MMath Mathematics

The aims of the Generalised Linear Model (GLM) part are to cover an important aspect of modern statistical modelling in an integrated way, and to develop the properties and uses of GLM, focusing on those situations in which the response variable is discrete. The Survival Analysis part aims to introduce some standard techniques in the modelling and analysis of survival data.

The aims of the Generalised Linear Model (GLM) part are to cover an important aspect of modern statistical modelling in an integrated way, and to develop the properties and uses of GLM, focusing on those situations in which the response variable is discrete. The Survival Analysis part aims to introduce some standard techniques in the modelling and analysis of survival data.

 On successful completion of the unit students will be able to:

• write down the fitted model, assess goodness-of-fit, test significance of parameters, compare models and use the chosen model to calculate various quantities of interest;
• write down a GLM with factors/covariates as appropriate, state the associated assumptions and constraints, derive the likelihood equation and algorithms for model fitting;
• define, derive and interpret the survival function, hazard rate and cumulative hazard, estimate them parametrically and non-parametrically, construct confidence intervals and test equality between groups;
• demonstrate that a given distribution belong to the exponential family, work out its mean, variance, variance function, and derive the canonical link.
  fit lifetime distributions and use proportional hazards (PH)  to analyse survival/lifetime data;
• use generalised linear models (GLMs),  including logistic regression and log linear models with a Poisson response, to analyse data with dependence on one or more explanatory  variables;
   

Week 1.Review of a Normal linear regression model. Maximum likelihood estimation of the parameters.. Limitations of the linear model. Some basic distributional results in statistics.

Week 2. The exponential family of distributions: Definition and examples. Derivation of mean and variance. Maximum likelihood estimation; the Fisher scoring algorithm.

Week 3. Generalized linear models: linear predictor, link function, canonical link, properties, the likelihood equation, the iteratively re-weighted least squares algorithm,

Week 4.  Goodness of fit, including deviance and scaled deviance, Pearson's chi-square. Residuals and residual plots. Examples of model fitting in R.

Week 5. Confidence intervals for parameter values, hypothesis tests for model reduction: chi-square or F-tests. Analysis of deviance examples.

Week 6.  Focus to be on students working on their coursework project.

Week 7. Logistic regression. Odds and odds ratio. LD50.

Week 8. Log linear Poisson models with an offset; contingency tables.

Week 9. Survival data; censoring; the survival, hazard and cumulative hazard functions; parametric lifetime distributions and fitting them to data with and without censoring.

Week 10. Kaplan-Meier estimate of the survival function. Nonparametric estimates of hazard and cumulative hazard functions. Confidence intervals.

Week 11. Proportional hazards models and Cox regression: assumptions and interpretation. Model fitting using partial likelihood. Inferential techniques.

Week 12.  Revision.
 

The lecture materials will be made available using videos accessed each week through Blackboard. The weekly review session and tutorial will be on campus.

• Coursework - data-based project involving fitting models, further analysis and inference: 30%
• End of semester examination: weighting 70%

Coursework: Individual feedback given on students’ submitted work.  Generic feedback statement to be provided for all students on Blackboard.

Exam: Generic feedback available after exam period
 

Dunn, Peter K.  Generalized Linear Models with Examples in R. Springer New York 2018 ISBN: 9781441901187

Dobson, Annette J., An introduction to generalized linear models  CRC Press Taylor & Francis Group 2018 ISBN: 9781351726214

McCullagh, P.,  Generalized linear models. Chapman & Hall/CRC 1989 ISBN: 9781351445856

Applied Survival Analysis Using R  Springer International Publishing AG 2016 ISBN: 9783319312439

Hosmer, David W., Applied survival analysis : regression modelling of time-to-event data. Wiley-Interscience 2008 ISBN: 9780470258019