A STOCHASTIC MODEL FOR THE STUDY
OF HUMUU~ FERTILITY
By
El-Sayed El-Sayed Nour
Department of Biostatistics
University of North Carolina
Chapel Hill, North Carolina
Institute of Statistics Mimeo Series No. 879
AUGUST 1973
•
A STOCHASTIC MODEL FOR THE STUDY
OF HUMAN FERTILITY
by
EI-Sayed EI-Sayed Nour
A dissertation submitted to the faculty of
the University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in the Department of Biostatistics.
•
•
Chapel Hill
1973
•
..
•
EL-SAYED EL-SAYED NOUR. A Stochasttc Model for the Study of Human
Fertility. (Under the direction of H. BRADLEY WELLS.)
Human fertility is described as a continuous time (age) discrete
state (parity) stochastic process.
Explicit general formulas are
derived for the transition probabilities which are used in establishing
recurrence relationships concerning the moments of the age-parity
distribution in both time-age marriage cohorts and time marriage cohorts.
A discrete approximation for the probability distribution of
the age at delivery of the ith live birth is also presented.
A numerical analysis based upon the age-parity distribution is
conducted to illustrate the dynamics of the fertility process.
The
estimation problem in the model is also discussed.
Finally, the model is used in evaluating the effect of family
planning on fertility.
Indices based on the number of births averted
by family planning programs are considered and formulas useful in
deriving its moments are given .
•
•
. ACKNOWLEDGEMENTS
I am greatly indebted to the late Professor Mindel C. Sheps who
stimulated my initial interest in working on this type of problem, and
who gave valuable assistance and council during the early stages of
the work.
Special thanks are due to my advisor, Professor H. Bradley Wells
not only for the interest he consistently showed in the progress of
this research but also for all the knowledge I gained simply by working
with him.
I would like to thank Professors Peter A.
Lachenbruch, Joan
Lingner,
Dana Quade, C. M. Suchindran, and Edward J. Wegman the other
(
members of my doctoral committee.
They were extremely cooperative and
provided encouraging and helpful suggestions that definitely expedited
the completion of the manuscript.
I wish to thank Ms. Carlie Carter for her assistance in writing
the computer programs for this study.
The financial support of the Population Council of New York is
greatly acknowledged.
Finally, lowe a very special debt of gratitude to Ms. Martyvonne
Morton whose support and encouragement has been immeasurable not only
in proofreading and typing this paper but also in her persistent effort
to make my stay in this country more pleasant.
•
rABLE OF CONTENTS
Page
LIST OF TABLES
.........................
vi
Chapter
INTRODUCTION AND LITERATURE REVIEW
1.1,
1.2.
1.3.
1. 4.
II.
1
Introduction. . . • • . • • • •
Models with the Demographic Approach .•
Models with the Biological Approach •
.•..
Objectives of the Study
•••..•••
1
3
5
12
15
DERIVATION OF THE MODEL • .
Glossary of Terms • . • • . . . • . . • • •
Assumptions and Definitions •
. • • •
Formulation of the Fertility Process in a
Time-Age Marriage Cohort • • • • • • • • • ••
Age-Parity Distribution in Time-Age Marriage
15
16
Cohorts. . . . . . . . . . . . . . . . . . . .
34
21
Age-Parity Distribution in Time Marriage
2.7.
III.
Cohorts. . . . . . . . . . . . . . . . . . . .
41
Fertility Formulation in Heterogeneous Cohorts.
2.6.1. Model A, Heterogeneous Birth
Probabilities. • • • • . . .
2.6.2. Model B, Heterogeneous Force of
Conception . • • • • • . . . . • • •.
Fertility in Cross Sectional Populations of
Married Women. • . • • • • • . • • • • • •
44
NUMERICAL INVESTIGATION OF EllMAN FERrILITY.
....
at Age :x; •
..
f
•
.,
•
•
•
•
•
•
t
t
•
•
•
,
•
•
•
50
•
•
47
50
Q (x) . . . . . . . . . . . , . . .
R~(X)~ the Unconditional Fecundability
J at Age x . , .
46
50
Choi.ce of Speci.fic Functi.ons • • • . • • • •
3.1.1. P(X)7 the Conditional Fecundability
3.1.2.
3.1.3 •
45
•
52
•
•
55
0(x), the Incidence of Fetal Loss
•
Associated with a Conception
at Age x . . . . . . . . . . .
....
55
v
Page
Chapter
x o' the Age at Marriage in a Time-Age
Marriage Cohort • • . • • • . • • • •
3.1,6. M(x), the Expected Number of Women
.Who Marry over the Age Month
(x, x+l). . . . . . . . . . . . . . .
3.1.7. q(x,x+l), the Probability of Dying over
the Age Month (x, x+1) for Women who
Are Alive at x. • • • • • • • • • • •
Techniques of the Numerical Analysis • • • • • •
3.2.1. Sequence of the Numerical Calculations •
3.2.2. Types of Output Considered •.
Numerical Analysis . • . . • • •
3.3.1. An Initial Example • • . • • • •
3.3.2. The Function p(x) . . •
3.3.3. The Function Qi(x) . • • • • • • • •
3.3.4. The Function q1x,x+l) • • . • •
3.3.5. The Age at Marriage Xo • • • • • •
3.3.6. The Function M(x). .
• ~ •
3.3.7. Conclusions • . . . . • .
3.1.5.
3.2.
3.3.
IV.
V.
56
58
59
59
61
62
62
66
73
81
86
97
108
A PRELIMINARY STUDY OF THE ESTIMATION PROBLEM ••
110
4.1.
4.2.
110
114
117
120
Estimation in Homogeneous Cohorts . • •
Estimation in Heterogeneous Cohorts ••
4.2.1. Estimation under Model A .
4.2.2. Estimation under Model B .•
THE NUMBER OF BIRTHS AVERTED BY FAMILY PLANNING
PROGRAMS • • • • • •
• • • • • • • •
·····
· ·· · · · · ·· · · · · ·
.
·
···· ·······
5.3.
··
·
·· · ··
·· ·· · · ·
SUMMARY Am> PLAAS :FOR FURrHER RESEARCH • · . . . · . .
St.UDnlary. • • , •
• • • • •
Plans for :Further Re~earch • . • • •
• •
6.2.1. Further Study of the Model.
,
6.2.2. Generalizations of the Model • •
t
f
•
•
•
•
•
..
•
· ..
BIBLIOGRAPHY.
•
t
•
•
....., ....
e
123
Introduction
Births Averted in Time-Age Marriage Cohorts.
5.2.1. Notation and Definitions
5.2.2. Derivations.
5.2.3. Discussion
Births Averted over a Specified Time Period
in Populations of Married Women
•
5.3.1. Notation and Definitions
• •
•
5.3.2. Derivat;i.ons.
•
•
•
•
•
5.1.
5.2.
VI.
56
t
•
•
•
123
126
126
127
131
138
139
141
146
146
149
149
151
.
153
..
LIST OF TABLES
Table
3.1.
Page
Area under the Probability Distribution of Xj ,
j=1,2, ••• ,10 for Different Values of the Age at
Marriage
3.2.
X
o . . . . . . . . • . . . . . . . . .
Area under the Probability Distribution of X~,
j=1,2, .•• ,10 for Different Patterns of the Age
at Marriage . . . .
..
•
88
.
99
4.1.
Estimates of Po o(x,x+l) and ¢(x)=({1-8(x)}R o (x»)
and Their Standard Error from a Sample of
Hutterite Women • • . • . • . . . • . • . . . . . . • 115
5.1.
Expected Number of Births Averted for Different
Combinations of £ and E{y(x)}, (Maximum
Fecundability=.ll) . . . . . • • • . . . . • • . • • • 136
5.2.
Expected Number of Births Averted for Different
Combinations of £ and E{Y(x)}, (Maximum
Fecundab iIi ty= •05). . . . . • • . . . . . . . . . . . 137
CHAPTER I
INTRODUCTION AND LITERATURE REVIEW
1.1
.
Introduction
Fertility as the positive force in the vital process is princi-
pally responsible for the current rapid acceleration in the growth of
human populations.
Implementation of family planning programs for the
purpose of altering fertility behavior is becoming a popular course of
population policy.
Awareness of this situation has led to the frequent
utilization of the tools of mathematics and statistics to investigate
the fertility process.
A number of mathematical models has been
proposed for that purpose.
fold.
The main interest in these models is two-
The first is to develop techniques, as well as schemes of
data collection, necessary to detect and evaluate the significance of
changes that take place in fertility.
The second is to enhance the
understanding of the fertility process by identifying the nature of
the interactions of various factors and fertility.
Two approaches to construction of models for human fertility can
be distinguished.
The first is the demographic approach which is based
on the conventional methods and habits of thought in population analysis.
Birth probabilities, which may be conditional on a variety of
.
selected characteristics, are the means of fertility analysis in the
demographic
appr~ach.
Such probabilities are usually derived without
considering the details of the biological basis of the process of human
.
2
reproduction.
Models using the demographic approach are usually
geared to the type of data which have long been available from civil
registration, censuses, and surveys.
The second approach to fertility
models, a relatively new one, is referred to as the biological approach.
This approach rests on the argument that social, economic, and psychological factors operate on' fertility through intermediate variables of
a biological nature.
Any changes in fertility patterns will be re-
flected in these biological factors before their effects become
apparent in fertility behavior.
Taking these intermediate variables
as the basis of the fertility models with the biological approach,
the theory underlying these models is developed by decomposing an
interval between successive births into its components .. Indices of
fertility change resulting from this procedure are receiving the Rarticu1ar attention of demographers.
This attention is justified by the
potentials of these indices in evaluating family planning programs
since such programs are expected to alter the distributions of intervals between births through contraception, sterilization, or induced
abortion.
The overall objective of this study is to develop a stochastic
model for the study of the age parity distribution, a demographic variable, using the biological approach to fertility.
The specific objec-
tives of the study are outlined later in section 1.4 after the current
status in fertility models is reviewed.
Section 1.2 reviews analytic
(as opposed to computer simulation) models that deal with fertility
within the framework of the demographic approach.
Analytic models with
the biological approach are reviewed in section 1.3.
3
1.2
---
~odels
t
with the demographic approach
I
_
~.
The stable population model, which was originally proposed for the
purpose of studying the dynamics of population growth, deals with fertility within the framework of the demographic approach.
This model
generally considers one sex and assumes time-homogeneous probabilities
of birth and death.
These probabilities are treated in a deterministic
fashion and are usually used in the sense of occurrence/exposure rates.
The stable population model has been used in fertility analysis in two
different directions. Firstly, it has been used as an effective tool
for studying the implications of the relationships between fertility,
mortality, and age distribution in a given population (Coale 1965, 1972
and Keyfitz 1968).
Secondly, it has been used as an assumed age struc-
ture in the analysis of fertility in cross sectional populations
(Sheps and Menken 1970, 1972).
Brass (1970) proposes a fertility model in which the reproductive
span of a woman is divided into a set of N unit intervals within each
of which only one live birth can occur with probability p.
Under the
assumptions of a binomial distribution, he describes the fertility
process in terms of Nand p.
Apart from the fact that the assumptions
underlying the fertility process are basically different from those of
a binomial distribution, the choice of N is arbitrary and practically
difficult.
The stochastic treatment of human fertility within the framework
of the competing risks model is another example of models employing
the demographic approach to fertility.
The competing risks model, also
known as the multiple decrement model, is theoretically appealing in
the analysis of duration variables.
In this model a woman starts in
an initial state
° at time 1:=0.
4
She is observed to leave that state
due to one of k causes of decrement.
a continuous function
j
..
W. ('()
J
at time '(, j=1,2, •.. ,k.
ro each cause, there corresponds
denoting the force of decrement for cause
The probability structure underlying the
competing risks model is that of a time continuous Markov chain with
one (or more) transient states (one of which is the initial state) and
k absorbing states (corresponding to k causes of decrement).
The
forces of decrement w ('() are the infinitesimal transition probabilj
ities of the process.
In this model the histories of the individuals
under study are regarded as independent sample paths.
The general form
of this model applied to the fertility process is to consider the initial state as denoting parity i, i=O,l, ••. , and 1:=0 to indicate the
time of delivery of the !th live birth.
The causes of decrement will
be either an (i+l)th birth or other causes such as death, widowhood,
sterility, or divorce.
Formulas for various probabilities of leaving
parity i to parity i+l are derived.
A total description of the fertil-
ity process, in terms of these probabilities, results when considering
all values of i=O,l, .•. simultaneously.
Fertility models proposed by Hoem (1968, 1970) and Sheps and Menken
(1972) are examples of one variation or another of this general description.
The formulas for various birth probabilities given in these mod-
e1s are in a perfectly general form.
To estimate these birth
probabilities, however, the use of these general forms requires a parametric representation of the forces of decrement.
Hoem (1968) in
dealing with this problem confines his discussion to the special case
where the individual's stay in parity i has a negative exponential distribution.
His estimation technique is based upon the use of aggregated
5
lifetime as operational time and the resulting estimators are the usual
occurrence/exposure birth rates.
Chiang (1968), on the other hand,
avoids the problem by assuming an underlying multinomial distribution
which, in turn, yields estimators in the form of occurrence/exposure
rates.
Occurrence/exposure birth rates have long been a standard technique of fertility analysis.
The use of these birth rates relies on
the argument that the fertility process is adequately represented by
its birth aspect without getting into the details of human reproduction
with conceptions t pregnancy outcomes, and infecundable periods.
This
underlying argument creates problems in defining the population at risk
at a given point in time and causes these rates to be insensitive to
short term changes in fertility patterns.
ConsequentlYt fertility rates
might not reflect adequately small short term changes in fertility due
to deliberate intervention in the reproductive process and the question
arises, therefore, as to whether more sensitive measures can be found.
1.3
Models using the biological approach
The nature of human reproduction suggests four basic variables that
should be considered for inclusion in fertility models using the biological approach.
1.
These variables are discussed below.
The effective reproductive period i.e.
the age span during
which reproduction is possible, is the first variable considered.
This
duration begins at a woman's time of marriage or the age of menarche
whichever occurs last, and ends at the time of her death, sterility, divorce, widowhood t or menopause whichever occurs first.
Apart from the
fact that women differ with respect to their effective duration of reproduction, the finite nature of the reproductive period causes what is
6
known in fertility analysis as the "truncation effect."
Because of
chance factors, women with the same duration for reproduction would
vary in parity.
A finite duration for reproduction implies that those
who achieve higher parity would necessarily tend to have shorter and
less variable birth intervals than those of lower parity.
·if
As a re-
sult of this situation women with longer birth intervals do not contribute higher confinements.
In other words, the observed fertility
behavior is a truncated sample of the behavior that would have been observed if the duration of reproduction were infinite.
2.
The time required for a woman who is susceptible, and at the
risk of conception, to conceive is also considered.
Treating con-
ception as a random variable implies that the time required to conceive
depends on a woman's fecundability.
Fecundability, as originally
defined by Gini (1924), is the monthly probability of conception outside
the gestation period and the temporary sterile period following the termination of pregnancy.
So defined, fecundability depends on many bio-
logical and behavioral factors that cause it to differ among women and
for the same woman under changing circumstances.
Considering the nature
of the fertility process and our inability to allow for the effects of
all different factors in the process, the age effect on fecundability
can be taken as an approximation for the combined effects of all other
factors (Henry 1957, 1965).
3.
The third variable considered is the various outcomes of preg-
nancy which depend upon the chance of a conception ending in a given
outcome.
The incidence of fetal wastage (a pregnancy outcome other than
a live birth) is correlated with several factors such as age, parity,
number of preceding conceptions, social status.
While women differ in
7
their probability of fetal wastage, the mean value of this probability is expected to increase as age of woman increases.
Studies done in
Taiwan (Jain 1969) have shown that the probability of fetal wastage as
a function of age attains. its minimum value through the age group
(20, 30) and then increases rapidly until it reaches its maximum level
at the end of the reproductive period.
4.
The duration of nonsusceptible
periods~
associated with vari-
ous pregnancy outcomes during which another conception is impossible
is the last variable considered.
This duration consists of two adja-
cent periods--a pregnancy period and a postpartum nonsusceptible period.
The length of pregnancy depends on its outcome.
Pregnancies that end
in a fetal loss may last from one to ten calendar months, but the mean
duration is 2 to 3 months.
Pregnancies ending in a live birth last on
the average for 9 months, and although the range is from 6 to 10
months, the variance is relatively small.
The postpartum nonsusceptible
period lasts, in the absence of lactation, for 1 to 2 months.
In the
presence of lactation, it may be prolonged by a year or even considerably longer.
It is possible that age or the order of pregnancy are
associated with variations in this duration.
Since age and the number
of preceding conceptions have some relation to the incidence of fetal
wastage, the duration of nonsusceptible periods in general is a function
of age and parity (Sheps and Menken 1971, b).
Several probabilistic models have been proposed to study the mode
of change in these variables and its relation to fertility performance.
The early work on these models appears to have been started by Henry
(1953) who utilized the concept of fecundability as described by Gini
(1924).
Dandeker (1955) proposed modified versions of binomial and
8
Poisson distributions to describe human reproduction under simplified
assumpt~ons.
The breakthrough along this line came with the 1964
papers by Perrin and Sheps (Perrin and Sheps 1964, Sheps and Perrin
1964).
In these models, two categories of interrelated variables are
usually studied--thedistribution of the random length of intervals
•
between events such as births, conceptions, etc., and the number of
events occurring in a specified time.
Henry (1965), Sheps (1969), and
Sheps et al (1969) have reviewed the basic features of these models.
In order to establish a starting point for some of the models derived
later in this paper, we shall review the current status of the human
reproduction models within the framework of three classifying systems:
(i)
the nature of the assumptions, used in the model, concerning
the four basic variables underlying human reproduction,
(ii)
whether the model describes a closed or an open group, and
(iii) whether time is treated as discrete or continuous.
Assumptions concerning the basic variables of human reproduction
Empirical research on the basic variables of human reproduction
is very limited due to problems encountered in making direct observations on these variables.
As indicated earlier, this limited empirical
research, together with the nature of the human fertility process,
seems to suggest that women are heterogeneous with respect to these
variables.
Further, these variables could be a function of other fac-
tors such as age, parity, duration of marriage, social status, and
coital frequency.
The existing models for human reproduction can be dichotomized
according to whether or not they conform to the definition of a renewal
process.
The major portion of the literature treats human reproduction
9
as a renewal process.
Briefly) a renewal process may be described as follows:
for a sequence of occurrences of a defined event, such as
conception or birth, let Xo be the length of the interval
up to and including the time of the first event and Xi
(i~l) the length of the interval following the ith event,
up·-to and including the time of the (i+l)th event. If it
is assumed that Xi are mutually independent for all i and
have identical distributions for i?_l, the sequence conforms
to the definition of a renewal process. Generalizations of
renewal processes include more than one type of event, as
in Markov chains and in Markov renewal processes. (Sheps
et al 1969)
Renewal theory is frequently used in constructing models for human reproduction because the theory of renewal processes is well developed
and the underlying mathematical analysis is more tractable.
Henry (1953), Dandekar (1955), Basu (1955), Singh (1963, 1964),
Perrin and Sheps (1964), and Sheps and Perrin (1963) are examples of
models for human reproduction derived under the assumptions of a renewal
process.
They vary, however, in the degree of complexity postulated.
Some of these models e.g.
Dandekar (1955) and Singh (1963, assume that
each conception must end in a live birth.
Other, e.g.
Sheps and
Perrin (1964) and Henry (1953), consider a variation of pregnancy outcomes.
Some of these models, e.g.
Singh (1963), Basu (1955), and
Perrin and Sheps (1964), postulate that women under study are homogeneous with respect to the basic variables of human reproduction while
others, e.g.
Sheps (1964), Potter and Parker (1964), Singh (1964), and
Brass (1958), introduce heterogeneity among women by assuming that a
woman is assigned a set of values for the basic variables according to
a specified probability distribution.
A well known example is the use
of a beta distribution to describe the distribution of fecundability
(Singh 1964, Potter and Parker 1964).
10
11
The
distinct~on
between closed and open groups described by human
reproduction models amounts to Whether the model is dealing with fertility in a cohort (closed) or a cross sectional (open) population setting.
Analytic models of reproduction are usually constructed for a
cohort of married women.
Henry (1953, 1957, 1961), Srinivasan (1966),
and Leridon (1969) are examples of the very few attempts to extend the
results to cross sectional analysis of cohorts.
The progress along
this line is very limited and covers only the special case of a stationary population.
Treatment of time
Another method of classification of the models depends on whether
time is treated as discrete or continuous.
The choice is partly a
matter of realism and partly one of convenience.
Since conception can
occur during only a small segment of the menstrual cycle and the mean
length of the cycle is about 30 days, many authors, e.g.
Dandekar
(1955), Sheps (1964), and Sheps and Menken (1971, a), have treated time
as discrete, each unit being considered equal to one month.
On the
other hand, the mathematical convenience offered by the use of continuous time has attracted other authors, e.g.
Henry (1953, 1957, 1961).
The results obtained with discrete and continuous time are not identical
but the differences have no substantial effect on conclusions.
As
Henry (1957, 1961) demonstrated, the discrete version of a model is
easier to
hand~e
in
~pplied
research where it is necessary to compute
solutions, but if the interest is in the mathematical analysis of the
model, the techniques for handling the continuous time versions are
more readily available.
12
Objectives of the study
'<
I
The above review of literature shows that existing analytic models
for human fertility are mostly derived under
~estrictive
assumptions
simplifying the aspects of the process for mathematical convenience.
As indicated earlier, the development of less restrictive and more
realistic models, though it will likely enlarge the understanding of
the fertility process, is faced by the prospect of being too complex to
be useful.
Obviously, the inevitable compromise should endeavor to
approximate the observed process in the most realistic way while maintaining the flexibility of the model as a tool of analysis.
The general objective of this study is to present a detailed analytic description of human fertility.
This description is based on the
age parity distribution as age and parity are considered to be the most
important variables associated with fertility.
Several variables asso-
ciated with the age parity distribution, namely the mean parity as a
function of age and the age at which a given birth occurs, are also considered.
Within this general objective, the specific goals of this
research include:
1) deriving birth probabilities, which are the means of analysis
in the demographic approach, in terms of the basic biological
functions underlying fertility, and studying the interrelation
between these functions,
2) studying the effect of changes in the pattern of input to the
marriage system on fertility, and
...
.
3) developing some indices useful in evaluating the efficacy of
family planning programs .
13
To achieve these goals, within a reasonable extent of mathematical flexibility, several concepts are employed.
1.
The concept of fecundability is redefined.
Natural fecunda-
bility as originally proposed by Gini (1924) is hereafter referred to
as conditional fecundabi1ity.
Conditional fecundabi1ity is defined as
the probability of conception per unit time conditional on a woman
being fecund at the beginning of that time unit.
The fact that being
fecund at any point of time is a random event, gives rise to another
version of fecundabi1ity hereafter referred to as unconditional fecundability.
Unconditional fecundability is defined as the probability of
conception per unit time.
~vo
2.
types of marriage cohorts are defined.
The first is a
time-age marriage cohort, referring to a group of women who marry at
the same age at the same time point.
The second is a time marriage
cohort, referring to a group of women who marry at the same time.
Obvi-
ously, a time marriage cohort is a weighted sum of time-age marriage
cohorts with the weights derived from the distribution of age at
marriage.
3.
An observed pattern of fertility behavior is thought of as a
sample realization of a stochastic process i.e.
as a sample of size one
from a universe of possible patterns.
4.
used.
For mathematical convenience, continuous time formulation is
A discrete approximation is outlined whenever appropriate.
The content of this study is arranged in the following chapters.
Chapter
~TO
provides the theoretical formulation of a stochastic model
for human fertility.
The assumptions underlying the model are listed
and the implications of different formulas are discussed.
In Chapter
14
Three a numerical analysis of the model is presented where effects and
interactions of different parameters are investigated.
Chapter Four
deals with the estimation problem in the model where only data from
complete samples are considered.
Chapter Five is devoted to an appli-
cation of the model to the evaluation of family planning programs.
In
this chapter the concept of births averted by family planning programs
is discussed and a numerical analysis of the dynamics of family planning
is outlined.
And finally, Chapter Six contains a summary and some
recommendations for further research.
CHAPTER II
DERIVATION OF THE MODEL
The content of this chapter will be arranged in the following
sections:
tation.
(2.1) is a glossary of terms which are used in this disser(2.2) is a list of the assumptions and definitions under
which the model is derived.
The actual derivation of the model is
presented in sections (2.3), (2.4), (2.5), and (2.6).
2.1
Glossary of terms
The following list is intended to define terms which are used in
the present study.
The terms relate to females as the convenient
notion of assigning a couple's characteristics to its female member is
adopted.
All definitions are consistent with their usual meanings
prevailing in the literature in so far as possibie.
GLOSSARY
1.
Fertility:
The actual reproductive performance of an
individual or a group of individuals.
2.
Fecundity:
The capacity of an individual to participate
in the reproductive process. The lack of this
capacity is called infecundity.
3.
Reproductive
Period:
The age span during which a woman can normally
reproduce.
4.
Terminal Events:
Events that cause a woman to become incapable
of participating in the reproductive process
e.g. sterility, death, divorce, and widowhood.
5.
Conception:
The fertilization of an ovum followed by implantation of the zygote in the uterine wall.
16
6.
Effective Reproductive Period:
The portion of the reproductive period which
begins with marriage and ends in a terminal
event. This is the period during which a
woman can conceive.
7.
Conditional
Fecundability:
The probability that a non-contracepting woman
will conceive over a unit time given that she
is fecund at the beginning of that time unit.
This probability is usually referred to in the
literature as natural fecundability.
8.
Unconditional
Fecundability:
The probability that a non-contracepting woman
will conceive over a unit time. This is a new
version of fecundability introduced in this
study and obviously is the product of natural
fecundability and the probability of being
fecund at the beginning of a time unit.
9.
Live Birth:
A confinement at which at least one live born
infant is delivered.
10.
Fetal Loss:
A pregnancy terminating in any outcome other
than a live birth.
11.
Nonsusceptible
Period:
The total duration of time during which ovulation is suppressed following a conception.
This is a period of temporary infecundity.
12.
Parity:
The number of live births a woman has had.
13.
Time l'farriage
Cohort:
A group of women who marry at the same point
in time.
14.
Time.. . Age
Marriage Cohort:
A group of women who marry at the same age at
the same time point.
2.2
As sumpt ions and . def init ions
The following lists of assumptions and definitions are meant to
provide a precise statement of the conditions under which the resulting
model is to hold.
..
..
In those cases where mathematical functions are de-
fined, these definitions implicitly assume that these functions exist
in the domain represented by a woman's life period .
ASSUMPTIONS
For any group of women the following assumptions are made:
17
1.)
The fertility histories of different members of a group are
mutually independent.
2.)
Fecundabi1ity is a function of age and parity.
3.)
The length of nonsusceptib1e periods associated with a conception
is a function of parity only and not of age.
Although age is
thought to have an effect on the length of these periods, this
effect is ignored for simplicity.
This action appears to be
justified by available results obtained by simulation (Venkatacharya
1969)
indicating that this variable (length of nonsus-
ceptib1e periods) is less important than the other variables in
determining the reproductive performance.
4.)
For any conception leading to a live birth, the associated pregnancy period is assumed constant.
5.)
Any conception ends in either a live birth or a fetal loss.
No
attempt is made to consider the different outcomes of a fetal
loss.
6.)
The incidence of fetal loss is a function of age only and not of
parity.
7.)
Terminal events are treated as a function of age.
8.)
The possibility of remarriage is not considered.
9.)
Multiple births are treated as one birth.
10.) Fertility performance can change with calendar time.
The specific
pattern of a woman's reproductive performance is related to the
calendar date of marriage.
DEFINITIONS
,
In the present formulation of human fertility, it will be necessary to consider age intervals
(x , x ).
1
2
It is assumed that such
18
intervals are always open to the left and closed to the right.
In the framework of the assumptions listed above, the following
is a list"defining the different functions considered in the present
model:
a woman's age at marriage
a woman's age at delivery of her ith live birth,
j=l,2, •.• ,a.
a woman's age at which the conception leading
to her ith live birth occurs, j=l,2, ..• ,a.
g
The pregnancy period associated with any live
birth.
p(x)~x
+
o(~x)
g = X - Y •
j
j
the conditional fecundability at age X; or the
conditional probability that a woman who is
fecund at age
interval (x,
Rj(x)~x
+
o(~x)
X
will conceive over the age
~~x).
the unconditional fecundability at age x when the
parity is j; or the probability that a woman will
conceive over her age interval (x,
x+~x)
when
her parity is j, j=O,1,2, ... ,a.
8(x)
the probability that a conception occurring at
age x will end in a fetal loss.
the length of the nonsusceptib1e period associated with the conception leading to a ith live
birth, j=1,2, .•. ,a.
the length of the nonsusceptible period associated with a conception leading to a fetal loss
for a woman whose parity is j, j=O,1,2, ••. ,a.
19
]J(x) 19c
+
o(L\x)
the conditional probability that a woman, who
is in the effective reproductive period at age
x, will experience a terminal event over the
ag~
z. (x,y)
J
interval (x, x+L\x).
the random number of consecutive fetal losses
(i.e.
uninterrupted by a live birth) over the
age interval (x,y) given that the parity at x
is j, j=0,1,2, ••• ,a and that x<y.
h. (x,y)
J,i
Pr{z. (x,y) = i},
i=0,1,2, •.• ;
J
j =0 , 1 , 2 , ..• a.
H.
J
(x,y,s)
the generating function of the probabilities
h. (x,y), which is defined as Lh
(x,y)si.
J,i
ij,i
the parity at age x for a woman whose age at
marriage is x o '
a
max max J(x/x o )'
Xo x
...
Also define the random variables S(x) and y(x) as:
1
If a woman is in the effective
reproductive state at age x
i.e. If she did not experience a
terminal event by age x;
o
otherwise.
S(X) =
y(x)
..
If a woman is passing through a
nonsusceptible period at age x
following a conception;
otherwise.
and
Q. (x)
J
Pr{S(x)=l, y(x)=O
I
parity is j},
...
j =0 ,1, 2 , ••• ,a.
Qj(x) is the probability that a woman will be
20
alive at the risk of, and susceptible to, conception at age x given that her parity is j,
j=O,l,ooo,ao
Define the following functions for a
(T, x o )
time-age marriage
cohort:
nT(xlx
)
j
0
the random number of women with parity j at age
x of those who married at age
j =0 , 1 , 2 ,
0
0
0
,
a
Xo
at time T,
0
the probability that a woman marrying at age
Xo at time T will have parity j at age x,
j=0,1,2, o. ,ao
0
the random number of women, which have parity i
at age x, who will move to parity j at age y out
of those who marry at age
Xo
at time T.
the probability that a woman with parity i
at age x will have parity j at age y, given that
she marries at age
•
q(x,y)
Xo
at calendar time To
the probability that a woman alive at age x
dies over (x,y).
This probability"is assumed
independent of marital status and parity.
Finally define the following functions for a T-time marriage
cohort:
the random number of women with parity j at age
x out of those women who marry at calendar
time T, j=0,1,2, ... ,ao
T
m
(x)~x
+
o(~x)
the expected number of women who marry in the
age interval (x, x+6x) at calendar time Tout
21
of those women who marry at calendar time T.
fX+l mT(u)du.
x
the expected number of women who marry at time
T over the age month (x,x+l).
2.3
..
Formulation of the fertility process in a time-age marriage
cohort
In formulating the fertility process in a time-age marriage
cohort, a woman who is a member of the cohort is considered.
The ran-
dom mechanism producing her fertility performance is subject to the
following postulates:
POSTULATE ONE.
J(xlx o)' the parity at age x for a woman whose
age at marriage is x o ' is always finite.
a finite constant "a" such that:
POSTULATE TWO.
In other words, there exists
P {J(xlx o) ;;;, a} = 1.
For any age interval (xj ' x), the probability
distribution, h. i(x.,x), of the random variable Z (x ,x) which denotes
J,
J
j
j
the random number of the consecutive fetal losses over (x. ,x), depends
J
only on the parity j and on the interval (x ' x) but not on the precise
j
ages within the interval at which the fetal losses occur.
POSTULATE THREE.
As x
x., the probability distribution
+
J
h . . (x ' x) converges to a limit, specifically,
J,1.
j
h. O(x;, x ')
= 1,
h
= 0 for all i
J,
J
J
and consequently,
POSTULATE FOUR.
.(x , x )
j ,1. j
j
AT least at x
are differentiable for all x
~
O.
~
1.
= x , the probabilities h
j
Specifically,
j ,i
(x, x)
j
..
22
(1)
X=X.
J
(2)
x=x
= 0,
for i
~
2,
j
where Rj(xj ) is the unconditional fecundability at age x '
j
This postulate follows when the assumptions of a time dependent
Poisson process are operative.
Equation (1) signifies the effect of
the age interval (x ' x) on the probability hj,O(xj , x).
j
to Postulate Three, this probability is unity when x
= xj
According
'
Equation
(1) implies that with the increase of x, the rate of decrease in this
probability depends on the unconditional fecundabi1ity at x '
j
POSTULATE FIVE.
The probabilities Qj(x), defined as the prob-
ability of being fecund at age x for women with parity j, j=0,1,2, ••• ,a,
are differentiable for every x
~
O.
Under these five postulates, the following derivations are
possible.
Lenuna 2.1
The unconditional fecundabi1ity Rj(x) and the conditional fecundabi1ity p(x) satisfy the relation:
j=0,l,2, ••• ,a.
Proof
The proof follows from the definitions of the functions Rj(x),
p (x), and Qj (x) .
Lemma 2.2
The probabilities Qj(x) satisfy the difference differential
equation:
23
---= dx
QJ.(x-c
2j
).8(x-c
2j
).p(x-c
).exp{-rX
X- C 2j
2j.
~(T)dT},
for all values of x such that the parity at age x is j, j=O,l, ••• ,a,
with the initial conditions:
Qj(x) = 0 for x in the nonsusceptible period following
the conception leading to the ith live birth,
W'here:
~(x)~
+ o(6x) is the conditional probability of a terminal
event over (x,
p(x) ty.{ +
o(~)
x+~),
is the conditional fecundability over (x, x+6x),
is the probability that a conception at age x
ends in a fetal loss, and
e(x)
is the length of the nonsusceptible period
associated with a conception leading to a fetal
loss for a W'oman whose parity is j, J=O,l, ••• ,a.
Proof
As indicated earlier, Qj(x) is defined by:
Qj (x) = Pr {S(x) = 1, y(x) = 0
I parity
is j},
where Sex) is an indicator assuming the value 1 when a woman is in the
effective reproductive state at age x, and y(x) is another indicator
which assumes the value zero when a woman is not passing through a
nonsusceptible period at age x following a conception.
For the sake of simplicity we will drop the suffix j from the
parameters involved:
Q(x+lgc)
P r{S(x+lgc) = 1,
y(x+~)
= O}
= P r{ B(x+ LX) = 1, l3<x) = 1, y(x+ &1:) = O}
by virtue of the faet that the two
events {S(x+ /)x.) = I} and
{s(x+ /)x.) = 1, l3Cx) = I} are
equivalent.
Z4
= Pr{a(x+llx) = 1,
y(x+8x)
= 1,
Pr{S(x+8x)
= 0,
y(x+8x)
sex)
= 0,
1, y(x)
=
= 1,
Sex)
= o}
y(x)
+
= I} .
• • • (2.3.1)
The first term of equation (Z.3.1) simpliHes to:
= 1,
Pr{S(x)
y(x)
=
O}.Pr{S(x+8x)
= 1,
y(x+Ax)
=
°
y(x)
Sex)
= 1,
= o}
= Q(x).Pr{neither terminal events nor conceptions occur
over (x, x+llx) given that a woman is alive and
fecund at x}
=
Q(x). {l-~(x)llx+o(llx)}{l-P(x)llx+o(llx)}
= Q(x).{l-~(x)llx-p(x)llx+o(llx)}.
• •• (2.3.2)
The second term of (2.3.1), namely,
Pr{S(x+llx)
=
1, y(x+llx)
= 0,
Sex)
= 1,
y(x)
= 1}
is the probability that a woman moves from the nonsusceptib1e state at
age x to the susceptible state at age x+llx, i.e.
the value of the
indicator Y(x) changes from a value of 1 at age x to a value of zero at
age x+6x, while she is alive in the effective reproductive state.
Given the postulates of the model, this event occurs in only 9ne possib1e way.
That is when a woman is alive and susceptible to conception
at age (x-c ), than has a" conception ending in a fetal loss over the age
Z
interval (x-c ' x-cZ+llx), then she stays alive over the age period
Z
(x-c '
Z
~x).
The probability of such an event is given by:
Pr{a woman is alive and susceptible at x-c Z}
.Pr{a conception ending in a fetal loss over
(x-c Z' x-c?+llx) given that a woman is alive and
susceptible at age x-c Z}
,Pr{no terminal occurring over
= Q(x-c Z).8(x-c
(x-c2'X+~x)}
),p(x-cz)llx.exp{-lX+llx~(T)dT}+o(llx).
Z
x-cZ
"
• • • (2.3.3)
25
And finally, equation (2.3.1) is written as:
Q(X+~x)
=
Q(x){l-p(x)~x-p(x)6x+o(&x)}
+
+ Q(x-c · ).8(x-c )'P(x-cz).exp{-!X+~X~(T)dT}~X+
Z
2
x-c2
+ o(&x).
Subtract Q{x), then divide through by
~x ~
~x
and evaluate the limits as
0; utilizing Postulate Five we obtain:
dQ{x)
------ = -Q{x){~{x) + p{x)} +
dx
.
x
+ Q{x-c ).8{x-c ).p{x-c ).exp{-!
~{T)dT}.
2
2
2
x-c2
Therefore lemma 2.2 is established. The initial conditions follow
immediately from the definition of Q.(x).
J
Lemma 2.2 implies that Q.(x) is generally a function of x o ' p(x),
J
8(x),
~(x),
c
~
, and c
~
•
This lemma is studied further in Chapter
Three in an effort to utilize it in the choice of a specific function
to represent Qj(x),
Theorem 2.3
Given that x is an age point such that:
(i)
the parity at x is j, j=O,l, ••• ,a, and
(ii) the conception ending in the j+lst live birth does not occur
before x,
then, for any age point y, y>x, we have:
H.{x,y,s) = exp{!y{s8(T)-l}R;(T)dT},
x
J
J
x<y,
j=O,l, ••• ,a.
Where Hj(X,y,s) is the generating function associated with the. random
variable Z.{x,y) which denotes the number of consecutive fetal losses
J
over the age interval (x, y) for a woman with parity j.
26
Proof
Using Postulate Four and proceeding in the familiar manner to construct a system of difference differential equations on hj,i(X'y),
we obtain:
h. 0 (x, y+~y)' = h. O(x,Y){l-R.(y)~y+o(~y)}
J,
h
J,
j,i
(x,
y+~y) =
J
(x,y){l-R.(y)~y+o(~y)} +
j,i
J
+ hj,i_1(x,y){e(Y)Rj(Y)~y+o(~y)} +
h
+
o(~y),
for
i~l,
j=O ,1, •.. ,a.
The logic behind the construction of these equations is simple.
The
first term corresponds to the situation where a woman already has i
consecutive fetal losses over (x, y).
The second term corresponds to
the case where the number of consecutive fetal losses over, (x, y) is
i-1.
The third term corresponds to the situation where the number of
fetal losses over (x, y) is less than i-1.
Subtracting h
(x,y) from both sides of these equations,
j,i
dividing through by ~y and passing to the limit as ~y + 0, we obtain:
-Rj (y). h
dh
_ ....
j .....=:i
(x,y)
=
ely
j,O
(x,y)
-R. (y) .h . . (x,y) + 8(y)R. (y) .h.
(x,y),
J
J,~
J
J,i-1
. . • (2.3.4)
Now, the generating function of the probabilities h.
J ,i
(x,y), H.(x,y,s),
J
is defined as:
H. (x,y.s)
J
00
= L h
i=o j . i
i
(x,y)s,
Isl~l
O~x<y<oo
. . . (2.3.5)
27
Differentiating (2.3.5) under the summation sign l and us~ng (2.3.4)
we obtain:
dHj(X,y,S)"
w dh i(x.y)
--"'------ = l.
dy
{
j,
ay
i=o
=-Rj(y)
}
si
E
E
h i(x,y)si + s8(y)R (y)
h " (x.y)si-l
j
i=o j ,
i-I j ,i-l
which can be written as:
with the initial condition, derived from Postulate Three.
H (x,x,s) = 1.
j
• • • (2.3.6)
A unique solution for (2.3.6) exists if the function R (y){s8(y)-1}
j
is continuous for all y (Bellman and Cooke
1963, p. 29).
This solu-
tion is found, by separation of variables. to be:
Setting y=x, the initial condition implies that the constant of integration is K(s)
= 1.
Therefore.
H (x,y,s)
j
Q.E.D.
lSince Ih
i(x,y)sil<lsli. the infinite series in (2.3.5)
•
ahj i (x,y)
converges uniformly in y for Isl<l; hence h
'
si converges
j
ay
in y and the term by term differentiation is justified for
s<l. The argument applies also for s=l since lim H (x,y,s) =
s-+l j
Hj(x,y,l) exists. (Chiang 1968, p. 48).
~iformly
28
Corollary 2.4
Pr(J{(ytg)lx o} = iIJ{(~g)lxo} = i) = H (x,y,1),
i
i=O,l, •.• ,a,
where:
J{Ylx o} is the parity. at age y for a woman marrying at age xo '
is the pregnancy period associated with a live birth
g
conception, and
=
H (x,y,l)
lim H (x,y,s).
8+1
i
i
Proof
From theorem 2.3 it can be shown that:
where Ai indicates the event that conception ending in the i+lst live
y
birth for women whose parity is i does not occur before age y.
In terms of parity, the event Ai is equivalent to the event that
y
parity is i for all ages between y and ytg.
A~
==
That is:
(J{ (y+s) Ix o} = i, for all e: such that O<e::;g).
This can be written as the intersection of several events as follows:
i
i
g
A
Y
= ()
B
e:=0
Y+e:'
i
where By+€: is the event (J{(Y+e:)lx o} = i). A count of the elements of
i
is a subset
each event Bi
, ~~E:~g, shows that if O.::e:l~e:~g, then B
y+e:2
g
y+e:
Using this result, the interseation ()
e:=0
is the event (J{ (y+g) Ix o} = i).
Therefore,
i
= Pr{By+g
IBi }
X+g
Bi
Y+e:
reduces to
29
Q.E.D.
Theorem 2.5
pl,j(x,ylx o)= exp{!y-g{e T(u)-l}Rj(u)du},
x-g
x<y,
j=O,l, ••• ,a,
where:
the index T indicates the calendar time of marriage,
Xo is the age at marriage,
pl,j(x,y/x o) is the probability for a woman who marries at age Xo
at time T to be in parity j at age y given that she is at parity .
at age x, xo< x<y,
j
eT(x) is the probability that a conception at age x ends in a
fetal loss for women who marry at calendar time T
and
T
Rj(u) is the unconditional fecundabi1ity at age u for women who
marry at calendar time T •
Proof
For simplicity, the index T will be dropped with the understanding
that formulas derived relate to women marrying at calendar time T.
The proof of this theorem follows from corollary 2.4 if we notice that:
Therefore,
H (x-g,y-g,l),
j
.
by corollary 2,4.
30
= exp
y g
ux-g
- {8(u)-l}R. (u)du},
J
by theorem 2.3.
Q.E.D.
Corollary 2.6
When age is measured in months, and under the assumption of no
multiple births, it follows that:
x+l-g
( expU
{8(u)-l}R (u)du},
i
x-g
{(x,x+l) /x o } =
~,j
P.
for j=i;
,.x+l-g
l-q (x,x+l)-exp {J
{8(u)-1 }Ri(u)du},
x-g
for j=i+l;
o
otherwise,
j=O,l, ••• ,a,
Where:
Pi . {(x,x+l)Ix o } is the probability for a woman marrying at age
,J
Xo
to be in parity j at the beginning of month x+l given that she
is in parity i at the beginning of month x, and
q(x,x+l) is the probability that a woman who is alive at the
beginning of month x will die during the month (x,x+l).
Proof
Under the assumption of no multiple births, the sample space over
(x, x+l) for a woman who has parity i at x is the union of the three
mutually exclusive events D, -fA ()Pi} and -fA ()P
i+l
}, where:
D indicates the event that a woman who is alive at x dies over
(x, x+l),
A indicates the event that a woman who is alive at x will still
be alive at x+l,
31
Pi indicates the event that a woman with parity i at x is still
in parity i at X+1, and
P + indicates the event that a woman with parity i at x moves to
i 1
parity i+1 over (x, x+l).
Theorem 2.5 implies that:
= exp{!x+1- g {8(u)-1}R (u)du}.
x-g
i
Since PdD} is given by q(x;x+l) and pr(DU{A()P }U{A()P + })= 1,
i 1
i
i t follows that:
x+1-g
= 1-q(x,x+1)-exP{!
{8(u)-1}R {u)du}.
i
x-g
Q.E.D.
Theorem 2.7
(Study of the Truncation Effect)
Let C denote the condition that the parity at age x is j and, that
the conception ending in the j+lst live birth does not occur before
x, then:
(i)
Assuming an infinite period for reproduction such that a
woman will eventually achieve any parity j, j=1,2, ••• ,a, then
PdY + >ylc} = Hj(x,y,l),
j 1
x<y
where Y + is the age at which the conception ending in the
j l
j+lst live birth occurs.
(ii) Assuming a finite reproductive period, and considering
only those women who will have a ith live birth before the
end of that period, j=1,2, •.• ,a, then
Hj (x,y,l){l-Hj (y,w,l)}
,
x<y<w
l-H (x,w ,1)
j
where Y*
is the age at which the conception ending in the
j+l
j+lst live birth occurs assuming that that conception
32
occurs before w, and w is the upper end of" the reproductive
period.
Proof
Case (i)
Assuming an infinite" reproductive period, the two events
{Yj+l>y} and {j{(y+g)lx o } = j} are equivalent where J(xlxo) is the
parity at age x for a woman marrying at x o '
But corollary 2.4 implies
that
Pr fJ{(y+g)lxo} = jlc) = H.(x,y,l).
" J
Therefore,
Q.E.D.
Case (ii)
Assuming a finite period for reproduction, and considering only
those women who will have a ith live birth before w, j=1,2, ••• ,a,
we have:
PdY;+l>ylc} = pr(J{(y+g)lx o} = jIJ{(w+g)lxo}~+l,C)
=
Pr (J{ (y+g) Ix o} = j ,J{ (w+g)
Pr (J{ (w+g)
IX o}~+llc)
IX o}2:j+ll C)
Pr (J{ (y+g) Ix o}= j IC) •Pr (:I{ (w+g) IX o }~j+ll J{ (y+g) Jx o}
= j ,C)
=---------------------l-Pr (J{ (w+g) Ixo}~ IC)
But the event {J{(w+g)lxo}~IC} is equivalent to the event
{J{(w+g)lxo}= jlc} since C places the condition that parity at w+g
is at least j.
Therefore
Now applying corollary 2.4 as well as theorem 2.5, we obtain:
33
x<y<w.
Q.E.D.
Remark:
Comparing the results of Case (i) and Case (ii) in theorem 2.7,
indicates that making the assumption of an infinite reproductive
period implies that:
lim
.Hj (x,y, 1) {I-Hj (y,w, I)}
~
l":'Hj (x,w,l)
= Hj (x,y, 1)
or,
l-Hj (y,w,l)
lim .
.
~ l-H (x,w, 1)
j
= l.
This implies that:
lim Hj(x,w,l)
~
=
lim H. (y,w,l).
~
J
But since x and yare any two age points such that x<y<w, it follows
that:
lim H.(x,w,l) is the same for all values of x such that x<w.
w~
J
• • • (2.3.7)
When assuming an infinite period for reproduction, inherent is
the assumption that the quantity {1-6(x)}R (x), which denotes the force
j
of a ith live birth at x, is always greater than zero i.e. there
exists a positive constant b such that:
. {I-8 (x) }R (x)
j
~
b,
Substituting this assumption
lim Hj(x,w,l)
~
=
for all ages x.
into (2.3.7), we obtain:
lim exp{fW{6(x)-1}Rj (x)dx}
x
~
~ lim exp{-bfwdx}
- w-lo<XI
x
34
= lim exp{-b(w-x)}
w~
=
o.
Recalling theorem 2.5, we conclude that:
;~ Pj,j(X'w/x o) = ;~ Hj (x-g,w-g,l)
=
o.
This final result implies that under the assumption of an infinite reproductive period, any parity j, j=O,l, ••. , is a transient state.
2.4
Age-parity distribution in time-age marriage cohorts
The objective of this section is to use the theory developed in
section 2.3 to" derive the moments of the age-parity distribution in
a time-age marriage cohort.
Inherent in this procedure is the assump-
tion that an observed pattern of fertility behavior is a sample rea1ization of a stochastic process.
The method suggested for this purpose consists of dividing the
reproductive age into a finite number of equally spaced age points.
A recurrence relationship is then developed between the parity distribution at any two successive age points.
The division of the repro-
ductive period into a finite number of age points is a rather subjective
matter and depends on the purpose of the study.
In this study the
reproductive period is divided into monthly intervals.
The choice of
a month as the constant period between any two successive age points
seems appropriate since the reproductive process has a cycle of a
monthly nature.
Lemma 2.8
Assuming no multiple births, we have:
35
n~(X+1Ixo)
=n~ , o(x/x o)'
j=1,2, ••• ,a,
where:
is the random number of women with parity j at the
beginning of month x out of those women who marry at
age Xo at time
nL (xlx o)
i,j
.
L, and
is the random number of women married at age Xo at
calendar time L who move from parity i to parity j
over the month (x, x+1).
Proof
The proof is immediate from the definitions of the functions
involved considering that the assumption of no multiple births implies
that the maximum change in parity over a period of one month is one.
Lemma 2.9
(i)
E{nTi j(x1x o )}= p~ o(x,x+1Ixo).E{nLi(xlxo)}.
,
1,J
(Note that under the assumption of no multiple births, this
is zero unless j=i or i+1.)
{OJOkPLi j(x,x+1 Ix ) - P: o(x,x+1I x o).pL (x,x+1I x o)}·
•
0
1,J
i,k
E{nI(xlx o )},
for i=~,
(Note that this is zero unless j=i or i+1 and
k=~
or
~+1.)
where:
pI,j(x,X+llx o) is the probability for a woman marrying at age Xo
36
and calendar time
~
to move from parity i to parity
j over the month (x, x+l) given that parity at x
is 1. ' Corollary 2.6 implies that under the assumption of no multiple births, this probability is
. zero for any value of j other than i or i+l, and
I1,
1°,
for j=k
for j#k.
Proof
The proof of this lemma follows from elementary principles.
Two facts are utilized in this proof:
(a) the first is the well known result about conditional expectations, namely,
E(X) = E{E(XIY)}, and
(b) the second is that given nI(x/x o )' and assuming that
nI,d(xlx o )
is
the number of women who die over (x, x+1)
out of nI(x/x o )' the joint distribution of the random variables nI,i(xlx o )' nI,i+l(xlxo ), and nI,d(x/xo) is trinomial
with probabilities PI,i (x,x+1/ x o )' PI,i+l (x,x+1 1x o )' and
1-PI,i (x,x+ll xO)-PI,i+l (x,x+ll x o ) respectively.
Considering these two results, the proof of the lemma is as follows:
Case (i)
From the properties of the trinomial distribution, it follows that:
Taking expectations on both sides, we obtain:
. ~ J'(x Ix o )} = p~I
~
Ix o)}.
E{ui
. . (x,x+1 Xo)·E{Ui(X
,
1. ,J
Q.E.D.
37
Case (it)
When iiR, , the joint probability distribution of nI,j (x Ix.o) and
nI,k(xlx o) conditional on nI(x/x o) and n~(xlxo) is given by:
Therefore,
and finally,
pL (x,x+llxo).pL (x,x+llxo).Cov{nL(xlxo )' n~(xlxo) •
i,j
R"k
i
~
When i=R, , the joint probability distribution of the random varia=
bles nl .(xlxo ) and nL (xlx o ) conditional on nLi(xlx ) is given by:
,J
i,k
o
38
ni(x/xo)!
=
n: ,(xlxo)! niT k(x/x O)! {niT(xlxo)-niT ",(xlxO)-n T k(xlx o)}!
i
1.,J,
, J ,
{p;
,(x,X+llxo)}nI,j(xlxo)~{PiTk(x,x+l/xo)}nI,k(x1x o).
1.,J "
'
{l-ptj (x,X+l/Xo)-P~,k(x,x+llxo) }n~ (x Ixo)-ni,j (x Ix o)ni,k (x/x o)'
From this distribution, it follows that:
Cov{n~
,(xlx o)' n~
k(xlx o) IniT(xlx o)}
1.,J
1.,
=
=
ir-n~(xlxo).pTi,J.(x,x+l/xo).pI , k(x,x+ljxo),
ln~ (x Ixo), p ~ ,j (x,x+ll Xo ){l-p ~ ,j (x,x+ll x o)},
1
Setting 0 =
jk
0
1
if
j~k
j=k.
j=k
if j~k
, we obtain:
T
Cov{n T
=
i ,J,(xlx o)' n i ,k (xlxo)lnT(xlxo)}
i
and finally, taking expectations on both sides of this equ&tion, the
required result follows.
Q.E.D.
Theorem 2.10
(Expected Age-Parity Distribution)
Under the present formulation, we have:
+ p; . (x,x+llx ).E{n;(x/x )}, j=1,2, .•• ,a,
J ,J
0
J
0
39
with the initial conditions:
where N~
is an initial size of a cohort marrying at age x
o
o
at calendar
time T.
Proof
The proof follows by taking expectations on both sides of lemma
2.8 and applying lemma 2.9, part (i).
Corollary 2.11
pT(X+llx ) = pT (x,x+llx ).pT(xlx ),
0,0
0
0
0
o
0
T
p.(x+llx) = p; 1 . (x,x+llx )p; l(xlx ) +
J
0
J- ,J
0
-
0
+ p; ,J.(x,x+1Ix 0 )p;(xlx 0 ),
j=1,2, ••• ,a,
with the initial conditions:
j=O
PjT (x Ix )
o 0
j;&O
where pl(x1x ) is the probability that a woman marrying at age X at
o
o
time T will have parity j at age x, and is defined by the relationship:
Proof
T
The proof follows from theorem 2.10 on replacing E{nTj(x/x )IN }
o Xo
,
T
T
T
l
by Nx •P . (x x ) and taking expectations over N •
xo
o J
0
Corollary 2.12
(A Discrete Approximation for the Probability Distribution of the Age at Delivery of the ith live
birth, X.)
J
The probability distribution of the random variable X satisfies
j
the relation:
40
= pT (xix ).pT
(x,x+llx ),
j-l
0
j-l,j
0
j=1,2, ... ,a.
Proof
The event {x < X, < x+l} is the intersection of two events A and
J
B, where:
= {parity at x is j-l}, and
B = {a woman with parity j-l at
A
x, moves to parity j over (x, x+l)}.
Therefore,
Pr{x < X, < X+llx } = Pr{A()B}
J
0
= Pr{A}.Pr{BIA}
= p~ l(xlx ).pT
,(x,x+1 Ix ).
J0
j-l,J
0
Q.E.D.
Theorem 2.13
(Covariance Matrix of the Age-Parity Distribution)
T
cov{nTj(x+llxo)' nkT(X+llx )} = Cov{nT
(xix ), n
(xix )} +
o
j-l,j
0 , k~l,k
0
+ Cov{n~J-,J
1 ,(xix ), nT
k(xlx )} +
k,
0
. 0
Cov{n~(x Ix ), nT(x Ix )} = 0,
with the initial conditions:
~
0
k
0
0
0
i,k=O,l, •.• ,a,
and where cov{n~ ,(xix ), n~
,J
0
x"
k
(xix)} is given by lemma 2.9, part. (ii).
0
Proof
The proof follows on substituting for n~(X+llx ) and nT(X+l/x )
J
0
k
0
using lemma 2.8.
With the respective initial conditions, the recurrence relations
displayed by theorems 2.10 and 2.13 can be completely solved.
The same
41
procedure can be used to find the higher moments of the age parity distribution, if required.
biological functions
These moments obviously are functions of the
und~rlying
human reproduction through the
probabilities pT . (x,x+l/x ).
i ,J
0
2.5
Age-parity distribution in time marriage cohorts
A time marriage cohort is a weighted sum of time age marriage
cohorts where the weights are derived from the probability distribution of age at marriage at the corresponding point of time.
The
results of section 2.4 can thus be utilized to derive the moments of
the parity distribution in time age marriage cohorts.
These moments
are to, be used later in Chapter Three to evaluate the effect of marriage
patterns on fertility.
The derivation of these moments is done under
the same assumption of section 2.4.
Definition 2.14
T
j=o,i, .•. ,a,
n (x+l) =
j
where:
is the random number of women with parity j at the
beginning of month x+l out of those women who marry
at calendar time
n:(x+llx)
J
i,
and
2S the random number of women with parity j at the
0
'beginning of month x+l out of those women who marry
at age x
o
at calendar time
i.
Le1IUlla 2.15
Assuming that the Markov property holds for the random variables
X., j=O,l, .•• ,a, it follows that:
J
42
pT j(x,x+1/x ) is independent of x •
i,
Theorem 2.16
0
0
(Expected Age Parity Distribution)
Under the assumptions of lemma 2.15, it follows that:
T
pT
(x,x+1).E{n
(x)} +
j-1,j
j-1
T
T
+ p, ,(x,x+1).E{n.(x)},
J ,J
J
j=1,2, ..• ,a,
where:
T
Pi .(x,x+1)
,J
is the probability for a woman marrying at calendar
time T to move from parity i to parity j over the
month (x, x+1) given that the parity is i at x.
According to lemma 2.15, this probability is independent of the age at marriage x , and
o
is the expected number of women in the age interval
(x, x+l) at calendar time T out of those women who
marry at time T.
Proof
Definition 2.14 can be rewritten as:
x-I
T
T
T
j=O,l, ..• ,a.
n. (x+1) = En, (x+ltx ) + n. (x+1Ix),
J
x o=1 J
0
J
Taking expectations on both sides of this equation, we obtain:
T
x-I
J
x o=1
E{n.(x+1)} =
T
T
r E{n.(x+1
Ix)} + E{nJ.(x+1Ix)},
J
j =0,1, ••• , a.
0
Applying theorem 2.10 to substitute for E{n:(x+1Ix ) , it follows
J
that:
0
43
x-l
I T T '
E{nT(x+l)} = L pT (x,x+l/x ).E{n (xix )}+ E{n (x+llx)},
o
xo=l 0,0
0
0
0
. 0
E{nj(x+l)}
x-I
T
(pT
(x,x+llx ).E{n
(xix)} +
xo=l j-l,j
0
j-l
0
L
T
+ pT(x,X+llx ).E{nj(X1x)}) +
j,j
0
0
j=1,2, •••
But the quantity E{n:(x+llx)
J
,a.
is the expected number of women with
parity j at the beginning of month x+l out of those women who marry
over the age month (x, x+l) at calendar time T.
The nature of the
fertility process implies that this quantity is zero for any value of
j different from zero.
For j=O, this quantity is the expected number
of women marrying over the age month (x, x+l) at calendar time T.
That
is:
j=O
j;&O.
We also recall from lemma 2.15 that the probabilities PI,j(x,x+l/X )
o
are independent of x and can thus be written as
j(x,x+l). Taking
P:
o
~,
these remarks into consideration, we have:
x-I
= pT
0,0
T
T
(x,x+l). L E{n (xIx )} + M (x),
x =1
0
o
x-I
T
1 . (x,x+l). L E{n. l(xlx )} +
J- ,J
x =1
J0
= P:
o
+
T
x-I
T
I
. (x, x+1). L E{ n (x x )},
j ,J
x o=1
j
0
P
Finally, utilizing definitiOn 2.14, we obtain:
j
=1 , 2, ... ,a.
44
= pl
j-l,j
(x,x+l).E{n: lex)} +
J-
1
..
l(
+ Pj·
. (x,x+1).E{n
x)},
,J
j
j=l,2, ••. ,a.
Q.E.D.
With appropriate initial conditions, the system of equations presented
by theorem 2.16 can be completely solved.
The higher moments of nJ(x) can also be derived in a similar
fashion.
These higher moments are not derived here however since the
expectation will suffice for the numerical investigation of the effect
of the function M(x) on fertility.
Corollary 2.17
(A Discrete Approximation for the Probability Distribution of the Age at Delivery of the Ith Live Birth,
xj, in a Time Marriage Cohort)
The probability distribution of the random variable Xl satisfies
j
the relation:
Pr{x <
xj
< x+l} = u~
J-1
(x).p~
. (x,x+l) ,
J-1,J
j=1,2, ••. ,a,
where:
j=1,2, .•• ,a.
Proof
Recognizing u:(x) as the probability that a woman, marrying at
J
calendar time.
1,
will be in parity j at age x, the proof of the
corollary follows.
2.6
Fertility formulation in heterogeneous cohorts
As indicated earlier, empirical research, together with the nature
of the human fertility process, seems to suggest that women are
heterogeneous with respect to their reproductive characteristics.
45
This heterogeneity can be introduced into the present model at two
different levels depending on the approach adopted in studying fertility.
When the demographic approach is employed, heterogeneity is·
introduced at the level of the function p
.
. (x,x+l).
j,J
On the other
hand, if the biological approach to fertility is adopted, heterogeneity is introduced at the level of the function h- Sex) }R (x) •
j
two
These
hereafter referred to as Model A and Model B respectively,
mode~s,
do not generally produce the same results.
2.6.1
Model A, heterogeneous birth probabilities
Under this model, p . . (x,x+l) is treated as a random variable
J ,J
among women whose value for any woman i is the sum of two components.
The first component is a constant. a.J (x) indicating the expected
value of P. . (x,x+1) among all women. The second is an error term·
J,J
{£j (x) }.; signifying the random element in P. . (x,X+I) which produces
...
J ,J
heterogeneity among women.
In other words, we assume that the value of P •• (x,x+l) for
J ,J
individual i, {Po . (x,x+l)}. say, is of the form:
J ,J
~
{P . . (x,x+l)}. = a. (x)
J ,J
1
J
+ {£ (x)} i'
j
j=O,l, ••. ,a,
i=1,2, ••• ,N,
where N is the cohort size, and
j=O,I, .•. ,a,
and from the assumption of independent individuals,
COV({E:
j
(x)}
i
, {£
(x)} ) =
j
i'
a~ (x),
i=i'
0,
i7'i'
1
J
j=O,l, ••• ,a.
46
It follows, under this representation, that:
E{p, ,(x,x+l)}
J ,J
= a. (x),
Var{p, j(x,x+l)}
J,
J
= a:(x),
J
and the form of the probability distribution of p. j(x,x+l) depends
•
J,
on the assumed probability distribution of {Cj (x)},
The family of probability distributions that are symmetric around
zero with a finite variance would serve as probability distributions
of cj(x).
The choice of such a distribution is however limited by
the fact that P . . (x,x+l) should lie in the interval (0, 1).
J ,J
To
.that end we might have to choose a distribution such that aj(x)/aj(x)
is "large," or we simply might have to use a truncated distribution.
Perhaps it is worth mentioning that for estimation purposes, the form
of the probability distribution of c. (x) does not have any effect on
J
the resulting estimator of a.(x).
The form of the distribution becomes
J
important of course when hypothesis testing is considered.
2.6.2
Model B, heterogeneous force of ·conception
Rj(X) is the unconditional fecundability at age x for a woman with
parity j.
8(x) is the probability that a conception at age x ends
in a fetal loss.
Therefore, {1-8(x)}R.(x) represents the force of a
J
conception ending in a live birth at age x.
Treating {1-8(x)}R.(x) as a random variable, we assume that
J
{1-8(x)}R. (x) for a given woman i is the sum of two components.
J
The
first component is a constant bj(x) representing the expected value of
{1-6(x)}~(x) and the second is an error term {OJ (x)}i indicating the
error term causing heterogeneity.
In other words, the value for
. {1-8(x)}R .(x) for woman i, {{1-8(x)}R.(x)}., is represented as:
.
J
J
1
\
47
j=O,l, ••• ,a,
where
and from the assumption of independence of individuals,
Cov{{o'(X)}i' {o (x)}.,}
J
j
1
=
V 2 (x),
i=i'
j
0,
i;'i' •
f
The relationship between {l-e(x)}Rj (x) and P . . (x,x+1) as displayed
J ,J
by theorem 2.2 implies that under this formulation p . . (x,x+1) is
J ,J
a random variable.
The probability distributions of {1-8(x)}Rj (x) and
Pj,j(x,x+1) depend on the probability distribution chosen for 0j(u).
The considerations discussed in section 2.5.1 concerning the choice
of a distribution for €.(x) apply, in general, to the choice of a
J
distribution for 0j(x).
Both Models A and B are studied further in Chapter Four while
investigating the estimation problem in the model.
2.7
Fertility in cross sectional populations of married women
The fertility performance of a cross sectional population of
married women is a weighted sum of the performances of time marriage
cohorts.
The weights are derived from the initial size as well as the
time pattern in fertility associated with each cohort.
Under these
circumstances, fertility in cross sectional populations may be described
by two random variables.
The first is the time pattern in the input
to the marriage system as reflected in the change in the size of the
population over time.
The second is the continuous internal change
in the population as expressed by the time trend in the age parity
distribution.
48
The dependence of these two random variables on calendar time is
. a problem since the nature of this dependence is hard to evaluate.
The
i~clusion
of this section in this paper is intended mainly to com-
plete the picture of the fertility process displayed in the previous
sections.
Keeping this in mind, no attempt is made to provide a com-
•
plete treatment of the effect of calendar time on fertility, but
rather, the concern is to point out the problems involved when attempting to provide such a treatment.
Recognition of these problems is
the starting point for any further research.
The model presented in the preceeding sections, as might be observed, started with considering a time age marriage cohort.
Averaging
these time age marriage cohorts over age resulted in formulas for time
marriage cohorts.
Now averaging time marriage cohorts over time would
result in a description of fertility in a cross sectional population.
This averaging procedure starts with the basic definition:
•
m. (x, t)
J
.
=f
t
nj(x)dT
t-x
. . . 2.7.1
where
mj(x,t) is the number of women with parity j at age x in the
population at calendar time T, and
is the number of women with parity j at age x out of those
women who marry at calendar time T i.e.
the members of
the time marriage cohort of time T.
From ~ection 2.5, the distribution of nI(x) , for all T, depends
..
on several functions:
(i)
the initial size of the cohort at the time of marriage
i.e.
49
at time T, 0
(ii)
~
T ~ t,
the probability distribution of the age at marriage at
time T,
(iii)
O~
T < t,
the birth probabilities associated with the cohort of
time T, 0
~
.
T ~ t.
The specification of a time trend in these functions is not an
easy matter.
The stable population model is a prevailing example of
such a specification in the literature.
This model, w.hich assumes that
these basic variables are homogeneous in time, is not suited for fertility analysis especially in the developing countries.
In their search
for acceptable fertility patterns, these countries experience fertility conditions that are far from stable.
is obviously required.
A more realistic treatment
In Chapter Six some ideas with regard to this
problem are presented as suggestions for further research.
An alternative way to describe the fertility in a cross sectional
population is to use a macro-analytic approach.
According to this
approach, both the population size and the age parity distribution are
regarded as continuous functions of calendar time and the time pattern
in fertility is expressed by the rates of change in these two variables
without going into the details of the averaging process underlying
these rates.
Studies available in the literature dealing with the
size and age structure of human populations (e.g.
Fergany
1970)
can be adapted in the present
Pollard 1966,
situation~
.
CHAPTER III
NUMERICAL INVESTIGATION OF HUMAN FERTILITY
In this chapter the results of a numerical study of human
fertility based on the present model are described.
The primary
purpose of this study is to evaluate the nature of the effects of each
of the functions included in the model on fertility performance.
The content of .this study is arranged in three sections.
In
3.1 the parametric representation of the functions included in the
model, which have been referred to only in general terms, is discussed;
3.2 is an outline of the approach adopted for this study and 3.3 is
a discussion of the effects of the individual functions on fertility •
..
3.1
Choice of specific functions
Human fertility, as described in Chapter Two, depends on several
functions whose forms have to be specified before any numerical work
can be done.
It must be understood that the forms that will be assumed
for these functions are at best approximations to a real pattern whose
existence is claimed.
3.1.1
p (x):
The conditional fecundabi1ity at age x
The function p(x) is defined such that
p(x)~x+o(~x)
is the
probability that a non-contracepting woman, who is susceptible and at
the risk of conception, will conceive over the age interval (x,
x+~x).
The precise form of this function is not known and furthermore, it is
/
.......
51
not possible to make direct observations on it.
The existence of such
a function however implies that it must have certain characteristics
which can be utilized in an effort to obtain a definite form for it.
The nature of the reproductive process implies that p(x) must
satisfy the property p(x)=O for x<w , where w is a parameter indicat1
1
ing the lower age limit for reproduction.
After W , p(x) is expected
1
to be an increasing function for some time until it reaches a maximum; then it starts to decrease up to a certain age w after which
2
p(x)=O.
The parameter W indicates the maximum age limit for repro2
duction.
Henry (1957, 1965), in an attempt to account for these properties,
chose to represent p(x) as a function of a trapezoidal shape.
choice has since prevailed in the literature.
This
Henry's choice seems
to have been influenced by the primitiveness of the computational
facilities that were available to him.
Modern computer facilities make
it possible to represent p(x) by more elaborate functions that take
into consideration the properties mentioned above and at the same
time, preserve a certain amount of flexibility.
Such a representation
is provided by the following form of a beta curve:
o<x<w
==1
p (x)
(x-w)
1
p-l
(w -x)
q-1
2
,
p>o, q>O,
W
<x<w
1== 2
x>w •.
= 2
This function is quite general and allows for many forms of the function
p(x) depending on the parameter pair (p, q).
Specifically, when
(p,q)=(l,l), p(x) is simply a constant over the interval (w ,w ).
1
2
If p=q#l, then p(x) is symmetric around the center of the interval
"'.
52
(w , w ), whereas if p>q, p(x) is skewed to the left, and if p<q,.
1
2
The choice of (ex., w , w ) has nothing to
it is skeweq to the right.
1
2
do with the shape of the curve but is reflected in the magnitude.
The fact that parity does not exceed a maximum integer, a, is a condition which forces the overall level of p(x) to decrease as the period
(w , w ) increases.
1
2
The mode of the curve p(x), suggested above, occurs at the age
point,
x =
(p-l)w
+ (q-l)w
-'=2'--
__=_1
p+q-2
• . • (3.1.1)
Empirical observations indicate that the mode of p(x).occurs around age
25 years (Henry (1965».
Taking age 25 years to indicate the mode of
p(x), 3.1.1 provides a relation useful in specifying p, q, w , and
1
w.
The value for ex. is determined, after p, q, w , and w , by the
2
1
maximum level required for p(x).
2
The choice of.these parameters will
be discussed further in section 3.3.
The interest in this numerical
study is to investigate the effect of the shape of the curve p(x)
on fertility as well as evaluating the effect of changing the .length
of the reproductive period (w , w).
1
2
This investigation is done
graphically as we wish to study general trends and the exact numerical
values are irrelevant.
The function Qj(x) is defined as the probability that a woman
with parity j at age x is in the
susceptible to conception.
~ffective
reproductive state and is
This function signifies the effect of
53
terminal events as described in Chapter Two,
The precise form of
Qj(X) is difficult to determine because of the problems involved in
obtaining direct observations on it.
The general properties of the
reproductive process however imply certain characteristics for this
function which can be used in an effort to obtain a parametric representation for it.
It can be argued that the nature of the reproductive process implies that a younger age for achieving a given parity j, j=1,2, ••• ,a,
is associated with a higher reproductive capacity.
Since Q (x) , which
j
is conditional on parity j, is an index of the reproductive capacity,
it follows that Qj(X) is a decreasing function over its domain.
This
assertion is justified by lemma 2.2, according to which the function
Q (x) satisfies the difference differential equation:
j
dQ (x)
j
----= dx
+ Q (x-c
j
Setting
d Q (x)
j
-~--
2j
).8(x-c
2j
).p(x-c
2j
X
).exp{-f
~(T)dT}.
X-C2j
= 0 to examine the curve of Qj(x), it follows that the
dx
stationary points on the curve occur when,
Or, after substituting Rj(x) for Qj(x).p(x),
x
= 8(x-c .).R (x-c .).exp{-f
~(T)dT}.
2J
2J
j
X-C2j
• • • (3.1.2)
Now the fact that 8(x) is usually of a small magnitude, that the
54
x
~cr)dT}
quantity exp{-!
is a fraction less than one, and that c 2j
X- C 2j
is of the magnitude of a few months implies that 3.1.2 is a result that
This means that
is impossible to occur.
d Q (x)
j
can never be zero.
dx
Furthermore, the same argument shows that the left hand side of 3.1.2
is always greater than its right, hand side.
This implies that the
function Qj(x) has a negative derivative for all x and therefore,
Qj(x) is a decreasing function over its domain.
To account for these properties, the following curve is chosen
to represent the function Qj(x):
d.>O, O<y.<l,
Yj.exp{-d.(x-a )},
J
j
J
x
~
-
J
a.,
J
j=O,l, ••• ,a.
otherwise,
°
The parameter a in this function represents the minimum age at which
j
reproduction is possible after attaining parity j.
Therefore, it
satistifes the relation:
a
o
=x +
g,
0
a j = aj -1 + mj + g,
j=1,2, ••• ,a,
where
x
o
g
is the minimum age at marriage,
is the duration of pregnancy associated with a live birth,
usually taken as 9 months, and
m.
J
is the minimum length of the post parttim nonsusceptible
period following theith live birth,
The parameter y. is the value of Q. (x) at a .•
J
it is observed that:
J
J
j=I,2, ••• ,a.
As' for the parameter d .,
J
55
dQ. (x)
= -d
J
dx
.Q (x)
j
j
••• (3.1.3)
and
.(3.1.4)
Equiation 3.1.3 implies that d
j
should always be positive in order to
preserve a negative derivative for all x.
EquatiOn 3.1.4 means that
if x is measured in months, as is the case in this study, dj must
be close to zero since the change in Qj(x) from one month to the next
is naturally small.
The choice of the specific numerical values for a., y., and d .
J
J
J
will be discussed further in section 3.3.
~(x):
3.1.3
The unconditional fecundability at age x
From lemma 2.1, R.(x) is given by
J
R. (x)
J
= Q. (x).p(x)
J
which, under the formulations of sections 3.1.1 and 3.1.2, leads to:
0,
R. (x)
J
=
y .• exp{-d (x-a )}.
J .
0,
j
j
a
(w -w )p+q-l
6(x):
1
2
21
-
2
w <x<oo ,
2==
j=O,l, ..•
3.1.4
(x-w )p-l(w -x)q-l, aj<x.:'S.w ,
,a.
The incidence of fetal loss associated with a conception
at age ~
No attempt is made to specify a parametric form for 8(x).
Rather,
sets of numerical values for this function that are found in the literature are used.
56
3.1.5
x o:
The age at marriage in a time age marriage cohort
-------~~~_.~_.~-_._----------
It is of primary interest in this chapter to evaluate the effect
of the age at marriage on fertility.
Values for Xo in the age interval
(15, 30) years will be used to investigate the effect of varying x
on fertility.
3.1.6
M(x):
The eXEected number of women who marry over the age month
(x, x+l)
The function M(x) is determined once the probability distribution
of the age at marriage is specified.
An investigation of several
observed distributions for the age at marriage reveals that the gamma
distribution would provide an adequate representation.
For this reason
the following form of a gamma density is chosen as a probability
density function for the age at marriage:
(0, .
m(x) =
i
>
.
K
.
(x-wl)N-l. exp {f(N)r N
(X-WI)
},
r
lV'here
1
K =
00
}-l .
v N-l e -v dv.
{1 - -- f
feN) (xo-w l
)
r
Clearly m(x) is a truncated gamma distribution.
cation Xo is the minimum age at marriage.
following relations hold:
E(X) =
AN
r. ( - - )
~-l
and
The point of trun-
It can be shown that the
57
Val' (X) =
;r2{~+1
~
_ (
~-1
)2}
~"l
. • . (3.1-.5)
where
00
Ai
=
!.
i - vdv.
v.e
X-WI
..,..Equation 3.1.5 implies that increasing x
results in an increase in
E(X) and a decrease in Var(X).
Given the density function m(x), the function M(x) is determined
by the relation:
M(x)
= Nx .!
x+l
m(u)du,
X
where N is the number of women available for marriage at age. x (at a
x
given point in calendar time).
To determine Nx ' it is assumed that
N is the number of women remaining at age x out of total of 1000 at
x
Therefore, Nx is given by:
N
x
= 1000{
x
IT {l-q(u,u+l)}}
U=WI
where q(u,u+l) is the probability of dying over the month (u, u+l) for
a woman who is alive at u.
The assumption of always having a 1000
women at age WI 'employs a concept similar to that of a current life
table.
The decision to use this assumption was motivated by the desire
to isolate the effect of the function M(x) from any
confou~ded
of the prevailing fertility and mortality conditions.
effects
58
The probability of dying over month (x, x+l) for
women who are alive at x
As a by-product of this numerical analysis, the effect of mortality level on fertility can be evaluated.
For this purpose the
function q(x,x+l) will be estimated using life table functions.
Choosing life tables with different values of life expectancy would
allow the study of mortality effect on fertility.
The problem with
this procedure is that life tables are not constructed in monthly age
intervals.
Regional model life tables
~omputed
by Coale and Demeny
(1962), for example, are constructed in five year age intervals.
Under
these conditions, the monthly probabilities q(x,x+1) can be calculated
in the following fashion.
Consider a life table which is constructed for age groups
(Yi' Yi+l) such that Yi+l-Yi
= c.
Let o(t) denote the force of mor-
tality and make the assumption that 0 (t) is a constant 0iover the age
group (Yi' Yi+l)'
Under these assumptions, the properties of· a
Poisson process imply that:
Pr{dying over (x,x+~x)}
= 0i·exp{-oi(x-y,)}~x,
1
Pr{surviving the whole period (Yi'Yi+l)}
Y'<X<Yi
1
+1
= exp{-co i }.
From these relations, it follows that:
x+l
q(x,x+l) = J
0i,exp{-<\ (u-Yi)}du
x
= eXP{-oi(x-y,)} - exp{-o.(x+l-y )},
1
1
i
To estimate 0i' we equate the survival rate calculated from the
L
life table which is given by
Yi+l Yi+2
L
Yi Yi+l
survival exp{-co i }, to obtain:
with the probability of
59
L
Yi+l Yi+2
L
Yi Yi+l
and therefore,
X~Yi
q(x,x+1)
= {Yi+1
L
L
{Yi+1 Yi+2}
Yi+2J
Yiryi+1
3.2
Yi<X<Yi+1'
Yi LY i+1
Techniques of the numerical analysis
It is instructive to discuss the approach adopted for this study
and to describe the specific techniques employed in it.
The purpose
here is to present the sequence of the numerical calculations and to
discusa the different types of output upon which the analysis is
based.
3.2.1
Sequence of the numerical calculations
The numerical evaluation of the different functions in the model
is carried out according to the following sequence:
1)
Given the forms of the functions p(x) and Qj(x), the unconditional fecundability Rj(X) is evaluated as:
R. (x) = p(x).Q.(x),
J
2)
J
j=O,l, •.. ,a,
O<x<oo.
Given specified values for the functions 8(x) and q(x,x+1),
the transition probabilities p . . (x,x+1), where x is measured
1.,J
in months, are calculated according to the equations given
by corollary 2.6.
3)
For a time-age marriage cohort starting at age Xo at calendar
Ume T, the expected age pari.ty distribution, E{n:(xlx )},
J
0
is evaluated using the recurrence relationship displayed by
60
theorem 2.10.
4)
Using the values of E{nJ<xlx o )}, calculated in step 3, the
mean expected parity at age x, J(x/x o )' is calculated according to the formula:
a
~ j.E{n:(xlx o )}
j=O
J
xo<x<oo.
J(xlx o) = - - - - - - - ,
a
T
~ E{ n (x X o ) }
j
j=O
I
This quantity is used later in Chapter Five.
5)
The probability distribution of X., the age of a woman in a
J
time-age marriage cohort at the delivery of her ith live
birth, is evaluated according to the formula given by
corollary 2.12.
6)
Given the form of the function mT(x), the probability density
of the age at marriage at calendar time T, the function
MT(x) is evaluated as:
= N.f
x x
7)
x+l
T
m (u)du,
O<x<oo.
For a time marriage cohort, marrying at calendar time T, the
expected age parity distribution, E{nj(x)}, is calculated
according to the recurrence relationship given by theorem 2.16.
8)
T
Finally, the probability distribution of X., the age of a
J
woman in a time marriage cohort at the delivery of her ith
live birth, is evaluated using the formula given by corollary
2.17.
Throughout these calculations only parities up to ten are con-'
sidered.
The integrals in steps 2 and 6 are evaluated by assuming that
the functions in the integrand are linear within each month.
61
3.2.2
TYE~s ~f o~t~ut consf,p~fed
In the previous subsection, the fUActions evaluated in steps 3,
4, and 5 relate to a time-age marriage cohort, while the functions
calculated in steps 7 and 8 pertain to a time marriage cohort.
As
indicated earlier, a time marriage cohort is a weighted sum of timeage marriage cohorts where the weights are derived from the function
M(x) calculated in step 6 in the previous subsection.
For this reason,
functions relating to time-age marriage cohorts will suffice in investigating the effect of the different functions included in the model,
with the exception of the function M(x), on fertility.
To study the
effect of M(x), functions pertaining to time marriage cohorts will be
considered.
Whether a time-age marriage cohort or a time marriage cohort is
considered, two types of output will be used in making the numerical
comparisons.
Type I:
The expected age parity distribution
Type II:
The probability distribution of the age at delivery
of the ith live birth.
The decision to consider Type I was motivated by a desire to formulate
the analysis in terms of a variable that is usually observed in
censuses and surveys.
Although Type II is related to Type I in a man-
ner explained earlier, Type II was included for the purpose of studying
the truncation effect.
The truncation effect expresses the fact that
those women who achieve a certain parity are a truncated sample of all
the women who are eligible to reaGh that parity.
It can thus be meas-
ured as the probability of never reaching a given parity.
This proba-
bility is given by the complement of the area under the curve of the
62
probab~~ity
distribution of the age at delivery corresponding to that
parity.
The numerical comparisons are done graphically by comparing the
curves corresponding to Type I for different specifications of the
underlying
f~ctions.
Type II is used to provide a measure of the trun-
cation effect associated with each curve of Type I.
3.3
NUlIl.erical analysis
In this section, the purpose is to study the effects of the
different functions included in the model on human fertility.
It
must be understood that this numerical analysis concerns natural
fertility, I.e.
planning.
fertility in the absence of any form of family
It should also be mentioned that the numerical values
assigned to the individual parameters are arbitrarily chosen for the
purpose of emphasizing the dynamics of the fertility process.
The
exact patterns observed therefore do not necessarily relate to any
particular real situation.
3.3.1
An initial example
To set a pattern for the analysis which follows in this section,
it is useful by way of an example to run through the different steps
which ultimately produce the two types of output considered in this
study.
At the beginning, a decision must be made concerning the numer-
ical assignment of the parameters involved.
Thus, suppose that the
decision is made to represent human reproduction as a renewal process.
This implies that all the basic functions included in the model,
except M(x), are assumed to be constant over the entire reproductive
life.
Assume that (w 1 , w2 ) = (180, 600) months, and that p(x) is
63
determined to be .20 (a value achieved by choosing (p, q, ~) to be
(1, 1, 84».
Further, assume that Q. (x) = .9 for all x (which is obJ
tained by setting d
= 0 for all j and Y = .9 for all j). If it is
j
also assumed that xo' the age at marriage which is required to evaluate
j
the parameters a , is 192 months, then there is sufficient information,
j
to complete the first step of the calculations, i.e. the evaluation
of R.(x), j=O,l, ••. ,lO.
J
Next, it is necessary to specify 8(x) and q(x,x+l) in order to
complete the second step.
To preserve the assumptions of a renewal
process, 8(x) is a constant assumed in this example to be .20 for all
x and q(x,x+l) is assumed to be zero throughout the entire reproductive
period.
After the completion of the second step, i.e.
the calculation of
the transition probabilities p1,J
. . (x,x+l), no additional information
is
.
required to execute steps 3, 4, and 5.
Figure 1 shows the expected
age parity distribution in a time-age marriage cohort.
To complete steps 6 through 8, the function m(x) has to be specified.
To that end, assume that
X
o = 192 months, r = 36.2, and N = 2.56.
The function M(x) is then calculated by completing step 6.
tion of steps 7 and 8 is then possible.
The comple-
Figure 2 gives the expected
age parity distribution in a time marriage cohort under' these numerical
specifications.
The curves illustrated in Figure 1 are a'representative example
of Type I output in a
time~age
marriage cohort.
The curves in Figure 2
are a representative graphical example of the Type II output in a time
marriage cohort.
It is the purpose of this section to examine the
effect of the parameter variation on these curves and to interpret the
e
·e
e
Figure 1
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,lO, in a TimeAge Marriage Cohort with x o=16 years,
p(x)=.2, Q.(x)=.9, and q(x,x+l)=O for all
x and the initial size of the cohort is 1000.
CQ
o
·
CD
-J
r",
·-'
,......,
""'o
~
~C1l
'-'-4'r
t--.,.."
s::cn
........,
J:>;.1
.-l
I
o
.-l
Cod
·...
OJ
....
Q)
l\J
0-1
D.
t
90.0
I
I
..
;
180.
f
.
I
f .. ··f
;-..;
,---'
;-'" - •.• -....... "'
i ·
2'0.
.,"
'.
"1
_ .. - "R
5eo.
I
'1S0.
r
$'tO.
I
'•
720.
&30.
Age in Months (x)
""\
0'\
-I::'-
Figure 2
....
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ..• ,lO, in a Time
Marriage Cohort with x o=16 years, r=36.2,
N=2.56, and the initial size of the cohort
is 1000 •
q;,
f\3
.......
~
_.....
C7
'0
,-...
~
........
l-'~~
'-r'
I't:l
...J
.-'
I\J
~
....
~
o.
·e
90.0
]50.
Z10.
3&0.
'iGO.
Age in Months (x)
--
s'tO.
6'0.
e
66
practical significance of these variations.
With this purpose in mind,
this example will be considered as a starting point for the analysis
in the remaining of this section.
3.3.2
The
f~ction
p(x)
According to section 3.1.1, the function p(x) is assumed to take
on the form:
p (x) =
0,
When this form of a beta curve was introduced, it was observed that
the shape of this curve is determined by the parameter pair (p, q).
It was further observed that the dependence of p(x) on
W
2
a, WI' and
has nothing to do with its basic shape but'is reflected strictly in
the magnitude.
The parameter pair (WI' W2 ) represents the age span over
which reproduction is possible.
The purpose of this analysis is to evaluate the effect of the function p(x), as a whole, on the fertility process.
This amounts to
investigating the effect of both the shape and the magnitude of p(x)
on fertility.
To accomplish this purpose, the analysis was designed
to provide for the following variations in the function p(x).
1)
Three levels of magnitude for p(x) are considered.
The first
is a high level with p(x) assuming values up to .20; the
secopd is a medium level with p(x) having a maximum of .11,
and the third is a low level where p(x) assumes values up to
only .02.
67
2)
Two basic shapes for p(x) are considered.
The first is that
of a constant function and the second is that of a unimodal
function.
Obviously, these variations are achieved by considering different
combinations of the parameters a, p, q, WI' and w2 •
For this purpose
we recall equation 3.1.1 which implies that the mode of the function
p(x) occurs at the age point x, such that:
(p-1)W 2 + (q-1)w I
X
= --~-----p+q-2
. . • (3.1.1)
Based on empirical observations (Henry (1965», it is assumed
.
throughout this chapter that p(x) achieves its mode at age 25 years
i.e.
300 months.
With the parameter pair (WI'
W )
2
taken to be
(15, 50) years, this assumption implies that, unless p=q=l, p is always
less than q.
In other words, when p(x) is not a constant function
it is represented by a unimodal curve which is skewed to the right.
As a starting point, consider the example introduced in section
3.3.1.
In that example we had p(x) = .2 for all values of x and the
function specification in the example conforms to the definition of a
renewal process.
Figure 1 shows the expected age parity distribution
corresponding to this example.
First investigation of this figure
indicates that almost all the women considered will reach their 10th
parity by age 375 months (approximately 31 years).
This indicates that
the numerical values assigned to the different parameters in this
example result in a high level of reproductive capacity.
The assump-
tions of a renewal process are reflected in the similarity of the age
distribution for different parities (except for parity zero).
To
68
measure the truncation effect, the area under the probability distribution of X., j=O,l, ••• ,lO, was calculated.
J
In this present example,
it was observed that this area is 1 for all values of j=O,l, .•. ,lO.
This implies that each woman in the cohort will have a 10th birth with
probability one and, therefore, the truncation effect for parities
one through ten is zero.
When reducing p(x) to a medium level of .11 for all x, the resulting output is shown in Figure 3.
As might be expected, the reduction
in p(x), through reducing the reproductive capacity, results in a
higher degree of dispersion in the curves of the expected age parity
distribution.
It was also observed that the area under the probability
distribution of X is 1 for all j=O,l, •.. ,lO, which indicates that the
j
reproductive capacity is high enough to produce no truncation effect.
To detect a possible pattern, a third example was considered
where p(x) was given a low value of .02 for all x.
the corresponding output.
Figure 4 shows
Here it is observed that this low value of
p(x) results in a high dispersion in the expected age parity distribution such that large numbers of women are still in parities less
than 10 at the end of the reproductive period.
This high degree of
dispersion travels to the probability distribution of Xj and results
in reducing the area under their curves over the reproductive period.
These areas are given as:
Variable X
j
Area
Xl
.997
.978
.924
.820
.672
.503
.342
X2
e
X3
X4
Xs
X6
X7
Figure ·3
The Expected Age Distribution for Women
Whose Parity is j, j=O,I, •.• ,IO, in a TimeAge Marriage Cohort with p(x)=.ll for all x.
--
CI
•CO
·t:J....
r-'-.
""'
_.
>:a.
0<1\
>: ...
'-"
;:'''
'-r'
~
r-l
I
0'"
r-lC"
...
•
·-'"
CIO
~"""
D.
I
90.0
" (I J f IN-P'&b-~~I~~i~
ISO.
210.
S60.
CiSD.
I
I
I
6'D.
63D.
72D.
Age in Months (x)
·e
e
e
'"
\0
e
e
e
J?l
to
Figure 4
I
l
I
.JI
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,IO, in a TimeAge Marriage Cohort with p(x)=.02 for all x •
\
I \
~.l
I
\
I
~~Ji
\
I
\
i
\
oCft
~~!
(J1 .
......,
~
rdr-)
'-0-'
I:il
...-l.
I
ow)
....
...-l'"
....
~~.,,,"
.--..'--.
.'"
~
=
:;:
~_.----
-
1
Q.
I
90.0
!
f
I
]80.
<
<
--~....---.
- ~-- -~.
===t..
Q'' ' ~'F===-~-:- ..
2"10.
3&0.
~SO.
Age in Months (x)
S~Q.
T
.:: -_.
-630.
1
120.
.....,
o
71
(Variable X.)
J
(Area)
.212
.120
.062.
The complements of these areas, as mentioned earlier, "express the truncation effect which results in the present example from the low reproductive capacity that requires a much longer reproductive period in
order for all women to achieve all different parities.
It is observed
that as parity increases, the effect of truncation becomes more serious.
Next, the effect of the shape of p(x) on fertility is evaluated.
For this purpose, p(x) is represented by a unimodal curve over the
age period (180, 600) months with its mode occurring at age 300 months.
The value of p(x) at its mode is taken to be .20.
output corresponding to this "example.
Figure 5 gives the
In this figure, it is observed
that fertility performance starts at a low level as a result of the
left tail of p(x).
It then increases to a high level around age 25
years where p(x) is at a maximum.
In other words, a unimodal shape
for p(x) creates a selection procedure according to which women tend
to choose the ages at which p(x) is at a maximum to achieve their fertility.
To investigate further the effect of the shape of p(x), con-
sider another example where p(x) in the current example is replaced by
a constant representing its average value over the reproductive period.
1
The average value of p(x) is calculated as
value is .11 for the current example.
w
----~! 2p(~)d~.
WI-W2 W 1
This average
The case where p(x) = .11 was
studied earlier and its corresponding output is shown in Figure 3.
figure, when compared with Figure 5, shows clearly the existence of
the selection procedure associated with the unimodal shape of p(x).
This
e
e
e
Figure 5
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, .•• ,lO, in a TimeAge Marriage Cohort with a unimodal p (x) "
where max p(x)=.20.
'D
'0
•
CO
·i\J
..,J
,-'-,
x.
...
_.!)CA
,.."
oCA
~.,..,
'-,-J
.
~
r-l
10
•
...
...-10
•
-
·
CD
f\.1
:0-\
O.
I
I' (·
90.0
180.
f •
Ci 'I
fi
j; -, . 1~
210.
$So.
I
I
I
I
~O.
S\o.
&30.
720.
Age in Months (x)
"-J
N
73
In conclusion, both the magnitude and the shape of p(x) has a
marked effect on the fertility performance.
The magnitude of p(x),
it is observed, affects the overall reproductive capacity while its
shape affects the" timing of births by causing women to have their
births at ages where p(x) is high.
Reducing the overall reproductive
capacity is reflected in a high degree of dispersion in the age parity
distribution while the selection procedure of ages corresponding to
high fecundabi1ity shows in the concentration of births around those
ages.
3.3.3
The function Q. (x)
,
----
The overall effect of the function Q.(x) is similar to that of
J
p(x) by virtue of the relation R.(x) = p(x).Q.(x).
J
The function
J .
Q.(x) however is suspected of being more effective since it is related
J
to the age at marriage.
In section 3.1.2, the function Qj(x) was assumed to take on the
form:
Q.(x)
=
J
y .. exp{-d (x-a.)},
j
J
d.>O, O<y.<l,
J=
J
J
x>aJ..
=,~
In this function, the parameters y. and a. are related by virtue of
J
J
the fact that Yj is the value of Qj(x) at a j .
It is also observed
that the parameter d. expresses the rate of change in Q.(x).
J
J
a
j
Since
is determined by the age at marriage, the effect of the parameter
pair (a., y.) is confounded in that of the age at marriage
J
J
discussed in section 3.3.5.
which is
The primary concern here then is to study
the effect of the parameter d.
j
considered.
Xo
Five patterns for the parameter d
j
are
74
1)
is the initial example of section 3.3.1 where all
Pattern I
the parameters in the model are constant and d.
J
is zero for all j.
2)
is the same as pattern I after taking dj to
Pattern II
.
be .001 for all j.
3)
Pattern III is the same as pattern II after taking dj to
be .01 for all j.
4)
is the same as pattern III after assuming a
Pattern IV
decreasing pattern for d
j
d. = .006 - .0005j,
J
5)
Pattern V
of the form:
j=O,l, •.• ,lO.
is the same as pattern IV after taking the function
p(x) to be unimodal with its mode at age 300
months and the value of p(x) at its mode is .20.
The effect of these different patterns is evaluated by taking pattern
I as an initial pattern and then comparing all the others with it.
Pattern I:
.
d .=0 for all j
J
The output corresponding,to this pattern is shown in Figure 1 •
This figure has b-een discussed be fore.
Pattern II:
d.=.OOl for all j
J
With all the other parameters included in the model held constant,
changing d. from zero to a positive value results in a negative exJ
ponential shape for the unconditional 'fecundability R. (x), j=O,l, ••• ,lO.
J
The value .001 for d
j
however seems to be too small to affect the
reproductive performance.
This is reflected in the output shown in
Figure 6 which is almost identical to the output corresponding to
pattern ·1 shown in Figure 1.
n
J
Figure 6
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,lO, in a TimeAge Marriage Cohort corresponding to
Pattern I I of Q.(x).
I
!
:=-/
,
<.oj
; I
l
J
j
i
-.1
I\)
oJ
I
~
~~J
...
~~ ~II\!~~~~~I.,
~
i.
.:tCJ\
r:::""
......,
, Ii
.
I
o
11.
..
cAlI
...
r-l 0'
I l\r\I\llf\!\ 1\
....
'jl"\ r: :
~ \~~
J
.
""'
ClO
o.
"
eo.. 0
...
I
I
\
.
JJ~J\\\~
\.
~\~""',
)\ I
.,. J1"t
<::I'
e
11\1
!
'11i":"
~\LI·ll
Ii'l
r-l
t
".
~ , -.' ::::~.
\.1\
~"iJ.
J50.
•
'3b 0 .
Age i n Months (x)
e·
~60 •
~v~a .
fJ30.
720.
e
v:
76
Pattern Ill:
The value of dj
~n
d =.01 for all j
j
this example is high and causes the functions
Q.(x), j=O,l, ••• ,lO, to decrease rapidly.
J
The output corresponding to
.
this pattern is shown in Figure 7.
In that figure the effect of dj
is reflected by the high degree of dispersion especially over the
•
older ages where the functions Qj(X)' j=0,1, .•• ,10, are at a minimum.
For this same reason, the degree of dispersion increases with parity.
It was also observed that the area under the probability distribution
of X , j=1,2, ••• ,10, is given by:
j
Variable X.
Area
Xl
X2
X3
X4
Xs
X6
X7
Xe
Xg
XIO
1.000
1.000
1.000
1.000
1.000
J
.999
.999
.997
.996
.993
The complements of these areas, though practically small, reflect the
truncation effect.
The truncation effect increases with parity as
a result of the shape postulated for Qj(x),
Pattern IV:
Decreasing d
j
In the three patterns considered above, the parameter dj was
assumed constant for all j.
This resulted in identical curves repre-
senting the unconditional fecundability R (x), j=O,l, .•• ,10, (except
j
for their point of origin).
....
Since a high parity usually indicates a
high reproductive performance, it can be argued that a pattern of
decreasing d
j
is closer to reality than a pattern of constant d •
j
In
this example, dj is taken to be .006 for j=O, then assumed to decrease
uniformly by .0005 over each parity until it is .001 for j=IO.
Under
Figure 7
The.Expected Age Distribution for Women
Whose Parity is j, j=O,l, •.. ,lO, in a TimeAge Marriage Cohort corresponding to
Pattern III of Q.(x).
(0
CJ
(0
J
-J
I\J
-J
,.......,
"....
_.....
oCJ1
~
~cn
........
Q•...,
'-,oJ
~
r-l
I
o
a:J
r-l 0-
....
CD
lU
~.4
D.
e
I
~~iiiiii~~~~~_"'
I
90.0
180.
270.
3&0.
'tSD.
Age in Months (x)
e
l
)
-
__"",
"", __
i
5~D.
•
I
,
630.
720.
e
::j
78
this specification, a woman with a high parity has higher fecundability
than a woman with a low parity.
ing to this example.
Figure 8 gives the output correspond-
As might be expected, the expected age parity
distribution displays a smaller degree of dispersion than that
corresponding to pattern III.
In fact, it is observed that the curves
in Figure 8 are almost identical to those in Figure I which represent
the case of a renewal process.
This observation is also valid to
some degree for all the previous variations of d .
The implication
j
of this result is that the effect of the function Q (x) can be conj
founded in that of p(x).
Pattern V:
Decreasing dj and a unimodal p(x)
Throughout the above discussion, it was assumed that all other
functions included in the model are constant for all ages x.
To
bring the analysis closer to reality, the pattern of dj used in
pattern IV above, together with a unimodal curve for p.(x) whose mode
occurs at age 300 months where p(x)=.2 at this point, is used in the
Figure 9 shows the expected age parity distribution
current example.
corresponding to this example.
The differences between this figure
and Figure 5, which corresponds to the same form for p(x) but with a
zero value for d
j
(for all j), seem negligible.
This observation
supports the conclusion that the effect of the function Q.(x) is
J
controlled by the effect of P(x).
In conclusion, the above examples, which are limited to studying
the effect of the parameter d
j
in the function Qj(x), reveal the
following.
1)
Changing the magnitude of the shape of Q.(x), given that all
J
other functions are constant, seem to have an effect on the
Figure 8
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,lO, in a TimeAge Marriage Cohort corresponding to
Pattern IV of Q.(x).
(Q
CJ
<0
J
.....Ai
-:-'-0
oCA
""'
>: ....
--"
>:cn
'-'
1=:""
'-0-'
~
...-i
I
04.:J
.....
...-idt
.
CD
~
0-\
o.
e
r
I I II I
90.0
180.
W:tp;t3?z--~---I----r---""'--~
_
i i i
1
270.
560.
'fSO.
Age in Months (x)
j
e
5'0.
72D. ~
630.
e
"
e
e
e
Figure 9
The Expected
Whose Parity
Age Marriage
Pattern V of
~
.o
<0
Age Distribution for Women
is j, j=O,l, ••• ,lO, in a TimeCohort corresponding to
Qj(x),
...I
I\J
....
--=-+
,...
0<11
>< •
........ CII
C::.,...,
'-r-'
f;I::l
r-l'
I
or-l1A
0'
+
-
.cu
I\l
04
o.
I
90.0
1'll(I(N5it;t:~~G~$~?~~~3~~~A~?1r~----rl----~I~----~l-----I
180.
270.
360.
'ISO.
Age in Months (x)
540.
630.
720.
~
81
fertility
pe~formance.
This effect is usually reflected in
changing the degree of dispersion of the age parity distributions (especially for later parities).
2)
It is observed that the effect of Q.(x) is dominated by the
J
function p(x).
When p(x) is high, a corresponding high Qj(x)
reinforces that by showing a more selection effect.
'But
when p(x) is constant the effect of changing Q.(x) seems miniJ
mal.
'This observation is supported by the relation
= Q.(x).p(x)
J
and by virute of the fact that Q.(x)
J
usually becomes small toward the end of the reproductive
period where the major part of the fertility performance has
already been realized.
3.3.4
The function q(x,x+l)
The function q(x,x+l) is the probability of dying over month
(x, x+l) for women who are alive at the beginning of that month.
Regional model life tables for females (West) calculated by Coale and
Demeny are used to evaluate q(x,x+l) in the manner explained in
section 3.1.7.
To study the effect of this function on fertility,
two mortality patterns are considered.
The first, indicating a high
mortality level, is based on a life table with life expectancy of 35
years.
The second, representing low mortality, is derived from a life
table with life expectancy of 65 years.
In evaluating the effects
of each of these two patterns, the other functions in the model are the
same as considered in pattern V in section 3.3.3 (i.e.
decreasing d
j
and unimodal p(x).
Figure 10 shows the output corresponding to the high mortality
level.
In that figure the expected age parity distribution is shown.
e
e
e
Figure 10
The Expected Age Distribution for Women
Whose Parity is j, j=O, 1, ••• ,10, in· a TimeAge Marriage Cohort with life expectancy
of 35 years •
co
CJ
<0
.....,
.
I\J
.....J
,.....
....
oC1)
~
XCn
y
.,..,
-.
s::
~
...-I
'OCIJ
...-10-
....
-
.I\J
QJ
0-\o.
I
90.0
I
I
I
I
~"~iiiI~Ioio·~_."~~---I-----'----"""f"'-~-180.
270.
630.
720.
3&0.
'ISO.
5'0.
Age in Months (x)
00
!',)
83
comparing
t~is
to Figure 9, which corresponds to the same conditions
except for mortality which is considered zero, the effect of q(x,x+l)
is apparent.
The shape of the curves,in both figures are the same,
but the magnitude is different.
When mortality is zero the curves
are higher than when mortality is high.
This expresses the fact
that some women in the later case are lost to death and do not have
the chance to contribute any more births.
It is observed that this
effect becomes more obvious as parity increases which is a natural
result since higher parities are usually associated with higher ages.
The same observations persist when studying the probability distributions of Xj , j=1,2, •.. ,10.
The areas under these distributions are
given as:
r
Variable X
j
Area
Xl
X2
Xg
X4
Xs
X6
X7
Xa
Xg
XIO
.982
.971
.963
.955
.946
.938
.930
.922
.914
.906
The complements of these probabilities signify the effect of mortality
since all the areas were 1 in the case when mortality was zero.
It
follows therefore that the effect of mortality on fertility is of a
truncation nature.
births.
Death prevents some women from achieving further
When mortality is high, as is the case in this example, the
effect of this truncation is not negligible especially for higher
parities where it is observed that 6.2 per cent of the women in the
cohort do not achieve a parity larger than 5 and that almost 10 per cent
of them do not have more than 9 children.
84
A ~u~ther result was
opse~ved ~n
parity {calculated in step 4 of
the values of the mean expected
sect~on
3.2.l}.
Comparing the mean
expected parity values in the case where mortality is high, with the
values 'corresponding to the case when mortality was zero, reveals
that there is always a difference of the order of 1/100 between the
two values at any given age with the mean expected parity in the case
of zero mortality always higher and with the difference increasing
with parity.
This may indicate that women who die before the end of
their reproductive period probably have a slightly less reproductive
capacity than those who live beyond their reproductive ages.
Careful
evaluation of this assertion requires an extensive analysis which is
beyond
t~e
concern of this study.
These observations are also recorded in Figure 11 which corresponds to the pattern of low mortality.
vious as in the case of high mortality.
They are not however as obIn the case of low mortality,
for example, the areas under the probability distributions of Xj
are as:
Variable X
Area
Xl
X2·
X3
X4
Xs
X6
X7
X8
Xg
Xlo
.995
.993
.990
.989
.987
.986
.984
.982
.981
.979
j
..
These numerical values show that the truncation effect
p~oduced
by
this low mortality level is small since women in the cohort have a life
expectancy of 65 which covers the entire reproductive period.
A
Fi.gure 11
rhe Expected Age Distribution for Women
Whose Parity is j, j=O,1, ••• ,10, in a TimeAge Marriage Cohort with life expectancy
of 65 years.
'D
·co
CII
-..J
I\.l
·
..,J
--c:.Jt-.
0
~
....
~CI1
.......
....,
~
'"""'
~
.-l.
Io
IM-
·....
.-l'"
-..
Q:I
·
I \.l
'0.'
?
e
"~
I
IJI'II'~t~~=.~~.~~C~"~~?wAA~~~~~~
90.0
180.
""
I
S&O.
'1$0.
Age in Months (x)
27D.
~
r
")
e
~
I
sq.o.
~
I
_
I
720.
630.
e
00
VI
86
mortality level between the two considered here is, of course,
expected to produce a truncation effect between the ones shown above.
This discussion of the effects of
common belief.
q(x,:~l)
seems to reinforce a
From this discussion, it is clear that a high mor-
tality level among mothers, as displayed by a life expectancy of 35,
does not reduce fertility in a substantial way.
This may suggest that
the balance between deaths and births observed in the first stage of
the demographic transition was caused by high infant mortality and
not by high maternal mortality.
Although the specific effects of
infant mortality on fertility can be evaluated through the present
model, no attempt is made to do so since this is beyond the scope of
this work.
3.3.5
The age at marriage
X
o
Given a finite period for reproduction, as indicated by (WI' W2)'
the age at marriage is an important variable in the fertility process
since it determines the portion of (WI' W2 ) over which a woman is at
the risk of conception.
The age at marriage xo' while it does not
have any effect on the conditional fecundability p(x), operates on
fertility through the probabilities Q.(x) by affecting the values of
J
the parameters a
j
and Y in that function.
j
Under the assumptions of a renewal process, the effect of age at
marriage is obvious as any delay in
Xo
is reflected in a location shift
to the right in the curves, representing the different types of output of the model, by exactly that much delay.
evaluate the effects of
X
The purpose here is to
o on fertility in a more realistic context.
To that end, the following postulates are made concerning the basic
87
functions in the model:
(i)
p(x) is assumed unimodal with its mode at age 300 months
and the value of p(x) at its mode is .20,
(ii)
the parameters in the function Q.(x) are chosen such that
J
Yj decreases as a
j
increases and that d
j
is decreasing in
a pattern identical to pattern V in section 3.3.4, and
(iii) a low mortality level is considered with life expectancy
of 65 years.
Six values for
X
o are considered in this analysis.
16, 18, 20, 22, 25, and 28 years.
They are ages
The areas under the curve repre-
senting the probability distribution of Xj corresponding to each value
of
X
o are given in Table 3.1.
An investigation of Table 3.1 indicates that the possible effect
of
X
o is to truncate the probability distritution of X.'
J
Specifically,
this table reveals the following.
1)
The effect of age at marriage on the total fertility is
negligible if
of
X
X
o is less than 25 years.
o becomes apparent.
the function p(x).
After age 25 years the effect
This is a result of the shape postulated for
In the curve assumed for this function, it is
apparent that a woman assumes her maximum reproductive capacity over
the ages (20, 30) years.
If the age at marriage is not well beyond
age 20 years, it is not expected to affect the total fertility especial1y since the reproductive period is assumed (15, 50) years which
is sufficiently long in relation to the numerical values assigned to
the basic functions in the model.
2)
The total fertility for women who marry at age 16 is very
slightly less than the total fertility for women who marry after that
88
TABLE 3.1
.
Area under the Probability Distribution of X., j=1,2, ..• ,10
J
for Different Values of the Age at Marriage x
0
I
x
20
22
25
28
.996
.996
.996
.996
.996
.994
.994
.994
.994
.992
.991
.992
.992
.992
.992
.988
4
.989
.991
.991
.991
.989
.982
5
.987
.989
.989
.989
.986
.974
I
I
6
.986
.987
.987
.987
.982
.961
7
.984
.985
.986
.985
.977
.939
I
8
.982
.983
.984
.982
.970
.906
9
.981
.982
.982
.979
.960
.860
10
.979
.980
.980
.975
.947
.796
....
I
I
1
j
16
1
.995
2
.993
3
I
18
e
"
i
0
~
I
I
89
age.
This difference may be taken as an indication that women who
marry at an early age usually experience a shorter effective marriage
duration (Henry, 1957).
The lack of real data from which estimators
for the different parameters in the model can be calculated, prevents
any realistic investigation of this assertion.
These observations are reinforced upon investigating the different
graphs representing the output of the model for different values of
xo '
Figure 12 which corresponds to a value of x o=16 years is consid-
ered as the initial point of thic investigation.
to xo=18.
Figure 13 corresponds
In that figure it is observed that the delay in.xo from
16 years to 18 years has a marked effect on the internal structure of
the age parity distribution as displayed.
The selection process,
which is discussed earlier, according to which women tend to select
their ages where fecundability is high to have their births is operative and its effect shows in earlier parities as a result in the delay
in xo'
This pattern persists if Xo increases to 20 years where the selection process is more obvious as shown in Figure 14.
The closer Xo gets
to the center of the maximum fecundabi1ity, the more obvious the
selection process appears in the different output of the model.
This
observation is supported by Figure 15 corresponding to a value for
Xo of 22 years.
When the age at marriage is increased to 25 years, the selection
process at lower parities continues but other changes in the shape of
the curves are also observed.
The degree of dispersion in the expected
age parity distribution shown In Figure 16 starts to increase as a
result of relying on fecundabi1ity values past the plateau of its curve.
90
This
~ncrease
in the degree of dispersion is what causes the trun-
cation effect since a high degree of dispersion could mean that the
available reproductive period will not be long enough to accomodate
the whole curve.
This trend continues and becomes more obvious, re-
suIting in higher truncation, when
Xo
is raised to 28 years.
Figure 17
gives the output corresponding to this case.
In conclusion, this analysis seems to indicate that as long as
the ages available for reproduction are sufficiently long, as is the
case in this analysis where (WI' W2) is taken as (15, 50) years, in
relation to the numerical values assigned to the different values
assigned to the different parameters in the model, the effect of the
age at marriage on total fertility appears to be minimal.
To produce
any effect on the total fertility through changing the age at marriage
requires a choice of
maximum.
Xo
that is beyond the age at which p(x) is at a
In this case women will reproduce according to the right
•
tail of p(x).
The dependence on these low values for p(x) increases
the degree of dispersion in the curves of X. thus creating a possibilJ
ity, depending on the level of p(x) and on the values of (WI' W2 ), of
the reproductive period being too short to accomodate the whole curve
and thus reducing the total fertility.
The effect of increasing the age at marriage seems to be more of
a redistribution of births rather than a limiting of them.
This effect
is achieved through a selection process according to which most births
are concentrated over the ages where the reproductive capacity is high.
./
Figure 12
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,10, in a TimeAge Marriage Cohort with x o=16 years.
-.0
·to
Q
-..J
I\J
·.....
_.
,-...,
oc:.lt-
~+
~tn
'-'
t=:"""
~
~
r-i
I
o~-
r-ic:Jt
+
....
·""
I\J
Q-\
?
I
1"("'~:~:&~'~~~~~~~R~3~W~~~~~_~_~I~
90.0
160.
5&0.
27D.
uSD.
"__~_-~I----~I
Slf-D.
630.
12D.
Age in Months (x)
e
e
t
"
e
\0
.....
.
e
e
~
e
Figure 13
'.
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,lO, in a TimeAge Marriage Cohort with x o=18 years.
~
·
<2
(G
·~
~
.......
0'"
><.
><
.......
"'"' CIII
-ell
-.
~
I:l
P<l
.-t
I
Cllt
...
°Ot
.-t •
·'"-c»
~_,
O.
I
I
90.0
180.
I Irr
«J:;C1lit£:-&-~
270.
S6D.
Age in Months (x)
•
,
I
I
I
"SO.
SltD.
530.
12D.
\0
N
Figure 14
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,10, in a TimeAge Marriage Cohort with x o=20 years.
co
·
G
(G
·~....
,.....,
"......
-.
aU
>: ...
>:cn
......
~..,
.
"'-0-'
l'::I
.-I
'0 d
·...
.-10'
·QI
N
~-'
D.
e
I
90.0
I ', ,
180.
~.[~J~-tii"'''*Iii=:t,
.....L;;''''~'·~iil7.~ij)iii~iiliiiiliii~ii._T
...
T
' "
270.
'60.
'ISO.
Age in Months (x)
e
p
6..0.
- - . , . - - - - - -..
6SD.
I
72D. ~
e
"
e
e
e
Figure 15
co
•
CI
CO
The Expected Age Distribution for Women
Whose Parity is j, j=O,I, ••• ,10, in a TimeAge Marriage Cohort with x o=22 years.
~
....
"
--
oCft
>:: ...
->::CIt
"
~
."
s::
......
f;I::l
.-I
I
.
o
CO
...
.-10'
"
CD
"I\J
o-\~---tl--~I=-~~
O.
90.0
J80.
2'7D.
S60.
qSO.
•
Age in Months (x)
I
$'0.
I
630.
I
720.
\0
~
Figure 16
to
The Expected Age Distribution for Women
Whose Parity is j, j=O,l, ••• ,lO, in a TimeAge Marriage Cohort with x o=25 years.
o
·
UlI
·~....
- ....
..........
oc.n
~
--
~CII
....,
s::
........
.
~
r-l
'0.
r-lc:l'
·...
-
·
CllI
f\.J
CI
·
.l
D.
~~fm'';':~----,I
I
I
I
90.0
180.
270.
e
•
S&O.
&tSO.
Age in Months (x)
e
~,.o.
630.
720.
e
\0
In
•
e
e
e
Figure 17
co
The Expected Age Distribution for
Women Whose Parity is j, j=O,1, ••• ,10,
in a Time-Age Marriage Cohort with
x o=28 years.
o
•tQ
·~....
r-'->
,-...
-.
aCft
:<.-
~Q
s::.,..,
......,
.
r>::l
.-I
10
Cd
.-10-
•
c.1
GIl
•N
I
~
:q
J
D.
I
90.0
I
180.
I
270.
IJ1Ji1m~FL
S60.
fi50.
Age in Months (x)
6'tD.
(.-- -=;
63D.
'720.
\0
0\
97
3.3.6
The funstion M(x)
As indicated earlier, a time marriage cohort is a weighted sum
of time age marriage cohorts where the weights are derived from the
function M(x).
The functionM(x) is the expected number of women in
a time marriage cohort who marry at the age month (x, x+1).
According
to section 3.1.6, M(x) is determined on the basis of x o ' the minimum
age at marriage, and the parameter pair (r, N).
To evaluate the effect the function M(x) might have on fertility,
consider as a starting point the example introduced in section 3.3.1
In that example, fertility was treated as a renewal process and we had
x o=16 and (r, N)=(36.2, 2.56).
Figure 2.
The area under the
The resulting output is shown in
~robabi1ity
distribution of X ,
j
j=1,2, ••• ,10, in this case is given as:
Variable X.
J
Area
LOOO
.999
. 999
.999
.999
.999
.999
•
.998
.998
.998.
The small deviations of these values from I indicate that M(x), in
this example, has a negligible effect on the total fertility.
The
effect of M(x) on fertility is reflected in the internal structure
of the expected age parity distribution.
This effect is apparent in
the points of inflection observed on the curves of Figure 2.
These
inflection points reflect the low input to the marriage system at the
early part of the reproductive period.
Since the number of women with
98
a given parity at a given age reflects the net balance between those
who move into that parity and those who move out of it at that age,
an inflection point indicates that the sign of this balance changes at
that point.
Since all functions in the model (except M(x»
stant, this change in sign is caused by the function M(x).
are conIt is
observed that these inflection points were initiated at parity zero
as a result of the early part of the left tail of M(x).
This effect,
then, travelled through all parities as observed in Figure 2.
An example postulating the assumptions of a renewal process,
though instructive, does not provide a realistic means for studying
fertility.
For this reason, it was decided to complete the analysis
in this section in a more realistic setting.
To this end, the basic
functions of the model are chosen as in section 3.3.5.
Further, it
was decided to fix the parameter pair (r, N) in the function M(x) and
to vary the minimum age at marriage xo.
The pair (r, N) was chosen to
be (36.2, 2.56) which gives a mean age at marriage of 22.7 years and
a standard deviation of 4.84 years.
considered.
Six different values for X were
o
They correspond to ages 16, 18, 20, 22, 25, and 28 years.
The output corresponding to these values is shown in Figures 18 through
23.
According to section 3.1.6, increasing the value of Xo results in
an increase in the mean age at marriage and a decrease in its standard
deviation.
Table 3.2 records the mean and standard deviation of the
age at marriage corresponding to each value of Xo as well as the area
under the curves of the probability distribution of
in each case.
xJ,
j=1,2, .•. ,lO,
In this table several observations are made.
TABLE 3.2
Area under the Probability Distribution of
.
~,
J
j=l,2, ••• ,lO,
for Different Patterns of the Age at Marriage
r----.
x~.=16
Mean=22.7
St.Dev.=4.84
x0 =18
Mean=23.6
St. Dev. =4.59
x0 =20
Mean=24.9
St.Dev.=4.34
x0 =22
Mean=26.5
St.Dev.=4.15
x0 =25
Mean=29.2
St.Dev .=3.95
x0 =28
Mean=32.0
St.Dev.=3.80
1
.967
.966
.962
.955
.940
.918
2
.951
.951
.947
.937
.915
.879
3
.935
.936
.932
.919
.889
.838
4
.918
.921
.916
.899
.862
.795
5
.902
.906
.900
.880
.833
.750
6
.887
.891
.883
.861
.802
.703
7
.871
.875
.867
.840
.769
.649
8
.857
.860
.850
.818
.736
.594
9
.842
.845
.834
.797
.701
.535
Marriage
Pattern
j
10
e
.829
.830
.817
•
e
.775
\0
\0
.665
.473
e
100
1)
The effect of the function M(x) on total fertility, expressed
in the form of a truncation effect, is apparent.
As the minimum age
at marriage increases, thus increasing the weights assigned to ages
beyond the plateau of the reproductive capacity, the truncation effect
becomes more serious.
2)
The total fertility for women whose age at marriage starts
at 16 years is slightly less than the total fertility for women who
start marrying at age 18 years.
A possible explanation for this obser-
vation is that women who marry at a very early age usually have a short
effective duration of marriage since they experience higher probabilities of sterility and divorce.
The mechanism through which the function M(x) produces a truncation effect is recognized upon studying the different graphs representing the output of the model for different values of
Xo
(and thus
different values for M(x».
The following remarks are made after
examining Figures 18 through
11.
1)
The effect of changing the shape of the function p(x), in
the presence of M(x), on fertility is shown by comparing Figure 18
with Figure 2.
While M(x) is the same in both figures, P(x) underlying
Fisure 1 is a constant while that underlying Figure 18 is unimodal.
No
inflection points are observed at the early ages in Figure 18 as a
result of the selection process which was discussed earlier.
It is
also noticed that the unimodal shape of p(x) resulted in a larger dispersion in the curves of that figure, as a result of low values for
p(x) toward the end of the reproductive period, and thus resulting in
a larger truncation effect.
It is clear, therefore, that in the pre-
sence of a unimodal shape for p(x), which is a realistic assumption,
101
the function M(x) produces the most truncation effect'by assigning
higher weights to older ages.
2)
The expected age parity distribution corresponding to
x o=l6, 18, 20, 22, 25, and 28 is shown in Figures 18, 19, 20, 21,
and
~
respectively.
1£~
Investigating these figures indicates the inter-
nal changes in the age parity distribution resulting from higher
values of M(x) at older ages.
The patterns observed in these figures
result from the interaction of two factors.
The first is the function
M(x), and the second is the selection process according to which women
tend to select the ages where their reproductive capacity is high,
usually the ages (20, 30) years, to have their births.
If the function
M(x) attains its maximum early in the age group (20, 30), the expected
age parity distribution acquires a small degree of dispersion as observed in Figures 18,
respectively.
~,
and 20 corresponding to xo =16, 18, 20 years
When the period corresponding to the plateau of M(x)
deviates from (20, 30), a larger dispersion in the expected age parity
is observed.
When this deviation occurs to the left, i.e.
at young
ages, the large dispersion appears in the distribution of low parities
as observed in Figure 18.
i.e.
When the deviation occurs to the right,
at older ages, the large dispersion is observed in the distri-
bution of high parities as shown in Figures 22 and 23 corresponding
to x o=25 and 28 years respectively.
Obviously, a high dispersion in
the distribution of high parities is important from a family planning
point of view since this dispersion is what causes the truncation
effect, thus reducing the total fertility.
•
e
~
e
e
Figure 18
-""
The Expected Age Distribution for Women
Whose Parity is j, j=O,1, ••• ,10, in a Time
Marriage Cohort with x o=16 years.
tn
.-.
~
o
'0
,......,
,...,
......~
1-'.,-,
~
......,
r::I-J
I\.J
-J
...,
..
~
~
,
I
i
o.
90.0
150.
((
r
« , , , (
I
I
I
l70.
I
'&0.
Age in Months (x)
I
I
I
I
~SO.
5'0.
630.
720.
f-I
o
N
Figure 19
.....
The Expected Age Distribution for Women
Whose Parity is j, j=O,1, ••• ,10, in a Time
Marriage Cohort with x o=18 years •
o:J
f\:J
.....
.....
~
.....
c=o
,...., .\0
,-...
x
'-'
f-'
r::""
.........
f;t:l
...J
""
-.J
..,
co
.....
~--=
.. ,
tJ •
I
"
90.0
180.
,
r
,
!
,
I'
I
J
I
,
eno.
ZZ
LESE
Ii!!!
I
I
1
I
I
'60.
CfSO.
s,o.
630.
720. ~
w
Age in Months (x)
e
•
e
~
e
«
e
•
e
e
Figure 20
--
TQe Expected Age Distribution for
Women Whose Parity is j, j=O,l, ••• ,lO,
in a Time Marriage Cohort with
x o=20 years.
~
<7:1
n)
--...
~
-e::t
'0
,......
,....,
><
'-'
1-"'"
I::
.......,
~-.J
'"oJ
~,
t:1'
....
~
Cr
o.
1l(!IIII/I!
I i i
2-'0.
160.
90.0
I
I
$&0.
~SO.
Age in Months (x)
I
I
5'0.
630.
I
720.
t-'
o
~
Figure 21
The Expected Age Distribution
for Women Whose Parity is j,
j=O,1, ••• ,10, in a Time Marriage
Cohort with xo=22 years.
<n
f\.:J
.-.
'(JO
~
-
'0
"~
'-"
l-' .,.,
I:l
........,
rz:l-.l
....J
l\J
..
".,
~
_ _\
o.
e
I
90.0
r
i
J&O.
I
~&O.
CfSO.
Age in'·Months (x)
! i! I ! ; ! I ! ! , ,
.f
210.
•
e
1
5'0.
I
720. ot-'
,.
630.
VI
e
~
"
e
e
e
Figure 22
....
The Expected Age Distribution
for Women Whose Parity is j,
j=O,l, ••• ,lO, in a rime
Marriage Cohort with
x o=25 years •
t1':I
r.:J
....
~
~
....
c
~
'"x'"'
....,
1-'.'"
s::
'-rJ
~...J
'"
oJ
IN
t:1'
~
_
j
I
o.
90.0
I
160.
11"11 i 1(((
I
'&0.
l"O.
1
I
"150.
Age in Months (x)
I
I
5'0.
630.
I
720.
.....
o
0\
Figure 23
-
The Expected Age Distribution
for Women Whose Parity is j,
j=O,l, ••• ,lO, in a Time
Marriage Cohort with
xo=28 years.
U-'
~
-...
(Jo
o
-
'0
,....
><
'-'
1-'....,
~
'-.-'
rz:I-.J
'"..J
'"
. CO
...
-
J
o.
I
90.0
I
180.
I
e70.
IIIUI'Il'J I
,
'1;0.
CfGO.
Age in Months (x)
I
5'0.
I
1
630.
7eO.
.....
e
e
e
~
~
108
3.3.7
Conclusions
In the previous sections the dynamics of natural fertility
(i.e.
fertility in the absence of family planning) was investigated.
The results of this investigation may be summarized as follows.
1.
Treating the fertility process as a renewal process does not
provide an adequate representation.
Although a renewal process can
produce the same level of total fertility as the real fertility process,
the internal structure of the age parity distribution in both cases is
basically different.
These differences result from the changes in the
timing of births which is caused by the selection procedure built into
the components of natural fertility.
2.
The functions p(x) and M(x) seem to be the most important
functions in the fertility process.
The level of p(x) determines the
overall reproductive capacity over the reproductive period, and its
shape determines the timing of births.
M(x) specifies the number of
women who are at the risk of fertility at different ages.
The inter-
action of p(x) and M(x) plays an important role in determining the
level of fertility performance.
3.
The function P(x) is so important that when it is confounded
with the function Q.(x), the result is that the function QJ'(x) can
J
be essentially overlooked.
.
For this reason the function 6(x), the
probability that a conception at age x ends in a fetal loss, was not
investigated in this analysis since its effect, in the setting of
natural fertility, is expected to be controlled by that of p(x).
function 6(x), however, can be very important in a family planning
setting where abortion is used as a means of birth control.
The
109
4.
The age at marriage affects the total fertility only if that
age is chosen beyond the ages of maximum fecundability.
A final observation concerns the high fertility level observed
in the different graphs of this analysis.
This high level of fertility
results from the interaction of the numerical values assumed for the
basic functions with the specification of (WI'
The effect of the parameter pair (WI'
W 2 ),
W2)
to be (15, 50) years.
though very important, was
not discussed in the present analysis simply because its effect is
obvious.
The parameters (WI'
tion effect.
W2 )
are directly related to the trunca-
Increasing the interval (WI' W2 ) reduces the truncation
effect while decreasing it results in a high degree of truncation.
Had the value of w2 ' for instance, been specified in the above analysis
to be 40 years, as observed in some developing countries, the total
fertility would have been significantly reduced.
In conclusion, the numerical analysis undertaken in this section,
despite its inherent inadequacies, provided some insights into the dynamics of natural fertility.
The results of this chapter will be
utilized in subsequent chapters.
CHAPTER IV
A PRELIMINARY STUDY OF THE ESTIMATION PROBLEM
In this chapter the problem of estimation in the model is briefly
considered.
The discussion is limited to data from complete samples
pertaining to time-age marriage cohorts.
A sample is considered com-
plete if the data are collected prospectively by observing all women
in the sample over their complete reproductive histories.
To make the estimation procedures feasible, it is assumed throughout this chapter that age is measured in integral months and that the
basic functions included in the model are constant within each month.
This assumption is convenient since the reproductive period has a cycle
of a monthly nature.
The content of this chapter is arranged in two sections.
Section
4.1 deals with the estimation problem assuming homogeneity in the basic
functions.
Section 4.2 provides a description of the estimation pro-
cedures when women in the sample are heterogeneous with respect to
the basic functions.
In both sections, estimators are derived for
birth probabilities as well as the unconditional fecundability and some
numerical examples are provided for illustrative purposes.
~
Estimation in h~m~gen~~~cohorts
Under the assumption that the basic functions included in the model
are constant within each month, we have from corollary 2.6:
111
t"l'({e<,..gHL'1 <,..g») ,
j=i
p-/ .(x,x+l) = "ll - q(x,x+l) - exp'({8(x- g )-t}.R (x-g»),
j
.I."J
j=i+l
otherwise,
0,
where Pi ,J.(x,xtl) is the probability that a woman with parity i at
age x will move to parity j over the month (x, xtl).
The estimators derived here are based upon data representing the
age-parity distribution.
The likelihood function for an observed
change in the age-parity distribution of a cohort over the month
(x, xtl) can be constructed as follows.
Assume for j=O, l, ••• ,a that n (x) is the number of women with
j
parity j at the beginning of month x, that nj . (x) is the number of
,J
women, out af nj(x), who are still in parity j at the beginning of
month xtl, and that nj,d(x) is the number of women, out of nj(x), who
die over the monthly period (x, xtl).
The portion of the likelihood
function corresponding to women with parity j at age x is given by:
L . {n j J' (x), n
J
,
0:
j
,
d(x) In. (x), p . . (x,xtl)}
J
J,J
a:
{ Pj ,j ( x,x+l) } n.J ,J. (x) • {q( x,xtl ) }n·J, d (x) .
j=O,l, ••• ,a.
By virtue of independence of these conditional probabilities, the total
likelihood over month x is given by:
a
L = 11 Lj"{nj .(x), n. d(x)lnj(x), p . . (x,xtl)}.
,J
J,
J,J
j=l
The log likelihood is therefore given by:
112
a
In L = constant + E ni,i(x).ln Pi,i(x,x+l) +
i=l
a
+ E ni,d(x).ln q(x,x+l) +
i=l
a
+ E {ni(x)-n .(x)-n
(x)}.ln {I-Pi i(x,x+l)-q(x,x+l)}.
i ,~i,d
i=l
,
• • • (4.1.1)
From equation (4.1.1), it follows that the maximum likelihood estimator
ofp . . (x,x+l) is given by
J ,J
p.. (x,x+l)
J,J
=
Pj ,J. (x,x+l)
where:
nj,j(x).{l - q(x,x+l)}
_
n. (x) - n. d(x)
J
J,
• • • (4.1.2)
By properties of the maximum likelihood estimators, it can be shown
that Pj,j(x,x+l) is a consistent estimator of ~j,j"(x,X+l) which implies
that Pj,j(x,x+l) is asymptotically unbiased.
.
For small samples, however, E{p. j(x,x+l)} is undefined since a
J,
nonzero probability is associated with the event Pj ,j (x,x+l) =0/0
which occurs when nj(x) = nj,d(x).
Two natural alternatives might be
considered to deal with this problem.
according to which" the quantity %
First is to adopt a definition
is considered zero, and second
is to modify the estimator Pj,j(x,x+l) to exclude the case Pj,j(x,x+l)
= %
by defining another extimator
p~
.(x,x+l) as:
J,]
Unbiased estimators for the probabilities p . . (x,x+l) result
J,J
from the method of moments. The estimators follow from theorem 2.10
which is written as:
113
E{n o (x+l)} = Po
0
(x,x+l).E{no(x)},
)
E{n j (x+l)} = Pj-i,j (x,x+l) .E{n _ (x)} + p. j(x,x+l).E{n. (x)},
j 1
J,
J
j=1,2, ••. ,a.
From this theorem it follows that:
p
0,0
P.
(x,x+1) =
.
E{n o (x+1)}
and
E{n (x)}
o
( x, x+l) =
J ,j
E{n (x+1)} - p. 1 . (x,x+l).E{n. lex)}
j
J- ,J
JE{n. (x)}
-......:<.-----.:::---::,.,<------.:-.--
J
Applying lemmas 2.8 and 2.9, we obtain:
E{n . . (x)}
p. j (x,x+1) =
J ,J
,
j=O,l, •.• ,a.
E{n.(x)}
J,
J
.(4.1.3)
From equation (4.1.3), it follows that the moment estimator of
Pj j(x,x+l), ~ . . (x,x+l), is given by:
J ,J
,
n . . (x)
~ . . (x,x+l) = J ,J
J ,J
j=O,l, ••• ,a.
,
n (x)
j
. • . (4.1.4)
It is noticed that
E{~. . (x,x+1)} = p. j (x,x+l), and
J ,J
J,
I
var{~j j(x,x+l)} = E{
}.P . . (x,x+l).{l - Pj . (x,x+l)) ,
,
n.(x)
J,J
,J
J
which is defined only when n.(x) is greater than zero.
J
pIes, it can be
sho,~
that
Var{Pj ,j (x,x+l)} ~
p. j (x,x+l){l - P . . (x,x+l)}
J,
J ,]
E{n (x)}
j
(Chiang 1968, p. 228).
,
For large sam-
114
The same methods we just
descr~bed
can be used to
biological functions underlying the model.
est~mate
the
Since p . . (x,x+l) is deJ ,J
fined as exp({8(x-g )-1}R (x-g»), it follows that {8(x-g)-1}R. (x-g)
j
.
J
can be estimated by the natural log of the estimator of p . . (x,x+l).
.
.
J.,J
It must be pointed out, however, that these methods are based upon
live births data.
The use of live births data does not allow estimat-
ing all the biological functions
simultaneou~ly
since some of these
functions underlie the whole reproductive performance and not just
birth.
In order to use live births data to estimate the basic biologi-
cal functions, the common practice of assigning given values to some of
theparameterswhiehal"e-though-tt0 beof-a stable natureucan be
u
adopted.
An Illustrative Example
The data used in this example (provided by C. M. Suchindran) is
obtained from 554 Hutterite women who reported their time from marriage
..
to first live birth.
All women were reported to have married before
the age of 25 years.
The data is used to estimate the parameters in
the model corresponding to parity zero.
Although this data does not
conform to the definition of a time-age marriage cohort, it is used
here only for illustrative purposes where it is assumed that all women
in the group married at age 20 years.
and {1-8(x)}Ro (x)
were obtained and their variances were calculated
using large sample formulas.
4.2
Estimation in
The estimates of p 0,0 (x,x+l)
The results are given in Table 4.1.
heter~&eneous
cohorts
In this section, we discuss estimators derived under both Model A
and Model B presented in section 2.6.
115
T,A.J3LE 4.1.
Estimates of PO,Q(x,x+1.) and ~(x)=({l-e(x)}Ro(x))* and Their
Standard Error from a Sample of Hutterite Women
(x) estimate of Standard estimate of Standard
Month n (x) n
0,0
Error
ep(x)
p
(x,x+1) Error
x*;I 0
°,°
1
554
554
1.0
0
.1206
.015
2
554
554
1.0
0
.3298
.029
3
554
554
1.0
0
.2139
.026
4
554
554
1.0
0
.2498
.032
5
554
554
1.0
0
.1931
.032
6
554
554
1.0
0
.2396
.039
7
554
554
1.0
0
.1414
.033
8
554
554
1.0
0
.1648
.038
9
554
491
.8863
.0135
.1095
.034
10
491
353
.7190
.0203
.1471
.042
11
353
285
.8074
.0209
.1581
.047
12
285
222
.7789
.0246
.2595
.068
13
222
183
.8244
.0255
.1177
.049
14
183
144
.7869
.0303
.1100
.051
15
144
125
,8681
.0282
.2058
.075
16
125
106
.8480
.0321
.0580
.053
17
106
95
.8963
.0296
.0310
.040
18
95
82
.8632
.0352
.3289
.112
19
82
70
.8537
.0390
.0444
.046
,
,
,
I
(
.
,
-
116
lABLE 4.1--continued
... -~ .....--... --- ""-'-::0-
.---~.-
r-.,. ...· _ ·...-_......-.
~~"""""""'.'-Mo"Utf; "noT;) no o(x) estimate. of Standard estimate of Standard
~
Nx)
Error
p
(x,x+l) Error
x**
o ,0
----..
•
20
70
54
.7714
.0502
.2577
.123
21
54
48
.8889
.0427
.1942
.119
22
48
43
.8958
.441
.1541
.116
23
43
35
.8140
.0593
.0869
.091
24
35
33
.9429
.0392
.2006
.153
25
33
32
.9697
.0298
--
26
32
23
.7187
.0795
--
---
27
23
22
.9565
.0425
--
--
28
22
17
.7728
.0893
29
17
14
.8235
.0924
30
14
12
.8572
.0924
----
----
31
12
11
.9162
.0798
--
--
32
11
9
.8182
.1163
>33
9
--
--
--
---
---
.
* g is taken as 9 months
** x is measured from the age at marriage
-
117
4.2.1
Estimation under Model A
- _ . . .
it
-
Under Model A, it is assumed that the probabilities p
. (x,X+l)
j,J
assigned'to the !th woman in the sample, {p, ,(x,x+l)} , is of the
J ,J
i
form:
i=1,2, ••• ,n, (x),
j=O, 1, ••• ,a~
= a,(x) + {£,(x)}i'
J
J
where
i=i'
i#i' ,
Estimators for
a~(x)
£,(x) is known.
J
J
cannot be obtained unless the distribution of
To estimate a,(x), the conditional likelihood function
J
for individual i, over the month (x, x+l) where the parity at x is j,
is given by:
n, (x») =
i ,J. (ui , Yil{P,J ,J,(x,x+l)}.,
J.
J
L
= {p
j ,j
•
u·
y
l-u -y
(x,X+l)} J..{q(x,x+l)} i,{l-p, ,(x,x+l)-q(x,x+l)}
i i,
J ,J
.
where:
if woman i whose parity at age x is j, still has
parity j at x+l,
u. =
J.
{:
y
{~
otherwise,
and
Hence,
i
=
n j (x)
L: u
if woman i who is alive at x dies over (x,x+l),
otherwise.
n, (x)
= n . . (x) and JL: y
i=l i
J,J
i=l i
= n,
J,d
(x).
Whatever the probability distribution of {po .(x,x+l)}., the unJ ,J
J.
conditional likelihood ,for woman i over the month (x, x+l) where the
118
parity at x is j' is given
where a (x)
j
= E{ p,J, j
~s:
(x,x+l)} •
i
The total likelihood function for women with parity j at month x, is
then given' by:
= {a, (x)}LUi.{q(x,x+l)}LYi,{l-a,(x)_q(x,x+l)}nj (x)-Lui-LYi ,
J
J
When considering all parites at month x, the likelihood function for
those who are observed over (x, x+l) is, finally, given by:
a
II L
L =
j=O j
•
=
•
n (X)-LU -Ly
II ({a,(x)} ui,{q(x,x+l)} Yi.{l-a,(x)-q(x,x+l)} j
i
i.
j=o
J
.
J
, •• (4.2.1)
a L L
Equation 4.2.1 implies that the maximum likelihood estimator of
a (x) is 8 (x) where:
j
j
a, (x)
J
{Lui}·{l-q(x,x+l)}
= ------n (x) - {LY }
j
i
• • • (4.2.2)
The estimator ~,(x) is a biased estimator of a,(x).
J
J
An unbiased
estimator of a,(x) results, however, by applying the method of mements.
J
.
~
This method gives a (x) as an estimator of a,(x) where:
j
J
119
• • . (4.2.3)
It can be shown that:
E{~j (x)} = a j (x),
Var{ a. (x)} = a j (x){ l-aj (x)} .E{
J
1
},
n (x)
j
and for large samples,
?\
Var{a. (x)}
J
a j (x){ l-aj (x)}
=~---~--
E{n (x)}
j
These estimators can be used to estimate the biological functions
underlying the fertility process as represented by the function
{l-e(X)}Rj(X).
By corollary 2.6, we have:
- ({l - e (x-g)}R. (x-g»). = In {P . . (x,x+l)} •
J
i
J ,J
1
This implies that ({I - e(X)}Rj(X)) is a random variable whose distribution depends upon the distribution of p . . (x,x+l).
J ,J
Therefore, the
exact moments of ({ I - e (X)}Rj (x») can be determined only i f the probability distribution of Ej(X), which determines that of Pj,j(x,x+l),
is known.
For large samples, however, it can be shown, using the
Taylor expansion of the function In Pj .(x,x+l), that:
,J
- In a j (x)
Var({l - e(x-g)}R. (x-g») ~
J
o~ (x)
]
+
oj (x)
, and
2aj (x)
,
j=O,l, ••• ,a.
aj (x)
It is noticed that these moments depend on
0
2
•
A discussion of their
estimators, therefore, requires a knowledge of the probability distribution of Ej(X).
120
4.2.2
Estimation.under Model B
~
I
.~
Under Model B it is assumed that the value of the function
"({l - e(X)}Rj (x») for the ith woman in the sample is given by:
({I - e(x)}R. (x»).
~
J
b. (x) +" {eS. (x)}i'
=
J
J
where
cov({o.(x)}i' {eS.(x)}.,)
J
=
~
J
l
V~ (x),
i-i'
J
0t
i'#! , •
By virtue of the relationship between ({I - e(X)}Rj(X»)i and
{P j ,j(x,x+1)}i' we have:
{Pj j(x,x+l)}
, i
=
exp(-{l - 8 (x-g)}R. (x-g»)
J
= exp{- b.(x-g)} • exp(J
i
{OJ (x-g)} ) •
i
Therefore,
E{p . . (x,x+1)}
J ,J
=
exp{- b (x-g) Lll . (x),
j
J
where
Consequently, it follows that:
1
= -
1n{
. a. (x)} ,
llj (x)
J
where
a (x)
j
= E{p . . (x,x+l)}.
J,]
The maximum likelihood estimator of b (x) is then given by:
j
b (x)
j
= -
1
lnf
.
;j (x+g)}
II (x+g)
where a.(x) is given by equation 4.2.2.
J
It is noticed that:
121
by Jensen's inequality
= exp{O}
=
l,
for all x.
Therefore, it follows that:
b.(x) > - In ~ (x+g).
j
J
It is also clear that b (x) cannot be evaluated unless
j
11
~j
(x) is known
which requires a knowledge of the probability distribution of 0j(x).
Maximum likelihood estimators, therefore, can be derived under Model B
only if a probability distribution is specified for 0 (x).
j
On the
other hand, the use of other methods of estimation, such as the method
of least squares, requires some measurements on the function
({I - 8 (X)}R (x»)i' It is not possible to make direct observations
j
on this function. Further, the use of the relationship
({I - 8(x-g)}R. (x-g»).
~
J
=-
In {Po
J ,j
(x,x+l)}
i
to observe the function
{I - 8(x)}R.(x) indirectly, is not possible since the observed values
J
of u
are either 1 or zero.
Additional information, besides
i
P . . (x,x+l), is required in order to measure the observed value of the
J ,J
function {I - 8(x)}R (x).
j
Remark:
The methods presented in the preceding sections are of a preliminary nature and deal with relatively simple situations.
These methods
are included to indicate the feasibility of obtaining estimators for
the different parameters in the model.
A complete treatment of the
estimation problem in the model was not attempted here because of the
122
lack of required data.
Such treatment should endeavor to study the
following problems:
1)
the estimation problem in time marriage cohorts where the
marriage patterns are to be considered,
2)
the problem of fitting the parametric representations
suggested for the basic functions in Chapter III, and
3)
the estimation problem in incomplete samples.
CHAPTER V
THE NUMBER OF BIRTHS AVERTED BY
FAMILY PLANNING PROGRAMS
5.1
Introduction
The purpose of this chapter is to use the present model to develop
some methods for estimating the number of births averted by family
planning programs.
Indices based on the number of births averted can
provide a suitable means for transforming data on family planning activities into measures of fertility change.
Although several attempts have been published, no consensus prevails in the literature as to an acceptable method for estimating the
number of births averted by a family planning program.
Broadly, the
proposed methods measure either the number of births averted per segment
of contraception or the number of births averted by contraception over
a specified period of calendar time.
The first approach, which measures the number of births averted per
segment of contraception, attempts to transform the length of time a
woman is protected against the risk of pregnancy through her adoption
of a contraceptive into a number of births averted.
This procedure,
though conceptually appealing, is only feasible in the presence of an
analytic model to underlie the transformation procedure.
The available
applications of this concept, proposed by Potter (1969) and Wolfers
(1968), assume a stationary population of homogeneous women i.e.
women
124
with identical characteristics.
Under this assumption, the biological
approach to fertility analysis is adopted and each contraceptive is
assigned two parameters.
The first is its use effectiveness defined as
the proportionate reduction in fecundability caused by the contraceptive and the second is the monthly probability of discontinuing the use
of the contraceptive for reasons other than accidental pregnancy.
Using these two parameters, life table techniques are employed to estimate the mean period of effective contraceptive use per woman per segment of contraception.
The number of births averted per segment of
contraception is then obtained by relating this mean period of effective
contraceptive use to the mean interval between any two consecutive
births.
Apart from the fact that the assumption of a stationary popu-
lation is not suitable for analysis of changing fertility conditions,
the concept of births averted per segment of contraception serves only
an.internal program need and could be important only transitionally.
On the other hand, the number of births averted over a specified
period of claendar time is the quantity of ultimate interest in the
evaluation of most family planning programs.
This concept provides a
means of viewing the performance of the program at any point of time.
The available methods for measuring the number of births averted over
time are based upon the concept of fertility rates (e.g.
ter 1966).
Lee and Isbes-
The formulas provided in these methods depend on the "poten-
tial fertility rate" over the specified time period.
:Potential fertility
is the fertility level which contraceptive users would have experienced
in the absence of the program.
In the published literature, estimates
for this potential fertility are derived either by an extrapolation of
recent fertility curves or through a procedure of matching the fertility
125
levels of users and nonusers of the contraceptive.
These procedures
are mainly designed to deal with empirical data, and the lack of a rigorous statistical formulation underlying them makes it difficult to
evaluate their efficiency.
In this chapter, a method for measuring the number of births
averted by a family planning program is developed.
The dual purposes
are:
1) to provide a means of studying the dynamics of family planning
for the purpose of clarifying the effects of varying the different elements of a family plannjng program, e.g.
the methods
used, their effect, and the number of users, and
2) to generate formulas that may be
~sed
to evaluate the number
of births averted by a family planning program over a specified
period of time.
To achieve the first purpose, family planning in time-age marriage cohorts is considered where formulas for the number of births averted per
woman in the cohort over a given age span are derived.
These formulas
can then be utilized in studying selected aspects of family planning.
The second purpose is accomplished by extending these formulas to cover
family planning in a cross sectional population of married women over
time.
The content of this chapter is arranged in two sections.
5.2 deals with births averted in time-age marriage cohorts.
Section
Section 5.3
is a derivation of the number of births averted in married populations
over a specified time period.
126
5.2
Births
av~rted
in
ti~e-~ge ~rriage
cohorts
For simplicity, it is assumed that only one contraceptive is used
in the program.
The resulting formulas can easily be generalized to
cover the situaton where more than one contraceptive is used.
5.2.1
Notation and definitions
The contraceptive under consideration is assigned two variables.
The first is the contraceptive use effectiveness denoted by £ and defined as the proportionate reduction in unconditional fecundability
caused by the contraceptive.
The second is a random variable y(x)
defined by:
if a woman, member of the cohort, is using the
contraceptive at age x,
otherwise.
The model, presented in Chapter Two, implies that parity is a random
variable which is a continuous function of age.
Parity, as a random
variable, is assumed to have the following relation to family planning.
1.)
In the absence of family planning, one parity curve is considered for all women in the cohort.
2.)
In the presence of family planning, three parity curves, two
hypothetical and one actual, are considered.
The first of .
the two hypothetical curves represents the parity for a woman
who starts using the contraceptive at the age of marriage and
continues using it over her entire reproductive\period.
The
second hypothetical curve indicates the parity for a woman
who never uses the contraceptive despite its availability
through the family planning program.
The third curve is the
actual parity curve of any woman in the cohort in the presence
127
of family planning.
This third curve falls somewhere between the first
two curves by virtue of the fact that, in the presence of family planning, women oscillate between the contracepting and noncontracepting
states.
Considering these assumptions, the following notation is used:
J(x)
is the parity at age x in the absence of a family planning
program,
rC(x)
is the parity at age x for continually contracepting women
in the presence of family planning,
rr(x)
is the parity at age x for continually noncontracepting
women in the presence of family planning,
lex)
is the actual parity at age x in the presence of family
planning, and finally
A(x,y) is the random variable representing the number of births
averted per woman in the cohort over the age period (x, y).
5.2.2
Derivations
Using the above notation, the interest in this subsection centers
around deriving formulas for the expectation of the random variable
A(x,y).
Definition 5.1
A(x,y)
where
Xo
= A(xo'y)
- A(xo'x),
is ,the age at marriage.
Definition 5.2
A(xo'x)
= J(x)
- r(x),
xo<x.
This definition is the analytic expression of the definition of births
averted per woman over a specified age group as the difference between
the number of births which would occur to a woman over that age period
128
if family planning were not available and the number of births which
would occur to her over that same period in the presence of family planning.
Lemma 5.3
Over the infinitesimal age interval (x,
x+~x),
the following rela-
tion hold:
~I(x)
~I(x)
where
= ~Ic(x) +
=
l(X+~x)
y(x).{~Ic(x) - ~lc(x)}
- lex).
Proof
Over the age interval (x,
x+~x),
the change in the actual parity
lex) depends on whether a woman is using the contraceptive at x;
if a woman is using the contraceptive at x,
~l(x)
if a woman is not using the contraceptive
at x.
This statement can be rewritten, utilizing the indicator variable y(x),
as follows:
Finally, arranging the terms on the right hand side, the lemma follows.
Q.E.D.
Lemma 5.4
For any function F that is continuous and one to one, the following
relation holds over any infinitesimal age interval (x,
x+~x):
Proof
Applying the same argument as in lemma 5.3, and since F is a one
to one function, we have
129
r~F{ IC(x)}
~F{I(x)}
-18FlI
if a woman is using the contraceptive
at age x"
CC';)}
if a woman is not using the contraceptive
at age x.
Using the indicator function y(x), this could be written after rearranging the terms, as:
Q.E.D.
This lemma, along with the next one, is useful in evaluating the moments
of l(x) through a suitable choice of the function F.
Lemma 5.5
(a) E{y(x).~F{lC(x)}}= E{y(x)}.E{~F{lc(x)}},
(b) E{y(x).~F{Ic(x)}}= E{y(x)}.E{~F{IC(x)}}.
Proof
This lemma expresses the fact, established by the definitions of
lC(x) and IC(x), that these variables are independent of y(x).
Lemma 5.6
~E{F{I(x)}}
=
~E{F{lc(x)}} + E{y(x)}.~E{lc(x) - lC(x)}.
Proof
The proof follows by taking expectations on both sides of lemma 5.2,
utilizing the fact that the two operators
~
and E are interchangeable,
and then applying lemma 5.5.
Definition 5.7
At this stage in the derivations, the function b(x) is introduced
and defined by:
dE{y(x)}
- - - = E{y(x)}.b(x).
dx
This definition implies that b(x) is the rate of growth (over age) in
130
Theorem 5.8
E{F{I(x)}}
where
-
= E{F{Ic(x)}} +
-
E{y(x)}.E{F{IC(x)} - F{Ic(x)}} -
o is the age at marriage in the cohort.
X
Proof
Lemma 5.6, by dividing by Ax on both sides and taking limits as
Ax+O, can be written as:
dE{F{I(x)}}
dE{F{Ic(x)}}
dE{F{Ic(x)} - F{Ic(x)}}
+ E{y(x)}.---------
----- =
dx
dx
dx
On integrating both sides over (xo, x) and using the initial condition
I(x o )
= I c (xo ) = IC(x
o)
E{F{I(x)}}
=
0, we obtain
= E{F{IC(x)}} +
x
+ ! E{y(x)}.d{E{F{Ic(x)} - F{Ic(x)}}},
x
o
which reduces, after integrating by parts and substituting for
dE{y(x) }
------- from definition 5.7, to the statement of the theorem.
dx
Q.E.D.
Theorem 5.9
E{A(xo,x)}
=
E{J(x) - IC(x)} +
131
Proof
The proof follows from the
de~~nition
of A(xo,x) given by defi-
nition 5.1 and applying theorem 5.2 after setting F(x)
=x
in theorem
5.8.
Remark:
Through an appropriate choice for the function F in theorem 5.8,
the higher moments of lex) can be obtained in terms of those of rC(x)
c
and r (x).
To obtain the higher moments of A(xo'x), however, requires
a knowledge of the joint moments
of I(x) and J(x).
With real data,
these joint moments can be estimated by calculating the function lex)
directly using lemma 5.3.
5.2.3
Discussion
A deliberate intervention in the fertility process for family
planning purposes aims at changing the reproductive capacity.
This
is usually achieved through one of the following approaches:
(i)
either by altering the conditional fecundability through
contraceptive practice,
(ii)
by changing the probability of a conception ending in a live
birth through changing the abortion rate, or
(iii) by changing the length of the reproductive period through a
delay in the age at marriage.
In Chapter Three, a numerical investigation of natural fertility
was presented.
According to that investigation, any approach to family
planning can be most effective when occurring over the age period corresponding to the plateau of the reproductive capacity (usually ages
20 to 30 years).
The question of determining the most appropriate ages
for family planning practice will not, therefore, be investigated any
132
further in this chapter.
The interest in the present discussion is to evaluate the nature
of the relationship between £, the. contraceptive use effectiveness,
and y(x), the indicator variable associated with contraceptive use at
age x, which are considered to be the determinants of the" demographic
effectiveness of a contraceptive.
The average value of y(x) in a cohort is a random variable whose
expectation is E{y(x)}, or the probability that a woman is using the
contraceptive at age x.
For a given contraceptive, the value of
E{y(x)} is determined by the interaction of two factors.
The first is
the fertility conditions which prevail in the cohort before introducing
the contraceptive and the second is the organization and effort spent
in recruiting contraceptive users.
The effect of the first factor is
reflected in the specifications of the objectives of the family planning
program as well as in the definition of its target population.
The
second factor determines the success of the program in reaching the
members of the target population.
The goals of a family planning program are usually expressed in
terms of reductions in the existing fertility level.
For a given set
of goals for a family planning program a target population is defined
as a pattern of contraceptive use which results in the fulfillment of
these goals.
A pattern of contraceptive use is composed of two elements
--the fertility characteristics of the users and their relative size at
different age points.
Assume for a given pattern of contraceptive use,
that v(x) is the proportion of women in the target population at age x.
The observed proportion of contraceptive users at age x, measured as
the average value of y(x) in the cohort, is a random variable defined
133
over (0, v(x».
The family planning program is a total succes if
Pr{y(x)
= v(x)} = 1
Pr{y(x)
= O} = I
for all x and is a total failure if
for all x.
The probability distribution of y(x),
or simply E{y(x)}, can be used therefore as an index of the adequacy
of a given family planning organization pattern in achieving its purpose.
On the other hand, the parameter E is a property of the contraceptive and is determined, therefore, in advance when the decision to use
a particular contraceptive is made.
The nature of the relationship between E and E{y(x)} is shown by
the next 1ennna.
Lemma 5.10
c
R (x) = R.(x) - p(x).E.A(x),
j
J
where
R;(X) is the unconditional fecundability in the presence of the
contraceptive,
Rj(X)
is the unconditional fecundability in the absence of the
contraceptive,
p(x)
is the conditional fecundability,
E
is the contraceptive use effectiveness, and
A(x)
is the probability that a woman is using the contraceptive
at age x while she is fecund and at the risk of conception.
Proof
According to lemma 2.1, the unconditional fecundabi1ity R.(x)
J
can be written as:
where
"
134
C is the event indicating that a conception occurs over (x, x+Ax),
and
F is the event indicating that a woman is fecund and at the risk
of conception at age x.
In the presence of contraception, R?(x) can therefore be written as:
J
Rj(x)
= pdc nF}
= Pr{C ()F nul + pdc nF nil}
Where U is the event indicating that a woman is using the contraceptive at age x.
R~(x) =
J
Therefore,
PdF nu}.pdclF nu} + PdF nu}.PdcIF nul.
Now, recall that:
pdFnu} = A(x),
Pr{CIF()U}
Pr{F()U}
= p(x)(l-£),
= Pr{F}
- Pr{F()U}
= Q.(x)
- A(x), and
J
Pr{CIF()U}
= p(x).
It follows that:
Rj(X)
=
A(x).p(x)(l-£) + {Qj(x) - A(x)}p(x)
~
p(x){A(x)(l-£) - A(x)} + p(x)Q.(x).
J
Using lemma 2.1 it follows that:
Rj(X)
=
R.(x) - p(x){£.A(x)}.
J.
Q.E.D.
The implications of lemma 5.10 are that £ and A(x) are inter....
changeable with respect to their effect on fertility.
But A(x) is the
/
"effective" part of E{y(x)} where the other part is the probability
that a woman is using the contraceptive at age x while she is infecund.
This means that the effects of £ and E{y(x)} are not interchangeable,
135
and further it implies that a value of E{y(x)} has less effect in reducing the fecundability than does an equal value for E.
The rela-
tionship between £ and E{y(x)} can be studied in detail through a
numerical approach.
lustrative
purpos~s.
Below are some examples that are intended for ilIn these examples the function E{A(x,y)}, which
is the expected number of births averted per woman in the cohort over
the age interval (x, y), is calculated using the formula of theorem 5.9.
The model of Chapter Two is used to provide the functions IC(x),
IC(x)~ and J(x) under the following assumptions.
1.)
The conditional fecundability p(x) is assumed unimodal with
its mode at age 25 years.
Two levels for p(x) are considered;
the first corresponds to a value of p(x) at its mode of .11,
and the second corresponds to a value of .05.
2.)
The parameters in the function Qj(x) are chosen to correspond
to an age at marriage of 16 years.
The parameter d. in this
J
function is assumed to have a decreasing pattern similar to
that of pattern V in section 3.3.2.
3.)
A low mortality level is considered with life expectancy of
65 years.
Tables 5.1 and 5.2 show the expected number of births averted, for
combinations of E{y(x)} and £, for the two situations where the maximum
conditional fecundability is .11 and .05 respectively.
The following
conclusions can be made from these tables.
1.)
Because of the postulated shape of p(x), the maximum number
of births averted is expected to occur over the ages at which p(x) is
maximum (20 to 30 years in the present example).
2.)
The parameters
£
and E{Y(x)} do not have the same effect on
e
•
11
~
e
e
TABLE 5.1
Expected
~umber
of Births Averted for Different Combinations of.£ and E{Y(x)}
(Maximum Fecundabi1ity=.11)*
~{y(x)}
= .5 for all x
E{y(x)}
= .3
for all x
Age Group
e:=.90
e:=.70
e:=.50
e:=.30
e:=~90
e:=.70
e:=.50
e:=:,.30
15 - 19
.859
.661
.491
.316
.576
.464
.372
.257
20 - 24
1.848
1.519
1.156
.775
1.262
1.056
.848
.613
25 - 29
1.989
1.699
1.342
.904
1.375
1.201
.987
.731
30 - 34
1. 723
1.504
1.182
.777
1.176
1.046
.854
.612
35 - 39
.803
.661
.416
.119
.461
.383
.227
.066
40 - 44
.283
.217
.067
0
.132
.091
.080
0
45 - 49
.091
.061
0
0
.032
.011
0
0
7.596
6.322
5.014
4.252
Total
4.654
2.901
3.368
2.279
*The number of children ever born to a woman in the cohort in the absence of family planning
is l4.900.
.....
'-oJ
0\
TABLE 5.2
Expected Number of Births Averted for Different Combinations of E and E{Y(x)}
(Maximum Fecundability=.05)*
E{y(X)} = .5 for all x
E{y(x) } = .3 for all x
Age Group
£=.90
E=.70
£=.50
E=.30
E=.90
8::.70
c-.50
c-.30
15 - 19
.420
.334
.245
.161
.280
.236
.187
.132
20 - 24
.780
.648
.480
.338
.540
.451
.360
.258
25 - 29
.707
.590
.473
.325
.496
.420
.360
.246
30 - 34
.572
.497
.400
.292
.401
.359
.301
.200
35 - 39
.407
.363
.303
.221
.292
.268
.225
.181
40 - 44
.232
.212
.180
.140
.169
.153
.142
.106
45 - 49
.071
.067
.059
.040
.052
.048
.040
.035
Total
3.189
2.711
2.140
1.517
2.230
1.935
1.615
1.158
*The expected number of children ever born to a woman in the cohort in the absence of family
planning is 5.889.
e
1I
~
e
~
I-'
W
-....J
e
138
the resulting reductions in fertility.
In both tables, for example, a
value of (.30, .50) for the parameter pair (8, E{y(x)}) results in
less births averted than a value of (.50, .30).
This is consistent
with lemma 5.10 and with a result obtained by Sheps and Perrin (1963)
under the assumptions of renewal process that more effective contraceptive methods used by smaller fractions of the population would produce a greater decline in fertility than would less effective methods
used by a large part of the population.
On the basis of both tables,
however, this conclusion does not strictly hold.
It is observed that
for values of E: greater than .50, an increase in E{-y(x)} produces more
births averted than would an equal increase in 8.
A hypothesis worth
testing, therefore, would be that for values of E: less than .50, a
larger decline in fertility is achieved by raising the value of E: rather
than raising the value of E{y(x)}, by the same quantity.
And for
values of E: greater than .50 this trend is reversed.
3.)
The proportionate reduction in total fertility is larger when
the level of fecundability is lower.
The practical implications of
this observation is that countries with high fertility need more family
planning efforts to achieve a proportionate reduction in fertility that
is achieved with less effort in countries with lower fertility.
5.3
Births averted over a specified time period in populations
of married'women
I
The purpose of this section is to derive a formula for the expected
number of births averted by a family planning program over a specified
time period in a given population.
The definition of population 'is
flexible enough to include any open group in the sense defined in section 1.3.
For simplicity only one contraceptive is considered.
139
5.3.1
Notation and definitions
Any population of married women is continually subject to vital
change affecting its size and characteristics.
For purposes of fertil-
ity analysis, such a change is adequately reflected in the behavior
of two random variables.
The first is the time trend in the size of
the population and the second is the time structure of the age parity
distribution.
The development of this section is similar to that of section 5.2.
A contraceptive is assigned an indicator function x(t) defined by:
x(t) -
t
if a woman , in the population, is using the contraceptive at time t,
otherwise.
The function x(t) operates on the population size of time t which is
denoted by M(t) and results in a random variable U(t) representing the
. number of women in the population who are using the contraceptive at
time t.
Time is measured from an initial point t=O which is assumed to
..
correspond to the time at which the family planning program was
initiated.
Parity is also a random variable which is a continuous function of
time.
Parity is assumed to have the following pattern in relation to
family planning.
1.)
In the absence of family planning, one parity curve denoted
by pet) is taken to represent the parity of all women in the
population.
2.)
In the presence of family planning, two parity curves are
assumed to exist.
The first is the parity among women who are
continually using the contraceptive, denoted by QC(t).
The
140
The second is the parity among women who are continually not
using the contraceptive, denoted by QC(t).
Any woman in the
population oscillates between the two curves by virtue of the
continuous movement between the contracepting and noncontracepting states.
The following is a complete list of the notation used in this
section.
M(t)
is the size of the ever married population of time t.
U(t)
is the number of married women using the contraceptive at
time t.
x(t)
is an indicator function associated with contraceptive use
at time t.
is the parity at time t in the absence of family planning.
is the parity at time t, among those women who are using
the contraceptive.
QC(t)
is the parity at time t for women who are not using the contraceptive in the presence of family planning.
Also define:
D(O,t)
is the total number of births occurring in the population
over the time period (0, t) had the family planning program
not been instituted,
DC(O,t)
is the total number of births occurring in the population
over the time period (0, t) in the presence of family
planning, and finally
B(t 1 ,t 2 ) is the number of births averted in the population over the
time period (t 1 , t 2 ).
141
A final remark is in order concerning the relationships between
the different functions defined above.
The parity functions defined
above are used in the sense of mean parity.
The observed parity
pattern is, therefore, conditional on the population size M(t), the
contraceptive users U(t), and the indicator function x(t).
5.3.2
Derivations
Using the above notation, the following derivations are possible.
Definition 5.11
B(O,t)
= D(O,t)
- DC(O,t).
This is the analytic expression of the definition of the number of
births averted by a family planning program over a specified time interval as the difference between the total number of births which would
occur in the population if the program were not instituted and the number of births which would occur during the same time period under the
influence of the program.
Lemma 5.12
t
E{D(O,t)} = E{M(t)}.E{P(t)} - E{M(O)}.E{P(O)}-
J
o
E{P(T)}.
dE{M(T)}
.dT
~
Proof
In the absence of family planning, the quantity
P(T+~T)
- peT)
represents the change in parity in the population over the infinitesimal
interval (T,
T+~T)
where O<I<t.
Condisering M(T,
of the married population over the interval (T,
T+~T)
T+~T),
as the size
it follows that:
for O<T<t,
which can be written as:
142
Taking expectations on both sides, utilizing the interchangeability
of the operators E and
~,
and using the remark made earlier that the
parity function pet) is conditional on M(t), we obtain:
Dividing through by
~T
and taking the limits as
lim
E{M(T)}, it follows that:
E{M(T,T+t.T)}
=
dE{D(O,T)}
dE{P(T)}
E{M(T)}.--dT
~T+O,
and assuming that
M+O
----=
dT
Integrating over (0, t), utilizing the inital condition D(O,O)
= 0,
we obtain:
E{D(O,t)}
Finally, integrating by parts, the lemma follows.
Q.E.D.
Lemma 5.13
dE{ QC (T ) }
- - - - - = E{U(T)}.----
dE{ QC <T )}
+ E{M(T) - U(T)}.----
dT
Proof
Over an infinitesimal time point, (T,
T+~T),
the change in the
overall parity in the population in the presence of family planning
depends on the proportion of women who are practicing the contraceptive
at time T.
The following statement holds:
( ~QC(T)
Increase in parity
=
over (T, T+flT)
j
l~QC(T)
for women who are using the contraceptive at time T,
for women who are not using the contraceptive at time T.
143
using the indicator variable x(t), this can restated as follows:
Increase in parity
over (T) T+AT)
=
X(T).AQc(T) + {l-x(T)MQc(T).
Now, assuming that M(T,T+AT) is the number of married women in the popu1ation over (T,T+AT), the change in the number of births occurring in
the presence of family planning, ADc(O,T), can be written as:
ADc(O,T)
=
M(T ,T+ln){x(T) .AQc(T) + {l-X(T) MQc(T)}.
where U(T,T+AT) is the number of women using the contraceptive over
(T,T+AT), it follows after substituting this in the above equation that:
Taking expectations on both sides (assuming they exist), dividing
through by AT, and taking limits as
AT~,
the lemma follows.
Q.E.D.
Corollary 5.14
Proof
Integrating lemma 5.13 over (O,t), utilizing the initial conditions
DC(O,O)
=
° and E{U(O)} = 0,
the proof of the corollary follows.
Definition 5.15
Define m(t) as the rate of growth in the population size E{M(t)}
by the relation:
144
didM(t) }
- - - = Eb·f(t) },m,(t).
dt
Definition 5.16
Define vet) as the rate of growth in the expected number of users
.,
E{U(t)} by the relation:
dE{U(t)}
--- =
E{U(t)}.v(t).
dt
Now, taking these definitions into consideration, lemma 5.12 and
corollary 5.14 can be rewritten as:
(1)
E{D(O,t)}
=
E{M(t)}.E{P(t)} - E{M(O)}.E{P(O)} -ftE{P(T)}.E{M(T)}.m(T)dT.
o
(2)
E{Dc(O,t)}
= E{U(t)}.E{Qc(t)}
- ftE{Qc(T)}.E{U(T)}.v(T)dT +
0_
_
+ E{M(t) - U(t)}.E{Qc(t)} - E{M(O)}.E{Qc(O)} -
These two final formulas, when used with definition 5.11, imply the
following theorem.
Theorem 5.17
E{B(O,t)}
= E{U(t)}.E{P(t)
- QC(t}} - E{M(O)}.E{P(O) - QE(O)} ~
C
+ E{M(t) - U(t)}.E{P(t) - Q (t)} -
This theorem clearly indicates that the number of births averted over
a given time period depends on the time pattern over that period in the
145
population size, contraceptors size, and the characteristics of the
contraceptive in use.
For example, this theorem indicates
tha~
the ex-
pected number of births averted increases when the contraceptors grow
at a faster rate than the" total population.
Observed data can be used to estimate E{B(O,t)} according to
theorem 5.17.
Under certain assumptions, it can be shown through the
likelihood function of an observed pattern that the observed values for
the functions M(t), U(t), QC(t), QC(t), met), and vet) are sufficient
statistics for the parameters included in theorem 5.17.
The maximum
likelihood estimators will, therefore, be in terms of these observed
functions.
C~~PTER
VI
SUMMARY AND PLANS FOR FURTHER RESEARCH
6.1
Summary
In the present study, a detailed analytic description of human
fertility in terms of the age parity distribution was developed.
This
specifically included first, the derivation of a stochastic model for
the fertility process, second, an analysis of certain aspects of the
model, and finally, the model was used in deriving some indices for
the evaluation of fertility changes due to contraceptive practice.
In developing the model (Chapter II), two types of marriage
cohorts which are related through the age distribution at marriage were
considered; the first is a time-age marriage cohort and the second is
a time marriage cohort.
Methods for deriving the moments of the age
parity distribution in both types of these cohorts were presented.
In
these methods, age and parity were taken as the most important variables
affecting fertility and the fertility process was considered as a continuous time (age) discrete state (parity) process.
The biological
approach was then adapted in order to derive the transition probabilities.
The basic biological functions underlying fertility were con-
sidered as functions of age and parity in a manner explained in section
2.2.
The basis of the biological approach, fecundabi11ty, as known in
the literature was redefined as conditional fecundability.
unconditional fecundability was introduced.
The term
Unconditional fecundability
147
is more closely related to the fertility performance and is linked to
the conditional fecundability through
a function Q.(x)
defined as
.
J
the probability that a woman is susceptible to, and at the risk of,
conception at age x given that her parity is j.
This function accounts
for the effects of the nonsusceptible periods whether they are permanent, such as those following death, divorce, or sterility, or temporary, such as pregnancy or ammenorhea.
For a given parity j, Qj(x)
was shown to be a decreasing function over its domain implying that
women who achieve a given parity at an early age tend to have a higher
reproductive capacity than those who reach that parity at a later age.
After deriving the transition probabilities, recurrence relationships concerning the age ·parity distribution were established.
These
relationships were also used to derive a discrete approximation for the
probability distribution of the age at delivery of the ith live birth.
The transition probabilities were also used to evaluate the truncation
effect.
j
It was suggested that the truncation effect for a given parity
at a given age x could be measured by the probability p . . (x,w)
J,J
where w is the upper age limit for reproduction.
It was further shown
that this probability tends to zero as w becomes infinitely large.
A decision was made to use a range of guesses about needed parameters to illustrate the use of the model rather than to seek real data
at this stage for the following reasons.
1.)
The present model was developed as a generalization of models
that already exist in the literature, particularly renewal
models.
Its contribution, therefore, is reflected in the more
general nature of its assumptions.
2.)
Because of the level of detail in the data required by the
148
model, it was felt that the effort spent in obtaining such
data and securing its accuracy would be better spent investigating other aspects of the model at this stage.
Despite the limitations of using only rough and arbitrary estimates of the parameters included in the model, the numerical analysis
presented in Chapter III served to illustrate the dynamics of the fertility process.
According to the results of that analysis, fertility
performance is determined by the interaction between the age distribution at marriage and the fecundability level.
It was further illustra-
ted that making the assumptions of a renewal process, though it might
produce the same total fertility, results in misleading differences in
the shape of the age parity distribution.
In Chapter IV some preliminary aspects of the estimation problem
in the model were studied.
This chapter was not intended as an exhaus-
tive study of the estimation problem but rather as an introductory one.
For this reason, only data from complete samples pertaining to time-age
marriage cohorts were considered.
Maximum likelihood estimators as
well as moment estimators for the transition probabilities and the unconditional fecundability were derived.
The resulting formulas were
illustrated using data relating to a sample from the Hutterite women.
No attempt was made to fit the parametric representations for the basic
functions proposed in Chapter III.
Rather, this was left for further
research.
In Chapter V, the output of the model was used for evaluating the
effect of family planning on fertility.
Indices based on the number of
births averted were considered as measures of that effect.
The methods
of this chapter regard the number of births averted as a random variable
149
and give formulas useful in deriving its moments.
The approach
employed in this chapter has the advantage of avoiding the problem of
whether an averted birth over a given time (age) peri.od results in a
r~duction
of the total fertility or merely represents a postponement
in the timing of births.
In this approach all the given formulas
express births averted as reductions in the total fertility.
In real
situations, the data required to evaluate the number of births averted
according to these formulas are the proportion of women using the contraceptive as a function of age (time) as well as the age parity distribution of the contraceptive users and nonusers.
6.2
Plans for further research
Because of time constraints, it was not possible to study all
aspects of the model in detail.
In fact, most of the work in this
dissertation can be characterized as preliminary in nature and consequently much work remains to be done.
Since the overall objective of
this study was to develop a stochastic formulation that adequately
describes human fertility, the things that remain to be done could be
categorized as follows:
1)
further study of the model in its present form, and
2)
modification
and further generalization of the model.
These two categories are discussed in the following sections.
6.2.1
Fu~ther
stufty of
~he
model
The numerical comparisons included in this study aimed at developing an understanding of the workings of the model in order to be able
to relate it to real data.
Assumed parameter values were used because
of the lack of knowledge of the exact numerical levels for some
150
functions such as Q.(x) for example.
J
It could be potentially useful
to re-examine the interrelationships between the different functions
in the model in a more 'realistic' context.
To do so, however, requires
a complete discussion of the statistical aspects of the model.
•
This
includes a study of the relationship between the model and reality and
should aim at answering the following questions
in relation to real
data.
1.)· Are the assumptions underlying the model satisfied in real
data?
2.)
Does the parametric representation proposed for the different
functions provide an adequate fit, and what is the range of
the different parameters included in these functions?
3.)
How do the methods for estimating the number of births
averted proposed in Chapter V perform in a real situation?
Once satisfactory answers are obtained for these questions, the use of
the model for the purpose of making inference becomes more meaningful.
With realistic specifications for the functions included in the
model, numerical comparisons could throw light upon the following questions that were raised during the course of the study:
1)
the extent of the confounding of the effects of the two functions p(x) and Qj(x),
2)
the relationship between mortality and fertility, in
particular,
(i)
comparing the reproductive capacity of women who die
before the end of their reproductive period with that
of women who live through the whole period, and
(ii) studying the effect of infant mortality on fertility,
151
3)
evaluating the effect of sterility and divorce on fertility,
~d
4)
studying the different aspects of family planning which include:
(i)
the relationship between the prevailing mortality
levels and family planning,
(ii)
the relationship between marriage patterns and family
planning, and
(iii) the interrelationship between the demographic effectiveness of different contraceptives employed in a
family planning program.
6.2.2
Generalizations of the model
Two directions for generalizaing the present model are recog-
nized.
The first direction is to generalize the model by modifying
the assumptions underlying the functions involved and the second is to
extend the model to allow for the study of the calendar time effects
on fertility.
Examples of generalizing the model in the first direction include:
1)
allowing the nonsusceptible periods c
lj
and c
2j
to vary with
age,
2)
stochasticizing some or all the parameters included in the
model, in particular the parameter pair (wI' w2) which represent the ages at the onset and at the termination of the
reproductive period respectively,
3)
considering the case where more than one contraceptive is
used in a family planning program.
The methods of Chapter V
can also be generalized to consider more than a single parity
152
curve for the women in the population to allow for different
behavior patterns.
Generalization of the model to allow for time effects starts with
equation 2.7.1.
A possible way to solve this equation is to formulate
the quantities involved as the convolution of the number of women
marrying at different time points and the probability of being in a
given parity j at a given age x, conditional on calendar time.
The
use of the Laplace transforms could then be potentially useful in providing a systematic treatment for the problem.
153
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k
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