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

1. 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@aueb.gr Byron Kotzamanis University of Thessaly, Greece email: bkotz@prd.uth.gr

2. 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

3. 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) 3

4. 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 . 2009Kostaki, 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. 3–4, pp. 37–58. 4

5. 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. 5

6. 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, No3, 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

7. 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. 7

8. Denote as nDxthe observed death count at the age interval [x, x+5) nDx is a random variable binomially distributed with E(nDx )=Ex .nqxand 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,x+n). 8

9. Let us now consider the observed death rate, nQx = nDx/Ex Since nDx~Bin, whileEx is large and nqxvery small, nDxcan also be considered as approximately ~Po , with E(nDx)=Var(nDx)= Ex nqx and also asymptotically normal distributed. Therefore nQxcan also be considered as asymptotically normal distributed, with E(nQx)= nqxandVar(nQx)= [nqx (1- nqx)]/Ex 9

10. Hence, the unknown probability of dying at the age interval [x, x+n), nqx is expected to have a value in the interval: or simpler 10

11. 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 5D10=0 . Thus the observed death rate5Q10 at age 10 is zero! Does it mean that 5q10=0 ? …. Unfortunately not! => 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, E55=1464 • Thus Q55=6/1464=> Q55=0,0041 • Thus calculating the confidence interval we conclude that: • -0,0009<q55<0,0091 !!! • Qx can be a highly inaccurate estimator of qxwhen Dx is small 11

12. 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

13. 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

14. Considering the previous example: The count of deaths at the age 55, D55 is6and the exposed population, E55is equal to 1464 Q55=6/1464=> Q55=0,0041 Putting these values inQ55 ± z1-a/2 √Q55*(1-Q55)/Ex we took:-0,0009<q55<0,0091, (95% CI: 0.0008, 0.0074) while now using the above defined CI we take0,0012<q55<0,0130 , (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! 14

15. 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. 15

16. 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, x+5), 5Dxx=0, 5, 10,… w-5. • Then the five-year death rates , x=0, 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.

17. 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, x+5), a constant multiple of the one underlying the standard life table in the same age interval, μ(S)(x), i.e 17

18. the one-year probabilities qx+i, 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:

19. Italy, Males 1990-91 19

20. Norway males 1951-55 20

21. Now for the ages that are multiples of five using will be calculated using while for the rest of ages we calculate using

22. It is interesting to observe that the resulting fulfill the desirable property

27. 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

28. Thank you!

29. 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 . 2009Kostaki, 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. 3–4, pp. 37–58.