SADC Course in Statistics

Basic Life Table Computations - II

(Session 13)

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Learning Objectives

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

• construct a Life Table or abridged Life Table from a given set of mortality data

• express in words and in symbolic form the connections between the standard columns of the LT

• interpret the LT entries and begin to utilise LT thinking in more complex demographic calculations

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Why compute nLx?

The concept behind nLx is of some interest in

its own right, but the main reason for its calculation as part of the Life Table is to contribute to the two remaining key columns found in most LT calculations, which look at the accumulation of years lived.

Note that to explain these we start at the end of the South African Male LT and work backwards from the highest age!

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Two more Life Table columns: 1

Ages lx nLx Tx ex

75-79 15704 61839 109634 7.0

80-84 9032 32315 47795 5.3

85-89 3894 12376 15480 4.0

90-94 1057 2686 3103 2.9

95-99 191 375 418 2.2

100+ 25 43 43 1.7

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Computing Tx

A relatively complicated calculation, which we shall see makes very little difference in the end, judges that the 25 (0.025%) who survive to 100 will thereafter live a total of 43 person-years. Accept this for now.

A standard calculation of nLx i.e. 5L95 - as

explained above - says we can expect the 191 males who survive to age 95 will live for a total of 375 years between them between ages 95 and 99 inclusive.

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Computing Tx

Thus the Total amount of living done by the LT population from age 95 onwards is (43 + 375) = 418 person-years.

This is T95.

In the same way, 5L90 = 2686 person-years are

expected to be lived between ages 90 and 94 inclusive, and the total from age 90 onwards is (43 + 375 + 2686) = 3103 = T90.

We continue totalling backwards in this way …

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Two more Life Table columns: 2

Ages lx nLx Tx ex

<1 100000 96175 4988823 49.9

1-4 95322 373818 4892649 51.8

5-9 93677 461637 4518830 48.7

10-14 92929 458216 4057193 44.1

15-19 92416 453845 3598977 39.4

20-24 91305 443736 3145132 34.9

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Calculating e0

When we finally get back to age 0, we find that T0 = 4,988,823. This is the total

number of years that we expect the LT population of 100,000 baby boys to live.

Averaged out, that is about 49.9 years each.

The South African Male life expectancy at birth is about 49.9 years, on these figures.

You can see the approximation in years lived after age 100 makes no difference to this answer!

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Observing ex

The age 1 figure e1 is 51.8. This is

described as the residual expectation of life, the further years expected to be lived by a survivor who reaches exact age 1.

At first sight it seems odd that after having lived a year, he can now expect to live longer than he could at birth.

Further values in the ex column go

steadily downwards, but observe that after each [4 or] 5 year period the ex

figure reduces by less than [4 or] 5.

Age ex

<1 49.9

1-4 51.8

5-9 48.7

10.. 44.1

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Explaining ex : 1

The life expectancy at birth, e0, is in fact a

weighted average over those babies who die before age 1, and all the others who survive beyond exact age 1.

e0 = 5465 x 0.3* + 94535 x 52.8** = 49.9

0.3* ~ as on slide 12, “this counts 0.3 of a year for each child that dies aged 0”

52.8** ~ 51.8 years future expected life, plus 1 year already lived.

Weights are no. dying and no. surviving.

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Explaining ex : 2

Same effect applies to all changes ex ex+n

The life expectancy at birth, e0, is 49.9 as

above ~ an overall average

BUT note that on reaching age 50, a member of this population still has future life expectancy of 18.6 years. Can you now explain this, verbally and arithmetically in the same way as on the previous slide?

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Life Expectancy at age x

0

10

20

30

40

50

60

0 50 100 150

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Life Expectancy at age x

0

10

20

30

40

50

60

0 50 100 150

Note that if future life

expectancy fell by 1 year for

every year lived, the curve would be replaced by

the diagonal line shown here!

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Practical work follows to ensure learning objectives

are achieved…