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School of Mathematics

# MATH48052 - 2011/2012

General Information
• Title: Generalized Linear Models and Survival Analysis
• Unit code: MATH48052
• Credits: 15
• This course unit cannot be taken as well as MATH38052 which is a level 3 version of the same course unit.
• Prerequisites: MATH20701, MATH20802. MATH20812 and MATH38011 are helpful but are not strictly required.
• Co-requisite units: None
• School responsible: Mathematics
• Members of staff responsible: Dr. J. Yuan and C. Charalanbous
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## Specification

This course unit consists of two parts, one is Generalised Linear Models (10 credits) and the other is Survival Analysis (5 credits). For the specification of Generalised Linear Models, see MATH38052.

### Aims

The Survival Analysis part of the unit aims to familiarise students with the methodology and practical applications of some standard techniques in modelling and analysing survival data.

### Brief Description of the unit

Survival Analysis
In many studies interests lie in the time from a well defined time point of origin to the time of occurrence of an event of interest. The difference between these two time points is referred to as the failure time or survival time or the lifetime. Examples from medical studies include time from cure of an illness to relapse or the time from one stage of a disease to another. Examples in criminology include the time from release from prison to the time of re-offence whereas examples in social studies include the time from wedding to the time of the failure of the marriage. Two typical characteristics of survival data are that the failure times have a skewed distribution so that the statistical methods based on the assumption of Normality are not appropriate for their analysis and that some of the failure times are censored i.e. their exact value is not observed but they are known to be in excess of a value which is then recorded as the censored failure time. Further, for each individual in the study, apart from his/her failure time, there is information on a number of other variables (sometimes called risk factors) e.g. type of treatment received, age, socio-economic status, clinical measurements etc, each of which may affect the individual's survival. Some of the methods of analysing and modelling survival data and their dependence on risk factors fall are studied in this course unit.

### Learning Outcomes

On successful completion of the survival analysis part of the unit students will be able to

• carry out exploratory non-parametric analysis of survival data;
• carry out more sophisticated analyses on survival data;
• explore and analyse survival data using using statistical packages;
• interpret the results of such analyses.

### Future topics requiring this course unit

This course unit is naturally related to another 4th year unit, Longitudinal Data Analysis.

### Syllabus

1. Introduction: background, review of linear models in matrix notation, model assessment, some pre-required knowledge. [2]
2. The exponential family of distributions: Definition and examples. Mean and variance, variance function and scale parameter. [2]
3. Generalized linear models (GLM): linear predictor, link function, canonical link, maximum likelihood estimation, iterative reweighted least squares and Fisher scoring algorithms, significance of parameter estimates, deviance, Pearson and deviance residuals, Pearson’s chi-square test and the likelihood ratio test, model fitting using R or S-Plus. [7]
4. Normal linear regression models: least squares, analysis of variance, orthogonality of parameters, factors, interactions between factors. [2]
5. Binary and Binomial data analysis: distribution and models, logistic regression models, odds ratio, one- and two-way logistic regression analysis. [5]
6. Poisson count data analysis: Poisson regression models with offset, two-dimensional contingency tables, log-linear models. [4]
7. Survival data. Censoring. The survivor, hazard, cumulative hazard functions.  Kaplan-Meier estimate of survivor function. [3]
8. Fitting exponential and Weibull distributions to survival data. Hazard plots and log cumulative hazard plots. [3]
9. Proportional hazards (ph) and Cox regression: assumptions and interpretation.. Model fitting and diagnostics. Hazard ratios and confidence intervals. [5]

### Textbooks

• Dobson, A. J., An Introduction to Generalized Linear Models, Chapman & Hall 2002.
• Krzanowski, W., An Introduction to Statistical Modelling, Edward Arnold 1998.
• McCullagh, P. and Nelder, J. A., Generalized Linear Models, Chapman & Hall 1990.
• Collett, D., Modelling Survival Data in Medical Research, 2nd edition, Chapman and Hall 2004.
• Klein, J. P. and Moeschberger, M. L., Survival Analysis, 2nd edition, Springer 2003.

### Teaching and learning methods

Three lectures and one examples class each week. In addition students should expect to spend at least six hours each week on private study for this course unit.

### Assessment

Coursework: 20%
End of semester examination: Three hours weighting 80%

## Arrangements

Online course materials for this course unit.