## MATH2775 Survival Analysis

### 10 creditsClass Size: 110

Module manager: Tim Heaton
Email: T.Heaton@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2715 or MATH2735 or equivalent

This module is approved as a discovery module

### Module summary

Survival analysis is concerned with data on how long it takes for a certain event to occur, such as: death of a patient after onset of a disease; breakdown of a home appliance after purchase; making a claim after taking out a car insurance policy; finding a job after graduation from University, etc. The appropriate statistical methods to analyse and model survival data are of critical importance in medical studies and actuarial work, as well as in other settings.

### Objectives

The aim of this course is to study statistical methods for the analysis of time-to-event data, with applications to medical, actuarial and industrial practices.

Learning outcomes
On completion of this module, students should be able to:
- describe the characteristic features of survival data including censoring;
- identify and discuss different types of censoring mechanisms;
- understand and explain different functions used to describe a distribution of failure times;
- define a number of probability distributions used for survival data and derive relevant properties of these distributions;
- use nonparametric methods to estimate a survival curve and to test for differences between survival curves;
- fit an appropriate distributional model to a data set of failure times;
- use a range of parametric and semi-parametric regression models to assess the effects of covariates on a survival distribution.

### Syllabus

1. The nature of survival data; censoring mechanisms.
2. Survival (failure) times as random variables; functions used to describe the survival distributions. Actuarial notation.
3. Nonparametic estimation of the survival function and hazard functions; the Kaplan-Meier (product-limit) and the Nelson-Aalen estimators, and their life-table analogues.
4. Comparison of survival curves; the log-rank tests.
5. Estimation of the variance (Greenwood's formula); the delta method.
6. Parametric survival models; exponential, Weibull, gamma, Gompertz, Makeham, log-logistic; different time dependence of the hazard.
7. Maximum likelihood estimation in the presence of censoring; exposed to risk.
8. Parametric hazard regression models; graphical methods; selection of significant covariates.
9. The Cox proportional hazards model; the partial likelihood.
10. The accelerated failure time (AFT) model.
11. Actuarial applications: complete and curtate future lifetimes; crude estimates; the principle of correspondence; graduation (smoothing) and statistical tests for comparison with a standard life table.

### Teaching methods

 Delivery type Number Length hours Student hours Lecture 22 1.00 22.00 Tutorial 10 1.00 10.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Private study

12 hours working on homework sheets (3 sheets, 4 hours each)
22 hours going over lecture notes and wider reading
10 hours preparation for tutorials (1 hour per tutorial)
22 hours revision and exam preparation
2 hours examination

### Opportunities for Formative Feedback

Regular homework sheets

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Problem Sheet 3 Homework Sheets 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 00 mins 85.00 Total percentage (Assessment Exams) 85.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated

The reading list is available from the Library website

Last updated: 13/09/2022

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