Acknowledgements References


1. The estimates for the life-table aging ratio have been smoothed using Kalman-Filter techniques.
2. Madame Jeanne Calment was born on the 21st of February 1875 and died on the 4th of August 1997 at the age of 122 years, 5 months and 14 days. Madame Jeanne Calment age at death is currently the highest verified age at death of a person.
3. In the period 5th of December 1992 – 31st of December 1993 116, 611 deaths occured in Bulgaria. A relatively small fraction of 8, 541 deaths (7.33 %), could not be linked to the census records (4,142 (48.50%) of the unlinked deaths are males, and 4,399 (51.50%) are females). Although the death certificate in Bulgaria contains some limited information on the socio-economic status of dead persons (e.g., education at death, marital status at death, place of residence at death), we omit the unlinked deaths from our analysis mainly for two reasons: (a) We do not know whether the persons corresponding to unlinked death certificates have participated the census, and their personal data were wrongly coded in the census records, or whether they are subject to census undercount. This latter possibility is supported by the fact that some relevant minorities in Bulgaria are more likely to be a subject of undercount, which is indicated by the fact that most of the unlinked deaths involve persons with low education (76.56% of the unlinked male, and 86.63% of the unlinked female deaths). (b) The unlinked deaths can also correspond to return migrants whose prevalence has increased after the first waves of rapid emigration in the early 1990s. We believe that census undercount and return migration are the most important factors leading to unlinked death certificates. If the persons corresponding to the unlinked deaths have not participated in the census due to the above reasons, then excluding them from the analyses is appropriate. If they have participated in the census, but the death cannot be linked due to incomplete identification numbers, our estimates about the mortality level are biased downward. However, we have no reasons to believe that this bias is highly sex-specific.
Therefore, the Bulgarian age and sex-specific patterns of mortality presented in this paper cannot be attributed to a bias caused by unlinked deaths because the overall numbers of unlinked deaths is quite small and is evenly distributed across sexes. Moreover, the age distribution of the linked and unlinked deaths is relatively similar for males, while it is slightly shifted to higher ages for females. For instance, the exact mean age at death of unlinked male deaths is 67.2 years, which is slightly below that of linked male deaths of 67.8 years, and female unlinked deaths have a mean age of 76.7, which is above that of linked deaths with a mean age of 73.2.
Age reporting is of relatively high quality in the Bulgarian census. The age of a person is coded in the personal identification number (PID), which was assigned to the individuals either during 1976–78 for those alive at this time, or at birth for those born afterwards. The age in the census form is taken from the age coded in the PID number. Since the old and oldest-old in 1992 have received their PID about 15 years earlier, when the propensity to age-misreporting may have been lower due to younger ages and careful administrative checking, the age information in our census data should be of high quality even for the old population.
4. The piecewise-constant proportional hazard model with unobserved frailty is specified as follows. Consider the age-intervals (c0, c1], ..., (cj-1, cj], ..., (cK-1, cK] that separate the observed age range into K disjoint intervals. Then assume that the mortality hazard, conditional on a frailty z = 1 and the observed covariates ya, is constant within each of these age intervals and equals a(ya)mj for x (cj-1, cj]. In this specification mj is the mortality hazard prevailing in the age interval (cj-1, cj], j = 1, ..., K, and a(ya) is the factor of proportionality for individuals with characteristics ya. Denote as sPW (x) the corresponding survival function at age x. The observed hazard at age x in a heterogeneous population with Gamma-distributed relative frailty then equals

Because of the numerical difficulties in estimating this piecewise-constant hazard function via maximum likelihood in the presence of many age-intervals and large data, we implement a slight approximation to the hazard function (12). In particular, the difficulties in the estimation arise because the hazard PW (x) is not constant within age intervals. This results from the fact that the value of the survival function sPW (x) in the denominator declines with age x. For sufficiently small age-intervals, however, the effect of this changing value of the survival function on the observed hazard PW (x) within an age interval is small. The piecewise-constant hazard function with relative frailty can therefore be approximated by replacing the value of the survival function sPW (x) in (12) with the value of the survival function at the mid-point of each age-interval. With this approximation, the observed hazard PW (x) is constant within age intervals and the MLE estimation is substantially simplified. We estimate this piecewise-constant hazard model with a constant mortality risk within two-year age intervals.


6. Acknowledgements References

Frailty Modelling for Adult and Old Age Mortality: The Application of a Modified DeMoivre Hazard Function to Sex Differentials in Mortality
Hans-Peter Kohler, Iliana Kohler
© 2000 Max-Planck-Gesellschaft ISSN 1435-9871