SADC Course in Statistics

A Life Table Discussion Topic

(Session 14)

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Learning Objectives – this session

At the end of this session, you will be able to

• explain some further summaries of survivorship data in Life Tables

• discuss some issues relating to the application of Life Table methodology

• think constructively about approaches to new practical applications of Life Table ideas

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Introduction

“Wikipedia” is a public-domain web resource with articles contributed voluntarily by the readership. In August 2007 search for “Life Table” produced definitions given below & an interesting diagram, reproduced below.

The summary description of the diagram was incomprehensible, so this session focuses on studying the diagram, & figuring out how to reproduce and explain what it shows.

See http://en.wikipedia.org/wiki/Life_table

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New LT summaries

As well as qx, px = 1-qx, lx and dx the article

introduces tpx: the probability that someone

aged exactly x will survive for t more years,

& t|kqx: the probability that someone aged

exactly x will survive for t more years, then die within the next k years.

Why would someone want to know this?

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Insurers’ interests

If a man of working age x takes out an annuity-based pension from an insurer, the insurance deal might be that:-

• he pays premiums for t more years till he reaches retirement age,

• if he survives longer the insurer pays him an annual sum for the rest of his life, or

• if he dies before retirement his premiums are lost

So insurer would want to estimate t|kqx

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The Wikipedia illustration

FOR CLASSROOM DISCUSSION BY YOU, facilitated by your trainer:-

The illustration shown in the web reference is the following diagram of various values of tpx with x-values being the age scale (x-axis) and each curve showing a different minimum survival time t in a different colour.

The data depicted draw on a 2003 Life Table for the total population of the US

http://upload.wikimedia.org/wikipedia/commons/b/be/Excerpt_from_CDC_2003_Table_1.png

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The added lines shown on the diagram read off a value of about 0.87 corresponding to x = 40. What is your interpretation of this point?

Q.1

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The added lines shown on the diagram read off a value of about 0.89 corresponding to x = 50. What is your interpretation of this point?

Q.2

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Q.3

Give your explanation as to why the second value above (0.89) is higher than the first (0.87). Discuss!

Note that the values given above are roughly read off from the graphs shown. They have not been computed.

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The added line shown on the diagram reads off 5 values corresponding to x = 60. What is your interpretation of this set of values?

Q.4

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The added line shown on the diagram reads off 5 values corresponding to tpx = 0.5. What is your interpretation of this set of values?

Q.5

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Q.6

In terms of LT “algebra” can you discuss, agree and write down the formula for:-

tpx: the probability that someone aged exactly x will survive for t more years, &

Q.7In terms of LT “algebra” can you discuss,

agree and write down the formula for:-

t|kqx: the probability that someone aged exactly x will survive for t more years, then die within the next k years?

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Q.8

Why would it be very difficult to produce a diagram like the above? for t|kqx?

Q.9An insurance company wants a set of

illustrations of t|kqx in order to familiarise its

annuity sales staff with the way its annuity policies work. Discuss what graphs you think it might be worth producing, and how you would explain them.

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Some practical work follows …