F79SU/F71AE Survival Models
Lecturer: I. D. Currie
Aims
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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
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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 multistate 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 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.
Learning outcomes
At the end of studying this course, students should be able to:
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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.
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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.
Reading
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.
Assessment
The courses F79SU and F71AE are examined in a single two-hour written examination paper
together with continuous assessment of practical assignments. F79SU is synoptic with F79SP
(Stochastic Processes).
Help
If you have any problems or questions regarding the course, you are encouraged to contact the
lecturer.
Course web page
Further information, course materials, past examination papers and illustrative R programs are
available at http://www.ma.hw.ac.uk/~iain/
Detailed syllabus
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Introduction, Notation and Revision:
o life time distributions, survival functions, rates and forces of mortality
Estimating the Lifetime Distribution:
o cohort studies
o censoring
o Kaplan-Meier estimate of the survivor function
o Cox regression model, partial likelihood, estimation
Markov Models: Theory:
o computation of tpx
o multi-state Markov models
o Kolmogorov forward equations
Markov models: Data and Estimation:
o 2-state model
o maximum likelihood estimate (MLE) of the force of mortality
o score function and the maximum likelihood theorem
o properties of the MLE of the force of mortality
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likelihood and estimation in the multi-state model
Binomial and Poisson Models of Mortality
o binomial model
o two assumptions: uniform distribution of deaths, constant force of mortality
o likelihood and estimation for the binomial model
o actuarial estimate of qx
o Poisson model
Graduation and Statistical tests:
o graduation process
o testing adherence to data
o χ2 test, standardised deviations test, sign test, change of sign test, grouping of
signs test, serial correlation test
Exposed to Risk
o Calculation of exact exposed to risk.
o Calculation of approximate exposed to risk using census data.
The course F79SU is synoptic with F79SP Stochastic Processes.