Sociology 433
2009-05-07
Samuel Clark
Problem Set 4
Fertility & Reproduction
Complete the assignment in an Excel spreadsheet and Word file. Email the
completed files as attachments to soc433@samclark.net by 5:00pm May 22nd.
Name your files using the following convention: <first name>.<last
name>.problemset4.<date sent>.<file extension>.
You will find Chapter 5 in Preston et al. very useful when completing this
assignment. When answering the questions on the biometric model of fertility use
the model I presented in class NOT the model described in the book.
Fertility
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
How is the term “fertility” used by Demographers; what does it mean and
what family of concepts are included in the study of fertility. [Don’t use too
many words; two or three sentences will suffice!]
Define:
a. Fecund,
b. Fertile,
c. Fecundability,
d. Primary sterility, and
e. Secondary sterility.
What are some of the reasons why the study of fertility is more complex
than the study of mortality? Again, use as few words as possible to
communicate your points.
Provide two representations of the crude birth rate (CBR); one as a simple
ratio, and one as a sum of two components. Your answers should take the
form of equations.
Is it possible to age standardize the crude birth rate; why is this not often
done?
What is the “general fertility rate” (GFR); write the equation that defines it.
How is the GFR related to the CBR? Write the equation that relates them.
What is an “age-specific fertility rate”? Write the equation that defines it.
What is the intuition behind an age specific fertility rate; define it briefly in
words. What results from exposing a woman to an age-specific fertility rate
for five years?
Define the “total fertility rate” (TFR):
a. In words (include your own intuitive understanding of what it is), and
b. By writing an equation.
What are typical values for the TFR?
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12.
13.
What is the “total marital fertility rate” (TMFR) and why would we want to
calculate it?
Using the data in Table 1 calculate:
a. The crude birth rate (CBR),
b. The general fertility rate (GFR),
c. Demonstrate the validity of your answer to question 7,
d. Age-specific fertility rates,
e. Age-specific marital fertility rates,
f. The total fertility rate (TFR), and
g. The total marital fertility rate (TMFR).
Table 1: Assignment 4 Data
Age
Person Years
Female
Male
0
1-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
45-49
50-54
55-59
60-64
65-69
70+
7,525
36,815
34,140
26,123
19,760
14,266
10,352
6,732
5,299
4,830
4,380
3,936
2,949
2,067
1,208
607
7,947
41,508
39,629
31,875
25,446
18,961
13,189
9,602
7,878
7,306
6,728
5,883
4,421
3,084
1,857
984
Total
180,989
226,300
Married Female
All
Births
Married
7,888
8,590
8,217
6,722
6,303
6,210
6,055
2,399
4,518
4,130
2,591
1,682
635
117
480
3,841
3,717
2,461
1,514
604
116
49,984
16,072
12,732
1. As used by demographers, fertility refers to the risk of producing a live birth. It
accounts for the biological components of fertility, that is the biological ability to
give birth, or fecundity. It also accounts for social behavior designed to reduce
or increase the chance of birth, known as contraceptive and proceptive
behaviors. Finally it accounts for abortions, or births terminated either from
intentional or unintentional causes.
2.
a. Fecund refers to the biological capability of a woman to conceive and
give birth.
b. Fertile refers to a woman who is actually bearing children.
c. Fecundability is the probability that a fecundable woman (a woman
who is capable of conceiving, but not necessarily giving birth) will
conceive during a given month.
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d. Primary sterility refers to a woman or man’s state of being
permanently unable to conceive (or sterile) throughout one’s entire
life.
e. Secondary sterility refers to the state of being sterile acquired during
the reproductive years.
3. Fertility is more complex than the study of mortality because fertility can
happen more than once to a single member of the population, not all members
of the population or even the female population are at risk of fertility, among
fecund women, there may be periods of infecunability, that is a woman’s risk of
fertility is not constant throughout an age-span, and fertility relies heavily on
other social behaviors such as intercourse and contraception, which may or may
not rely on behaviors such as marriage.
4. As a ratio:
CBR[0, T ] 
B[0, T ]
PY [0, T ]
where B[0,T] is the number of births in a population between time 0 and time T
and PY[0,T] is the number of person-years lived by the entire population during
the same interval.
As the sum of two components:
I
I
CBR[0, T ]   Fi  C  
S
i
i 1
where
i 1
BiF
PYi
F
 CiS
BiF is the number of births to women in age interval i and PYi F is the
S
number of person years lived by the women in age interval i, and C i is the
proportion of the total population (male and female) comprised of women in the
age interval i.
5. Yes, crude birth rate can be standardized between several populations in the
S
same way as crude death rate, by choosing standard values for C i . However
this is uncommon because such standardization introduces an element of
arbitrariness into the already complex situation especially since the rate will be
heavily susceptible to gender imbalances or large numbers of young and old
women, whereas other measures such as TFR present neater interpretations
and do not account for these infecund populations.
6. The general fertility rate (GFR) is the fertility rate of women of child bearing age
(typically 15 to 50) and is expressed by the equation:
B15F [0, T ]
GFR[0, T ] 
F
35 PY15 [0, T ]
35
Page 3 of 8
Where
35
35
B15F [0, T ] is the number of births to females aged 15 to 50 and
PY15F [0, T ] is the number of person years lived by females aged 15 to 50.
7. Assuming no births occur outside ages 15 to 50, B[0, T ] for the whole
population should equal
35
B15F [0, T ] defined above. Thus GFR is simply a
refinement of the ‘exposure’ element of the rate, that is the person years lived
by those at risk giving birth. Accordingly,
CBR[0, T ]  GFR[0, T ] 35 C15F [0, T ]
where
35
C15F [0, T ] is the proportion of women aged 15 to 50 out the total
population (males and females) during the interval 0 to T.
8. An age-specific fertility rate is the fertility rate for only women in a given age
interval, given by the equation:
n Fx [0, T ] 
with
35
B15F [0, T ] and
35
BxF [0, T ]
F
n PYx [0, T ]
n
PY15F [0, T ] analogous to above for the age interval age x to
age x+n.
9. An age-specific fertility rate is the number of births per year an average woman
surviving from age x to x+n would have each year during that age interval.
Thus, the number of births a woman would most likely have during the five year
age interval (x,x+5) is 5  n Fx [0, T ] .
10.
a. The total fertility rate (TFR) is the total number of births expected by
a women surviving through her reproductive span if she was
subjected to the age-specific fertility rates of that population. Thus,
it is the sum of the age specific fertility rates at each age (the
integral of the fertility density function), or discretely the sum of the
age-specific fertility rate over an interval multiplied by the length of
the interval for age intervals covering ages α to β (making the
assumption that no births occur outside this interval).
b. The equation for TFR (defined discretely) is:
 n
TFR[0, T ]  n   n Fx [0, T ]
x 
11. Typical values for TFR range from around 1.2 to 7 or 8, with a TFR of around
2.1 necessary for replacement in a population with low mortality. Low TFRs
(below 2.0) are found much of Western Europe and Eastern Asia and usually
indicate heavy use of effective contraceptive measures. Especially high TFRs
(above 4.0) are found in most of Africa. High TFRs are needed to replace the
Page 4 of 8
population if mortality at young and middle ages is high. TFRs for the rest of
the world range between these extremes usually reflecting differences
contraceptive habits.
12. Total marital fertility rate (TMFR) is the number of births expected by a woman
who is married throughout her reproductive span and experiences the period
age-specific fertility rates of married woman in that population at each age
interval. This is useful to calculate because it shows the contribution of
marriage to child-bearing. In most societies, married women are much more
likely to give birth than non-married women, also this varies greatly, and this
effect is diminishing in societies where out-of-wedlock births are becoming more
common and socially acceptable, such as Northwestern European countries.
13.
CBR 
16,072
 0.039462
180,989+226,300
See sam.clark.problemset4.2006-05-11.xls
Reproduction
14.
15.
16.
17.
18.
19.
20.
How is the term “reproduction” used by Demographers; what does it mean
and what family of concepts are included in the study of reproduction. [My
favorite refrain  – don’t use too many words; two or three sentences will
suffice!]
Define the gross reproduction rate (GRR) using both words and an equation.
Define the net reproduction rate (NRR) using both words and an equation
and compare it to the GRR, what important component is added to create
the NRR?
What do the following NRR values indicate about reproduction of a
population:
a. NRR < 1,
b. NRR = 1, and
c. NRR > 1.
How is the NRR related to the GRR; write an equation and explain it.
How is the TFR related to the GRR and NRR; write equations and explain
them.
Assuming NRR = 1, what is the relationship between the TFR and p(A m)?
Explain how increases in mortality work through this relationship to dictate
the TFR necessary to maintain replacement reproduction.
14. The term reproduction is used by demographers to describe the process by
which a population replaces itself, thus reproduction accounts for both the
fertility and mortality and is analogous to the natural growth component of the
balancing equation.
15. The gross reproductive rate (GRR) calculates the total number of female births
born to a woman who lives through her entire reproductive span and is
subjected to the age-specific female birth rates of that population. It is
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calculated analogously to the TFR substituting age-specific female-birth fertility
rates for overall age-specific fertility rates, given by the equation:
 n
GRR[0, T ]  n   n FxF [0, T ]
x 
where
F
n
Fx [0, T ] is the age-specific female birth rate for ages (x,x+n), or the
number of female births per year by a woman in the age interval.
16. The net reproductive rate (NRR) is the number of female births a woman can
expect to have if she is subjected to the period age-specific female fertility rates
at each age between α and β and the age-specific female mortality rates at each
age between 0 and β. It is calculated as the sum of the products of the agespecific female fertility rates and the average number of person years lived by a
woman born into the cohort during the age interval (x,x+n):
 n
L
NRR[0, T ]   n FxF [0, T ]  n x
l0
x 
Where GRR assumes that all women born into a cohort survive through their
reproductive span, the NRR accounts that some women die before their
reproductive span or during their reproductive span, thus giving an accurate
number of the female births that can be expected by a woman born into a
cohort. Under this assumption made by GRR that
l0  lx  n ,
Lx n  l0

 n , so
l0
l0
n
under this (unrealistic) assumption NRR=GRR.
17.
a. NRR<1 indicates that on average each women born into a cohort
bears less than one female into the next cohort, thus the next female
cohort will be smaller than the preceding.
b. NRR=1 indicates that on average each woman born into a cohort
bears exactly one female birth into the next cohort, thus the next
female cohort will be exactly the same size as the preceding
c. NRR>1 indicates that on average each woman born into a cohort
bears more than one female into the next cohort, thus the next
female cohort will be larger than the preceding.
18. NRR is related GRR by the equation:
NRR
where
p( Am )  GRR
p( Am ) is the probability of surviving to the mean age at birth (Am). This
approximation will be equality if the probability of a newborn dying during a
given age interval is constant during the reproductive period (indicating a linear
survivorship), since then the mean age at birth is equal to the average number
alive at each age weighted by age-specific maternity. In words, this means that
the net reproductive rate is approximately equal to the gross reproductive rate
times the probability of surviving to the mean age at birth, since this will
Page 6 of 8
decrease the GRR by the number of births that would occur for the average
woman after she dies.
19.
TFR  (1  SRB )  GRR 
(1  SRB)
 NRR
p ( Am )
where SRB is the sex ratio at birth (males/females). First, the second equality
is derived by assuming the equality of the approximation given in 18 and
dividing by p(Am). The first equality is because the TFR is the number of
female births expected per woman plus the number of male births expected per
woman. The expected number of male births is the ratio of males to females
born multiplied by the number of female births expected. Factoring this
appropriately gives the desired equality.
20. Assuming NRR = 1, a perfect replacement rate, gives TFR 
(1  SRB )
.
p ( Am )
Assuming that the sex ratio at birth remains constant as female mortality
varies, as the probability of surviving to the mean age at birth increases, TFR
decreases, and vice versa. If mortality increases for ages less than p(Am),
p(Am) will decrease, so according to the relationship above, TFR must increase
to maintain a NRR = 1, or replacement reproduction.
Biometric Model of Fertility
21.
22.
23.
Define L, p, e, S, K and S* and write the TFR as a function of these
quantities.
In your own (few!) words describe what the equation you wrote to answer
question 21 means; how does it work, how would changes in each
parameter affect the TFR?
Assuming L=240, p=0.2, S=22, K=0.1, and S*=4, draw a plot of the TFR as
a function of e where the value of e runs from 0 to 0.999. Change the value
of K to 5.0 (an average of 5 miscarriages per live birth) and draw another
line on your plot. Compare these two lines and try to explain the differences
you see.
21.
TFR 
L


1
1
 S  K 
 S *
  (1  e)
   (1  e)

where L is the average lengths of a woman’s reproductive span (in months), ρ is
the probability of a fecund woman conceiving in a given month, e is the
effectiveness of a contraceptive methods being employed, S is the number of
months of sterility associated with a live birth, K is the ratio of conceptions
leading to abortion to those leading to live births, and S* is the average number
of months of sterility associated with a conception leading to abortion.
Page 7 of 8
22. The above equation asserts that the number of births to a woman is the number
of months is of her reproductive span divided by the number of months between
births. It further distinguishes the interval between births into the period of
sterility resulting from conception and the waiting period between conceptions.
Finally it asserts that the waiting period between conceptions is the inverse of
the probability of conceiving in a month, which decreases proportional to the
effectiveness of contraceptive methods. It accounts for the variability of period
of sterility S associated with a live birth and an abortion by adding this period
and the probability of an abortion. While biologically unrealistic, increasing L
would increase TFR. As noted, decreasing ρ or increasing e would decrease
TFR. Finally increasing S or S* would decrease TFR, likewise increasing K would
decrease TFR.
23.
10
9
8
7
TFR
6
K=.1
K=4
5
4
3
2
1
0.99
0.96
0.9
0.93
0.87
0.84
0.81
0.78
0.75
0.72
0.69
0.66
0.6
0.63
0.57
0.54
0.51
0.48
0.45
0.42
0.39
0.36
0.3
0.33
0.27
0.24
0.21
0.18
0.15
0.12
0.09
0.06
0
0.03
0
e
We see that both lines decrease to zero as e goes to 1, that is as contraceptive
methods become 100 % effective, the number of births go to zero. However,
the TFR is much higher in the K=.1 situation than the K=4 situation for all
values of e. This asserts that with many more abortions per live birth the
number live births per woman would be less. This makes sense, because with
of the probability of conception (weighted by contraceptive efficacy) held
constant, more conceptions aborted would imply that less go to live births.
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