Survival Analysis

The details
Mathematical Sciences
Colchester Campus
Postgraduate: Level 7
Monday 13 January 2020
Friday 20 March 2020
01 October 2019


Requisites for this module



Key module for

DIP G30009 Statistics,
MSC G30012 Statistics,
DIP N32309 Actuarial Science,
MSC N32312 Actuarial Science

Module description

Calculations in clinical trials, pensions and life and health insurance require reliable estimates of transition intensities/survival rates. This Survival Analysis module covers the estimation of these intensities. This module covers the related units of CS2 (Risk Modelling and Survival Analysis, Core Principles), Institute and Faculty of Actuaries CS2 syllabus.

Module aims

The aims of this module are:
1. to critically outline the distinctive characteristics of non-parametric estimation procedures for the lifetime distribution;
2. to critically analyse and derive maximum likelihood estimators for the transition intensities;
3. to examine in detail the Binomial and Poisson models of mortality and compare with the Markov models;
4. to analytically describe the estimation procedure for transition intensities depending on age;
5. to critically analyse and carry out tests for the consistency of crude estimates with a standard table or a set of graduated estimates;
6. to describe in detail the process of graduation and the advantages and disadvantages of the various methods.

Module learning outcomes

On completion of this module, students should be able to:

• explain the concept of survival models (4.1)
• describe estimation procedures for lifetime distributions (4.2)
• estimate transition intensities dependent on age (exact or census) (4.4)
• understand graduation and graduation tests (4.5)
• understand mortality projection (4.6)

Module information


1. Concepts of actuarial modelling [CS2-4.1]:
* Describe why and how models are used, their benefits and limitations.
* Explain the concept of survival models.
* Describe the model of lifetime or failure time from age x as a random variable.
* State the consistency condition between the random variable representing lifetimes from different ages.
* Define the distribution and density functions of the random future lifetime, the survival function, the force of mortality or hazard rate, and derive relationships between them.
* State the Gompertz and Makeham laws of mortality.
* Compute life tables.
* Define the expected value and variance of the complete and curtate future lifetime and derive expressions for them.
* Define the curtate future lifetime from age x and state its probability function.

2. Exact or approximate estimations of transition intensities [CS2-4.4]:
* Describe the principle of correspondence and explain its fundamental importance in the estimation procedure.
* Specify the data needed for the exact calculation of a central exposed to risk (waiting time) depending on age and sex, and calculate a central exposed to risk.
* Explain how to obtain estimates of transition probabilities, including in the single decrement model and the actuarial estimate based on the simple adjustment to the central exposed to risk.
* Explain the assumptions underlying the census approximation of waiting times.
* Explain the concept of rate interval.

3. Explain the concept of survival models [CS2-4.2]:
* Recognise the characteristics of survival data, e.g. censoring and truncation
* Describe the various ways in which lifetime data might be censored
* Describe the Kaplan-Meier (or product limit) estimate of the survival function and the Nelson-Aalen estimate of the cumulative hazard rate; compute it from typical data and estimate its variance
* Determine the proper method to be used in analysing time-to-event data (e.g. parametric, semi-parametric or non-parametric method)
* Describe the Cox proportional hazard model, derive the partial likelihood estimate, and state its asymptotic distribution
* Perform survival analysis using a computer statistical software package, and interpret computer outputs

4. Method of graduation and statistical tests [CS2-4.5]:
* Describe and apply statistical tests of the comparison of crude estimates with a standard mortality table testing for:
a. the overall fit
b. the presence of consistent bias
c. the presence of individual age where the fit is poor
d. the consistency of the 'shape' of the crude estimates and the standard table
* For each test describe:
a. the formulation of the hypothesis
b. the test statistic
c. the distribution of the test statistic using approximations where appropriate
d. the application of the test statistic
e. how tests should be amended to compare crude and graduated sets of estimates
f. how tests should be amended to allow for the presence of duplicate policies
* Describe the reasons for graduating crude estimates of transition intensities or probabilities and state the desirable properties of a set of graduated estimates

5. Mortality projection [CS2-4.6]:
* Describe the approaches to the forecasting of future mortality rates based on extrapolation, explanation and expectation, and their advantages and disadvantages.
* Describe the Lee-Carter, age-period-cohort, and p-spline regression models for forecasting mortality
* Use an appropriate computer package to apply the Lee-Carter, age-period-cohort, and p-spline regression models to a suitable mortality dataset.
* List the main sources of error in mortality forecasts.

Learning and teaching methods

The module consists of 30 lectures, 5 classes, 5 labs, in the spring term. In the summer term 3 revision lectures are given. A project is undertaken in groups.


This module does not appear to have any essential texts. To see non-essential items, please refer to the module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework Group Research Proposal 25%
Coursework Group Assignment 75%
Exam 180 minutes during Summer (Main Period) (Main)

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr Hongsheng Dai (, Jackie Wong (, Dr Yanchun Bao (
Dr Hongsheng Dai (



External examiner

Dr Dimitrina Dimitrova
Cass Business School, City, University of London
Senior Lecturer
Available via Moodle
Of 37 hours, 37 (100%) hours available to students:
0 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

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