MTH6157 - Survival Models - 2023/24
Topic outline
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The particular focus of this module is on mathematical models for Survival and Mortality which are widely used in Life Assurance, demographics and medical statistics. We will consider the mathematical foundations, a number of modelling approaches and a range of model applications. The syllabus is designed to meet the requirements of the CS2 examination of the Institute & Faculty of Actuaries, "Risk Modelling and Survival Analysis".
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Welcome to Survival Models. In this first week we will introduce the module and its place in both the study of actuarial science and a wide range of practical applications. We will look at the general shape of human survival probability distributions and consider heterogeneity in mortality. This will give us an excellent foundation for the module.
There will be lectures on Monday (MB203) and Thursday (Eng2016) at 11.00am
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module introduction
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Heterogeneity and the actuarial concept of Selection
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short article here, full ONS report below
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link to full Office for National Statistics report here, BBC News reporting above
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article here, full Swiss Re report below
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link to full Swiss Re Institute report here, article from The Actuary above
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Why linear regression modelling does not work for survival models, example using National Life Tables data
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This week we will cover Survival Model Concepts - the key elements of actuarial mathematics that underpin the models we will examine later in the module. Much of the material from this week will build upon Actuarial Mathematics II (MTH5125) but we will also make sure that anyone who hasn't studied actuarial mathematics before is fully caught up. Once again there will be 4 hours of timetabled activities on Monday and Thursday at 11.00.
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survival model concepts - you will also need to access the "demonstration" pdf's below to complete the notes
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using R commands and functions for week 2 survival model concepts including exponential and Gompertz models
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Having developed general survival model concepts last week, we are now ready to move on to the first of our Survival Models - the Kaplan Meier estimate. This is an example of a 'non-parametric' estimate of survival which is used in medical trials amongst other applications. Once again lectures will be Monday and Thursday morning. We will also include a session on how to implement the model in R.
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non-parametric approaches, censoring, Kaplan-Meier Estimate
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Kaplan Meier in R
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This is the question we went through live in the Thursday lecture - full R code and plot given here
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In this section you will find a PDF document with the coursework question and instructions [to be released after the week 3 lectures on Thursday 12th October], a CSV file with the dataset for the question and a submission point to upload your solution in a MS Word document by 5pm UK time on Wednesday 18th October (week 4). This assessment counts towards 15% of your module mark and is based on the material in week 3 using R programming.
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Please use this submission point to upload your solution to the first assessed coursework. You should upload one MS Word document that includes your R code, R output and typed answers to the question.
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You will need to download this csv file and load it into R for the coursework
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We now move from non-parametric to parametric models of survival and the use of likelihood estimates. First we will look at proportional hazard models which can be used to model mortality and survival in relation to a number of factors such as age, health, socio-economic conditions.
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parametric modelling, covariate data, proportional hazards, Cox's PH model
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maximum likelihood estimation in the exponential model
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Cox's Proportional Hazard interpretation and calculations
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CoxPHdataset.csv
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questions on covariate data and Cox's PH model
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We now move to a different framework for parametric modelling - the Markov Process, which will lead us into Multi-State models. Material in this section will link to your studies in Random Processes. These Models can be used for quite complex scenarios such as the social security system of a country.
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Markov processes, 2-state and multi-state models
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this is a challenging question on multi state models and their use in social security
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We will complete the set of survival models this week moving on from Multi State models to look at approximations to these that are derived from assuming well known Statistical distributions - the Binomial and Poisson. We will then look at ways to compare the different models we have considered over the last four weeks in this module.
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Binomial and Poisson approximations to the Multi State model and model comparisons
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for more details on this question and alternative solutions see the recording of the lecture on Monday 30 Oct
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Having looked at different survival models in weeks 3,4,5,6, we now move in the next 3 weeks to consider how these models are used in practice and are tested in a life assurance context in particular. We begin with Exposed to Risk. In previous years, students have found exam questions on this topic to be the most troublesome ones - so I would recommend taking time to go through the material carefully this week.
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exposed to risk
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Past paper exposed to risk questions from this module and IFoA exam papers
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will be used in Thursday lecture to practice calculating exposed to risk by census and then estimating the transition intensity by age
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In this section you will find a PDF document with the coursework question and instructions , 3 CSV files with the data needed for the question and a submission point to upload your solution in a MS Word document by 5pm UK time on Friday 24 November (week 9). This assessment counts towards 15% of your module mark.
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Please use this submission point to upload your answer for the second assessed coursework. You should upload one MS Word document that includes your R code, R output and typed answers by 5pm on Friday 24 November.
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We now come to how actuaries use survival model outputs for practical work in life assurance. The key task here is called 'Graduation' which takes force of mortality or rate of mortality from the models we considered in weeks 2-6 and uses them to construct the mortality tables like those used for Life Assurance premium and reserve calculations in Actuarial Maths II. We will spend two weeks on this area. In week 10 we will look at techniques for doing a Graduation but before that we consider some statistical tests of the model output.
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graduation and statistical tests
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example problem question on the Chi-Squared test in R
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This question was actually set as assessed coursework in 2022
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you will need this CSV file for the Additional R question above
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We continue with Graduation - moving from statistical tests of how well a graduated table or standard table fits observed mortality on to questions of how actuaries actually do the graduation.
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weeks 9 and 10
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The question above should be answered as an exam-style question with calculator rather than Excel or R but the data is here in excel in case helpful
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Our final topic in this module is Mortality Projections - models of how survival and mortality rates might change in future years. We will give an overview of different methods used. The implementation of these projections links to Time Series which many of you will study next semester.
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interpretation of the Lee Carter parameters
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The length of people’s lives is of crucial importance in the Insurance and Pensions industry so models for survival must be studied by trainee Actuaries. This module considers a number of approaches to modelling data for survival and mortality. These include parametric and non-parametric statistical approaches and methods developed by actuaries using age-specific death rates. We will also look at tests of the consistency of crude (not exact but useful) estimates with a standard table using a number of non-parametric methods.
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Having completed the course content, the last week will be given over to revision lectures and some past paper and exam-style questions.
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This is the 2020 LSR paper from which we will use Q1,2,3,4 in the Monday revision session (Q5 is on GLM which has moved from this module to Stats Modelling 2 / Actuarial Statistics since 2020)
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Note this was a 2 hour exam on campus
Q5 relates to a topic no longer on the syllabus for this module - Generalised Linear Modelling
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