1. Estimating mortality from defective data
Rob Dorrington
Director of the Centre for Actuarial Research

2. 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

3. National vs sub-population
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

4. Industry data
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, underwriting)

5. 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)

6. Childhood mortality
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 can be used to derive q(1), 20-24, q(2), 25-29, q(3), 30-34, q(5), 35-40, q(10), etc.

7. Childhood mortality
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 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

8. 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 age interval

9. 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

10. 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
Problems:
Adoption effect
Absent fathers
Age misstatement
Bias introduced by HIV/AIDS

11. Adult - widowhood and sibling methods
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.

12. Adult - vital registration
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

13. 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 and allow for migration (usually not the case)

14. Adult - Preston-Coale Rationale:
Closed population at time t, aged x = sum of deaths in future arising from this cohort i.e. =
Obviously don’t know
If population is closed and stable, growing at r p.a. then
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

15. Male deaths corrected for under-registration

16. Female deaths corrected for under-registration

17. Adult - indirect methods
Problems:
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

18. Adult - model life tables
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 experience with them yet.

19. Extrapolation of old age mortality
Problem: Only have ‘reliable’ rates to age 85, but wish to extend the life table beyond that age
Solution: One could always extrapolate using e.g.
But shape not necessarily correct at advanced ages
However, Coale and Guo have suggest that there is evidence to assume that declines by a constant decrement for ages above 80
Thus we can use and by setting
, which often seems to be reasonable
Derive mortality rates at higher ages