Page 21 - IJPS-5-1

Gu and Feng

Px t,)
AR xt(, ) = 5 *( , (1)
t ,
Px( − 2 t , ) + P x( −1 t ,),+ Px t( ,) + P x( +1 t , ) + Px( + 2 ))
where t is the census year, P(x,t) is the enumerated number of population at the age x in the census year t, and AR(x,t)
is the age ratio at the age x in the census year t, also called as the age concentration index. If numbers of births in these
five cohorts are similar (or change in a stable manner) and the following changes are even, the age ratio should be equal or
close to 1. If the age ratio is larger than one to a significant degree, the population at the age x should exceed populations
at the other adjacent four ages (concentration); if the age ratio is smaller than 1 to a significant degree, the population at
age x should be smaller than populations at the other adjacent four ages.
However, it is possible in reality that the population at a certain age may vary greatly in size as compared to populations
of neighboring ages (e.g. in the baby boom and baby bust periods). That means that the value of the age ratio may deviate
from one, even though there is no age misreporting. On the other hand, when age misreporting presents, it is yet possible
that the age ratio is close to one at certain ages, because populations at these ages might be lower (or higher) at birth in
contrast to other ages. Therefore, in the cases when the number of births varies greatly by cohort and when these data are
available, it is better to further adjust the age ratio by the number of births and/or survival rates of the neighboring cohorts:
Px t,)
AR xt'( ,) = 5 *( /( (1a)
Cx),
Px( − 2 t , ) + P x( −1 t ,),+ Px t( ,) + P x( +1 t , ) + Px( + 2 ,tt)
where
Bt −
Cx()= 5 *( x)* L x() ,
x
x
x
x
L x − 2
Bt( −+ 2 )*( ) + B t( −+1 )* Lx( −1 ) + Bt( − x)* L(() x + ( B t − −1 )*( Lx +1 ) Bt+ ( − − 2 )* ( L x + 2 )
B(t−x) is the number of births in year (t−x), and L(x) is the number of survivors relative to 100,000 (or survival ratio from
birth to age x) for the cohort born x years ago. When there are no epidemics or wars, it is acceptable to assume that survival
ratios are the same or very close to each other for neighboring cohorts. If this is the case, C(x) would be fully depending
on the size of neighboring birth cohorts; and when the size of birth cohorts is stable or changes smoothly, AR’(x,t) would
be very close to AR(x,t). Of course, numbers of births of these cohorts need to be accurate or have the same pattern of
under- or over-estimation. As the vital registration system in most developing countries is usually underdeveloped or
incomplete, the availability of the number of births may be a challenge. In such cases, the value of such an adjustment
would be depreciated. However, this method is still useful as it would not produce substantial biases as long as the
coverage of vital registration is stable (or improved gradually) and the quality of data is generally acceptable over time.
If population changes are not even by age (e.g., large scale migration due to epidemics or wars for some specific
periods), we may consider further adjustments through adding the parameters of population changes Z(x) of each cohort
in C(x). Unfortunately, in most cases, such parameters are hard to obtain, particularly in developing societies. Thus, when
the number of births at adjacent years is roughly close to each other, and the subsequent population changes did not suffer
from a substantial fluctuation, the age ratios without adjustment for births may be considered as acceptable.
• Step 2: Calculate the APAR
For the age range to be studied, we could add up all age ratios of the same ending digit of ages, and calculate APAR
for each ending digit:
AR xi(, ) + AR x( +10 i ,) + + AR xn( + *10 i ,)
APAR i() = PAR i() = , (2)
n +1
where i is an ending digit of age, APAR(i) or PAR i() is the APAR for the ending digit i, (n+1) is the number of age
ratios with the same ending digit. When numbers of births vary greatly by cohort, we could further adjust the age ratio by
the birth size for a more accurate estimation (Equation 1a). Based on our preliminary investigation (see Appendix
Figure A), we recommend using the following criteria for assessment of digit preferences or age heaping: A ratio ranging
from 0.97 to 1.03 indicates almost no ending digit preference/avoidance of age, a ratio of 0.95-0.97 or 1.03-1.05 indicates
a mild digit preference/avoidance, a ratio lower than 0.95 or higher than 1.05 indicates a moderate digit preference/
avoidance, and a ratio lower than 0.90 or higher than 1.10 indicates a severe digit preference/avoidance. For a ratio lower
than 0.85 or higher than 1.15, it is necessary to adjust and correct the raw data.
For a given census, we could also sum all APARs for a general description of age heaping across all ending digits,
namely, the Total Period Age Concentration Index (ToPACI):
9
∑ 2 (3)
ToPACI = 100* ( APAR i −1() ).
i=0

International Journal of Population Studies | 2019, Volume 5, Issue 1 15