A method for estimating the age-specific mortality pattern in limited populations of small areas

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A method for estimating the
age-specific mortality pattern in limited
populations of small areas

• Anastasia Kostaki
• Athens University of Economics and Business
• email kostaki_at_aueb.gr
• Byron Kotzamanis
• University of Thessaly, Greece
• email bkotz_at_prd.uth.gr

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Need of analytical and reliable mortality data

• e.g.
• for providing population projections
• for construction of complete life tables
• for calculation of net reproduction rates
• for Age-specific mortality comparisons
• for construction of complete multiple decrement
  tables

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Problems in mortality data of small population
areas

• A. Limited information
• Aggregated data or
• Incomplete data sets
• B. Misleading information (Data of low quality)
• Data affected by sources of systematic errors
  (age misstatements heaping, under registrations
  of deaths, etc).
• C. Unstable documentation (small exposed-to-risk
  population, and low age-specific death counts)

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Limited information

• Incomplete data sets
• POSSIBLE WAYS OUT
• Fit a parametric model to the existed one-year
  values in order to produce estimates for the
  missing values and also smooth the rates,
  providing closer estimates to the true
  probabilities underlying the empirical rates.
• e.g. 1992 Kostaki A. "A Nine-Parameter
  Version of the Heligman-Pollard Formula".
  Mathematical Population Studies, Vol. 3, No. 4,
  pp. 277-288.
• or
• Apply a nonparametric graduation technique to
  existed one-year values
• e.g. 2005 Kostaki, A., Peristera P.
  Graduating mortality data using Kernel
  techniques Evaluation and comparisons Journal
  of Population Research, Vol 22(2), 185-197 .
• 2009 Kostaki, A., , Moguerza,M.J.,
  Olivares, A., Psarakis, S.  Graduating the
  age-specific fertility pattern using Support
  Vector Machines. Demographic Research., Vol
  20(25) 599-622.
• 2010 Kostaki, A., , Moguerza,M.J.,
  Olivares, A., Psarakis, Support Vector Machines
  as tools for Mortality Graduations to appear in
  Canadian Studies in Population 38, No. 34, pp.
  3758.

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B. Misleading information(data of low quality)

• A Typical problem
• HEAPING At age declaration, a preference of the
  responder to round off the age in multiples of
  five.

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• A way out
• Group death counts in five-year groups
  with central ages the multiples of five. Form
  the five-year death rates for these groups. Then
  apply an expanding technique in order to estimate
  the correct rates and/or counts.
• TECHNIQUES FOR EXPANSION
• 1991 Kostaki, A. "The Helignman - Pollard
  Formula as a Tool for Expanding an Abridged Life
  Table". Journal of Official Statistics Vol. 7, No
  3, pp. 311-323.
• 2000 Kostaki A., Lanke J. Degrouping mortality
  data for the elderly Mathematical Population
  Studies, Vol. 7(4), pp. 331-341.
• 2000 Kostaki A. A relational technique for
  estimating the age-specific mortality pattern
  from grouped data . Mathematical Population
  Studies, Vol. 9(1), pp.83-95.
• 2001 Kostaki, A., Panousis, E. Methods of
  expanding abridged life tables Evaluation and
  Comparisons Demographic Research, Vol 5(1) pp
  1-15. http//www.demographic-research.org/volumes/
  vol5/1/
• These techniques can also be used if the
  data are provided in five-year age groups

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• C. Small exposed-to-risk populations - low death
  counts
• (a sneaky problem)
• On the contrary of problems of types A and B
  (incompleteness bad quality), problems of type C
  is easy for the researcher to neglect, though
  their impact can lead to seriously misleading
  results.
• Let us consider this problem, which is highly
  actual when we deal with spatial (small area)
  population analysis or limited population samples.

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• Denote as nDx the observed death count at
  the age interval x, x5)
• nDx is a random variable binomially distributed
  with
• E(nDx )Ex .nqx and Var(nDx) Ex . nqx
  .npx
• where Ex is the exposed-to risk population at
  age x,
• and nqx is the unknown probability of dying in
  x,xn).

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• Let us now consider the observed death rate,
  
• nQx nDx/Ex
• Since nDxBin , while Ex is large and nqx
  very small,
• nDx can also be considered
  as approximately Po , with
• E(nDx)Var(nDx) Ex nqx
  
• and also asymptotically normal distributed.
  Therefore nQx can also be considered as
  asymptotically normal distributed, with
• E(nQx) nqx and Var(nQx) nqx (1- nqx)/Ex

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• Hence, the unknown probability of dying at the
  age interval x, xn), nqx is expected to have a
  value in the interval
• or simpler

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• What do all these mean in practice?
• Consider a real example in Eurytania in
  Greece, the exposed population at age 10 is E10
  1888 while the death count at age 10 5D100
  . Thus the observed death rate 5Q10 at age
  10 is zero!
• Does it mean that 5q100 ? . Unfortunately
  not!
• gt We are not able to provide estimates for the
  value of 5q10.

• Another example from the same population
• The death count at age 55, D55 6 and the
  exposed population, E551464
• Thus Q556/1464gt Q550,0041
• Thus calculating the confidence interval we
  conclude that
• -0,0009ltq55lt0,0091 !!!
• Qx can be a highly inaccurate estimator of
  qx when Dx is
• small

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Alternatively a more properly defined CI might
be derived from the following

• We know that

this is equivalent to
The inequality in the probability statement is
satisfied for those values of which lie
between the roots of the
quadratic equation
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We therefore have
and forms a CI for
where are given by 2002
Garthwaite, P.H., Jolliffe, I.T., Jones, B.,
Statistical Inference 2nd edition, Oxford Science
Publication
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Considering the previous example

• The count of deaths at the age 55, D55
  is 6 and the exposed population, E55 is equal
  to 1464
• Q556/1464gt Q550,0041
• Putting these values in Q55 z1-a/2
  vQ55(1-Q55)/Ex
• we took -0,0009ltq55lt0,0091, (95 CI
  0.0008, 0.0074)
• while now using the above defined CI
• we take 0,0012ltq55lt0,0130 , (95 CI
  0.0014, 0.0112)
• which is better in the sense that the limits
  are always positive but still too wide to be
  useful!

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Possible ways out

• Use wide age intervals (five-year ones, or ten
  year ones)
• Consider wide periods of investigation (up to
  ten-year periods)
• Utilize an expansion technique for estimating
  the unknown age-specific probabilities from
  grouped data.
• The later can be a cure for problems of
  Types A and B too
• At the outset a technique for estimating the
  age-specific
• death counts from data given in age-groups is
  presented.

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A technique for estimate age-specific death
counts data given in age groups

• Consider the empirical death count for the
  five-year age interval x, x5), 5Dx x0,
  5, 10, w-5.
• Then the five-year death rates , x0, 5, 10,
  are calculated by

where the summation is restricted to multiples
of five. Obviously the consideration of the
exposed-to-risk population of a given age as the
sum of deaths after that age is precisely valid
when the data concern a closed cohort. However,
as it will be demonstrated, this procedure
produces excellent results.
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• Let us consider the set of the abridged
  5qx-values as calculated before. The next step
  is to expand them. For that
• Let us also consider a set of one-year
  probabilities, qx(s) (S for Standard) of a
  standard complete life table.
• Under the assumption that the force of
  mortality, µ(x), underlying the target abridged
  life table is, in each age of the 5-year age
  interval x, x5), a constant multiple of the
  one underlying the standard life table in the
  same age interval, µ(S)(x), i.e

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• the one-year probabilities qxi , i 0,1,..., 4,
  for each age in the five-year age interval can
  be calculated using
• 
  
• 
  (1)
  
• 
  
  
  
• 
  

where
An inherent property of the new technique is that
its results fulfill the desired relation
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Italy, Males 1990-91
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Norway males 1951-55
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• Now for the ages that are multiples of five
  using will be calculated using
• while for the rest of ages we calculate
  using

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• It is interesting to observe that the resulting
  fulfill the desirable property

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Comments and guidelines

• The results are nice in the sense that they are
  very close to the
• true values and also fulfill desirable
  properties.
• The choice of the standard table does not affect
  the results . Use every complete life table you
  have at hand!
• Easiest to apply using a simplest software

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• Thank you!

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• References
• 2002 Garthwaite, P.H., Jolliffe, I.T.,
  Jones, B., Statistical Inference 2nd edition,
  Oxford Science Publications
• 2001 Karlis D., Kostaki A. Bootstrap
  techniques for mortality models Biometrical
  Journal, Vol. 44(7) pp 850-866.
• 1992 Kostaki A. "A Nine-Parameter
  Version of the Heligman-Pollard Formula".
  Mathematical Population Studies, Vol. 3, No. 4,
  pp. 277-288.
• 2005 Kostaki, A., Peristera P.
  Graduating mortality data using Kernel
  techniques Evaluation and comparisons Journal
  of Population Research, Vol 22(2), 185-197 .
• 2009 Kostaki, A., , Moguerza,M.J.,
  Olivares, A., Psarakis, S.  Graduating the
  age-specific fertility pattern using Support
  Vector Machines. Demographic Research., Vol
  20(25) 599-622.
• 2010 Kostaki, A., , Moguerza,M.J.,
  Olivares, A., Psarakis, Support Vector Machines
  as tools for Mortality Graduations to appear in
  Canadian Studies in Population 38, No. 34, pp.
  3758.