## MAS474 Extended Linear Models

 Semester 2, 2019/20 10 Credits Lecturer: Prof Richard Wilkinson uses MOLE Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

The module will further develop the general theory of linear models, building on theory taught in L3. The first extension is to the use of mixed-effects models for situations in which variation arises from several sources: from different life-style choices, for example, in relation to patients' responses to medical treatment, or from variations in field fertility as well as local micro-climate in the growth of crops. The second extension is to the case of linear modelling with partially observed or missing data, and imputation methods for making full use of the available data. The methods will be implemented using R.

Prerequisites: MAS367 (Linear and Generalised Linear Models) or MAS467
No other modules have this module as a prerequisite.

## Outline syllabus

S
emester 2: Extended 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.

## Aims

1. review and extend the students knowledge of the standard linear model, introducing the concept of mixed effects modelling, and methods for missing data;
2. develop enough of the theory to allow a proper understanding of what these methods can achieve;
3. show how these methods are applied to data, and what kinds of conclusions are possible;
4. demonstrate the implementation of the methods using the statistical computing language R.

## Learning outcomes

1. obtain a technical understanding and appreciation of mixed effects modelling methods;
2. be able to identify circumstances in which mixed effects models can be used for data analysis, and understand what conclusions and inferences can be drawn;
3. be able to analyse partially observed data with linear models where appropriate;
4. know how to implement the methods using R, and interpret the output.

## Teaching methods

There will be two formal lectures per week, which will involve the formulation of new theory and worked examples. Lectures will include practical demonstrations of analysis using R. Students will submit coursework based on both theory and R implementation.

20 lectures, no tutorials

## Assessment

There will be a two hour written exam (restricted open book) worth 100% of the module mark, which will test understanding of the theory, the ability to draw conclusions from the results of model fitting.

## Full syllabus

1. Introduction and examples to illustrate the module contents.
2. Motivating example for mixed effects models, illustrating the classical approach for estimating variance components.
3. Fitting a mixed effects model in R.
4. Parameter estimating using restricted maximum likelihood (REML).
5. Predicting random effects using best linear unbiased predictions.
6. Further examples: multilevel models (split plots and nested arrangements).
7. Further examples: repeated measures.
8. Checking model assumptions.
9. Comparing random effects structures with the GLRT.
10. Bootstrapping for comparing fixed effects structures.
11. Mechanisms for missing data (missing at random, missing completely at random etc.). Naive methods (i.e. analysing complete cases only).
12. Exact missing data methods for linear models.
13. EM algorithm (i).
14. EM algorithm (ii).
15. EM algorithm (iii).
16. Single imputation methods.
17. Estimation of single imputation uncertainty.
18. Multiple imputation methods (i).
19. Multiple imputation methods (ii).