the general theory.
• carry out logistic regression and log-linear analysis of contingency tables.
• assess the fit of a model to data, and make suggestions as to how to improve it if it is unsatisfactory.
• understand basic techniques of mixed effects modelling.

## Teaching methods

Lectures, with a complete set of printed notes, plus task and exercise sheets and 4 computer classes in semester 2.

36 lectures, no tutorials, 4 practicals

## Assessment

One project in semester 2 (15% overall), and a three hour restricted open book examination (85%). Exam format: 5 questions from 6.

Type Author(s) Title Library Blackwells Amazon
A Dobson, A.J. An Introduction to Generalized Linear Models Blackwells Amazon
B Christensen, R. Log-linear models and Logistic Regression Blackwells Amazon
C Atkinson, A.C. Plots, Transformations and Regression Blackwells Amazon
C Cook, R.D. \& Weisberg, S. Residuals and Influence in Regression Blackwells Amazon
C Draper, N. and Smith, H. Applied Regression Analysis Blackwells Amazon
C McCullagh, P J and Nelder, J A Generalised Linear Models Blackwells Amazon
C Montgomery, D.C. and Peck, E.A. Introduction to Linear Regression Analysis Blackwells Amazon
C Pinheiro, J.C. and Bates, D.M. Mixed-Effects Models in S and S-Plus Blackwells Amazon
C Seber, G.A.F. Linear Regression Analysis Blackwells Amazon

(A = essential, B = recommended, C = background.)

Most books on reading lists should also be available from the Blackwells shop at Jessop West.

## Timetable (semester 2)

 Mon 09:00 - 09:50 lecture Arts Tower Lecture Theatre 3 Fri 12:00 - 12:50 lecture Diamond Building LT9