Estimating mortality from defective data

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Estimating mortality from defective data

• Rob Dorrington
• Director of the Centre for Actuarial Research

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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

• 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

• 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)

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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

• 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 15-19 can be used to derive q(1), 20-24,
  q(2), 25-29, q(3), 30-34, q(5), 35-40, q(10), etc.

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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. 15-19 are young and children have
  lower survival which doesnt 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 age interval

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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
• Problems
• Adoption effect
• Absent fathers
• Age misstatement
• Bias introduced by HIV/AIDS

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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.

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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

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Adult - Brass Growth Balance

• Rationale
• For a population close to migration P2 - P1B - 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)

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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.
• Obviously dont 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

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Male deaths corrected for under-registration
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Female deaths corrected for under-registration
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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

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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.

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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