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| 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 |
MATH20800
|
| Co-requisites |
None
|
| School/department |
School of Mathematics |
| Faculty |
Faculty of Science |
Description including Unit Aims
This unit explores the role of linear models as a statistical tool for modelling data. Theoretical aspects of such models are explored but the emphasis is on strategies and methodology for modal 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-square criterion, exploiting the Gauss-Markov theorem, but connections will also be made with likelihood-based approach. The use of R for modelling data via linear models will be integral to the course.
Aims
- To provide students with a systematic treatment of statistical linear model theory;
- To enable students to formulate in various ways (R syntax, matrix notation etc.) a linear model appropriate to particular contexts;
- To demonstrate the optimality of least squares inference for linear modelling;
- 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;
- To enable students to use R to handle the computational aspects of model fitting and to interpret sensibly the corresponding output;
- To illustrate connections between least-squares inference and likelihood inference, and to set linear modelling in the wider context of statistical modelling.
Syllabus
- Model formulation: applications.
- Model expression: analytical, algebraic and model syntax. Model parameterization.
- Least-squares estimation: inference and properties.
- The Gauss-Markov Theorem.
- Model-fitting in R; Simple diagnostics.
- Analysis of variance - strategies for model selection and checking.
- Analysis of variance for factorial experiments: one-way, two way and interactions.
- Use of R for performing analysis of variance.
- Likelihood-based inference for linear models.
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.
Intended Learning Outcomes
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
The assessment mark for Linear Models is calculated from a 1½-hour written examination in April consisting of THREE questions. A candidate's best TWO answers will be used for assessment. Calculators of an approved type (non-programmable, no text facility) are allowed.
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.
