lThe research was supported primarily by National Institute of
Child Health and Human Develqm:mt Grants number 5-ROl-IID072l4
and l-ROl-IID09028.
2 DepartJrent
of Biostatistics, School of Public Health.
3DepartJrent of Biostatistics, SChool of Public Health and Curriculun
in Operatict1s Research and System Analysis.
4DepartJrent of Statistics.
THE COMPARISON OF POST-ABORTUM AND POST-PARTUM
TIME-TO-DELIVERY USING A DATA-BASED MARKOV CHAIN l
by
Carol J. Hogue2, Richard H. Shachtman 3
John R. SChoenfelder 4
University of North Carolina at Chapel Hill
Institute of Statistics Mineo Series No. 1034
November 1975
THE C'CMPARISCN OF POST-AOORTUM AND POST-PARIUM TIME-'ID-DELIVERY
USING A DATA-BASED W\RKOV CHAm l
Carol J. Hogue2, Richard H. Shachtm:m 3 ,
John R. Schoenfelder 4
1.
INTroIXCI'ION
Between rrenarche and nenopause, \\OITeIl' S reproductive histories consist
of periods of pregnancy, pregnancy outcomes--births, fetal deaths, induced
abortions, etc. --and inter-conoeptional non-pregnant intervals of varying
length, depending upcn sexual activity, COI1traceptive practioe, and fectmdity.
Eeproductive histories may thus be COIlpletely described within a finite set
of states which includes all pregnant and non-pregnant periods as well as
pregnancy outcares.
To consider individual or group rrovenent fran state
to state as a stochastic process, we observe that at any given t.ine a wanan
will be found in one of the defined states, and her novemen.ts to other states
are governed by a probabilistic law.
state
(i)
to a state
(j)
When the probability of a transition fran
at sooe ti.ne
(n)
depends en
n,
the process
is said to be t.ine-dependent.
The class of stochastic processes known as Markov Chains (M.e.) reflects
tOO state-to-state rrovenent and ti.ne-dependency of reproductive histories
while allowing sate mathematical si.nplifications.
It will be an appropriate
nodel if the state definitions are so structured that a visit to any state
"carries" sufficient infornation about the prior history to ascertain the
transitien probability for any irmediately succeeding state (the Markov property).
lThe research was supported primarily by National Institute of Child Health
and Hunan Developrent Grants nunber 5-ROI-HD07214 and I-ROI-HD09028.
2Depart:m:mt of Biostatistics, School of Public Health
3Departrrent of Biostatistics, School of Public Health and curriculum in
Operations Research and Systems Analysis.
4[)epart:m:mt of Statistics
-2-
Since many reproductive events are dependent on predecessor events, the chain
rrnJSt be structured in such a way that a
(j)
\\OlEl1
going fran state
(i)
to state
has a comron reproductive history, in a probabilistic sense, with any
other
\\UIIlaI1
going fran
(i)
to
(j).
This structuring also rrnJSt take into con-
sideration the need for hcnogeneity of cohorts.
It is further assured that
the process is stationary, i. e., over a SPeCified Period of ti..Ire, the probability
of a transition fran state
(i)
to state
(j)
is not dependent on the length
of ti..Ire from the beginning of the process to the achievenent of state
(i).
It is well knCMn that by using a sufficiently large set of states we may achieve
a M.C. approximation to the underlying stochastic process.
The procedure is
to create a M.C. rrodel of sufficiently high order to reflect any state to state
dependence in the process.
Then we define a second, equivalent M.C. IOOdel of
low, preferably first, order but necessarily having many states to account
for all FOssible transitions in the first M.C. (6).
a IOOdel which is unrealistic in tenns of data
and sheer complexity of canputation.
This procedure generates
availability, interpretation
Hence we rrnJSt find a state set definition
for which a Markov property of low order is a reasonable statistical representation
for the flow of wanen.
Markov Chains have been utilized in mathematical derrography by Chiang (2)
to describe human fertility, by Perrin and Sheps (10) to IOOdel pregnancy
and FOstpartum periods and by Sheps and Perrin (13) and Sheps and Menken (12)
to IOOdel birth rates as a function of fecundability, contraceptive effectiveness
and other parameters.
Sheps and ~ (12) discussed the potential for M.C.
m::xieling as well as other mathematical IOOdels of conception, birth, pregnancy
...
wastage and interrelated variables.
The present paPer introduces a M.C. IOOdel which was originially designed
for inferential study of the effects of induced abortion on subsequent pregnancy
Qutcares. The m:x1el and its data base are described in Section 2.
A discussion
-3-
of testing the fit of the nodel may be found in section 3, and section 4
c:x:mtains a discussion of an application of the nodel canparing
tirre-to~livery
after an initial pregnancy is terminated by induced abortion to ti..ne-to-nextdelivery following a first pregnancy delivery.
This application has inq:>lications
for determining sarrple size in an epidemiological stooy of pregnancy outcc.me
following indu:::ed abortions.
-4-
2.
THE mDEL AND DATA BASE
During the course of an investigation of incidence of premature
(low-birth weight)
deliveries in pregnancies following induced abortion
canpared with suspected premature deliveries in non-aborting wcm:m, a M.C.
m:xlel was developed by "b.o of the authors (II) to investigate related t:iIredependent research questions.
'!his m:xlel, its structural justification
and developnent have been described in detail in (II) and will only be briefly
presented here.
'!he overall m:xlel is a 79-state, one-m::>nth discrete tine
interval Markov Chain with transition probabilities which are assurred to be
stationary over a limited time period.
States were defined subsequent to
data gathering and processing but do not reflect potential m:xlifications as
a result of testing (see Section 3).
'!he state set exhaustively defines a
•
reproductive path fran the onset of m:marche through the fecundable period
in three groups:
Gl--prior to the first pregnancy outcare, G --following a first
2
pregnancy but prior to the first delivery; and G --follawing the first delivery.
3
Within each group, states can be placed into three broad categories--susceptibility (to pregnancy), pregnancy, and post-event.
By post-event, we rrean
the nonth in which a pregnancy tennination occurred, since births and other
events are relatively instantaneous.
'!his convention was required to translate
the occurrences into events for the constant interval transition probabilities.
A typical ne"b.ork for particular subclasses is diagrarrm:rl in Figure I.
FIGURE I
e
"
-5-
•
\
•
\
\
\
\
\
\
\
\
<::>
denotes single state
c=J
denotes colection of states
FIGURE 1: TYPICAL NET\o!ORK FnR SUBCLASSES OF THE MARKOV CHAIN
-6-
In addition to the possible transitions shown in Figure 1, self-loops nay
be nade for the states, or collections, rsv and rsi but not for the states
si'
Ilk'
ia, sa, pd,
ro.
Susceptibility:
= Early Period of susceptibility, first eight nonths,
possibly following sene other reproductive event
= Ninth
rronth of susceptibility, prior to pregnancy
or rsi
rsv
= Collection
rsi
= Involuntary
of states which are defined by various
types of c6ntraceptive usage (reduced susceptibility,
voluntary)
reduced susceptibility (low fecundity)
Pregnancy:
= First
= Last
six rronths of a pregnancy
three rronths of a pregnancy
Post-event:
ia
= Induced
sa
= Spontaneous
m:1
= Mature
abortion
abortion
delivery
= Prerrature
delivery
Data observations were translated fran event codes, Appendix 1, into
state definitions, Appendix 2, which are shown in a network flow diagram,
ApPendix 3.
The full 79 x 79 transition matrix has the following block fonn:
G3
I
0
I
0
------r-----~---
H23
II
G2
I
,
0
------t-------~---I
I
H13 I H12 I Gl
-7-
~
= transitions
for individuals within group
H
jk
= transitions
for individuals from group
States in
and ergodic.
G
l
and G
2
k; k
j
= 1,2,3.
to group k; j
are transient and those in
There could be sane wanen who stay in
rsi
G
3
2
= 1,2,
= 2,3.
k
are Persistent
or rsi3 or in the
contracepting states over the full tine horizon due to infecundity or contraceptive effectiveness but, in general, these states are not absorbing.
The transition ma.trix is in canonical fonn with transitions for the 27 ergodic
states in the UpPer left and the two groups of 27 and 25 transient states in
the middle, and lower right.
are through
23 '
C
to G
2
ia2'~'
p\;
Since all transitions in the
k
= 1,2;
jk
sul:matrices
these are adjoined as one additional colunn,
and two additional columns, Cl ,23'
fonn of the transition ma.trix, Appendix 4.
follCMs in the Appendix in the order:
H
[G3 l,
to
Gl
in the partitioned
The resulting three part partition
[C
23 ; G2l,
[C ,23;
l
GIl.
The testing and results given in this paper refer to the subclasses of
the ma.trix denoted G
3
the initial delivery
delivery
and
(G )
3
G which represent the groups of states after
2
and after the initial abortion which precedes a
(G ).
2
As previously rrentioned, this rrodel was develoPed for inferential
rather than descriptive purPOses.
For a canplete discussion, see (11).
a data-based rather than a simulation rrodel, states
operationally with a definite population in mind.
~re
Since it is
necessarily defined
As a result, the scope
and size of the rrodel are considerably larger than rrost previous M.C. rrodels
in rrathanatica1 derrography.
This size increases the flexibility of the rrodel
for answering diverse darographic and epidemiologic questions, but it also
rrakes interpretation of the results rrore ccmplex.
Furtherrrore the rrodel requires
a rather extensive data base for estimating the nurerous Pararreters.
-8-
Our data base is fran an historical prospective study of Yugoslavian
wanen, residing in Skopje, Macedonia, whose first pregnancy was tenninated
by induced abortion
Both
~re
(N
=
217)
or delivery
(N
= 711)
during 1968-1969 (3, 4).
cohorts carre from an harogeneous ethnic and residential background and
similar with respect to location of the tennination of their first pregnancy.
Interviews of these wanen conducted in the fall and winter of 1972 obtained
a full reproductive history for each wanan to that date.
Birth
~ights
deliveries were verified as much as possible through hospital records.
regarding reproductive history were therefore tiIre-censored.
of
Data
For each wanan,
the data were coded so that a chronological path of relevant events for each
subject could be obtained, thus compiling the optimal tyPe of observation
for a stochastic process analysis (9).
Not all potentially positive transitions were observed in this data base.
The ratio of observed to expected positive transitions (denoted
Table 1) ranged fran 0.640 in
G
l
to Q.885 in
r/r
in
G .
3
Cell frequencies are given in categorical fonn in ApPendix 4.
In general,
frequencies were adequate for the est.i.rration of transitions to pregnancies
and post-events but were marginal for some of the particular contraceptive
states.
These states are essentially aggregated over tiIre in the tiIre-to-delivery
application discussed in section 4.
The extent to which this data-based IOOdel
satisfies the Markov assumption will be discussed in the next Section.
-9-
3.
TESTING THE FIT
Before this Markov Chain, or any stochastic
rrode~
may be used for
statistical inference it should first be sul:mitted to several goodness-of-fit
tests.
The purpose of such tests is to insure that the nodel does, indeed,
depict reality.
In particular it must be verified that the assurrptions made
in formulating the rrodel are reasonable
Fbr this nodel
tv.Q
in light of the available data.
crucial assurrptions, the basic Markov assurrption and a
stationarity assurrption, soould be checked before the data-based estimates
are used to answer specific research questions.
A.
The Markov Property
The basic Markov assurrption which was made is that of first-order de-
p:mdence, i. e., giVeIl the present, a wanan 1 s future rrovement in the chain is
stochastically independent of her
past history.
Assuning that the chain is
stationary this may be expressed as:
P {Xn
= j n Ixn-1 = j n-l'
where
jo' jl,···jn
Xn- 2
= j n- 2'···'X= jO} = P
-1)
{Xn
= j n Ix
'n-l
="
In-l
are states of the chain.
A slightly rrore general assurrption would have been that of secnnd-order
dependence; mder this assurrption a woman I s next transition is detennined,
probabilistically, fran both her current state and the i..Ime:liately preceding
one, i.e.,
P {Xn --
J'
n
Ixn-l --
J"
n-l' Xn-2 --
J'
n-2'··· X0 --
J'
0}
=P
{Xn
= J" n Ixn-l = J' n-l'
X
="
}
n-2
I n- 2 ·
It is against this assunption of second-order dependence that the assurrption
of first-order dependence will be tested.
Thus, the null hypothesis is that
first-order dependence is equivalent to second-order dependence.
If this
}
-10-
hypothesis is accepted we may then have confidence in using the chain for
statistical inference as the state definitions i.rrply that higher order dependence will be absent.
For example, in
G ,:rrovement to a particular
2
amtracepting state is assured to depend only on the current state (post-event,
susceptibility, or reduced susceptibility) and not on the previous history of
contraceptive choice.
that the
\\UITBI1
(The strocture of the rrodel carries the information
has had at least one abortion but no delivery.)
Consider a stationary Markov Chain with states l,•.. ,m and let
i, j, k E {l, ... m}.
n = klxn-1 = j},
P"k
= P {X
J
The mathematical expression of the null hypothesis becares :
H :
O
for all
~
1 < j
i
m,
1 < k < m
= l, ... m.
Let
C
= set
of cell indices in the second order chain
(i.e. the
m3 triplets
i,j,k).
oc = observed
frequency in cell c,
c
E
C
= expected
frequency in cell c,
c
E
C
E
C
'!hen the t\\O statistics which will be used to test this hypothesis are the
ordinary chi-square statistic:
(0
and
c
- E )2
c
-11-
a statistic suggested by Kullback, Kuppennan, and Ku (8) which is based on the
concept of a minimum discrimination infonnation statistic °
this latter concept is given by Kullback (7)
frequency of transitions fran state
(i)
Denote by
°
to state
(j)
A discussion of
f. 'k
1)
the observed
to state
(k);
it
rray be shown that these statistics are expressable as:
2
2
m
m
m
(e.1)'k - f.1)'k)
I
Xo = 1.
i=l
2
XK
Y.
j=l
=2
k=l
e ijk
m
m
m
I
I
I
i=l
j=l
f ijk
f"
1)k
k=l
in. e
ijk
m
I
f ... =
1)
fO jk
=
f
=
0.0
)
f 1')'k
k=l
m
I
f. 'k
i=l
1)
m
m
i=l
k=l
\'L
f.1)'k
\'
L
An equivalent ca:nputational form for
2
(1/2) X
K
m
1.
=
i=l
m
m
I
I
i=l
j=l
m
1.
j=l
2
X
K
to reduce rounding error is
m
m
f"
in. f"
+
1) k
1) k
1.
k=l
f 1)
.. ° in f ..
0
1)
-
2 in.
Using the approxirration
m
I
j=l
I
f ... in. f
j=l
)
° •°
)
m
I
fO
k=l
0c
E
jk
in f. jk
°~2E_ Ec2 , see
=
(8), it may be shown
c
c c
that these statistics are asymptotically equivalent and that both have a
limiting distribution which is chi -square with
17
degrees of freedano
The
basic fonmlia for the degrees of freedan, assuning that all transitions i
are possible, is given by
17
= m(m - 1) 2 °
If, however, there are
c
t
~
j
~
expected
k
-12-
zero cells this must be adjusted to
r
= m(m - 1)2 - Ct' Letting Pt
number of theoretically positive cells it may be shown that
c
t
= m(m
be the
- 1)2 - Pt'
'fuus the expression for the degrees of freedan reduces to
r
= m(m
- 1)2 - c
= m(m
- 1)
2
t
- [m(m - 1)
2
- Pt]
=Pt
Some theoretically positive cells have small transition probabilities and
thus may not be observed with data fran a sarrple.
will be
Co
In these instances there
observed zero cells which are theoretically positive.
reduce the degrees of freedan to reflect these
r
= pt
- c
Co
We may
cells to
0
= po
where
Po
is the nurrber of observed positive cells.
This adjusted paraneter
is chosen since it is not only roore conservative but also canputationally
necessary when using the Kullback, Kuppenna.n and Ku statistic.
B.
Stationarity
It should be noted that the assumption of stationarity was irrplicit in the
previous testing problem as well as in the estimation
of the transition matrix.
Although this assunption has not yet been fonnally tested future plans do
include such a test to be based on the likelihood ratio statistic suggested
by Anderson and Goodman (1).
Based on pre-data considerations, however, it
appears to be quite reasonable to assure that this property holds over a short
ti:rre period, say three to five years. 'fuese considerations include such factors
e
-13-
as the average (typical) age span of the waren sanpled and the actual fonnulation
of the state definitions and choice of tine interval; for further details on
this subject see (11).
C.
Numerical Results
For the purpose of statistical inference concenling the tima-to-delivery
questiOIt the original 79-state chain has been divided into three subchains
\\hich, due to their structure, may be tested separately.
These subchains
correspond to the partition of the transition matrix in Appendix 4, which we
call groups I, II, and III.
For our purposes, we need the maxiIm.m likelihood
estimators of the transition probabilities for Group III only for waren wh::>
delivered their first pregnancy.
(The original estimators include post-delivery
paths for wanen who al:x>rted their first pregnancy but who Ultimately had a
delivery.)
Call this group IIID and note that all transition probabilities
enployed below are for group IIID, not group III.
'Ib date tests have been
ccmpleted on two of these subchains:
(a)
Group II:
(b)
Group IIID:
The post-al:x>rtion pre-delivery subchain of first-pregnancy
al:x>rters.
The post-delivery subchain of first-pregnancy deliverers.
Due to the fact that states
~
and m1
2
in the group II subchain were
nade artificially absorbing a slight nodification of the degrees of freedan
is necessary for the test of fit.
Eight of the observed cells in this mxlified
chain are actually irrpossible cells in the real chain, e. g • ,
Hence the actual degrees of freedom will be r
= PO
- 8.
~P2 + ~ + ~.
Since no such, mxlification
is necessary for testing the other subchain, the degrees of freedom for
that test will be merely r
= PO'
Results of the actual canputation are given in Table II.
-14-
TABLE II
Using :the nomal approxination to the chi-square distribution we obtain the
following approxirrate p-values for the tests:
Group II:
2
p
hK
(138) >
90.62}
>
.999
P
h O2
(138) >
-
259.52}
<
.001
p
h K2
(257) >
249.62}
=
.618
p
h o2
(257) >
2038.69}
<
.001
Group IIID:
-
Clearly if we base the test on the Kullback, Kuppennan, and Ku statistic we
nay
accept the equivalence of first and second order dePendence for both
subchains.
If we base the test on the ordinary chi-square, however, we appear
to be forced to reject that same hypothesis.
A cell by cell examination of the contributions to the respective statistics
explains this discrepancy.
Consider the test in group II.
Out of the 138
cells with p:>sitive observed frequencies, 6 of them (4.35%) oontribute 200.94
(77.07%) of the total ordinary chi-square.
Each of these cells had a p:>sitive
observed frequency but, based on this data, the expected frequency was much
less than one; hence, the very large contribution.
These same cells also made
large contributions to the Kullback, Kupperrnan, and Ku statistic;
the use of
the natural logarithm function, however, acted to "snooth" these contributions
to a magnittrle rrore C<JITlre11surate with those fran the other cells.
we
can,
therefore, draw two obvious conclusions:
(1)
The Kullback, Kupperm3Il, and Ku statistic tends to be rrore robust
than the ordinary chi-square with respect to the infrequent occurrence
of highly inprobable transitions, and
e
-15-
we
(2)
may, for this subchain, accept the hyPOthesis of first order
dependence for all, except possibly 6, of the cells.
Further investigation reveals that all of these 6 troublesare cells involve
contraceptive states.
In fact, each involves a visit of only one rronth's
duration to the state rsv .
24
These cells must, of course, be nore closely
studied in the future but they will not significantly affect the
Thus for the purpose of investigating this Particular research
question.
question
tiIre-to~livery
~
may accept the hyPOthesis.
The sane tyPe of situation is responsible for the large ordinary chi-square
value for testing the group IIID subchain.
In this case 5 of the 417 observed
positive cells (1.20%) contributed 1425.13 (69.85%) to the total value.
cells are all characterized by a one nonth visit to state rsv
31
These
and, again,
it is felt that deviation fran the hyPOthesis in such cells will not affect
the
tiIre-to~livery
investigation.
-16-
4.
ONE APPLICATION:
THE TIME-'ID-DELIVERY QUEsrION
As has now been indicated, the Markov rrodel that has been presented may
be used for statistical inference as well as for description and prediction.
A.
The Research Question
we now consider an application to the following research question: What is
the relationship between the length of observation and the proportion of a
designated semple expected to deliver?
In Particular, what is the difference
between post-al:x>rtum (first pregnancy al:x>rters) and post-Partum (first pregnancy
deliverers) designated samples?
The tircE interval between a teI:m delivery,
or an abortion, and the next subsequent delivery depends on a number of
variables.
One is the length of the period of anenorrhea i.rtnaiiately following
the delivery or al:x>rtion.
It has often been noted, see for example, Jain (5),
that this period is shorter for an abortion than for a tenn delivery.
Other
variables include the propensity to ch<x:>se contraceptive techniques and which
techniques are actually employed.
One objective here is to detennine a function, t (.), which gives the percentage, t (n), of a specified cohort which has delivered by tine n.
In cohort
studies it is essential to have a sufficient initial sample size and to follow it
over a sufficient period of time in order to obtain adequate estimates of the
variables under study, in this case relative incidence of prematurity.
The
function, t (.), provides an estimate of the denaninator for the incidence rate-the total expected number of deliveries--and thus may be used to calculate the
necessary number in each cohort for a given period of follow-up.
-17-
Paraneterization for Tine to Delivery
B.
Fortunately, we can detennine percentages like t (n) directly fran the
Markov Chain using functions of sul:::matrices of the
(p .. (n))
1)
= pn. we
will use calculations for groups II and IUD.
cases we will need the
(f .. (n» •
n-step transition rratrix
(po. (n))
1J
In both
and/or the first-PaSsage titre probabilities,
The relationship between these is well-known and given by:
1)
0
1)
for all i, j, where po. (0)
JJ
==
n
L
=
po (n)
f
0
•
mFl 1)
(m) po
0
JJ
1, for all j.
(n - m),
n > 1
First passage tines are then
iteratively determined by:
f
0
0
1J
(n)
=
n-l
l
p .. (n) -
1J
p.
mFl
f. . (m) p. (n-m)
1J
n > 2
0
JJ
n=l
0
1J
To answer the tiIre-to-delivery questions,
we will consider the following
two subcha.ins:
CA
==
{ ia , sa , S2j (j=1, ... ,9), rsv
(j = 1, ... ,4), rsi , m P (j=1, ... 9).
2
2j
2
j 2
2
nrl , pd2},
2
I
Co
== {
(j = 1, ... ,4), rsi , m P (j=1, .•. 9),
ia , sa , S3j (j = 1, ..• ,9), rsv
3
3j
3
j 3
3
nrl , pel3}·
3
CA
is the subchain representing the abortors and
post-delivery rrovenent.
The transition rratrix
the corresponding sul:matrix of
infonnation, the states nrl 2
P
C;
is the subchain representing
PA for
CA
is the sane as
for group II, except that, without loss of
and pd2
are made absorbing.
See Appendix 4.
-18-
Since entrance to group 1110 is through states
mil
or
pdl'
these
must be adjoined to group 1110 to calculate tirres to the next subsequent
delivery.
Hence the subchain used
Po
The transilion matrix
for
for group 1110 will be:
Co
is the sarre as the corresponding sub-
matrix for group III with rows for states
rrrl
and
3
pel")
made absorbing;
mil
and
pdl
adjoined and states
see Appendix 4.
To simplify the notation, we write:
TA
-
(j = 1, ... ,4), rsi ,
{ia , sa , S2j (j = 1, ... ,9), rsv
2
2j
2
2
ffi
j P2
(j = 1,. .. ,9)}
To
-
{ia3' sa 3 , S3j (j = 1, ... ,9), rsv3j (j = 1, ... ,4), rsi 3 ,
ffi
j P3
(j = 1, ... ,9),
nrl , pd1}
1
and
CA =
{
TA U {mi2 '
Co=
{
To U
Since
~,
e
IX1:2} }
{m::13 ' pd3} }
IX\ are
= 2,
absorbing for k
3,
the states in
TA' To are transient.
Let
a.
-
a group II induced al:xJrtion
=
ia
S
-
a group I mature delivery
=
mil
Y
-
a group I premature delivery
=
pd1
11 k -
a group
k
mature delivery
=
~
k -
a group
k
premature delivery =
P'\
TI
and write
typical for
5
For
k
=
j = 7, 8, 9 for
k
=2
ffi
or 3 with
jtlc
llk'
(j = 7, 8, 9), where
k = 2, 3
5
•
2
The subne~rk
k , absorbing is shown in Figure 2.
TI
3 here, we refer to group IIID.
-.L~-
T
FIGURE 2:
GroUP II (OR HID) NE'IW)RK \VIT.H ABSORBING STATES.
To detennine the percentage of the specified cohort who deliver by tine
n,
CA:
t (n), we first note that for
f
aj
(n)
=
==
JP [X
==
F [delivery of type
n
j; Xm
=l=
j, 1 :: m :: n - 1
j
at nonth
and induced abortion at nonth
where
j
=
Jl2
or
1120
==
JP [X
n
= j
I
= f
=
n-l
. (n) +
L
aJ
IIFl
n
L f . (m)
IIFl aJ
for all
t,
for
j = Jl2
or
j
at nonth
n
I Xo --
a]
f . (m) p " (n - m)
aJ
JJ
= JP [delivery of type
p .. (t) = 1
JJ
o. ]
X = a]
0
= JP [delivery of type
since
for the first tirre given
n
For the network soown in Figure 2:
P . (n)
aJ
(401)
I Xo = a]
j
ex. nonth
n\ X0 = a]
11 0 Due to the absorbing
2
-20-
states, the ordinary
n-step
transition probabilities give the cumulative
percentages of the cohort to have a delivery by roc>nth
an asstllTed initial distribution on
CA
n,
at roc>nth zero.
for
The latter, for our
purposes, is taken to be the probability distribution:
~
=I
~
= 0,
for all
j
E
CA
with
j 4= a.
If first passage probabilities are desired, the expression in (4.1) makes a
recursive solution very simple in tenns of required cemputations.
The ti.Ire to delivery for the aborters is then:
(4.2)
t <n} :: JP [{X
A
n
= qa (:I'
=
[X
n
~2}
n
{X
n
=
'IT2} ]
= ~2 I Xo = a] +
= p a~2 (n) +
where the Paraneter
U
p
a'IT2
F [X
n
= 'IT 2 I Xo = a])
(n),
nms through the months of interest.
Hence,
{~(n}}
is the distribution of the ti.Ire-to-delivery random variable for the cohort
represented by
CA.
Fractiles may be obtained directly fran a graph since
is a cunulative distribution function;
p
a~2
(.) + P
k
=2
(. )
see Figures 3 and 4.
There is an alternate approach which recognizes that entrance to
for
a'IT 2
or 3, insures a delivery within three rronths.
Il7I1<'
To avoid extra
carrputation the subnetwork is Irodified; see (11).
There is also an alternative derivation for the (transient) first passage
tirres for the roc>dified subnetwork to be obtained directly for a vector/sul::natrix
equation; see (II).
e
-21-
To detennine the tine-to-delivery percentages for the cohort defined by
Co,
the pertinent equation is:
Then
~ (n)
= :P
I
(E
BUG)
= ]I? [(EnB) U (EnG) lI{]I? (B)
+ JP (G)},
B n G = </>.
since
But the
ini tial distribution for group 111D will always assune, for our pllqX)ses, that
8 = m:ll
all the individuals are distributed be'bJeen the states
=1
Hence, JP (B) + :P (G)
(4.3)
=F
tn(n)
where
0
= JP
[X
O
(B) JP (EIB) + JP (G) :P (EIG)
=
0 JP
=
0 [Po
(Elxo =
~~3
+
8)
(n) + Po
~~3
(1 - 0):P
(Elxo = y)
(n)] + (1 - 0) [p
y~3
(n) + p
YTI 3
(n) ]
= 8].
let
0,
we could silrply use the prenaturity rate
f1
be the probability of a prerraturity in group I ,
r
Using the Skopje data,
then, see (11),
and
y =
and,
To detennine values for
for group I.
and
0:: 1- f
0
= 0.9494.
tx1..
-22-
A nore refined analysis might be to corrpute
6(n)
= P(rsv
14
),S(n) / [P(rsv ),S(n) + P(rsv ) ,y(n)]
14
14
and nonmlize (or average) the
{6 (n) :
n
= 1, ... ,60}
for an estinate of
In either case, sensitivity analysis soould be perfonred on
varying in, say, the interval
~ (n)
for
6.
6
[0.92, 0.98].
The t.irre-to-delivery results for the two groups
C and
A
CD
as
derived by fonnula should be canpared to ti.rre-to-d.elivery canputed directly
fran the
~
for each
n,
state to
tx\
's paths.
The latter is a curmiLative frequency histogram where,
the nurrber of worcen who have made the transition fran $eir initial
or
In:\.
by nonth
n are counted for k
= 2,
3.
A fit test, like the
Kolnogorov-Srnirnov (K-S) test, is then used to canpare the resulting cumulative
frequency histograms with, respectively,
C.
t (n)
A
or
~ (n)
.
N1.1lTErical Results
Using the notation introduced above,
'We
e
present tables and graphs of
cumulative probability distributions for the t.irre-to-d.elivery randan variables
for the cohorts represented by
CA and
The graph of the distribution
Figure 3 and that for
{~ (n) : n
{t
A
CD.
(n) :
= 1, ... ,60},
n
= l, ... ,60}
in Figure 4.
is shown in
The
p-value
of the statistic of the K-S test for fit between incidence rates of
~
(n) is 0.0162.
t
A
(n) and
Divergence between the incidence rates does not begin until
after rronth 50, and the fit rerrains gocx:1 until after rronth 55 (see Table VI).
Be~
the nodel rates of
t A (n)
and
~(n),
when
~(n)
is defined with 6
= .9494,
the p-value equals 0.99+.
D.
Discussion
The incidence rates are based on data fran the total sample through rronth
35 (Le., observations from December, 1969 through October, 1972), but
-23-
I
\
I
------,
.<
o
._u..::
.K
o
\
If.
'..I.
\~
,x.
\x.
o
c'
_l..[;
,>(
X
\
>(
><
\
:x.
\:
.f.
~.
c
C-'
'?;
CO'
~:
_.
l""I
"-l
.z
~
::>
8
:"..
c·
c
o
...:I
~
Q
0
::E:
~
Q
G
u..
~
f-o
en
~
~
~
t.J
Z
r:.l
t.J
H
Q
Q
H
~
H
Po<
t.J
Z
X
X
X
X
X
~
<'
_.
T[Mf TO NUT
:; r
Dfl.:rVf~Y FOU.OW~NG r!F!~
/fP,Y
----------,
n
- - - - - - - - - -
~)
co
co
to
~ PREDICTED BY MODEL
x
-.)
X
X
X
X
INCIDENCE RATES
-.)
co
G'
0'
-.)
U1
'n
w'
-"
X
(.-)
tl'
.1'
,n
,.;
j
.,
-"
,.;,
":1
e'
.",X
~q
.-
,
.
f~'
_.j
L.
I
'
I
>~
( ..'j
//
;<
~)
X·'<
><
1'.
/./
f\j
f\j
; .....
//
x
"
'}
Xl',
)(
r'
"
.
XX
o
I ,., :. :". :( j(
o.
e
,. X X
)( )( j( ) (-¥.
5.1)1)
x.
10.1)
x
x
x
J',
.
/'-.
A
XX
i
15.0
!
:)0.0
I
,
it;
i i i
.lO.O
f)
e
M:W!!i
3'i.0
~~TNl:r
nF!fJFRf
F Il:I. I RE 4
If 0.0
'15.1)
I
tv
~
//~/X
(.,
CD
.J
.x....
/--'"
, , c..,:.
-,
..
X
X
X X
("1
:li
:1:,.,
x
XX~-<~
,XxY~~
·0
I
I
~
, ) ' , X · . . .I
XXX,"
50.0
i
55.0
e
I
600
-25-
thereafter the available data base becanes progressively smaller, since
the samples were drawn from first-pregnancy abortions dating from January, 1968
through December, 1969 for group II and fran first-pregnancy deliveries dating
tram July, 1968 through Decanber, 1969.
Only those aborters of January, 1968
who were interviewed as late as December, 1972
were observed for 60 IIOnths.
It is therefore not surprising that the rrod.els diverge from incidence rates
in the latter rronths, even though the divergence is not significant.
The
difference only becorres apParent when the incidence rates of the two groups
are ccmpared;
see Table VI.
Based on the
data, the probability of delivery by rronth
(n)
does not
dePend on the outcorre of the first pregnancy, when comparing induced aborters
with deliverers (Tables III and IV) or deliverers with a premature outcare
with those having a full-tenn delivery (Table V).
Regardless of first pregnancy
outcare, one-fourth of the cohort will deliver wi thin the next 30 IIOnths.
second one-fourth will take at least 50 IIOnths to deliver.
The
As our estirrates are
not very good beyond this period, we cannot detennine the titre necessary for the
remainder to deliver.
Being that sore waren will have no future deliveries,
the probability of a future delivery will never reach 1.0.
These estirrates are based on a particular sample which had high incidence
of induced abortion, law incidence of contraceptive use, and lCM family-size
expectations (3).
Furthernore, the aborter group was predominantly unrrarried for
a period of tilre post-abortion while the deliverer group was, with few exPeCtions, married throughout their POSt-delive:ry period.
The interval to next
delivery thus encompasses different exposure patterns for each group and, necessarily,
-26-
required different contraceptive utilization to achieve the unifonnity of
predicted time to next delivery.
It is quite possibly coincidental that the
e
curves are similar, and in other cultures this similarity may well not exist.
sensitivity analysis could be perforrred using this M.C. nodel to detennine the
effect of these variables on the tine to next delivery.
In sl.IDlYarY, Markov Chains may contribute much to mathematical denography
because of their ability to reflect the stochastic, state-to-state, and
time-dependent nature of human reproduction.
This particular rrodel, which
was developed to explore the epidemiologic relationship between induced abortion
and subsequent reproductive performance in a Yugoslav population, may be used
to answer additional fertility questions, such as the one presented here.
For the sample studied, there was no detectable difference in the tiIre to next
delivery between groups of wrnen with different prior reproductive histories.
By the end of the censored observation period of five years, approximately
one-half of the waren had delivered.
behavioral carponents
This rate reflects biological and
such as minimal tine required to deliver, utilization
of abortion and contraception for child-spacing, and desired family size.
'Ihe M.C. rrodel presented here may be applied to other data-based and si.ml1ated
samples to compare the robustness of estimates rulder varying behavioral
settings.
e
REFERENCES
1.
Anderson, T.W., Goodman, L.A., "Statistical Inference About Markov
Chains," Ann. Math. Stat. Vol. 28, (1957), pp. 89-110.
2.
Chiang, C.L., "A Stochastic r-bd.el of Human Fertility,"
Vol. 25, (1969), pp. 17-69.
3.
llog\E, C. J ., "lDw Birth weight Subsequent to Induced Abortion:
An Historical Prospective Study of 948 Wcrren in Skopje, Yugoslavia,"
Am. J. Obstet, and Gynec.
(In press)
4.
Hogue, C.J., "Prerraturity Subsequent to Induced Abortion in Skopje,
Yugoslavia: An Historical Prospective Study. II Dissertation,
University of North Carolina at Chapel Hill, 1973.
5.
Jain,A.K., "Pregnancy Outcane and the Tine Required for Next Conception, II
Population Studies, Vol. 23, (1969), pp. 421-33.
6.
Keneny, J.G., Snell, J.L., Finite Markov Chains, VanNostrand, 1960.
7.
Kullback, S., Infonna.tion
8.
Kullback, S., Kuppenna.n, M., Ku, H.U., "Tests for COntingency Tables
and Markov Chains," Technanetrics, Vol. 4, (1962), pp. 573-608.
9.
Lee, T.C., Judge, G.G., Zellner, A., Estimating the Pararreters of
Th~D.l
Bianetrics,
and StatiE tics, Wiley and Sons, 1959.
the Markov Probability Model fran Aggregate Tine-series Data, North
Holland Publishing Co., 1970.
10.
Perrin, E.B., Sheps, M.C., "Human Reproduction:
Biometrics, Vol. 20, (1964), pp. 28-45.
A Stochastic Process,"
11.
Shachtman, R.H., Hogue, C.J .R., "Prerraturity and Other Event Rates
Subsequent to Induced Abortion I: A Markov Chain r-tJdel," Institute
of Statistics M:iIreo series No. 1009, University of North Carolina,
Chapel Hill, 1975.
12.
Sheps, M.C., Menken, J .A., "A Model for Studying Birth Rates Given
Tine Dependent Changes in Reproductive Pararreters," Biometrics,
Vol. 27, (1971), pp. 325-43.
13.
Sheps, M.C., Perrin, E.B., "Changes in Birth Rates as a Function of
Contraceptive Effectiveness: SCire Applications of a Stochastic Model,"
Am. J. Public Health, Vol. 53, (1963), pp. 1031-46.
TABLE I:
MATRIX DENSITY - GENERAL MARKOV CHAIN AND
DA.TA ll1PLEMENTION
Groups
1
2
3
full chain
Number of states
=
s
25
27
27
79
Number of cells
=c
625
729
729
6241
Potentially positive
cells*
=r
114
130
130
374
73
85
115
273
0.640
0.654
0.885
0.730
0.182
0.178
0.178
0.060
0.117
0.117
0.158
0.044
Data irnplerrented
positive cells*
A
Ratio of r to r
Potential Density
Data irnplerrented
Density
A
=r
A
= rlr
= ric
A
= ric
I.
*The numbers r and r may be considered to be the di.m:msion of the matrix
rrodel, pre and post-data.
I
TABLE II:
RESULTS OF TEST FOR FIRSI'-QRDER DEPENDEN::E
Pt
Co
Po
Group II
613
467
146
Group 1110
695
438
257
2
2
XK
d.f.
259.52
90.62
138
2038.69
249.62
257
Xo
TABLE III:
POST-AOORl'UM TIME-'ID-DELIVERY
tA(n}
•
.
•
.
Incidence
M:>nth
n
Model
m
Rates
i
Difference
(m-i)
5
.0000
.0000
.0000
10
.0034
.0000
.0034
15
.0819
.0876
-.0057
20
.1538
.1429
.0109
25
.2169
.1935
.0234
30
.2750
.2811
-.0061
35
.3282
.3272
.0010
40
.3769
.3687
.0082
45
.4215
.4194
.0021
50
.4625
.4424
.0201
55
.5000
.4516
60
.5344
.4516
.0484
.0828
K - S test for fit, p
=
.1813
•
TABLE N:
POST-PARTUM TIME-'ID-DELIVERY
~(n)
Incidence
Rates
i
Difference
zronth
M:xlel
n
m
5
.0000
.0000
10
15
.0035
.0830
.0014
.0352
20
.1633
.0578
25
.2309
.1055
.1744
30
35
40
45
.2906
.3441
.3925
.4366
.4771
.5144
.5488
.2461
.0445
.3432
.4065
.4641
.4880
.4909
.4909
.0009
-.0140
-.0275
-.0109
.0235
50
55
60
K - S test for fit, p = .5095
For this table,
<5
= 0.9494
(m-i)
.0000
.0021
.0478
.0565
.0579
I.
TABLE V:
SENSITIVITY ANALYSIS OF POST-PARIUM TIME-'ID-DELIVERY
t=n(n)
.92
.93
.94
.95
.96
.97
.98
5
10
15
20
25
30
.0000
.0035
.0832
.0000
.0035
.0000
.0035
.0830
.0000
.0035
.0829
.0000
.0034
.0829
.0000
.0034
.0829
.1635
.2311
.2908
.1633
.1631
.2309
.2905
.2307
.2903
.1630
.2304
.2901
.0000
.0034
.0827
.1628
35
40
45
.3448
.3440
.3924
.4366
.4771
.3438
.3922
.4364
.4769
.3435
.3919
.4361
.4766
.4359
•4764
.5144
.5488
.5142
.5139
.5137
.5486
.5484
.5482
.1638
.2315
.2912
.0831
.1636
.2313
.2910
.3932
.4373
.3445
.3929
.4371
.3443
.3927
.4368
50
55
.4778
.5150
.4775
.5148
.4773
.5146
60
.5494
.5492
.5490
.2302
.2898
.3433
.3917
I
I
.
TABLE VI:
K - S TESTS FOR
INCID~E
RATES OF AOORI'ERS AND DELIVERERS
Months
P-values
1-37
0.8881
1-38
1-39
1-40
1-41
0.8973
0.9057
0.9135
0.9206
1-42
1-43
0.9272
0.9333
1-44
1-45
0.9389
0.9440
1-46
0.9487
1-47
1-48
0.8384
0.8475
1-49
1-50
0.6994
0.5441
1-51
1-52
1-53
0.4051
0.2914
0.2037
1-54
0.1389
1-55
0.1458
0.0978
1-56
1-57
1-58
0.0642
0.0413
1-59
1-60
0.0261
0.0162
I.
APPENDICES
Appendix 1:
Event Codes
Appendix 2:
State Definitions
Appendix 3:
Network Flo.v Diagram
Appendix 4:
Partitioned Transition Matrix
Frequencies for the theoretically r;ositive
cells are given with respect to the categories:
category
Observed Frequency
a
a
1
1 -
2
5 - 19
3
4
20 - 49
50 - 99
5
> 100
4
•
•
APPENDIX 1
Event Codes used in the path coding for the Markov chain analysis.
Code
Definition
01
Oldest age
Age at marriage
Age at separation
Age at reunion
Age began smoking
Age discontinued smoking
Age commencing menses (001 if less than 15 years)
02
03
04
05
06
10
11
12
13
14
15
16
17
18
19
20
21
22
25
26
31,32
33-36
37
38-46
47,48
57,58
68-76
77
Contraceptive events:
Age beginning abstention
Age beginning douching
Age beginning coitus interruptus
Age beginning rhythm method
Age beginning paste, jelly, etc.
Age beginning condom
Age beginning diaphragm
Age beginning rhythm + condom, etc.
Age beginning IUD
Age beginning birth control pills
Age beginning other method (primarily same as coitus interruptus)
Age ceasing contraception
Pregnancy events:
Age at a conception (Last Menstrual Period), not smoking
Age at a conception (LMP), smoking
Induced abortion, first (second) commission
Induced abortion, illegal, different methods
Spontaneous abortion
Mature live (still) birth, occurring at various locations
Twins (both live or one live, one still)
Abortion, equivocal whether spontaneous
Premature live (still) birth, corresponding to 38-46
Woman's age at loss to follow-up, if not interviewed (2 cases)
APPENDIX 2
STATE DEFINITIONS FOR THE CURRENT MODEL
State
rsv
rsv
rsv
3l
32
33
Definition
Operational Definition 6
reduced susceptibility,
involuntary, post-delivery
(After at least one 38-46 or 68-76),
10th consecutive month within
marriage, in absence of code 11-26.
reduced susceptibility,
voluntary 1, post-delivery
(unsafe method)
(After at least one 38-46 or 68-76),
code 21 or 12-14.
reduced susceptibility,
voluntary 2, post-delivery
(moderately safe method)
(After at least one 38-46 or 68-76).
code 15-16.
reduced susceptibility,
voluntary 3, post-delivery
(safe method)
(After at least one 38-46 or 68-76),
code 11 or 17-20.
reduced susceptibility,
voluntary 4, post-delivery
(assumed abstinence)
(After at least one 38-46 or 68-76),
code 03 or 31-76 in absence of
code 02, 04 or 11-26.
first (to ninth) month
susceptibility, post-delivery
(After at least one 38-46 or 68-76),
within marriage and in absence of
code 11-26. Progression continues ~
until code 11-26 or until 10th
~
month, when transition is to rsi •
3
first (to sixth) month
pregnancy, post-delivery
(After at least one 38-46 or 68-76),
code 25 or 26. Progression continues
until code 3l-3~ or 57-58, when
transition is to ia or sa 3 . :f no
3
code 31-37 or 57-58, progress10n
continues to m7P3.
induced abortion, post-delivery (After at least one 38-46 or 68-76),
code 31-36 or 58 (only after code 25
or 26).
6
spontaneous abortion.
post-delivery
(After at least one 38-46 or 68-76),
code 37 or 57 (only after code 25 or
26) .
seventh (to ninth) month
pregnancy. post-delivery
(After at least one 38-46 or 68-76),
continuation-of progression begun by
code 25 or 26. Progression continues
until code 38-46 or 68-76, when
transition is to md or pd 3 •
3
Numbers correspond to event codes
in Appendix 1.
State
Definition
Operational Definition
~3
mature delivery,
post-previous delivery
(After at least one 38-46 or 68-76),
code 38-46 (only after code 25 or 26).
premature delivery,
post-previous delivery
(after at least one 38-46 or 68-76)
code 68-76 (only after code 25 or 26).
mature delivery,
first pregnancy
(In absence of previous code 31-76),
first code 38-46 (only after code
25 or 26).
premature delivery,
first pregnancy
(In absence of previous code 31-76),
first code 68-76 (only after code
25 or 26).
first delivery, mature,
post-abortion
(In absence of previous code 38-46 or
68-76 but after 31-36 or 58), first
code 38-46 (only after code 25 or 26).
first delivery, premature,
post-abortion
(In absence of previous code 38-46 or
68-76 but after 31-36 or 58), first
code 68-76 (only after code 25 or 26).
seventh (to ninth) month
pregnancy, first pregnancy
(In absence of previous code 31-76),
continuation of progression begun by
code 25 or 26. Progression continues
until code 38-46 or 68-76, when
transition is to md or pd 1 •
l
(In absence of prev10us
code 38-46 or
68-76 but after 31-36 or 58),
continuation of progression begun by
code 25 or 26. Progression continues
until code 38-46 or 68-76, when
transition is to md or pd •
(In absence of prevtous co~e 38-46 or
68-76 but after previous 31-36 or 58),
code 25 or 26. Progression continues
until code 31-37 or 57-58, when
transition is to ia or sa • If no
2
2
code 31-37 or 57-58, progression
continues to m P .
7 2
seventh (to ninth) month
pregnancy, post-abortion,
pre-delivery
first (to sixth) month
pregnancy, post-abortion,
pre-delivery
reduced susceptibility,
involuntary, post-abortion,
pre-delivery
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58),
10th consecutive month within marriage,
in absence of code 11-26.
reduced susceptibility,
voluntary 1, post-abortion,
pre-delivery (unsafe method)
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58),
code 21 or 12-14.
reduced susceptibility,
voluntary 2, post-abortion,
pre-delivery (moderately safe)
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58),
code 15-16
State
ia
2
Definition
Operational Definition
reduced susceptibility,
voluntary 3, post-abortion,
pre-delivery (safe method)
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58~
code 11 or 17-20.
..,
reduced susceptibility,
voluntary 4, post-abortion,
pre-delivery (assumed
abstinence)
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58),
code 03 or 31-76, in absence of code
02, 04, or 11-26.
first (to ninth) month
susceptibility, post-abortion,
pre-delivery
(In absence of previous code 38-46 or
68-76 but after previous code 31-36 or
58), within marriage and absence of
code 11-26. Progression continues
until code 11-26 is met or until 10th
month, when transition is to rsi .
2
induced abortion, pre-delivery
(In absence of previous code 38-46 or
68-76), code 31-36 or 58. (only after
code 25 or 26).
spontaneous abortion,
pre-delivery
(In absence of previous code 38-46 or
68-76, code 37 or 57. (only after code
25 or 26).
first (to sixth) month
pregnancy, first pregnancy
(In absence of previous code 31-76).
code 25 or 26. Progression continues
until code 31-37 or 57-58, when
transition is to ia or sa . If no~
2
2
code 31-37 or 57-58, progression
continues to m7P1.
reduced susceptibility,
involuntary, pre-pregnancy
(In absence of previous code 25 or 26)
10th consecutive month within
marriage, in absence of code 11-26.
reduced susceptibility,
voluntary 1, pre-pregnancy
(unsafe method)
(in absence of previous code 25 or 26)
code 21 or 12-14.
reduced susceptibility,
voluntary 2, pre-pregnancy
(moderately safe method)
(In absence of previous code 25 or 26)
code 15 or 16.
reduced susceptibility,
voluntary 3, pre-pregnancy
(safe method)
(In absence of previous code 25 or 26)
code 11 or 17-20.
reduced susceptibility,
voluntary 4, pre-pregnancy
(assumed abstinence)
(In absence of previous code 25 or 26)
This is each woman's initial state,
and code 03 in absence of code 11-26.
first (to ninth ) month
susceptibility~ pre-pregnancy
(In absence of previous code 25 or ~
within marriage,and in absence of code
11-26. Progression continues until
code 11-26 or until 10th month, when
transition is to rsi •
l
~5
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