1
2
'
Department of Statistics#
University of Cairo
Department of Statistics# University of North Carolina
*This research was done while on a fellowship from the Population
Council of New York. It was partially supported by research
grant HD03441 from the National Institute of Child Health and
Human Development.
A STOCHASTIC fvODEL FOR HUMtW FERTILITY, I
E1-Sayed Nour1 #* 4 Edward J. Wegman 2
University of North CaPoZina at ChapeZ HiU
Institute of Statistics Mimeo Series No. 894
September~
1973
A STOCHASTIC MODEL FOR HUMAN FERTILITY, I
El-Sayed Nour
1.
&Edward J.
Wegman
Introduction and Summary
Fertility as the positive element in the vital process is largely responsible for the current rapid acceleration in the growth of human populations.
Implementation of family planning programs for the purpose of altering fertility
behavior is becoming a popular course of population policy. Awareness of this
situation has led to the freqhent utilization of the tools of' mathematics and
statistics to investigate the fertility process. A number of mathematical
models has been proposed for this purpose. The main interest in these models
is twofold: the first is to develop techniques~ as well as schemes of data
collection~ necessary to detect and evaluate the significance of changes that
take place in fertility; the second is to enhance the understanding of the
fertility process by identifying the nature of the interactions of various
factors and fertility.
The nature of human fertility suggests four basic variables that should be
considered for inclusion in the fertility models:
The effective reproduct i ve period, i. e.. the age span during which reproduction is possible. is the first variable considered. We will assume this period
begins at a woman's age of marriage or her age of.menarche whichever occurs last.
and ends at her age of death~ sterility~ divorce. widowhood, or menopause whichever occurs first. The finite nature of the effective reproductive period implies
that the observed fertility behavior is a truncated sample of the behavior that
_ would have been observed if that period were infinite.
1.
2. Fecundability. as originally defined by Gini~ (1924) is also considered.
This concept of fecundability is hereafter referred to as conditional fecundability. Conditional fecundability is defined as the probability of conception
per unit time conditional on a woman being susceptible, and at the risk of
conception~ at the beginning of that time unit.
The fact that being susceptible
at any point of time is a random event, gives rise to another version of
2
fecundability hereafter referred to as unconditional fecundability. Unconditional fecundabi1ity is defined as the probability of conception per time unit.
3. The third variable considered is the various outcomes of pregnancy
which depend upon the probability of a conception ending in a given outcome.
4. The duration of nonsusceptib1e periods, associated with various pregnancy outcomes, ,during which another conception is impossible is the last
variable considered for inclusion in fertility models. This duration consists
of two adjacent periods--a pregnancy period and a postpartum nonsusceptib1e
period. The length of these periods depend on the pregnancy outcome.
Several models have been proposed to study the mode of change in these
variables and its relation to ferti1~ty performance. These models' can be
dichotomized according to whether or not they conform to the definition of
a renewal process. The major portion of the literature treats human fertility
as a renewal process (e.g. Henry ~953, Singh 1963, 1964, Perrin &Sheps 1964)
since the theory of renewal processes is well developed and the underlying mathematical analysis is more tractable. However, the incongruity of the assumptions
underlying human reproduction with those of a renewal process introduces inadequacies that will make the analysis inconclusive. To treat human reproduction
as a renewal process implies that the basic variables underlying the process
are c9nsider~d independent of age, parity and marriage duration. Further,
since results for instantaneous time in a renewal process are not available in
simple forms, treating human reproduction as a renewal process usually assumes
that the reproductive period is sufficiently long such that a state of equilibrium can be achieved.
In the present paper, it is recognized that age and parity are the most
important variables associated with the basic variables underlying fertility.
Human fertility is thus described as a continuous time (age) discrete state
(parity) stochastic process. Such a description is considered in relation to
two types of marriage cohorts. The first is a time-age marriage cohort,
teferring to'a group of women who marry at the same age at the same time point.
The second is a ti~~ marriage c~hort, ~eferring to·a·g~oup of women who marry
at the same time. The relationship between a time marriage cohort and a time-
"
.,
3
age marriage cohort which is derived from the age distribution at marriage
will be utilized in subsequent work in studying the effect on.fertility of
changes in the pattern of input to the marriage system.
~'
..
2. Definitions and Assumptions: In this study, we shall follow the convenient
practice of assigning a couple's reproductive performance to its female number.
In this section we adopt the following formal definitions and assumptions as the
basis for the remainder of: this work. So far as possible, all definitions are
consistent with their usual meanings prevailing in current literature.
DEFINITIONS
2.1. FERTILITY is the actual reproductive performance of an individual or a
group of individuals.
.2.2 FECUNDITY is the capacity of an individual to participate in the repro~
ductive process. The lack of this capacity is called infeaundity.
2.3 The REPRODUCTIVE PERIOD is the age span during which a .woman can normally
. reproduce.
2.4 TERMINAL EVENTS are those which cause a woman to become incapable of
participating in the reproductive process. These include sterility,
divorce and widowhood.
2.5 CONCEPTION is the fertilization of an ovum and implantation of the
resulting zygote in the uterine wall.
2.6 The EFFECTIVE REPRODUCTIVE PERIOD is the portion of the reproductive
period which begins with marriage and ends in a terminal event or death.
This is the period during which a woman can conceive.
2.7 CONDITIONAL FECUNDABILITY is the probability that a non-contracepting
woman will conceive over a time interval given that she is fecund at the
beginning of that time interval. This probability is usualiy referred
to in the literature
feoundabiZity.
. . as naturaZ
.
2.8 UNCONDITIONAL FECUNDABILITY is the probability that a non-contracepting .
woman will conceive over a time interval. This is a new version of
fecundability introduced in this study and is the product of natural
fecundability and the probability of being fecund at the beginning of a
time interval.
' .
:
4
2.9
LIVE BIRTH is a confinement at which at least one live-born infant is
delivered.
2.10
FETAL LOSS refers to a pregnancy terminating in any outcome other than
a live birth.
2.11 The NONSUSCEPTIBLE PERIOD is the total time period during which ovulation
is suppressed following a conception. This is a period of temporary
infecundity.
2.12 PARITY is the lUmber of live births a woman has had.
2.13 TIME-MARRIACJE COHORT is a group of women who marry at the same point in
time.
2.14 TIME-AGE MARRIAGE COHORT is a group of women who marry at the saJ!te age
and at the same point in time.
ASSUMPTI ONS
2.15
2.16
2.17
2.18
2.19
2.20
2.21
2.22
2.23
The fertility histories of different members of a group are mutually
. independent.
Fecundability is a function of age and parity.
The length of nonsusceptible periods associated with a conception is a
function of parity only and not of age. Although age is thought to have·
an effect on the length of these periods, this effect is ignored for
simplicity. This action appears to be justified by available results
obtained by simulation (Venkatacharya, 1969) indicating that this variable
(length of nonsusceptib1e periods) is less important than other variables
in determining the reproductive performance.
For any conception leading to a live birth, the associated gestation
period is assumed constant.
Any conception ends in either a live birth or a fetal loss. No attempt
is made to consider the different outcomes of a fetal loss.
The incidence of fetal loss is a function of age and parity.
.Termina1 events are treated as a function of· age.
Multiple births are treated as one birth.
Fertility performance can change with calendar time. The specific
pattern of a woman's reproductive performance is related to the calendar
date of marriage.
5
The Stochastic Model.
3.
good deal of notation.
We shall, regrettably introduce in this section a
For this we now apologize.
Let the random,variables
Xo and T denote the age and calendar time respectively at which an individual
or a group of individUals marries.
The probabilities discussed in this section
shall be conditioned upon the events
[X
o = .xO] and [T = T]., For simplicity,
explicit reference to such dependence shall, for the most part, be suppressed.
Bearing this in mind, we shall let p • . (x,y) denote the conditional probability
1.,)
that a woman is at parity j
age
y given that she is at parity i
at
In this section 'w~ desire to relate this probabl1i"ty to quantities
x.
which
at age
a~e
either estimable, measurable, or for which a reasonable mathematical
form may be assumed.
Three basic probabilities are assumed to be quantities in
this category:
3.1
Pj
(x)llx + o(llx)
is the conditional fecundability at age
x.
This is
the conditional probability that a woman will conceive exactly once over
the interval (x,x+llx) given that she is fecund at age
parity j
at age
x.
Of course,
x and that she is at
l-p.(x)llX + o(llX) is the conditional
)
probability that she will not conceive and
o(llX), that she will conceive more
than once.
3.2
ej(x) is the conditional probability that a conception at age
end in a fetal loss given that the parity at age
we assume
3.3
is
For convenience,
e.(x) is a continuous function of x.
)
Q. (x) is the conditi..onal probability that a woman is alive, ·at the risk
)
of, and susceptible to conception at age
x
x is j.
x will
j.
x given that her parity at age
6
We first examine the structure of Qj(x) in Theorem 3.1.
To do this
effectively we introduce 4 random variables.
(l(x) =
3.4.
C
1
B(x) =
3.5.
{
0
3.6.
If a woman is alive at age x
otherwise.
If a woman is in the effective reproductive
state at age x
otherwise.
1
If a woman whose parity is j is passing
through a nonsusceptib1e period at age x
following a conception g"iven that the conception leading to the j+lst live birth
does not occur before age x.
o
otherwise.
y. (x) =
J
J(x) = the parity at age x
3.7.
Theorem 3.1:
where
Q(l) (x) = Pr[a(x) = 1]
(3.l.i)
(can be estimated from a life :~',.,
table)
and'
( 2)
(3.l.ii)
with
X
o
Q
-f
x
p(t)dt
x .
(x) = Pr[B(x)=lla(x)=l] = .~e 0
go the probability of being in the effective reproductive state
and il(t)8t +
over (t, t+~t).
(3.l. iii)
age
the probability that a woman in the effective repro- .
o(~t)
ductive state at age
a~
t
will experience a terminal event (other than death)
Here we also assume Q(2)(x) is dj..fferentiable,
and finally,
Q~3)(x) = Pr[y.(x)=O!a(x)=l,B(x)=l, J(x)=j].
J
satisfies the differential equation
J
7
dQ~3)(x)
.J
dx
= _Q(.3) (x)p.(x) + Q(3)
(x- c 2' )e , (xc
'
- 2' )p , (xc)
- 2' ,
'. .
J
J.
J . J
J
J . J
J
Q~3)(x)
Here, of course, we assume
is differentiable and we let c 2 , be the
J
length of the nonsusceptible period associated with fetal loss while at parity
J
j.
Proof:
In terms of the random variables, a, S, Yj
and J,
Q, (x) ~ Pr[a(x)=l,l3(x)=l, y, (x)=oIJ(x)=j].
J
.
.
.
J
Q (x) =Q (1) (x) x Q(2) (x) .x Qj3) (x) , is immediate.
Hence, that
j
To prove that
_Jx ll(t)dt
X
o
, we note that
1 -
+
ll(t)~t
woman in the effective reproductive state at age
interval (t, t+
age
t+At
~t).
o(~t)
is the probability a
will not leave over the
t
Hence a woman is in the effective reproductive state at
if she is in the effective reproductive state at age
does not leave it over (t,
t+~t).
t
and she
That is,
Q(2) (t+~t) = Q(2) (t) (l-ll(t)At + 0 (~t» •.
Rewriting,
Letting , ~t
Q2(t+~t) ~tQ(2) (t)
-+
0 yields
= _Q(2) (t)ll(t) + 0
dQ(2) (t)/dt = _Q(2) (t)ll (t).
(~~)
Separating variables
and integrating,
x
log Q(2)(x)_ log Q(2) (x )
O
= -J
ll(t)dt
X
The initial condition Q(2) (x )
O
o
=~,
,
yields (3.l.ii).
8
Qj3)(X) = pr[y j (x)=Ola(x)=l, S(x)=l, J(x)=j] ,
Finally, noting that
we obtain
QJ3)(X+6X)= Pr[Yj(x+6X) = O/a(x+6x)=l, (3(x+6x)=l, J(x+6x)=j]
(3.1.1)
= pr[y j (x+6x)=O, yj (x)=l!a(x+6x)=l, (3(X+6x)=1,J(x+6x)=j]
+ Pr[y.(x+6x)=O, y.(x)=lla(x+6x)=l, (3(x+6x)=1, J(x+6x)=j] .
J
J
.
Now the first term on the right-hand side of (3.1.1) is
Pr[Y.(x+6x)=O/y.(x)=O, a('x)=l, S(x)=l, J(x)=j]
J
(3.1. Z)
.
J
x Pr[y.(x)=O/a(x)=l, S(x)=l, J(x)=j]
J
.
= Qj (x) (1 - Pj (x)6x + 0(6x)).
The event, [y.(x+6x)=O, y.(x)=lla(x+6x)=l, l3(x+6x)=l, J(x+ x)=j], appearing in
J
J
the second term on the right-hand side of (3.1.1) can occur only if a woman was
passing through a nonsusceptible period at age x occuring after a fetal loss
and then moves to a susceptible state at age
x+6x.
This can occur only when
a woman is susceptible at age x-c Zj ' then has a conception ending in fetal loss
over the interval (x-c 2j , x-c 2j + 6x).
Thus this second term on the R.H.S.
of (3.1.1) is
Pr[y.(x-cz·)=O/a(X-CZ·)=l, l3(x-c z ·)=l, J(x-cz·)=j]
J
J
J.
J
J
x
Pr[a conception ending in fetal loss over
•
(x-c ZJ·'X-C ZJ·+6X) IJ(x-cz··)=j]
J
.
Symbolically, this can be written
(3.1.3)
(3)
(x-C z.) [P.(x-C Z·)6x + 0(6x)]6.(x-C Z·) .
J
J
J
J
J
J
Q.
9
Combining (3.1.2) and (3.1.3) yields
QJ3)(X+~X)
= QJ3)(X)[1-Pj(X)~X]
+ Qf3)(X-C2j)[Pj(X-C2j)8j(X-C2j)~x] + o(~x).
Subtracting Q~3)(x) from both sides and then dividing through by ~x,
J
Q~3) (X+Ax)-Q~3)(x)
J
ax J
Letting
~x +
=
Qj3) (x)Pj(x) + Qj3) (x-C2j)Pj(x-C2j)8j (x-c 2j ) +
0, we obtain the differential equation in (3.1.iii).
may us~ initi~l conditions
o(~~)
Note we
~3)(Xo)= 1 and Qj3) (x O) = 0, j ~. 1 to determine
the solution to this system of differential equations.
We may now relate the quantities in 3.1 and 3.3 to the unconditional
fecundability mentioned in 2.8.
3.8.
R.(x)~x
J
parity is
over (x,
+
j.
x+~x)
o(6x) 'is the unconditional fecundabi1ity at age
x when the
This is the probability that a woman will conceive exactly once
given that she is at parity j
is the probability she will not conceive and
at age
o(Ax)
x.
1-Rj(X)~x
+
o(~x)
the probability of more than
one conception.
Lemma 3.2.
Proof:
R. (x) = p. (x)Q. (x) •
J
J
J
Rj(X)AX + o(ax)
Pj(x)ax + o(ax)
= Pr[exact1y one
= Pr[exact1y one
conceptionIJ(x)
conceptionIJ(x)
= j]
= j,
a(x)
Y (x) = 0] •
j
Q.(x)
J
= Pr[a(x)
= 1, Sex)
= 1,
y.(x) • O,J(x)
J
= j]
.
Therefore,
Rj (x) ax +
Dividing through by
0
(ax)
= [p j (x) ax
Ax and then letting AX
+
+
0 (~x)] x
0 yields
Qje.x) .
= 1,
Sex)
= 1,
10
which is the desired result.
We now relate the structure of fetal losses to the quantities Rj(x) and
ej(x).
Let
Z(x,y) be the number of consecutive fetal losses over the age
interval (x,y).
We may define
3.9.
h . . (x,y)
J,l.
= Pr[Z(x,y) = i,
= jIJ(x+g) = j]
J(y+g)
.
Note that hj,i(x,y) is the probability that a woman will experience.i
consecu-
tive fetal losses over the interval (x,y) and not yet experience the conception
leading to the (j+1)st live birth by age
ienced it by age x.
y given that she had not yet exper-
The quantity g is the length of the gestation period
which we assume constant.
It is somewhat more convenient to express results in
terms of the mathematically equivalent probability generating function.
co
3.10.
H.(x,y,s)
J
:
_
= I
h . . (x,y)s].
i=O ),].
Theorem 3.3:
If h . . (x,y) is a differentiable function of y, if RJ.(x) and
J,l.
e).(x) are continuous functions of x and if h . . (x,y) has the limiting properJ,l.
ties
(3.3.i)
h. O(x,x)
J,
=1
and h . .'(x,x)
J ,l.
=0
for i
~
then
Hj (x,y,s)
= exp[ JY{se j (t)-l}Rj (t)dt],
x < y, j
1,
~
O.
x
Proof:
We construct a system of differential-difference equations in the usual
manner.
(3.3.1)
and
h.
J,
O(x,y+~y)
= h.J, O(x,y)[l-R.(y)~y
+
)
o(6y)]
11
(3.3.2)
h . . (x,y+6y) = h. . (x,y)[l-R.(y)6y + 0(6y)]
),1
),1
)
+ h . . 1(X,y)[R.(y)6y + 0(6y)]a.(y)
),1-
+
J
).
0(6y) .
The logic behind the construction is relatively simple.
The R.H.S. of (3.3.1)
and the first term on the R.H.S. of (3.3.2) correspond to the situation where
a woman already has experienced
again over (y,y+6y).
i
fetal loss over (x,y) and does not conceive
The second term on the R.H.S. of (3.3.2) corresponds to
the casedthat·she has experienced i-I fetal losses over
(y,y+6y).
The o(6x)
(x~y)
and another over
corresponds to fewer than i-I fetal losses over (x,y)
and hence more than 1 fetal loss over (y,y+6y).
Subtracting h . . (x,y) from both sides of (3.3.1) and (3.3.2), dividing
),1
through by
'6y and passing to the limit as
6y
0, we obtain
+
(3.3.3)
and
ah . . (x,y)
),~y
(3.3.4)
= -R.(y)h . . (x,y) + e.(y)R.(y)h . . l(x,y)
J
J,l
J)
J,li
~
1 .
MUltiplying (3.3.4) by s i and then summing with (3.3.3) yields
I
ah . . (x,Y)
i=O
ay
00 .
(3.3.5)
J
co
si = -R.(y)
,1
J
I
•
h . . (X,y)Sl
i=O J ,1
co
~
l
+ se.(y)R.(y)
h . . l(x,y)s i-I •
J
J
i=l J,l00
Now since
•
Ih . . (x,y)si l ~ Isil , the infinite series I h. . (X,y)Sl converges
J,l
i=O J,l
uniformly in y for
lsi
<
1.
Hence the L.H.S. of (3.3.5) converges uniformly
12
converges uniformly in
Is I
Y for
co
a t..r
~-
h . . (x,y)s
oY i=O 3,1
<
1, so that
•
=I
ah . . (x,y)
co
1
. 0
1=
--"3"",,,,-:1~_
a. y
Combining (3.3.5), (3.3.6) and (3.10), we have
= R.(y)
[se.(y)-l]H.(x,y,s).
)
J
J
=1
With the initial condition, H.(x,x,s)
)
if the function
Cooke 1963, p.
(from 3.3.i), a unique solution exists
R.(y) {s6.(y)-1} is continuous for all
)
29)~
y
J
(see Bellman and
This solution is £ound, by separation of. variables,to be:
Hj(X,y,S) = exp[.JY{S$j(t)-l}Rj(t)dt] .
x
We now relate the £etal loss distribution to the parity structure in the
following two theorems.
p1,)
. . (x,y)
3.11
= Pr[J(y)=jIJ(x)=i]
.
.
That is, Pi,j(x,y) is the probability that a woman at parity i
will move to parity j
Theorem 3.4:
Proof:
by age
at age
y.
Under the hypothesis of Theorem 3.3,
y-g
Pj,j(x,y) = exp[J
f6j (t)-1}R j (t)dt], x <y, j
x-g
We first note that
x
H.(x,y,s) is analytic in
)
there is a unique analytic extension to
Is/ = I
s
on
Is/
~
O.
<
1.
Hence
and this is given by
co
H .. (x,y,l)
)
= I
h . . (x,y).
i=O J,1
Now the sequence of events, [Z(x,y)
= i]
form a
13
disjoint sequence of events so that
r h . . (x,y)
i=O J,1
~
~
= Pr[{ u [Z(x,y)=i]} n [J(y+g)=j] IJ(x+g)=j]
i=O
= Pr[J(y+g)=jIJ(x+g)=j]
That is,
H.(x,y,1) = Pr[J(y+g)=jIJ(x+g)=j] .
(3.4.1)
J
~.~-'f. -~
~,
By definition then
p. . (x,y) = H.(x-g, y-g, 1) ,
J,J
J
so that by Theorem 3.3
•
y-g
P. . (x,y) = exp[
J,J .
Theorem 3.5:
f
x-g
.
{e.(t)-1}R.(t)dt]
J
J
"
Under the conditions of Theorem 3.3,
x+l-g
exp[
f
{e i (t)-1}R i (t)dt]
x-g
Pi,j(x,x+1) =
x+1-g
l-q(x,x+l)
exp[
J
for j = i
_
{Si(t)-l}Ri(t)di] for j = i+l
x-g
o
otherwise.
where q(x,x+1) is the probability that a woman who is alive at the beginning
of month x ,- will-die during the month (x,x+l).
Proof:
Let D be the event that a woman alive at age
Let A be the event that a woman alive at age
x+1.
x
x dies by age
x+l.
is still alive at age
14
Let
age
P.
be the event that a woman at age
1
x is still at parity
i
at
x+l.
Let
P. 1 be the event that
a woman at parity i
1+
at age
x is at parity
i+l at age x+l.
D u {AnPi } U {AnP i +1 } has probability one.
By assumption 2.22 the event
Hence
=1
Pr(AnP + )
i 1
By thE' previous
- Pr(D) - Pr(AnP i ).·
theorem
x+1-g
Pr (Anpi) = exp [
J
{e i (t) -l,lR i (t)dt]
x-g
We shall make use of this result in our discussion in section 5, where age.
increases are considered to occur in discrete increments.
4.
The Truncation Effect.
The non-negative quantity, {l-e.(x)}R.(x),
J
J
force of a j-th live birth.
be given the interpretation as the
can
It is clear
that any realistic model of fertility would require this force to approach 0
as the age
x became large.
The rate at which this quantity approaches 0
becomes critical as the following theorem demonstrates.
Theorem 4.1:
Under the assumptions of Theorem 3.3 if
00
f {l-e.(t)}R.(t)dt =
x
then the, parity
J
00
J
o
j" is' a transient state.
Moreover if
f
X
then a woman achieving parity j
at age
00
probability exp[l' {e. (t)-l}R.(t)dt].
x. J
J
J
x.
J
00
{e. (t)-lJR. (t)dt
o
J
J
wi 11 remain in parity j
with
<
00
'
15
will be transient if p . . (x. ,(0) = o.
Since' {I-e. (t)}R. (t)
J,J J
J
J
is a continuous function, over any finite interval, it is bounded. Hence
Proof:
f
Parity j
x.
00
Ju-e.(t)}R.(t)dt < 00 implying f {e. (t)-llR.(t)dt = -00
That is
X
J
J
x J
J
o
j
p . . (x. ,(0) = O. The probability of remaining in parity j is p . . (x. ,(0) =
J ,J
J
J ,J
J
exp[! {e. (t)-l}R.(t)dt].
. x. J
J
J
Notice, in particular, that if the force of a j-th birth is bounded below
.....
by a constant, as for example with models based in renewal theory, the parity
j is transient.
In general, a good procedure will be to construct our model so that the
force of a j-th live birth is 0 for all ages greater than some fixed age, say
w, which is then taken to be the .end of the effective reproductive~eriod.
Theorem 4.2 gives a measure of disparity due to this "truncation effect."
Theorem 4.2:
Y + denote the age at. which conception ending in the (j+l)st
j l
live birth occurs. Then
4.2.i
Let
Assuming an infinite reproductive period,
Pr[Y. I >YIJ(x+g) = j] = H.(x,y,l)
J+
J
4.2.ii
Assuming a finite reproductive period and considering only those women
who will have at least a (j+l)st live birth before the end of that period,
Pr[Y. l>yIJ(x+g) ~ j+l, J(x+g) = j] =
J+
H.(x,y,l)[l-H.(y,w,l)]
J
J
I-Hj(x,w,l)
x<y<w.
Noticing that the events [Yo l>y] and [J(y+g)=j]
J+
have by (3.4.1)
Proof:
are equivalent, we
16
Pr[Y. l>yIJ(x+g)=j] = Pr[J(y+g)=jIJ(x+g)=j] = H.(x,y,l) •
J
J+
Consider now 4.2.ii.
Let
Pr[Y. l>yIJ(w+g)~j+l,J(x+g)=j] = p, say.
J+
p =
Pr[J(y+g)=jIJ(w+g)~j+l,
Then
J(x+g)=j] .
By the definition of conditional probability,
P
= Pr(J(y+g)=j, J(w+g)~j+lIJ(x+g)=J]
P[J(w+g)~j+lIJ(x+g)=j]
By Bayes' theorem,
Since
Pr(AIB) = 1 - pr(AcIB),
Since parity is known to be j at x+g and less than j+l at w+g, it must be
at w+g also.
Thus
H.(x,y,l) {l-Pr[J(w+g)=jIJ(y+g)=j, J(x+g)=j]}
p = J
1 _ H.(x,w,l)
J
Now consider
Pr[J(w+g)=jIJ(y+g)=j, J(x+g)=j]
But
J(w+g)=j
and J(x+g)=j implies
J(y+g)=j, x
<
Y
<
w.
j
17
Pr[J(w+g)=j IJ(y+g)=j, J(x+y)=j]
_ RtiJ(w+g)=jIJ(x+g)=jt _ Hj (x,w,l)
- Pr[J(y+g)=j jJ(x+g)=j
- H. (x,y,l)
J
Now since Hj (x,y,l) = exp[
JY{e j (t)-l}R j (t)dt]
x
Hj(X,w,l)
H.(x,y,l)
w
= exp[
J
y
J{e j (tJ-l}Rj (t)dt - J {e j (t)-I}Rj (t)dt]
x
·x :
w
= exp[
J {e j (t)-I}Rj (t)dtJ = Hj (y,w,l).
Y
This completes the proof of Theorem 4.2.
I-H. (y,w,l)
The implication of Theorem 4.2 is then that the ratio I-H:ex,W,I)
is an
J
adjustment factor for converting the results of a model with no upper bound on
the effective reproductive period to one with such a bound •
5.
Age-Parity Distribution in Time-Age
section is
~1arriage
Cohorts.
TIle objective of this
to use the theory developed in Section 3 to derive the moments of
the- age parity distribution in a time-age marriage cohort.
To do this we will
introduce explicit notation indicating the dependence of p . . (x,y) on the
1,]
events [Xo = xO] and [T = T]. We shall use the notation, Pi,jex,y" IXO,T), but
we shall feel free to supress the XO,T condition in proofs where explicit
notation is not used.
The method suggested for the derivation of the moments of the age-parity
distribution consists of dividing the reproductive period into a finite number
of equally spaced age points and then developing a recurrence relation between
18
the parity distribution at any two successive age points.
The division of
the reproductive period into a finite number of age points is a rather subjective.
matter and depends on the purpose of the study.
Here the reproductive period is
divided into monthly intervals which is appropriate since the reproductive process
has a cycle with a monthly nature.
We now introduce
several.nece~sary random
variables.
nj(xlxo.~ is the random number of women with parity
5.1
j
at the begin-
ning of mohth x out of all tho'se who marry at X and at time T.
o
·"S:Z·'rii,j(xlxo,T)' is the ·random number
women married at age Xo at time
T who move from parity i to parity jover the month (x,x+l).
at
In both 5.1 and 5.2 we shall suppress the
XO,T condition in proofs
not explicitly requiring that notation.
Lemma 5.1:
and
n.(x+1I xo ,T) = n. 1 .(XIXO,T) + n . . (X/XO,T)
J
Proof:
J~
,)
),J
Immediate from Definitions 5.1 and 5.2 and Assumption 2,22 that mUltiple
births increase parity by one.
Lemma 5.2:
S.2.i
E{n . . (XIXO,T)}= p. . (x,x+1I x ,T)E{n. (XIXO,T)} •.
1,)
1,)
o
1
(Note that under Assumption 2.22, this is zero unless j = i or i+l.)
5.2.ii
. (XIXO,T); noN, k(xlxOT)} =
.. Cov{n.
1,J
{O'k
. . (x,x+1I xo,T)-p 1,)
. . (x.x+llxo,T)p.1., k(x,x+llxo,t)}
J P1,)
E{n i (XIXO,T)}
(Note that this is zero unless j=i or i+l and k=R, or R,+l.)
where
<5
Proof:
jk
={
1 j
19
=k
0 j ,. k
Two facts are used in this proof:
5.2.1
= E{E(XjY)}
E(X)
and
5.2.2 Given n.(x) and assuming n. d(x) is the number of women who die
1
1,
over (x,x+1), the joint distribution of the random variables
; ... ,~i.,i (x).' ni,i+l (x) and ni,d(x) is trinomial with probabilities
p 1,1
. . (x,x+1), p 1,1+
. . 1 (x,x+1) and l-p 1,1
. . (x,x+l) - p.
. 1 (x,x+1).
1,1+
To prove S.2.i, by the properties of the trinomial distribution
E{n . . (x)
1,J
In.1 (x)}
= n. (x)p . . (x,x+l) .
1
1,J
Taking expectations on both sides,
E{n . . (x)}
1,J
= p 1,J
. . (x,x+I)E{n.(x)}
1
For 5.2.ii and when i ,. t, it similarly follows that
E{n 1,J
. . (x)nnN, k(x) In.1 (x) ,nnN (x)}
= n.1 (x)p.1,J. (x,x+l)nn (x)Pn
N
N,
k(x,x+l).
Thus
E{n 1,J
. . (x),nnN, k(X)1
= p.1,J. (x,x+l)Pn
k(x,x+1)E{n.(x)nn(x)}
1
N
Cov{n 1,J
. . (x),nnN, k(X)}
= p.1,J. (x,x+I)Pn
k(x,x+l)Cov{n.(x),nn(x)}.
1
N
so that
l\1hen
i
= ~,
N,
N,
the joint probability distribution of n . . (x) and n. k(x) condition-
ed on n.(x)
is given by
1
1,J
.1,
20
fen 1,J
. . (x),n.1, k(x)ln.(x))
1
=
n.1 (x)!
fi.. (
In. . (X) ~
1 )~n.1, k(X)
I
I
·
I
p.1,J··
. x,x+ 1) . 1,J
p.1,k (x,x+
n1,J
. . (x).n.
k(x).{n.(x)-n
.
.
(x)-n.
k}.
.
1,
1
1,J
1,
-
[
n. (x)-n: . (x)-n. k (X)
I
1-p. . (x,x+1)-p. k(x,X+1)
1,J
1,
1
1,J
1,
It then follows that
Cov{n..1,). (x) ,n.1, k(X) In.1 (x)}
=
. . (X,X+1). p.1, k(X,X.+1)
-no1 (x)p1,J
j" k
. . (x,x+1)U-p . . (x,x+1)} j = k··
{ n.1 (x)p 1,J
1,J
Using
we can rewrite as
~jk
Covin1,J
. . (x),n.1, k(x)\n.(x)}
1
=
{O.kP. . (x,x+1) - p . . (x,x+1)p. k(x,x+1)}n.(x).
J
1,J
1,J
1,.
1
Taking expectations on both sides furnishes the required result.
Theorem 5.3:
(Expected Age-Parity Distribution)
5.3.i
and
5.3.ii
E[n.(x+1 \xo,T)IN T ]
x o.
J
= p.J-,J
1 . (x,X+1/X O,T)E{n. l(xlxo ,T)}
.
J-
+
p . . (x,x+1Ixo,T)E{n.(x/xO,T)},
j ~ 1
J,J
J
with the initial conditions
T
E[n.(xO/xo,T)IN
]
J
X
o
NT
Xo
j = O·
0
j ,. 0
=
X
where
NT
time
T.
X
Proof:
21
Xo at calendar
The proof follows by taking expectations on both sides of the 'expres-
sions in
l~
o
is the initial size of a cohort marrying at age
Lemm~
5. 1 and· then appl ying 5.2. L
Corollary 5.4:
5.4.i
and
5.4.ii
j
~
1
with the initial condition
j
=0
j ;. 0
~
where
Pj(X/XO,T) is the probability that a woman marrying at age
T will have parity j
E{nj(x/xo,T)}
Proof:
with
= E{N~
at age
Xo at time
x and is defined by the relationship:
}Pj(XIXO,T).
o
The proof follows from Theorem 5.3 by replacing E{n.(x/xo,T)/NT }
J
o
NT p.(XIXO,T) and then taking expectations over NT •
X
oJ
Corollary 5.5:
X
Let X.
J
o
be the age at delivery of the j-th live birth, then
Pr{x<x.<X+l/X O}
J
Proof:
X
= p.J- l(xlxo,T)p.J- 1 ,J. (x,x+l/XO,T).
Obvious.
We note here that the probabilities p.{X/XO,T) and p. 1 .(x,x+l/xO,T)
J
J- ,3
can be evaluated with and without the truncation effect mentioned in Section 4.
Consequently the discrete approximation of the probability distribution of Xj'
derived in Corollary 5.5 can be calculated with and without this effect, hence
also the moments of X.•
J
22
Theorem 5.6:
(Covariance Matrix of Age-Parity Distribution).
cov{nj(x+IIXo,T), nk(x+llxo,T)} =
cov{nj_l,j(x1xo,T), nk_l,k(xlxo,T)} + cov{nj_l,j(x1xoT), nk,k(xlxo,T)} +
Proof: _._P'!?V!Pll..s!. '.
With respect to initial conditions, the recurrence relations displayed in
Theorems 5.3 and 5.6 can be completely solved.
The same procedure can be used
to find the higher moments of the age-parity distribution, if required.
6.
Age-Parity Distribution in Time Marriage
Cohorts~
A time marriage cohort
is a weighted sum of time-age marriage cohorts where the weights are derived
from the probability distribution of age at marriage at the corresponding point
in time.
The results of Section 5 can be used to derive the moments of the
parity distribution in time marriage cohorts.
is done under the assumptions of 5.
n. (x+ 11 T)
6.1
J
where
The derivation of these moments
We give the following definition.
=
x
In. (x+ 11 xo ' T)
x =1 J
o
n.(x+lIT) is the random number of women with parity j
J
of month
x+l
out of those who marry at calendar time
at the beginning
T.
Lemma 6.1:
Assuming that the Markov property holds for the random variables
Xj' j ~ 0,
Pi,j(x,X+1Ixo,T) is independent of xO•
23
4It
Proof:
Pi,j(x,x+l/XO,T) = P[X j +1 > x+I/Xi +1 > x, Xo = xo ' T = T]
= P[X.J+ I > x+l/X.~+ I > x, T =T]
by the Markov assumption.
Theorem 6.2: (Expected Age-Parity Distribution).
Under the Markov assumption,
6.2.i
and
E{n. (x+I/ Tn = p. I . (x,x+I/ T)E{n. 1 (x h)}
J
J- ,)
J-
6.2.ii
j ~ 1,
+ p. '. (x,x+I/T)E{n. (X/T))
J,J
J
where
p . . (x,x+IIT) is the probability for a woman marrying at calendar time T
~,J
to move from parity i
parity is
i
at
x.
to parity j
over the month (x,x+l) given that the
According to Lemma 6.1, this probability is independent
of the age at marriage xo ' and
MT(x) is the expected number of women in the age interval (x,x+l) at
calendar time
Proof:
T out of all those women who marry at time
T.
Definition 6.1 can be rewritten as
x-I
n.(x+1/T) =
n.(x+1Ixo,T) + n.(x+1Ix,T) •
J
x =1 J
J
r
o
Taking expectations on both sides,
E{n.(x+1IT)}
J
=
x-I
r
x =1
o
E{n.(x+llxo,T)} + E{n.(x+llx,T)}
J
J
24
Applying Theorem 5.3,
and
x-I
E{n. (x+11-r)}
J
= L
xO=1
[Pj_l,j(x~x+1Ixo,-r)E{nJ'_I(xlxo,-r)}
+ p . . (x,x+llxo,-r)E{n.(xlxo,-r)}]
J,J
J
+ E{n.(x+1Ix,-r)},
J
j ~ I .
But the quantity E{n. (x+llx,T)} is the expected number of women with parity j
J
at the beginning of month x+l out of those who marry over the age month
(x,x+l) at calendar time
The nature of the fertility process implies this
T.
quantity is zero for j ,. O.
For j = 0, this quantity is the expected number of
women marrying over the age month (x,x+l)at calendar time
T.
That is,
E{n.(x+llx,-r)}
J
=
MTO(X)
{
j
=0
j ,. 0
We also recall from Lemma 6.1 that the probabilities p 1,J
. . (x,x+1I xo ,T) are
independent of Xo and so can be written as p 1,)
. . (x,X+1IT).
Thus
and
E{n.(x+IIT)}
J
x-I
.(x,x+IIT)
E{n ·_ 1 (xlxo ,T)} +
J
J- ,J
x =1
= p.
1
r
o
x-I
p . . (x,x+IIT)
J,J
r
x =1
o
E{nJ.(xlxo,T)}
j ~ 1.
25
~
With appropriate initial conditions, the system of equations, 6.2.i and
6.2.ii, can be completely solved.
The higher moments of n.(xIT) can also be
J
derived in a similar manner.
7. Acknowledgement. We thank the late Mindel Sheps who was most instrumental
in drawing our attention to this problem.
8.
References.
[1]
Bellman, R. and Cooke, K., (1963), Differentia'l-Differenae Equations"
Academic Press, New York.
.
[2]
Gini, Corrado, (1924), "Premi~res recherches sur la f~condabilit~ de la
femme," Froc. Internationa'l Mathematics Cong:Pess" VoZ. II" 889-892.
[3]
Henry, L., (1953), "Fondements th~oretiques des measures de la f~condit~
naturelle, Ii Revue de 'l'Institute IntemationaZe de Statistique" 21"
135-151.
[4]
Perrin, E.B. and Sheps, M.C., (1964), "Human reproduction: a stochastic
process," Biometrics" 20, 28-45.
[5]
Singh, S.N., (1963), "Probability models for the variation in the number
of births per couple," J. Am. Statist. Asso<;?" 58, 721-727.
[6]
Singh, S.N., (1964), "A probability model for couple fertility,"
(B)" 26" 33-40.
[7]
Venkatacharya, K., (1969), "Certain implications of short marital durations
in the analysis of live birth intervals," Sankhya (B), 31, 53-68.
Sankh'[Ja