Page 12 - IJPS-2-2

Assessments of mortality at oldest-old ages by province in China's 2000 and 2010 censuses

In the second set of our approach, the Kannisto model was applied by fitting the age-specific
death rates from ages 80 to 98 by sex for China (as a whole) and for each province in the 2000 and
2010 censuses. The Kannisto model is a special case of the Gamma-Makeham model (see Thatcher,
1999). The purpose of this application is to investigate whether the age trajectories in the censuses
matched the well-established model. The Kannisto model was based on single years of age of death
rates that were split from five-year age groups the piecewise cubic Hermite interpolating polynomial
and were smoothed with iterations under a constraint that the new five-year age group probabilities
of dying calculated from split and smoothed single years equal the original five-year age group
probabilities of dying (United Nations, 2013). A relational technique for estimating the age-specific
mortality pattern from grouped data (Kostaki, 2000; Kostaki and Lanke, 2000) was also applied and
ae bx
the results was very similar. The Kannisto function is µ ()x = c + , where µ ()x denotes the
1 ae+ bx
force of mortality at age x, also known as the central death rate, or simply death rate, and a, b, and c
are parameters to be estimated. Maximum likelihood estimation procedures were used to fit death
rates for the Kannisto function, which is the same used by Thatcher, Kannisto, and Vaupel (1998:36)
and Zeng and Vaupel (2003: 236). The logarithm of the maximum likelihood function is

Dx
() 1 e
())ln (1 q x−
L = ∑ [ ()ln ( ()) ( () x+ q x N − Dx ())] , where q x = − − x ∫ x+ 1 µ () t dt is the probability
x
for an individual who ages x and would die before age x+1, N(x) is the number of individuals who
live to age x, and D(x) is the number of people who die before age x+1. To a sufficient approxima-
tion µ ()x = θ (x + 1 , )α for all t between ages x and x+1, we estimate the parameters
2
α ∈ (, , )abc by maximizing L with ( ) 1 exp(q x = θ − (x + 1 2 , )) with a convergence criterion of
α −
-8
-4
1*10 for q(x) and 1*10 for each parameter.
Given the inaccuracy of mortality at oldest-old ages in China, we did not directly apply the
Kannisto model to the observed data. To examine the accuracy of the age trajectory, we assumed that
the death rates at ages 60–70 were accurate and then applied the Gompertz model to the observed
death rates from ages 60 to 84 and then extrapolated to age 98. Finally, we applied the Kannisto
model to these rates from ages 80 to 98. The selection of the inclusion of the observed death rates
from ages 60 to 84 in the Gompertz model was because the average error between the fitted and ob-
served death rates for ages 60 to 80 was smallest in comparison to other cut-points, such as
bx
ages 70, 75, 80, 90, and 95. The formula of the Gompertz model is ()xµ = ae . The maximum
likelihood formula is the same as noted above.
3. Results

3.1 High Underestimation of Death Rates at Oldest-old Ages in Most Chinese Provinces

q against
q in the 2000 and
The upper panel in Figure 1 presents the logit-transformed 10 60 25 70
2010 Chinese censuses in comparison with corresponding values for Sweden, Japan, and the 11 oth-
er HMD countries with high quality of mortality data in the period 1950–2014. The results show that
many Chinese provinces were below the lower boundary of the confidence ellipse. The ellipse in-
cludes 95% of data points from the 13 HMD countries based on the linear associations of all data
points in these 13 countries (see below). Provinces in northeastern and western China such as Xin-
jiang, Hainan, Tibet, Guangxi, Ningxia, Gansu, Jilin, and Heilongjiang were far below the low-
er boundary of the ellipse, indicating a substantial underestimation of mortality after age 70 in these
provinces. The results further show that the underestimation was more dramatic in the 2010 cen-
sus compared with the 2000 census, although provinces in the 2010 census had lower mortality than
in the 2000 census.

6 International Journal of Population Studies | 2016, Volume 2, Issue 2