Pre-requisite: This two-semester-long double module is only available to students on the Mathematics and Statistics & Operational Research MSci pathway.
A substantial investigation of a statistical/operational research problem incorporating literature survey,, use of relevant statistical packages and when necessary the construction of computer programs to solve specific stages of the problem, presentation of the work in the form of a technical report, a sequence of oral presentations culminating in a 30-minute presentation which is assessed. Each student will work under individual supervision of a member of staff.
The mathematical contents of the project will depend on the nature of the research problem.
80% Dissertation 20% Oral Presentation.
Survival analysis is an important tool for research in medicine and epidemiology. It is that part of statistics that deals with time-to-event data. For example, in a clinical study the data might consist of the post-treatment survival times of patients with hypernephroma (i.e., a malignant tumour of the kidney). Survival analysis might address questions such as:
How does the patient's survival time depend of her age at treatment?
What is the affect of kidney removal on the survival times of patients compared with others who are treated just with chemotherapy?
Is the size of the tumour an equally good predictor of survival for patients under 60 years of age as for the over 60s?
The module introduces the student to the special features of survival data such as censoring (e.g. where a patient is lost to follow up but is known to have survived to a particular time) and positive skew in the distribution of survival times. Fundamental concepts of survival analysis will be introduced including the survivor function, the hazard function and the hazard ratio. The course will build from some elementary nonparametric techniques such as the Kaplan-Meier estimate of the survival curve to the Cox proportional hazards model - one of the most flexible and widely used tools for the analysis of survival data.
An important element of this module will be two hours a week of survival data analysis classes using statistical software packages, in particular SAS and R. These will be used to demonstrate the theory taught in the lectures. Previous experience of SAS is required.
Introduction to survival data: Features of survival data, distribution of survival times, survivor function, hazard function, cumulative hazard function.
Some nonparametric procedures: Estimating the survivor function - life-table, Kaplan-Meier, Nelson-Aalen, confidence intervals. Estimating the hazard function, estimating median and percentile survival and confidence intervals. Comparing two groups of survival data, the log-rank and Wilcoxon tests. Comparison of k-groups of survival data.
The Cox proportional hazards model: The Cox proportional hazard model (Cox model), baseline hazard function, hazard ratio, including variates and factors, maximum likelihood for the Cox model. Treatment of ties in the Cox model. Confidence intervals for the Cox model regression parameters and hypothesis testing. Estimating the baseline hazard. Model building, Wald tests, likelihood ratio tests and nested models.
Evaluating the proportional hazards assumptions.
The stratified Cox procedure.
Extending Cox proportional hazards models for time dependent variables.
Recurrent events survival analysis.
Competing risks survival analysis.
Design issues for randomised trials.
Parametric models for survival data, time-dependent variables and non-proportional hazards, accelerated-failure-time models. Fitting parametric distributions.
Exam 75% Coursework 15% Presentation 10%
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