F79SU Survival Models

Peter RidgesDr Alastair WallisMarjan Qazvini

Course co-ordinator(s): Peter Ridges (Edinburgh), Dr Alastair Wallis (Malaysia), Marjan Qazvini (Malaysia).

Aims:

  • To understand the use of mathematical models of mortality, illness and other life history events in the study of processes of actuarial interest.
  • To be able to estimate the parameters in these models, mainly by maximum likelihood.
  • To apply methods of smoothing observed rates of mortality and to test the goodness-of-fit of the models.

Summary:

  • Estimation for lifetime distributions: Kaplan-Meier estimate of the survival function, estimation for the Cox model for proportional hazards.
  • Statistical models for transfers between multiple states (e.g., alive, ill, dead), the multi-state Markov model, relationship between probabilities of transfer and transition intensities, estimation for the parameters in these models. The binomial and Poisson models of mortality.
  • Methods of projecting future mortality rates to allow for improving longevity.
  • Methods of graduation: parametric and standard table. Tests of consistency of crude estimates of rates of mortality and their graduated values.
  • Computing facilities, especially R, will be used extensively and this work will be assessed by practical assignments.

Detailed Information

Course Description: Link to Official Course Descriptor.

Pre-requisite course(s): F78AB Actuarial and Financial Mathematics B & F78PB Probability and Statistics B .

Location: Edinburgh, Malaysia.

Semester: 2.

Syllabus:

  • Introduction, Notation and Revision:
    • life time distributions, survival functions, rates and forces of mortality
  • Estimating the Lifetime Distribution:
    • cohort studies
    • censoring
    • Kaplan-Meier estimate of the survivor function
    • Cox regression model, partial likelihood, estimation
  • Markov Models: Theory:
    • computation of tpx
    • multi-state Markov models
    • Kolmogorov forward equations
  • Markov models: Data and Estimation:
    • 2-state model
    • maximum likelihood estimate (MLE) of the force of mortality
    • score function and the maximum likelihood theorem
    • properties of the MLE of the force of mortality
    • likelihood and estimation in the multi-state model
  • Binomial and Poisson Models of Mortality
    • binomial model
    • two assumptions: uniform distribution of deaths, constant force of mortality
    • likelihood and estimation for the binomial model
    • actuarial estimate of qx
    • Poisson model
  • Graduation and Statistical tests:
    • graduation process
    • testing adherence to data
    • χ2 test, standardised deviations test, sign test, change of sign test, grouping of signs test, serial correlation test
  • Exposed to Risk
    • Calculation of exact exposed to risk.
    • Calculation of approximate exposed to risk using census data.
  • Mortality Projection
    • Approaches to projecting mortality
    • The Lee-Carter model
    • The Cairns-Blake-Dowd model
    • The P-spline model
    • Age-period-cohort models
    • Sources of forecast error
  • The course F79SU is synoptic with F79SP Stochastic Processes.

Learning Outcomes: Subject Mastery

At the end of studying this course, students should be able to:

  • Estimate a survival function using the Kaplan-Meier method.
  • Find the partial likelihood function in the Cox model.
  • Use the partial likelihood to estimate the parameters (with standard errors) in the Cox model.
  • Write down an appropriate Markov multi-state model for a system with multiple transfers.
  • Obtain the Kolmogorov forward equations in a Markov multi-state model.
  • Derive the likelihood function in a Markov multi-state model with data.
  • Use the likelihood function to estimate the parameters (with standard errors) in a Markov multi-state model with data.
  • Obtain the likelihood function in the 2-state model with states Alive and Dead under the binomial or Poisson models.
  • Use any of two assumptions (uniform distribution of deaths, constant force of mortality) to reduce the binomial likelihood to a function of a single parameter, and estimate the parameter.
  • Obtain graduations of mortality data using generalized linear models in R, and interpret the results.
  • To apply the χ2 test, the standardised deviations test, the sign test, the change of sign test, the grouping of signs test, the serial correlation test to testing the adherence of a graduation to data.
  • Calculate exposed to risk using both exact and census methods.
  • Fit stochastic models (Lee-Carter, Cairns-Blake-Dowd, P-spline) to mortality data
    and make projections of future mortality.

Learning Outcomes: Personal Abilities

At the end of the course, students should be able to:

  • Demonstrate the ability to learn independently
  • Manage time, work to deadlines and prioritise workloads
  • Present results in a way which demonstrates that they have understood the technical and broader issues of modelling mortality and morbidity data
  • Communicate findings effectively in the actuarial and financial services industry

Reading list:

The course book is: I D Currie, Survival Models. The book is essential reading and is available from the department. It contains outline copies of the lecture material and all tutorial material. Copies of past examination papers and illustrative R code will be available through the course web site.

Supplementary reading is provided in Macdonald, A.S.,
Richards, S.J &  Currie, I.D. (2018). Modelling Mortality with Actuarial 
Applications. Cambridge University Press.

Assessment Methods:

2 hour exam (80%-90%), project work (10%-20%).

SCQF Level: 9.

Credits: 15.

Other Information

Help: If you have any problems or questions regarding the course, you are encouraged to contact the lecturer

VISION: further information and course materials are available on VISION