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Estimating mortality from
defective data
Rob Dorrington
Director of the Centre for Actuarial Research
Overview
National vs sub-populations (e.g. life
offices, group schemes)
 Childhood
 Adulthood

Population survival and direct census question
 Orphanhood, widowhood and sibling methods
 Using vital registration records

Model life tables
 Extrapolation to older ages

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National vs sub-population
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In order to measure the mortality of a sub-population one
needs to gather data specific to that population
Methods:
 For life assurance and group schemes - survey
companies with these records (e.g. CMI in UK or CSI in
SA)
 For socio-economic groups (sometimes other surveys
can be used to get a handle on this)
 Problem with some of the methods applied to national
is that they make assumptions (e.g. closed population)
which are not applicable to sub-populations
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Industry data
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Lack of priority in contributing companies (inability, lack of
importance, etc)
Quality of company records often poor
Data often not available on one database
In SA:
 Data good enough to produce standard tables for lives
assured but not to measure:
 Impact of smoking
 Impact of HIV
 Produce sensible select rates
 Problem of changing mix of business (market, products,
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Industry data

In SA:
 Annuitant mortality investigation for first time
 Have been trying to investigate mortality of group
schemes for some time without success (lack of
industry enthusiasm)
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Childhood mortality
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This method (originally proposed by Brass) derives
survival rates of children by asking mothers in different
age groups (or duration of marriage, etc) about how many
children they have ever given birth to, and how many of
these are still alive.
By making use of fertility schedules one can derive an
expected age distribution of the children of women in
specific age groups and hence estimate the implied
average survival of children to the time of the survey.
On average it has been found that response of women 1519 can be used to derive q(1), 20-24, q(2), 25-29, q(3), 3034, q(5), 35-40, q(10), etc.
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Childhood mortality
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Usually q(5) is taken as a measure of the level and an
appropriate shape is taken from a set of model life tables
Various problems which can lead to inaccurate estimates:
e.g. 15-19 are young and children have lower survival
which doesn’t represent that of all children for their first
year, women may not wish to talk of children who have
died, or have forgotten them, particularly at the older ages
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Adult mortality - survival rates

Method: calculate cohort survival rates from censuses
at two points in time

Problems:
Populations may not be closed to migration (particularly
at some ages)
 The quality of enumeration of the censuses unlikely to
be equal, particularly at corresponding ages
 Focus is on survival and not mortality and a small error
in survival = large error in mortality
 Censuses often 10 years apart and two years in the
release and thus estimates a good 6 years out of date,
and rates would only be given as average of 10-year
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Adult mortality - question on
deaths in census

Method: Ask in the census about people who have died
in the household over the last 12 months - age, sex,
whether natural/non-natural/connected to childbirth

Problems: In practice this question has not produced
very reliable results, because:
 Memory
 Dissolution of households on some deaths
 Uncertainty about who is in which household (and who
reporting)
 Uncertainty about cause of death
 Time frame
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Adult - orphanhood method

Rationale: As with estimating child mortality so too, if
we have an average age of mothers/fathers at birth of their
children, we can, by asking respondents about whether
their mothers/fathers are still alive, derive an estimate of
survival from that age to that age plus the age of the
respondent. These proportions can then be turned into
survival probabilities. Question commonly included in
African censuses
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Problems:
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Adoption effect
Absent fathers
Age misstatement
Bias introduced by HIV/AIDS
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Adult - widowhood and sibling
methods
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Similarly one could ask widows/widowers about when they
got married and the survival of their spouses. Here the
major problems are definition of marriage and memory.
Or in the case of the ‘sibling method’ ask respondents
about the age and survival of their siblings. Not as
common, and some doubt about accuracy. Similar
problems about definition of ‘sister’ and ‘brother’ and recall
and loss of contact.
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Adult - vital registration
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If vital registration complete (and accurate estimate of
population) then can estimate rates directly. Problem is if,
as is commonly the case in Africa, there is significant
under-registration of deaths. Methods have been
developed which attempt to estimate the extent of under
registration of the deaths RELATIVE to the population,
commonly on the assumption that under-recording is
constant with respect to age (for adults at least)
Methods:
 Brass Growth Balance method
 Preston-Coale method
 Bennett-Horiuchi method
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Adult - Brass Growth Balance

Rationale:
For a population close to migration P2 - P1=B - D
 Dividing through by the mid-period population one get
the same relationship in rates, i.e. r = b - d
 This rationale applies for any sub-population aged x+,
i.e. rx+ = bx+ - dx+ where bx+ = ….
 Now if we can assume that a proportion, C, of deaths
are reported (constant wrt to age), and that the
population is ‘stable’, i.e. grows at a constant rate, r, we
can estimate both r and C by regressing bx+ on dx+
 If data support it one can relax the stability assumption
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and allow for migration (usually not the case)
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Adult - Preston-Coale

Rationale:
Closed population at time t, aged x = sum of deaths in
future arising from this cohort i.e. =  D  x  s, s  ds
0
 Obviously don’t know D  x  s, s  ds
 If population is closed and stable, growing at r p.a. then
D  x  s, s   D  x  s, t  er ( s t )
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Thus one has two estimates of the population one
derived from deaths the other from census and one can
use the ratio to estimate the completeness of deaths
Again if data support can relax the stability assumption
(Bennett-Horiuchi) and use 5rx and allow for migration
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Male deaths corrected for
under-registration
25000
Proj1999
20000
1996
15000
1997
10000
1998
5000
1999
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85+
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Female deaths corrected for
under-registration
18000
Proj1999
16000
14000
1996
12000
10000
1997
8000
6000
1998
4000
2000
1999
0
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85+
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Adult - indirect methods
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Problems:
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Rough
Number of assumptions which often do not hold these
days
However, robust in different ways
Often good for deciding on level, but not so useful for
estimating ‘shape’
Shape often derived by using model life tables
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Adult - model life tables
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Patterns of mortality by age derived from all known life
tables
Princeton tables (Coale and Demeny):
 A bit long in the tooth but still most widely used.
 Four families North, South, East, West.
 West, residual, similar to average, often used as default
pattern when nothing to suggest one of the other
patterns should be used
 Not representative of developing countries and Africa in
particular
UN tables: not very widely used
WHO tables: recently released, very flexible, not much
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experience with them yet.
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Extrapolation of old age
mortality
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Problem: Only have ‘reliable’ rates to age 85, but wish to
extend the life table beyond that age
bx
Solution: One could always extrapolate using e.g.  x  ae
But shape not necessarily correct at advanced ages
However, Coale and Guo have suggest that there is
evidence to assume that ln  5 mx 5 mx5  declines by a
constant decrement for ages above 80
Thus we can use ln  5 m80 5 m75  and by setting
5 m105 5 m75  0.66, which often seems to be reasonable
Derive mortality rates at higher ages
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