Unit information: Linear Models in 2015/16

Unit name	Linear Models
Unit code	MATH35110
Credit points	10
Level of study	H/6
Teaching block(s)	Teaching Block 1A (weeks 1 - 6)
Unit director	Professor. Beaumont
Open unit status	Not open
Pre-requisites	MATH11300 Probability 1 and MATH 11400 Statistics 1; MATH 20800 Statistics 2 is desirable but not essential; what is actually needed is a thorough grasp of basic ideas of estimation, hypothesis testing and confidence intervals, from either course.
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including Unit Aims

Unit aims

The aims of this unit are:

1. To provide students with a systematic treatment of statistical linear model theory;
2. To enable students to formulate in various ways (R syntax, matrix notation etc.) a linear model appropriate to particular contexts;
3. To demonstrate the optimality of least squares inference for linear modelling;
4. To develop the tools with which to discriminate between different linear models for given data and to make decisions on whether a final model is adequate or not;
5. To enable students to use R to handle the computational aspects of model fitting and to interpret sensibly the corresponding output;
6. To illustrate connections between least-squares inference and likelihood inference, and to set linear modelling in the wider context of statistical modelling.

General Description of the Unit

This unit explores the role of linear models as a statistical tool for modelling data. Theoretical aspects of such models are explored, and most proofs require familiarity with basic results in linear algebra. The emphasis is on strategies and methodology for model selection, estimation, inference and checking. Models covered include simple and multiple regression, and one- and two-way analysis of variance for factorial experiments. Inference will be based largely on the least-squares criterion, exploiting the Gauss-Markov theorem, but connections will also be made with likelihood-based approaches. The use of R for modelling data via linear models will be integral to the course.

Relation to Other Units

This unit builds on the basic ideas of linear models introduced in Statistics 1 and 2. Generalisations are studied in Generalised Linear Models.Other related units are Bayesian Modelling and Theory of Inference.

Further information is available on the School of Mathematics website: http://www.maths.bris.ac.uk/study/undergrad/

Intended Learning Outcomes

Learning Objectives

By the end of the unit the student should be able to:

• Express a range of linear models in different forms: R syntax, matrix notation, etc.
• Prove fundamental results from linear model theory: calculation of least-squares estimates and their sampling properties, the Gauss Markov theorem, partitioning results for analysis of variance, etc.
• Calculate least-squares estimates in low-dimensional problems.
• Carry out simple analyses of variance.
• Fit models in R, discriminate between models and check goodness-of-model fit.
• Use analysis of variance techniques to check goodness-of-fit in the simple linear model case with replicates.
• Formulate simple factorial experimental results as an analysis of variance.
• Derive standard results for linear models from the likelihood function.

Transferable Skills

Computing skills (use of an advanced package, simple programming, interpretation of computational results in problem context). Relation of numerical results to mathematical and statistical theory. Building models for uncertain phenomena. Data analysis. Self assessment by working through examples sheets and using solutions provided.

Teaching Information

Lectures supported by problem and solution sheets.

Assessment Information

100% Examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

Reading and References

• S.Weisberg, Applied linear regression, Wiley, 1980
• J.J. Faraway, Linear models with R, Chapman & Hall, 2005
• I. Guttman, Linear models :an introduction, Wiley, 1982
• G.A.F. Seber and A.J. Lee, Linear regression analysis (Second Edition), Wiley, 2003
• W. J. Krzanowski, An Introduction to Statistical Modelling, Arnold, 1998.

Additionally, the following may be useful to consult: W. N. Venables and B. D. Ripley, Modern applied statistics with S-Plus, Springer, 1994