## Generalised Linear Models and Survival Analysis

 Unit code: MATH68052 Credit Rating: 15 Unit level: Level 6 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

None

Students are not permitted to take more than one of MATH38052 or MATH48052 for credit in the same or different undergraduate year.  Students are not permitted to take MATH48052 and MATH68052 for credit in an undergraduate programme and then a postgraduate programme.

#### Aims

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.

#### Overview

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.

#### Learning outcomes

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

• 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;
• fit lifetime distributions and use proportional hazards (PH) and accelerated failure time (AFT) models to analyse survival/lifetime data;
• 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 nonparametrically, construct confidence intervals and test equality between groups;
• prove that a given distribution belong to the exponential family, work out its mean, variance, variance function, and derive the canonical link.

#### Assessment methods

• Other - 20%
• Written exam - 80%

#### Assessment Further Information

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

#### 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, Pearsonâ€'s chi-square test and the likelihood ratio test, model fitting using R. [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]

• 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.

#### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

#### Study hours

• Lectures - 33 hours
• Tutorials - 11 hours
• Independent study hours - 106 hours

#### Teaching staff

Jingsong Yuan - Unit coordinator