Paper on Special Property of TFR (72) - Final

On a Special Property of the Total Fertility Rate
Lu Lei *
21 April 2012
* The author is from China and was a former research fellow at the Institute of Population Research, Renmin University of China, Beijing, China. The author has a B.Sc. degree in computer science and applied mathematics, and an M.A. degree and a Ph.D. degree in demography.
E-mail: lulei321@https://www.wendangku.net/doc/2a961951.html,

On a Special Property of the Total Fertility Rate – by Lu Lei (China)
1. Introduction
In demographic analysis, the (period) total fertility rate ( TFR ) is one of the most widely used summary indicators for measuring the period fertility of a population. The total fertility rate is a period indicator, i.e. it is defined based on variables for the same period (e.g., a calendar year). However, when the demographic meaning of the total fertility rate is interpreted, demographers have to refer to a hypothetical (synthetic) birth cohort of women because of the nature of the definition of the total fertility rate. By definition, the total fertility rate is the sum of the age-specific fertility rates in a given year. The standard demographic interpretation of the total fertility rate is that it represents the average number of children that a hypothetical (synthetic) birth cohort of women would bear during their entire reproductive life span (normally between ages 15 and 50) if (i) all the women (of the birth cohort) survive through to the end of their reproductive life span, and (ii) they follow the age-specific fertility rates of the year in question.
So far, there has been a huge amount of demographic research on the total fertility rate. Now, researchers and demographers are very familiar with the advantages and disadvantages of the total fertility rate (e.g., Shryock, Siegel and Associates (1980); Ní Bhrolcháin (1992); Bogue, Arriaga, and Anderton (1993); Bongaarts and Feeney (1998); Ní Bhrolcháin (2007)). In demographic analysis, the total fertility rate is usually used for measuring and comparing (the level of) period fertility over time and/or across regions. The total fertility rate is also commonly used in population projections.
The present paper discusses a special property of the total fertility rate as a statistical indicator per se, i.e. the (quantitative) relationship between the total fertility rate of a total population and the total fertility rates of two sub-populations.
Suppose that we have a total population (denoted as P ), which is divided into two subpopulations P 1 and P 2 , and the two sub-populations satisfy the following conditions:
P 1∩P 2 = ? (null) and P 1∪P 2 = P . Mathematically, it can be easily proved that the crude
birth rate ( CBR ) of the total population for a given year is the weighted average of the crude birth rates of the two sub-populations for the same year, with the weights being the respective proportions of the two sub-populations in the total population. Now, the question is: Does the
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
total fertility rate have a similar property?
Let TFR denote the total fertility rate of the total population ( P ), and TFR1 and TFR2 denote the total fertility rates of the two sub-populations ( P 1 and P 2 ) respectively. In the present paper, we will look at the relationship between TFR and [ TFR1 and TFR2 ].
Let W ( x) represent the number of women aged x at the midpoint of a given year t and B( x)
represent the number of live births delivered by women of age x in the same year, then the (period) age-specific fertility rate for age x of year t is defined as f ( x) = B( x) W ( x) ,
x = 15, 16, K , 49 , and
{ f ( x)
x = 15, 16, ..., 49} is called the (period) age pattern of fertility.
The corresponding total fertility rate is then defined as TFR =
∑ f ( x) . The total fertility
x =15
49
rates of the two sub-populations are TFR1 =
∑
x =15
49
f1 ( x) and TFR2 =
∑ f ( x) ,
2 x =15
49
where f1 ( x)
and f 2 ( x) are the (period) age-specific fertility rates of the two sub-populations of year t. Mathematically, the calculation of the total fertility rate is simple and straightforward.
2. A graphical analysis of the relationship between TFR and [ TFR1 and TFR2 ]
Since the total fertility rate is the summation of the corresponding age-specific fertility rates, let’s first look at the relationship between the age-specific fertility rates of the total population and the age-specific fertility rates of the two sub-populations. According to the definition of the age-specific fertility rate for age x, we can easily arrive at the following:
f ( x) = u1 ( x) ? f1 ( x) + u2 ( x) ? f 2 ( x) ,
(1)
where f ( x) is the age-specific fertility rate of women aged x of the total population, f1 ( x)
and f 2 ( x) are the age-specific fertility rates of women aged x of the two sub-populations respectively, u1 ( x) = W1 ( x) W ( x) and u2 ( x) = W2 ( x) W ( x) are the proportions of women aged x of the two sub-populations in the women aged x of the total population. It is obvious that
u1 ( x), u2 ( x) > 0 and u1 ( x) + u2 ( x) = 1 , for all ages x = 15, 16, K , 49 . Therefore, equation (1)
shows that for each age x ( x = 15, 16, K , 49 ), f ( x) is a weighted average of f1 ( x) and
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
f 2 ( x) . In other words, for each age x ( x = 15, 16, K , 49 ), f ( x) always falls between f1 ( x)
and f 2 ( x) . Specially, if for an age x? , there is f1 ( x? ) = f 2 ( x? ) (i.e. the two fertility curves
f1 ( x) and f 2 ( x) intersect at age x? ), then we have f ( x? ) = f1 ( x? ) = f 2 ( x? ) , i.e. curve
f ( x) must pass through the point of intersection. Next, we discuss different situations based
on the relative relations between the two fertility curves f1 ( x) and f 2 ( x) .
2.1 First type of relative relation between f1 ( x) and f 2 ( x) (Figure 1)
In this situation, the age-specific fertility rate of all ages ( x = 15, 16, K, 49) of the subpopulation 1 is lower than the corresponding rate in sub-population 2, i.e. f1 ( x) < f 2 ( x) ,
x = 15, 16, K , 49 . Therefore, we have f1 ( x) < f ( x) < f 2 ( x) , x = 15, 16, K , 49 .
Figure 1. First type of relative relation between f1 ( x) and f 2 ( x)
f ( x)
f 1 ( x) f 2 ( x)
15
Age
49 49
50
By taking summation with respect to age x, we obtain
∑
x =15
f1 ( x) <
∑
x =15
f ( x) <
∑f
x =15
49
2
( x) , i.e.
TFR1 < TFR < TFR2 . It is obvious that, in this situation, the relationship TFR1 < TFR < TFR2
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
always holds regardless of the values of u1 ( x) or u2 ( x) . However, it is not possible to know if TFR is a weighted average of [ TFR1 and TFR2 ], as this will require knowledge of u1 ( x) or u2 ( x) .
2.2 Second type of relative relation between f1 ( x) and f 2 ( x) (Figure 2) In this situation, the three fertility curves f ( x) , f1 ( x) and f 2 ( x) intersect at age α and
form five regions, i.e. A, B, C, D and E. In age interval (15, α ) , we have
f 2 ( x) < f ( x) < f1 ( x) , and in age interval (α , 50) , we have f1 ( x) < f ( x) < f 2 ( x) . Let A, B, C,
D and E represent the areas of the corresponding regions. When the two curves f1 ( x) and
f 2 ( x) are fixed, areas A, B, D and E change with u1 ( x) and u2 ( x) , while area C remains
constant. It is obvious that, no matter what values u1 ( x) and u2 ( x) take, A + B and D + E are constant. Let X = A + B and Y = D + E , then X and Y are not affected by u1 ( x) and u2 ( x) .
Figure 2. Second type of relative relation between f1 ( x) and f 2 ( x)
f ( x)
f 1 ( x) f 2 ( x)
A E
B
D
C
15
α
Age
50
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
The following table provides a summary of the theoretical upper and lower limits of areas B and D, depending on u1 ( x) and u2 ( x) .
Table 1. Theoretical upper and lower limits of areas B and D
Area Upper limit B Lower limit Age Interval X (when
(15, α )
Age Interval
(α , 50)
u1 ( x) →1 or u2 ( x ) →0)
0
(when
u1 ( x) →0 or u2 ( x ) →1)
Y (when
Upper limit D Lower limit
u1 ( x) →0 or u2 ( x ) →1)
0
(when
u1 ( x) →1 or u2 ( x ) →0)
Since
TFR = B + C + D
(2) (3) (4)
TFR1 = A + B + C = X + C TFR2 = C + D + E = Y + C
we have
TFR = TFR1 + TFR2 ? (A + C + E)
(5)
It is obvious that area A is affected by u1 ( x) and u2 ( x) (15 < x < α ) and area E is affected
by u1 ( x) and u2 ( x) (α < x < 50) , while area C is not affected by u1 ( x) and u2 ( x)
(15 < x < 50) . From equation (2) and Table 1, we know that the TFR has a theoretical upper limit of X + Y + C and a theoretical lower limit of C. From equations (3) and (4), we can obtain X + Y + C = TFR1 + TFR2 ? C . Thus, the theoretical upper limit for the TFR is
TFR1 + TFR2 ? C .
It is obvious that area C is completely determined by the relative relations of the two fertility curves f1 ( x) and f 2 ( x) . Specially, when C = 0 (i.e. the two fertility curves f1 ( x) and
f 2 ( x) do not have an overlap), the TFR has a theoretical upper limit of TFR1 + TFR2 and a
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
theoretical lower limit of zero.
2.3 Third type of relative relation between f1 ( x) and f 2 ( x) (Figure 3)
In this situation, the three fertility curves f ( x ) , f1 ( x) and f 2 ( x) intersect at ages α and
β and form seven regions, i.e. A, B, C, D, E, F and G. In age interval (15, α ) , we have
f 2 ( x) < f ( x) < f1 ( x) ; in age interval (α , β ) , we have f1 ( x) < f ( x) < f 2 ( x) ; and in age
interval ( β , 50) , we have f 2 ( x) < f ( x) < f1 ( x) . Let A, B, C, D, E, F and G represent the areas of the corresponding regions. When the two curves f x1 and f x2 are fixed, areas A, B,
D, E, F and G change with u1 ( x) and u2 ( x) , while area C remains constant. It is obvious
that, no matter what values u1 ( x) and u2 ( x) take, A + B , D + E and F + G are constant. Let X = A + B , Y = D + E and Z = F + G , then X, Y and Z are not affected by
u1 ( x) and u2 ( x) .
Figure 3. Third type of relative relation between f1 ( x) and f 2 ( x)
f ( x)
f1 ( x) f 2 ( x)
F
G
A B C D E
15
α
β
Age
50
The following table provides a summary of the theoretical upper and lower limits of areas B,
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
D and G, depending on u1 ( x) and u2 ( x) .
Table 2. Theoretical upper and lower limits of areas B, D and G
Area Upper limit B Lower limit Age Interval X (when
(15, α )
Age Interval
(α , β )
Age Interval
( β , 50)
u1 ( x) →1 or u2 ( x ) →0)
0
(when
u1 ( x) →0 or u2 ( x ) →1)
Y (when
Upper limit D Lower limit Z Upper limit G Lower limit (when
u1 ( x) →1 or u2 ( x ) →0)
0
(when
u1 ( x) →0 or u2 ( x ) →1)
u1 ( x) →0 or u2 ( x ) →1)
0
(when
u1 ( x) →1 or u2 ( x ) →0)
Since
TFR = B + C + D + G
(6) (7) (8)
TFR1 = A + B + C + D + E = X + Y + C TFR2 = C + F + G = Z + C
we have
TFR = TFR1 + TFR2 ? (A + C + E + F)
(9)
It is obvious that area A is affected by u1 ( x) and u2 ( x) (15 < x < α ) , area F is affected by
u1 ( x) and u2 ( x) (α < x < β ) , and area E is affected by u1 ( x) and u2 ( x) ( β < x < 50) ,
while area C is not affected by u1 ( x) and u2 ( x) (15 < x < 50) . From equation (6) and Table 2, we know that the TFR has a theoretical upper limit of X + Y + Z + C and a theoretical lower limit of
C.
From
equations
(7)
and
(8),
we
can
obtain
X + Y + Z + C = TFR1 + TFR2 ? C . Thus, the theoretical upper limit for the TFR is TFR1 + TFR2 ? C .
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
It is obvious that area C is completely determined by the relative relations of the two fertility curves f1 ( x) and f 2 ( x) . Specially, when C tends to 0, the TFR approaches the theoretical upper limit of TFR1 + TFR2 .
From the above graphical analysis, we can conclude that the TFR has a theoretical upper limit of TFR1 + TFR2 and a theoretical lower limit of zero. Here, we have an interesting observation. Although the TFR is standardized for the age-sex structure of the total population, it is affected by the relative age distributions of the women of reproductive ages of the two sub-populations. In a special situation where TFR1 = TFR2 (i.e. the two subpopulations have the same total fertility rate), we cannot guarantee that TFR = TFR1 = TFR2 because the TFR is affected by u1 ( x) and u2 ( x) , while TFR1 and TFR2 are not.
3. A mathematical analysis of the relationship between TFR and [ TFR1 and TFR2 ]
From the above graphical analysis, we have noticed that the relationship between the TFR and the [ TFR1 and TFR2 ] is complex. For given TFR1 and TFR2 , the TFR may range between zero and TFR1 + TFR2 , depending on u1 ( x) and u2 ( x) . Now, let’s look at the relationship between the TFR and the [ TFR1 and TFR2 ] from a mathematical perspective.
Let g1 ( x) = f1 ( x) TFR1 and g 2 ( x) = f 2 ( x) TFR2 , x = 15, 16, K , 49 , then it is obvious that
g1 ( x), g 2 ( x) ≥ 0 ,
∑ g ( x) = 1
1 x =15
49
and
∑ g ( x) = 1 .
2 x =15
49
Sequences
{ g ( x)
1
x = 15, 16,K, 49} and
{ g ( x)
2
x = 15, 16,K, 49} are called the standardized (period) age patterns (schedules) of
fertility of the two sub-populations. From the above definitions, we have f1 ( x) = TFR1 ? g1 ( x) and f 2 ( x) = TFR2 ? g 2 ( x) . By taking summation on the two sides of equation (1) with respect to x, we obtain
TFR = TFR1 ? ∑ [u1 ( x) ? g1 ( x)] + TFR2 ? ∑ [u2 ( x) ? g 2 ( x)]
x =15 x =15
49
49
(10)
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
Equation (10) gives the general mathematical relationship between TFR and [ TFR1 and
TFR2 ],
49
which
provides
the
49
theoretical
2 2
basis
for
the
discussions
below.
Let
k1 =
∑[u1 ( x) ? g1 ( x)] and k2 =
x =15
∑[u ( x) ? g ( x)] .
x =15
Since 0 < u1 ( x) < 1 , and 0 < u2 ( x) < 1 , it
follows that 0 < k1 < 1 and 0 < k2 < 1 . Equation (10) can be written as
TFR = k1 ? TFR1 + k 2 ? TFR2
(11)
Equation (11) shows that the relationship between TFR and [ TFR1 and TFR2 ] is completely determined by the two coefficients k1 and k2 . Suppose that TFR1 ≤ TFR2 , then from equation (11), we have (k1 + k2 ) ? TFR1 ≤ TFR ≤ (k1 + k2 ) ? TFR2 .
Let
u1max = max{ u1 ( x) x = 15, 16 , ..., 49} u1min = min{ u1 ( x) x = 15, 16 , ..., 49}
(12) (13)
and
max u2 = max{ u2 ( x) x = 15, 16 , ..., 49} min u2 = min{ u2 ( x) x = 15, 16 , ..., 49}
(14) (15)
min Since u1 ( x) + u2 ( x) = 1 , x = 15, 16, K , 49 , it can be proved that u1max + u2 = 1 and max max min u1min + u2 = 1 , and further u1max ? u1min = u2 ? u2 . In addition, it can also be proved that min max min max u1min ≤ k1 ≤ u1max and u2 ≤ k2 ≤ u2 . Therefore, we have u1min + u2 ≤ k1 + k2 ≤ u1max + u2 .
According to the definition of the two coefficients k1 and k2 , we have
k1 + k2 = 1 +
∑ { u1 ( x) ? [ g1 ( x) ? g 2 ( x)]} = 1 + ∑ { u2 ( x) ? [ g 2 ( x) ? g1 ( x)]}
x =15 x =15
49
49
(16)
Equation (16) shows that the two coefficients k1 and k2 do not necessarily constitute a pair of weights. Whether the two coefficients k1 and k2 form a pair of weights depends on the
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
second term on the right-hand side of equation (16). Therefore, the TFR may not be a weighted average of TFR1 and TFR2 with respect to k1 and k2 .
Let k =
∑ { u ( x) ? [ g ( x) ? g ( x)]} , then we have
1 1 2
x =15
49
k1 + k2 = 1 + k . It is obvious that whether
the two coefficients k1 and k2 constitute a pair of weights depends on the value of k. From the discussion above, we have ? 1 < k < 1 . Specially, when k = 0 , then the TFR is a
weighted average of TFR1 and TFR2 , with k1 and k2 being the two respective weights.
Next, we look at the relationship between TFR and [ TFR1 and TFR2 ] under three special situations.
3.1 First situation If the two sub-populations have the same standardized age pattern (schedule) of fertility (denoted as
{ g ( x)
?
x = 15, 16, K, 49 ), i.e. g1 ( x) = g 2 ( x) = g ? ( x) , x = 15, 16, K , 49 , then
}
from the definition of k, we have k = 0 . In this situation, the TFR is a weighted average of
TFR1 and TFR2 , with k1 =
∑[u1 ( x) ? g ? ( x)] and k2 =
x =15
49
∑[u ( x) ? g
2
x =15
49
?
( x)] being the two
respective weights. Obviously, in this situation, u1 ( x) and u2 ( x) do not affect the relationship, but affect the two weights.
3.2 Second situation If the two sub-populations have the same total fertility rate (denoted as TFR ? ), i.e. TFR1 = TFR2 = TFR ? , then from equation (11), we have TFR = (k1 + k2 ) ? TFR ? = (1 + k ) ? TFR ? . Therefore, the relationship between the TFR and the TFR ? is dependent on k. Specifically, when k > 0 ， we have TFR > TFR ? ; when k = 0 , we have
TFR = TFR ? ; and when k < 0 , we have TFR < TFR ? . Here, we see that even if the two subpopulations have the same total fertility rate, the TFR (of the total population) may not necessarily be equal to the total fertility rate of the sub-populations.
3.3 Third situation Suppose that u1 ( x) can be represented by the following nth-degree polynomial of x:
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
u1 ( x) = λ0 + λ1 ? x + λ2 ? x + L + λn ? x = ∑ (λi ? x i )
2
n i =0
n
(17)
where n is a non-negative integer. Then from the definition of k, we have
k=
n n 49 ? ? ? i ? i [ g ( x ) ? g ( x )] ? ( λ ? x ) = λ ? ? 1 ? ∑ ? i ∑ x ? [ g1 ( x) ? g 2 ( x)] ? ∑ ∑ i 2 x =15 ? i =0 ? i = 0 ? x =15 ? 49
[
]
(18)
Define the rth absolute moment (about zero or origin) of g1 ( x) and g 2 ( x) as follows:
? g = M r 1
? g = ∑[ x r ? g1 ( x)] and M r 2
x =15
49
∑[ x
x =15 n
49
r
? g 2 ( x)]
(19)
? g ?M ? g )] . It is obvious that Then equation (18) can be rewritten as k = ∑ [λi ? ( M i i 1 2
i =0
? g = 1 and M ? g = 1 . Define the mean age of g ( x) and g ( x) as follows: M 0 1 0 2 1 2
μ g1 =
∑[ x ? g1 ( x)] and μ g 2 =
x =15
49
∑ [ x ? g ( x)]
2 x =15
49
(20)
? g then we have μ g1 = M 1 1
? g . Define the variance of g ( x) and and μ g 2 = M 1 2 1
g 2 ( x) as follows:
v g1 =
∑[( x ? μ g1 )2 ? g1 ( x)] and v g 2 =
x =15
49
∑[( x ? μ
x =15
49
g 2 ) 2 ? g 2 ( x)]
(21)
? g ? ( μ g ) 2 and v g = M ? g ? (μ g )2 . then we have v g1 = M 2 1 1 2 2 2 2
Next, we look at three special cases.
Case (1): u1 ( x) is a constant with respect to age x. This is equivalent to taking n = 0 in
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
equation (18), i.e. u1 ( x) = λ0 , x = 15, 16, K, 49 . In this case we have
49 49 ? 49 ? k = λ0 ? ∑ [ g1 ( x) ? g 2 ( x)] = λ0 ? ? ∑ g1 ( x) ? ∑ g 2 ( x)? = 0 x =15 x =15 ? x =15 ?
(22)
Therefore, in this case, the TFR is a weighted average of TFR1 and TFR2 , with k1 = λ0 and k2 = 1 ? λ0 being the two respective weights.
Case (2): u1 ( x) is a linear function of age x. This is equivalent to taking n = 1 in equation (18), i.e. u1 ( x) = λ0 + λ1 ? x , where λ1 ≠ 0 . In this case we have
k = λ1 ? ( μ g1 ? μ g 2 )
(23)
If μ g1 = μ g 2
(denoted as μ ), then we have k = 0 . In this case, the TFR is a weighted
average of TFR1 and TFR2 , with k1 = λ0 + λ1 ? μ and k2 = 1 ? (λ0 + λ1 ? μ ) being the two weights respectively. It is interesting to note that in this situation, the value of k is affected by the relative positions (as measured by the mean age of fertility) of the two standardized fertility curves g1 ( x) and g 2 ( x) , but not affected by the shapes of the two curves.
Case (3): u1 ( x) is a quadratic function of age x. This is equivalent to taking n = 2 in equation (18), i.e. u1 ( x) = λ0 + λ1 ? x + λ2 ? x 2 , where λ2 ≠ 0 . In this case we have
k = λ1 ? ( μ g1 ? μ g 2 ) + λ2 ? [( μ g1 ) 2 + v g1 ] ? [( μ g 2 ) 2 + v g 2 ]
{
}
(24)
Equation (24) shows that, in this situation, the value of k is not only affected by the relative positions (as measured by the mean age of fertility) of the two standardized fertility curves
g1 ( x) and g 2 ( x) , but also affected by the shapes of the two curves (as measured by the
variance). It is obvious that if the two standardized fertility curves g1 ( x) and g 2 ( x) have the same mean (denoted as μ ) and the same variance (denoted as v), then k = 0 . The TFR is therefore a weighted average of TFR1 and TFR2 , with k1 = λ0 + λ1 ? μ + λ2 ? ( μ 2 + v) and
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
k2 = 1 ? [λ0 + λ1 ? μ + λ2 ? ( μ 2 + v)] being the two weights respectively.
If μ g1 = μ g 2 , then from equation (24), we have k = λ2 ? (v g1 ? v g 2 ) . In this case, k = 0 is equivalent to v g1 = v g 2 . If μ g1 ≠ μ g 2 , then k = 0 is equivalent to
[( μ g1 ) 2 + v g1 ] ? [( μ g 2 ) 2 + v g 2 ] λ =? 1 . μ g1 ? μ g 2 λ2
Next, we establish the general criteria regarding the relationship between TFR and [ TFR1 and TFR2 ]. From equation (11), we can obtain the following:
(a) TFR1 < TFR < TFR2 is equivalent to
? k2 1 ? k2 ? ? k1 TFR1 TFR2 1 ? k1 ? ? ? < min? , or > max? , ? ? ? TFR2 k1 ? TFR1 k2 ? ? 1 ? k1 ? 1 ? k2 ?
(25)
(b) TFR = TFR1 = TFR2 is equivalent to
TFR1 = TFR2 and k1 + k2 = 1
(26)
(c) TFR > TFR1 and TFR > TFR2 is equivalent to 1 ? k 2 TFR1 k < < 2 k1 TFR2 1 ? k1 (27)
(d) TFR < TFR1 and TFR < TFR2 is equivalent to k2 TFR1 1 ? k2 < < 1 ? k1 TFR2 k1 (28)
From (c) and (d) above, we notice that, theoretically speaking, it is possible that the TFR is larger or smaller than both TFR1 and TFR2 . This seems to be a “paradox”. Regarding Simpson’s paradox in demography, Cohen (1986) has made a comprehensive analysis of the commonly used crude rates in demography, e.g. the crude death rate ( CDR ).
Here, we need to add one point regarding the relationship between TFR and [ TFR1 and
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
TFR2 ]. For convenience of discussion, we suppose that TFR1 < TFR2 .
When k1 + k2 > 1 , from equation (11), we have TFR > (k1 + k2 ) ? TFR1 > TFR1 . But it cannot be guaranteed that there is also TFR > TFR2 . For example, if TFR1 = 2 , TFR2 = 5 , k1 = 0.8 and k2 = 0.4 , then we have k1 + k2 = 1.2 > 1 and TFR = 3.6 . In this case, the TFR falls between TFR1 and TFR2 .
When k1 + k2 < 1 , from equation (11), we have TFR < (k1 + k2 ) ? TFR2 < TFR2 . But it cannot be guaranteed that there is also TFR < TFR1 . For example, if TFR1 = 2 , TFR2 = 5 , k1 = 0.3 and k2 = 0.5 , then we have k1 + k2 = 0.8 < 1 and TFR = 3.1 . In this case, the TFR falls between TFR1 and TFR2 .
From the discussions above, we know that even if coefficients k1 and k2 do not constitute a pair of weights (i.e. k1 + k2 ≠ 1 ), it is still possible that TFR falls between TFR1 and
TFR2 . It is obvious that when k1 + k2 = 1 , TFR always falls between TFR1 and TFR2 .
4. A real case with regard to the relationship between TFR and [ TFR1 and TFR2 ]
Now, let’s look at a real case vis-à-vis the “paradox”. The data used are from the 1% Population Sampling Survey of China 1987, which was conducted by the National Bureau of Statistics of China. The total population here is the population of Shanghai Municipality and the two sub-populations are urban Shanghai and rural Shanghai. Table 3 shows the numerical values of the total fertility rates of Shanghai in 1986.
Table 3. The total fertility rate of Shanghai, China, 1986
Shanghai Urban Shanghai Rural Shanghai
(TFR )
1.371
(TFR1 )
1.255
(TFR2 )
1.356
It is obvious from Table 3 that the 1986 total fertility rate of Shanghai (as a whole) was larger than the total fertility rates of both urban and rural shanghai. Table 4 shows the corresponding numerical values of the related factors.
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
Table 4. Numerical values of related factors, Shanghai, China, 1986
k1
0.657
k2
0.403
k1 + k2
1.060
k
0.060
TFR1 TFR2
0.926
1 ? k2 k1
0.908
k2 1 ? k1
1.176
This is a case that meets criterion (c) of section 3, i.e.
1 ? k 2 TFR1 k < < 2 . k1 TFR2 1 ? k1
Now, let’s take a look at the age patterns of fertility of Shanghai in 1986 (Figure 4). The two (period) fertility curves of urban Shanghai and rural Shanghai intersect at around age 25. An interesting phenomenon from Figure 4 is that the age patterns of fertility of urban and rural Shanghai are both single-peaked curves, while the age pattern of fertility of Shanghai (as a whole) has two main peaks, which correspond to the rural peak (at age 23) and the urban peak (at age 26) respectively. The fertility curve of Shanghai (as a whole) has a local valley at ages 24 and 25.
Figure 4. The age patterns of fertility of Shanghai, China, 1986
Shanghai Rural Shanghai Urban Shanghai
The means and the standard deviations of the age at childbearing are as follows:
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
Table 5. Mean age at childbearing and standard deviation, 1986
Shanghai Mean age of childbearing Standard deviation 26.20 3.66 Urban Shanghai 27.65 3.37 Rural Shanghai 24.35 3.19
Figure 5 shows the relative age structures of urban and rural Shanghai in 1986.
Figure 5. Relative age structures of urban and rural Shanghai, China, 1986
Rural Shanghai -
u2 ( x )
Urban Shanghai -
u1 ( x)
Age
5. Summary
In demography, the total fertility rate ( TFR ) is a very important measure of period fertility. In this paper, we analyzed the relationship between the total fertility rate of a total population and the total fertility rates of two sub-populations. The analysis shows that the relationship is complex and a “paradox” may occur in the relationship. Although we have only discussed a scenario of dividing a total population into two sub-populations, the results of this paper should apply to situations with multiple sub-populations.
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On a Special Property of the Total Fertility Rate – by Lu Lei (China)
References
Bogue, Donald J.; Arriaga, Eduardo E.; and Anderton, Douglas L. (1993). Reading in Population Research methodology, Volume 3, Fertility Research. Chicago: United Nations Population Fund and Social Development Center. Bongaarts, John; and Feeney, Griffith. (1998). “On the Quantum and Tempo of Fertility”. Population and Development Review, Vol. 24, No. 2. Cohen, Joel E. (1986). “An Uncertainty Principle in Demography and the Unisex Issue.” The American Statistician, Vol. 40, No. 1. Ní Bhrolcháin, Máire. (1992). “Period Paramount? A Critique of the Cohort Approach to Fertility”. Population and Development Review, Vol. 18, No. 4. Ní Bhrolcháin, Máire. (2007). “Why we measure period fertility”. Applications & Policy Working Paper A07/04, Southampton Statistical Sciences Research Institute, University of Southampton. Shryock, Henry S; Jacob S. Siegel and Associates. (1980). The Methods and Materials of Demography. US Bureau of the Census.
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