Page 22 - IJPS-5-1

Average age ratio method and age heaping in Chinese censuses

If ToPACI is < 25, age heaping at ending digit could be considered minimal; if ToPACI is between 25 and 50, age
heaping could be considered mild, between 50 and 100, moderate, between 100 and 200, substantial; and above 200, the
aging heaping could be considered severe, calling for an adjustment and a correction on the raw data.
In use of APAR, without information on the number of births by cohort, cautions are needed to assess age
heaping in that the irregular births across cohorts may occur due to specific historical events such as wars, natural
disasters, fertility policy, and migration. In addition, APAR could only be used for investigating age heaping in
a single census. When the data of two and more censuses are available, ACAR to be introduced in the following
session could be applied, and APAR and ACAR could be jointly used to compare and verify findings of age heaping
in censuses.
2.2. Average Cohort Age Ratio (ACAR)

There are three steps to calculate the ACAR. The first step is to calculate the age ratio, which has been previously
described. We thus start with Step 2.
• Step 2: Calculate the cohort age ratio
We could calculate the cohort age ratio by comparing age ratios of the same cohort at the two censuses:
AR xt(, )
CARx t(, ) = , (4)
( −
, −
AR xk tk)
where t is the current census year, k is the interval of the two censuses under study, t−k is the year of the previous
census, x is the age, and CAR(x,t) is the cohort age ratio at the age x and the census year t.
Assume S(x−k) is the survival ratio for individuals aged x-k in the previous census to the current census when the
same cohort reaches the age x. If there are no age misreporting and no population migration, P(x,t)=P(x−k,t−k)*S(x−k).
P(x−k,t−k) is the number of population at age x−k in the first census in the year t−k, and P(x,t) is the number of population
at age x in the second census in the year t.
And Equation (4) could be transformed as:
 2 
 ∑ Sx ki tk( −+ , − )* ( − + , −
Px ki tk)
CARx t(, ) = S xk tk( − , − ) /   i=−2 2   . (5)
 ∑ Px ki,,tk− ) 
( −+
  i=−2  

Evidently, if Sx kt k( − , − ) , Sx kt k( − , − ) , Sx kt k( − , − ) , and Sx kt k( − , − ) are close to one, or the survival rates of adjacent
Sx k , t k)− Sx k , t k)− Sx k , t k)− Sx k , t k)−
( −−1
( −−2
( −+2
( −+1
age groups changes evenly, CAR(x,t) should be equal or close to one. That is, if the digit preference/avoidance does not
exist, as long as the survival rates of adjacent age groups change similarly, the cohort age ratio should be close to one,
even though the birth numbers of these cohorts may show irregularities. To clarify such issues, it is necessary to examine
the age ratio, preferably those adjusted by the cohort size (i.e., number of births).
As long as, the survival rates of adjacent age groups change similarly, CAR(x,t) significantly higher or lower than one
would indicate the existence of age misreporting, either age heaping in the current census at the age x or age avoidance in
the previous census at the age x−k. To clarify which case it is, it is suggested to examine the age ratio adjusted by numbers
of births if the number of births is available and reliable or by the APAR as introduced in the previous session.
• Step 3: Calculate the ACAR
For the age range to be studied, we could add all the cohort age ratios of the same ending digit of ages, and calculate
an ACAR for each ending digit:
( +
CARx i(, ) + CAR x( +10 i ,) + + CARx n* ,)
i
10
ACAR i() = CAR i() = , (6)
n +1
where i is the ending digit of age, APAR(i) or PAR i() is the ACAR for the ending digit i, (n+1) is the number of age
ratios with the same ending digit.
As long as, the censual interval is not exactly equal to 10 years and the population change over the censual interval is
even (if not evenly, extra data are needed in analyses), with ACARs for certain ending digit being significantly higher or
lower than one, it will be relatively easy to evaluate the presence of digit preferences/avoidances. If the censual interval
is exactly equal to 10 years and when number of births is not available by cohort or by year, ACAR may be less effective.
In this situation, we need to combine the use of the APAR method.

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