Survival Analysis

The details
Mathematical Sciences
Colchester Campus
Undergraduate: Level 5
Monday 13 January 2020
Friday 20 March 2020
01 October 2019


Requisites for this module



Key module for

BSC N233 Actuarial Science (Including Placement Year),
BSC N323 Actuarial Science,
BSC N324 Actuarial Science (Including Year Abroad),
BSC N325 Actuarial Science (Including Foundation Year),
BSC GN13 Finance and Mathematics,
BSC GN15 Finance and Mathematics (Including Placement Year),
BSC GN18 Finance and Mathematics (Including Foundation Year),
BSC GN1H Finance and Mathematics (Including Year Abroad),
BSC 5B43 Statistics (Including Year Abroad),
BSC 9K12 Statistics,
BSC 9K13 Statistics (Including Placement Year),
BSC 9K18 Statistics (Including Foundation Year),
BSC I1G3 Data Science and Analytics,
BSC I1GB Data Science and Analytics (Including Placement Year),
BSC I1GC Data Science and Analytics (Including Year Abroad),
BSC I1GF Data Science and Analytics (Including Foundation Year)

Module description

Calculations in clinical trials, pensions and life and health insurance require reliable estimates of transition intensities/survival rates. This module covers the estimation of these intensities. The syllabus includes the following: Concepts underlying actuarial modelling; distribution and density functions of the random future lifetime, the survival function and the force of hazard.

The module also covers estimation procedures for lifetime distributions including censoring, Kaplan-Meier estimate, Nelson-Aalen estimate and Cox model; statistical models of transfers between multiple states; maximum likelihood estimators for the transition intensities; Binomial and Poisson models of mortality; estimation of age-dependent transition intensities; the graduation process; testing of graduations and measuring the exposed-to-risk.

Module aims


Concepts of actuarial modelling
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 and 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.

Binomial and Poisson models of mortality
Describe an observational plan in respect of a finite number of individuals observed
during a finite period of time, and define the resulting statistics, including the waiting
times, derive the likelihood function for constant transition intensities in a Markov model of transfers between states described. Derive maximum likelihood estimators for the transition intensities and state their asymptotic joint distribution.
Describe the Poisson approximation to the estimator.
Describe the Binomial model of the mortality of a group of identical individuals subject to no other decrements between two given ages.
Derive the maximum likelihood estimator for the rate of mortality in the Binomial
model and its mean and variance and describe the advantages and disadvantages of the multiple state model and the Binomial model, including consistency, efficiency, simplicity of the estimators and their distributions, application to practical observational plans and generality.

Exact or approximate estimations of transition intensities.

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 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.
Describe how to test crude estimates for consistency with a standard table or a set of graduated estimates, and describe the process of graduation by parametric formula, standard table or graphical and state the advantages and disadvantages of each.

Explain the concept of survival models.
Recognize 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 analyzing 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. Interpret computer outputs.

Module learning outcomes

On completion of the course students should be able to (learning outcomes):

* Describe the principles of actuarial modelling
* Describe non-parametric estimation procedures for the lifetime distribution, including censoring, the Kaplan-Meier estimate, Nelson-Aalen estimate and Cox regression model (proportional hazards model);
* Derive maximum likelihood estimators (and hence estimates) for the transition intensities in models of transfers between states with piecewise constant transition intensities;
* Describe the Binomial and Poisson models of mortality, deriving maximum likelihood estimators for the probability/force of mortality and compare with the Markov models;
* Describe how to estimate transition intensities depending on age, exactly or using the census approximation, including calculation of exposed to risk and specification of census formulae based on various age definitions;
* Describe and carry out tests for the consistency of crude estimates with a standard table or a set of graduated estimates;
* Describe the process of graduation and the advantages and disadvantages of the various methods.

Module information

No additional information available.

Learning and teaching methods

The module consist of 18 lectures, 4 classes, 5 labs. 3 hours (2 lectures, 1 class or lab) per week for 9 weeks in the spring term. In the summer term 3 revision lectures are given. A project is undertaken in groups. Coursework consists of problem sheets, a project report and presentation.


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 11/02/2020 25%
Coursework Group Assignment 17/03/2020 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 (, Dr 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 32 hours, 32 (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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