•
e.....,
STCcHASTIC MODELS OF BIRTH INTERVALS ACCORDING
TO DATA ASCERTAINMENT METHOD AND RELEVANT
FERrILILTY INDICES
by
Mohamed Futuah Abul-ata
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1841T
December 1987
STOCHASTIC MODELS OF BIRTH INTERVALS ACCORDING
TO DATA ASCERTAINMENT METHOD AND RELEVANT
FERTILITY INDICES
by
Mohamed Futuah Abul-ata
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
1987
Approved by:
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ABSTRACT
MOHAMED FUTAUH ABUL ATA. Stochastic Models of Birth Intervals
According to Data Ascertainment Method and Relevant Fertility Indices.
( Under the direction of Dr. C. M. SUCHINDRAN.)
Three stochastic models for the ith-order birth interval are developed
corresponding to each of the follOWing ways for considering birth
interval data:
(a) All Closed Birth Intervals (Retrospective and Prospective Studies);
(b) Last Closed Birth Interval;
(c) Prospective Next (Straddling) Birth Interval;
(d) Prospective Interior Birth Intervals; and
(e) Open Birth Interval.
The first model assumes that the fertility hazard for the
/h
birth,
A. (x), of women who have reached parity (i -1) is independent of parity
1
and time (age), Le., Ai(x)=A. The second model regards fertility as
strictly a parity-dependent process: Ai (x)=A i ' Ak* Al for all k*1=1,2, ... ,i.
The third model takes into account that reproductive behaVior is more
likely to be independent of eXisting parity after some level of parity is
attained (e. g., 4 or 5). In this model:
Ak*A l for all k*1=1,2"",r, and
A 1=A 2=···=A.=A*A k
k=1 2 ... r
r+
r+
1
""
The second model was utilized to investigate how truncation bias in
the first two moments of the birth interval varies with birth order, the
II
length of the observation period (marital duration), fertility hazard
function chosen, and data ascertainment method. The sensitivity of the
birth interval measure to detect fertility changes was also investigated.
MLE produces estimates of fertility hazards that are recovered
(purged) from truncation bias for 1.\11 Closed, Last Closed, and Open
Birth Intervals. These hazard estimates are further utilized in deriving
three important length-unbiased cohort fertility quantities, viz., mean
length of inter! ive birth interval, parity progression ratio, and birth
interval survival function. The models were applied to data from the
Egyptian Fertility Survey (1980) for a marriage cohort with 5-9 years
of intact marital life. While the recovered birth order-specific hazard
estimates are generally of the same magnitude irrespective of the
ascertainment method of considering birth interval data, they are largely
below the corresponding estimates derived from Straddling Birth
Intervals. Comparison between model-based and life-table-estimated
survival probabilities of the second birth interval indicates that the
former is more likely to be better adjusted for truncation bias.
III
ACKNOWLEDGMENTS
I wish to express my sincere appreciation to prof. C. M. Suchindran
for his help and encouragement during each and every step of the preparation of this study. It was a rewarding privilege to work closely with
Dr. Suchindran who first stimulated my interest in this research problem.
Special thanks are due to my academic advisor prof. Richard E.
Bilsborrow not only for the interest he consistently showed in the progress of this research but also for the generous care, gUidance, and help
he offered me throughout my study program.
I would like to thank professors Michael Symons, Shrikant
Bangdiwala, and Moy Freymann the other members of my doctoral
committe. They were extremely cooperative and provided encouraging
and helpful suggestions that definitely expedited the completion of this
research.
Special thcmks and appreciation are due to prof. Amy Tsui for her
academic, professional, and personal advice during my graduate work.
Discussions with Dr. Pandey, A. during the progress of this study
were extremely helpful.
The financial support of both the USAID Mission in Cairo and the
Egyptian Cultural Bureau in Washington, D.C. is grately acknowledged.
I owe a very special debt of gratitude to Dr. Awad Mokhtar Halloda
and Mr. Asmael Rafat; the President and former First Under-secretary
of the Central Agency for Public Mobilization and Statistics of the
IV
Government of Egypt. The special care and the continued understanding
and encouragement they offered me are rea11 y immeasurable.
Finally, though words are far below from sufficiency, I avail this
opportuinty to express my sincere gratitude to my wife Sonia, my son
Tarek, and my daughter Yasmien for being a wonderful family to me
and for their patience, understanding, and moral support.
v
TABLE OF CONTENTS
LIST OF TABLES
:""IST OF FIGURES
X
XII
CHAPTER
.
1. BACKGROUND AND RESEf\RCH PROBLEM
1. 1 Intoduction
1..2 Errors In Birth Interval Data
1.3 Life Table Analysis of Birth Intervals
~.4 Specification of the Research Problem
1.5 Review of Pertinent Literature
II. BIRTH INTERVAL MODELS WITH CONSTANT
FERTILITY HAZARD
2. 1 Introduction
2.2 Stochastic Models of Birth Intervals in the
Absence of Fertility Competing Risks
1
" . . ..
~
.4
11
:5
18
34
34
38
2.2.1 Prospective Studies
38
2.2.2
2.2.3
2.2.4
2.2.5
2.2.6
42
42
44
50
53
Retrospective Studies
Retrospective Last Closed Intervals
Prospective Next (Straddling) Intervals
Prospective Interior Intervals
Retrospective Open Intervals
2.3 Stochastic Models of Birth Intervals in the
Presence of Fertility Competing Risks
2.3.1 Prospective Studies
2.3.2 Retrospective Studies
54
54
58
2.3.3 Retrospective Last Closed Intervals
59
2.3.4 Prospective Next (Straddling) Intervals.................... 60
VI
2.3.5 Prospective Interior Intervals
2.3.6 Retrospective Open Birth Intervals
65
68
III. PARITY-DEPENDENT MODELS OF BIRTH
INTERVALS
69
3. 1 Introduction
3.2 Stochastic Models of Birth Intervals and Related
Quant it ies
3.2.1
3.2.2
3.2.3
3.2.4
3.2.5
All Closed Intervals
Last Closed Intervals
Prospective Next (Straddlii,g) Inte!"'vals
Prospective Interior Intervals
Open Birth Intervals
3.3 Patterns of Bias in the Mean and Variance of Birth
Intervals and Sensitivity of the Mean to
Fertility Change
3.3.1
3.3.2
3.3.3
3.3.4
All Closed Intervals
Last Closed Intervals
Straddling Intervals
Prospective Interior Intervals
Introduction
All Closed Inter'vals
Last Closed Intervals
Straddling Intervals
Prospective Interior Intervals
Open Birth Intervals
VII
72
72
75
80
87
89
92
93
95
96
97
IV. QUASI PARITY-DEPENDENT MODELS OF BIRTH
,
INTERVALS
4. 1
4.2
4.3
4.4
4.5
4.6
69
"
103
103
105
109
113
120
124
V. PARAMETER ESTIMATION AND FERTILITY
INDICES.. .
. .. ..
...
5.1 Introduction
5.2 All Closed Intervals
127
127
129
5.2. 1 Constant Hazard Model......................................... 132
5.2.2 Parity-Dependent Model.
134
5.2.3 Quasi Parity-Dependent Model. . . . . . . .. . . . . .. . . . .. . . .. . .. . . . .. 136
5.3 Last Closed Intervals
139
5.3. 1 Pari ty-Dependent Model........................................ 139
5.3.2 Quasi Parity-Dependent Model.
141
5.4 Prospective Next (Straddling) Intervals
~
144
5.4.1 Costant Hazard Model.
146
5.4.2 Pari ty-Dependent Mode1. . . . . . . . . . . . . . . . .. . . . . .. . .. . .. . . . . .. . ... 148
5.4.3 Quasi Parity-Dependent Model.
151
5.5 Prospective Interior Intervals
160
5.5. 1 Constant Hazard Model......................................... 16 i
5.5.2 Parity-Dependent Model.
162
5.5.3 Quasi Parity-Dependent Model................................ 162
5.6 Open Birth Intervals
167
5.6. 1 Parity-Dependent Model........................................ 167
5.6.2 Quasi Par'ity-Dependent Mode1.
168
5.7 Relevant Fertility Indices
170
5.7.1 Mean and Variance of the
5.7.2 Parity Progression Ratio
5.7.3 Survival Function ~
VIII
/h Birth Interval.
170
171
178
VI. APPLICATrONS
6. 1
6.2
6.3
6.4
187
Introduction
Data Source and AppraisaL
187
188
Assessing Exponentiality
Hazard Estimates and Relevant Fertility Indices
191
200
VII. SUMMARY AND SUGGESTIONS FOR
FUTURE RESEARCH
7.1 Summary
7.2 Suggestions for Future Research
7.2. 1 Theory
,.,
7.2.2 Further Application
210
,
,
210
213
2 13
215
REFERENCES
217
APPENDIX A Derivation of the general expression for the p.d.f
of Straddling Birth Intervals
221
APPENDIX B Observed Distribution of Order-Specific Birth
Interval of the Marriage Cohort with 5-9 Years
of Intact Marital Life- Egyptian Fertility
, Survey, 1980
224
IX
LIST OF TABLES
TABLE
3. 1 Patterns of Variation in Mean and Variance of
All Closed Birth Intervals for Different Values of
t and Varying Fertility Levels
99
3.2 Patterns of Variation in Mean and Variance of
Last Closed Birth Intervals for Different Values of
t and Varying Fertility Levels
100
3.3 Patterns of Variation in Mean and Variance of
Straddling Birth Intervals for Different Values of
(t,T) and Varying Fertility Levels
3.4 Patterns of Variation in Mean and Variance of
Prospecti ve Interior Birth Intervals for Different Values
of (t,T) and Varying Fertility Levels
6.1
Parity Distribution of the Marriage Cohort with 5-9
Years of Intact Marital Life
:01
:02
: 90
6.2 Testing the Linearity of the Cumulative Hazard
with Birth Interval
: 92
6.3 Graphical Estimates of Birth Order-Specific
Fertility Hazal~ds
193
6.4 Distribution of Birth Intervals Straddling the End of the Second
Year of Marriage by Birth Order
204
6.5 Parity-Dependent Hazard Estimates According to
Ascertainment Method
205
6.6 Length-Unbiased and Observed Mean Birth Interval
According to Data Ascertainment Method
206
6.7 Length-Unbiased and Observed Parity Progression
Ratio
207
x
6.8 Model-Based and Life Table Survival Probabilities
of the Second Birth Interval Ascertained from the All
Closed Strategy
XI
208
LIST OF FIGURES
FIGURE
1. Cumulative Hazard Plot for First Birth IntervaL
194
2. Cumulative Hazard Plot for Second Birth Interval.
195
3. Cumulative Hazard Plot for Third Birth Interval.
196
4. Cumulative Hazard Plot for Fourth Birth Interval.
197
5. Cumulative Hazard Plot for Fifth Birth Interval.
198
6. Cumulative Hazard Plot for Sixth Birth Interval.
199
7. Model-Based and Life Table Survival Probabilities
of the Second Birth Intervals Ascertained from All
Closed Strategy
209
XII
CHAPTER I
BACKGROUND
AN)
RESEAROi PROBLEM
1.1 Introdtrlion
In recent years enquiries into childspacing patterns have increasingly focused upon the number of months between successive births (Le.,
closed birth intervals) and/ or the open interval since the last birth.
Studying such intervals is of interest for several reasons. First, models of the birth process have indicated a close relationship between
length of birth interval and birth rates (Perrin and Sheps, 1964).
Birth intervals play an important role in determining birth rates and
rates of natural increase. For example, consider an ·increasing population, where each generation is "k" times the size of the previous one.
Assume that, with no other changes, the interval between births become
longer, producing an increase in the mean age of women at childbirth.
Since in this situation it takes longer for the population to increase by
k times, annual birth rates are reduced, leading to lower rates of
natural increase.
Second, current interest in evaluating family planning programs, especially in developing countries, has necessitated developing indices
sensitive enough to detect current or very recent changes in fertility
patterns of women still in the reproductive ages. Variations in either
the number or timing of births are produced by modifying one or more
behavioral and physiological factors that determine fertility, ultimately
by modifying the effective reproductive period, fecundability, pregnancy
wastage, and the duration of non-susceptibile periods. Since changes
in these factors are immediately reflected in the distribution of birth
intervals, it has been suggested that analyses of data on closed and
open birth intervals might detect changes in underlying fertility behavior earlier than period birth rates, which are not highly sensitive to
these changes (Ryder, 1965).
A crucial value of closed birth interval data (Srinivasan, 1979)
stems from the pOSSibility of carrying out component analyses of such
interval data. Broadly speaking, the closed birth interval can be decomposed into the following four components:
(a) the period of postpartum amenorrhoea following birth;
(b) the duration of menstruation between two live births;
(c) the periods of pregnancy and post-termination amenorrhoea
of any abortions, miscarriages, and stillbirths intervening
between the two live births; and
(d) the period of pregnancy associated with the last Ii ve birth.
The first component, the duration of post-partum amenorrhoea,
during which conception is generally impossible, is influenced by the
health of the woman, her innate fecundity, and breastfeeding. As a
result, sociologists, demographers, and reproductive physiologists and
epidemiologists have become increasingly interested in this component
of the birth interval. The second component is the total waiting time
during which the woman is susceptible for conceiving following a birth.
2
The time at which conception may occur can never be predicted with
certainty and is determined by a variety of biological and sociological
factors. The fecundity of the woman (biological capaCity to conceive),
the frequency and timing of sexual intercourse, the sperm count and
mobility of sperms in the ejaculations of the male partner, contraceptive practice, and the health status of the couple are among the more
important factors. The duration of menstruation is determined basically by the above mentioned and possibly other factors.
The distribution pattern of these menstruation interval and the
identification of the effect of various factors on thiS component of the
birth interval have become of increasing interest to scientists in various disciplines, including demography. The parameter usually considered for characterizing the length of the menstruation interval is fecundability, Le., the monthly (instantaneous) probability of conception.
The duration of menstruation intervals between two live births depends
on the number of fetal losses, which, in turn depends on the probability that a conception will result in a live birth.
The third component of the birth interval is the effect of fetal losses as they occur to the woman between two live births. The number and
gestational length of pregnanCies that terminate in fetal losses and the
accompanying periods of amenorrhoea are important factors to be taken
into account.
The fourth component is of course the gestation period that ends in a
live birth. Although it is the least variable component, it is not possible to determine the exact time of deli very for a pregnant woman.
3
1.2 Errors In Birth Interval Data
Information on birth intervals and the family-building process is
ascertained mostly from retrospective household surveys and, to a
much lesser extent, from prospective studies of females of reproducti ve age.
Retrospective last closed birth intervals, prospective next
birth intervals (straddling intervals), and prospective interior intervals
are alternative data ascertainment strategies that can be used based on
these types of data. A detailed description of each of these strategies
is given in section 1. 4.
A major source of error in retrospective data is what is known as
memory failure or recall bias. Time misallocation of events, especially those occurring long before the survey date, and underreporting of
events themselves are consequences of memory failure. A strategy of
detecting digit preference biases, reflecting temporal misallocation of
date data, has been developed by Srini vasan for birth interval data
(1979). The major underlying assumption of thiS method is the uniform distribution of the residues 1,2, ... ,6 or 1,2, ... ,12 resulting from
dividing the reported monthly birth intervals by 6 or 12 respectively.
Additionally, most fertility surveys, irrespective of the method of
data collection, produce only cross-sectional data, where the information pertains to the experience up to the date of the survey (or the end
of the prospective study) of women still of reproducti ve age. In this
situation the analyst is confronted with a set of Incomplete maternity
histories. With these, she/he cannot proceed to a direct estimation of
4
all birth interval distributions, or calculate complete cohort fertility
indices such as parity-specific cohort fertility rates and cohort parity
progression probabilities. In this context, it is important to distingUish three problems caused by the incomplete observation of birth
intervals, viz., selectivity, censoring, and truncation.
Selectivity refers to the fact that the transition from parity i-1 to
parity i can only be studied for women who reached at least parity i-1
by the survey date. This group of women tends to be selected on a number of characteristics, for ex?mple, age, and is thus not representative
of the entire population. Rodriguez et al [1980] exemplified this situation by saying,
"... the transition from parity 2 to parity 3, for example,
can only be studied for women who have 2 or more children at
the time of the survey. For the cohort aged 20-24 the subset
with two or more children consists of women who married
early and had two children in relatively qUick succession. Such
women will tend to be more fertile, and hence less educated and
modern, than the average member of the cohort aged 20-24".
Al though most demographers use the terms censoring and truncation
interchangeably to refer to the incomplete observation of failure time
variables, there exists, from the statistical standpoint, a significant
difference between them (Suchindran, 1972 and Srinivasan, 1979). In
the context of birth interval data, Suchindran (1972) has defined censsoring and truncation as follows:
If a cohort of women N. of parity i, with marriage considered parity
1
zero, are observed from the birth of the i th child for a time period of
length t , the exact time to the (i + 1) th live birth is known only for n
1
out of N. women who experience such an event during the observation
1
5
Period. For the remaining N.1- n.1 women, what is known is only that
their time to the (i + 1) th Ii ve birth exceeds t. Data of this nature are
generally known as censored data. There are three types of censorship
that usually affect birth interval data, as follows: (1) Type 1, where
observations censored at a single point t: the exact time to the event is
known for the n.1 women, but for the remaining N.n.1 women the time
1
to the occurrence of the event is known to exceed t. (2) Type 2, where
a cohort of N. women is observed for a maximum duration t time units,
1
women are allowed to withdraw from observation before having given
birth to the 0+ l)th child. The number withdrawn is treated as fixed
or random. The number is considered random if women drop out from
observation as a result of death, Widowhood, divorce, or are lost to
follow up. The withdrawals are fixed only if a preassigned number of
women are excluded from the study to reduce the data collection work
load. (3) Type 3, where each woman has a distinct potential censoring
point: For every woman of parity i we know either an exact time to
her 0+ 1) th birth or know that the time to the occurrence of this birth
exceeds some period of time. This situation occurs when a study is
conducted for t time units and all those who experienced the
/h
birth
during the study period are observed. Thus, the maximum potential
observation period (censoring time) differs among women.
A truncated observation, on the other hand i:;volves a truncated probability distribution, which is formed from the corresponding complete
probability distribution by cutting off and ignoring the part lying in
some finite or infinite range. Symbolically, suppose F('), and f(·) are
6
distribution and density functions of a positive random variable X, and
we consider values of X only below some level C. Then the density
function, h(x), of the corresponding truncated variable is given by:
h(x)
f(x)
, 0
F(C)
=0
~
x
<c
x~
C.
Then, h(x) is said to be a truncated density function of X, with a truncation at C. A truncated sample is obtained by sampling from a truncated
population, Le., for failure time variables, this population haVing an
under! ying failure time distribution truncated at a specific point in
time. In a censored sample, although the exact times until the (i + 1) th
live birth of the N.1- n.1 censored observations are not known, the number of women whose observations are censored ( N.-n.) is known. But
1
1
in a truncated sample, the number of women for whom the time until
the occurrence of the 0+ 1) th birth exceeds the truncated point is not
known. This type of sample can result from a retrospective study in
which only women married for t years and haVing at least one live
birth are asked to give their time to the first live birth.
In light of the above discussion, we conclude that birth interval data
are generally subject to a
tl~uncation
effect that may mask changing
parameters, such as changes in fecundability, lactation practices, etc.
This truncation effect can be thought of as haVing two aspects. The
first relates to the fact that if women are followed until they become
7
secondary sterile, their birth intervals are truncated as compared with
theoretical results derived from reproductive period of infini te length.
The second aspect arises when the observation is terminated at some
point before the intervention of secondary sterility, creating distributions of birth intervals different from the complete (final) distributions of the same women.
In order to appreciate the presence of the biological truncation effect on estimates of fertility from truncated cohorts, Poole (1973) and
Sheps et al (1970) examined the follOWing rather Simple situation.
Suppose, in the absence of all risks competing with fertility, a sample
of n women from the same age-marriage cohort, defined as a birth
cohort of fertile women married at the same age, is observed throughout its reproductive life. The date of birth of each child is recorded.
This, in effect, represents the situation where a number of n general
renewal processes are observed for the entire reproductive life (0 ,A),
where A is age of menopause, with age at marriage is set to zero.
Mathematically, under the assumption of infinite reproductive life, let
X 1 ' X , X ,... , represent the birth intervals between marriage and
2
3
first birth, first and second birth, etc., for a typical woman, if and as
successive births occur.
Let:
, k=1,2, ...
and
So=o
Then the sequence { Sk } is a general renewal process. If C(O, A) is
8
the counting process which records the number of births (renewals) in
(O,A), Le., C(O, A) is the largest k such as Sk
~
A, then the following
duality relation holds:
Sk
~
A
~
C(O, A)
~
k
, k=i ,2, ...
Let Xi (A) be the random variable indexing the length of the interval (if
it
exists) between the (i_1)th and i th births when the process is truncat-
ed at A. Then obViously;
~
X. (A)
1
A
which shows that X. (A) and X. ((X)) , do not generally have the same
1
1
distribution. To clarify this, define the biologically truncated distribution function,
r( (x), of Xi (A)
as:
H~(x)=Pr[
X. (A) ~ x ]
1
1
=Pr [ X.
1
Pr [ X.
1
~
~
I S.1
~
A]
n s.1
~
A]
x
x
G (A)
i
where G. (A)=Pr[
1
Here
Pr[ X.
1
~ x n s.1 ~
x
J
A ]= Pr[ S.
0
1
~
A
I X.=y
]
1
s.1
9
A ].
f(y) dy, which,
assuming independence of Xi' X 2 ,··.,
x
=oJPr[ Si-i ~ A-y ] f i (y)
~
dy
x
=of G i -1 (A-y) fi (y) dy
Thus
x
of G i- 1 (A-y) fi(y)
H~(x)
1
dy
(1.2.1)
G (A)
i
Where f (y) is the p.d.f of Xi' the length or me i·" birth intervai,
i
assuming an infinite length of the reproductive life. The p.d.f of X. (A)
1
is given by:
A
h. (x)
d
A
= -d
H.
X
1
1
(x) •
G. 1 (A-x) f. (x)
A
1-
h. (x)
1
G
1
(1.2.2)
i (A)
It is clear that
ft(x)= lim
A--+-oo
Equivalently, f. (x) =
h~(x)
h~(x).
1
iff G. (A)=l, but G. (A) is always less than
I I I
<
1
1 whenever G i - 1 (A-x)
1 for x >x O and for some x > xo' f i (x) > O.
This is because G (A) is the convolution:
i
A
G. (A)
1
=fG.
0
1-
1 (A-x) f. (x) dx.
1
It then follows that
E [ X r (A) ] ~ E [ X:] , for all i ~ 1 , r ~ 1,
1
1
where the inequality strictly holds iff G (A)
i
10
< 1.
(1.2.3)
This result arises because for some x
> xO'
the weight given to f (x) in
i
(1.2.2) decreases with increasing x. Hence the mean length of the i th
birth interval can be overestimated if it is based on an analytical method that assumes a reproductive life of infinite length.
1.3 Life Table Analysis Of Birth Intervals
The simple analysis of data on closed birth intervals, irrespective
of how they are ascertained, is subject to serious limitations when the
observations are truncated at some point before the age of menopause.
Since for any woman in the age-marriage cohorts under study, the interval recorded between marriage and any birth cannot be longer than
the duration of the marriage at the time its history is terminated, a
tendency for selection against long intervals exists. With respect to the
first birth, however, this tendency could largely be dealt with by limiting the analysis to the records of women who had been married for a
sufficiently long time. With truncated data the bias against selecting
(observing) long intervals is greater the higher the order of the birth.
To minimize thiS bias resulting from incomplete observations of birth
histories, a life table analysis of successive birth intervals is usually
,
applied. In such an analysis a woman who has had i-l births is a member of cohort i birth until she has her
/h
birth or her record termi-
nates (e.g., by date of interview). If she does not have the i th birth
she is "withdrawn" from risk, and from the cohort, at the end of the
observation period. The analysis method yields estimates of the probability of not having the i th birth at each specified time interval after
11
entry into the cohort, giving an attrition curve (survival curve) over
time. At any time since the (i_1)th birth, the probability of not having
another birth is the complement of the parity progression probability
at that same time. The life table also yields estimates of the conditional probability of an
/h
birth during a specified time after the
(i -1) th birth, of the probability distribution of the i th birth per a wom-
an in the original cohort, and of the relative frequency distribution of
the length of birth intervals. From the latter, adjusted means and variances (or higher moments) may be obtained. Al though these estimated
moments are not highly reliable, for reasons to be explained shortly,
they provide less biased estimates than those calculated directly from
the data.
A number of different methods are applied to derive estimators of
q. (x), the conditional probability of an i th birth at interval x after an
1
(i -1) th birth. Given such an estimator, ~. (x), an estimator of S. (x), the
1
1
unconditional probability of not having an i th birth before the end of an
interval x after the (i -1) th, is then derived as follows:
'"
x
'"
S.(x)=I1 { i-q. (y) }
1
where
t'
y= 1
, x=1,2,3,
1
t',
is the last interval in the life table.
An estimator often used for qi (x) incorporates the person-years concept. Let N -1 (t) be the number of women who have had at least (i -1)
i
births by time t. For N (t) among them, there is an observation Xi (t);
i
th
the length of the i birth interval, and for N - - Ni ' there is an
i 1
12
observation on the open interval (censoring time), u. (t) .
Let n be the
x
the number with u. (t) = x.
1
number of observations with x. (t) =x, and c
1
X
1
Then the person-years estimator is:
q. (x)
x-l
1
N i - i (t)
- 2: (n
+ c
y=l y
The life table mean and variance of the
Y
)
/h
-1
c
x
birth interval may be calcul-
ated as :
"[
A
A
2: x { S. (x-l)
x=i
- S. (x) }
1
1
'"
1 - S. (r)
1
r
"'2
(7.
1
'"
2: {x- x.}
x=l
2
A
'"
{S. (x-l) - S. (x) }
1
1
1
'"
1- S. (r)
1
Simulation studies (Sheps et ai, 1969) on the efficiency of life table
analysis of birth intervals show that such an analysis compensates, to a
degree, for the truncation effect on closed birth intervals between
births, but does not fully succeed in eliminating this effect. Serious
shortcomings appear especially in the presence of heterogeneous fecundability, which surely, characterizes human (and other) populations.
Thus, women with longer intervals have a greater probability of not
having the next birth and thus of being considered withdrawals. Hence,
a fundamental assumption of life table analysis is violated in the presence of heterogeneity. In general, the magnitude of the adjustment
13
introduced by life table analysis depends on both the proportion of
women who complete an interval and the distribution of open intervals.
Morever, life table analysis does not account for the fact that the reproductive span is limited and hence Xi ' the length of the i
th
birth in-
terval, cannot exceed a certain value. In other words, Xi belongs to
some truncated probability distribution (Sheps
et~,
1973; Srinivasan,
1979). To see this,
let X. be the random variable indexing the length of an i th birth ir:tervl,
1
and
A. (x)6(x) = Pr { x< X.
1
1
~
I X.1 > x }
x+6x
, i=1,2, ... ,
be the fertility risk function of the i th birth, given the
occurrence of the (1-1) th at some time point within the
reproducti ve span.
It follows that
x
F. (x)= 1- exp[1
0
fA,1 (y)
and
dy J
x
f. (x)= A. (x) exp[1
0
1
fA,1 (y)
dy ]
, x
>0
where F. (x) and f. (x) are, respectively, the probability distribution and
1
1
density functions of X..
1
This is the probability model estimated with
the life table method.
For F (x) to be a proper distribution function,
i
it should satisfy the condition that:
, for some Xo E (0 , A),
F.(xo)=1
1
where A is, as defined before, age at menopause, with age at marriage
set to zero.
14
In other words, the condition required for the above model to be valid
is that the transition probability from parity i-i to parity i is certain,
which is not generally true. Hence, such a model, which can reasonably
approximate the intervals of the first few births when the observation
period is sufficiently long, may introduce biases for births of higher
order. In this respect, it is worth mentioning that the adjustment factor, i-So Cr), that usually used to properize life table distribution of
1
birth interval ( see the above expressions of mean and variance) does
not entirely sol ve the truncation problem because the truncation point '[
is di ctated by the data, and hence may differ from sample to sample
drawn from the same population. Also, thiS adjustment does not clearly account for the fact that the total length of successive birth intervals
has a natural upper limit which is the duration of reproducti ve life.
Despite the inadequate adjustments prOVided by life table analysis,
they are preferable to simple calculations of birth interval parameters
based only on the observed closed intervals.
1.4 Specification of the Research Problem
The ultimate goal of the present research is to devise an analytical
method for dealing with birth interval data that may be more effective
than the life table method in alleviating biases resulting from the two
aspects of truncation as well as selectivity problems, as explained in
th
sections 1.2 and 1.3. Towards this end, stochastic models of the i
birth interval will be derived for several data ascertainment strategies.
They are described as follows:
15
(a) Prospective Studies. A sample from an age-marriage cohort of
fertile women is observed from their marriage until t time units have
elapsed. The dates of birth, death, widowhood, and divorce are
recorded for each woman.
(b) Retrospective Studies.
The age rnarriage cohort is sampled t time
uni ts after marriage where the reproductive histories of selected individuals are ascertained.
(c) Studies on Retrospective Last Intervals. This scheme is similar
to retrospective studies except only the dates of the last closed birth
interval (if existant) are recorded. The open birth intervals are
measured as well.
(d) Studies on Prospective Next (Straddling) Intervals. t time units
after marriage, a sample of currently married women is drawn and
observed until the next event occurs. The interval between the last
event before the observation period and the first occurring during the
observation period is recorded. Hence, intervals straddling the
hypothetical survey point t are observed.
(e) Prospective Interior Intervals. The age-marriage cohort is
sampled at time t and observed until T,all intervals interior to (t , T)
are observed. For example, if there is an i th interval interior to (t, T),
then observations are made on X.=t.-t.
l' t.1- 1- t, and T- t.,
where
1
1 11
t. is the time (measured from marriage) at which the
1
(f) Open Birth Intervals.
/h
birth occurs.
The open birth interval is the interval,
measured at the survey point, since the last live birth.
16
SCHEME OF
DATA ASCERTAINMENT PLANS
i-Prospect! 'Ie Stooies.
I
o
Xl
*----*----*---r----*-----*--I-1
X
2
2
X
3
3
d,w,m
r-l
r U +
r l
X
r
A
t
2-Retrospeeti'le Stooies.
'------.:*
0
Xl
1
*
X2
2
X3
*-------.:*---*
3
r-1
Xr
IA
r Ur+ 1 t
3-Last Closed Birth Interval.
\-------------*-----*
o
i -1
Xi
i
!---Ui + 1 t
A
4-Prospeetive Next (Straddling) Intervals.
* - - - - - tI---:-d,-w-,-m---*--I
i- l \
/ i,
T
A
----...,.---~
S-Prospeetlve Interior Intervals.
1---1---*-----*---.----
b
t
i-l
Xi
i ci, w,m
17
T
--------~
A
Three different models of the i th birth interval will be considered in
this study according to data ascertainment method. The first model is
based on the simplistic assumption of a parity-and time (age) -independent fertility hazard. Subsequently, this restrictive assumption will be
partly relaxed in two stages, producing the other two models. In the
first, fertility is regarded as strictly parity-dependent process, while
in the second considerations are given to the pOSSibility that after a
certain birth order fertility is governed by biological factors rather
than by how many live births a woman has ever had. In all three cases,
however, age is assumed to play no prime role in determining the likelihood of having a new birth (see Chapter III). For reasons explained in
Chapter III , forces which compete with fertility, e.g., mortality, widowhood' divorce, are conSidered only in the first model but skipped
in others.
The parity-dependent fertility models will be used to study how truncation bias and sensiti vity of birth interval to detect fertility change
are influenced by the duration of the observation period, birth order,
fertility hazard, and data ascertainment method.
The Maximum Likelihood Method will be followed in estimating
model parameter(s). Apart from its appealing asymptotic properties,
a crucial advantage of thiS method for the current situation is that its
resultant estimates are recovered from truncation bias in the three
most common ascertainment methods, namely, All Closed (prospective
and retrospective studies), Last Closed, and Open Birth Intervals.
18
e-
The recovered estimates of the fertility hazard are further utilized
in estimating three important length-unbiased cohort fertility indices,
viz., mean length of birth interval, parity progression ratio, and birth
interval survival function. Fertility differentials among age-marriage
cohorts can be analysed via the estimated fertility indices. The availability of variance estimates of each also makes statistical inferences
possible.
The birth histories of a sample of Egyptian women, obtained retrospectively in the Egyptian Fertility Survey (1980) as a part of the
World Fertility Survey Program, will be analysed applying the proposed
analytical approach. The credibility of the suggested method relies, to
a large extent, on the tenability of the underlying assumption. The
Egyptian data, as will be seen in Chapter VI show no major deviation
from the important assumption of an age invariant parity-dependent
fertility hazard.
1.5 Review of Pertinent Literature
A brief review of pertinent literature comprises two related categories. The first describes important work in the area of modelling the
fertility process, from which the current work is a natural extension.
The second focuses on several studies which have given some attention
to methods of data ascertainment for analysing birth intervals.
1.5.1 First General maternity models
Thanks to Henry (1953), the distribution of birth intervals was
19
initiially derived under various models. Henry devised a rather simple
mathematical model to estimate natural fertility, i.e., fertility in the
absence of del iberate measures to regulate births. Birth intervals
were assumed to be a function of: (a) pregnancy duration, (b) the nonsusceptible period following a live birth, and (c) the fecundability of the
woman. Based on the assurnptions of no foetal losses, constant fecundability, and constant duration of the non-susceptibile period associated
wi th a Ii ve birth, Henry showed that the instantaneous birth rate, B(x) ,
converges to the asymptotic fertility rate, p"', with increasing duration
of marriage, where p'" is the inverse of the mean birth interval. This
result is analogous to the one obtained in renewal theory (Chiang,
1968). An expreSSion for the expected number of births to a woman
wi th a marital duration of X years was also deri ved, as was an expression for the expected number of births in a population where fecundability and non-susceptible periods were allowed to vary.
An extension of the above model included foetal losses and nonsusceptible periods associated with a pregnancy, varying with the outcome of pregnancy, is given by Henry (1957). Fecundability was conSidered as a function of age in a non-contracepting SOCiety.
Under certain limiting conditions, Perrin and Sheps (1964) introduced the notion of treating the reproductive process as a Markov renewal process. Consequently, the distributions and moments of birth
intervals were derived.
used right after birth and practiced without interruption. Considering
20
e-
two groups of women, identical in all respects except that one is contracepting while the other is not, Potter concluded that if a contraceptor
Srinivasan (1966) assumed a beta distribution for fecundability among
women, constant fecundability for a given woman over time, and no secondary sterility in deriving probability models for the closed birth intererval. An empirical illustration showed that the expected frequency
under given models for parities 2 and 3 were consistent with observed
frequency. Later, in (1967), he extended the model to include open intervals, assuming the open interval (terminated by the survey date) is a
random portion of the closed interval. Based on this assumption, he obtained expressions for the first two moments of open intervals.
Leridon (1969) questioned thiS above assumption of Srinivasan
(1967), pointing out that the distribution of a closed interval which
straddles a fixed point (survey date) is not the same as that of any
closed interval. Leridon also showed that the longer the closed
intervals, the more likely it is to be interrupted by the survey date.
Hence, the mean length of an
/h
interval which straddles the survey
point is greater than the mean length of all
/h
intervals.
Venkatacharya (1969) investigated the effects of short marital durations on live birth intervals. The bias, defined as the difference between the mean of the i th birth interval estimated from retrospecti ve
studies terminated at some date before menopause and that corresponding to a complete observation, Le., terminated at the end of the reproducti ve life, is viewed as an important element which ought to be
21
accounted for in the context of fertility change assessment studies. The
author showed that this bias is not necessarily small, and is notably
serious in low fertility situations.
Suchindran
~~
9.L
(1977)) addressed the problem of comparing the
distribution of the length of birth-order-specific intervals for two or
more groups when the distributions are obtained using life table techniques. Mantel-Cox, Gehan, and Peto & Pyke's significance tests have
been applied using data from the 1965 U.S National Fertility Survey.
The performance of these tests in various situations is discussed.
Mishra, R. N., et ol (1983) developed a probability distribution for
the first birth interval taking into account the folloWing two factors:
(a) Two groups of women with different levels of fecundability.
(b) Some women may not be exposed to the risk of conception immediately after marriage due to temporary separation for a specific period.
The proposed model was judged to approximate empirical observations
well.
Nour (1984) proposed parity-specific fertility tables, in which the
parity of women, instead of their age, is used as the criterion to
summarize the fertility experience of a synthetic cohort. The theoretical underpinning of the proposed fertility table incorporates developing
an analytical model of the family-building process. The author assumed
that the parity of a woman at age x, [ I (x) ], is a Markov process,
where for every x E X, I(x) has a finite space S={ 0,1,2, ... ,k }, with
k the maximum attainable parity. An expression for the parity trans-
22
ition probability was developed.
Chiang (1985) developed a general stochastic model of human reproduction which accounts for the truncation of reproductive life at the age
of menopause. He assUmes that the fertility hazard is related to
parity and age according to the following specification:
A1 (x) = A1 e(x).
where i and x represent parity and age respectively. No specific
functional form of
e(x)
was given.
1.5.2 Second Models emphasizing ascertainment plans
Sheps and Menken (1973) discussed the lifetime variables under
various data ascertainment methods. Using very general forms of
competing risks, the authors derived the relevant probability distributions for cohort as well as stable and stationary populations. The present study will be based on the general stochastic expression developed
by Sheps et al (1970) and Sheps & Menken (1972). For this reason, a
detailed expOSition of some of these models is given below.
Let:
PJ.
Pr
be the conditional risk (hazard) of occurrence of event E. at time
'
J
t, where j=1 ,2, ... ,k.
r• E.J
E. has not occurred before t 1=
occurs between t and t+At
J
•
pj+O(At)
Pr [E.
J
n
E."
J
j
* j'
,occurs in (t, t+At) ]= o(~t); and
23
Pr [ none of the events E ,j=l ,2, ... ,k occurs in (t,
j
1-2: p.
J
t+~t)
(t)
~t
] =
+
o(~t),
where lim
o(~t) =0
~t--.o ~t
Define y. (t) to be the unconditional probability that E. occurs in
J
(t,
t+~t),
J
i.e., the p.d.f. of E .. Then, this p.d.f. can be written as the
J
product of the hazard and survival functions,i.e.,
f
co
y . (t) = P. (t) Y . (x) dx.
J
J t
J
If we let
co
P . (t) =
J
t
f
Y. (x) dx,
J
then
d
p . (t) = - -dt log P. (t) .
J
e J
Define the following notation. Let:
be a r.v defining the length of an /h birth interval, i=1,2, ... ;
rJ>i (x,a) be the conditional risk of an i th birth at age (a+x) to a married
Xi
woman who had an (i-l)th birth at age a and has not had yet an
th;
1.th b'l r
F i (a,b) be the probability that no births occur to a woman in the age
interval (a,b), given she had an 0-1) th birth at age a, then
b-a
J
F. (a,b)= exp [ - cp. (x,a) dx] ;
1
0 1
24
PL (a) be the conditional risk of death at age a, given that death has not
occurred before a;
P (a) be the conditional risk of widowhood or divorce to a married
w
woman at age a , given neither event has occurred before age a;
P L (a,b) be the conditional survival probability from age a to b given
surviving up to age a
b
=exp
[- fPL(T) dT] ;
a
Pw(a,b) be the probability that a married woman at age a will not get
di vorced or widowed in the age group (a,b)
b
= eXP[-!pw(T)dT].
a
Also let
gi (x,a) be the conditional density of an i th birth at age (a+x) given the
occurrence of the (i-i) th birth at age a, Le.,
gi (x,a) = <Pi (x,a) F i (a,a+x) P w (a,a+x) P L (a,a+x)
(1.5.i)
assuming independence of the events.
Then w. (a), the density of an
1
/h birth at age a,
is given by:
a-iS
w. (a) = g. (x,a-x) w. 1 (a-x) dx
1
where
W
o(a)
0
J
1
1-
is the density of marriage at age a.
25
(1.5.2)
The unconditional density of an i th birth at age a+x and an (l_1)th birth
at age a is, therefore given by:
h.1 (x,a) = g.1 (x,a) w.1- 1 (a).
(1.5.3)
this yields, from (1.5.2):
a-i5
w. (a)
1
=
0
fh. (x,a-x}dx.
(1.5.4)
1
The unconditional density of X. irrespective of the age at birth of
1
the (i -1) th child is given by:
50-x
hi (x) = fh i (x,a) da
15
50
=
f
(1.5.5)
hi (x,a-x) da
15+x
Equations (1.5.4) and (1.5.5) imply no births occur before age 15 or
after age 50.
The above densities pertain to improper r. v's, since the integrals of
h. (x,a) and h. (x) do not equal 1. We present here the properized dist1
1
ributions as given by Sheps (1972), details can be found in the quoted
reference. The truncated joint density function of Xi and the occurrence
of an (i -1) th birth at age a is therefore given by
h. (x,a)
1
50 50-a
h. (x,a) dx da
i5
0
1
f f
and the unconditional truncated density function of X. is
1
26
(1.5.6)
e-
hi (x)
=~~---
50
15
f
=
(1.5.7)
w. (a) da
1
50
where w.=f
w.1 (a) da.
1
15
Since the above densities consider all intervals occurring to all
women in the cohort, the ascertainment method is clearly based on
prospective data, with truncation at age 50.
The interval Xi for surviving married women who have had an i th
birth at age (a+x)
<b ,
Le., a retrospective study with truncation at
age b, has the following p.d.f.:
b-x
ei
f
h. (x,a) PL(a+x,b) P (a+x,b) da
15 1
W
(x Ib) = ..;;:;...,-.b----------15
f
(1.5.8)
w. (a) PL(a,b) P (a,b) da
1
W
, x
<b-15.
The retrospective last interval X. for women of exact parity i at age b
1
has the following density:
b-x
f
hi (x,a) P L (a+x,b) Pw(a+x,b) Fi+ 1 (a+x,b) da
~i (x Ib) = -=..1.; . 5
_
(1.5.9)
27
where
b-15
7T. (b)= fw. (b-u) PL(b-u,b) P (b-u,b) F'+ (b-u,b) duo
1
0
1
W
1 l
And the straddling interval Xi which straddles age b has the following
p.d.f:
b
f
vi(xlb) =
hi (x,a) da
b-x
-~:-'-_-----,
x(35
(1.5.10)
b
35
fo f
hi (x,a) da dx
b-x
An interval Xi' which is interior to the age group (b,b+y), has the
following p.d.f;
y-x
f
I = _0
r . (x b)
hi (x,b+v) dv
y.
1
o
_
y-x
Jf
0
, x(y.
(1.5.11)
e-
h. (x,b+v) dvdx
1
In retrospective studies terminated at age b, let U + =b-(a+X)
i 1
th
be the open interval following an i birth. Then the p.d.f of U. at
1
age b is given by
lJI i (ulb) =
w. 1 (b-u) PL(b-u,b) P (b-u,b) F. (b-u,b)
1w
1
-....=......;;;;---=------..,;...:---=-----
,u(b-15. (1.5.12)
b-1S
f
o
w. 1 (b-u) P (b-u,b) P (b-u,b) F. (b-u,b) du
L
1W
1
Poole (1973) used models of the p.d.f's of Xi analogous to those
28
•
developed by Sheps and Menken, to estimate non-parametric probability
distributions of Xi free from the truncation effects introduced by various data ascertainment strategies. He approached this problem byadjusting the observed distribution of Xi resulting from a specific strategy. The adjusting factors are composite functions of the survival
probabilities of different competing risks and of the truncation point
imolied bv a oarticular strategv. Poole relied on the relative frequen.I
1.
1
J
cy concept in estimating survival probabilities.
Sehgal (1971) investigated the sensiti vi ty and robustness of birth
intervals obtained through various ascertainment plans. He defined the
sensitivity of a fertility indicator, such as the birth interval, as the
extent to which it changes in a short period of time as a consequence
of a change in the underlying fertility component parameters, such as
in fecundability, incidence of foetal loss, and duration of post partum
amenorrhoea, or the age specific fertility rate. The earlier in time a
particular fertility indicator, such as the closed birth interval obtained
from a particular ascertainment plan, can pick up and reflect changes
the more sensitive the fertilitv inin underlving fertility oarameters.
l
J
J
I
dicator. On the other hand, it is also important that the indicator be
robust, Le., not change as a result of changes in factors other than
the underlving fertilitv 1oarameters. Thus, the less the indicator is
J
J
influenced by non-fertility factors, the more robust it is. Sehgal
investigated the sensitivity and robustness of data on birth intervals
using computer simulation, where comparisons were made between
29
closed and open birth intervals obtained through various ascertainment
plans under different fertility assumptions. He studied how changes in
fertility and non-fertility input parameters have affected birth intervals.
The criterion used to investigate sensitivity is such that, if in a particular ascertainment of birth interval, J..ll = mean length of birth interval without changes in fertility input parameters, and J..l2 = mean length
when fertility parameters are changed in such a way as to induce an increase in the birth interval by 25%, say. Then, the hypothesis of
interest is:
Ho :
J..l2 ~ 1.25 J..ll
vs.
H :
a
J..l2
>
1.25 fll
A rejection of Ho lends credence to the sensitivity of this type of birth
interval.
With regard to robustness, assume J..l a is the mean birth interval in
a particular ascertainment under a certain set of parameters describing various competing risks (Le., fertility, mortality, marriage, etc.),
and flb is the mean birth interval under a different set of parameters,
where the only parameters that differ are for variables not directly
affecting fertility. Then the test criterion is
vs.
Rejecting Ho indicates this type of birth interval measure is not robust.
The follOWing findings were obtained by Sehgal:
(a) The length of the open birth interval is highly sensitive to fertilility changes when analysed by the age of mother at the survey date,
30
remaining at the same time robust to changes in other non-fertility
parameters.
(b) The last closed interval is both sensitive and robust when the anal v,
,
J
sis is made by parity.
(c) The straddling interval is qUite sensitive and fairly robust.
(d) All closed intervals, taken together irrespective of parity and age,
were judged to be not sensitive to fertility changes. For example,
changes in fertility input parameters did not affect the mean closed
birth interval noticeabll y until about 15 years after the change.
(e) Prospective interior intervals were shown to be neither sensitive
nor robust, especially when the observation period starts at relatively
older ages.
Hoem (1970) developed stochastic models, Within the framework of
a Markov process, to estimate marital fertility. He argued that in
retrospecti ve studies the birth histories for a given cohort of women
are not representative of the actual experience of the entire cohort,
since the women included in such studies are a select subset of their
cohort by virtue of survival. In this contex t, Hoem provided an expression for the transition probability from parity m at age x to parity
n at age y conditional on survival to age z (the survey date). It is
clear that
x :::;; y :::;; z , and m :::;; n. This transition probability is shown below:
31
Pr { N(y)=n, D(y)= 0
{
I N(x)=m
Pr{ D(z)=O
P* (x,y,z)
mn
Pr {D(z)=O
I N(x)=m,
, D(x)=O} x
I N(y)=n,
}
D(y)=O
D(x)=O }
Pmn (x,y) Pn (y,z)
=--------Pm (x,z)
where if
D(x) = {
0 if the woman is ali ve at age x
1 otherwise
and N(x) is the parity attained at age x.
Then
P
mn
(x,y)=Pr { N(y) =n , D(y)=O
Hoem enphasized that
*
~n(x,y,z)
I N(x)=m,
D(x)=O }, and
cannot be the same as the transition
probability of the entire cohort, Pmn (x,y), unless the fertility experience of those who died or emigrated befOI'e the time z is included. The
term purged fertility index was given to
*
~n(x,y,z)
by Hoem. The idea
behind this terminology is that data of the kind collected in retrospective studies have the same property as data that result from
purging a continuous population register of all information concerning
members deceased or emigrated.
32
More recently, Pathak (1983a) developed a stochastic model for
describing the variation in any closed interval of women of marital
duration t. The model incorporates the possibility of the woman being
sterile at the time of marriage consummation. An estimate of truncation bias in mean closed birth intervals is given. In another article.
Pathak et af (1983b) proposed a modified stochastic model of closed
birth intervals that allows for the possibility of varying periods of post
partum amenorrhoea among women. The major assumption implied in
both models proposed by Pathak and Pathak et af is the exponential
survival of all risks under study.
33
CHAPTER II
BIRTH INTERVAL MODELS wrrn CONSTANT
FERTIUlY HAZARD
2.1 Introduction
In thiS chapter the probability distributions of X., the interval from
1
an (i-1) th birth to an /h birth, will be given for a rather simplistic
situation, that is in which the fertility hazard function is viewed as
constant with respect to parity and age. The probability models will be
shown for each of the data ascertainment strategies, defined in section
1.4, and for the open birth interval as well. Section 2.2 deals with
the imaginary situation in which marriage dissolution forces or equivalently fertility competing risks (e.g. divorce, Widowhood, and death)
do not exist. The models are then extended in Section 2.3 to accommodate such competing risks. ExpreSSions for the distribution function
and the first two moments of X. corresponding to each ascertainment
1
strategies will also be developed.
2.1.1 Notation and Definitions
In all definitions below "i" is used to denote the order of a birth
interval: i=1,2,3,....
Xi
Also,
is the random length of the interval from the (i -1) th
birth to the i th birth when the potential duration of
e-
a fecund marriage is assumed to be infinite; that is,
every married women will experience birth interval X.
1
with certainity.
5.
is the total waiting time from marriage till the
1
i
/h
birth, 5.=2: X ..
Ij=l J
is the duration of marital life that terminates by death,
occurrence of an
y
widowhood, or divorce.
A..
1
(x)~x+o(~x) is the probability that an
/h
birth occurs in (x ,
x+~x)
given being at parity i-l at time x.
jJ(x)~x+o(~x)
is the probability that a marriage terminates in
(x,x+~x)
given being married at x.
jJ (x) =jJ 1 (x) +jJ2 (x) + jJ3 (x) , where
jJ 1 (x) is the risk of mortality,
f12 (x) is the risk of widowhood, and
,u3 (x) is the risk of divorce.
is duration of marriage measured at survey (study
t
termination) date .
A
C(O, t)
. . is age of menopause, with age at marriage being zero.
is the counting process that counts the number of births
in (0, t).
iJ. (t)
1
is pr [ C(O,t)=i ].
is the random length of the interval from the (i-1) th
to
/h birth, conditional on duration of marriage
(observation period) of length t.
is the random length of the open interval from the
35
(i-1) thbirth, as being the most recent one, up to the
survey date.
The notation used for density, distribution, and survival functions of
the variables given above is shown in the following table:
r.v
y
X.
1
Function
--------------gi (x)
t
hi (x)
o~(x)
r(x)
Distribution F. (x)
G (x)
i
t
Hi (x)
t
0i (x)
R(x)
Survival
G (x)
i
H.t (x)
O.t (x)
R(x)
Density
f. (x)
1
1
F. (x)
1
1
1
1
2.1.2 Assumptions
i. The fertility hazard function is assumed to be ir.deper;dent of
parity and age; that is Ai (x) = A.
2. Fertility and the competing forces ( e,g., mortal ity, widowhood,
di vorce) are assumed to operate in a statistical! y independent
fashion. As a result, such competing forces can be represented
by a single risk parameter equivalent to total individual risks,
which, in turn, is independent of parity.
3. The risks of the competing forces combined are assumed to be
constant over the age interval (15, 49). This assumption relies
on the fact that while the risks of mortality and widowhood are
slightly and steadily increasing in this age segment, empirica 1
observations generally indicate downward trends of divorce rates
36
e·
with age. The resulting rate of change over time (age) of these
three risks combined can therefore be considered negl igible.
4. The forces of temporary sterility during periods of gestation,
pospartum amenorrhoea, and contraception are assumed to affect
women through .\ i (x) .
5. A Ii ve birth is defined here as a confinement producing at least
one live birth.
The assumption of a time-and-parity-invariant fertility hazard gi ves
rise to the congruity of human reproduction with 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 X o be the length of the interval up to and including the time of
the first event and X. (i~ 1) be the length of the interval follOWing the
i'th event. If it is a~sumed that X. are mutually independent for all i
and have identical distributions forI i~ 1, the sequence conforms to the
definition of a renewal process. The generalization of renewal processes include more than one type of event, as in Markov chains and Markov
renewal processes (Sheps et aI, 1969) .
37
2.2 Stochastic Models of Birth Intervals in the Absence of
Fertility Competing Risks
2.2.1 Prospective Studies
In prospective studies, we follow an age-marriage cohort of fertile
women, defined as a birth cohort of women married at the same age,
from the date of marriage until a time period of length t has elapsed,
where 0
~
t
~
A.
Let X, (t) be a random variable denoting the length of the interval
1
(if exists) between the ( i-l )th and /h births, where the fertility
process is being truncated at t. Then, obviously,
X. (t)
1
~
t.
Let H~ (x) be the distribution function of X. (t):
1
e-
1
= Pr [ X.1
~
Pr [ X.
1
~
x
IS.1
x
~ t ]
n s.1
~
t ]
Then we have
Pr [ X.
1
~ x n s.1 ~
x
t ]
=0 JPr
of
( S.
1
~
x
t
I Xl'
= Y ) f 1, (y)
pr ( Si-l ~ t-y) f (y) dy.
i
38
dy
Thus, the distribution function for X. (t) is
1
x
H~(x)
J G i _1 (t-y)
= _0"---
fi(y) dy
_
, xSt.
G (t)
1
i
And the density function of Xi (t) is
G.1- 1 (t-x) f.1 (x)
(2.2.1)
G. (t)
1
According to the model specification for a time-and-parity-invariant
fertility hazard, the underlying density function of Xi is exponential
with parameter (A), Le.,
f. (x)
1
= A e-AX
, A)O , x )0, i=1 ,2,.···
And 5. has a gamma distribution with location and scale parameters
1
A and i respectively. The density function of 5 is
i
where i is a positive integer and A)O, s)O.
It follows from the duality relationship between gamma and Poisson
distributions that:
t
G. (t)
1
=Pr ( 5.1 S t ) =°Jg.1 (s)
-
ds - 1 -
39
i-1
2
j=O
e
-At (At)j
'1.
J.
Similarly,
i-2
G. (t-x) = 1 - 2: e -A(t-X) JA( t-X)~
.,.
J.
1-1
j=O
Therefore, from equation 2.2.1 on p. 39
Ae- AX { 1 -
h~ (x)
1
ij=O.f:
e-A(t-x) [A(t-X)]j/j! }
=------.,----------i -1
.
1- 2:
e- At (>.t)J/j!
j=O
for
i~2,
x
~
t, A>
o.
(2.2.1.1)
And for the first birth interval,
t
hi (x) =
A e->'x
1- e
-At
' x~t ,>'>0.
(2.2.1.2)
The distribution function of Xi (t) is
x
f
H~(x)
= h~(y)dy
1
0 1
for
40
i~2, x~t,
A>O. (2.2.1.3)
An d for the first birth,
The expected value of X. (t) ,
1
,
x~t,
t
t
\)0.
f x h.1 (x)
0
(2.2.1.4)
dx , is evaluated as:
i-2
\-1 _ e-\t [t+\-1+\-1 j~O (\t)j+2/(j+2)!]
E [ X. (t) ]
1
=
1 _e-\t
i -1
2:
.
(\t)J /j!
j=O
for
i~2.
(2.2.1.5)
And for the first birth,
\ -1 _ ( t _ \-1) e -At
E [ X 1 (t) ] = -----""'\-:-t- - - - ,
1 - e
(2.2.1.6)
for
i~2.
(2.2.1.7)
And for the first birth,
(2.2.1.8)
2
2
By definition the variance of X. (t) is: V[X. (t)]=E[X. (t)]- { E[X. (t)] } .
1
1
41
1
1
2.2.2 Retrospective Studies
In this strategy, the survivors of the cohort are traced retospectively t time units after marriage so as to solicit information on the
number and timing of their live births in the period (O,t).
The probability distribution of X. (t) is the same as that derived in
1
prospective studies, as are the first two moments. This is because the
fertility behavior of those who dropped out of observation are discarded
in both ascertainment strategies.
2.2.3 Retrospective Last Closed Intervals
In this situation, it is envisaged that an age-marriage cohort of
women has been sampled at time t for information about the dates of
their last two births, with marriage considered as birth order zero.
The distribution function of X. (t) is defined as,
1
H.t (x) = pr[ X. (t)
1
~ x ]
1
~
x
I exactly
Xi~
x
n C(O,t)=i]
=pr[ X.
1
pr[
i births in (O,t)]
Pi (t)
x
pr [ C(O,t)=i IX.=y ] f. (y) dy
°f
1
1
Pi (t)
f
o
Xpr[ C(O,t-y)=i-l] f(y) dy
1
----::~---...,....,.------
Pi (t)
42
e·
Thus,
t
Pi_l(t-x)
h. (x) = ---=--=-..,.(-,.)---- f. (x)
1
Pi t
1
,i~ 1, x~t.
The fact that Xi are LLd exponentially distributed with parameter A
implies that C(O,t) has a Poisson distribution with parameter At, thus,
·>2 ,
, 1-
<t .
x-
(2.2.3.1)
And for the first birth,
t
1
hi (x) =T
(2.2.3.2)
It is interesting to note that the density function does not involve the
parameter A.
Consequently, the distribution function of Xi (t) is
H~(x)=
1
1 _ (t-x)i
ti
,i~2, x~t.
(2.2.3.3)
And,
(2.2.3.4)
The first two moments of Xi (t) are
E [ Xi (t) ] =
t
for
i+ 1 '
43
i~2
, and
(2.2.3.5)
E [ X 1 (t)] =
+.
(2.2.3.6)
Similarly,
2t 2
(i+1) (i+2)
2
E[ Xi (t) ] =
(2.2.3.7)
, for i~2 , and
t2
2
E[X 1 (t)] = -3- .
(2.2.3.8)
2.2.4 Prospective Next (Straddling) Intervals
!n this ascertainment method, the cohort under consideration is
sampled at time t and asked about the preceding open interval. Women
are then followed until the next birth occurs. Consider the i th ir.ter:ive birth interval among those women in the sample who had one in the
interval (t, T), assuming that observation terminates at T « A), its distribution can be defined as follows:
Let X. (t) be the length of the i th birth interval for those women who
1
have had the (i -1) th birth before t and /h birth after t (and before T).
Thus the distribution function is
= pr [ X.1
_
O
S x IS.1- 1S tnt
fpXr [So1-
<S. < T
1
J
l~t n t < s.s T/X.=y] f.(y) dy
Pr[S.1- 1 <t
1
n
44
1
t<S.~T]
1
1
_0
f;r[
~ T-y)] f; (y) dy
5 i-l <t n (t-y < 5;-1
Pr[Si-l <t n t <Si ~T]
,
x~T.
Differentiating, the density function is
r[S. 1 <t n (t-x <So 1 ~T -x)] f. (x)
h~,T (X)= P
111
,x~T.
pr [Si-l <t n t<Si~T]
1
In order to evaluate the intersected event of the numerator, it is necessary to consider the relationsip of t with its complement (T-t). It
turns out that the random variable X. (t,T) has two densities according
1
to whether t
~T -t
or t
>T
(see Appendix A for details).
First conSider the situation t
e.-11 pr[t-x<S.1- l~t]
e.-1
pr[S.1- 1~T-x]
1
where
Second, for t
The density function is as follows:
f.1 (x)
, for
f.1 (x)
for
x~t
,
T-t<x~T,
e.1=pr[S.1- 1 <tnt <S.1 <T] .
> T-to
The density function is
e.-1
pr[t-x<S.1- 1~t]
1
h~,T(x)=
1
~T-t.
, for
f.1 (x)
x~T-t,
e~1pr[t-x<S.
1~T-x] f.(x)
1
11
for T-t <X~t,
e.-1
pr[S.1- 1 ~T -x ]
1
for,t<x~T.
f.1 (x)
45
In the following are the formulae of e i and other terms included in the
above densities,
e.=pr[s.
1 <t
1
1-
n t<S.~T]
1
t
=0
f pr[t<S.~TIS.
1
1-
1=z] g.1- 1 (z) dz
t
~f pr[t-z<Xi~T-z]
gi-l (z) dz
Applying the assumptions employed in this chapter we find,
t AZ -At -AT
f
ei =
o
e
(e
-e
Ai - 1z i - 2e-Az
r (i _1)
dz
)
_ ( -At -AT) (At) i-1
- e -e
(i -1 ) ! .
Also,
pr[t-x<Si_1 ~t] = G i - 1 (t) - G i - (t-x)
1
= pr[C(O,tP~i-1] - pr[C(O,t-x):?:i-1]
=pr[C(O,t-x)~i-2]
pr[C(O,t)~i-2]
-
= ~2 e-A(t-x) [A(t-X)]j- e-At(At)j
L.
.,
J.
j=O
And,
pr [ Si-1 <t ] = pr [C(O,t) ~i-1]
_ i-2 e-At(At)j
-1- 2: ----\-.,-..:..:..j=O
J.
Following the same logic, we find,
46
e·
Pr[t-x<5.1- l~T-x] = G.1- 1 (T-x) - G.1- 1 (t-x)
= ~2 e-A(t-X) [A(t-X)]j _ e-A(T-x) [A (T-x)l:!
L.
'I
J.
j=O
And
pr[ 51' -1
<T -x ]
_
- 1 -
i-2 e -A (T -x) (A(T -x)
2:
.I
l:!.
•
J.
j=O
Accordingly, the density function, of X. (t,T) ,
1
Case 1
t
~
i~2
, is as follows:
T -to
for
x~t,
for t <x~T -t,
for T-t <x~T.
(2.2.4.1)
Case 2
for t
> T-t.
for
h~' T (x)= e~ 1 ~2
1
Ae-At(A(t-X)]j
J.
j=O
1
i-2
e~ 1 [ Ae-AX _ 2: Ae
1
~ IAe-AT (A (T -x) ~
j=O
-AT
j
~~ (T-x) 1::..
J.
]
x~T-t,
for T-t <x~t,
for
t<x~T.
(2.2.4.2)
47
The density function of the first birth interval is shown below
f i (x)
pr[tsX i sTI
h\'T(x)
={
o
-AX
Ae
= e -At -e-AT
for, t$;x$;T,
(2.2.4.3)
otherwise.
i~2
are given next.
_At i - 2 e-At[A(t-X)]j+i
e-A(t+x) (At)j
The expressions of the distribution function for
Case i t s T-t.
-1 (At)i-ie-At
ei
[
(i - i)!
-e
-,~{
U+ i)l
-
t"T(x) =
H1
e~1 [(At),
1
i-i -At
~
(1-i).
i-2 -At
j
-e-Ax{i-2: e
(,~t)
j=O
J.
j!
})
,x~t
J-
}]
, t<x~T-t
i-i
i-2
1+1
-L(At)
( -At_ AT)+ -AT(i+~ (A(T-x)]J ) _ -AX]
le , L ('-i)1 e
eeL.
e
1
1
•
'-0 ('+i)1
J
.
J, T-t<xST.
(2.2.4.4)
Case 2
t> T-to
-i (At)i-1 e- At -At i-2 e-At[A(t-X)~ e-A(t+x) (At)j
e'1 [ ('1
_1).I
-e - 2: { (' + i) I
,I
})
'-0
J
.
J.
J,xsT-t
H~,T(x)=
1
i-i
i-2
]j+1 -At
e~i[{ ~At) I _l}{e- At _e- AT}_2:{1A(t-X~
Ie
'-0
(J+i).
'+i -AT
1 (1-i).
JJA(T-x)]J e
}]
U+ i)!
-1 (At)i-1 -At -AT -AX -AT
i-2[A(T_x)]j+1
,T-t<xst
[ (i -1)! (e
-e
)-e
+e
{i +j~O U+ 1) ! }]
ei
,t<xsT.
(2.2.4.5)
48
e·
The distribution function of the first birth interval is
, for t<x<T,
otherwise.
(2.2.4.6)
The moment expressions of straddl ing birth interval, i
~2,
are the
same for both densities given above. The first two moments are
shown below:
,
i~2.
(2.2.4.7)
And for the fiiSt Oiith,
Erx
(t T)] =,\
• 1 '
-1
[e
-'\t
(1+,\t).: e
-'\t
e
-e A I
-'\T
(1+,\T)L
(2.2.4.8)
Also,
e-,\TA-2 {( 1+'\t) 2_(1 +AT) 2}+2,\-1 e-,\T (T-t) {1 +'\l-
~:l~/ - (~~)i}
+(T-t)2 e-,\T{1-
,
And for the first birth,
49
i~2
.
\~:l~~\]
(2.2.4.9)
(2.2.4.10)
2.2.5 Prospective Interior Intervals
The cohort is sampled at time t and observed until T. All birth intervals interior to (t , T) are recorded along with the parts of other intervals that occur within (t, T). For example, if there is an
/h
interval
interior to (t,T), then observation is made on X.1=t.1- t.1- 1 ' t.1- 1- t, and
T- t., where t. is the time (measured from marriage) at which the i th
1
1
birth occurs. Suppose,
X. (t , T) = length of the i th interli ve birth interval for women who have
1
such an interval in (t , T).
e·
Then let
.. T
H~'
1
(x)
= pr [ X.1 (t, T)
= pr [ X.1
Pr lr . X.1
=--~
Pr [
~ x
~ x ]
IS.1- 1 ~ t n s.1
~
T ]
n S.1- 1 ~ t n s.1 ~ T ]
..:..._-S.1- 1 ~ t n s.1 ~ T]
~
x
x
oJpr [Si-1
~ t n Si ~
T
I Xi = y]
fi(y) dy
=.=..-_--,"""='"--::::--:--:::""'.--~,......------
pr [ S i -1 ~ t
n Si
~ T]
x
oJ pr [Si-l ~ t n Si-1 ~ T-y] fi(y) dy
Pr [ S.1- 1 ~ t
n s.1 ~
T ]
50
Thus,
h~,T (x) =
1
n
S. 1 ~ T -x ] f. (x)
Pr [S.1- 1 ~ t
11
--pr~[ S. 1 ~ t n S. ~ T ]
1-
1
Pr [ t ~ S.1- 1 ~ T -x] f.1 (x)
=--~
Pr [ S. 1 ~ t n s. ~ T ~ .
1-
1
With reference to the constant risk assumption, we find that,
pr [ t :s; S i-1
~
T -x ]
= pr
[ C ( 0, t )
~
i- 2 ] - pr [ C (0 ,T -x ) :s; i- 2 )
i-2
=2:
j=O
And
pr [ Si-1
~ t n Si ~ T ] ~
f
T
pr [ Si
~ T I Si-I = y]
gi-I (y) dy
T
=f pr [Si-1+ Xi ~ TI
Si-1=y] gi-! (y) dy
t
T
=tf pr [Xi ~ T- y] gi-1 (y) dy,
1
i
where g.1 (y) = r'1 'A y
i-I
e
-'Ay
,
f f
T T-y
=
r (i1 -1) t
0
51
'A e
-AX
'A
i -1
i -2
y
e
- AY
dx dy
Thus,
h;,T(x)=e;lAj~ {e-A(l+X) (Al)j
- e- H [ A (T - x) jj }/j!,
for
x~T-t, i~2,
(2.2.5.1)
where,
for
x~T-t, i~2.
And
...
<~
'> 2 .
rorxj -t,l-
(2.3.5.2)
The first two moments of the i th interior birth interval
follows:
E [ X. (t, T) ]
1
= e~1{
. .1 e-At - e-A"'"
1
[
1+A(T-t)]
A
-
e~~-AT~2 t:~~1
'-0 J
J-
1
.
,i~2, are as
*
2: r>.. t,j
}i-2
j=O J.
[Tj+2_(j+2)Tt j + 1+(j+1)t j + 2 ]. (2.2.5.3)
And
E
[X~(t,T) 1 = e2;1{2e-\t _e-\T[ {\(T-t) +1}2 +1] }~2( \t)j
1
j=O J.
\
-1 _AT i - 2 \j+1{ '+3 '+1
.'
2 }
- e i e j~ (j+3)! 2TJ -t J {(T-t)(J+3)[J(T-t)+2T]+2t}. (2.2.5.4)
52
2.2.6 Retrospective Open Intervals
The open birth interval is defined as the interval between the birth
date of the most recent Ii ve birth and survey date.
Let Ui (t) be the random variable indexing the length of the open interval between an (i_1)th birth and the survey date, and t the woman's
age at the time of data collection, with age at marriage set to zero.
Then the density function of U (t) is
i
g. 1 (t-u) F. (u)
o.t (u)=
1-
<t , 1'>2 •
,u-
1
Pi-1(t)
1
With reference to the assumptions employed in this chapter, the
functional forms of g.1- 1 (t-u) , F.1 (u), and p.1- 1 (t) are as follows:
gi-1 (t-u)
1
= r(i-1)
F. (u) =1- e
1
-Au
i-1
i-2 -A(t-u)
A
(t-u)
e
Fi (u)
,
<t
,u-
. >
,1
2,
= e -AU ,and
Thus the density function is
= .l.. (i -1) (1t
J:!..) i - 2
t
,u~t
which does not contain the parameter A.
And the distribution function is defined as follows:
53
,i~2,
(2.2.6.1)
o~ (u)
= 1- (1-
f )
, u ~ t
,i
~
2.
(2.2.6.2)
The first two moments are shown next:
_
t
, i
E [ Ui (t) J - - i -
, i
~
~
2, and
2.
(2.2.6.4)
2.3 Stochastic Models Of Birth Intervals In The
Fertility
(2.2.6.3)
~
Of
Competi~ Risks
2.3.1 Prospective Studies
As before, let the distribution function for Xi (t) be
H~ (x) = pr [ X. (t) ~
1
1
= pr [ X.1
~
x
x ]
I survival
to S.
1
~
t J
, x
~
t.
Here,
Pr [ X.1 ~x
n s.1 ~t n Y~S.1
x
] =fpr [ S, ~t
0
1
54
n y~s.1 I Xl' =z
J f , (z) dz
1
x
=fpr
o [S.1- 1~t-z
n y~s.1- 1+z
t
Therefore, Hi (x) is as follows:
x
t
H.(x)
f pr [ S'_l~t-z n Y~S'_l+Z
=O
1
pr [ S. ~t
1
1
n
] f. (z) dz
1
1
Y~S. ]
1
and
h~ (x) =
Pr [S. 1 ~t-x
1-
n
pr [ Si ~t
1
y~s.
1+x ] f. (x)
1-
n
1
Y~Si ]
Thus
t-x
t
ni (x) = f i (x)
0
f
R(y+x) dG i _ (y)
1
-:-t-------"~--
, where
of R(y) dG i (y)
-fro)
Gi(y) -0
y1
i
A z
i-1
- AZ
e
It follows that,
55
dz
, and
] f. (z) dz.
1
And
t
J
-- 0 Je
t
o R (y)
dG. (y)
1
-jJ.y 1
r
i i-1
(i) >.. y
e
->..y
dy
>..i
i-l,
- - - [ 1-exp{-(>"+jJ.)t} 2:{ (>..+jJ.) t }Jjj!].
(>"+jJ.)i
j=O
Thus, the density function is
(>..+jJ.) exp{-(>..+;..I)x} - exp{-(\+jJ.)t}
hi (x)
i-2
2:
'+1
[(\+jJ.)J
'
(t-x)Jjjl)
j=O
= - - - - - - - : - i_-;1----,--W---"------
1- exp{-(\+,u)t}
2: [ (\+,u)t
)Jjj!
j=O
,
x~t, i~2.
(2.3.1.1)
,
x~t.
(2.3.1.2)
And the density of the first birth interval is
t
h1 (x) =
(\+jJ.) exp{-(\+jJ.)x}
1 - exp{-(\+jJ.)t}
These density expreSSions are the same as expressions (2.2.1.1) and
(2.2.1.2) in which the parameter>.. is replaced by >"+jJ..
The distribution function is shown as follows:
56
i-2
l-exp{-(A+,u)x} - eXp{-(A+,u)t}
(A+,u)j+1 (t j + 1_[t-X]j+1)
2:
j=O
U+1)!
,- 1
I J.
•
1 - exp{-(.\+,u)t}j~ [ (.\+,u)t]J/j!
,
x~t, i~2.
(2.3.1.3)
And for the first interval,
1- exp{-(A+,u)X}
H~ (x)
x~t.
,
1-exp{-(.\+,u)t}
(2.3.1.4)
The first two moments are shown next. For the first,
~
,:or
1'>
- 2,
where
e. =
i -1
1 -exp{-(A+,u)t}
1
2:
j=O
(2.3.1.5)
.
[(>-'+,u)t ]J/j!.
And for the first birth interval we find that
(.\+,u)
-1 - exp{-(.\+,u)t} [t+(A+,u) -1 ]
(2.3.1.6)
1 - exp{-(A+,u)t}
The second moment is derived as follows:
57
E
[x: (tl 1=2e~ 1{ (A +)Jl -2 -exp{- (A+)J)t)[~H(A+)Jf 1 + (A+)Jf 2 +
2 (A+,u) -2
~ \ (A+,u) t}j+3 / U+ 3 )!]
}
j=O
,fori~2.
(2.3.1.7)
And for the first interval,
2
-2
{
t
2 (A+,u)
- 2exp -(A+,u)t}[2+t(A+,u) -1 +(A+j1) -2 ]
(2.3.1.8)
1 - exp{-(A+,u)t}
2.3.2 Retrospective Studies
The distribution function of Xi (t) is defined as follows:
H.t1 (x) = pr [
X.~
1
x ) = pr [
I S.~t
n
1
X.~x
1
pr [ S. ~t
1
_0
x
pr [ S. ~t
f
1
n
Y~t
Y~t
]
]
n Y~t Ix.1=z]f.1 (z)
dz
pr[S.~tnY~t]
1
Hence,
pr [ Si-1 ~t-x
n
n
Y~t
] f (x)
Y~t
]
i
h~(x) = ---~'-------=----,
X~t.
pr [ S i ~t
58
Following the assumption that fertility is independent of its competing
risks (mortality, widowhood, divorce), we have
h~ (x)
=
! ~t-x ) f. (x)
Pr (S.
--=1_-"'--_ _--=1
_
,
x~t,
pr ( S. 5: t)
1
1
which is identical to the density function for the prospecti ve/retrospecive studies in the absence of these risks compete with fertility. Therefore, all expressions given in section 2.2.! are applicable as well to
the current situation.
2.3.3 Retrospective Last Closed Intervals
The distribution function of X. (t) is defined as follows:
1
t
Hi (xl = pr [ Xi (t)
= pr [
X.~
1
x
~
x)
I exactly
i births in (O,t) and survival from
other decrements to t )
= pr [
X.~x
1
= pr [ X. ~x
1
I C(O,t)=i n
I C (0, t)
Y
~t
)
=i ) , when survi val from other decrements is
independent of fertility.
It follows that all formulas presented in section 2.2.3 for non-competing risk situation are also applicable here.
59
2.3.4 Prospective Next (Straddling) Intervals
The distribution function of X. (t,T) is defined as
1
H~,T(x) = pr [ X. (t,T) ~ x]
1
1
x
ofpr [ Si-l ~t
n ( t-y<Si_l <T-y) n Y~Si-l +y]
Pr [S.1- l~t
n
n
(t <S.<T)
1
f i (y) dy
Y~S.]
1
Hence,
h~,T(x)=e~1pr
[S.1- 1~t
1
1
e.=pr[s.
1~t
1
1-
n (t-x< S.1- 1<T-x) n Y~S.11+x
1 f.(x),
-· 1
n
(t(S.<T)
1
n
where
e·
Y)S.].
1
Following the same reasoning as for the non-competing r:sk situation
above, the density funcion is as follows:
Case 1 t
~
T-t.
e.-1
pr[t-x<S.1- 1<t n
1
h~'
T (x)
1
=
Y~S. l+x]
1-
f.1 (x)
for x
~t,
e~1 1pr[S.1- 1 <t n Y~S.1- 1+x] f.1 (x)
for
t<x~T-t,
e.1-1 pr[S.1- 1 <T-x n
for
T-t<x~T.
Y~S.
1-
60
1+x] f.1 (x)
Case 2
t) T-t.
e.-11 pr[t-x<S.1- 1 <t n Y~S.1- 1+x]
h~'
T (x)=
1
for
x~T-t,
e~1 1 pr[t-x<S.1- 1 <T-x n Y)S.1- 1+x] f.1 (x)
for
T-t<x~t,
-1
e.1 pr[S.1- 1 <T-x
for
t<x~T.
n
f.1 (x)
Y)S.1- 1+x] f.1 (x)
Applying the assumptions followed in this chapter, we find that the
value of e. is
1
e.=pr[S.
1~t
1
1-
n
(t<S.<T)
1
n
Y)S.]
1
t
~f Pr[(t<Si <T) n Y)Si IS i - 1=z]
gi-1 (z)dz
t
~f pr[(t-z<X i <T-z) n
T-z
t
~
Y)Xi+z] gi-l (z)dz
fJ
f i (s) R(s+z) gi-l (z) ds dz
t-z
i i-1
_ A t
[-(A+j1}t -(A+j1)T]
- (A+j1)(i-1)! e
-e
.
Similarly,
pr[t-X<Si_1 <t
n
Y)Si-l +x]
=
Ai-1e-~x ~2 e-(A+j1) (t-x) [(A+J.L) (t-x)]j-e-(A+j1)t[(A+J.L)t~
(A+j1)1-1 j=O
j!
61
Ai - 1 -jJ.x
i-2 e-(A+jJ.)t((A+,u)tlJ
Pr(S'_1 <t]=
e '-1 (1- ~
.,
],
1
(A+jJ.) 1
j=O
J.
Pr[Sl'-1 <T -x]
Ai - 1 e -jJ.X [1~~ e-(A+jJ.)(T.-1x) [(A+tJ) (T-x)l:!1
L.
(A + jJ.) i j =0
J.
And pr(t-x<Si_1 <T-x]=
Ai-1e~jJ.x ~2 e -(A+jJ.) (t-x) ((A+,u) (t.~x)]j-e-(A+,u) (T-x) ((A+hl) (T-x)~
(A+,u) 1-1 j=O
J.
Accordingly, the density function of X. (t, T) , i
1
~2,
is evaluated as
shown below,
Case 1
t~T -to
(2.3.4.1)
62
Case 2
t> T-to
~
,~
<x-<.....1 ,
(2.3.4.2)
The density of the first interval is
, t<xsT.
(2.3.4.3)
The expressions for the probability distribution functions of X. (t, T),
1
i~2,
are shown as follows: Case 1
63
tsT-t,
Case 2
t> T-to
• rr".
". ,. i-2 -f.\+uhr" , ) (·t_ )\1~j +l
-t;\+,u;t_","{ e'
ll;\+jJ_~_
,.,i-i -f.\+uh
e'
"
-ljll;\+jJJtJ
ai
l
(i-1)!
J~O
-e
I'
v
U+ 1H
e -A (t+x) [('\+jJ)tl.:! }
- - - - ' 'I
}
J.
, x:S;T-t
i-l
-1 {l('\+jJ)tL -1}{ -(.\+,u)t -(.\+,u)T}_
Ht,T()=
i
x
a i { (i-i)!
e
-e
~~[(.\+I-L) (t-x)]j+1 e -(.\+,u)t-[(A+jJ) (T_x)]j+l e -(A+,u)T }
U+ 1 )!
j=O
, T-t<x:S;t
i-l
-1 [(.\+jJ)t]~ [-(A+,u)t -(A+,u)T] -(A+,u)X~ -(.\+,u)T[l+
a,1
('-i)'
e . 2 -e
-e
.e
1.
11}
2: [(.\+,u) (T-x)]J' +
/(j+l)!]
:S;T
j=O
'
x .
~
t<
(2.3.4.5)
The distribution function of the first interval is
e
-(.\+,u)t -(.\+,u)x
-e
, t<x:S;T.
The expressions of the first two moments are shown next
and for the first interval,
64
(2.3.4.6)
e-
(>-'+/-I) -1 e -(>-'+j..l) t( 1+ (>-.+,u)t) -(>-'+,1.1) -1 e -(>-'+j..l)T [1 + (>-'+,1.1) TL
E[X (t,T)]
1
-(>-,+/J)T
e -(>-'+j..l)t ~,...
•
(2.3.4.8)
e- (>-'+ j..l) T}+ (>-'+j..l) -2 e -(>-'+j..l) T{[ 1 + (>-'+ j..l) tJ 2_[ 1 + (>-'+ j..l) T] 2} +
,i~2.
(2.3.4.9)
The second moment of the first interval is
(2.3.4.10)
2.3.5 Prospective Interior Intervals
.'
The distribution function is defined as follows:
t T
H.'
(x)= pr [X.(t,T)
1
1
= pr
[
X.~x
1
~
x]
I S.1- l~t n S.~T
n Y~S.1 ]
1
,
65
x~T
- t.
Hence,
pr [t:S:S i _1:S:T-x n Y~Si_l+x J f 1(x)
h~' T (x)= ---..:.....::-----=--~--=--1
pr [S.1- l~t n S.~T
n Y~S.]1
1
Here, pr [t:S:S.1- l~T-x
n
Y~S.
1-
l+x]
, x:S:T-t
,i~2.
=
And
Pr [S.1- 1~t
n S.~T
n Y~S.1 ] = pr
1
[So1- 1~t
n S.1- 1~T-X.1 n Y~S.1- 1+X,]1
= pr [t:S:S.1- 1:S:T-X.1
n Y~S.1- 1+X.1 ]
T-t T-x
=o f t
f g'-1
(s) f. (x)
I l
R(s+x) ds dx.
With reference to the assumptions adopted in this chapter, the above
expression is equivalent to
Hence, the density function is
h~'
T (x) =
1
e.-1
1
(A+I-J~ [e-(A+J1)(t+X)[(A+J1)t]j-e-(A+J1)T[(A+}i)(T-x)]j
] /j!
j=O
,
66
i~2, x~T-t,
(2.3.5.1)
e-
where the value of
ei
is
And the distribution function is
The first two moments are shown next:
E [Xi (t,T)]=
And the second moment is as follows:
67
2
E [Xi (t,T)J=
e~
1
[{ 2e -(A+fl)t _ e -(A+fl) T [{ (A+fl) (T -t) + 1}2+ 1
2
(A+fl)
J} ~2 [(A.~,U) t~ lJI _
j=O J.
e~ k(A+I')\~(t:~)~+
1
[ 2Tj+3 -t j + 1((T-t) U+3) (j (T-t)+2T]+Zt Z)]
,i~2.
(2.3.5.4)
2.3.6 Retrospective Open Birth Intervals
The density function of the open birth interval is
t
o. (u)=
1
gi-l (t-u)F i (u)R(t)
0
f
t
=
g.1- 1 (u) F.1 (u)
-~----=:.--
gH (t-u)Fi (u)R(t)du
The above d.f. is identical to that given in the non-competing risk situation. Thus, all expressions given in section 2.2.6 are also applicable
here.
68
CHAPTER III
PARITY-DEPENDENT tv10DELS OF
BIRTI-I INTERVALS
3.1 Introdlction
The key assumption of all models described in Chapter II is that fertility is independent of a woman's age and parity. such a simplistic
description of human reproduction is inappropriate for populations
where fertility is Widely under voluntary control. It may, however,
approximate the reproduction process in the exceptional cases where
fertity is totally governed by biological forces.
Traditionally, human reproduction has been viewed as an age-parity
dependent process, where a woman's age and parity, in order of importance, are recognized as the most important variables associated with
her reproductive behavior. But given current levels of effective contraceptive practice, age ceases to play the prime role in determining reproduction in most populations. Instead, birth order or parity has emerged as the most influential factor in the formulation of reproductive
patterns of human populations. [Chiang & Vanden Berg, 1982; Nour,
1985] .
In recognition of this fact I present, in this chapter, stochastic
models of birth intervals specific to each ascertainment plan, using a
strictly
parity~ependent fertility
hazard. Age is considered here to
play no direct role in the likelihood of having a birth. Thus any effects
of age are assumed to be exerted through birth order. This postulate
is based on the fact that age at birth and birth order are almost always
highly positively correlated. For example, if we consider the extreme
case in which all cohort members have births at the same age, then the
step function that relates the fertility hazard to birth order is the same
function that relates the former to age, but in segmented fashon. Thus,
the higher the association between birth order and age at birth, the
more the relation of the fertility hazard with birth order will also
reflect the influence due to age.
It is important to point out for the probablistic cierivations below
that all assumptions of Chapter II are also applicable here, but after
letting A. (x) = A., i=1,2, ... , there is no need to repeat them. However,
1
1
the condition AI*Ak for alll*k=1,2, ... ,i, is crucial in all expressions
given in this chapter, otherwise the probability models are not defined.
Chapter III is organized into three sections. Following this introduction, Section 3.2 includes derivations of the probability distributions
and related quantities of the i th birth interval corresponding to each
ascertainment plan. Section 3.3 discusses how the truncation bias in
the mean and variance of the birth interval ( and the sensitivity of the
former to fertility change) are influencd by the type of ascertainment
70
e-
former to fertility change} are influencd by the type of ascertainment
method, the length of observation period, birth order, and the fertility
input parameter ( hazard).
It is evident in Section 3 of the preceding chapter that models of
birth intervals and expressions that account for fertility competing
risks are exactly the same as in the non-competing risk situations. The
only difference is that the parameter in the first case is (A+Jl), while
it is only A in the second. For this reason, the competing risk will be
disregarded in the remaining work of this study. Consequently, the prospecti ve and retrospective studies coincide; in the sense that only the
survi vors enjoying their first marriage are the units of analysis under
both ascertainment methods. The closed birth intervals ascertained by
either method will be termed henceforth by "All Closed Intervals".
71
3.2 Stochastic Models Of Birth Intervals And Related Quantities
3.2.1 All Closed Intervals
The general expression of the density function of the
/h birth
interval has been shown in equation 2.2. 1 as
G. (t-x) f. (x)
1
1
,
x~t.
Here the density function of the waiting time till the ith birth, gi (.), is
obtained as shown below:
t t
t
f r········f
gi(t)=
o
f 1 (x 1) f2(x2)· .. ·fi(t-xl-x2-· .. xi_l)dxi_l .. ·dx1'
t-x 1 t-(x 1+x 2 +·· .+x i - 2 )
i
i
e-Akt
A 2: ---.,;:;...-1=1 lk=1
ITi (AfAk)
1=1
l*k
= IT
, t)O.
t
It follows that the denominator of
.
G. (t)=
1
.
1
1
1=1
k=l
[Chiang, 1968]
h~1 (x);
G. (t) =
1
0
-Ak t
1
IT Al 2: ---:-.-_e
_
IT (A 1-Ak)Ak
1=1
l*k
72
f
Similarly,
i-1
G. (t-x)=
n
>-
Al
1=1
1
-A (t-x)
1- e k
i-1
k=1
Substituting for G (t), G - (t-x), and f (x) in the general expression of
i
i
i 1
h~ (x) yields,
-A (t-x)
1-e k
h~1 (x) =
i
n
Al
1=1
-1
= e.
\
e
1
L.
k=1
2: ----:i~----
n
k=1
. 1
-I\i x ~
-Akt
1 - e
i
(A -A )A
1= 1 1k k
l*k
-A (t-x)
1-e k
---==--i-...;:;,.1----
n (Al-Ak)A k
1=1
l*k
where
e.1
is
73
<t
,x-
·>2 ,
,1-
(3.2.1.1)
The density function of the first birth interval is given next,
A
t
hi (x) =
-A x
1
1e
(3.2.1.2)
x~t.
,
The distribution function of X. (t) takes the following form,
1
-A.x
H~(x)= e~1
1
1
i -1 (A .-A ) ( 1- e
2:
k= 1
1
1) - A. e
k
-A t
-(A.-A )x
k [ 1- e l k ]
1
AiA k
i
IT (AI-A k)
1=1
l *k
And for the first birth interval,
<t , 1'>2 •
, x-
(3.2.1.3)
Hi (x) is
ex~t.
,
(3.2.1.4)
The first two moments of Xi (t) are
,
74
i~2,
(3.2.1.5)
and
-\ t
-\ t
}
\~ [2e k - {1+[1+t(\(\k)]2} e i]
,i~2.
(3.2.1.6)
And for the first interval,
E[X (t)]
1
=
\"11 [1 - (1 +t\ 1) e
-\ t
1]
(3.2.1.7)
-\ t
1- e
1
and
\~2 [ 2 - {1+(1+t\.)2} e
2
E[X 1 (t)]=
-\.t
1
1-e
-\. t
1]
(3.2.1.8)
1
1
3.2.2 Last Closed Intervals
The general expression of the density function of Xi (t), for
h~ (x)
1
1>t-x ) f. (x)
= Pr( S.1- 1 <t-x n S.1- 1+X.+
1
1
pr( S i <t n S i + 1 >t )
The analogous expression for the first interval is
75
,
i~2,
x~t, i~2.
is
t
_ pr(X 2 >t-x) f 1 (x)
h 1 (x) - pr( S1 <t n S2>t )
,
x~t.
The denominator of hi (x) is evaluated as:
pr( S.1 <t
n S'+1>t
) =
1
pr ( S.1 <t ) pr ( S'+1>t
IS.1 <t )
1
t
o=fg·1 (y)
=
f
o
pr ( S'+1>t IS.=y ) dy
1
1
t
gi(y) pr (X i + 1 >t-y) dy.
Applying the assumptions of the current model to the above expression
yields
ft 2:
pr ( S. <t n S'+1>t ) = IT Al
i
1
1
1= 1 0
e- AkY
i
--:-1'- - - " - - -
k= 1
e
-A i + 1 (t-y)
dy
IT (AI-A k)
1=1
1*k
1
i
\.
l'
l'
-Akt
=ITA -/\i+1 t 1
-1
e
k~1-i-+"'-1~-- + IT Al 2: i+1
.
1=1 1 e
IT (A(A )
1=1 k=1 II (A - A )
1k
k
1= 1
1= 1
1*k
1*k
78
e-
Using the result given by Chiang (1968) that
we found,
i
2:
k=1
Thus the value of pr ( Si <t
pr (S.<t
1
n
n Si+ 1)t )
reduces to
i
i+1
-""k t
e
S'+1)t) = n""l2: '+1
.
1
1= 1 k= 1 1
1=1 1 k
l*k
n ("" _"" )
Also the first term of the numerator of h~ (x) is evaluated as
pr (Si-1 <t-x) pr ( [Si-1 +X i + 1 ])t- x l si-1 <t-x)
77
t-x
~f gi-1(y)
pr (X i + 1 )t-x-y) dy
f
t-x
i-1
i-1
1=1
k=1
i-1
i-1
=IT .1.. 1 2:
=IT.I.. 2:
1=1 lk=1
i-1 1
(.I.. -.I.. ) 0
1=1 1 k
1*k
IT
-.I.. ky -.I.. i + 1 (t-x-y) d
e
e
y
i -1
1[I1(.1..(.I..k) (.I.. i + 1 - .I.. k)
l*k
Accordingly, the density function of X. (t) is
1
e(3.2.2.1)
The density of the first interval is shown below
,
78
x~t.
(3.2.2.2)
The distribution function of X. (t),
1
t
H. (x)=
i~2,
has been evaluated as follows:
-1 i-1
e. 2:
1
1
k=1
-'\.
1+ 1
t
e
[1-e
'\i-'\i+1
- ('\. -'\. + 1) x }
11
,
]
x~t, i~2.
(3.2.2.3)
And for the first interval the distribution function may be seen to be
-(,\ -,\ )x
t
_
H1 (x) -
1- e
1-e
1
2
,
x~t.
(3.2.2.4)
-(\ -\ )t
1 2
The Expressions for the first two moments can be shown to be as
follows:
E[X. (t)]=e~1
1
1
i -1
2:
k=1
,
79
i~2;
and
(3.2.2.5)
(3.2.2.6)
And for the first interval,
E[X (t)]
i
-(A -A )t
1- e 1 2 [1+t(A -A )]
1 2
-(A -A )t
t 2
(A 1-A ) (1 - e
2
(3.2.2.7)
)
and
2
(3.2.2.8)
E[X 1 (t)]
3.2.3 Prospective Next (Straddling) Intervals
It has been shown in Chapter II that this strategy of collecting data
on birth intervals gives rise to two density functions of Xi (t, T), for
i~2,
depending on whether t
~
T-t or t) T-t. The general expressions
of the p.d.f in both cases are shown over
80
Case 1 t
~
T-t.
for
t T
-1
h.'
(x)= a.1 pr (S.1- 1 (t) f.1 (x)
1
for t (x~T -t,
-1
for T-t (x~T.
a. pr (S. 1( T-t) f. (x)
1
1-
x~t,
1
Case 2 t> T-t.
-1
a. pr (t-x(S. 1 (t) f. (x)
1
t
T
1-
for
1
-1
for T-t (x~t,
h.'
(x)= a.1 pr (t-x(S.1- 1 (T-x) f.(x)
1
1
-1
a. pr (S. 1 (T-x) f. (x)
1
1-
for
1
where a.1 is pr (S.1- 1 (t
x~T-t,
n
t(x~T,
t(S.1 (T).
According to the parity-dependent assumption, the quantities involved
in the above two densities are evaluated as shown next,
t
a.= pr (S. 1 (t
1
1-
n t(S.1 (T)
=0 fpr (t(S.1 (T IS.1- 1=z) g.1- 1 (z) dz.
81
The formula of gi-1 (z) is given in section 3.2.1 as
, 1
1-
IT
gi-1 (Z) =
' 1
1-
2:
\1
1=1
-\k Z
-ie~_1~--
IT (\
k=l
1=1
l*k
-A )
1 k
It follows that,
ft
i-1
-AkZ
=IT \1 2: -i-_1,.:;.e_i-1
a
i
1= 1 0 k= 1
-A. (T-z)
-\. (t-z)
1
(e
-e
1
) dZ
IT (\I\fl\k
\)
1=1
l*k
i-1
=
-A.T -(\k-A.)t
i-1 e -Akt -e -\.t
1 -e
1 e
[
1 -1 L
IT \1 2:
1= 1 k= 1
i
e-
IT (\l-A k)
1=1
1*k
i-1
Similarly, pr (t-x<S. -1 <t)
1
i-1
= IT A 2:
-A (t-x)
e
k
-A t
-e
k
1
1=1 "k=1
and
i-1
pr (5'_1 <t)
1
i-1
= G'_1 l (t) =1=IT1Al k=2:1
Hence, the density functions of Xi (t, T) ,
82
i~2,
take the following forms:
Case 1
t~
T -to
e~1A. e
1
-A.xi-1
1
2:
k=1
1
for
x~t,
for
t<x~T-t,
for T-t <x~T.
(3.2.3.1)
Case 2 t
> T-t.
e.-1 A.e
1
1
-A. x i -1 -\k (t-x) -Ak t
1 2: e
-e
. 1
1k=1
(A -A ) \
1= 1 1k k
n
for
x~T-t,
for
T-t<x~t,
for
t<x~T,
l*k
ht,
. T (x )
1
= e-1. /\.\
1
1
-A.X i-1
e
1
-Ak(t-x) -Ak(T-x)
~ e . 1
-e
L..
k=l
(A -A ) A
1=1 1 k k
IT
l*k
. 1
-Ak (T -x)
-/\\ x 1-1
i ~ 1-e
ei Ai e
L.. -7"-i--:-1--:"'--k=1
A
1=1 1 k k
n (\ -\ )
l*k
where
(3.2.3.2)
i -1
e.= ( n AI)
1
1= 1
83
-1
a ..
1
The pdf of the first interval is shown as follows:
f 1 (x)
t T
hl'
(x)
=
{ pr (
,t<x~T
t~X 1 <T )
o
(3.2.3.3)
Otherwise.
The probability distribution function of Xi (t, T), for
i~2,
is evaluated
as shown next:
Case 1
t~T -to
-Akt
e~ 1
1
i-1 A.e
2:
-(,).\.-Ak)X
[1-e
1
J-
-Akt
(A.-Ak)e
i
1
k=1
-A.X
[1-e
1
J
1
IT (A(Ak)A k
,x~t
1=1
l*k
-Akt
Ake
-Ait
-Ake
-A.X
- (Ai -Ak)e
1
(l-e
-A t
k)
i
IT (AI-Ak)A k
,t<x~T-t
1=1
l*k
e.-1
1
i -1
'"
L..
k=l
,T-t<x~T.
(3.2.3.4)
84
A .
..
Case 2 t> T-to
(A.-Ak)e
-A.X
1
1
+A.e
1
-\, T-(A.-Ak)X }
I<
1
,
t<x~T.
(3.2.3.5)
The distribution function of the first interval is
e
-A t -A x
1 -e 1
,
t<x~T,
Otherwise.
The first two moments of Xi (t,T),
i~2,
are exactly the same for both
densities given above. They are shown next:
85
(3.2.3.6)
-1 i-1
E[X. (t,T)]=e.
2:
1
1
k=1
-\. T
-\. t
\.(2+t\.)}e
1
1
1
+{\.(2+T\')-\k(1+T\.)}e
1
1
1
1
+
-\. (T-t)-\ t }
{\k(1+\. (T-t)]-\. (2+\. (T-t)]}e
1
1
1
1
,i~2.
k
(3.2.3.7)
2
-1 i-1
1
{3
3 -\k t
E[X. (t,T)]= e. 2: ----=---.--- 2[\. -(\'-\k) e
+
1
1 k= 1
2
2 1
1
1
\. (\. -\k) IT (\(\k) \k
1
1
1=1
l*k
'>2 .
,1-
(3.2.3.8)
The first two moments of the first interval are given by:
, and
E{X (t,T)]
1
86
(3.2.3.9)
2
E[X (t,T)]
1
3.2.4 Preopecti ve Interior Intervals
The general expression of the density function of X. (T,t),
1
h~,T(x)
1
Pr (t < S.1- 1 <T-x) f.1 (x)
pr (S.1- l>t n S.<T)
1
,
x~T-t.
T-x
Here, pr (t<Si-l <T-x)
~
f
gi-l (x) dx
i-l
i-l
=TI
1=1
A
2:
lk=l
T-t
And pr(S.1- 1>t
f
n s.1 <T) =0 pr (t <So1- 1 <T-x)
87
f. (x) dx
1
i~2,
is
Therefore, the density function of Xi (t, T),
h~,T(x)
1
= e~l
1
i~2,
is
-\.x i-l
\.e
1
1
2:
(3.2.4.1)
k= 1
e·
The probability distribution function of Xi (t, T),
i~2,
,
is as follows:
x~T-t.
(3.2.4.2)
The first two moments of this distribution are given in the following:
88
[l+A (T-t)] e
i
-A. (T-t)J
1
-A
-
T[
A~ e k l-[l+(A(A k) (T-t)]e
,
-(A.-A) (T-t)
1
(3.2.4.4)
3.2.5 Open Birth Intervals
The general expression of the density function of U.1 (t) is
1
=
gi_l(t-u) Fi(u)
---=--=~---=---
pr [ C(O,t)
= i-l
]
89
,
u~t
J}
(3.2.4.3)
i~2.
'>2
,1
- .
t
o. (u)
k
,
i~2.
According to the assumptions of the present chapter, the different
terms involved in
0i (u) take the folloWing forms:
. 1
. 1
-Ak(t-U)
gi-1 (t-u) = IT Al 2: -i--:_1-=e_-1=1 k=l IT (A -A )
1= 1 1k
l*k
1-
1-
, F i (u)
=e
-A,U
1,
and
Accordingly, the p.d.f of the open birth interval is
t
o. (u)
1
-1 i-1
= e,1 k=1
2:
-Akt - (A 1, -A k)u
---'::'---:-1'_'"7'1---e
IT (ACAk)
1=1
1*k
.
where
e.=
1
1
~
L..
e
-Akt
-,1 ~-­
k= 1 IT (A -A )
1=1 1 k
l*k
And the distribution function of U.1 (t) is as follows:
90
e·
,
u~t, i~2,
(3.2.5.1)
t
O. (u)
1
-1 i-1 e
= e. 2:
1
k=1
-A t
-(A. -A )u
k [1 - e l k
i
L
,u~t, i~2.
(3.2.5.2)
IT (AI-A k)
1=1
l*k
The first two moments of the open birth interval are as follows:
,
2
E[U.(t)]=e~
1
1
1 i-1 2e
2:
k= 1
i~2,
-A t
-A. t
k - {1+[1+t(A -A )]2} e 1
i k
(A. -A )
k
1
2
i
IT (AI-Ak)
1=1
l*k
91
and
,
i~2.
(3.2.5.3)
(3.2.5.4)
3.3 Patterns of Bias in the Mean and Variance of Birth Intervals and
Sensitivity of the Mean to Fertility Olange
The manner in which truncation bias affects the mean and variance
of order-specific birth intervals, and the sensitivity of the mean to detect fertility changes are the focus of the present section. Elaboration
will be made on how the bias and sensitivity are influenced by the
length of the observation period (marital duration), birth order, fertility hazard function, and data ascertainment method. For each ascertainment method the bias is measured as the difference in the moments
from those attained based on the complete reproductive history of the
cohort. In an other dimension, sensit ivi tY is assessed as the proportionate increase in the mean interval relative to the level and pattern
of decline in fertility hazards
Apparently expressions for the first two moments of the i
th
birth
interval gi ven in Section 3.2 above are so involved that one cannot
determine analytically any pattern of relationships among the moments
pertaining to the different ascertainment methods. For thiS reason the
analysis will be made on the basis of numeric values of the mean and
variance, which are calculated for the second, third and fourth birth
intervals. Fertility parameters (Ai' A2' A3' A4 , AS) are assumed to
take intial values of (.9 .8 .7 .6 .5), level (H). Then, thiS set of
values is allowed to decrease to four different levels in two different
manners. First the decrease is proportionally by 10% for level (Ip) and
92
by 20% for level (Lp), in all fertility hazards. Second, the decline is
disproportionately by (10%, 15%, 20%, 25%, 30%) for level (Id), and
by (20%, 25%, 30%, 35%, 40%) for level (Ld), in A , A , A , A , AS'
1 2
3 4
respectively. The results of this numerical illustration are summarized below.
3.3. 1 All Closed Intervals
Assuming that age at marriage is 15 and age of menopause is 50
for all cohort members, the complete observation period is 3S years.
In addition to the mean and variance corresponding to the complete reproducti ve history, which represent the true population values ( Le., at
t=3S years), table 3.1 shows such summary statistics for two incomplete observations of length 10 and 20 years. As expected, the bias,
measured as the deviation from the true population values, increases
with the shorter the observation period, the lower the fertility level,
and the higher the birth order interval (table 3.1). The absolute relati ve bias is consistently higher in the variance than the mean
In general, 20 years of observations are sufficient to yield summary
statistics of birth intervals very close to the true values. This is evident even for the fourth interval at the lowest fertility level (Id), where
the relative biases are -2.7% and -12.2% in the mean and variance
respect ivel y.
When observations are truncated 10 years after marriage, the bias
in the second interval is almost negligible at the highest fertility (H),
93
but increases steadily the lower the level of fertility, peaking at the
lowest fertility level (Ld) at -4.2% and - 8% for the mean and variance,
respectively. The bias, however, is relatively large in the third and
fourth intervals. We found that the relative biases in the mean and
variance of the third interval range from (- 4.2%, - 15.7%), at the highest level of fertility, to (-13.2%, -38.9%), at the lowest level. The
corresponding figures of the fourth interval are (-11. 4%, - 33.8%) and
(- 28.5%, - 62.5%).
With respect to sensitivity, Table 3.1 below reveals strong eVidence
of high sensitivity, particularly when the observation period extends to
20 years or more. This is true irrespective of the level or manner of
fertility decline. In fact, the percentage increase in the mean interval
of the three birth orders, at t=20, exceeds the percentage decline in the
respective fertility hazards. For instance, at t=20 and with a 20%
proportional decline in all fertility hazards, the gain in mean interval
length is about 25% for all births. In the case of disproportional decline in fertility according to level (Ld), mean birth intervals increase
by 34%, 42%, 50% in the second, third, and fourth intervals, respectively. But when observations continue only for 10 years or less, the
means, particularly for the higher order intervals, do not possess the
same detective power. For instance, the proportionate increase in the
mean length of the fourth interval is about half the proportionate decrease in the respective fertility hazard (e.g., the difference 1.48 to
1.56 is only 5%).
94
3.3.2 Last Closed Intervals
Controlling for the length of the observation period, the numbers in
Table 3.2 are larger than their counterparts in Table 3. 1. The disparity increases the longer the observation period, the higher the fertility, and the lower the birth order. This upward bias is due to the
additional influence of left truncation caused by discarding women of
parity i+ 1 and higher at time t, when studying intervals of order i or
less. Of course such women have had their births in qUick succession
relative to others. At t=20, the total relative bias in the mean and
variance of the second interval ranges from (246%, 629%) at the
lowest fertility, to (371 %, 1214%) at the highest fertility. It ranges,
for the fourth interval, from (89%, 113%) to (182%, 385%). At t=10,
the corresponding numbers are (93%, 96%), (158%, 250%) for the
second, and (-10%, 51 %), (36%, 13%) for the fourth intervals.
The sensitivity of the last closed birth intervals is minimal especially when fertility risks decline proportionately by the same amount.
The response of the mean is either null or in the opposite direction.
This insensitivity is manifest irrespective of the level of fertility dec,
line, length of observation period, or order of birth interval. Table 3.2
shows that the best possible improvement in mean last closed interval
is about 4.4%, corresponding to birth order 2, t=20, and fertility level
Lp.
95
3.3.3 Straddling Intervals
This type of birth interval is similar to the last closed one with
respect to liability to both right and left truncation. Also, the positive
bias of left truncation here more than offsets the negative bias of right
truncation, leading to an overall positive bias in the moments, with the
exception of the variance of the fourth interval corresponding to the
observation period (5,10). In this respect, it is important to point out
that thiS result is not always generalizable because the direction of
bias, in thiS and in the last closed interval, is contingent upon, among
other factors, the prevailing level of fertility. If fertility was much
lower than the levels specified in the tables, the net bias might move in
the reverse direction. Table 3.3 affirms the conjecture that the farther point t is from marriage and/or the longer the observation period,
the greater the effect of left truncation. When the observation period
is (5, 10), the total relative bias in the mean and variar.ce of the
second interval ranges from (148%, 24%), at the lowest fertility level
to (209%, 101 %) at the highest. Similarly, the bias in the fourth interval at the same fertility levels ranges from only (25%,57%) to (71 %,
10%). Similar figures corresponding to the observation period (10,20)
are (356%, 270%), (466%, 507%) for the second and (123%, 43%),
(197%, 167%) for the fourth intervals.
The sensitivity of straddling intervals is, to some extent, better
than that of last closed ones. Generally, the proportionate increase in
the mean is positively associated with the birth order, the decline in
96
e·
the current hazard relative to the decline in lower order hazards, and
especially the length of the observation period.
The sensitivity also
shrinks remarkably as the survey point t moves towards the upper age
of the reproductive span. As an illustration, with the observation period (5, 10), the proportionate increases in the mean of the second interval at fertility levels (Lp) and (Ld) are 4.1% and 7.5%, respectively,
while similar numbers for the fourth interval are 4.9%, 11.9%. When
the observation period becomes (5,15) the corresponding gains in the
mean birth interval, at the same fertility levels, are 6.6% and 11.1 %
in the second, and 10.4% and 23.6% in the fourth intervals. But if the
observation period changes to (10, 20), the gain in the mean second
birth interval, at fertility levels (Lp), (Ld), drops to 2.0% and 7.8%,
respectively, and in the fourth interval, to 4.0% and 15%.
3.3.4 Prospective Interior Intervals
This type of birth interval is very similar to that labeled as All
Closed Intervals. In fact, they are identical if observations start with
marriage instead of t ()O). The overall bias, in absolute value, is generall y less than that of the last closed and the straddling intervals, particularly for the mean. The relative bias decreases with the length of
observation period, lower birth order, and higher fertility. Shifting
some observation periods along the reproductive span does not result
in major change in total bias. It is noteworthy that the second interval
remains entirely unchanged if the location of an observation period is
97
altered. see Table 3.4. When the observation period is (5. 10). the
total relative bias in the mean and variance of the second birth interval ranges from (-15,2%,- 44.2%) at the highest fertility to (-26.9% ,
-62.6%) at the lowest. For the fourth birth interval, the relative bias
at the same fertility levels ranges from (- 29.3%,-65.1%) to (- 48.1%,
- 82.9%). The corresponding numbers for the observation period
(10,20) are (-8%, - 5.1%) and (-4.2%, - 16.5%) for the second,(- 4.8%,
-18.0%) and (-16.4%,-46.6%) for the fourth birth intervals.
The sensitivity of prospective interior intervals is more dependent
upon the length rather than the location of the observation. With shorter observation periods, sensitivity is reasonably high only for lower
order intervals. But with a longer observation period it is relati vel y
high for all birth intervals. For instance, at the observation period
(5, 10), the gains in mean second interval corresponding to fertility
levels (Lp) and (Ld) are 11.3% and 15.1% respectively, while the gair.s
in the fourth interval are only 5.9% and 12.7%. When the observation
period is (10, 20), the gains in the mean interval, corresponding to
the same fertility levels above, are (21.8%, 29%) for the second and
(17.6%, 34.6%) for the fourth birth intervals.
98
99
100
Table 3.3
PATTERNS OF VARIATION IN MEAN AND VARIANCE OF
STRADDLING BIRTH INTERVALS FOR DIFFERENT VALUES
OF (t,T) AND VARYING FERTILITY LEVELS
(t,T)
Order of Birth Interval
Fertility
3
2
Level
(5,10)
H
Ip
Id
Lp
Ld
3.86
3.94
4.07
4.02
4.15
3.14
3.27
3.31
3.40
3.44
3.15
3.22
3.39
3.30
3.48
(5,15)
H
Ip
Id
Lp
Ld
3.95
4.07
4.23
4.21
4.39
3.59
3.92
4.09
4.34
4.58
(10,15)
H
Ip
Id
Lp
Ld
6.98
6.99
7.35
7.01
7.38
H
(10,20)
Ip
Id
Lp
Ld
1
1
Years
Mean
Var.
2.75
2.85
3.00
2.93
3.11
2.85
2.91
3.11
2.99
3.19
2.49
2.57
2.76
2.66
2.84
3.30
3.42
3.68
3.59
3.88
3.46
3.84
4.35
4.31
4.93
3.09
3.23
3.59
3.41
3.82
3.62
4.05
4.92
4.56
5.56
9.02
9.21
8.94
9.41
9.16
5.48
5.48
5.87
5.49
5.89
7.60
7.67
7.92
7.72
8.00
4.73
4.73
5.09
4.73
5.11
6.29
6.29
6.68
6.28
6.69
7.07 9.46
7.12 9.86
7.51 9.72
7.21 10.34
7.62 10.29
5.63
5.69
6.16
5.78
6.29
8.32
8.65
9.27
9.08
9.83
4.96
5.04
5.57
5.14
5.72
7.42
7.76
8.84
8.19
9.41
Mean
Var.
4
Mean
See footnote in table 3. 1.
101
Var.
102
CHAPTER IV
QUASI PARITY-DEPENDENT MODELS OF
BIRlli INTERVALS
4.1 Introcftrlion
As mentioned before, the models in Chapter III have been set up on
the basis of a crucial assumption: the strict dependence of the fertility
hazard on birth order or parity. In fact those probablistic models are
undefined if two or more of the parit y~ependent hazards are of the
same value. In real situations, however, we are more likely to accept
, apriori, the notion that after a certain parity, say 4 or 5, the risk of
fertility for higher birth orders is almost independent of achieved
parity. The reproductive behavior of women who have already attained
high parity is likely to be governed more by fecundity rather than
conscious regulating of fertility. Therefore, for this group of women
achieved parity have little influence on their reproduction. However,
the average risk for lower birth orders among all cohort members is
still regarded as strictly parity-dependent.
Morever, suppose as a result of a formal testing of hypothesis,
based on the parity-dependent model, we were not able to reject the
null hypothesis that after a certain parity, r, the fertility hazard is
invariant with birth order. The models proposed in Chapter III are not
of any practical use for reestimating the parameters to improve the
precision that would result from considering the reduced parameter
space (Ai' A2" .. ,A , A) of dimension r+ i, instead of the i th dimenr
sional space (Ai' A ,. . " A , . . . ,Ai)' where i~3 and i-r ~2. In
2
r
response to these factors, we believe it is necessary to extend the
parity dependent models of the i th birth interval and of the open interval since the most recent birth so as to accommodate the assumption of
the equality of the parameters after a designated parity.
In this chapter we present the distribution of the i th birth interval,
for each ascertainment method included in this study, accounting for
this situation of reduced parameter space. The open birth interval will
be covered as well. Apart from this newly introduced assumption, all
others in the preceding chapters are still in effect. FolloWing this
introduction, we deal with each of the different ascertainment methods
and the open birth interval, each in a separate section.
104
4.2 All Closed Intervals
The key assumption throghout this chapter can be stated formally as,
~1 *~2*' . '*~r
where
~*~k
~r+l =~r+2=' . '=~i=~
and
for all k=1,2, ... ,r,
3~i~m,
i-r~2,
and m is the highest birth order under study.
In All Closed Intervals we know that the general expression of the
p.d.fofX.(t) (expression 2.2.1 p. 39) is
1
t
hi (x)
G. ! (t-x) f. (x)
11
= -~~G:::;-1--;'
(~t)~--
In the current situation the waiting time from marriage until the occurence of the i th birth, S., is comprised of two independent components,
1
Si!' Si2' Here Sit is the summation of r independent random variables (X! ,X ,' . " X r ) , each with an exponential distribution with
2
parameter ~. such that ~.* ~k for all j*k=!,2,- . ·,r. And S'2 is the
J
J
1
summation of i-r identically exponentially distributed random
variables with
parameter~,
where
~*~k
' k=1,2,' . ·,r. The density
function of S.1 is, therefore, the convolution of the densities g.1 ! ('),
gi2(') of Sit and Si2' respectively, where
,s)o,
105
And
Therefore,
g. (s) =
0
1
f
s
g'1 (y) g'2(s-y) dy
1
1
r
.
A Al-r
IT 1
1=1
r
~
= r(i-r)
k-1
1
JS -AkY
i-r -A (s-y)
e
(s-y)
e
dy
r
IT (A
1=1
-A ) 0
1 k
1*k
r
IT
.
A Al-r
_ 1= 1 1
-
r(i-r)
r
2:
k=1
e-/\k s
S
r
IT (ACAk)
1=1
f
o
e
-(A-Ak)Z.l-r- 1
Z
edz.
1*k
The integral in the above expression has been evaluated by successive
integration by parts. So, this expression becomes
.
gi (S)=A l-r
r
r
1
{
IT
Al 2: -r-=---
1=1
k=1
IT (A -A
1=1
)
1 k
l*k
, S20.
106
And the distribution function of S. is:
1
s
G. (s) =
1
0
f
g. (y) dy.
1
}.
Similarly, we can find the expression of G i - l (.). Substituting for Gi_
(t-x) , G. (t), and f. (x) in the general expression for h~ (x) , we get the
1
1
1
l
following functional form of the density function:
h~ (x) =e ~ 1
1
1
2:r
k=l
-AX
r
[ l-e
e
IT (A
1=1
-A)
e.1
\
Ak(/\-Ak)
-2:
[A(t-X)~
al
. - 2: _--:.:a'--=....:;;.,O_ _.,..--__·~l-r-l. 0
(\ _\ )j+l \ i-r-j-l
J=
/\ /\k
/\
1 k
l*k
where
-Ak(t-X).l-r- 2 1
.
.
-A(t-X)
_1 r - J - 2 e
,
i~3, i-r~2
is gi yen by:
107
,
x~t,
(4.2.1)
The probability distribution function of Xi (t) is found to be
-AX
-A t -At
A-1 (1-e
)-(A-A ) -1 (e k -e )
k
A (A-A )i-r-1
k
k
-1 r
1
=
e.
2:
---'--1 r
11
t
H. (x)
k
- 1=1
n (A 1-A k)
l*k
-1
i-r-2 A
-Ax i-r-j-2 e
) - ~O
(1-e
-At
Aa t a + 1
(a+ 1) !
2:
j=O
,
i~3, i-r~2, x~t.
(4.2.2)
The first two moments of X. (t) are shown below:
1
,
i~3
108
,
i-r~2;
and
(4.2.3)
J
'>3 , 1-r. >2 •
, 1-
(4.2.4)
4.3 Last Closed Intervals
The general expression of the density function of X. (t) is
1
r[S·l<t-xn(S·t+X·+l)t-x]f.(x)
h.t (x) = P
111
1
1
pr [ Si <t n Si+l)t ]
t
where, pr [Si <t n Si+l)t]=
Io
gi(y) F i + 1 (t-y) dy.
The expression of gi (0) is as given in section 4.2, and
109
,x~t,
It follows that:
Pr[Si <t
.
r
1
r
n Si+1 )t]= ,\ l-r IT 2: -r--=--1=1 k=1
IT (,\
1=1
1*k
[
-A )
1 k
i-r-1
2:
j=O
Similarly, the first term of the numerator of h~ (x) is
1
t-x
pr [ Si-1 <t-x
n
(Si-1 +X i + 1)t-x] =
fo
r
. 1 IT
1
~e
r
=A l-rA 2:
r
1
1=1 k=1 IT (A -,\ )
1=1 1 k
l*k
gi-1 (y) F i + 1 (t-x-y) dy,
-A (t-x) -A(t-X)
k
- ~
l-r
(A-A k)
Accordingly, the density function takes the following form:
110
e-
i-r-2
e
-A(t-X)
(t-x)
i-r-J'-1 ]
,i~3, i-r~2, x~t,
where
e,1
(4.3.1)
is
r
e.= 2:
1
k=1
The distribution function of the
/h
-Akt
t
-1 r _--=-1_ _ [e
Hi (x) = ei 2: r
[
k=1 IT (A -A )
1=1 1 k
l*k
last closed interval is
-(A-Ak)x
-At
{1-e
.} - t(A-A k) e
('\_'\ )1-r+1
/\ /\k
,
111
i~3, i-r~2, x~t.
(4.3.2)
The first two moments are shown below:
r
e~1 2:
E[X i (t)] =
r 1
k=1 IT (A -A )
1=1 1 k
1*k
'>3 ,1-r-;
. >2
,1-
(4.3.3)
and
2
E[Xi(t)]=e~
1 r
1
[-Akt
-At
2: - r - - - " - - - - - - - - 2e
- 1/3 e {3 +
k=1 IT (A -A ) (A-A )i-r+3
1= 1 1k
k
1*k
. 2
-At.. 2
[l2:-r - _-=-~---:._l-_r-_J_+~___:_
r __~__
-1 ~
1
2 e
t
ei
L..
k=1
r
IT (A
1=1
-A)
1 k
j=O
. 1
J
(i-r-j+2)! (A-Ak)J+
1*k
(4.3.4)
112
e·
4.4
Straddli~
Intervals
It has been shown previously that this strategy of collecting data on
birth intervals gives rise two density functions, according to whether
t~
T-t or t
Case 1 t
>T-to
The general forms of these pdf's are as follows:
~T ;-t.
-1
a. pr (t-x~s. 1 ~t) f. (x)
1
1-
1
, for
x~t,
for t <x~T -t,
-1
a.1 pr (S.1- 1 ~T -x)
Case 2 t
for T-t <x~T.
> T-to
, for
h~'1 T (x) = a ~1 1pr (t-x ~S.1- 1 ~T -x) f.1 (x)
-1
a.1 pr (S.1- 1 ~T-x) f.1 (x).
where a.=
pr[ S.1- 1~t
1
n
(t<S.~T)
1
x~T-t,
for T -t <x~t,
for
t<x~T,
].
Applying the assumptions of the current models to the above densities
gives the following expressions of the p.d.f.:
113
Case 1
t~T -to
[e
[
-A (t-x)
k
.
-e
-A t
k
A (A-A )i-r-1
k
,
k
,
l-r::J- 2 e
-A (t-x)
i-r-2 a=O
2.
[A(t-X)]a- e
a!
.
-At
(At)a
]
. 2
l-r~-
e
-At
(At)a
r
-AX
-Akt
i-r-2
1 - a=O a!
h.t, T(x)=e.-1 '" A e
1 - e.
_~
1
1 ~
r
I
l-r-1 '_
(A-A )j+ 1 Ai-r-j-1
k-1
(A -A
Ak (A-A k)
Jk
1=1 1 k
r
r
IT
l*k
,t <x~T-t,
-AX
r
h~,T(x)=e~1 '"
1
1
Ae
r
k=l
(A -A)
1=1 1 k
~
IT
J
-A (T-x)
1 -e
.
l-r-1
Ak (A-A k)
[k
l*k
i-r.:j-2
2.
-A (T -x)
e
a
(A(T-x)L
i-r-2 1 - a=O
a!
2:
(A-A ) j + 1 Ai-r- j - 1
j=O
J
k
,
for,
114
T-t<x~T.
i~3, i-r~2,
(4.4.1)
e-
where,
=t
eo1
r
_
-At
e
-1
-AT
- e
nr (A -A)
[e
1=1 1 k
1*k
-(A -A)t
.
2
k
- 1 l-r0-
-
(A-A k )l r
2:
j=O
t i -r - j - 1
(i-r-j-l)l (A-Ak)j+ I).
Case2 t) T-t.
--
i-r-j-2
i-r-2
2:
2:
a=O
j=O
, x:S;T -t,
h~' T (x) = e~ 1
1
1
:f ~Ae:::--_AX
_ _ [ e-A k (t-x) _ e-A k (T-x)
k--1
r
Ak(A-Ak)i-r-1
IT (A1-Ak)
1=1
l*k
i-r-j-2
i-r-2
2:
2:
-A (t-x)
-A (T-x)
e
[A(t-X))a- e
[A(T-x)J~
a=O
(A-Ak)j+l
j=O
a~i_r_j_l
J
, T-t<x:S;t,
115
i-r.:j-2 e
1-~
a=O
i-r-2
2:
-\(T -x)
[\(T-x)J~
J
a!
j=O
,t<x~T.
'>3'
>2 .
, 1-rf or, 1-
(4.4.2)
The distribution functions corresponding to the above densities are
gi ven next below:
Case 1
t,T
H.
t~T -to
_ -1 r
(x)-e.
11
2:
k =1
[e
1
-\ t
k
-Ax
{e
r
IT (\ -\ )
1=1
l*k
(\-\k)+\k-\e
.
\
1-r
/\k (\ -\k)
-(A-A )x
k
e}
1 k
i-r-2 _ _---'1~
j~O (\-\k)j+1 \i-r-j-1
fe-At (\t)i-r-j-1 -1 _
L [(i-r-j-l)! ]
\t'
. 2
+1
-AX
e-/\
l-r~- :..J\(t-x)]a
- (a+l) (\t)a e
a=O
(a+ 1)!
}
J
,
118
x~t,
[
1
r
1
----=--~.-_
A (A-A )1 r
k
k
IT (ArAk)
{
-Akt -At
e
[e
(A-A ) +A ] k
k
1=1
1*k
-AX
Ae
-At
-Ak(e
-AX
-e
.-At -AX -A (T-x) }
k
) - (A-Ak)(e -e)e
-
,t<x~T-t,
t,T _ -1 r
1
[1
{-Akt
-At
-A(T-t)-Akt
H.1(x)
-e.
2:
.
Ake
-Ake
-Ake
1k
r
l-r
= 1 IT (A -A) Ak (A-A k)
1=1 1 k
l*k
-AX
-(A-A k)X- AT}
i-r-2
{
-AX
k
1
- (A-Ak)e
+ Ae
- 2:
.+ 1 l._ _. -1 - e +
j=O (A-Ak)J
A r J
-At (At)i-r-j-1
(At)i-r-j-1 -AT
-AT i-r-j-~A(T_X)]a+1} ]
(i-r-j-1)! +(1- (i-r-j-1)!)e
+ e
a~O
(a+1)!
e
,T-t<x~T.
for
117
i~3, i-r~2.
(4.4.3)
Case 2 t>T-t.
i-r-2 e
2:
-At (\t)i-r-j-l
-At i-r-j-2 1\(t-x)]a+1
-AX (\t)a
[-l]-e
""
{
-e
-}
(i-r-j-l)!
~O
(a+l)!
a!
j=O
J
(\_\ )j+1 \i-r-j-1
k
,x~T -t,
t,T _ -1 r
H.1 (x)-e.1 T 2:
_
K-l
[\k e
1
-\ t
k
-\(T-t)
[1-e
r
IT (\ -\ )
1=1
l*k
-(\-\)x -\ T -\ t
k
k
k
]+\e
[e
-e
]
.
\k(\-\k)
l-r
e·
1 k
,T-t<x~t,
H~,T(x)=
e~1
~
1
1 ~
k=l
r
1
IT (\ -\ )
1=1
l*k
1 k
118
-AX
e
-AT i-r-j-2
a+l }
~~
+e
~O
(a+l)!
for,
i~3, i-r~2.
J
,
t<x~T.
(4.4.4)
The first two moments of case 1 & case 2 distributions are identical.
They are:
{ t i - r -j - 1
At i - r - j -At -AT
j~O (A-A )j+l [(i-r-j-l)! + (i-r-j)! He -e ]k
-AT
i -r- j - 1 }
(4.4.5)
A(T-t)e
(i-r-]-l)!
' i~3, i-r~2.
i-r-2
1
J
119
-At
+e
[(A-Ak)3{1+(1+At)2} - A3 {1+[1+(\-A )t]2}]
k
-AT
+e
[\3{1+[l+(A-A )T]2} - (A-A )3{1+(1+AT)2}]
k
k
-A(T-t)-A t
+e
- Ae
k [(A-A )3{1+[1+A(T-t)]2} - \3{1+[1+(A-A )(T-t)]2}]
k
k
-AT
}
t i-r - j - 1
At i - r - j
(T-t) { [2+\(T-t)] (i-r-j-l)! + 2 (i-r-j)!}
.>3 , 1-r-.
. >2
, 1-
4.5 Prospecti'le Interior Intervals
The general expression of the density function of Xi (t, T) is
h~' T (x) =
1
Pr [ t~S. 1 ~T -x] f. (x)
1-
pr [ Si -1 ~t
1
n Si ~T ]
120
,
e·
J
x~T-t.
(4 .4. 6)
Applying the current assumptions to the terms of the above expression
yields
T-x
pr(t~Si_1~T-x)
f
=
( where gi-1 (0) is as shown in 4.2)
gi-1 (x) dx
t
.
=\1-r-
1 r
1
n \1 2:r -r~-1=1 k=1 n (\ _\ )
1=1
1*k
1 k
\ (\_\ )i-r-1
k
.
.
-At
-\(T-x)
1-~y2 e (\t)a_ e
[\(T-x)t::..
~
a=O
i-r-2
2:
a!
-
k
J.
j=O
And
Pr (S.1- 1 ~t
n S.1 ~T)
T-t
= pr
f
o
(t~ S.1- 1 ~T -x)
{
-\k t
-\(T-t)
.
(\-\ )e
(1-e
) \ (\ _\ ) 1-r
k
. 1 r
r
1
[1
= \1-r\1 2: -r--=---
n
=
1 1
=
k 1
n
(\
-\)
1=1 1 k
f.1(x) dx,
/\k /\ /\k
l*k
121
i -r- 2
i-r-j-2 (a+1)(At)a(e
2:
a=O
-At -AT
-AT
-e
) _ Aa + 1 (T a + 1 _ t a + 1 )e
-->-=......;;.."'"'-'--~->...;;...--:(:--a....;;;.+-:-1":""':) !~---'-'---->.,;,.---'------'-"--
]
j ~ ----.;;;;-.=.----(A---A-)-j-+-:-1--i--r--J-'- - : - 1 - - - - - - - A
k
.
Accordingly, the density function of Xi (t, T) is
-AX
-Ak t -Ak(T-x)
A
h~,T(x)
e~1 2:----:.....:...::e:...--__ [e
-e
11 k 1 r
[l.k(1.-l.k)i-r-1
r
=
= IT (A -A)
1=1
l*k
2:
j=O
L..
a=O
/\ /\
-A (T-x)
(At)a- e
[A(T-x)~
a!
(A-A )j+1 Ai - r - j - 1
k
i-~j-2 e
i-r-2
/\
1 k
-At
,i~3, i-r~2, x~T-t,
where
e.1
is as follows:
e.
1
= [ Ai - r - 1
IT
Alr1 pr (
1= 1
S'_l~t n S. ~T)
1
1
The formula of the probabilty distribution function is given next:
122
e-
J
(4.5.1)
.
i-r-2
. 2
l-r2: J -
2:
(a+ 1) (\I\t ) a e
a=O
-At
(1 -e
-AX
1 -AT
)__ 1\
\ a+ e
1
1
a + _(T-x) a+ ]
[T
..•
(a+ 1)!
]
(A-A )j+1 Ai - r - j - 1
k
j=O
'
i~3, i-r~2, x~T-t.
(4.5.2)
The first two moments of this distribution are evaluated as follows:
E[X.1(t,T)]=
e.1-1 2:r
k= 1
1
r
II (A -A)
1= 1 1 k
l*k
-A (T-t)
e
[1
AA k (A-A k)
{2 -Akt {1-
. + 1 (A-A k) e
l-r
.
2 -AkT
-(A-A k) (T-t)
}
[l+A(T-t)]} - A e
[l-e
{1+(A-A )(T-t)}] k
123
-1 r
1
e.
2:
-~1 r
11
2
E[X. (t,T)] =
k
= IT (A -A)
1=1
l*k
2
}
(A-A k) (T-t)J}]
\
._ +2 (A-A ) e
k
\ (\ _\ ) 1 r
2
/\
[2 -
/\k /\ /\k
1 k
i-r-2
-
{ 3 -Akt
[1
2:
1
"+1
j=O (A-Ak)J
(a+2) (a+3) (At)a[2e
i-r- "+1
A
J
{i-r- j -2 1
2:
a=O (a+3)!
[(a+1)
-At -AT
-AT
-e
{1+[1+A(T-t)]2}] _A a + 3 e
[2T a + 3 _
(4.5.4)
4.6 Open Birth Intervals
The general expression of the density function of the open interval
for the i th birth, U. (t), is as follows:
1
t
o. (u)
1
=
gi(t-u) F(u)
pr(S.1- 1~tns.>t)
1
,U~t.
With reference to the current assumptions, the denominator of the
above expression is found to be:
124
Pr(S,1- 1~t
n
At
Akt
r
r __--=--__ {e_ ei
r
1
S.)t)= A - - IT A ~
1
1
1=1 lk~1
r
IT (A -A ) (A -A k) i-r
1=1 1 k
l*k
i-r-2
2:
j=O
It follows that the density function is
t _
0, (u) 1
r
-1 ~
e, ~
1 k=1
-Ak(t-U)
i-r-2
i-r- '-2 -A(t-U)
{e
_ ~ (t-u)
J e
}
. 1 ~
'1
'
(A -A) (A-A )1-r- j=O (i-r-j-2)! (A-Ak)J+
k
1=1 1 k
l*k
i~3, i-r~2, u~t,
(4.6.1)
e
-AU
IT
where
e.1 = [A i-r-1 1=ITr 1 AI] -1
pr (S'_1~t
1
n S.)t).
I
And the probability distribution function of U, (t) is
I
O~(t)= e~1 ~
I
1
1 k 1 r
IT (A -A )
1=1 1 k
l*k
=
{e
-A t
-(A-A)U
k [1-e.
k I_
I-r
(A-A k)
L}
. 2 -At.
. 1
.. 1
I-r[ 1-r-J- - (t-u) I-r-J- .
~
e.t
~
'+1'
j=O
(i-r-j-l)! (A-Ak)J
(4.6.2)
125
The first two moments of the open interval are given below:
_ -1 r
E[U i (t)]-e i
k-1
:?
1
{e
-A t
k
-At
- e
r
IT (A
1=1
l*k
-A )
1 k
[1+t(A-Ak)]
i-r+1
(A-A k)
i-r-2
2:
e
-At.
.
t 1-r- J
}
j=O
·>3 , 1-r. >2 .
,1-
-1 r
E[U 1" (t) ] = e1 k2:
1
2
0
r
=1 IT (A
1=1
l*k
(4.6.3)
{2e-Akt-e -At [1+{1+t(A-A k)} 2
1·
'\ - '\ ) (1\
I\k
-A )
1 k
r+2
e,i~3,
126
i -r~2.
(4.6.4)
CHAPTER V
PARAMETER ESTIMATION AND FERTILITY
INDICES
5. 1 Introdoction
This chapter deals with procedures for estimating the parameters
of birth interval models developed in previous chapters. It also illustrates how the parameter estimates can be uti 1ized in quantifying various aspects of the reproductive process.
Due to its favorable asymptotic properties, the Maximum Likelihood
Estimation method (MLE) will be used. More important, MLE makes
possible deriving a single set of estimates of parity-specific fertility
hazards generalizable to an entire cohort. Using other methods, say
moment estimation, may result in different estimates of the same
parameter, depending on the order of the birth interval employed in the
estimation process. For example, in All Closed Intervals, the moment
estimation based on parity dependent models of X i - 1 (t) and Xi (t) produces estimates for {~1'~Z'~3,... ,Ai-l}, {A 1 ,A Z,A 3 ,"·,A i } respectively.
It is not guaranteed that the common points in the corresponding two
sets of parameter estimates are of the same magnitude, nor is it clear
how to merge them into a single set of estimates. Furthermore, in
some instances MLE would enable us to combine information on closed
and open birth intervals for the sake of recovering the hazard estimates
from the bias resulting from right truncation. In other words, estimating Ai from the closed birth intervals of those members with achieved
parity i jointly with the open interval for the birth order i results in
an estimate which better reflects the underlying propensity of shifting
to parity i from parity i-l for all cohort members. Also, MLE alleviates the effect of left truncation impacting Last Closed, Open, Straddling, and Prospective Interior Birth Intervals because the experiences
of women with different parity are Simultaneously incorporated in estimating the parameters. Yet, it was still not possible to purge the
true hazards of right truncation in two ascertainment methods, namely
for the Straddling and Prospective interior intervals.
In addition to the inferential value of the hazard estimates which are
recovered from right and left truncation, they will be further utilized
to produce other important length-unbiased fertility quantities such as,
the survival function of the
/h birth interval,
mean interval length,
and parity progreSSion ratios.
This chapter is comprised of seven sections. The five sections follOWing thiS introduction present the likelihood and log likelihood functions for the five ascertainment methods dealt with in this study. All
three models developed in previous chapters will be covered. The first
and second partial derivatives of the log likelihood function with respect to the parameters estimated will not be shown because they are
very lengthy and, in most cases, do not result in expressions compact
enough to be of practical use in estimating the parameters.
128
e·
A derivative-free MLE method will be applied instead. Section 7 is
devoted to describe the various fertility quantities that can be built
using the hazard estimate .
5.2 All Closed Intervals
The data on all closed intervals are ascertained from the truncated
fertility history of all cohort members, where the truncation point t,
t~A,
is common among them. If we consider the random vector,
X (t)
.... a
= ( Xl a (t)
X
2a
(t) '"
X
ra
(t))
as representing the successive birth intervals up to the r
th
birth for a
woman a who is of at least parity r at time t, then under the independence assumption of Xi' X 2 ,"', X r , the joint density function of
~a (t)
can be shown to be,
f 1 (xl) f 2 (x 2 )'" fr(x r )
ht(x) r ....
, where
Gr(t)
r-l
O<x 1 <t, O<x 2 <t-x 1 ,", O<x <t- 2: x ..
r
j=l J
Obviously, for a given woman a, Xl (t), X 2 (t),", Xr (t) are not
independent random variables. Since
r t
IT
h. (x)
i=l
1
*htr (x).
....
....
129
Le.,
IT
i=l
G i - 1 (t-x) fi(x)
* f 1 (xl)
Gr(t)
f 2 (x2} ... fr(xyJ
Gr(t}
As a result, the product of the univariate densities of X. (t) over all
1
women of at least parity i and over all parities cannot be taken as a
valid likelihood function of the data.
Furthermore, it is known that in single failure problems the MLE
derived from truncated probability distributions overestimates the underlying parameter [Johnson and Johnson, 1980]. Hence, to recover the
estimate it is important to combine the information on closed and open
birth intervals in the estimation process. By doing so, we compensate
for the effects of right truncation that characterizes this type of birth
interval. This is because the MLE of A. i derived in this way is not
solely based on the experience of women who achieved parity i in (O,t).
The open interval since the occurrence of the (i -1) th birth is also utilized. The rational behind this proposed approach rests on its analogy
with the conventionial MLE of the hazard function in single failure
problems. For instance, if , in the later situation, the underlying distribution is untruncated exponential, we know that the MLE of the hazard
is a rate in terms of occurrence/exposure, where exposure consists of
two components, one for the unfailed (censored) observations and the
second for the failed ones. Thus, the derived MLE of the hazard utilizes all available information, and hence is generalizable to all subjects
included in the study.
In order to achieve thiS target for a multiple failure situation, such
as the one understudy using truncated probability models, we need to
130
e·
modify the condition of truncation in such a way as to include the
open interval along with all previous closed intervals in the multi variate probability model. The appropriate condition is the event that
(Sr:S:;t
n
Sr+ 1 >t ), which is exactly eqUivalent to the event that
a cohort member is of exactly parity r at time t, instead of (S :S:t).
r
To see that, define the random vector X (t), of dimension r+ 1,
.... a
slightly different from before, as including all closed intervals for a
woman a of exactly parity r at t, and the open interval for the birth
order r+1, Le.,
X
(t) =( X 1 (t) X (t) . . . X (t) U( + 1) (t)), where
2a
.... a
a
ra
r
,a
r
obviously, U + 1 (t) = t - 2: X. (t).
r
i=1 1
The multivariate density function of X (t) conditional on
a
(5 ra :S:;t
n
'"'"
5 r + 1 ,a >t)
can be shown to be:
r
f (x ) f (x ) . . . f (x ) [ 1 - F +1 (t-2: x. )]
ht (x ) = 1 1a 2 2a
r ra
r, ,a i = 1 1 a
r .... a
--------------------=---=--Pra (t)
r-1
where Q<x 1 <t, Q<x 2 <t-x l' ..
, Q<x <t r
2:
i=1
x..
1
Correspondingly, the likelihood function of the data pertaining to a
specific age-marriage cohort is as follows:
131
m
IT
m
L(
x , A) =
nr
h~(xa'
IT IT
A)
,
r=1 a=l
where r~m , and nr is the number of women of exactly parity r at t.
This is the likelihood function that will be followed in estimating the
r=1
parameters of All Closed Intervals for parity and quasi parity-dependent models given in Chapters III and IV. Unfortunately, we cannot
apply it in the constant hazard situation because, as shown below, the
likelihood expression turns out to be free of the parameter of interest.
Sections 5.2.1,5.2.2,5.2.3 include, respectively, MLE procedures
for the constant hazard, parity-dependent, and quaSi parity-dependent
models.
5.2.1 Constant Hazard Model
The joint density function of the random vector X (t) of a woman
_a
a of exactly parity r at time tis,
r
r
ITf.(x.)
h t (x )
r a
-
[1 -F +l(t
i=1 1 1
r
-2: x.)]
i= 1 1
=.....;;...--=----------=~:...---
Pr(t)
-AS -A(t-S)
r e
A
e
= ---,....-----At
e
(At)r
r!
=
r!
r
t
-~..::---
r-l
r
. , xr <t - . ~ x.1 , s=.2:1x..
1=
1= 1
Clearly, the probability model is independent of A.
132
To overcome this problem, we resort to the univariate density of
X. (t) , i=1,2, . . . , m, where m is the highest birth order under
1
study. We expect that the MLE of A derived from birth intervals of
different order will not be exactly the same, so some fluctuation is
anticipated. Although the resulting estimates should be in the same
vicinity if the model fits the data adequately, we propose to take a
weighted average of all resulting estimates as an overall MLE of
A~
with the weights being the proportions of births of order i, i=l ~2~ ... ~m.
The density function of X. (t), for a woman a of at least parity i at t
1
is
i-2
-AX
Ae
h.t (x ) =
1
a
a[l_
2:
e
-A(t-X)
.
a [A(t-X ) ]J/j!J
,'= 1
i-l -At
a
_
~_e..-.::...
1 -. "5
J~
e
.
(At)J/j!
x ~t.
The likelihood function of the data is
n.1
n. -A 2: x n i
i-2 -A(t-X )
n
Ale a=l a IT [1 - 2: e
a [A(t-Xa)]j /j!]
i t
a=l
j=O
L i = 1 hi (x a) = -------:i:----;-l--'A-:-t- - .---'-n-.- - - - - - - - [1 - 2: e (At) J/ j! J 1
j=O
(5.2.1.1)
JJ
and the log likelihood function, l=Log L, is shown as follows:
133
ni
ni
i-2 -\(t-x)
l.=nlog\-\~x +~ log [1-~ e
a [\(t-x )]j/j!J1
a= 1 a a= 1
j =0
a
i-1
n. log [1 - ~ e
1
j=O
-\t
(\t)j/j!].
(5.2.1.2)
5.2.2 Parity-Dependent Model
According to this model, the joint density function of the random
vector Xa(t)
= (X (t) X a (t) . . . X ra (t) U r +1 ,a (t)) is
_
1a
2
r
r
11\.
ht (x ) = ~i=_l~l
r_a
-\
(t-s)
- ~ \. x.
ra
e i=l 1 la e r+1
_
r
r+1
11 \ 2:
1=1 lk=l
r
- ~ \. x.
i=l
1
la
- \r+1 (t-s ra )
e
e
= ----'---------"-----,
,
r+l
~
k=l
134
e-
The likelihood function is
L =
n
m
r
t
hr (x a )
IT IT
r=l a=l
.... ""
where n is the number of women of exactly parity r at time t, and m
r
is the highest parity attained by any cohort member. Then
m
L
=
nr
r
m
-2: 2: 2:
r=l a=l i=l
-2: 2:
A.x.
la
1
...;:e::...-
r=l a=l
----:e::..-
And the log likelihood function, 1
m
1 =-
nr
r
m
2: 2: 2:
r=l a=l i=l
A. x.
m
2:
1
la
nr
-
= log L,
A +l(t-s
r
ra
)
_
(5.2.2.1)
is
nr
2: 2:
r=l a=l
Ar+ 1 (t - s
ra
) -
r+l
n
r=l r
log
2:
(5.2.2.2)
k=l
135
5.2.3 Quasi Parity-Dependent Model
According to this model, the random vector of successive closed
intervals and the open interval since the most recent birth for a woman
a of exactly parity r at time t ,
X
+ 1, a (t)··· xra (t) U r
+,
1 a (t)) ,
_ a (t) = ( x1a (t) x2 a (t) . . . xva (t) xv
is such that the first subset of intervals up to Xv follow the specification of the parity-dependent model, while the remaining intervals have a
constant fertility hazard different from all preceding parity-dependent
hazards.
Accordingly, the multivariate p.d.f of X (t) of dimension r+ 1, for
a
a woman a of exactly parity r (~v) is
-
v
-'5A.X.
i~ 1 la
e
v
2:
k=1
-A(t-s
*
ra
)
e
1
v
IT (A -A
1=1
l*k
)
1 k
. ,x
ra
<t -
r-1'
* v
2:
x. ,s = 2: x. ,r~v.
i=1 la
ra i=1 la
And the p.d.f. of Xa (t) for r<v is as follows:
-
136
e·
r-1
r
where x 1a <t , x 2a <t-x 1a , . .. ' r
x a < t -.~
" xia' s ra ="
x
.~ l' a
r<).I
1= 1
1= 1
'
.
It follows that the likelihood function of the data is
n
).1-1
L=
r
n
m
*
r
[IT IT h~ (x a) ] [IT IT
h t (xa) ],
r=).I a=1 r "-
r=1 a=1 "- "-
).1-1 nr
-2 2 A + 1 (t-s )
er=1 a=1 r
ra
1 nr r
-2 2 2 A. x.
= [ er=1 a=1 ;=1 1 1a
).I-
).1- 1
r+ 1
IT [2
r=1
k=l
-Ak t
n
J
r+1 e
IT
(A -A )
1=1
1 k
J*
r
l*k
m nr
).I
- 2
2
2 A. x.
r=).I a=1 i=l 1 1a
[e
m nr
*
-A2 2 (t- s )
r=).I a=1
ra
~At
J.
-A t
-At.
k
r-).I-1
e
-e
e
t r-).I-J
J }nr
[
I] ~ ).I
r-).I+ 1- ~
j +1
r-).I k-1 IT (A -A) (A-A k)
J-O (r-).I-j)! (A-A k)
1=1 1 k
l*k
(5.2.3.1)
m {
).I
1
The log likelihood function of the data, 1 = log L, is shown as follows:
137
v-l
I =-
nr
r
J/-l
2: 2: 2:
r=l a=1 i=1
m
nr
A.X. -2:
1 la r =1
J/
- 2: 2: 2:
m
r=J/ a=l i=l
J/
m
- ~ nr log ~
r-J/
k-1
nr
*
A.X, - A 2: 2: (t- s )
1 la
r=J/ a=l
ra
J
[-Akt -At r-J/-1 -At r-J/-j'
e
-e
e
t
J/
r-J/+l- ~ ------j'+-1 .
II (A -A) (A-Ak)
j-O (r-J/-j)! (A-A k)
1=1 1 k
l*k
(5.2.3.2)
1
It should be noted that the ML estimate of A1 for parity-dependent and
quasi parity-dependent models is based exclusively on the closed interval for the first live birth. In other words, cohort members who have
not yet had the first birth contribute no information towards estimating
A1. This limitation will not have a major effect if the observation
period (0, t) is long enough for almost every woman to have a first
birth. Analogously, the estimate of Am + l' the fertility hazard for the
birth that immediately follows the highest observed birth order attained
by some cohort member, is solely based on the open interval since the
occurrence of birth order m. For this reason, the reliability of the
resulting estimate of Am + 1 may be doubtful, particularly if it is likely
that women will eventually shift to parity m+ 1 with higher probability.
138
e·
5.3 Last Closed Intervals
In this ascertainment method, every woman under study is only represented by one live-birth interval, the space between the two most
recent births. As such, the MLE procedures of the univariate distributions are applicable for the current situation.
Let h~ (x
ia
) be the p.d.f. of the last closed interval , Xi (t), for a
woman a of exactly parity i at time t
(~A).
Then the likelihood funct-
ion of the data is:
m
n.
1
t
,i=1,2,. . ',m , a=1.2, .. ·,n.,
L= IT IT hi (\a)
1
i=1 a=1
where m is the highest birth order considered in the analysis and n is
i
the total number of cohort members of exactly parity i at t.
The likelihood functions are given below for parity- and quasi paritydependent models. The constant hazard model is skipped because its
truncated density function is no longer dependent on the parameter \.
5.3.1 Parity-Dependent Model
The likelihood function is
n.
1
t
IT
h.(x. )
a= 1 1 la
139
-A.X.
ella
m
i -1
~1
ni
i-l
lU1(ArAk) (A + 1-Ak)
i
l*k
IT IT
i=2 a=l
i+1
~
k=1
*
m
-~
.L.
e 1=2
ni
A.
~
x.
1a m
a=1
IT
i=2
1 L.
(5.3.1.1)
The log likelihood function, 1. =log L, is shown next as:
140
m
ni
i-1
+ 2: 2: log 2:
i=2 a=1
k=1
-Ak(t-x ia )
-A i + 1 (t-x ia )
e
----=-~_-:"1-----..:::...----
191 (Af Ak)(A i + 1-A k)
l*k
m
i+l
-Akt
- 2: n. log 2: -.--=+e=:-l-i=2 1
k= 1
(A _,)... )
1k
1= 1
l*k
h
(5.3.1.2)
5.3.2 Quasi Parity-Dependent Model
The assumption employed here is that all birth intervals for lower
order live births up to and including birth order
l/
follow the specifica-
tion of the parity-dependent model while birth intervals of order higher
than
l/
have a constant hazard model. As such, the last closed intervals
of the cohort members of parity
l/
or less at time t have the following
density function:
e
-A. x
1
i=2,3,4, ... ,l/,
141
X~t.
For the first interval this is
h\ (x)
,
x~t.
The p.d.f. of the last closed interval for birth order i, i> v, is
v
2
k=1
v<
i ~
m,
x~t.
It follows that the likelihood function of the data on last closed
intervals is
n
n.
n.
1 t
v
1
t
m
1
L= IT h 1 (x 1a ) IT IT h.(x.) IT
IT
a=1
i=2 a=1 1 1a i=v+1 a=1
*t
hi (x ia )·
Substituting for the expressions of the densites in L yields,
142
e·
n.
v
1
2 x.
-2 cr=1
la
v
e i=2
i-1
n.
2
1
IT
e
IT
-A (t-x. ) -Ai + 1 (t-x )
k
la -e
ia
i-1
EL (A(A k) (A i + 1 -A k)
k=1
a=1
i=2
1*k
-Akt
i+ 1
v
IT
[
i=2
2
IT
1= 1
l*k
m
-2
e
]
i+ 1
k=1
1*
n.
e
.J
1
(A(A k )
1
n.
1
2
A
X
ia
i=v+ 1 a=1
n
m
v
i
IT
IT 2
i=v+ 1 a= 1 k= 1
i - v- 2
_2
j=O
{e
1
-A, (t-x.)
-A (t-x. ) I
K
l~ _ e
la I
V
IT
(A -A )
1=1 1 k
1*k
-A(t-X. )
1a
(t -x. )
(A-A k )
I-V
i-v-J'-l
e
la}
(i-v-j-1)! (A-A )j+l
k
)
m [V
.rr '5
1=11+ 1
1(=1
1
rr (A -A)
11
1=1
1*k
1 k
-At
{e-Ak_t e-v+
1
(A-A k)
i
J
(5.3.2.1)
143
The log likelihood function is
m
- 2:
ni
m
n.
1
J/
1
>.. 2: x.
+ L.. 2: log 2: J/
i=J/+1 a=l la i=J/+ 1 a= 1
k= 1 II (1\ -1\ )
1=1 1 k
l*k
~
-1\ (t-x.)
->"(t-x.)
{
k -lae
la
e
i-J/
(I\-I\k)
.
2 -I\(t-X·la ) (t -x. )i-J/-J'-l
e
la
j=O (i-J/-j-1)!(>"-l\k)j+1
I-J/_2:
(5.3.2.2)
5.4 Prospective Next (Straddling) intervals
It hds been shown that this ascertainment method gives rise to two
different density functions of Xi (t, T), for
i~2,
depending on whether the
length of follow up period (T-t) is less than or greater than t, the marital duration of cohort members at the time of observation ( e.g.,
144
interview date). It was also demonstrated that each density function
has three different components for three disjoint ranges of the random
variable X. (t,T) .
1
Let
h~dT(xir)
be the p.d.f. of the
/h
birth interval ,
the r th range, where r=1,2,3, and d=1 for
i~2,
that lies in
t~T-t, d=2 for t>T-t. The
definition of r is as follows:
For
t~T-t;
r=1
~ x.~t,
1
And for t>T-t; r=1
r=2
~ t<x.~T-t,
1
~ x.~T-t,
1
r=2
r=3
~ T-t<x.~T.
1
~ T-t<x.~t,
1
r=3
~ t<x.~T.
1
Thus, the likelihood function of the data on straddling birth intervals is:
where m is the highest birth order under study and \ra .is the observed
length of the i th birth interval that 1ies in the range r for a woman a.
3
m
The aggregate number of such women is n. ,and n. = 2: n. , n= 2: n..
1r
1 r= 1 1 r
i= 1 1
Sections 5.4.1, 5.4.2, 5.4.3 present the likelihood and log likelihood functions for the constant hazard, parity-dependent and quasi
parity-dependent models respectively.
145
5.4. 1 Constant Hazard Model
t~T -to
Case 1
The likelihood function is given by
n1
L =
II
a=l
t,T
h l (x la )
n
nil
t,T
ni 2 t,T
ni3 t,T
II [II h, (x· ) II h. (x· ) II h, (x'3 )],
i=2 - a=1 1 11 a a=l 1 12 a a=l I 1 a
m
1
-A~l x 1a
An e
L= - - - - - - - - m [-At
i[Il
(e
n'1 l
. 2
1-
a=l
[2:
j=O
II
-AT (At)i-l
-e
) (i-l)l
m
J i=2
A(t-x'l )
.
e
-At,
("\/\t )J
\t
'2-/\ (\t)J ]n '
12
][ 1- >" e
'1/\
j~
J.
1-
J.
n'13
II
-A(T-x.
. 2
1-
e
)
13a
[1 - 2:
a=l
j=O
'I
J.
*
_
a [\/\ (t -x'
)] J - e
l
'1 1 a
1
e
[
II
3 n ir
-A2: 2: x,
r=l a=l Ira
,
[A(T-x'3 )]J
1 a]
J.
(5.4.1.1)
The log likelihood function, 1. =log L, is therefore
m
1.= -i~l ni log [(e
-At
-AT
-e
(At)i-l
n1
) (i-l)! ] +n(1og A) -A ~1 x 1a
m nil
i-2 e
- A >" 2 2 Xira + 2: 2 log [2
i~ r=l a=l
i=2 a=1
j=O
m
3
nir
i-2 em
+1,7 n i 2 log[l-J~O
~
At
(At)J
'1
J.
-A(t-x' )
. -At
'
1l a [A(t-X' )]J- e
(At)J
l
'I
I a
]
J.
-A(T-x' 3 )
.
1 a [A(T-x'
)]J
3
]:->" 2 log[1 ~2
j!
1 a ].
l~ a=l
J=O
(5.4.1.2)
m ni3
146
i-2 e
e·
Case 2
t> T-to
L=
-=-1
m [
IT
i= 1
_
-At
(e
-AT (At)i-1
- e
) -('-1)1
n
3
n.
ir
-A '5 :2 :2 x.ira
i~ r=l a=1
e
m
i
J
1._
-A (t-x. 1 )
,-At
1 a [A(t-x,
)]J -e
(At)j
IT IT [2:
'I da
]
i=2 a=l j=O
J.
m
nil i-2 e
-A(t-x'2 )
m
n i2 i-2 e
1
IT IT [2:
i=2 a= 1 j =0
,-AT
'
a [A(t-x'2 )]J - e
[A(T-x'2)]J
a
1 'I
1 a
]
J.
-A(T -x,
e
)
[A(T-x,
13a
\)J
13a i
(5.4.1.3)
J.'I
.J
The log likelihood function, l=log L, is
n
1
m
3 n,ir
l=n(log A) - A 2: xl - A '5:2 :2 x,
a=l a
i~ i=l a=1 ira
m nil
+i~ ~log
to
-A(t-x'
i-2 e
1
j!
-A(t-x·
i-2 e
~
~
j=O
)
.
-At
.
l a [A(t-x, 1 )]J - e
(At)J
1
1
a
)
,-AT
'
2 a [A(t-x'2 )]J - e
[A(T-x'2 )]J
1 a
1 a
'(
J.
147
m ni3
i-2 e
-A (T-x.
+2: 2: log [1- 2:
i=2a=1
j=O
m
- 2:
i=l
.
J.
-At
n. log [(e
)
13a [A(T-x.
)]J
'1
13a
]
-AT
-e
(At)i-l
)
J.
(i-1)!
1
(5.4.1.4)
5.4.2 Parity-Dependent Model
t~T-t
Case 1
3
m
-2:
e
i=2
A.
1
n.Ir
2: 2:
r=1 a=1
x.
Ira
em
II
i=2
iI
1
a=1
-A k (t-x. 1)
e -1
[~
-k=l
II
i
1= 1
1*k
-Akt
1 a - e
-Akt
n. 2
1. 1
}: .....;;;~_1"""-.,; ;,.,e- 1
1=2lk= 1 II (A -A ) A
1= 1 1k k
1*k
(A -A ) A
1k
k
n' . 1
-Ak(T-x·
)
. 1 -Akt -A.t
3
3 ~- 1
m
aj
m
cII II 2:. 1
2:_ e -.e
._
1
.
_
1=2 a=1
-1
1
~
1
IT (A1k
-A ) A
1= 1
1*k
k
lr1
I
-1
J
J.IT r
IT
1= 1
1*k
-A.T
1 ~e
1
-(Ak-A.)t
[e
1
-no
1]J
.
1
(A -A )
1k
(5.4.2.1)
148
The log likelihood function, 1
= log L,
is
n1
-A T
[ -A t
n.logA. -A 2 Xl -n log
e 1 -e 1
1 a= 1 a
1
i= 1 1
1
m
1=2
J
· 1 -A k (t-x·1 1a ) -/\k t
A.
x.
+ '5
log
e. -1
- e
i=2 1 r=1a=1 lra i~ a=1
k=1 lIT (/\ -A ) /\
1=1 1 k k
m
-2
3 n.lr
m
2 2
n·
11
1-
2
2
l*k
t
1'-1 1 -e -/\k
n i2 log 2 i-1
k=l IT (A -/\)
1k
1= 1
1*k
1 -e -A k (T-x:.3a )
3 1'-1
m n'1
+.2 2
log 2 -i-=-l;--=-----1=2 a=l k=1 IT (A -/\) /\
1= 1 1k
k
1*k
. 1 -Akt
-A. t
-A. T -(A -/\·) t
k 1
11
1e
[
_'"
1og ",e
-e
-e
-1
~ n.
~.
i=2 1
k=1
-A )
m
L
IT (/\
1=1
.
1 k
l*k
(5.4.2.2)
Case 2 t> T-to
3
m
IT IT
i=2 a=l
lr
- 2 /\. 2 2
m n.
L= IT /\.1
i=l 1
m nil
n.
e
i=2
. 1
-A k (t-x. 1)
-Akt
e
1a_ e
i-1
=1 IT (A -A ) A
1=1 1 k k
l*k
-
~
149
x.
1 r=1 a=1 1ra
J*
, 1
1-
[
m
n'13
IT IT
i=2 a=1
m
IT
i=2
5"
K=1
e
J*
-'\k(t-x'2) -'\k(T-x'2 )
. 1 a -e
1 a
1-1
IT (,\ -,\) ,\
1=1
l*k
1 k
k
i-1 -:1:..,...'----=-e-_,\_k_(T_-x_i_3 a_)_J
[ 2.: i-1
J
k=1 IT (,\ -,\ ) ,\
1=1 1 k k
1*k
[ i~
k=!
-'\kt
e
-'\.t
;e
II
1=1
l*k
1
*
-'\.T -('\k-,\.)t
1 e
[
1 -1
-e
LJ-no
1
('\r'\k)
(5.4.2.3)
The log likelihood function, l=log L, is
t
n.!,'
. 1 -'\k(t-x.!)
1
11 a
e -'\k
+2.: 2.:1og2.:ei_1
i=2 a=!
k=! IT (,\ -,\ ) ,\
1=1 1 k k
1*k
m
.! e -'\k(t-x·12 a ) -'\k(T-x·1 2 a )
-e
log 2.: ~-;-i-'!----=~--k=1 IT (,\ -,\ ) ,\
1= 1 1k k
1*k
1-
150
e-
n i3
-A k (T-\3a)
+2 2 log 2 --'7-i _-;--1..;;..e----i=2 a= 1
k= 1
(A -A ) A
1= 1 1k k
l*k
m
i-1
1
n
m
-2
i-l
n. log
i=2
1
2:
k=l
-Akt
e
-A.t
- e
1
- e
-A.T -(Ak-A.)t
1 r
1
he
- 1L
i
n (AI-A k)
1=1
l*k
(5.4.2.4)
5.4.3 Quasi Parity-Dependent Model
The underlying key assumption of this model is that X. has an expon1
ential distribution with parameter A for all i=l.I+1,l.I+2,"·,m, such that
*'
A*A 1 I\k for all l*k=1 "
2 ...
> 2"
i=3"
4 ... m .
,l.I and i-l.I Hence, the straddling birth intervals of order l.I or less follow a
parity-dependent model, while the intervals of order l.I+ 1 ,l.I+2,···,m
follow a quasi parity-dependent model.
Accordingly, the li:<elihood function of the data is
n.
lr
n
a=1
t T
h ' (x. )
1d
lra
J
*tT
h· d' (x. )
1
lra
where n. =
1
3
2:
n. ,i~J.I
r=l lr
3 n.* ,J.I( i ~ m.
and n.* = 2:
1 r= 1 lr
151
Substituting for h~d,T(x. ), h~dt,T(x. ) in the above equation for L ,
1
1ra
1
1ra
we get the expressions for the 1ikelihood and log likelihood functions,
l=log L, as follows:
t~T -to
Case 1
n
L=
a=1
1- 2
e
i-1
1- e
IT
- 'k=-1
1=1
l*k
nir
3
v
IT
i=2
-Akt -A. t -A. T -(Ak-A.)t
-no
1
1 [
1
1
'5~ e
-~
-e
e
-1
*1
J.Il
IT
v
J
. 1
v [1-
1
-2
A. 2 2
i=2 lr=1 a=1
x.
1ra
IT IT
i=2 a=1
-Akt
(A -A )
1=1 1 k
l*k
-At
-AT
-(A -A)t
[V e -e {k
e
.
. IT
v
1=V+ 1 = 1 IT (A -A)
(A-A )
m
t
k
1 k
*
-A 2: 2: 2:
m
2:m n.*
i=v+ 1
(A)
-1
l-V
1=1
l*k
3
n ir
i=v+ 1 r= 1 a= 1
1
x.lra
e
*
152
i -1
v n.1 1
'5 i-1
- k= 1 IT
[
(A -A )
1 k
-t=1
[
e
-A k (t-x i 1a)
i -1
-\kt
-e
IT (\ -\ ) A
1= 1 1k
l*k
k
J*
*
-A k (t-x i 1a)
-Akt
_--=1:....--_ { e
- e
2: v
i-v-1
k = 1 IT (A -A )
("A-Ak
" )
"
Ak
nil
m
IT
IT
i=v+l a=l
JI
[
1=1
1*k
1 k
i-~V'-2 e
-A (t-x.
lia [A(t-X.
l1a
)].8_ e
-At
(At).8
i-v-2
-
m
.D
I-V+
j~
1
[V
~_
1 - K-1
{1
IT (A -A)
JI
1=1
l*k
1 k
-
-A r t
-e
(A-A k)
K
i - v-1
i-~-2
i-v-2
1-
-L
2.8=0
Ak
-At
e
(At).8
.8!
j=O
m
IT
*
n i3
IT
J/
[
L
i=v+1 a=l -k=l
1
J/
IT (A1k
-A)
{l-e-Ai -)..1-1k (T-x'3l a )
("A-Ak
" )
"
Ak
1= 1
l*k
(5.4.3.1)
153
The log likelihood function is
- 2:v n. log
i=2 1
[1-.2:1
1 J
-Akt -A. t -A. T -(Ak-A.)t
e
-e. -e
e
: 1L
IT
k= 1
1 1[
(A -A )
1=1 1 k
.
l*k
v
+.~
. 1
1
[1~ log >-
1-2 a-1
1=1
l*k
m
-2:
i=v+l
*
n.log
1
m
J
1
-A (t-x· 1 ) -Akt
a -e
e k
i-1
-k=-1
IT (A -A ) A
n' 1
~ e
-At
-AT
-e
=1
(A -A )
1=1 1 k
1*k
*
1 k
IT
m
3
k
-(Ak-A) t
( e
i-
-1
(A-Ak) v
n.*
1r
+log A 2: ni -A 2: 2: 2: xira
i=v+ 1
i=v+ 1 r= 1 a= 1
154
.
1-1.1-
-2:
2
.
. 1
t 1-V-J-
.+
j=O (i-v-j-1)! (A-Ak)J 1
}
J
*
nil
+2:
2:
log
i=v+ 1 a= 1
.
. 2
I-V-J-
~
i-v-2
e
-,A. (t-x 1, 1 a)
[,A. (t-x,
I1a
p=O
)]
/3
- e
/31
-,A.t
(,A.t)
/3
}J
-2:-----j=O
(,A.-,A. )j+l
,A.i-v-j-l
k
i -v-2
}J
2:
j=O
-,A.k(T-x·
{
1 -e
I 3a
)
(5.4.3.2)
155
t>
Case 2
n
L=
T -to
-A x
A
1
IT
[
a=1
1 1a
1e
-A t
e
-A T
1 - e
JIT Cv
. 1
1=2
t
.
1
e
J
-Akt -A.t
1 -A.T -(Ak-A.)t 1 -no
_e. -e 1 [e
1 - ] ~
1
IT (A
=1
1=1
1*k
3
v
- 2:
e 1=2
Ai
-A )
1 k
nir
2: 2:
r=l a=l
x ira
[i-l -A k (t-x i1a) -Akt
eo
-e
.
1-1
1=2 a= 1 =1 IT (A -A) A
1= 1 1k
k
1*k
v nil
t
J*
1.n rr
J*
rrn
. 1 -Ak(t-x'2 ) -A k (T-x· 2 )
n'3
-A (T-x'3 )
1- e,
1 a -e
1 a v
1 [.1 - e k
1 a
[ >"
1-1
1-1
i=2 a=l k=1 IT (A -A) A
-1~L a=l
(A -A) A
1k
1= 1
k
1= 1 1k
k
1*k
Uk
n'2
1
v
rr rr
rr
V- 2
-At -AT {e-(Ak-A)_1_i -2:
ti-v-j-1
}
m
[~_V e v -e
L..
i-v
J'+l
I-v+1 -k-1 IT (A -A)
(A-A k)
j=O (i-v-j-l)! (A-A k)
1=1 1 k
1*k
,JI
2: n.*
m
-A
i=v+ 1 1
m
*
nil
IT
IT
i=v+l a=l
2: 2: 2:
i=v+ 1 r= 1 a= 1
e
(A)
3 n.lr
m
V
1
-k=1
IT (A
[
2:
v
1=1
1*k
xira
*
{ e
-A)
1 k
-Ak (t-x. 1)
1
(A - Ak)
156
a -e
i -v-1
-Akt
Ak
J-ni** e·
*
IT
m
n i2
IT
i=J,I+ 1 a= 1
[f
1
IJ
IT (~ _~ )
k= 1
1=1
l*k
i 2_a--;)_----::e:....-_~k_(_T_-x_i_2 _a)
{ . .: :e:. . -~_k_(_t-_x_
(~-~k) i -J,l-l ~k
1 k
)
-~(T-x.
)
12a [~(t-x. )]~- e
12a [~(T-x. )]~
12a
12a
-~(t-x.
'i-J,l-J'-2 e
1=0 -------;::;~.--! --~}]
i -J,l- 2
j~
(~-~
*
n i3
m
k
IT IT [ 2:
i=J,I+ 1 a= 1 -k=l
J,I
IT
{
(~-~)
1k
1= 1
~i-J,l-j-l
)j+l
1
J,I
*
-~
(T-x. )
k
13a
i -J,l-l
(~-~k)
'\k
1 _e
l*k
-~(T-x.
)
i-j./-i-2 e
13a (~(T-x.",
)it?
1-50
i - j./- 2
1 - 2:
Q!
{3=0
I-'
- 2 ---'-----,+--;-1-,---'-i--j./---i- --=-1-'----j=O
(~-'\k)J;\
v
~
j
}
1
I·
-
(5.4.3.3)
The log li kelihood function is
n1
J,I
1 =
2:
n, log
i=l
1
~. - ~1 2: xl
_2:
J,I n, log
. 2
1=
a=l
1
1
[1-.2:1
k-l
-
a
-~kt
e
- n
log (e
l
-~.t
,- e
IT1
1= 1
l*k
-~.T
-~ t
-~ T
1 - e 1 )
1- e 1[e
(~-~
)
1k
157
1- 1 ]J
-(~k-~·)t
11
- "5
i~
3
A.2:
. 1
1
[1+. "5 2: log
2:
n ir
n' 1
11
2:
x.
1r=1 a=1 1ra
1~ a=1
1
-A (t-X· 1 )
-Akt
a -e
e k
i-1
J
II
k=1
(A -A) A
1=1 1 k
k
l*k
. 1
1
[12 log 2
11
n' 2
1=2
a=l
+;2:
1 J
1
-Ak(t-x'2) -A k (T-x'2)
a
a
e _
-e
i 1
k=l
II (A -A) A
1=1 I k
k
l*k
m
~
- L
.
n.*
log
1
-At -AT
[t
1l
e
=1 n
1=11+ 1
- e
11
-(A -A)t
{ e k , - 1
\
\
(A -A) (I\-I\k)
-i -1122
1-11
t
i - v- j - 1
}
j=O (i-v-j-1)! (A-A )j+l
k
1~1 1 k
l*k
m
+(
2
i=v+ 1
*
n.)
2
X
i=v+ 1 r= 1 a= 1
1
n'*
1
m
[V
.~2 ~ log ~
1
1
I-v+1 a-1
2
log A - A
*
lr
2
n.
3
k-1
v
II
1=1
(A -A )
1 k
i-v-J'-2 e
-2
j=O
1=0
1)
{e-Ak (t-x. v-1'
a - e-Akt
1
l*k
i-1I-2
ira
(A-Ak)
i-
Ak
-A(t-X. )
-At
l1a [A(t-X. )]p - e
(At)P
l1a
- - - = - P !~--}J
(A-A )j+1
k
158
A i - v- j - 1
J
-A(t-X.
i-v-J"-2
i-v-2
-2:
1=0
j=O
e
)
12a [A(t-X.
)]f3 - e
-A(T-x.
)
12a [A(T-x.
]f3
12a
------:f3-:--!- - - - - } ]
12a
(A-A, )j+1
A i - v- j - i
K
m
n*
i3
2:
2:
1
log [ ~ -v---::o-- - - {
i=v+1' a=1
-k-1 IT (A -A )
1=1 1 k
l*k
+
1
-A (T-x' )
13 a
-e k
i-v-2
-2:
j=O
(5.4.3.4)
159
5.5 Prospective Interior Intervals
The probability models of this ascertainment method dealt with thus
far are basically concerned with an i
th
birth interval occuring in the
observation period (t , T), where i=2,3,' .. m and m is the highest
birth order under study. In other words, we have been only considering
the univariate probability distribution of a prospective interior birth
interval. However, if a woman bears more than twice within the interval (t, T), the intervals spacing these sequence of live births are not
statistically independent. Hence, it is necessary, for the estimation
purposes, to derive the multivarite distribution of the random vector
x
.....
(t,T) = ( Xi (t, T) Xi + 1 (t,T) Xi +2 (t,T) . . .
where j ~ i
,2 ~ i
~
X.(t,T) )
J
m.
With reference to the condition imposed by this ascertainment
(t, T)
method, the multi variate density of X(t, T) ,h.. (x), can be shown
1,j
.....
as:
pr [
h~t,.T)
(x) =
1,J.....
.
where z=
t~ Si-l <T-z]
f l (Xl)
_
pr[Si_l~tnSj~T]
~ xl' xi~T-t, x i + 1~(T-t)
l=i
rl
-.;;1_="'-1
-Xi' . . . , X .~(T-t) J
It follows that the likelihood function of the data is
160
. 1
~
l=i
xl·
where, n.. is total number of cohort members who gi ve the
IJ
(i_l)th, i th , (i+l)th, . . . , /h live births in the interval (t,T).
Sections 5.5.1, 5.5.2, and 5.5.3 include the likelihood and log
likelihood functions of the constant, parity-dependent, and quasi paritydependent models for this ascertainment method .
5.5. 1 Constant Hazard Model
The likelihood function is
e
-A (t+z 6)
(A t) r - e
-AT
[A (T -z 6) (
1
,
r!
I
'
J
(5.5.1.1)
where
. 2
e.=~ e
1 r=O
-At
(At(-e
-AT
(AT(
r!
.
1 -AT. a l' + 1
t. 1. iA1+re
(_1)a T r-a(T + - _t a- )
1
r~
a=O
1
a! (r-a)! (i-2)! (i+a-l)
The log likelihood function, l=log L, is
[ (j-i+1)
[~-2 e
-A (t+z6)
(At(-e
r!
log A + log lr~o
-i
~
i=2 j=i
161
-AT
n.. log
IJ
e.
1
J.
[A (T-z 6)] r}
(5.5.1.2)
5.5.2 Parity-Dependent Model
The likelihood function in this case is
where
e.=~
!
k=l a=i
1
The log likelihood function, l=log L, is therefore
-i
~
n.. log
i=2 j=i
IJ
e..
(5.5.2.2)
1
5.5.3 Quasi Parity-Dependent Model
To recall, this model implies that the fertility hazards for birth
orders
11
or less are parity-dependent, while those for higher order
births are constant.
162
Here, we are faced with three different types of birth experiences
that may take place in the study period (t, T) . They are described as
follows:
(1) The
/h birth order is less than or equal to V;
(2) The birth order
(3) The
/h
V
lies somewhere between i and j, i.e,
i~v<j;
birth order is greater than v.
Let h~\t,T)(x), h~\t,T) (x) , h~\t,T) (x) be, in order, the multivariate
1J
"1J
"1J
"density functions of the above three sets of birth intervals. Then the
likelihood function of the data is:
(I)
Ul IT re
n ..
L=
j=i 6=1
(1)
(2)
ii
(2)
lJ
_6
h~\t,T)(x
1J
n ..
[
ITV ITm
i=2 j=v+1
2( Tl (x l
h..t,
=1 1J
_6
J*
(3)
where n.. , n.. , n.. are the total cohort members classified according
1J 1J
1J
to the three categories of birth intervals indicated above. It is clear
(1)
that h.. (x) is exactly the same as in the parity-dependent model. Subs1J "tituting for the density functions in L yields the follOWing likelihood:
(2)
v
i~
m
IT
j=v+l
~1 [? {
*
1
163
IT Al
[ ~{
e. 1=1
j
A -
V
*
1
v
[2:
k=1
i-v-a-2
i-~2----l:l=;..---::=Q~
where z~ l~ x16 ' and
ei
-
1.
2: 2..
k=1 a=l
}] }],
Ai - v- a - 1
(5.5.3.1)
1.
i-1
f3!
(A-A )a+1
k
a=Q
(1) _
-At
f3 -A (T-z 6)
f3
e (At) - e
(A (T-z )]
6
(1)
(2)
(3)
ei ,e i ,e i
are defined as follows:
-A t
-A T
-A (T-t)-A t
(\a-\k) e k - Aae k + Ake a
k
i-l
.
IT (\1k
-A)
1= 1
rl (\ -\ )\a Ak (A a -Ak)
1=i 1 a
1*k
1*a
164
i-j./-l
(3)
e. = A
1
~
j./
-A t
j./
1
IT Al ~
1= 1
j./
IT (A
k-1
1=1
k
e
-A)
1 k
-A T
k
i-j./-1
-e
Ak (A-A k)
l*k
i-j./-a-2
i-j./-2
{
1=
e
-At
R
(At)"" - e
-AT
Q
(AT)""
=0
01
2:=O ---"'-_--""-~-~---"'tJ;...;..·
-~--i
A -j./-a-l (A-A )a+1
a
}
k
J/J.::l
- 2: 2.
k=l a=O
j./
'A
a
.
IT (A
{
e
-A T
k
-A (A-'A ) 1-j./-1 (A-Ak)
1= 1 1k
k
l*k
a+l -
}
+
J/
.t:.!
i-J/-2 a
2: 2. 2: 2:
e
k=l a=O y=O 1jJ=0
-AT a
IjJ a-1jJ
i-J/+IjJ-y-l i-J/+IjJ-y-l
: (- 1) T
(T
t
)
IT
1=1
l*k
('A -A ) a!
1 k
165
(i-J/+IjJ-y-l)
J
.
The log likelihood function, 1=10g L, is given next:
.
1
t
(1)
.
J..
1I
(1)
= -.2 ~. ni · log
1=2 J=1 J
(1)
ei
-.
n...
1I
IJ..l
.~. ~
~ A l x16 +
J=1 0= 1 1=1
1=
(2)
1I
n..
.
J..
1I
1J
m
??
~ [~A 1 x 16 + A 2.
x1oj
1=2 J=lI+ 1 0= 1 1= 1
1=1I+ 1
+
(2)
V
i?;
~
m
tv+ 1
f='
i-1 e
log [ ~
1
k=l
-A t
-A (T-z.d
k _e k
0
. 1
1-
IT
J -
(A -A)
1=1
1,
(3)
!
.
~
i=lI+ 1 j=i
n ..
1J
1 k
*k
.
v
.
J..
1
.
2.
1=1I
A
m
(3)
~ n.. log
+1' . IJ
J=1
(3)
e.1 +
k
e·
(3)
m
n ..
~
~ 1 1~1 log Al ~=~+ 1 t i ~ 1 (j-1I)
log A +
(3)
n..
If
~i=lI+ 1 ti ~ 1 log 4<~i
m
1J
i-v-a-2
1'-v-2
~
{3=O
-At
-A(T-z )
e
(At)f3 - e
<5 [A(T-z )J
6
f3!
a~-~~-(\-_-\-)-a-+-1-------1\
}
J.
Ai -v-a-1
I\k
(5.5.3.2)
166
5.6 Open Birth Intervals
It was shown in Chapter II that the density function for open birth
intervals under the assumption of constant hazard does not include the
parameter of interest
>... The likelihood functions is therefore confined
to parity-dependent and quasi parity-dependent models.
5.6.1 Parity-Dependent Model
The likelihood function is
m
ni
n IT
l~~ a=l
L~
where u.
la
t
o. (u. ) ,
1
la
is the observed length of the open interval for the birth
order i experienced by a woman a (= 1,2,· .. ,n ), and (m- 1) is the
i
highest birth order attained by the cohort members.
Substituting for the parity-dependent density function in L gives
m
(5.6.1.1)
L=.n
l~~
167
The log likelihood functio, 1=log L, is
n.
. 1 -Akt-(A.-Ak)U.
.
e-Akt
1
la
m
1
log 2
~-1
- 2 n. log 2 -.--';:~1=2 2
i=2 a=1
k=1 lIT (A -A )
i=2 1
k=1
(A -A )
1k
1k
1= 1
1= 1
l*k
l*k
m
1
1-
IT
(5.6.1.2)
5.6.2 Quasi Parity-Dependent Model
The assumption of this model is that all birth intervals for lower
order Ii ve births up to and including birth order J/ follow the specification of a parity""i:lependent model, while intervals for births of order
higher than J/ have a constant hazard. Therefore, the p.d.f's of the open
birth intervals for birth order J/ or less and of order higher than J/ are
as given in expressions (3.2.5.1) and (4.6.1) above, respectively.
The likelihood function can therefore be shown to be
n.*
1
IT-1
a-
*t (u.) ] ,
o.
1
la
where o~(·) is the p.d.f of U i (t) , i=2,3,4,.··,J/, and
o~t (.) is the p.d.f. of U. (t), i=J/+1,J/+2,.··,m.
1
1
168
i -1
2:
k=l
v
L=n
n.
1
IT
i~ L a=1
*
i
e-Akt
2:--i-~--­
n (A1-Ak)
k= 1
1=1
l*k
-AU.
la
v
~
n*
i
m
IT
k-l
1-V-
Vet e
IT (A
1=1
l*k
IT
J -Ak (t-u.la ).
-A)
1 k
(A-A k)
i - v- 1-
2
.~
J-O
i=v+l a=l
(5.6.2.1)
And the log likelihood function, I=log L, is
v
. 1
n.
1
I=.2: 2:
1=2 a=1
-Akt- (A. -A )u.
.
e- Akt
e l l ka
2: i-1
-.2: ni log 2: - i - - k=l IT (A -A )
1=2
k=1 IT (A -A )
1k
1k
1= 1
1= 1
l*k
l*k
VI}
{ 1-
log
169
+
*
-AU.
n1
{vela
2: 2: log 2: - - i=v+l a=l
k=l IT(A -A)
1=1 1 k
l*k
m
-Ak(t-U. )
-A(t-U. )
.. 2
la i-v-2 e
la (t-u. )l-V-J- }
[
. - 2:
la. ]
(A-A )l-V-l j=O (i-v-j-2)! (A-A )J+l
k
k
e
(5.6.2.2)
5.7 Relevant Fertility Indices
The estimated fertility hazard has an important descriptive value as
it reveals the intensity of fertility and how it varies with birth order.
The asymptotic variance of hazard estimates, obtained via maximum
likelihood estimation, makes statistical inference feaSible. In addition,
recovered hazard estimates, Le., derived from All Closed, Last Closed,
or Open Birth Intervals with incomplete observation periods, can be
further utilized in deriving several useful length-unbiased cohort fertility indices. They are described below:
5.7. 1. Mean and Variance of the ith Birth Interval
We know that when an age-marriage cohort of women is followed
from marriage until menopause, the resulting distribution of the ith
birth interval is the population distribution which we attempt to
170
e.
estimate unbiasedly from truncated fertility information. Lengthunbiased estimates of the moments of this distribution can be obtained
by substituting the recovered hazard estimates in the relevant moment
expressions of all Closed Birth Intervals with complete observations.
5.7.2. Parity Progression Ratios (PPR)
The parity progression ratio for the i th birth order is defined as the
probability of ever shifting from parity i-1 to parity i. Let Si -1 be the
random variable of the waiting time until the occurrence of the (i -1) th
live birth. Thus, the PPR.1 conditional on a certain value of S.1- 1 is
*1
PPR.=pr[X.~A-sIS·l=S)
1
1-
and the PPR.1 over all pOSSible values of S.1- 1 is
A
of
pr [Xi ~ A-si Si_l=s] gi-l (s) ds
PPR.
,i~2.
A
1
of
gi -1 (s) ds
Using the above expression, the recovered hazard estimates, A, can
be utilized to derive length-unbiased estimate of PPR 1.. Expressions of
,.,
PPR. estimate, PPR., and its variance approximation will be shown
1
1
next for Constant, Parity-dependent, and Quasi Parity-dependent models.
The Taylor series expansion are used to approximate the variance of
,.,
PPR., where A. is replaced by A. in the partial derivative expressions.
1
1
1
171
First, Constant Model
"
Using the relevant expressions given in chapter 2, PPR is
"
-AA"
1
j(i-1)!
. 2 -AA"
.
e '/ (A A} J
1j=O
J.
Based on linear approximation,
(>\Al
e
PPR= 1 -
1.
,i~2.
-
(5.7.2.1)
S:
V [ PPR]
~
E [
-i(PPR) ~ _,
aA
a
"
~
= [ - " (PPR) , _,]
aA
a
(~- A) ]2
A-A
2
"
V(A),
A-A
~
where, -,,- (PPR) -
aA
+_ _--:;1~___ [A e-AA
,[
[ 1 _ ~ e '/ (AA) J] 2
j=O
J.
. 2
"
2AA
('\A)2i-3 ]
(i-1)! 0-2)!
'>2 .
1'
Second, Parity-Dependent Model
"
Using the relevant expressions given in chapter3, PPR. is
1
"
i-1
k~
e
-AkA
"
-e
-A1·A
i~2.
172
(5.7.2.2)
where A=(A 1 A2 A3 ·· .A i )·
""
For j=1,2,3"",i-1,
~ (PPR.) is
aA.
1
J
a(PPR)
" = ---=---0:-,,--1
---;=;-
aA.
J
i
i-1
~ ~- e
-AkA
" "
k - 1 1-1
IT (A1k
-A ) A
1= 1
k
l*k
"
"
. 1 -AkA -A.A
11
{~e
. -e
}
k=1
n(\ -\ )
1= 1
1*k
"
Ae
"
-A.A
J - (1 -e
-A.A
1
i-1
J )(,,--- _ ~ "1,, )
A.1=1A -A.
1
i -1
J 1*;
i-1 ""
"
1[11 (AfA} Aj
{ 2: ------:---- +
k= 1 " " i -1 "
k*j (AfA k) l[I1(Af Ak}A k
1*j
l*k
173
J }
1k
And
"
""
a'"
-1
""" (PPR.)=
"
1. 1
-AkA
aA.1
1~ 1~ e
k=1
-A.A
[~~1 [l+(~(YA: e • 1. - e
"
-AkA
l
(Ai-Ak) 1[11 (A 1-A k)
IT (~ -~ )~
1=1 1 k k
l*k
l*k
Third, Quasi Parity-Dependent Model
"
Using the relevant expressions given in chapter 4, PPR is
i
...
"
"
PPR.=1- - - - - - - - - - - - - - - - - - . . , , , , . . . - - - - 1
-AA "
...
i-V-j-2 e
(AA) a
1-2
a!
v
-AkA
i-v-2
a=O
2
1
[ 1-e
- 2
k=1
(~
(~_\ )i-v-1~ j=O
1=1 1 k
k
k
IT _\)
1*k
for
i~3, i-v~2,
A/A for all j*k=1,2,"',v, and
k
A
v+ 1=A v+ 2 ="·=A.=A*A.forallJ·=12"·v.
1
J
"
,
"
The variance approximation of PPR. is as follows:
1
"
v+1 {
V[ PPR.] ~ :2
1
tJ=1
a("
[ --"..
PPR.)...
aAf3
1
~
=" ]
:.
174
2"'}
V (Af3) +
(5.7.2.3)
e·
a
P=1 ,2,3, "',v,
where for
..
.
--;; (PPR.)
a>-..
IS
1
P
" ) = - [[--a (num.)][deno.] - [num.][ - a..- (deno.)]
- ..a
- (PPR
i
a>-..p
- a>-..p
and
..
v
num .
a>-..p
= 2:-1
k-
ITv (>-........->-..)
1=1
..
A
e ->-"k -e ->-..A
1
[ .. ...
(""
/\-/\k )l-V
-e
-/\;A'1- v- 2
2:
1 k
...
a
a>-"/3
-1
2:v --...::....-..----
IT
..
->-"kA ->-"A
e
-e
k=l (~ _~)
(~ _~)
k*P P k 1= 1 1k
1*k
[
175
Al.
- v-.J - 1
.. .. .
'-0 (.I-V-J. 1)1. (\/\-/\k
" )J+
J-
1*k
- ..- (num.)=
J/[deno.] 2
......
(>-"->-"k) I-V
1 ],
"
JI
deno.=~
k~l
2:
1
IT (~ _~)
1=1
-AA '"
'"
i -j/-j-2 e
(AA) a
-AkA
1·-j/-2 1I
1- e
a=O
a.
[ " "'. 1'" - ~ ----.,.''.....:::;'':...-::....~1--,,-"· ---=-1-],
(A-A ) 1-j/- A
j~O (A-Ak)j+ A1-j/-jk
k
1 k
l*k
i -j/-2
2:
j=O
"
The partial derivative of PPR with respect to A is, similarly,
i
176
where num. and deno. are the same as shown above, and
Ae
-'AA
- (e
-'AkA
-e
-'AA
) ( i" - :' )
'A-'A k
a
jI
1
[
"" (num.) 2: --"--- -----::,,:---:,,::--...,...----------:.-'-- +
a'A
k=1
(~ _~
('A- 'Ak)i-jI
1=1 1 k
1*k
=
IT
)
]
i -jl-2
2:
j=O
i-jl-2
"
1
J~O 6'-\k)j+l ~i-lJ-j-l
i-jl- ;-2 e
{
,,~
-AA
,
~
A a \ a-l
1\
~O
(a-'AA) +
a!
+
The Parity Progression Ratio for the first live birth, in both
constant and parity-dependent models, is
"
'Ai A
PPR = 1- e
1
Finally, the variance estimate of PPR 1 is approximated as
177
5.7.3. Survival Function
In the context of birth interval analysis under complete observation,
the survival function,
H~
(x), is defined as the probability of not having
1
the i th birth within x time units after the occurrence of the (i_l)th
birth. When the observation period is incomplete, a length-unbiased
estimate of the survival function can be obtained by substituting the
recovered hazard estimates in the expression for the survi val function
of the i th birth corresponding to the All Closed ascertainment method
with complete observation. In the following sections, we show the
estimated survival functions and their variances for the Constant,
Parity-dependent, and Quasi Parity-dependent models.
First, Constant Model
The estimated survival function is
1- e
H.
"A (x) = 1 -
->.x
- e
->'A i-2
2:
~j+1 [Aj+1_(A-x)j+l]/(j+l)!
---.1_--=;,;
j=~..: --- >.A
1
1-e
_
i-1
2:
j=O
" .
(>.A) J/ j !
i~2,
and its variance approximation is as follows:
178
(5.7.3.1)
~A
V[ Hi (x)]:::
[a [Hi
~A
(x)]
a~
~=A
J2 V(A).
~
Here the partial derivative of H.A(x) with respect to
1
a "A
-~ [ H. (x)] =
aA
~
-1
[-AX
x e
-'AA i-l
1
~ . 1
. 1
. 1 -AA
AA 1- [(A-x)l- - A 1- ] e
(i-1)!
A
(~~)
2:
1- e
-
~ is
j=O J.
-AA
- x e
~
i-2
[1+
~ [A(Aj-r))~]
J-1
[1- e
-AX
- e
Second, Parity-Dependent
.
J+
-AA
e
{
l-e
~
i-1
~A(~A) ~
!(.i-1)!
-AA 1-1 (AA)J}2
2:-,
j=O J.
-AA i-2 ~ . 1
. 1
. 1
2: AJ + [AJ+ - (A-x)J+ ]!(j+1)!
j=O
I
I.
~
Mo~el
The estimated survival function of the i th birth is
,
179
i~2.
(5.7.3.2)
The variance of HiA(X) is approximated as
v [~
H. (x)
1
.
1
] ~ 2:
J'=1
2 2:
[a-.... [ H.~(x) ] -A J V (A. . .) +
aA.
J
~
1
J
A-
-
-
[a
i
~A (x) ] .\. _\
'5'
-" [H.
.<k~
J
2
aA.
J
1
a
- .... -
aA
A-A
-
--;;-A....
""]
[H.
(x) L _\ COV(A. ,A ) ,
k
1
k
-
And the partial derivative of Ht(x) with respect to
A-A
J
--
~j'
j=!,2,3"",i-!,
is as follows:
-a
"[~
H. (x)
aA.
J
e]
1
....
....
1
-A.X . . "
,,-AjA
-(A.-A .)x
+ - - - " ' - . - - [(1-e 1)(A .-A.-1)+A.e
(1-e 1 J )
1""
J 1
1
A
AiA j
(AC }
IV!
l*j
1 i
1
A .A - (A .-A .) x
(A+----2:
)+A.xe J e
1
J]
A. 1=1 AI- AJ'
1
J l*j
A
A
A
180
-
..
J+
-AkA
-(1-e
Ae
-A .A
J - (1-e
+ --
[kt. ..
k*j
-A .A
)
i..
i
""
.*.
1 J
l*k
A
and the •oartial derivative with respect to A.1 is
A
1=1
A
1 k
A
A
TI(AfA.) A.
1=1
J
J
Ak(AfAk) 1V1AfAk)
2 i
k= 1 IT
(A ->. )A
1
i
1
J )('" - 2:
A. 1= 1 A.-A.
J -----::._1.1..-H'
1
j
A
k
l*k
181
A
)
1
1
J
+
.....
.....
-A.A
Ae
1
-A.A
(1- e
-
1
1
) ( -.. . -
i-1
-
2: . .
1.. .
1=1 AI-Ai
Ai
]
- - - -.. .~i--1:--. .-. -,.-..;;......---~--=--} .
A.
1
IT
1=1
(AfA.)
1
Third, Quasi Parity-Dependent Model
The estimated survival function is
.....
H:A.
(x)
1
= 1-
H:A.
(x)
1
, where
.....
-AX -AA i-j/-j-2 (~A)a+1
1
e
i_j/_2 -e
2:
(a+1)!
a=O
]
4:=O -~---~~-(~_~ )j+ 1 ~ i-j/-j
J
k
.....
!
2: --..;:,..-k=!
IT (AI-A k)
j/
j/
..........
...
-A A
[1 e k
.. - ..... .
Ak(A-Ak)l-j/
-
,
-AA . . .
i-j/-j-1
(\A)a
1 - 2:
-.:::....e--:-..l.,;../\~
i-j/-!
a!
2: ~. ~
. . . a=O
~.'-:--;-1-::....-:-.- . : - - j=O (A-Ak)J+ A1-j/-J
J
1=1
I*k
(5.7.3.3)
where A .*A for all J'*k=1 2 ... j/ A
=A
=,,·=A.=A*A. J'=1 2 ... j/
J k
"
"
j/+ 1
j/+ 2
1
J'
"
,.
182
The variance of H':A(x) ia approximated as
1
~A
~A
J/+l
V [H; (x)]= V [H. (x)] ~:2
..
P=1
1
+2
i <L:= [a
[a--.;- [H.~ A (x) J;-\ J2 v (A~
OA
1
fJ
A-A
......
a
J/+ 1
:.A
~
~A ~
~ ~
-~ [ti. '(x) ] \ _\ -~-[ H. (x)) \ _\ Cov. (AQ,A)
1 aA
1
A-A
aA
1
A-A
JJ
Y
Y
fJ
--
Y
a
[a
~A (x)] = [ - (num.)][deno.]- [num.][ -~- [ H.
aAfJ
1
J,
......
~fJ' fJ=1 ,2," ',J/
The partial derivative of Hi'(x) with respect to
aAfJ
Q)
JJ
a
(deno.)]
, is
J/[deno.]~'"'
aAfJ
where, num. and deno. are, respectively, the numerator and denominator
a~ (num.) , ~ (deno.) are as
aAfJ
aAfJ
of the expression given in (5.7.3.3), and
follows:
i-J/-2
+
L:
j=O
-AX i-J/-j-2
1 - e
- L:
a=O
e
-AA
~
(AA)a+l
(a+1)!
-~~~~-----
(~-~k)j+ 1 ~ i-J/-j
183
J+
.....
{ - (1-e
.....
(1- e
-\x
) - \ (e
-~ A
f3
-\x
...
-\A
1
-e
)] [..---
i- v
(~_~f3)j+1 ~i-v-j
A
... ... )
- [ (\-\
f3
f3
v
i-v-1 1 ..,
2:
1=1 \ -\
1*f3 1 f3
-\A "
e
(\A) a
(~-\)j+1 ~i-v-j
a=O
a!
J+
184
1
}{ 2:" " }
"
i - v- j -1
j=O
f3
"
.....
-\x i-v-j-2 -\A . . .
i-v-2 1 - e
- 2:
e
(\A)a+1
2:
a-O
(a+1H
2:
-~
]}-
""
\-\
\f3
j=O
) + \A e
J.
'
and
i-v-l
2:
j=O
i-v-l
2:
j=O
The partial deri vati ve of
~
aA
H~(x)
with respect to A is
1
[ H.A(x))= [[..£. (num.)] [deno.]-[num.] [~ (deno.)] ] /[deno.]2
1
3A
3A
where num. and deno. are as above, and..£. (num.), -.£.(deno.) are shown
3A
3A
next:
a
a/\\
---(num.) =
1
v
2: ---=--k 1 v
t
= IT (A. -A, )
A
1= l'
l*k
A
1
A
1
A
A
\
\
A
A
(\
A
_
\
/\/\k /\ /\k
)
A
{(A-A, )xe
•
I-V
-AX
+ (l-e
-AX
)-
K
K'
A
-AA
-AkA
(e
- e
-AA
A
)- AAe
-AX
- [(A-A )(l-e
k
A
)- A(e
A
.
.2
l-~-
a~O
(a+l) AaA a
A
+ 1 AA
1 -\A
e
- A(AA)a+ e
_
A
(a+l)!
185
-AA
-AkA
-e
)]
.....
...
-\A . . .
-\x i-v-j-Z e
(\A) a+ 1
j+ 1
[(l-e
) - 2:
(a+1)!
][.......... +
a=O
\-\k
(deno. ) = ~
~
a\
k-l
a
-~
-\kA
1
...
- ITv.....(\ -\)
1=1
l*k
i-v-l
_-(i~v)Sl-e.)
~ n
1 _ ~
~
~
[\
k
i-~-j]}
]
; and
A
__ 1 _. .
~~-:::n---"'--:::~-,--~
'-0 (\_\) \l-V-J
k
(\_\ )l-V+
k
J-
1 k
i -~- j
\
The estimated survival function for the first birth, in the Constant and
Parity-dependent models, is
Its variance is approximated as
"'"
'"
A
"'"
-\ x
-\ A
-\ x
-\ A
""
[
1
1
1
where ~ [H~(x) ] = - [xe
( 1-e
)~- (1 -e
)Ae 1 ] .
a\l
-\1 A Z
[1 ~
188
]
tit -
CHAPTER VI
APPLICATIONS
6.1 Introdu:tion
This chapter deals with estimating parity-specific fertility hazards
8ased on birth interval data pertaining to different ascertainment methods. The parity-dependent models described in Chapter III for All
Closed, Last Closed, Straddling, and Open Birth Intervals are utilized
for this purpose. In addition, estimates are obtained from the quasi
parity-dependent model for the Last Closed Interval. As explained in
Chapter V, maximum likelihood estimation method is followed. Evidently, MLE hazard estimates derived from All closed, Last Closed,
and Open Birth Intervals are recovered from truncation bias ( see
Chapter V). However, the estimates cannot be fully recovered from
truncation bias in the cases of Straddling and Prospective Interior
Intervals; estimates derived from those strategies are likely to be
recovered from left but not from right truncation. The recovered
hazard estimates are further utilized in estimating three important
length-unbiased cohort fertility indices, namely, mean length of orderspecific birth interval, parity progession ratio, and birth interval
survival function--the probability of not having the i
th
birth within a
specified time period from the occurrence of the (i -1) th Ii ve birth.
The crucial assumption underlying all models in this study is the
exponentiality of live birth intervals when reproduction is regarded as
an infinite process. Assessing this assumption is necessary before
applying any of the proposed models to real data. Inferences based on
these models are only meaningful if this exponentiality assumption is
tenable.
Data from Egyptian Fertility Survey, 1980 (EFS) , are used to demonstrate the plausibility of the proposed models.
The present chapter is organized into four subsequent sections.
Section 2 deals with data sources and appraisal of quality; Section 3
addresses the issue of assessing the exponentiality assumption; and
Section 4 shows the maximum likelihood estimates of parity-dependent
fertility hazards and the resulting relevant fertility indices.
6.2 Data Source ani Appraisal
The data used for this application are extracted from the 1980 Egyptian Fertility Survey conducted by the Central Agency for Public
Mobilization and Statistics (CAPMAS) of the Government of Egypt.
The survey was a part of the World Fertility Survey Program and was
executed in collaboration with the World Bank (CAPMAS, 1983a).
The EFS was designed to be representative of the entire nation, excluding only the Sinai, nomads, and non-Egyptian nationals. A selfweighted sample of 10,000 households was selected from 200 Primary
Sampling units (PSU's), Shiakha and villages, the smallest adminstrative units in urban and rural areas, respectively. Within each selected PSU, a second stage cluster sampling was performed, in which the
188
Ultimate Area Units (UAU's) were selected. In the final stage of the
sampling process, a systematic sample of dwellings from each selected
UAU was drawn.
The EFS data include pregnancy, birth, and marital histories, as
will information on infant/child mortality, contraceptive practice, and
socioeconomic characteristics of 8788 ever-married women under age
50 (CAPMAS, 1983b).
Consideration should be given to the quality of the survey data before
drawing any inferences based on them. In particular, since thiS study
deals with dates of retrospecti ve events--dates of marriage and consecuti ve live births, the accuracy of reporting of these events is of special
interest. Indead, any survey of similar magnitude as the EFS undertaken in a society which is not highly literate cannot be expected to be
error-free. However, evaluation of the quality of the EFS data determined that "omission of births was negligible and that misplacement of
dates of birth was of a modest magnitude" (CAPMAS, 1983c).
Bearing in mind that this application is merely for illustrative purposes, a rather young marriage cohort of 5-9 years of women in their
first intact marriage was selected. Fertility histories of the members
of thiS cohort were used to estimate fertility hazards and related
fertility quantities. For this cohort, data on the detailed reproductive
history, the observed interval between the two most recent live births,
the open interval since the last Ii ve birth, and the birth interval
straddling the end of the second year of marriage were compiled from
the original EFS dataset. Since twins and multiple births were treated
189
in the original data file as if they were consecutive births, minor
editing was undertaken so that multiple birth dates representing
multiple live births could be reduced to only one date. Hence, there is
no closed birth interval of length zero. The cohort size amounts to
1473 ever-married women whose parity distribution is shown in Table
6.1 below.
Table 6.1
PARITY DISTRIBUTION OF THE MARRIAGE COHORT
WITH 5-9 YEARS OF INTACT MARITAL LIFE
190
With reference to the proposed models, childless women contribute
no information for estimating the fertility risk for the first live birth.
They are, therefore, omitted in the estimation process. Observed distributions of All Closed Birth Intervals by birth order are given in
Appendex B. No Significant heaping or digit preference can be detected
from those distributions.
6.3 Assessing Exponentiality
A common method of assessing the plausibility of a presumed parametric model to failure time data is to plot some function of the nonparametrically estimated survival probabilities against time. The life
table method, as detailed in Chapter I , is usually used for estimating
survival probabilities. If an exponential distribution is hypothesized,
the plot of negative cumulative hazard (log of the survival probability)
should be linear with time, with the line passing through the origin.
This can be stated formally as follows:
It is known that the survival function is defined as
F(x)
exp {-
x
\(y) dy}
J
o
x
==>- - Log
F (x) =of A(yl dy .
191
When exponentiality is assumed, the above equation becomes
- Log F(x) = AX.
This assessment approach has been followed with the order-specific
birth interval data under study. But, as there exists a protection period
of about 9 months for a live birth to occur after the preceding one, the
survival probability was plotted against x-9, where x is the length of
the order-specific closed birth interval. Thus a line passing through
the origin would be expected if exponentiality was satisfied. Ordinary
Least Square Regression was performed to fit the model
E [- Log F(x)
J= a + A(x-9).
2
The R for the above model and the results of testing the null hypothesis that a=O for all birth orders are shown in Table 6.2.
e-
Table 6.2
ASSESSING THE LINEARITY OF CUMULATIVE HAZARD
WITH SIRTH INTERVAL
Order of birth
interval
1
2
3
R
.93
.99
a
.358
see (a)
2
1
p-value
4
5
6
.99
.98
.95
.96
.006
-.102
-.107
-.076
-.061
.182
.066
.057
.066
.069
.051
.09
.92
.10
.13
.30
.28
lFor testing Ho : a=O
VS.
H·
a' a*O.
192
The above results suggest that the linearity of negative Log survival
probabilities versus birth interval, with the line passing through the
origin, is reasonable. As a result, a linear model without intercept
was fitted to the data. The resultant fit as well as observed cumulative
hazard are plotted aguinst x-g. Figures 1-6 confirm the appropriateness of the exponentiality assumption to model this set of birth interval
data. The estimated regression coefficients of the latter model are
simply graphical estimates of birth order-specific fertility hazard
when fertilitv is regarded as an infinite orocess. Table 6.3 disolavs
.I
.I.
.I.
J
the graphical estimates of fertility hazard.
Table 6.3
GRAPHICAL ESTIMATES OF BIRTH ORDER -SPECIFIC
FERTILITY HAZARDS
Order of Birth
interval
1
2
3
4
5
6
.0413
.0534
.0438
.0467
.0346
.0324
lEstimated monthly.
193
Figure 1
Cumulative hazard plot for
first birth interval
4I
I
-....
~
:a- 3 ~
l'I
..c
•
0
a..
-....: 2 I
~
~
a..
::J
•
/.
•
•
I
III
•
~ 1L
~
I
0-'-
o
I
I
8
\6
I
24
I
I
I
32
40
48
I
56
I
64
I
72
I
80
I
I
88
First birth interval in months
194
e
e
e
Figure 2
Cumulative hazard plot for
second birth interval
5
r - ,- - - - - - - - - - - - - - - - - - - - - - - - - - - ,
.......
-=4
~
....
...0...
l'I
..0
o
;'3
/
-....
l'I
..
?y
~
...
~
-
; 2
b/)
o
~1
I
o·
3
•
•
I
I
I
I
8
13
18
23
L-....------L__ ~
28
33
38
I
I
I
I
I
~J
48
53
58
63
L
68
Second birth interval in months
195
I
I
73
78
I
196
e
e
e
Figure 4
Cumulative hazard plot for
fourth birth in terval
3,
•
-.....-....
•
~
.c
l'S
.c 2
o
•
/
...
c..
-....
...
l'S
//
>>-
='
//.
lIll
•
bf.)
o
~
I
•
/'
o./~
o 5
•
I
10
'-
15
I
20
L_
2~)
J
30
I
J
I
35
40
45
J
50
Fourth birth interval in months
197
I
55
60
I
Figure 5
Cumulative hazard plot for
fifth birth interval
2,
I
....-........
-..c
~
l'OS
..c
...
•
I
0
Q.,
-....
...
l'OS
I>
I>
1
~
iii
b/)
0
H
I
.-./
•
I
0-
•
I
I
I
1
6
I1
16
I
I
21
26
I
31
I
36
I
41
Fifth birth interval in months
198
e
e
e
Figure 6
Cumulative hazard plot for
sixth birth in terval
•
-..........
>...
....
..0
cos
..0
o
....
(:l,.
....cos
....
////
-
•
~
....~
='
II)
bI>
•
o
~
I
0--
o
•
I
I
.1
5
10
15
-l....--
('0
I
?~)
Sixth birth interval in months
199
I
30
I
6.4 Hazard Estimates and Relevant Fertility Indices
The likelihood functions corresponding to each ascertainment and
model developed in Chapter V were used to estimate the parameters.
The computer program followed in finding the maximum likelihood
estimates of parity-dependent fertility hazards is BMDP ( PROG =
BMDPAR ). The first derivative of the Log likelihood function with
respect to each parameter being estimated is not reqUired. The graphical estimates of fertility hazards given in Table 6.3 represent a
reasonable initial set of estimates to start With.
Hazard estimates are derived from parity-dependent models for All
Closed, Last Closed, and Open Birth Intervals. Assuming that fertility
hazards for birth order 5 and higher are constant, the quaSi paritydependent model of the Last Closed Interval is applied to estimate the
relevant set of parameters. Estimates of the first 3 hazards are derived from the parity-dependent model of Straddling Birth Intervals,
where the point of straddl ing is defined as the last day of the second
year of marriage. The distribution of birth intervals straddling the
end of the second year of marriage according to birth order is given in
Table 6.4.
It is worth noting that during the process of estimating the parameter set (A 1 A2 ... A7) based on parity-dependent model of All Closed.
Last Closed, and Open Birth Intervals, the asymptotic variances were
given conditional on some of these parameters fixed at their initial
values. This situation is caused by high correlation among parameter
estimates (Singularity of the variance/covariance matrix). A way out
200
of this problem was to find the maximum likelihood estimates of the
seven parameters without paying much attent ion to the variances; then
the computer program that produced the MLE estimates was rerun with·
a new instruction that the least important parameter \7 be held fixed
at its preViously attained MLE value. HaVing done that, the program
produced variance/covariance estimates for the remaining 6 parameters conditional on \7 fixed at its MLE (initial) estimate. The resulting new set of estimates as well as the likelihood value are about the
same as in the first run. Also, it should be noted that with thiS set of
data the likelihood function for the Open Birth Intervals was apparently
very flat around its maximum value since different combinations of resulting estimates were accompanied by almost the same value of the
log likelihood function at which convergence was obtained.
Table 6.5 displays MLE estimates of fertility hazards jointly with
their asymptotic coeffiCients of variation. The important finding is
that while the set of recovered hazard estimates do not vary tangibly
with the ascertainment method, they are considerably below the corresponding estimates derived from Straddling Bbirth Intervals. The type
of truncation introduced by the latter ascertainment method clearly
leads to overestimating the underlying fertility hazards.
Knowing that the mean age at marriage of the marriage cohort under
study is about 18 years, we roughly estimated the longevity of the effective reproductive life of this cohort as of 26 years. The following
gUidelines were considered in approximating the effective reproductive
span:
201
a. Age at menopause of the Egyptian women is about 47 years.
b. The expected number of years to be survived in the age span 20-50
by a married female is about 28.8 years ( level 18 of the South
Region, Coale & Oemney Model Life Table).
c. Husband's age is , on average, 5 years higher than wife's.
d. The expected number of years to be survived in the age span 25-55
by a married male is about 28.2 ( same source as in b).
e. Remarriage of a widowed or divorced woman is uncommon.
The estimated effective reproductive span of 26 years, jointly with
the recovered hazard estimates, produces length-unbiased estimates of
the moments of birth intervals, parity progression ratios, and survival
probabilities. Substituting for t and other parameters in expreSSion
3.2.1.5- 3.2.1.8, we get the mean and variance of each order-specific
birth interval. Similarly, expreSSions 5.7.2.2,5.7.3.2 and related
variance apprOXimations are used to estimate the parity progreSSion
ratios and survival function along with their variances. The lengthunbiased estimates of the order-specific mean birth interval and the
standard error of the mean for each data ascertainment method are
displayed in Table 6.6. For purposes of comparison, the observed
means and standard errors derived from the same ascertainment
methods are also given. Table 6.7 gives the observed and lengthunbiased parity progreSSion ratios and their standard errors. The
parity-dependent model-based survival probabilities of the second birth
interval derived from the All Closed ascertainment method jointly with
the life table estimate of the same function are shown in Table 6.8 and
202
Figure 7. EVidently, the median of model-based survival function (36
months) is about 12 months larger than the corresponding life table
estimate, which is very close to the observed (23 months). This again
confirms what has been indicated in Chapter I , that life table analysis
of birth intervals does not fully succeed in eli minating truncation
biases. The proposed analytical approach performs better.
203
Table 6.4
DISTRIBUTION OF BIRTH INTERVALS STRADDLING
THE END OF SECOND YEAR OF MARRIAGE BY
BIRTH ORDER
Birth
order
Number
%
1
434
29.5
2
880
59.7
3
34
2.3
125
8.5
1473
100.0
1
NA
Total
includes childless and women of parity one who have had
their births in the first two years of marriage.
1
204
Table 6.5
PARITY-DEPENDENT HAZARD ESTIMATES ACCORDING
TO ASCERTAINMENT METHOD
Ascertainment
method
1
A
1
A2
A3
A
4
All closed
2
(CV)
.0281
(.903)
.0187
.0148
.0278
(1.333)
(1.672)
Last closed
(CV)
.0329
.0232
(.121)
Last closed
(CV)
AS
A
6
.0481
(.899)
.0403
(.607)
.0177
.0326
.0329
.0483
(.083)
(.032)
(.122)
(.014)
(.030)
.0303
.0233
(.0002)
.0241
4
(.000)
.0522
(.001)
.0189
(.009)
Open interval
(CV)
.0202
.0200
.0184
.0394
.0549
.0643
(1.237)
(1.246)
(1.325)
(.618)
(.451 )
(.427)
Straddl inginterval (CV)
.0717
.06617
.1209
(.0001)
(.0017)
(.0118)
3
I
2
3
4
per month.
Coefficient of variation.
Based on quasi parity-dependent model
The figure is .000038.
205
(.515)
(.042)
Table 6.6
LENGTH-UNBIASED AND OBSERVED MEAN BIRTH
INTERVAL ACCORDING TO DATA
ASCERTAINMENT METHOD
( in months)
Ascertainment
method
Birth Order
1
3
2
4
5
6
a. Length-Unbiased
All closed
1
(Se)
35.5
(.95)
51.6
(1. 48)
58.2
(1. 74)
32.9
(1.68)
23.1
(2.52)
19.3
(5.07)
Last closed
(Se)
30.4
(.81 )
42.7
(1. 20)
52.1
(1.62)
29.3
(1. 52)
28.3
(3.06)
19.6
(5.14)
48.1
(1. 29)
49.0
(1.52)
24.2
(1.26)
17.5
(1. 94)
15.0
(3.96)
Open interval 49.0
(Se)
(1. 28)
b. Observed
All closed
( Se)
22.0
(.38)
25.2
(.33)
26.3
(.37)
24.1
(.54)
21.7
(.84)
20.3
(2. 12)
Last closed
(Se)
39.9
(2.01)
31.07
(.72)
29.2
(.50)
25.4
(.63) ,
22.7
(.94)
20.3
(2.12)
Straddlinginterval (Se)
38.2
(.70)
26.3
(.41)
28.0
(2.59)
1
Standard error of the mean.
208
e·
Table 6.7
LENGTH-UNBIASED AND OBSERVEO PARITY
PROGRESSION RATIO
Ascertainment
method
Birth Order
1
2
3
4
5
6
All closed
(Se)
.9998
(.001)
.9917
(.057)
.9407
(.331)
.9454
(.233)
.9476
(.189)
.9435
(. 175)
Last closed
(Se)
.9999
1
(.000)
.9976
(.001)
.9735
(.004)
.9736
(.002)
.9561
(.004)
.9588
(.004)
Open interval. 9981
(Se)
(.014)
.9880
(.078)
.9545
(.241)
.9679
(.135)
.9699
(. i 12)
.9683
(.107)
Observed
.9100
.6969
.4009
.2260
.1750
1
.945
The variance is 1.89*10- 9
207
Table 6.8
MODEL-BASED AND LIFE TABLE SURVIVAL PROBABILITIES
OF THE SECOND BIRTH INTERVAL ASCERTAINED
FROM ALL CLOSED STRATEGY
Model
Life table
x
surv. prob.
0
12
24
36
48
60
72
84
96
108
120
132
144
156
168
180
192
204
216
228
240
252
264
276
288
300
312
1.000
.797
.635
.506
.403
.320
.254
.202
.160
.126
.099
.077
.061
.047
.036
.028
.021
.016
.011
.008
.006
.004
.002
.001
.001
.000
.000
Se.
Surv. prob.
.000
.229
.364
.433
.458
.453
.429
.394
.354
.312
.271
.232
.196
.163
.134
.108
.086
.067
.051
.038
.027
.018
.011
.006
.003
.000
.000
1.000
.943
.512
.221
.103
.057
.031
.023
.019
Se.
.000
.006
.014
.011
.009
.007
.006
.006
.006
e~
208
Figure 7
Model-based and life table survival probabilities of the
second birth interval ascertained from all closed strategy
1.0·
,
0.90>.0.8
.........
:.c 0.7
~
..c
o 0 .6
5-f
P-.0.5
-....
~ 0.4
t
0.3
tI)
0.2
='
Model
0.1
···Life Table
0.0 I
o
I
24
,i····
48
72
I
,
I
96
120
144
,-==:-r-::168
\92
I
'
,
216 240 264
Second birth interval in months
209
,I
288 312
,
CHAPTER VII
SUMMARY AND SUGGESTIONS FOR
FlmJRE RESEARai
7.1 Summary
Irrespecti ve of the method of data collection, most fertility surveys
produce only cross-sectional data, where the information pertains to the
fertility experience up to the date of the survey (or the end of the prospective study) for women still of reproductive age. In this situation the
analyst is confronted with a set of incomplete maternity histories.
With these, (s)he cannot proceed to a direct estimation of all birth interval distributions, or calculate complete cohort fertility indices such
as birth order,,-specific fertility rates, parity progreSSion ratios, and
birth interval-survival functions. The magnitude and pattern of truncation bias in birth interval data vary considerably with the prevailing
fertility level, birth order, parity, marital duration, and the ascertainment method of considering birth interval data.
Estimates of birth interval distributions and related moments can be
partially adjusted for truncation bias by applying life table analysis
for censored data to birth intervals of different orders. Nevertheless,
life table techniques do not produce satisfactory estimates, if at all, of
parity progression ratios from incomplete fertility histories.
The present study has as its major purpose to bridging this gap. It
proposes an analytical approach to deal with birth intervals viewed in
different ways, Le., different data ascertained strategies. Based on
these strategies, various cohort fertility quantities recovered from
(purged of) truncation bias are shown to be possible. This approach is
bUil t upon a stochaStiC formulation of the i th -order birth interval obtained from each of the following ascertainment methods of considering
birth interval data:
Birth Interval;
(a) All Closed Birth Intervals; (b) Last Closed
(c) Prospec~ive
Next (Straddling) Birth Interval; (d)
Prospecti ve Interior Birth Intervals; and (e) Open Birth Interval.
Three models are proposed for the interval between an
(i -1) th
and an
/h live birth. The first assumes that the fertility hazard for the i th
birth, A; (x), of women who have reached parity i-1 is independent of
~
parity and time (age), Le., A;(X)=A for all i=1,2,· ... The second model
~
regards fertility as a strictly pcrity-dependent process: \ (x) =\'
Ak*A for all k*I=1,2,···,i. The third model takes into account that repl
roducti ve behavior is more likely to be independent of the existing parity after some level of parity is attained (e.g., 4 or 5). In thiS model:
Ak*A l for all k*1=1,2,''',r ; and Ar + 1 =A r + 2 ="'= Ai=A*A k ,
for k=1,2"",r.
The second model was applied to hypothetical age-marriage cohorts
with specified fertility risks to investigate how truncation bias affects
the first two moments of the ith-order birth interval ascertained from
different strategies. The truncation bias, measured as the deviation of
211
the length-biased observed moments of the i th-order birth interval
from those resulting from a complete fertility history of the same
cohort, is generally low for the All Closed Birth Intervals strategy,
especially at higher fertility levels, longer marital durations, and
lower birth orders. Controlling for the length of the observation
period (marital duration), the moments derived from the Last Closed
and the Straddling Birth Intervals are generally higher than from the
All Closed Birth Intervals ascertainments. This disparity increases
with the longer the observation period, the higher the fertility, and the
lower the birth order. This upward bias is due to the additional influence of left truncation characterizing these types of birth intervals,
which apparently more than offsets the regular downward bias of right
truncation. The moments of Prospective Interior Birth Intervals are
generally less biased than those of Last Closed and Straddling Birth
Intervals.
Investigating the sensitivity of the mean length of inter-live birth intervals, measured as the proportionate increase in the mean resulting
from several levels of decline in fertility hazards, leads to the general
conclusion that the higher the truncation bias inthemean birth interval,
the lower the sensitivity of the mean to fertility change.
Maximum likelihood estimation is a very important procedure for
estimating the parameters of the proposed models for the following
reasons: (a) its asympototic properties are known, hence hypothesis
testing is possible; (b) alternative estimation methods, such as method
of moments, may result in different estimates of the same parameter
212
depending on the order of birth interval involved in the estimation
process; and (c) it produces estimates that are recovered from truncation bias in All Closed, Last Closed, and Open Birth intervals.
The second and third models were applied to data from the Egyptian
Fertility Survey (1980) for a marriage cohort with 5-9 years of intact
marital life. While the recovered birth order-specific hazard estimates are generally of the same magnitude irrespective of the ascertainment method of considering birth interval data, they are largely below
those derived from Straddling Birth Intervals. In addition to the descriptive value of the recovered hazard estimates, as refleCting the underlying fertility levels of all women exposed to the risk of haVing the
i th birth since the occurrence of the preVious one, they are used as an
input to estimate three important length-unbiased cohort fertility indices, viz., mean length of interlive birth interval, parity progression
ratios, and the birth interval survival function. A comparison between
model-based and life-table-estimated survival probabilities of the
second birth interval suggests that the former is likely to be better
adjusted for truncation bias. The extent of thiS adjustment is greatly
profound in the corrected set of parity progression ratios as contrasted
with its observed analogue particularly for higher parities.
7.2 Suggestions For Future Research
Possible theoretical improvements in the proposed models and
broader avenues of application can be outl ined here.
213
7.2.1 Theory
Various potentially interesting research topics are stimulated by
the present study, including the following:
...
(a) A common assumption of all models developed in this study is
that cohort members are homogeneous with respect to the risk of
having a new live birth. This assumption may not be strictly valid in
real populations with a vast diversity across socioeconomic classes.
For this reason, giving some consideration to potential heterogeneity
among cohort members may result in models that more adequately
describe the underlying fertility process. Heterogeneity can be either
observed or unobserved. Observed heterogeneity is basically attributable to dissimilarities in the socioeconomic determinants of fertility
at the individual, household, and community levels. A way to deal with
this type of heterogeneity is to assume that Ai is a log linear function
of an array of socioeconomic variables under consideration; Le., that
X.8.
1
A.= e"" ....
1
1
where, for each woman involved in estimating A. , X. is a row vector
1
....1
of the fertility predictors chosen, and Biis the column vector of corr-
....
esponding parameters to be estimated. Although this model speCification of Ai may not be very manageable from a computational standpoint,
the likelihood functions developed in Chapter V can be used to obtain
maximum likelihood estimates of the vector of parameters 8 1. Hence,
....
the relative importance of each of Xi components can be assessed .
....
214
On the other hand, the unobserved heterogeneity can be independently
handled by considering Ai a random variable from some probability
distrbution (e.g., gamma). Then, the resulting compounding probability
model of the birth interval can be applied to estimate E[A.), the mean
1
of A., as representing the underlying fertility hazard for the i th birth
1
accounting for unobserved heterogeneity. Unfortunately, It is not clear
how to cope with both types of heterogenity simultaneously. Existing
methods require dealing only with either independently ( however, see
Lawless (1987) for a possible approach).
(b). The assumption that fertility is strictly parity-dependent or quasi
parity-dependent (without any direct effects of age) is plausible to the
extent that fertility is either widely under voluntary control or that age
at birth and birth order are highly correlated. OtherWise, it may be
reasonable to include age in the stochastic formulation of the interli ve
birth intervals. Research in this area is probably worthwhile.
(c) Further research is needed to modify the proposed models of
Straddling and Prospective Birth Intervals so that recovered hazard
estimates may be attainable from those ascertainment methods.
7.2.2 Further Applications
The utility of the proposed models of birth intervals, particularly
those which produce recovered estimates, can be greatly enhanced by
using them to formally assess cohort differentials in fertility risks.
Evidently, the models avoid any loss of information due to controlling
for marital duration which would otherwise be inevitable if comparisons
215
among cohorts were based on observed birth interval data. Statistical
multiple comparisons can be conducted among cohorts of the same fertility surveyor between surveys. In either case changes in fertility
over calendar time, can be detected based on the under! ying truncated
maternal histories, which may not be possible following other approaches. In this context, it is important to emphasize that the application of the family rejection region (e.g., Bonferroni) is necessary for
drawing meaningful statistical inferences out of multiple comparison
procedures.
e-
216
Refererees
CAPMAS.
(1983a). Egyptian Fertility Survey, 1980. Standard
Recode, Version 3. Cairo, Central Agency for Public Mobilization and Statistics.
(1983b). Egyptian Fertility Survey, 1980. Volume 1: Survey
Design. Cairo, Central Agency for Public Mobilization and
Statistics.
---------- (1983c). Egyptian Fertility Survey. 1980. Volme 2:
Fertility and Family Planning. Cairo, Central Agency for
Public Mobilization and Statistics.
Chiang, C. L. (1968). Introduction to Stochastic Processes in
Biostatistics. John Wiley and Sons, New York.
Chiang, C. L.
(1985). A Stochastic Model Of Human Reproduction.
paper presented at the annual meeting of American Statistical
Association, Biometrics Section.
Chiang, C. L. and Vanden Berg, B. J. (1982). A Fertility Table for
the Analysis of Human Reproduction. Mathematical
Biosciences 62: 237-51.
Henry, L. (1953). Fondements Theo'riques des Measures de la
Fe'condite' Naturelle. Revue de I' Institut International de
Statistigue 21: 135-51.
Henry, L. (1957). Fe'condite' et Famille-Mode/les Mathe/matiques.
Population 21: 413-44.
Hoem, Jan M. (1970). Propablistic Fertility Models Of Life Table
Type. Theoritical Population Biology 1: 12-38.
217
Johnson, Regina C. Elandt, and Johnson, Norman L. (1980). Survival
Models and Data Analysis. John Wiley and Sons, New York.
Leridon, H. (1969). Some Comments On Article By Srinivasan.
Population Studies 23: 1, 101-04.
Lawless, J. F., (1987). Regression Method For Poisson Process
Data. Journal Of American Statistical Association 82:808-15.
Mishra, R. N., Yadava, K. N. S., Singh, K. K., Singh, S. R. J.(1983).
A Modification Of Probability Distribution For First Birth Interval. Health And Population 6 (2): 95-99.
Nour, E. (1984). Parity Specific Fertility Tables. SOCial Statistical
Section, Proceedings Of The American Statistical Association.
Pathak, K. B. (1983a). An Extension Of A Probability Model For
Closed Birth Interval. Health And Population 6 (3): 133-42.
Pathak, K. B., and Sastry, V. S. (1983b). A Modified Stochastic
Model For Closed Birth Interval. Journal Of Mathematical
SOCiology 9: 155-63.
Perrin, E. B., and Sheps, M. C. (1964). Human Reproduction: A
Stochastic Process. Biometrics 20: 28-45.
Poole, W. K. (1973). Fertility Measures Based On Birth Interval
Data. Theoritical Population Biology 4: 357-87.
Rodriguez, G. and Hobcraft, J. N. (1980). Illustrative Analysis:
Life Table Analysis Of Birth Intervals In Colombia.
Scientific Report; World Fertility Survey.
Ryder, N. B. (1965). The Measurement Of Fertility Patterns. in
Public Health And Population Change (eds. Sheps, M. C., and
218
•
Sehgal, J. M. (1971). Indices Of Fertility Derived From Data On
The Length Of Birth Intervals, Using Different Ascertainment
Plans; Department Of Biostatistics, University Of North Carolina Chapel Hill, Institute Of Statistics Mimeo Series No.768.
Sheps, M. C., Menken, J. A., Ridley, J. C., Lingner, J. W. (1969).
Problems Of Birth Interval Analysis In Survey Data. Mimeo.
Carolina Population Center, RO 1080.
Sheps, M. C., Menken, J. A., and Rakick, A. P.
(1969). Probability
Models for Family BUilding- An Analytical Review.
Demography 6: 161-83.
Sheps, M. C., and Menken, J. A. (1973). Mathematical Models Of
Conception And Birth, The Uni versi ty Of Chicago Press.
Sheps. M. C., Menken, J. A., Ridley, J. C., Lingner, J. W. (1970).
The Truncation Effect In Closed And Open Birth Interval Data.
Journal Of American Statistical Association 65: 678-93.
Srini vasan, K. (1966). An Application Of A Probabili ty Model To The
Study Of Inter-Live Birth Interval. Sankhva B 28: i 75-82.
Srinivasan, K.
(1967). A Probability Model Applicable To The Study
Of Inter-Live Birth Intervals And Random Segments Of The
Same. Population Studies 21: 1,63-70.
Srinivasan, K.
(1979). Birth Interval Analysis. Scientific Report,
World Fertility Survey, London
Suchindran, C. (1972). Estimation Of Parameters In Biological
Models Of Human Fertility; Department Of Biostatistics,
University Of North Carolina, Chapel Hill, Institute Of
Statistics Mimeo Series, No. 849.
Suchindran, C., and Lingner, J. W.
219
(1977). On Comparison Of Birth
Suchindran, C., and Lingner, J. W. (1977). On Comparison Of Birth
Interval Distributions. Journal Of Biosocial Science 9: 25-31.
Venkatacharya, K. (1969). Certain Implication Of Short Marital
Durations In The Analysis Of Live Birth Intervals. Sankhya B:
31, 53-68.
220
APPENDIX A
Derivation of the General Expression for the p.d.f of
Straddling Birth Interval
The distribution function is defined as:
H~'
T (x) = pr [ X. (t, T) ~x]
1
1
= pr [
X.~x
1
I
(S.1- l~t n t<
S.~T)]
1
x
of pr [Si-l ~tn (t<Si ~T
I Xi=yl] fi(yl dy
=-----------------Pr [ S.1- 1 <tnt <S.1 ~ T]
Differentiating with respect to x, the p.d.f is
pr [ Si-l ~t n (t-x <Si-l ~T-xl] f i (xl
T]
Pr [S.1- 1 <t n t< S.~
1
= e~lpr
[S.1- l~t n (t-x < S.1- l~T-xl] f.(xl
1
1
, x~T,
where e.1= pr [ S.1- 1 <tnt <S.1 ~T] .
Define the event e to be:
e = [S.1- l~t n (t-x <So1- l~T-x)], then e can be written as:
e= [S.1- l~t n (S.1- l)t-x n S.1- l~T-x)]
= [ (S.1- 1~ t n S.1- l)t-x) n (S.1- l~t n S.1- l~T-Xl]
= (t-x<S.1- l~t ) n (S.1- l~t n S.1- 1~ T-x)
=(AnB),
here A= ( t-x < Si-l ~t) and B = (Si-l ~t n Si-l ~T-x).
221
A ={
Here
t-X(S. ~t
for
x~t
for
x>t
1
S.1- 1 (t
,and
•
B-
{
S.1- 1~t
for
S.1-1 ~ T-x
for T-x(t or x>T-t.
t~T-x
x~
or
T-t
Therefore the event ( A () B ) can be written as follows:
[(t-x(S.1- 1~t) () S.1- l~t] = (t-x(S.1- l~t) ,for
C= (A () 8) =
x~t
and
x~T-t
x~T-t
S.1- 1 <t
for x> t and
[(t-x(Si_1 (t) () Si-1 ~T-x]
for x~t and x>T-t
[ Si-l (t () Si-1 ~T-x]
for x>t and x>T-t
Thus, in order to determine disjoint ranges of x it is necessary to
consider two cases: case 1;
t~T-t,
and case 2; t>T-t. They are given
next
Case 1
C=
t~
T -to
(t-x ( Si-l ~ t)
for
<t
for
S.1- 1
~
x~t
t(x~T-t
for x>t and
S.1- 1 ~T-x
for T -t
222
x~T-t
<x~T.
Case 2
t
> T-t.
(t-x <Si -1 ~t)
for
~
c=
x~T-t
for x>t and x:S;T-t
(t-x<S.1- 1 <T-x)
for T-t <x:S;t
(Si-1 :S;T -x)
for t<x<T.
.
Accordingly, the general expressions for the p.d.f of X; (t,T) given on
page 45 and onwards follow.
223
APPENDIX B
CLOSED BIRTH INTERVAL (CBIJ*) BY BIRTH ORDER FOR
THE MARRIAGE COHORT WITH 5-9 YEARS
OF INTACT MARITAL LIFE
(IN MONTHS)
CBn
FREQUENCY
•
PERCENT
.-_.--_.-_.-_._.-.-_._ .. _.
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
3S
36
37
38
39
40
41
42
43
44
4S
46
47
48
81
3
28
85
86
70
87
72
77
55
73
64
50
53
49
47
38
21
33
30
21
25
25
15
21
19
15
14
13
11
13
9
12
5
7
16
8
12
5
7
4
6
6
224
0.2
2.0
6.1
6.2
5.0
6.3
5.2
5.5
4.0
5.2
4.6
3.6
3.8
3.5
3.4
2.7
1.5
2.4
2.2
1.5
1.8
1.8
1.1
1.5
1.4
1.1
1.0
0.9
0.8
0.9
0.6
0.9
0.4
0.5
1.1
0.6
0.9
0.4
0.5
0.3
0.4
0.4
e
,
•
...
(Continued)
49
54
55
9
0.6
1
1
0.1
0.1
0.2
56
3
57
58
59
60
4
0.3
2
3
1
2
1
3
1
0.1
0.2
0.1
0.1
0.1
0.2
0.1
0.1
0.1
0.1
0.1
0.1
61
62
63
64
65
66
69
70
71
72
74
75
76
77
2
1
2
2
1
4
1
6
1
2
2
2
2
81
84
85
88
*
3
0.5
0.5
0.1
0.2
7
7
2
50
51
52
53
2
92
1
107
1
J stands for the order of birth interval.
225
0.3
0.1
0.4
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
CLOSED BIRTH INTERVAL (CBIJ) BY BIRTH ORDER FOR
THE MARRIAGE COHORT WITH 5-9 YEARS
OF INTACT MARITAL LIFE
(IN MONTHS)
CBI2
FREQUENCY
..
PERCENT
-------------_._._-------8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
206
7
13
20
39
45
42
54
42
49
52
49
54
50
47
44
59
57
39
38
33
29
30
26
33
26
32
25
19
32
20
12
12
9
15
7
8
8
9
6
8
8
5
5
7
4
226
0.6
1.0
1.6
3.1
3.6
3.3
4.3
3.3
3.9
4.1
3.9
4.3
3.9
3.7
3.5
4.7
4.5
3.1
3.0
2.6
2.3
2.4
2.1
2.6
2.1
2.5
2.0
1.5
2.5
1.6
0.9
0.9
0.7
1.2
0.6
0.6
0.6
0.7
0.5
0.6
0.6
0.4
0.4
0.6
0.3
a
e
,
•
(Continued)
54
55
56
57
58
59
60
61
62
63
5
4
1
3
2
3
3
1
3
1
1
1
64
66
67
68
69
70
1
77
1
2
1
2
1
1
78
87
1
1
71
75
227
0.4
0.3
0.1
0.2
0.2
0.2
0.2
0.1
0.2
0.1
0.1
0.1
0.1
0.1
0.2
0.1
0.2
0.1
0.1
0.1
0.1
CLOSED BIRTH INTERVAL (CBIJ) BY BIRTH ORDER FOR
THE MARRIAGE COHORT WITH 5-9 YEARS
OF INTACT MARITAL LIFE
(IN MONTHS)
CBB
FREQUENCY
•
PERCENT
------_._._--_._-----_._-8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
590
2
10
14
17
29
27
16
26
21
33
26
28
34
35
33
37
40
35
45
29
30
34
35
20
18
20
15
11
13
17
20
9
10
13
12
0.2
1.1
1.6
1.9
3.3
3.1
1.8
2.9
2.4
3.7
2.9
3.2
3.9
4.0
3.7
4.2
4.5
4.0
5.1
3.3
3.4
3.9
4.0
2.3
2.0
2.3
1.7
1.2
1.5
1.9
2.3
1.0
1.1
1.5
1.4
11
1.2
1
9
5
0.1
1.0
0.6
0.6
0.6
0.3
0.2
0.3
0.2
5
5
3
2
3
2
228
e.
•
(Continued)
53
54
55
57
59
60
61
62
64
67
70
72
75
2
4
1
1
2
2
1
2
2
2
2
1
1
229
0.2
0.5
0.1
0.1
0.2
0.2
0.1
0.2
0.2
0.2
0.2
0.1
0.1
CLOSED BIRTH INTERVAL (CBIJ) BY BIRTH ORDER FOR
THE MARRIAGE COHORT WITH 5-9 YEARS
OF INTACT MARITAL LIFE
(IN MONTHS)
CBI4
FREQUENCY
PERCENT
•
-------------------------8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
43
44
45
46
47
48
50
51
53
56
65
66
72
1119
2
4
5
7
16
16
7
16
18
16
13
11
11
16
11
22
18
10
11
9
14
8
8
8
13
7
8
5
5
6
2
7
6
1
2
1
3
1
1
1
1
1
2
1
1
1
1
230
0.6
1.1
1.4
2.0
4.5
4.5
2.0
4.5
5.1
4.5
3.7
3.1
3.1
4.5
3.1
6.2
5.1
2.8
3.1
2.5
4.0
2.3
2.3
2.3
3.7
2.0
2.3
1.4
1.4
1.7
0.6
2.0
1.7
0.3
0.6
0.3
0.8
0.3
0.3
0.3
0.3
0.3
0.6
0.3
0.3
0.3
0.3
e
~
•
CLOSED BIRTH INTERVAL (CBIJ) BY BIRTH ORDER FOR
THE MARRIAGE COHORT WITH 5-9 YEARS
OF INTACT MARITAL LIFE
(IN MONTHS)
CBI5
FREQUENCY
PERCENT
-------------------------8
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
34
36
38
39
50
CBI6
1393
1
1
2
4
2
2
6
6
2
3
2
6
3
6
3
6
3
4
4
1
1
5
1
2
1
1
1
1
FREQUENCY
1.3
1.3
2.5
5.0
2.5
2.5
7.5
7.5
2.5
3.8
2.5
7.5
3.8
7.5
3.8
7.5
3.8
5.0
5.0
1.3
1.3
6.3
1.3
2.5
1.3
1.3
1.3
1.3
PERCENT
-------------------------9
11
14
16
18
20
21
24
25
30
39
1459
1
1
1
3
1
1
1
1
2
1
1
231
7.1
7.1
7.1
21.4
7.1
7.1
7.1
7.1
14.3
7.1
7.1