MMath Mathematics and Statistics

MATH48311 Requisites: A) Pre-Requisites: MATH27720 and MATH27711; (B) Anti-Requisites - Students who took MATH38171 are not permitted to take MATH48311

The unit aims to:

Review and extend student’s knowledge of Classical Linear Models (CLM).

Introduce generalisations of these models, such as Generalised Linear Models (GLM), penalised regression models, building on the familiar notion of linear models.

Develop the theory of regression models to support a deeper understanding of the model applicability, inferential techniques, methods for model selection, validation and limitations.

Exemplify a range of model applications and possible conclusions and insights into real world problems, all implemented using the statistical software R.
 

• Identify and formulate appropriate linear and generalised linear models and statistical hypotheses to investigate real-world questions involving different response variables, using appropriate methodologies
• Explain and implement penalisation techniques in different contexts covered (e.g. collinearity, variable selection, and spline regression smoothness)
• Investigate using appropriate diagnostic techniques whether the assumptions underpinning models and analyses are justified
• Explain and interpret the resulting models and conclusions, with reference to the original real-world questions where appropriate, and explain key underpinning ideas, assumptions, procedures, and theoretical results
• Prove underpinning mathematical and theoretical results for CLMs and GLMs
• Use R to implement the methods, and evaluate their performance
   

Syllabus:

Part A – Regression Models

•    Review of Classical linear models (CLM) and inference, diagnostics, prediction and cross-validation

•    Model selection (AIC, Cp, stepwise variable selection)

•    Penalised regression models:
-   Ridge regression in context of CLM
-   LASSO in context of CLM
-   splines, penalised regression splines

Part B – Advanced Regression Models

•    Generalised Linear Models (GLMs)
      -   defining GLMs, canonical links, MLE, Deviance
      -   examples of models, including Poisson, Binomial, Gamma
      -   some diagnostics for fitted models 
      -   contingency tables and Log-linear models, higher (than two) dimensional tables, conditional independence
      -   overdispersion (for both discrete and continuous distributions)
      -   mixed effects models

•    Introduction to Generalised Additive Models (GAM)
 

Teaching is composed of two hours of lectures per week and one tutorial/computer class per week. One week is reserved for assessment related activities. Teaching materials will be made available online for reference and review.

Written exam - 3 hours; Weighting 70%

Generic feedback will be provided after marks are released.

Data analysis project (including in class and take-home elements) - 9 hours; Weighting 30%

Generic feedback will be provided after marks are released. 

Primary
•    A.J. Dobson, An Introduction to Generalized Linear Models
•    P.K. Dunn and G.K. Smyth, Generalized Linear Models with Examples in R
•    J.J. Faraway, Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models
•    S. Wood, Generalized Additive Models an Introduction with R
•    A.C. Atkinson, Plots, Transformations and Regression

Additional

•    R. Christensen, Log-linear models and Logistic Regression
•    P.J. McCullagh, and J.A, Nelder, Generalised Linear Models 
•    G. James, D. Witten, T. Hastie, R. Tibshirani, Introduction to statistical Learning with R 
•    D.C. Montgomery and E.A. Peck, Introduction to Linear Regression Analysis
•    J.C. Pinheiro and D.M. Bates, Mixed-Effects Models in S and S-Plus

For Information and advice on Link2Lists reading list software, see: 
http://www.library.manchester.ac.uk/academicsupport/informationandadviceonlink2listsreadinglistsoftware/