lTha research was supported primarily by National Institute of Child
flealth and Human Development Grants number 5-ROl-HD072l4 and
l-ROl-HD09028.
2Department of Biostatistics, School of Public Health and Curriculum
in Operations Research and Systems Analysis
3Department of Biostatistics, School of Public Healtil
PREMATURITY AND OTHER EVENT RATES
SUBSEQUENT TO INDUCED ABORTION I:
A MARKOV CHAIN MODELl
by
Richard H. Shachtrnan 2 , Carol J. R. Hogue 3
lfuiversity of North Carolina at Chapel Hill
Institute of Statistics Nlimeo Series No. 1009
/
MAY 1975
PREMATURITY AND OTHER EVENT RATES SUBSEQUENT TO
INDUCED ABORTION I: A MARKOV CHAIN MODEL
ABSTRACT
A Markov chain model is presented which incorporates several reproductive variables in a conceptual and a data-based modeling approach
to questions of possible sequelae to induced abortion.
It will be
applied to data collected on 948 women who participated in an historical
prospective study conducted in Skopje, Yugoslavia.
The current 79 state
model evolved from consideration of the criteria for Markov chain development as well as an emphasis on epidemiologic research questions.
Intennediate models helped to clarify the particular problems of this
technique and suggested methods of dealing with these problems.
Indications
are given as to how the chain, in its current form, may be used to answer
specific questions of interest to epidemiologists, physicians and health
policy makers.
Table of Contents
............... .....
1.
Introduction. . . . . . . .
II.
Research Methodology
A. The Substantive Problem. . . . . .
B. Research Design . . .
C. Data Considerations . . . . . .
D. Preliminary Analysis .
Statist ical Modeling .
Markov Chain Models
A. A Pilot Model. . . . . . . . . . . . . . . . . .
B. Specific Criteria for Markov Chain Construction
1. lIomogeneity of the cohorts . . . • .
2. State flow choices
.
3. Time parameter and time interval .". •
4. The Markov property. . . . . . . .
5. Time-homogeneity (stationarity). .
6. Equil ibrium distribution. . . .
7. Aggregation/disaggregation. . . .
8. Closed classes of the Markov chains . . . .
9. Implication for data needs . .
10. Testing . . . . • • . • •
C. The Current Models
1. The network flow diagram . .
2. Current state definitions
3. Justification for the state definition~
4. ~mtrix and matrix density classification .•
Uses of the Model and Parameterization for the Research Questions
A. Aims of the Modeling. . • • . . • . • . . . . • . . . . . . . .
B. Parameterization for the Research Questions
1. Direct computations .
2. Time to delivery. . . .
3. Mean time to delivery .
4. Intervening conditions . . .
5. A ratio for time-dependent prematurity rates. .
III.
IV.
V.
Acknowledgements .
Bibliography.
Appendices
1
5
7
8
10
13
19
21
21
23
26
27
29
30
30
31
31
32
36
36
42
45
45
47
52
54
61
65
66
PREMA1URITI AND OTHER EVENT RATES SUBSEQUENT TO
INDUCED ABORTIONS I: A MARKOV CHAIN MODEL
Richard H. Shachtman1 , Carol J.R. Hogue 2
I.
INTRODUCTION
Questions concerning the long-range biological effects of induced
abortion have gained prominence with the legalization of abortion for
indications other than medical necessity.
One of the suggested consequences of induced abortion is an increased risk of low birth weight 3
(less than 2501 grams) in subsequent pregnancies.
Positive association
between low birth weight and a history of induced abortion has been reported from some retrospective, or case-control, studies (2,13,23,35),
but other investigators have found no such association in a cross-sectional
study (47 ) or an historical-prospective study (22).
If an association
does exist, it may be related to the timing of the subsequent pregnancy as
well as to the occurrence of intermediate, related events.
To investigate the
time-dependent factors surrounding these possibilities and related questions,
a Markov chain model has been developed which incorporates several reproductive
variables.
This paper presents the model along with an explanation of its
development and potential uses.
The results of the analysis for a particular
data base will be discussed in a later paper (70 ).
1 Department of Biostatistics and Curriculum in Operations Research and
Systems Analysis.
2
Department of Biostatistics
3 Low Birth Weight and Prematurity are not synonymous since births of greater
weight occur before eight months of gestation and low-weight births occur
at full-term. Nevertheless, for simplicity, the terms will be used interchangeably in this paper.
2
Markov chains have been utilized for questions in the reproductive
areas of human fertility, Chiang (69 ), pregnancy and postpartum periods
of a fertility model, Perrin and Sheps (73), and birth rates as a function
of fecundability, contraceptive effectiveness and other parameters, Sheps
and. Menkin (88 ), Sheps and Perrin ( 89 ).
The most comprehensive study
available on mathematical models of conception, birth, pregnancy wastage
and interrelated variables is the treatise by Sheps and Menkin (87 ).
Other uses of Markov chain models for biological and health-status
problems are given in the third section of the bibliography and the fourth
section contains some general references.
The models proposed in this study are part of the analysis and design
modules of a large complex field study.
The modeling approach differs
from many earlier Markov chain applications in its scope, its dimension,
the degree of detail in its structural justifications and in its motivating
purposes.
One goal was to create an overall reproductive model with
particular attention to substantive abortion research questions.
Another
goal was to serve a didactic purpose of presenting a critical use of Markov
modeling for an Epidemiological problem and a large field study.
In particular
it is both a conceptual and a data-based modeling approach as opposed to
many models which are conceptual only or, at best, use artificial data.
structural justifications are discussed in Sections III and IV and
II
goodness-of-fit ll tests are planned for
d~ta:..oriented
C70). Typical,but specific,
research questions include:
a.
The
What is the time-dependent probability of a first birth being
low-weight, subsequent to a first-pregnancy induced abortion?
3
b.
How does the sequence of probabilities defined above compare with
the probability of a low-weight birth of the first pregnancy (i.e.,
with no previous spontaneous or induced abortions)?
c.
What are the probability distributions of time-to-delivery for
aborters and deliverers?
d.
What are the distributions
of the length of contraceptive use
for aborters and deliverers, pre- and post- abortion?
Due to the complexity of the phenomena under investigation and a realization
that the data would be adequate to meet the requirements for a more detailed
analysis, the dimension of the current model is larger than other reported
Markov chain applications.
As
will be described, this dimension evolved
from careful adherence to criteria for Markov chain development linked with
an emphasis on epidemiologic 'meaningfulness" which led to a laborious but
necessary consideration of numerous steps in the modeling process.
The actual
number of observations for post-abortion prematurities prevented the design
of a data-implemented chain to fUlly answer all time-dependent questions
on low birth weight; however, other time-dependent information is available.
It appears that the only way of reducing the dimension of the model
as currently aggregated is to attempt to use a semi-Markov process, an
approach which is presently under investigation.
An
understanding of the current model cannot be complete without a
brief discussion of the research design and data base (Section II), the
rationale for using a Markov chain modeling in this instance (Section III),
and the considerations which led from a small pilot model to the more complex
current form (Section IV).
Section V outlines how the current model may be
used to answer specific research questions within the scope of the population
studied, the sample collected, and the particular states delineated in the
current model.
5
II •
A.
RESEARGi METIfODOLCX:;Y
The Substantive Problem.
Induced abortion, the deliberate removal of a living fetus from a
woman's womb, is one of the oldest known methods of birth control.
It
has recently been observed legally and openly in great numbers; for example, in Skopje, Yugoslavia, there are over 70 legal abortions reported yearly
for every 1000 women in the age range 15 to 45 (18).
Many of the women
obtaining an induced abortion will subsequently give birth at a time perceived to be more appropriate for them to rear a child (22).
This practice
places such women in potential risk of future pregnancy complications due
to the previous abortion procedure.
The actual risks are not known but
are believed to include increased probability of experiencing a low-weight
birth ( 61).
Retrospective studies (2,13,23,35) investigating this risk have, in general,
found a positive association between induced abortion and low birth weight.
They have, however, suffered methodologic problems that place their results
in question.
Women are sampled at the time of delivery; in classifying them
according to prior abortion status, this design depends on the women's reported abortion history as opposed to documented abortions.
If a proportion
of subjects is unwillingito admit the previous abortion and if this proportion
differs according to the outcome of the present pregnancy, the resulting
bias (termed "selective recall") may well be significant enough to affect
the retrospective studies' conclusions.
The cross-sectional study (47 )
which found no association between induced abortion and low birth weight
depended not only on subjects' recall of abortions but also on their memory
of childrens' birth weights.
Although selective recall has not been proven,
evidence of its existence emerged from our own study which, because of its
6
design (see below) documented prior abortions and, independently, asked
subjects about prior pregnancy history.
Using documented histories eliminated
the selective recall problem to the extent that the records were accurate.
Our study has attempted to correct for the methodologic problems referred to above.
Other methodologic considerations -- appropriateness of
the population, degree of coverage of the population, sample size, etc.
which are not great problems in the above-mentioned studies, do, in the
first two instances, pose a concern for our study which suffered from a
low rate of interviews at the follow-up period (23.7%' for "aborters" and
46.5% for "deliverers") and which involved a relatively small number, 101,
of interviewed aborting women who had subsequently delivered.
The Markov
chain model has maximized the usefulness of the data from the small sample
( 22, 72 ), but it cannot correct for biases which may have entered as a
result of incomplete follow-up.
in~ortant
Although these problems are felt to be less
than those introduced by recall bias, they may affect the results
to be reported in (70).
Hence these results should not be considered
as conclusive evidence of the nature of the relationship between induced
abortion and subsequent prematurity in an arbitrary
population~
None of the previous studies considered the interval between abortion
and subsequent delivery.
It
would seem that such an interval could be im-
portant in that shorter intervals might trigger an effect whereas the effect
would "wear off" after longer periods.
Through sensitivity analysis the
Markov chain model may be useful in determining the impact of pregnancy interval on subsequent complications, given that there is sufficient information
available for disaggregation; see Section V.
7
B.
Research Design.
In our study, two methodological approaches were considered for data-
gathering, namely retrospective and historical-prospective designs.
The
latter design was chosen, both for its independence from patient recall and
for its ability to directly estimate incidence.
Utilizing the retrospective
design, one would identify a group of women who had delivered a premature
infant and a "control" group who had delivered a full-term infant.
The
groups would then be interviewed to ascertain their previous obstetric
histories, especially the number of induced abortions.
The rationale is
Bayesian in that the association between the condition of prematurity (C)
and the characteristic of induced abortion (CH) is(CHIC) and the estimate
of incidence, given a rare condition, is (9
lP (CICH)
= 1P~) H(CHI9.. ,
IP
where
(CH)
IP (CH) = IP (C) lP (CH IC) + lP (C) lP (CH IC) ,
IP(C)
1S
lP (01 IC)
lP
)
(CH IC)
estimated from prior data,
is estimated from the study group, and
is estimated from the control group.
Following the historical-prospective approach, we identified through a search
of records a cohort of women who had undergone induced abortion of a first
pregnancy and a similar cohort who had not aborted prior to delivery of a
child.
These were located to ascertain their subsequent obstetric histories,
especially low-weight births.
The rationale is that incidence of (CICH)
can be measured directly from the data (38).
The historical-prospective
method differs from the purely prospective design in that subjects are recruited from records rather than at the time of the initiation of the study.
8
When such records are available, they may shorten the necessary period of
observation and thus shorten the duration of the study.
C.
Data Considerations.
In addition to the research design, certain other features of the study
are of particular relevance to the choice of statistical models.
For a
Markov chain analysis care must be taken to provide homogeneity within groups
and to collect sufficient data to fulfill the state definitions as well
as to provide estimates of the initial transition matrix.
To maximize within-cohort homogeneity, selection of the abortion cohort
was limited to the population of those women whose first pregnancies were
legally terminated in one recognized hospital during a two-year time
period and, likewise, the control cohort included only those subjects whose
first
pregnancies were delivered in the same facility during the same time
frame; see (
22, 72 ) for details of the study location and timing.
eligible women were sought for interviewing.
All
Since it was felt that data
from both cohorts could be utilized for estimation of some properties,
between-cohort homogeneity was also desirable.
Consequently, various socio-
economic characteristics of the study population were held constant, such
as ethnic group and residence.
Other characteristics which were known from
the intake records, such as education and occupation, were allowed to vary
because it was felt that within the chosen population, variation would not
be great enough to affect the results.
Still other characteristics, such
as smoking habits, were determined only upon interview and could therefore
not be used in selection of the populations.
All characteristics relevant
to the outcome measure (in this case, prematurity) should be tested to
determine their effect on reducing homogeneity.
9
Collection of sufficient data is dependent upon the sample size, the
nature of the information obtained, and the care which is taken to insure
accuracy and consistency of the information.
For this study, sample size
was estimated using the formula for a prospective study (125)
and 200 grams as the expected birth weight differential ( 22 ).
This is
considered to be a conservative estimate since a Markov chain should require less data for statistical accuracy than does a "traditional" incidence
measure.
This relative efficiency comes through the use of the Markov chain
equational relationships over a time period [d, tol to estimate the transition
from some state (i) to another state (j), by considering a cross-section of all
possible transitions from i to j over {t: t
£
[0, toll in computing p .. (t);
1J
the incidence measure makes use of the number actually making the transition
during that period, without benefit of the Markovian asstnnptions.
Further-
more, because more information may be included in the Markov chain, any increase in the number of observations has a higher multiplier effect on the
Markov chain analysis
than on the incidence analysis (see Sections IV and
V and ( 70)).
The optimal type of observation for a stochastic process analysis is
a chronological path of relevant events for each subject (132 ),
which leads to maximum likelihood estimators for the transition probabilities
in a Markov chain.
To achieve this type of data, each individual's reproductive
history was chronologically coded with the exact date of such occurrences
as obstetrical events, martial events, and contraceptive practice from age
fifteen to the date of the interview.
With very little additional effort,
these bits of information were coalesced into a monthly event-coded history;
see Appendix 1 for a listing of event codes.
In addition to the optimal
nature of chronological paths, detailed chronologies increased the flexibility
10
of choice for state definitions and thus broadened the potential scope
and dimens ion of the Markov chain model.
An exposition of the steps taken to assure quality of data is given
in (22 ).
Checks and counter-checks were built into questionnaire design-
ing and pretesting, interviewer training and on-the-job review of all
interviews, coding of medical records and interviews, and "cleaning" and
editing the coded data.
An elaborate computer program was written to
translate the event-coded data as accurately as possible into specific
transitions codes.
The resulting data set will be described in depth in
( 70 ).
D.
Preliminary Analysis.
Contingency tables were developed to test, using Chi-square analysis,
(a) the comparability of the group of interviewed first-pregnancy aborters
with the group of interviewed first-pregnancy deliverers and (b) the
comparability of the interviewed women in each group with their respective
non-interviewed groups ( 22,72).
Further Chi-square tests were conducted to determine overall incidence
rates as a substantiation of the forthcoming Markov chain results.
Finally,
a Discriminant Function model which included variables of maternal age,
1
education, work history, per capita income, pre-pregnancy weight, smoking
history, and sex of the child, as well as abortion history, was constructed to
determine the relative importance which these variables had in predicting low
birth weight for the first births in the interviewed sample.
are also reported in (22,72).
These results
They indicate that induced abortion is not
a significant factor when considered either alone in a contingency table
or together with other relevant variables.
The conclusions now need to be
substantiated with the time-dependent relationships of the Markov chain analysis.
11
The smoking status of a pregnant woman was one of several important factors found upon initial analysis to differ between abortion and
control groups.
It is envisioned that if sufficient data exists such
variables will be included in future modifications of the current model
to determine their effect on the estimated rates.
13
III.
STATISTICAL MODELING
For any woman in the designated population group, one of a number
of sequences of reproductive events may occur.
As a first analysis the
component events might be drawn from Table 1.
TABLE 1: REPRODUCTIVE EVENTS
pregnancy
mature birth
reduced susceptibility to pregnancy
abortion
complications associated with pregnancy, abortion or morbidity:
prematurity
spontaneous abortion
stillbirth
However, these events do not fully characterize a woman's status at
any given time.
Hence it is necessary to add other states for an ex-
haustive representation in terms of time and the research questions.
Further-
more, a selection of the various events and states must be made which are
pertinent to the substantive questions, e.g., concerning prematurity.
This
is especially true for susceptibility to pregnancy and for reduced susceptibility
which will include types of contraception, amenorrhea and infertility.
In
addition to the variables defined by the above reproductive events there
are other variables associated with certain of these events which were included in the data acquisition; for example, age, marital status, smoking
and age at menarche.
features.
The pilot model is aggregated so as to ignore these
The current model includes marital status, age and age cannnensing
menses in the event but not state definitions; they are felt to be of minor
14
importance for the key research questions.
The smoking question is being
delayed for a later modified analysis.
In addition to occurrences of the events themselves, account must
be made for the time intervals between occurrences.
For example, there
are periods of susceptibility to pregnancy which vary with respect to other
characteristics of the woman.
There are also post-partum and post-abortum
amenorrheic intervals; in the former case the woman could be lactating or not.
These events and period designations must be combined in such a way
as
to completely describe the reproductive path of a woman as it correspnds
to the abortion sequelae questions.
The next step is to use this description
to define a set of states which adequately reflects the various periods
and events of interest.
The particular events adopted were a result of con-
centrating on pregnancy outcomes, as opposed to such events as complications
arising during pregnancy or the puerperium.
'fhe flow of women through these states may be viewed as a stochastic
process; that is, at any given time a woman is in one of the defined states
and the possibility of making a transition to another state is governed by
a probabilistic law.
Since a woman in a given state at a particular time
may make a transition to anyone of a number of succeeding states, the ability
of a model to reflect the multi-state possibilities is as important as its
capacity for statistical representation.
Hence, given the possible event
sequences experienced by women in the selected population group, a breakdown
of events leading to a preliminary set of states is possible.
At any given
time, a fertile woman is in one of several states of pregnancy, post-pregnancy
periods, susceptibility to pregnancy, reduced susceptibility or states which
have a bearing on subsequent complications.
made to reflect
Some state definitions must be
a post-event period which accurately models a state-to state
movement over fixed time periods.
15
This stochastic, multi-state feature of the reproductive process,
and any corresponding model, makes other types of statistical analysis
complex, especially if they are to take into account a multiplicity of
time periods.
Suppose the aggregation of states may be structured in such a way
that the following is true:
a visit to any state Ilcarriesll sufficient
information about a woman's previous reproductive history to predict the
transition probability for any immediately succeeding state.
Then the set
of all states, indexed by a judiciously selected time parameter, may satisfy
the assumptions of a Markov chain (M.C.).
It
is well known 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. model
of sufficiently high order to reflect any state to state dependence in the
process.
Then we define a second equivalent M.C. model of low, preferably
first, order but necessarily having many states to account for all possible
transitions in the first M.C. (130 ).
This approximation rapidly gets out of hand from the viewpoint of data
availability, interpretation and sheer complexity of computation.
Hence
it is desirable, in fact necessary, to find a state set definition for which
a Markov property of low order is a reasonable statistical representation
for the flow of women.
Another requirement is that the state representation reflect the fact
that some events are defined for time periods with interval lengths, e.g.,
pregnancy and susceptibility or reduced susceptibility, whereas some events
are Ilinstantaneous ll such as an abortion or a premature birth.
Hence for a
M.C. interval, periods must be designated which are subsequent to such events.
16
Many of the research questions are answerable with the initial matrix
of transition probabilities and other parameters derived directly from
this matrix and its P9wers.
A prototypical scheme for using transition
probabilities to answer biological questions is as follows.
baseline probability for a complication given the
is established.
First, a
absence of a characteristic
Then curves, or sequences of points, generated by the time-
dependent higher order transition probabilities for the complication given
the presence of the characteristic are compared to the baseline.
In some
cases, it is necessary to derive other functions of these transition
probabilities; see Section V.
In our case one complication is prematurity,
a characteristic is first-pregnancy aborted and the absence of the characteristic
1S
first pregnancy delivered.
Questions of prematurity given multiple abortions, or of the occurrence
of abortions given varying conditions on pre- and post- first delivery (e.g.,
contraception), or about particular sequences using first order transition
probabilities and low-for-dates versus premature labor, allover time, may
be answered with functions of the initial transition matrix.
Also, questions
requiring mean length-of-stay times in states may be answered by using a
transform of the initial transition matrix.
Section
Some details are given in
v.
In general the questions concerned with incidence rates 4 are answered
by cwnulative functions of the n-step transition probabilities, p .. en), and
1.)
first passage time probabilities, f .. (n), whereas the questions answered by
1)
4 The period in which incidence is measured varies withn and is cumulative
over all n transitions.
17
prevalence rates relate to the state probabilities
7T.
J
en)
=l
iEI
7T.
1
en-I) p ..
1J
=I
i~I
f3. p .. en)
1
1J
where {So1 : iEI} is the initial state distribution.
For an episodic condition like prematurity, the ordinary n-step transition
probabilities are appropriate, whereas chronic diseases may require first
passage probabilities.
In addition, statistics representing the number
of visits to the states are pertinent.
19
IV.
A.
MARKOV mAIN MODELS
A Pilot Model
We will assume for both the pilot model and later models that the time
periods are constant; a discussion is given in Section IV. B. 3.
The pilot model state definitions are given below (Figure 1) and, as
designated, the individual states may be viewed as representing an aggregation
of states for the first full model (Figure 2).
State
Symbol
Definition, event or class
Aggregation
5 ,5 ,5
0 1 2
Sl
5
susceptible
S2
p
pregnant
S3
ch
completed pregnancy, healthy
pI, pn
S4
aa
post-abortion
ab, as, am
S5
rs
(other) reduced susceptibility
0,
S6
pp
post-premature birth
S7
ps
post-stillbirth
S8
id
infertile, death
u, c, a
i, d
FIGURE 1: A PILOT MODEL MARKOV CHAIN FOR THE ABORTION STUDY
20
State
Cl.:lSS
susceptible
to pregnancy
State
Syr::bol
El
60
E2
sl
Defi~1tion
or Event
sU3cepCible, no completed pregnancies
SU5ceptible, one - three completed
pr~gnan~ies \~thout complic~ions
•
susceptible, any number of completed
pregnancies with complications or four
or more without complications
E3
62
pregnaut
Elt
P
pOGt-pr~gnancy,
E5
pl
E6
pn
post-p3rtum,
£7
ab
5post-abortion, "social u indications
ES
u
£9
c
unexposed
5
contracepting
EIO
a
amenorrhea
Ell
&3
post-abortion, spontaneous
E12
8.:lU
post-abortion, medical
L13
0
oth~r
5pregnant
rc.:l ..\ccd sus-
ccptibility
complica tions
non-lact~ting
i~dication8
reduced susceptibility, morbidity
(to:tero.a, etc.)
E11+
pp
poat-prematura birct (under 2500 g.)
E15
ps
post-stillbirth (lata fetal los.)
EIG
i
Sinfertile
El7
d
death
5 Each of these, at least, could be further dis aggregated into E , E ••• }; for
jl
jZ
example, infertility, i, decomposes into involuntary and voluntary infertility.
Other states were left out due to previous indications of rare prevalency.
FIGURE 2:
FIRST FULL MODEL MARKOV CHAIN FOR THE ABORTION STUDY
~
21
B.
Specific Criteria For Markov Chain Construction
1.
HOl1X>geneity of the cohorts
The two cohorts should both be as homogeneous as possible with respect
to the variables which interact with the complication:
age, obstetric
history, pre-existing medical conditions, smoking, marital status, education,
residence, race, religion.
Difficulty and lack of reliability in measure-
ment of psychological and social effects prompted removal of these as possible
control variables in the study.
As discussed in Section II, homogeneity
was achieved for prior obstetric history, race, residence, and religion.
For pre-data purposes, other factors were assumed to be equally distributed
to groups of aborters and deliverers and, furthermore, not to affect homogeneity enough to require disaggregation, except in the instance of birth
order.
Por post-data analysis, these assumptions should be tested, especially
when preliminary results (see Section II) indicate that effects are present.
In general any variable not matched, or controlled for, in the two
cohorts
should be used to define new states in the chain to account for
all effects.
Separate chains have been constructed for the cohort of deliverers,
the cohort of aborters and a combination of the cohorts based on follow-up
percentages; using these we will be able to test for differences in rates
of sequelae to induced abortions due to the three groups.
2.
State flow choices
An investigation of potential event sequences for typical women led
to the construction of particular state visit sequences.
For an individual
woman, any such sequence is a possible path of the Markov chain.
These paths
were used to define potentially positive transition probabilities and to
review other criteria for the goodness of fit of the model.
22
For the pilot model the flow possibilities led to the following
transition matrix in Figure 3.
s
s
p
*
*
ch
aa
rs
W
ps
*
*
p
*
*
id
*
*
+
ch
*
*
+
aa
*
*
+
rs
*
*
+
W
*
*
+
ps
*
*
+
+
id
I
symbol
*
+
meaning
positive probability
positive, but small,probability
FIGURE 3: TRANSITION MATRIX FOR THE PILOT MODEL
For example, (rs)
+
(P) represents a contraceptive failure; otherwise (p)
cannot be entered from states besides (s).
It is seen from the initial model that certain states should be disaggregated to reflect further characterizing conditions; e.g., rs may be
partitioned into voluntary reduced susceptibility, rsv, and involuntary
reduced susceptibility, rsi.
In another example, division of the susceptibility
state was a result of the different complication rates, especially prematurity
rates, associated with different pregnancy order and birth order classes.
The problem of non-unique assignment of biological status to states
was solved by verifying, for the pre-data stage, that the set of states did
23
represent a mutually exclusive and exhaustive description of the wamen1s
status.
For the post-data stage, the actual paths generated were examined
to determine the need for additional states;
see Section IV.C.
Even though
the description is complete, it is not as detailed as it could conceivably
be.
The problem is that one cannot separate some states of health.
For
example, if the health status of a newborn child is considered for determination of possible complication states, one particular set of outcomes is
prematurity and congenital malformation, which are related.
A solution is
to either aggregate to one state (less information, but less data demand),
to disaggregate to prematurity with and without congenital malformation
and maturity with and without congenital malformation, or to design separate
matrices to achieve rates for these two complications.
In our case, rate
prevalency for the latter condition made the problem for this condition insignificant.
Another interacting variable was the possible presence of twins in
later births of either the deliverers or aborters.
Firstly, twins
have a higher chance of prematurity, hence could significantly affect measurement of rates, particularly since there are small numbers of premature
infants to begin with.
Secondly, even if the twin births could be matched
and numbers of premature births were large, how should single or double
prematurities be counted in a twin birth? As will be seen, the incidence of
twins was very low and those few paths were omitted from the analysis.
3.
Time parameter and time interval
This particular Markov chain has a natural finite state space depiction,
as outlined above.
The immediately succeeding question regards the choice
of time parameter; should it be discrete or continuous and, if discrete,
24
what is the appropriate period length? A more sophisticated modeling alternative is the possibility of using a
the time-parameter to be random.
semi~Markov
process which allows
This contingency is presently under
investigation.
The assumption of a continuous time-parameter implies that any state
change may occur instantaneously, although governed by a rate or intensity
which helps to define the transition probabilities.
Pregnancy state stays
alone would require an awkward modification of the design should a continuous
parameter be used, let alone post-partum and other post-event stays caused
by some of the other state definitions.
Furthermore, a choice of a continuous
time parameter makes the design to meet time-homogeneity assumptions much
more difficult to achieve.
Finally, data gathering efforts almost always
result in recordings taken at certain periods.
If we choose a discrete time parameter, and later assume time-homogeneity, then we must define a constant time interval between transitions.
The interval rust be short enough to avoid having multiple state changes
during a time period and to allow, to as great a degree as possible, the
structure of state definitions for stationarity.
The interval must be
long enough to make data gathering and Markovian property testing meaningful.
There are a sufficient number of possible reproductive event changes
to make any period longer than three months an unreasonable assumption.
The initial selection of the time period was three months; this appeared
consistent with a breakdown of pregnancy into trimesters, the fact that
most abortions occurred within twelve weeks of conception and assumptions
relating to possible differences in susceptible states or contraceptive
usage; it also seemed justifiable from the viewpoint of stationarity and
25
Markovian considerations.
This choice provoked changes in the state
definitions, such as analysis of the pregnancy trimesters and time considerations for various divisions of susceptibility and reduced susceptibility states.
For example, the susceptibility state separation was
necessary to represent a particular sequence of, say, three or more
periods 6, after which if no pregnancy occurred, then a transition
was made into a state of involuntary reduced susceptibility.
Let
(s) represent the three-month time intarva1 with
sl := (s),
s2:= (s,s), s3 := (s,s,s).
Also suppose that four successive quarters of exposure defines the state
of involuntary reduced susceptibility; i.e.,
(s,s,s,s)
==
rsi.
Figure 4 shows resulting changes in the transition probabilities.
sl
s3
*
rsv
rsi
*
*
s2
rsi
s3
*
sl
rsv
s2
*
*
*
*
*
*
FIGURE 4: SUBMATRIX OF TRANSITION PROBABILITIES
FOR DISAGGREGATION OF THE SUSCEPTIBILITY STATE
6 A period of exposure (to conception) is one during which the woman is
assumed to be sexually active (e,g., she is married and living with her
spouse) but is not using any method of contraception.
26
The state of yolWltary reduced susceptibility (contraception) cannot lead
to rsi except through sl' s2' s3 and rsi cannot lead to sl except through rsv.
Of course the interactions of sl' s2' s3' rsv, rsi with all other states must
be analyzed for the full transition matrix.
to one month;
4.
Later the time interval is reduced
see Section IV.C.
The Markov property
The key assumption for the stochastic model proposed is that the Markov
property is satisfied.
Since many reproductive events are dependent on
predecessor events, the chain must be structured in such a way that a woman
going to state (j) from state (i) has a common reproductive history, in
a probabilistic sense, with any other woman going from (i) to (j).
As
suggested in Section III, we might define a chain of sufficiently large order,
say k
> >
1, to approximate accurately the Wlderlying stochastic process
and transform the resulting state space to a new state space, necessarily
much larger, which corresponds to a first order chain.
A direct definition
for a first order state space must reflect all past reproductive histories
in each state to as fine a detail as the data and other assumptions allow.
In the pilot model each of the possible first and second order transition probabilities was compared for each triple of states;
e.g. ,lP [X3
=
aalX 2
=
p, Xl
=
s, Xo
=
s] versus lP [X3
This led to further disaggregation of states.
=
aalX 2
=
p, Xl
=
s, Xo
=
ch].
For example, it is immediately
recognized that probabilities for transitions from pregnancy states to an
induced abortion state will have to take into accoWlt a woman's pregnancy
order and/or birth order 7 class; see Tietze and Dawson ( 57 ) for conparative
rates of births and abortions by birth order.
The birth order class also
7 Birth order is defined as the number of previous live births or, as in
this case, as the number of previous deliveries (live births and/or stillbirths).
27
affects the probabilities of a healthy (mature) delivery versus a premature
delivery.
The first birth is at higher risk of being premature than is
the second; see Shapiro et.al., (_ 53).
Birth order did not exceed three
ln this study, and the vast majority were birth order 1 or 2.
Further details on rationale for possible transitions will not be
given for this interim model.
DoclDllentation for some of the transitions
is given for the current model; see Section IV.C.
5.
Time-homogeneity (stationarity)
Stationarity is a much used criterion for Markov Chain analysis,
certainly necessary when equilibrilDll results are desired.
In many cases
it is simply supposed that it obtains over an unlimited time horizon.
In
our case, due to (a) the reproductive span of the women in the study, (b)
the average (or typical) time-interval of observation for each individual
in the reconstructed cohort and (c) the choice of state definition and time
interval, it seems quite reasonable to aSSlDlle that this property holds
over a short time period -- e.g., five to ten years.
Again, we must judge
its merits, for pre-data analysis, on the basis of the probability law
for a homogeneous ensemble of women rather than for an individual.
Each of
the pairs of states for possible transition in the pilot, and succeeding
models, was examined for the legitimacy of the stationarity asslUIIption. For
example, IP[X = aalXs = p] where (s, t) is a time interval early in a woman's
t
reproductive life was compared to the same expression where (s, t) is later
on.
Note that, again, disaggregation to reflect birth order helps sub-
stantiate the asslUIIption.
Since maternal age is at least as important a
factor in prematurity as is birth-order, see (53), post-data analysis
plans include testing for age effects on stationarity by isolating
certain age groups of women for separate analysis and comparing the resulting
rates.
28
In the case where little or no stationarity is assumed we would need
estimates of the transition probabilities p .. (5, t) :: lP [X
1J
which may be interpreted by treating
5,
t
£
t
= j
Ixs
=
i],
T = [0,360J 8 as age parameters
for typical women in the target population and using only those women whose
paths include the appropriate ages; i.e., for s
= 60 months and t = 63 months,
we consider the aggregate paths of women between the ages of 20 years and
20
years, three months.
intervals were used, a "k - step" transition probability
would be computed as p .. (s,s+k) from the matrix 9
If
three~month
1J
k
pes, s + k) = IT
v=l
pes + v - 1, s + v),
1 'i k -.:: z,
where each pes + V-I, s + v) contains the transition probabilities
p 1J
.. (s + V-I, s + v) given by the maximum likelihood estimators:
n .. (s
+
v - 1, s + v)
n.1 (s
+
v - 1, s + v)
1J
n. (s + V-I, s + v) =
1
n .. (s, s+l)
1J
=
j
I
&
n., (5
S
1J
+ V -
1, s
+
v).
number of transitions from state i to state
during the time period (s, s+l) for the
particular cohort under observation.
J
S = set of all possible states.
,--------8 Months are numbered starting with 0, at age 15, up to at most 360, at age 45.
9 z is a bound on k for either stationary or non-stationary cases and may
be derived as follows:
(i)
The restriction for the stationary case is not severe:
k < min (to' 360) :: z
where to is the deemed length for the 5ubperiod of stationarity.
.
(ii) For a non~stationary version of the model, the bound will be
defmed by the length of the period for which a sufficient amount of non~tatio~ary data is available. For T1 =[60, 240] and s £ T1' Pi'(s, s+k)
IS ~stlffiable for k 'i 240 - s :: z.
J
29
The k-steps would be multiples of three month intervals using the corresponding
month segments of the paths,
An bmmediate caveat for this approach is to recognize the dearth, and
possible inaccuracy, of data (transitions) particularly at the I'tails" of
the paths, ages 15-20 and ages 35-45.
In our case, there are a sufficient
mnnber of paths for the early years but those segments for older women,
prior to the first reported pregnancy, may not be as well-substantiated.
However this approach will be investigated for interim years, e.g., 20 to
35 or T = [60, 240]. This provides a test of overall stationarity for
l
the period of observation, T, and can be used to guide the choice of stationarity assumptions for subperiods of Tl ,
6. Equilibril.D1l distribution
Stationarity is assumed to hold only over a limited time period;
hence we cannot guarantee existence of an equilibrium distribution valid
for representing movements between our reproductive states for a "stable"
population lO .
probabilities,
However the key to existence of limits of the transition
lim p .. (t), especially in practice, is how soon the tails
t
1J
of the sequences get within a small neighborhood of the limits.
Hence, if
the entire set of sequences "enter" their respective neighborhoods by the
time to' where to is no longer than the period over which we believe
stationarity to hold, the limits will represent an interpretable equilibrium
distribution.
This contingency, of course, will depend on the values of the
initial transition probabilities and will be examined in C 70 ) for various
values of to'
10 The existence of the equilibrium distribution if stationarity holds is
insured since a subchain of our (finite) Markov Chain is irreducible and
aperiodic, hence ergodic. (See section IV.C" Current Model.)
30
Another possible use of equilibrium probabilities is to posit bounds
for limits of the original (non-stationary) transition probabilities;
e.g., suppose that we have documented an increasing probability for a
transition from a susceptible state, after a "long" visit there, to a
state of involuntary reduced susceptibility in an "older" woman.
The
derived equilibrium probability for rsi may then represent a lower bound
on the "true" equilibrium probability for that state.
7.
Aggregation/disaggregation
It can be seen fram the various considerations above for state flow
choices, time parameter and interval selection, the Markov property
criterion, stationarity and cohort homogeneity, that the restructuring
of the model for a more accurate reflection of the underlying event sequences was achieved primarily through reaggregations of the state definitions.
Hence the feedback process involved consideration of the
individual criteria given above, which proyoked iterative reaggreation
of the states and criticizing of the successive'aggregate versions from
the viewpoints of the given criteria.
8.
Closed classes of the Markov chains
Formally, death and (irreversible) infertility would be absorbing
states in the M.C. models.
However, rare prevalency of the first state and
inability to fully document the second condition indicated that these be
omitted from the chains ll .
Involuntary reduced susceptibility is included
and serves the purpose of the infertility state; see Section rv.C.3.
11 Surgical sterilization is not performed in Yugoslavia except for
extreme medical necessity.
31
There are no reflecting states and no
p~riodicities.
There is a
closed class of states for post-delivery transitions and certain states
may be made absorbing for separate analysis, e.g., premature and mature
deliveries, see Section IV.C.4.
the existence of at least one
The aperiodic property is equivalent to
non~zero
diagonal element.
Moreover, there
is a possibility of using a retention model for further analysis of "stays"
in states of contraception and susceptibility; see Henry (127 ).
9.
Implication for data needs
The above considerations, especially as they relate to both the
number and type of state definitions necessary for an accurate representation
of the flow of women, have implications for the quantity and quality of
data needed; see Section II for a brief discussion.
Details on the design
for data gathering, questionnaire, interviews, and quality control are
given in (
22
) and detail on event coding, path coding, data pre-
paration for the transition matrices, computer program structure 1 parameterizations for many of the research questions, sensitivity analysis,
testing and other specifics for the Skopje, Yugoslavia data base will appear
in (70 ); see also Section V.
Some discussion on the dimension of the
current models is given in Section IV.C.4.
10. Testing
Categorical and other "fit n tests for the models are part of the important testing criterion, albeit for post-data use.
linp1ementab1e tests is deferred to (
70 ).
Consideration of data-
32
C.
TI1e Current Models
1.
The network flow diagram
As presently aggregated, the current models have been constructed
sw)sequent to data gathering, coding and preparation for the transition
matrices but do not take into consideration any modifications resulting
from initial runs or as yet unperformed testing.
The models presented below are finite state, discrete tilne interval
Markov chains with transition probabilities which are stationary over a
limited time period.
As indicated above, the length of the stationarity
subperiod will be investigated.
Furthennore, we note here that the sub-
period limitation will not be a result of truncating the data but of
testing for modeling accuracy.
The possibility of multiple event occurrences within three-month
periods provoked a decision to use one-month intervals.
The path data
was coded by month of occurrence and cell data for the resulting enlarged
chain actually increased or stayed the same for all states.
The original design for state definition came from the logical possible
reproductive sequences and resulted in a typical network for particular
subclasses which looked like that in Figure 5..
INSERT FIQJRE 5 ABOUT HERE
33
•
s 1 - s8
\
•
\
\
\
\
\
\
\
\
FIGURE 5:
c=:>
denotes single state
c=J
denotes colection of states
TYPICAL NET'·!nRK FnR SUBCLASSES OF THE MARKOV CHAIN
34
~
Early period of susceptibility, first eight months,
possibly following some other reproductive event
rsv
=
Ninth month of susceptibility, prior to pregnancy or rsi
=
Collection of states which are defined by various types
of contraceptive usage (reduced susceptibility, voluntary)
rsi
""
Involuntary reduced susceptibility
m - m6
1
m7 mg
=
First six months of a pregnancy
=
Last three months of a pregnancy
ia
=
Induced abortion
sa
=
Spontaneous abortion
md
=
Mature delivery
pd
=
Premature delivery
In addition to possible transitions shown above, self-loops can be made
for the states, or collections, rsv and rsi but not for the states Sj' mk ,
ia, sa, pd, md.
The data observations were not coded as these transitions but were
reproductive paths of the women coded by events.
Hence the precise definitions of the states are required to be a function of the events 12 .
'Dle set of event codes is given in Appendix 1.
The event codes are a
key to accurately determining membership for a given state but, for the
particular questions here, are not needed independently as outcome measures.
A network flow diagram including the current model is presented in Figure 6.
Since some states are in collections, it is necessary to look at the transition matrix in Appendix 2 to determine the class of all possible (positive)
transi tions.
12 A rather complex computer program was needed to translate the event-coded
data into transition paths, including test questions, definition and error
checks. Further details will be available in (70 ).
35
LEGEND:
State, or group
of states
Possible
transition
Weak
transition
)
1
/
FIGURE 6.
EXPANDED FLOW DIAGRAM INCLUDING
AGGREGATION OF STATES FOR THE CURRENT MODEL
36
INSERT FIGURE 6 ABOOT HERE
---------------------------------~----------------
2.
Current state definitions
Table 2 lists the formal state definitions giving both descriptive
and operational characterizations.
INSERT TABLE 2 ABOOT HERE
3.
Justification for the state definitions
The current state flow choices are necessary for an exhaustive representation of all possible state visits
and for an accurate selection
of the states corresponding to prematurity rates and other rates differing
by birth order of the mother.
'that is, one subclass of states is required
for first pregnancy deliverers, another for first pregnancy aborters and
at least one other for reproductive events following first deliveries,
whether these are first deliveries of first pregnancies or first deliveries
for women with a history of abortion.
These three subclasses are referred
to as groups 1,2, and 3 respectively and are so numbered in Figure 6.
The presently proposed categorization may be somewhat limited by the
assessed scope of available data; e.g., we have not currently reflected
smoking pattems. 13
13 A continuation grant has recently been funded and part of the subsequent
work will investigate smoking patterns.
37
TABLE 2:
State
rsv
rsv
rsv
rsv
3l
32
33
34
STATE DEFINITIONS FOR THE CURRENT MODEL
Definition
Operational Definition 14
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-3J or 57-58, when
transition is to ia or sa 3 . If no
3
code 31-37 or 57-58, progression
continues to m7P3.
induced abo~tion, post-delivery (After at least one 38-46 or 68-76),
code 31-36 or 58 (only after code 25
or 26).
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 rod or pd 3 •
3
14Nurobers correspond to event codes
in Appendix 1.
38
State
Definition
Operational Definition
mature delivery.
post-previous delivery
(After at least one 38-46 or 68-76)_
code 38-46 (only after code 25 or 2 .
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 •
~
1
(In absence of preVl0US coae 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 2 •
Z
(In absence of preVlOUS
coae 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 m7P2.
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)
e
(In absence of previous code 38-46 or
68-76 but after previous 31-36 or 58) •
code 15-16
39
State
ia
sa
2
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 saZ. If no
2
code 31-37 or 57-58, progression
continues to m P1.
7
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 Z5 or 26)
This is each woman's initial state,
and code 03 in absence of code 11-26.
first (to ninth ) month
(In absence of previous code 25 or 26)
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
susceptibility~ pre-pregnancy
40
Within these subclasses, provision must be made for different rates
associated with states of susceptibility and both voluntary and involuntary
reduced susceptibility.
The characterization must reflect the case when
a wonmn has been susceptible and exposed for a sufficiently long period
to infer that she is "infertile"
~~.
whether temporary or not - - which in
our case is designated as involuntary reduced susceptibility.
The current
assumption for the transition into a state of involuntary reduced
susceptibility is a stay of nine months susceptibile and exposed.
defines the limit on the stay in susceptibility states.
This
Ultimate versions
of the model will reflect sensitivity analysis on the time at which continuing susceptibility and exposure lead to involuntary reduced susceptibility.
Distinction between the various types of con~raceptive devices and
birth avoidance experience led to the selection of state definitions within the rsv collection for each group.
As may be seen in the event code
list, there is allowance for eleven possible contraceptive usages.
These
were structured into four reduced susceptibility states based on a
graduated scale of assumed effectiveness of the contraceptive device or
practice.
(See the event code list and operational state definitions for
rsv .. with respect to groups i=l, 2, 3 and contraceptive collections
1J
j=l, 2, 3, 4.)
As an example of a state for which the definition is not as straight-
forward, and perhaps somewhat less precise, consider rsv
, i = 2, 3. A
i4
woman may have demonstrated that she is sexually active by having tenninated
a pregnancy, but during an interview may report that she was not living
41
with a man at the time in question, nor was she using any kind of contraception, nor was she pregnant; hence there is no event code for her.
If
she states that she is indeed abstaining, the event code llllll is entered
and she is placed in rsvi3 (safe contraceptive method) . Without such a
statement we feel we cannot consider her to be continuously abstaining-perhaps most of the t:iJne, but probably not continuously--so we have a
contraceptive state of asswned abstinence.
In the instance of rsv l4 ,
the asswned abstinence period continues to the time of first conception,
unless there is a marriage before it or the woman reports use of a contraceptive.
It
was deemed important to include the various contracepting states
not only for the initial and substantial reason of including all interim
periods for the women but also to test potential interactions between
these and other states.
If no relationships with other states exists,
there is no loss since the extra states don't affect data availability
\
,
except for themselves and data demands in terms of cell count and other
checks are being made. 15
In ensuing proposed research, we hope to
investigate interactions between contraceptive states and abortions.
By referring to the three groups in the network flow diagram,
operational state definitions and the appropriate submatrices containing
potentially positive transition probabilities (see Appendix 2),
possible reproductive paths were checked for structural justification of
the Markov puoperty with respect to the current set of state flow choices
and the choice of a monthly unit for the time parameter interval.
15
The numbers of cell observations for most of these states in the Skopje
data exceed the minimum required for reasonable confidence intervals on
the corresponding maximum likelihood estimators.
42
By construction 1 appropriate Markovian dependence has been built into
the currently specified states and their organization.
This type of con-
ditional dependence for pregnancY1 susceptibility and reduced susceptibility
states is inherent and leads to the first order Markovian dependence
structure assumed for the outcome states of induced and spontaneous abortion,
premature and mature delivery and loops back into susceptible and reduced
susceptible states.
4.
Matrix and matrix density classification
States in groups 1 and 2 of the network flow diagram are transient
and those in group 3 are persistent, in fact, ergodic.
There could be
some women who stay in rsi 2 or rsi 3 over the full time horizon due to infertility but, in general, these states are not absorbing. The transition
matrix (Appendix 2) is in canonical form with transitions for the 27 ergodic
states in the upper left and the two groups of 27 and 2S transient states
In the middle, and lower right.
The full transition matrix P has 79 x 79 = 6241 cells of which 374
are potentially positive, a density of 6.0 percent.
made absorbing and an initial distribution with
If pd
(0)
TI.
la
2 and md 2 are
1 is assumed,
2
then group 2 (the aborters) may be studied in isolation.
The corresponding
submatrix of P has 27 x 27 = 729 cells of which 130 are potentially positive,
a density of 17.8 percent.
Figures for all groups are given in Table 3.
43
TABLE 3; MATJHX DENSnx
~
GENERAL J-1ARKOV CHAIN
Groups
1
2
3
full chain
25
27
27
79
Number of cells
=s
=c
625
J29
]29
6241
Potentially 16
positive cells
=r
114
130
130
374
Density
= ric
0.182
0.178
0.178
0.060
Number of states
records the figures for the Skopje,
The continuation in Table 4
Yugoslavia,data base, using the actual transitions.
TABLE 4: MATRIX DENSITY - SKOPJE MARKOV CHAIN
1
Actual positive celll 6 = ~
~
= rIc
Density
Gro,s
73
85
114
272
0.117
0.117
0.156
0.044
Cell frequency and related figures will be given in ( 70 ).
a particular
for any £
+j
p ..
1J
+
1; J
has the value 0 by definition.
= 1,
..... ,8;
k
Full Chain
3
= 1,2,3.
In many cases
For example,
Furthermore,
p £, (ia ) = 0
Z
j
= 1, ..... ,6; k
= 1, 7} .
In other cases, a particular p.. has been assigned the value 0 by
1J
structural as Sl.UIlp t ion ; e. g., we may have
16
A
The numbers r and r may be considered to be the dimension of the matrix
n~del, pre- and post-data.
44
for ~
£
{ski'
rSYkj
i:::: 1" .... ,8; j :::: 1, .... .,4},
k:::: 2,3;
but we assume
p Cia
since entry to
rs~,
), (xsi ) :::: 0
k
k
k :::: 2,3, can be only after 9 months of susceptibility.
Roughly speaking, we assume immediate successor states of ski' rsvkj
after all outcome states: iak , sak , pdk , m~. This determines positivity
for many of the transition probabilities.
As the Skopje data show, there may be many more actual zero transitions
for a given data base than the theoretical model predicts.
Most of these
observed zeros are interaction transitions with respect to contraceptive
choice from other rsvkj states or Skj states; some are for late aborters.
For the estimation problem, one needs to consult the data. However,
as we did when designing the study, it is possible to give some idea of
the available observations for the maximum likelihood estimates.
We had
interview information and event-coded transition paths for 948 women.
The
(weighted) average path length for the women was approximately 11.3 years
or over 135 transitions per path.
we used 927 paths.
Due to some deletions, explained in (70 ),
Hence there are approximately 927 x 135 :::: 125,145 tran-
sitions for cell estimation.
17
Of course, these are not distributed uni-
form1yover the.273 positive cells in the study, but there should be sufficient
data for any statistical tests we may later make for most particular cells.
lienee, at least for this data base and chain structure, the
pre~
or post-data
dimension of the matrix is not too large for the number of observations.
17 1ne actual numbers of transitions were 26,607 for the aborters and 99,055
for the deliverers or a total of 125,662.
e
45
V.
A.
USES OF TIffi MOnEL AND PARAMETERIZATION
FOR TIffi RESEARlli QUESTIONS
Aims of the Modeling
Ultimately we expect to use the chain representation for all three
groups (deliverers, aborters and post-deliverers) as currently defined.
With this structure, and later modifications, we want to obtain rate estimates and other cohort percentages for the reproductive variables of
interest to epidemiologists, physicians and policy makers.
Use of the re-
sulting parameterizations has an essential statistical inference objective
as well as being descriptive and predictive. Maximum likelihood estimators
are obtained from the path-coded data; functions of these may be used
for point estimation, interval estimation and hypothesis testing.
Prototypical
constructions are given below.
The design of the current model
IS
tempered by the quantity of data
available from the Skopje study which has limited our ability to obtain
time-dependent rates for prematurity; this difficulty is indicated in
Section V.B.5.
Many other research questions, including time to delivery,
mean stays in contraceptive states, etc., are apparently estimable from
the Skopje baseline data.
B.
Parameterization for the Research Questions
1.
Direct computations
Some questions of interest are answered directly from the estimated
transition
18
rnatri:x:
See Appendix 2.
18
pregnancy ;rates fo;r given Jllonths of susceptibility,
46
contraceptive failure rates, rates for our de.finition of infertility (involuntary reduced susceptibility), abortion rates and others _. for all three
groups in the network for a single time period.
Prematurity rates, for a
single period, must take gestation differences into account.
If Yk represents
the probability of a prematurity in group k, given that a delivery has
occurred, then the correct expression is
3
Yk = ill [
U {X n = (pdk )}
Ixo
= (m 7Pk)]
n=l
since entrance to m7Pk assures an outcome of md or pd .
k
k
ponding subnetwork is given in F~gure 7 .
•
.
.. .
FIGURE 7:
···
TYPICAL DELrVERY NETWORK
The corres-
47
z.
Time to delivery
Other questions require components of the n-step transition matrix,
(Pij (n))
= pll,
or first-passage time probabilities, (f ij (n)).
known relationship between the above is given by
The well-
n
Pl"J'
where p .. (0)
1J
en)
=1,
t
:=
1ll=1
n ~ 1,
f .. (}n)p .. (n - m),
IJ
JJ
for all j.
Hence, we may iteratively determine
first passage times by
n-l
f .. (n) =
1J
p .. (n) 1J
l f .. (m)p IJ.. en - m)
m=l 1J
n > Z
n
p ..
1J
=1
Questions of particular interest here do provoke same simplifications.
For
example, suppose we are interested in certain time-dependent probabilities
for the aborters only.
We may assume some initial distribution on group
Z of the network and, without loss of information, make the states
(md Z) and (Pd Z) absorbing.
T = {ia Z' sa Z'
i
2j
To siIiIplify the notation, we write
(j=1, ... ,9), rsv Zj (j=1, ... ,4), rSi Z' mj PZ (j=1, ... ,6)},
a subset of the group Z states, denote
a
= a group Z induced abortion
n = a group Z premature delivery
11
;:
iaZ
~
pd Z
= a group Z mature de1iyery
::; md Z
(j = 7,8,9). 12 is the corresponding submatrix of P.
and write j = 7,8,9 for m P
j Z
This subnetwork, with n, 11 absorbing, is shown in Figure 8 •
48
FIGURE 8: GROUP 2 NETWORK WITH ABSORBING STATES: N,
Expressions for transitions to the prematurity state are:
W [premature delivery at month n for first time
at month 0.]
-
I
an induced abortion
for the network NI :
Pan(n)
= IP[premature
delivery at month n
I Xo = a]
n-l
fan (n)
L
+
m=l
fan (m)Pnn (n - m)
n
L
m=l
=
fan (m)
W [premature delivery ~ month n
I Xo = a] ,
since pnn (t) = I for all t. Thus the ordinary n-step transition probabilities
give the cumulative percentages of premature deliveries by month n.
This, of course, makes a recursive solution for first passage probabilities
very simple in terms of required data.
If we are interested in times to delivery, we could compute the corresn
ponding p
a~
L f (m) and calculate pan (n) + pa~ (n), for the months
m=l a~
An alternate approach recogn:j.zes that entrance to m PZ insures
7
(n) =
of interest.
49
delivery within three months.
Pa7(n)
=
=
n-l
f a7 (n) + mIl f a7 (m)P77(n - m)
f
a7 (n)
= n> [birth (mature or premature) at n+l, n+Z, or n+3 for the
first time I Xo = a] ,
since P77(t) = 0 for all t > O.
In this case, to obtain cumulative per-
centages, we need
n
L P 7(r)
r=l a
= lP
[birth ~ month n+l, n+Z or n+3
I Xo = a]
In general, this would require cumulative computer storage of all powers
of the matrix p~, r=l, ... ,n; to avoid this, the network may be modified,
as in Figure 9.
FIGURE 9: GROUP 2 NETWORK WITH ABSORBING DELIVERY STATE; N2
With m7PZ absorbing, and the tilde representing calculations for NZ'
so
n-l =
f
(n)
+
L f (m)P77(n - m)
Pa7 (n)
a7
m=l a7
n
L
f
m=l
a7 (m)
= lP [birth Ql. month n+1, n+Z or n+3
I Xo = a]
since P 77 (t) = 1 for all t; in addition,
Consequently,
p
a
n
7(n) =
n
f 7(m) = L p 7(r) and the required probability
m=l a
r=l a
L
can be obtained directly from
P~ for network NZ' avoiding the cumulative
storage required by network Nl . (P Z is the stochastic submatrix of Pz
with m7PZ absorbing.) Pa7(n) gives the percentage of births occurring by
Fractiles may be obtained directly from a graph for either N
l
or NZ since pan (.) + Pa~ (.) and Pa 7(') are cumulative distribution functions.
time n+3.
For studies of rare conditions (low prevalence percentages) it is important to have a sufficient number of individuals in the cohort at intake
and to perform follow up studies for a sufficiently long time to reduce
variances in the estimators of prevalence and incidence rates.
In particular,
the time to delivery estimators derived above were used to select the period
of follow up and to estimate the number of probable deliveries for cohorts
to be used in another proposed historical prospective study of long term
complications subsequent to induced abortion. 19
In this particular case, the
estimates indicated the need for a longer follow-up period than that originally called for in the request for proposals.
More detail will appear
ln ( 70).
19 "Low Birth Weight and Other Complications Subsequent to Induced Abortion"
contract proposal submitted to NIH, Carol J. Hogue, Principal Investigator.
51
An alternative derivation for the (transient) first passage times
for network NZ is obtained directly.
a slightly modified version of PZ:
Q.z
=
I
-
I
0
~
- -
Select j arbitrary in T and write
0
- - - - -
p
0
.
.
I
I
Q.
T
0
the top two rows in Q.Z are for ID7PZ
P = P(ID PZ), (ID PZ)·
7
6
LHvMA
and
ID6P Z' respectively, hence
QT is the matrix corresponding to' transitions in T.
5.1.
=
L Pjv
f V7 (n-l) ,
n >
Z
vsT
n
=I
sz
Proof:
= JP[Xn = 7; Xm + 7,1
f j7 (n)
= L F [X = 7; X
n
m
+7,
< m < n-l 1210
= j]
2 ~ m ~ n-l; Xl = v IX = j]
0
!
+7
v
2:
ll' [Xn
= 7;
Xm
f
7,2:e m :e n-l; Xl
lP
[Xl = v;
= V; X = j]
o
=
V
£
=L
v
£
T
T
p.
JV
X
= j]
IF [X
o
j]
n > 2
f 7 (n - 1),
v
by the ~hrkov property.
fj7(n) is the absorption probability from
=
j
£
D
T to the absorbing
state m7PZ. Since the latter state is fixed and T has cardinality 22,
we write the column vector:
and have
i(n) = QTf(n-l), n
i(l)
=
> 2
(p,O, . . . , 0)'
or
3.
Mean time to delivery
Let Aj7 be a random variable representing the time to absorption
in m7P2 from j £ T By a standard result, see (123 , p. 73) or (130,P.46 ),
53
the mean number of times the process visits a state k
£
T before being
absorbed in m7PZ' having initially started in state j £ T is the (j, k)
-1
element of the fundamental matrix (I - QT) • Denote this by Ajk and
note that the mean time to absorption from j
£
T is
= L
AO
k£T Jk
This result, however, is for the limiting case.
If to is a bound
on the period over which it is reasonable to assume stationarity then
we must compute truncated means
= E[AoJ 7 I A0J 7 -<
t
0
]
using the (transient) first passage times.
compare
Tj
(to)
It will be of interest to
and A for selected values of to and appropriate
j
initial distributions on group
z.
The limit values may be useful since
we believe that the bias resulting from the stationarity assumption
may be predominantly in one direction; namely, there would be an
overestimate of the number of births.
Absorption probabilities also come from the limit case but by
the structure of the network NZ' for JET,
00
f j7
= L f j7 (n) = IP[ultimate absorption in m7PZ Ixo = j] = 1
n=l
and for ultimate absorption in pd Z or md Z' Yz and l-yZ give the respective
probabilities; see network NZ. As with mean times to absorption, similar
truncated expressions using f. (n),. f. {n) and an initial distribution
J7T
on group Z should be used.
Jl1
S4
Ntunber of visits to, or mean times in, contracepting states prior
to deliveries also come from the ftmdamenta1 matrix.
Time in susceptibility
states is presently limited to nine months, so fracti1es for the
distribution of times in those states prior to pregnancy will be of
interest.
4.
Intervening conditions
We are interested in the rate of term pregnancies as a ftmction of
n, but also wish to impose intervening conditions.
For example, we may
restrict the ntunber of intervening abortions to zero, one, two, etc.
There are two ways to interpret this restriction in terms of the n-step
transition probabilities.
We again, refer to group 2 for a concrete
example and use network N .
2
First we need expressions for taboo probabilities, see (123, p. 109 ).
Suppose that we wish to avoid state h
N2 . We may define taboo
probabilities for either ordinary transitions or first passage times.
= lP [Xn = k; Xm +h, 1
= lP[Xn
=
£
-< m -< n-1
IX0 = j]
k; Xm J.t {h,k}, 1 < m< n-1
Ix0
=
j]
An expression which looks at the group of aborters at time zero and
later looks at those who have had no intervening abortions and who have
carried to term is the taboo probability
due to the structive of N2 . If we are interested in a subsequent abortion
for group 2 at a given time n but with no intervening abortions, then
55
these expressions reduce to ordinary first return probabilities
= f aa (.n) -- apaa (n) .
f
(n)
a aa
Other index definitions are used for non-contraceptive use, etc.
There is, however, a second interpretation of the question.
What
if we wish to look at the set of aborters who have had no subsequent
abortions up to a certain time.
The taboo restriction then becomes a
prior condition and the expression, for k either a delivery or an
abortion, is:
n-l
(5.1)
IF [X
n
= k i n {X
m
f
a}; X = a] ,
o
n > 2.
m=l
We note that the intersection would be implied by the event
n
n-t
{Xm
f
a}
m=3
since a period of susceptibility and pregnancy would have to follow to
allow for an abortion and t > 3 depends on whether k
is another
abortion or a delivery.
It turns out that these two viewpoints have corresponding interpretations
in epidemiological studies. The probability hfjk(n) may be used for computation
of rates in prospective or historical prospective studies and the formula
in (5.1) is the corresponding probability for retrospective or case-control
studies.
This will be discussed in detail in (71).
The special type
S6
of prior condition first passage taboo probability
n-l
IP[Xn = k i n {Xm 4 (h,k)}; Xo = j]
m=l
is related to the usual first passage taboo probability by the following
h9- jk (n)
==
formula, where S is the set of states for an arbitrary (finite)
Markov chain.
Theorem 5.2 20
where
n-l
hg · . (n)
)1
and
=
I
hf .. (m) ,
1
m=l)l
= h,k
n > 2.
A similar result holds for ordinary (non-first passage) taboo
probabilities and a corresponding definition of prior condition taboo
probabilities.
Proof
n-l
Let Eo
-n
{X
m
4 (h,k) }.
= IP [~ = k I
Then
m=l
Eo; X
o
= j]
n-l
and
lP (E )
o
= IP [
n u
m=l
{X
m
= r
}].
m
rES
m
r m +h,k
20 If the probabilities of ultimate absorption exist, hfjh
+
hfjk = 1
then the denominator of the right hand side is the tail of the ultimate
absorption probability distribution.
57
After commutation with the intersection, the union may be decomposed
so that
n-l
lP (Eo)
=
U
lP [
rES
m
rm
{Fr(n)
n
U
m=1
We then note that for
n-l
q
q
+h,k
= (r1 ,
~(n)
q =1
{X = r }].21
... , r - ) the events
n 1
: r m E S, r m + h,k, 1 < m < n-l}
are disjoint, where
n-l
F
!(n)
-n
q=1
{X
q
=r
q
}
Consequently,
n-l
IP(EO) =
L
IP[H
I I {X
L
m=1 rmES
rm+h,k
q=1
q
= r q }]
and
(5.2)
L
m=1
n
n-l
n-l
IP [XO = j;
q=1
{X = r }]
q
q
S8
n-2
JP [XO = j] PJo r
l
(II Pr T' )
q=l q' q+l
o
by the Markov property , with the convention II, ( ... ) :: l.
q=l
A snnilar argument with the post-condition tXn = k} yields:
n-2
n-l
(5.3) JP [Xn = k; EO; Xo = j] = 1:
JP [XO = j] p
(II P s
) Ps
k
JSl w=l sw' w+l n-l'
t=l
°
and from (5.2), (5.3) the conditional probability for h£jk(n) is
lP [Xn = k
I
EO; Xo = j]
n-l
(5.4)
n-2
1:
PJosl (II Ps s
)ps
k
w=l w' w+l n-l'
t=l
n-l
n-2
I
(II
q=l
m=l
Pr r
)
q' q+l
The definition of the first passage taboo probabilities results in
JP[X
n
=
k·
'
MU
t=l
StSS
stfh,k
n-l
IP [X
n
= k·'
U
U
t=l StES
stfh,k
n-l
= 1:
t=l
{Xt = St}
I Xo =
j]
59
by an argument like the above.
Furthennore, by the Markov property,
n-l
I
(5.5)
n-2
I
t=l St eS
stfh,k
PJ·sl (IT Ps s ) Ps
k
w=l w' w+l
n-l'
Observe that (5.5) is the numerator of (5.4) and we must verify the
theorem statement for the denominator.
We do this by induction. Denote
the denominator of (5.4) by G(n) and let H ~ {h,k}.
where we use the conventions:
o
11
( ... )
For n = 3,
.. (l) = f .. (l) = p .. for i= h,k and
hf J1
J1
J1
= l.
w=l
We then assume that
G(n) = 1 - hgjh(n) - hgjk(n)
for all n > 2.
(With the conventions, the case for n = 2 is trivial.)
Now consider
n
G(n + 1) ==
L
n-1
I
PJ"r
m==l r fH
m
(IT Pr
1 q==l
r
)
q' q+1
n-1
==
==
I ... r I
r1fH
~
r1 H
P "r (IT Pr r
)
nfH J 1 q==l q' q+1
n-2
···r
n-1
tH PJor1 (IT
Pr r
)
~ Pr r
q==l q' q+1 r H n-1' n
n
== [1 - hgjh(n) - hgjk(n)]
n-2
- r tH ... r
1
n-1
~ pO r (IT Pr r ) Pr n-1h
H J 1 q==l
q' q+1
by (5.5); hence we get
G(n+l) == 1 - hgjh (n+l) - hgjk(n+l).
Since the numerator for h£jk(n+l) is hfjk(n+l), we are finished.
0
COROLlARY 5.3
Corollary 5.3 has an interpretation for bias estimation in epidemiological
studies which will be discussed in (71).
61
5.
A ratio for time-dependent prematurity rates
The fOTIllula for y
for group k,
gives an incidence measure for prematurity rates
k
k = 1,Z,3.
If we are interested in comparing prematurity
rates for aborters to deliverers as a function of time, we must employ
the transition probabilities
aTT (n) = IP [Xn = 'IT I Xo = a].
But we need to compare the ratio of prematurities to total deliveries.
p
1bis is due to the fact that, at any time n, individuals who started 1n
a = 1a
the abortion state
~
may not have reached either TT
Z
~
pd Z or
= md Z' Hence we postulate that the ratio is
YZ(n) = PaTTen)
/
[PaTTen) + Pa~(n)]
for n > 9 (to allow time for the intervening susceptibility and pregnancy
states).
To show that this is correct, let {B.: 1
J
~
j
distribution for group Z, with md Z and pd Z absorbing.
aborters having subsequent premature births is
sz(n) =
~(n)
/
for i
=
TT,
~
and n
>
1.
J
27} be an initial
The percentage of
[~(n) + q~(n)]
where q.(n)
is a state probability and
1
Z7
27
q.1 (n) = L q. (n-l) p .. = L
j=l
~
J1
j=l
B·p·· (n)
J J1
For aborters, B = 1 and B. =
a
J
a for
all j
+a.
Hence q.1 (n) = pa1. (n) and Sz (n) = yz (n) for all n.
In general the sequence YZ(n) would be compared to Yk' k = 1,Z,3,
although the particular interest is for YZ(n) versus Yl'
62
The behavior of YZ(n) is of interest over n
Ll~
~
9.
5.4 The sequence YZ(n) is increasing (decreasing) if and only if
the rate of increase for prematurities is not less than (not greater than)
the rate of increase for mature births.
n
Proof.
p. (n) = L f. (m) for i = n, ~; hence the two sequences
en
m=l en
Thus
vZ(n) is a ratio of increasing
Pan (n) and pa~ (n) are increasing.
,
sequences.
yz(n)
~
YZ(n+1) <=>
pan (n) [pan (n+1) + pa~ (n+1)] <- Pan (n+1) [pan (n) + pa~ (n)]
<=> P
an
(n+1) / pan (n)
~
pa~ (n+1) / pa~ (n).
o
The number of observations of prematurities subsequent to abortion available in the Skopje data base, although sufficient to allow incidence measure
estimates and non-time-dependent statistical tests, was not sufficient to
reaggregate the Markov chain formulation to account for time differences.
The existing chain structure reflects this; i.e., there is sufficient cell
data for this structure but not for a structure which would better
characterize time dependency.
As a result the YZ(n) sequence does capture
a jump in the prematurity rate due to gestation, but becomes constant as n
goes much beyond month 11 or lZ.
LEMMA 5.5
YZ(n) = yz for all n
structure within T.
~
nO' where nO depends on the network
63
Proof.
Consider the network N3 in Figure 10.
n
1-8
FIGURE 10: GROUP 2 NETWORK FOR Y2(n) RATIO ESTIMATION; N3
U
= {mj Pz :
j
= 7,8,9} and since we are assuming that
can be seem that n = YZ'
IP [Xo
= a] = 1, it
There are, of course, transitions within T and U.
However, the limiting behavior of YZ(n) may be considered by looking at
the transitions for N3 , involving only the probabilities 8 and
which we assume to be in (0,1).
n = Yz,
Moreover, there can be at most two tran-
Let m = men) be an increasing subsequence of the number
sitions within U.
of transitions within T and
is obvious that for i =
U
determined by transitions in
N •
3
Then it
TI, ~
S m-Z (1 - 0) £,
f Ti (m) = ( :
m
~
Z
m = 0,1
where
E:
TI =
Y2'
E: ~
=
1 - YZ.
From our previous argument, for m ~ 2
64
m
PTi (m)
= L
f - (k)
k=l T1
=
f:.
(I_om-I)
1
and YZ(m) = En that is, YZ(m) is constant for m ~ Z or for some n
~
nO
where nO depends on the sequence m = men).
We also see that lim PT-(m) = E .•
J'Tt+<X>
1
1
D
SO, ultimately, each of the time-to-
delivery transition probabilities (premature or mature) converges to the
probabilities for immediate transition,
yz
or 1 - YZ.
For our data
nO is approximately 13.
To reduce the problem we may use separate chains for groups of different birth
order , groups defined by prematurities within time intervals and groups
of multiple or spontaneous aborters versus individuals having exactly
one abortion.
For example groups may be defined for each year sub-
sequent to an induced abortion and prematurities recorded for each of
these.
The current data base will provide a start on restructing the
chain for these estimations but apparently does not provide a sufficient
number of observations for statistical accuracy for the time-dependent
prematurity ratio.
Currently under consideration are two studies which
may possibly add useful data for this question.
lienee, we have a sufficient number of observations for certain time
dependent rate questions other than prematurity and time-to-prematurity rates
but not time-dependent prematurity rates.
Details on the possibilities of statistical testing of the Markovian
property and stationarity (lZ], 131) as well as sensitivity analysis results
for transition matrix components, time to infertility, equilibrium distribution
comparisons and the other parameterizations suggested above are deferred
to (70 ).
e
65
ACKNOWLEDGEMENTS
1his report
is being issued simultaneously as Biostatistics, Institute
of Statistics Mimeo Series number 1009 and Operations Research and Systems
Analysis Technical Report number 75-4.
The research was supported primarily by National Institute of Child
Health and. Human Development Grants number 5-ROl-1ID07214 and l-R01-HD09028,
llidmrd II. Shachtman, Principal Investigator.
We acknowledge gratefully our debt to the late Mindel C. Sheps who
encouraged our research at its beginning and critically reviewed our first
grant proposal.
Thanks also to John Schoenfelder for building a complex
computer program for transfering event-coded data paths into state transibon paths.
66
BIBLIOrnAPHY
For ease of reference, articles and books are broken down
into four categories. Parts I, II, and III of the list are intended to
be reasonably thorough but not exhaustive.
PART I
Substantive references on abortion,
prematurity, smoking, etc.
PART II
Markov chain and other stochastic models
concerned with reproduction.
PART III
Other Markov chain applications for health
system problems.
PART IV
Some general Markov chain and testing
references.
67
REFERENCES
1.
Substantive references on abortion, prematurity, smoking, etc.
1.
Anonymous, "Smoking, Pregnancy and Publicity."
(1973), p.61.
2.
Barsy, G., Sarkany, J., "Impact of Induced Abortions on the Birth
Rate and Infant Mortality." Demografia, Vol. 6, (1963) 'pp.428-67.
3.
Bracken, M.B., et. al. "Correlates of Repeat Induced Abortions,"
American Journal Obstet. and Gynec, Vol. 40 (1972), pp. 816-25.
4.
Burch, P., "Smoking and Pregnancy,"
p. 177.
5.
Burch. P., "Smoking Hazard to the Fetus," British Med. J., Vol. 1,
(1974), p. 40.
6.
Butler, N.R., Bonham, D.G.,
London: Livingstone.
7.
Butler, N.R., Goldstein, H., "Smoking in Pregnancy and Subsequent
Child Development," British Med. J., Vol. 4, (1973), pp. 573-81.
8.
Comstock, G. W., et al. "Low Birth Weight and Neonatal Mortality
Rate Related to Maternal Smoking and Socioeconomic Status,"
American Journal Obstet. and Gynec. Vol. 111 (1971), pp. 53-59.
9.
Cornfield, J., "A Method of Estimating Comparative Rates from
Clinical Data. Applications to Cancer of the Lung, Breast, and
Cervix," Journal of the National Cancer Institute, Vol. 11, (1951),
pp. 1269-75.
Nature, Vol. 245,
Nature, Vol. 246, (1973),
Perinatal Mortality,
(1963),
10.
Czeizel, A., Bognar, Z., Tusnady, G., Revesz, P., "Changes in
Mean Birth Weight and Proportion of Low-Weight Births in Hungary,"
Brit. J. Prevo Soc. Med., Vol. 24, (1970), pp. 146-53.
11.
Ford, C.V., Castelnuevo-Tedesco, P., Long, K.D., "Women Who
Seek Therapeutic Abortion: A Comparison with Women Who Complete
Their Pregnancies," Amer. J. Psychiat., Vol. 129 (1972), pp. 58-64.·
12.
Fuchs, V., Houdek, J., Kral, J., Sistek, J., "Influence of
Artificial Interruption of Pregnancy on the Future Pregnancy,"
f~Gynek.Vol. 34, (1970)
,pp. 163-64. Reviewed in Excerpta Medica,
Vol. 23, (1970), p. 58.
13.
Furusawa, Y., Koya, T., "The Influence of Artificial Abortion on
Delivery", Harmful Effects of Induced Abortion, (1966), pp. 74-83.
Tokyo: Family Planning Federation of Japan.
68
e14.
Goldstein, H., "Birthweight and the Displacement Hypothesis."
Vol. 95, (1972), p.l.
Am. J. Epid.
15.
Goldstein, H., "Cigarette Smoking and Low-Birth-Weight Babies,"
Am. J. Obstet. and Gynec. Vol. 114 (1972), pp. 570-1.
16.
Goldstein, H., "Smoking, Pregnancy and Publicity," Nature, Vol. 245,
(1973) pp. 467-8.
17.
Goldstein, H., "Smoking, Pregnancy and Publicity,"
Vol. 246 (1973), p. 540.
18.
Gjorgov, A., Rogleva, K., "Informacia za Prekinite na Bremenosta
vo Skopje vo 1970 gadina kako del ad Politikata za Planiranje na
Cemejetvoto," Skopje: Zavod za Zdravstvena Zastite.
19.
Hayashi, M., Momose, K., "Statistical Observation on Artificial
Abortion and Secondary Sterility", Harmful Effects of Induced
Abortion, (1966), pp. 36-43. Tokyo: Family Planning Federation of
Japan.
20.
llellman, L.M., Pritchard, J .A., Williams Obstetrics,
Appleton-Century-Crofts, 1971.
21.
Hickey, R.J., Boyce, D.E., Clelland, R.C., Hamer, E.B., "Smoking
and Pregnancy," Nature, Vol. 246 (1973), pp. 177-8.
22.
llObJUe, C.J., "Prematurity Subsequent to Induced Abortion in
Skopje, Yugoslavia: An Historical Prospective Study." Dissertation,
University of North Carolina at Chapel Hill, 1973.
23.
Imngarian Central Statistical Office. Perinatal Mortality.
Budapest: Hungarian Central Statistical Office, 1972.
24 .
IIyde, G., "Abortion and the Birth Rate in the USSR,' I
Sci. (1970), Vol. 2, pp. 283-92.
25.
James, W.H., "Smoking in Pregnancy,"
p. 235.
26.
Johnston, F.D., Vincent, L., "Factors Affecting Gestational Age
at Tennination of Pregnancy," Lancet, Vol. 9, (1973), pp. 717-9.
27.
Klinger, A., "Demographic Consequences of the Legalization of
Induced Abort ion in Eastern Europe," Int' 1. J. of Gynaec. and
Ohstet. Vol. 8, (1970), pp. 680-91.
28.
Kozponi
Kozponi
"Latent
(1973),
Nature,
J. Bio -soc.
Nature, Vol. 246, (1973),
Statistikal Hivatal. Parinatalis Ha1alozas, Budapest:
Statistika1 Hivata1. 1972. Reviewed in Potts, D.M. 1973.
Morbidity After Abortion." British Med. J. Vol. 1,
pp. 739-40.
69
29.
Kurciev, K., Lazarov, A., Kismanoc, M. "Abortusi Kako
Pricine na Ektopicnata Bremenost. II Godisen Zbornik na Medicinskiot
Fakultet vo Skopje. Vol. 16, (1970), pp. 405-11.
30.
Lembrych, Von St., "Schwangerschafts - , Geburts - lIDd Wochenbettverlauf nach KlIDstlicher UnterbrechlIDg der ersten Graviditat,"
Zbl. Gynak. Vol. 94, (1972), pp. 164-68.
31.
Lindahl, J., Somatic C0InElications Following Legal ~ortion~.
Translated by Stanley H. Vernon, Scandinavian U. Books, 1959.
32.
Liu, D.T.Y., Melville, H.A.H., Martin, T., "Subsequent Gestational
Morbidity After Various Types of Abortion," Lancet, Vol. 2, (1972)
p. 431.
33.
Margolis, A.J., Davison, L.A., Hanson, K.H., Laos, S.A., Nikkelsen,
C.M., "Therapeutic Abortion Follow-up Study," Am. J. Obstet. and
Gynec. Vol. 110, pp. 243-9.
34.
Miltenyi, K., "Demographic Significance of Induced Abortions,"
Demografia. Vol. 7, (1964), pp. 419-28.
35.
Miltenyi, K., "On the Effects of Induced Abortion,"
Vol. 7, (1964), pp. 73-87.
36.
Moriyama, Y., Hirokawa, 0., "The Relationship Between Artificial
Termination of Pregnancy and Abortion of (sic) Premature Birth,"
Harmful Effects of Induced Abortion, pp. 64-73. Tokyo: Family
Planning Federation of Japan.
37.
Murphy, J. F., Mulcahy, R., "The Effect of Age, Parity, and Cigarette
Smoking on Baby Weight," Am. J. Obstet. and Gynec. Vol. 111, (1971),
pp. 22-25.
38.
MacMahon, B., Pugh, T., Epidemiology - Principles and Methods,
Little, Brown and Company, 1970.
39.
Nisbet, I.C. T., "Smoking, Pregnancy and Publicity,"
Vol 245, (1973), p. 468.
40.
Novak, F., "Effect of Legal Abortion on the Health of Mothers in
Yugoslavia," IPPF 7th Conference, 1963. Proceedings: Changing
Patterns in Fertility. Feb. 10-16, 1963, Singapore, pp. 223-24.
New York: Excerpta Medica F01mdation.
41.
Omran, A.R., "Abortion in the Demographic Transition", Rapid
Population Growth,Vol. II Research Papers, Johns Hopkins Press, 1971.
42.
Panayotou, P.P., Kaskarelis, D.B., Miettinen, O.S., Trichopoulos, D.B.,
Kalandidi, A. K., "Induced Abortion and Ectopic Pregnancy,"
Am. J. Obstet., and Gynec. Vol. 114, (1972) pp. 507-10.
Demografia
Nature,
70
43.
Pantelakis, S.N., Papadimitriou, G.C., Doxiadis, S.A., "Influence
of Induced and Spontaneous Abortions on the Outcome of Subsequent
Pregnancies," Am. J. Obstet. and Gynec.Vol. 116 (1973), pp. 799-805.
44.
Potts, D.M., "Latent Morbidity Mter Abortion," British Med. J.
Vol. 1, (1973), pp. 739-40.
45.
Rantaka11io, P., "Groups at Risk in Low Birth Weight Infants and
Perinatal Mortality," Acta Paed. Scan. Suppl. Vol. 193, (1969).
46.
Rochat, R.W., Tyler, C.W., Schoenbucher, A.K., "An Epidemiological
Analysis of Abortion in Georgia," Am. J. Public Health , Vol. 61,
(1971), pp. 543-52.
47.
Roht, L.H., Aoyarna, H., "Induced Abortion and its Sequelae:
Prematurity and Spontaneous Abortion," Am. J. Obstet. and Gynec.,
Vol. 120, (1974), pp. 868-874.
48.
Ross, E.M., Butler, N.R., Goldstein, H., "Smoking Hazards to the
Fetus," Brit. Med. J. Vol. 4, (1973), p. 51.
49.
Rovinsky, J.J., "Abortion Recidivism: A Problem in Preventive
MeJicine," Am. J. Obstet. and Gynec. Vol. 39, (1972), pp. 649-59.
50.
Royal College of Obstetricians and Gynaecologists, Report by;
"Legalized Abortion: Council of the Royal College of Obstetricians
and Gynaecologists',' British Med. J., Vol. 1, (1966), pp. 850-54.
51.
Rush, D., Kass, E.B., ''Maternal Smoking: A Reassessment of the
Association with Perinatal Mortality," Am. J. Epid. Vol. 96, (1972),
pp. 183-96.
52.
Sawazaki, C., Tanaka, S., "The Relationship Between Artificial
Abortion and Extrauterine Pregnancy", Harmful Effects of Induced
Abortion, (1966) pp. 49-63. Tokyo: Family Planning Federation of
Japan.
53.
Shapiro, S., Schlesinger, E.R., Nebitt, R.E.L., Infant, Perinatal,
Maternal and Childhood Mortality in the United States, Harvard
Univ. Press, 1968.
54.
Shinagawa, S., Nagayarna, M., "Cervical Pregnancy as a Possible
Sequela of Induced Abortion. Report of 19 Cases." Am. J. Obstet.
and Gynec. Vol. 105, (1969), pp. 282-84.
55.
Sung-bong Hong. Chan in Patterns of Induced Abortion in Seoul,
Korea. (1971) Seoul, orea: PopulatIon OunCI .
56.
Terris, M. , Gold, E.M., "An Epidemiologic Study of Prematurity,"
Am. J. Obstet. and Gynec. Vol. 103,(1969), pp. 358-79.
71
57.
Tietze,C., Dawson, D.A., "Induced Abortion: A Factbook,"
Report on Population/Family Planning, Vol. 14, (1973).
58.
Valaoras, V.G., Trichopoulos, D., "Abortion in Greece"
Abortion in a Changing World, Vol. 1, (1970), pp. 284-290.
Edited by Robert E. Hall, New York: Columbia University Press.
59.
Van Den Berg, B.J., "Smoking and Pregnancy,"
(1973), p. 540.
60.
Venkatacharya, K., "Reduction in Fertility Due to Induced Abortions:
A Simulation Model," Demography, Vol. 9, (1972), pp. 339-52.
61.
WHO Expert Connnittee. "The Prevention of Perinatal M:>rtality and
Morbidity: Report of a WHO Expert Connnittee," WID Technical
Report Series Number 457. Geneva: World Health Organization.
62.
Wolfers, D., "The Demographic Effects of a Contraceptive Program,"
Population Studies, Vol. 23, (1969), pp. 111-40.
63.
Wright, C.S.W., Campbell, S., Beazley, J., "Second-Trimester
Abortion After Vaginal Termination of Pregnancy," Lancet
Vol. 1, (1972), pp. 1278-79.
64.
Wyrm, M., Wyrm, A., "Some Consequences of Induced Abortion to
Children Born Subsequently," Marriage and Family Newsletter,
Vol. 4, (1972), pp. 1-24.
65.
Yerushalrny, J., "Infants with Low Birth Weight Born Before Their
M:>thers Started to Smoke Cigarettes," Arn. Jo. Obstet. and Gynec.
Vol. 112, (1972), pp. 277-84.
66.
Yerushalrny, J., ''The Relationship of Parents' Cigarette Smoking to
Outcome of Pregnancy--Implications as to the Problem of Inferring
Causation from Observed Associations," Am. J. Epid. Vol. 93, (1971~
pp. 443-56.
67.
Yerushalrny, J., "Reply to Mr. Goldstein,"
Gynec. Vol. 114, (1972), pp. 571-4.
Nature, Vol. 246,
Arn. J. Obstet. and
72
II.
Markov chain and other stochastic models concerned with reproduction
68.
Bosso, J., Sorarrain, O.M., "Application of Finite Absorbent Markov
Chains to Sib Mating Populations with Selection," Biometrics,
Vol. 25, (1969), pp. 17-69.
69.
Chiang, C.L., "A Stochastic Model of Htnnan Fertility,"
Vol. 27, (1971), pp. 345-56.
70.
Hogue, C.J., Shachtman, R.H. ,"Prematurity and other Event Rates
Subsequent to Induced Abortions II: Results of the Study for
Skopje, Yugoslavia," in preparation.
71.
Hogue, C.J., Shachtman, R.H., "A Model-based Comparison of
Prospective and Retrospective Rates Using a Markov Chain," in
preparation.
72.
Hogue, C. J ., "Low Birth Weight Subsequent to Induced Abortion:
An Historical Prospective Study of 948 Women in Skopje, Yugoslavia,"
Am. J. Obstet. and Gynec. ( In press)
73.
Perrin, E.B., Sheps, M.C., "HLunan Reproduction: A Stochastic
Process," Biometrics, Vol. 20, (1964), pp. 28-45.
74.
Perrin, E. B., Sheps, M. C., "A Mathematical Model for Human
Fertility Patterns," Archives of Environmental Health, Vol. 10,
(1965), pp. 694-698.
75.
Potter, R.G., "Estimating Births Averted in a Family Planning
Program." In Behrman et al. (eds .), Fert iIity and Family Planning:
A World View. University of Michigan Press, 1969.
76.
Potter, R.G., "Renewal Theory and Births Averted,"
Conference, I.U.S.S.P., Aug., 1969.
77.
Potter, R.G., "Births Averted by Contraception: An Approach
Through Renewal Theory," Theoretical Population Biology 1
(1970), pp. 251-272.
78.
Potter, R.G., Parker, M., "Predicting Time Required to Conceive,"
Population Studies, Vol. 18, (1964), pp. 99-116.
79.
Potter, R.G., McCann, B., Sakoda, J.M., "Selective Fecundability
and Contraceptive Effectiveness, " Milbank Memorial Fund Quartely,
Vol. 48, (1970), pp. 91-102.
80.
r~dley, J. C., Sheps, M. C., "An Analytic Simulation Model of
HlUUan Reproduction with Demographic and Biological Components,"
Population Studies, Vol. 19, (1966), pp. 297-310.
81.
Sheps, M. C., "Pregnancy Wastage as a Factor in the Analysis of
Fertility Data, " Demography, Vol. 1, (1964), pp. 111-118.
82.
Sheps, M.C., "A Review of Models for Population Change," Review
of International Stat. Inst. Vol. 39, (1971), pp. 185-96.
Biometrics,
London
•
73
83.
Sheps, M.C., Menken, J .A., "On Closed and Open Birth Intervals in
a Stable Population," Sagtlllda Conferencia Regional de Poblacian,
I.U.S.S.P., Mexico City, Aug., 1970.
84.
Sheps, M.C., Menken, J .A., "Distribution of Birth Intervals
According to the Sampling Frame," Theoretical Population Biology,
Vol. 3, (1972) pp. 1-26.
85.
Sheps, M. C., Menkin, J .A., Ridley, J. C., Lingner, J. W., "Birth
Intervals, Artifact and Reality," Sydney Conference, 1. U. S. S. P .
(1967), pp. 857-67.
86.
Sheps, M. C., Menkin, J .A., Ridley, J. C., Lingner, J. W., ''The
Truncation Effect in Birth Interval Data," J. Am. Stat. Assoc.
Vol. 65, (1970), pp. 678-94.
87.
Sheps, M.C., Menken, J .A., Mathematical Models of Conception and
Birth. Univ. of Chicago Press, 1973.
88.
Sheps, M.C., Menkin, J .A., "A Model for Studying Birth Rates
Given Time Dependent Changes in Reproductive Parameters,"
Biometrics, Vol. 27, (1971), pp. 325-43.
89.
Sheps, M.C., Perrin, E.B., "Changes in Birth Rates as a Function
of Contraceptive Effectiveness: Some Applications of a
Stochastic Model," Am. J. Public Health, Vol. 53, (1963), pp. 1031-46.
90.
Venkatacharya, K., "Reduction in Fertility Due to Induced Abortions:
A Sinnllation Model." Demography, Vol. 9, (1972), pp. 339-52.
74
III.
Other Markov chain applications for health system problems
91.
Alling, D. W., ''The After-History of Pulmonary Tuberculosis:
A Stochastic Model," Biometrics, Vol. 14, (1958), pp. 527-47.
92.
Balintfy, J., "A Hospital Census Predictor Model," Smalley, H.E.,
Freeman, J.R., Hospital Industrial Engineering, pp. 312-316,
Reinhold, 1966.
93.
Bithell, J.F., "A Class of Discrete-Time Models for the Study of
Hospital Admission Systems," Operations Research, Vol. 17,
(1969), pp. 48-69.
94.
Bithell, J. F. ,"Some Generalized Markov Chain Occupancy Processes and
'111eir Applications to Hospital Admission Systems." Review of the
International Statistical Institute, Vol. 39, (1971), pp. 170-184.
95.
Bithell, J. F., "The Statistics of Hospital Admission Systems,"
Applied Statistics, Vol. 18, (1969), pp. 119-29.
96.
Bush, J.W., Chen, M.M., Zaremba, J., "Estimating Health Program
Outcomes Using a Markov Equilibrium Analysis of Disease Development,"
Am. J. Public Health, Vol. 61, #12, (1971), pp. 2362-75.
97.
Choi, S.C., "Truncated Sequential Designs for Clinical Trials
Based on Markov Chains," Biometrics, Vol. 24, (1968), pp. 159-68.
98.
Eyman, R.K., Tarjan,G., McGunigle,D., "The Markov Chain as a
Method of Evaluating Schools for the Mentally Retarded," Am. J. of
Mental Deficiency,Vol. 72, (1967), pp. 435-44.
99.
Fitzsirrnnons, J .A., "A Methodology for Emergency Ambulance Deployment,"
Manage. Sci. Vol. 19, (1973), pp. 627-36.
100.
Fix, E., Neyman, J., "A Simple Stochastic Model of Recovery
Relapse, Death and Loss of Patients," Human Biology, Vol. 23,
#3, (1951), pp. 205-41.
101.
Gani, J., Jerwood, D., "Markov Chain Methods in Chain Binomial
EpidemicModels," Biometrics, Vol. 27, (1971), pp. 591-603.
102.
Jennings, J.B., Kolesar, P., "Connnents on G!- Blood-Bank Inventory
Model of Pegels and Jelmert," Operations Research Vol. 21, (1973),
pp. 855-857.
103.
Kao, E.P.C., "A Semi-Markovian Population Model with Application
to Hospital Planning," IEEE Transact. on Systems, Man, and Cybernetics,
Vol. SMC 3, (1973), pp. 327-36.
104.
Kao, E.P.C., "A Semi-Markov Model to Predict Recovery Progress of
Coronary Patients," Health Services Research, Vol. 7, (1972),
pp. 191-208.
75
105.
Kao, E.P.C., "Study of Patient Admission Policies for Specialized
Care Facilities," Yale Univ. Institution for Social and POli~
Studies Health Services Research Program, Working Paper No.,
Noverilber, 1973.
106.
Kolesar, P., "A Markovian Model for Hospital Admission Scheduling,"
Manage. Sci., Vol. 16, (1970), pp. B-384-396.
107.
Kreyberg, H.J.A., "Empirical Relationship of Lung Cancer Incidence
to Cigarette Smoking and a Stochastic Model for the Mode of Action
of Carcinogens," Biometrics, Vol. 21, (1969), pp. 839-57.
108.
Lu, K.H., "A Path-Probability Approach to Irreversible Markov
Chains with an Application in Studying the Dental Caries Process,"
Biometrics, Vol. 22, (1966), pp. 791-809.
109.
Marshall, A. W., Goldhannner , H., "An Application of Markov Processes
to the Study of the Epidemiology of Mental Disease,"
J. Arner. Stat. Assoc. ,Vol. 50, (1955), pp. 99-129.
no.
Meredi th, J., "Markovian Analysis of a Geriatric Ward," Manage.
Sci. Vol. 19, (1973), pp. 604-612.
Ill.
Navarro, V., Parker, R., White, K., "A Stochastic and Deterministic
Model of Medical Care Utilization," 11ealth Services Research,
Vol. 5, (1970), pp. 342-57.
112.
Pegels, C.C., .Jelmert, A.Eo, "An Evaluation of Blood-Inventory Policies:
A Markov Chain Application," Operations Research, Vol. 18, (1970),
pp. 1087-1098.
113.
Rodda, B.E., Miller, M.C., Bruhn, J.G., "Prediction of Anxiety and
Depression Patterns Among Coronary Patients Using a Markov Process
Analysis," Behav. Sci., Vol. 16, (1971), pp. 482-9.
114.
1homas, W.H., "A Model for Predicting Recovery Progress of Coronary
Patients," Health Services Research, Vol. 3, (1968), pp. 185-213.
ll5.
Weiss, G.H., Zelen, M., "A Semi-Markov Model for Clinical Trials,"
J. Appl. Probability, Vol. 2, (1965), pp. 269-85.
ll6.
Weiss, G.H., Zelen, M., "A Stochastic Model for the Interpretation
of Clinical Trials," Proc. Nat. Acad. Sci., Vol. 50, (1963),
pp. 988-94.
117.
Whitmore, G.A., Neufeldt, A.H., "An Application of Statistical
Models in Mental Health Research," Bull. of Math. Biophipics
Vol. 32, (1970), pp. 563-79.
118.
Yang, M.C.K., Hursch, C.J., "The Use of a Serni-MarkovModel for
Describing Sleep Patterns," Biometrics, Vol. 29, (1973),
pp. 667-76.
76
119.
Ztulg, W., Naylor, T., Gianturco, C., Wilson, W. P ., "Computer
SilTRllation of Sleep EEG Patterns Using a Markov Chain Model,"
BioI. Psychiat. Vol. 8, (1965), pp. 335-55.
120.
Ztulg, W., Wilson, W., "Sleep and Dream Patterns in Twins: Markov
Analysis of a Genetic Trait," Recent Advances - BioI. Psychiat.
Vol. 9, (1967) pp. 119-29.
77
IV.
Some general Markov chain and testing references
121.
Anderson, T. W., Goodman, L.A., "Statistical Inference About Markov
Chains," Ann. Math. Stat. Vol. 28, (1957), pp. 89-110.
122.
Bartholomew, D.J., Stochastic Models for Social Processes,
Wiley and Sons, 1967.
123.
Bhat, U.N., Elements of Applied Stochastic Processes, Wiley and
Sons, 1972.
124.
Chiang, C.L., Introduction to Stochastic Processes in Biostatistics,
Wiley and Sons:-I968.
125.
Dixon, W.J., Massey, F.J., Introduction to Statistical Analysis,
McGraw Book Co., Inc., 1969.
126.
Guthrie, D., Youssef, M.N., "Empirical Evaluation of Some
Chi -Square Tests for the Order of a Markov Chain," J. Am. Stat.
Assoc., Vol. 65, (1970), pp. 631-4.
127.
Henry, N.W., ''The Retention Model: A Markov Chain with Variable
Transition Probabilities," J. Am. Stat. Assoc., Vol. 66, (1971),
pp. 264-7.
128.
Howard, R.A., Dynamic Probabilistic Systems Vol. I: Markov Models,
Wiley and Sons, 1971.
129.
II:
Semi-Markov
130.
Kemeny, J.G., Snell, J.L., Finite Markov Chains, Van Nostrand, 1960.
131.
Kullback, S., Kuppennan, M., Ku, H.U., "Tests for Contingency
Tables and Markov Chains," Technometrics, Vol. 4, (1962), pp. 573-608.
132.
Estimating the Parameters of
e ate Time-Series Data, North
133.
Lucas, H.L., "Stochastic Elements in Biological Models, Their
Sources and Significance," Stochastic Models in Medicine and Biology,
Gurland, ed., Univ. of Wisconsin, 1964, pp. 355-83.
134.
Regier, M.H., "A Two-State Markov Model for Behavioral Change,"
J. Amer. Stat. Assoc. Vol. 63, (1968), pp. 993-9.
135.
Tan, W. Y., "Applications of Some Finite Markov Chain Theories to
Two Locus Selfing Model with Selection," Biometrics, Vol. 29,
(1973), pp. 331-46.
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 ~
A Partitioned Form of
The 79 x 79 Transition Matrix
The full 79
x
79 transition matrix has the following block fonn:
I
G3
:
_______
L
H23
I
I
I
I
:
I
~
0
I
L
_______ IL
H
13
0
I
I
I
:
G2
:
_
0
IL
_
I
I
H
12:
I
G
1
= transitions for individuals within group k; k = 1,2,3.
Hjk = transitions for individuals from group j to group k;
j = 1,2, k = 2,3.
Since all transitions in the Hjk submatrices are through ia 2 , m~, JX1<;
k = 1,2; these are adjoined as one additional column, C23 , to G2 and two
additional columns, Cl ,23' to Gl : the resulting three part partition
follows in the order:
[G 3],
[C 23 ; G2],
[C l ,23:
Gl ].
I--------------------~--------------------------------l-J-J-J-J_J_J-J-J-J-J_J_J_J_J_J_J_J_J_J_J_J_J:J:J:J_J__
~
I I I I I I I ! I I I ! 1 I I ! I ! 1 I ! I*!* I*! -I__I .
r-------------------------f
I I I I I I I I I I I I I II! I I I
1
I
1 1 1* 1 1 1
I·
I=J=J=J=J=J=J=J=I=J=J=J=J=J=J=J=J=J=J=J=J=J=J:J=J=J=J=J
I I I I I I 1 1'·1 1 I 1 1 1 I ! I I 1* I 1 I I 1 I 1 I
I-----------------------------------------------------I 1 I I 1 I I 1 I I I I 1 I 1* 1* 1* 1* 1* 1* I I I I I I I I
-1-1-1-1-1-'-1-1-1-1-1-----1;1;1;1;1;1;1-1---1-1-1-1-1
------------------------------------------------------
I I I I I I 1I I I I 1 I 1 1
1 1* 1 I I
1I 1I
1-----------------------------------------------------I
I
1 1 I
1 1 I
I
I
I
I· 1 I
1 I
I
I
1* I
I
I
I
I
I
I
1 I
I
I
I
'-'-'-'-1-1-1-'-'-'-'-'-'-'-'-'-';'-'-'-'-'-'-'-'-'-'-I
1-----------------------------------------------------f
f 1
1 1
f I
1
1
1
1
I If
1
1 '*
1 1
1 f f 1
ff
1 1
f 1 1
1 ,
, f f
1-----------------------------------------------------I
I f '1
I
I
1*
I
,
I
I
I
1
1
I
I
1
I
I
I
I
I
I
I
I
I
I
•
I
I
I
1
I
I
I
I
I
I
1
til
I
t
I
,
I
1
I
I
I
I
I
I
1
I
'I
I
I
I
I
1
I
I
I
1
I
1
I
1
I
1
1
I
1
I
I
I
I
I
I
I
I
I
,
I
I
,
1
I
,
I
I· I
f
II
I
I
I
I
t------------------------------------•
I* 1* 1* .* 1* 1* 1* 1* 1* I* 1*
f* 1* '*
I
1
1
I
f
I
I
I
t
I
I
I
I
I
I
I
1
't
I
I
I
I
I
I
I
I
1
I
I
I
I.
I
1
I
I
I
,
·f
I.
I
~-------1
I
I
I
1
f
I II
• I1 1
II I
I f
I '1 II I1 I1 1*
I1 II II :f l
I i: tI II I1 II I, I• I1 II I1
1
, 1'1
I
I
I
,-----------------------------------------------------I
1 1*
I
I
I
1 I I
' l l
fI I 1
I
I'
1
Ie
"I II II 1
,I II tII 1
If 1
I
1
I
I
I
1
f
1
1
I
i
I
I
1 ,, 1
1 II 1
I
1
f 1
I
f ,I 1
I
1
f .
1
I
1
fI
f
,-----------------------------------------------------I
I
t
I
I
I
II
II
l
i
I
I tI
II
I . I
I
I
II
I
,-----------------------------------------------------I 1 1 11 1* 1
I
I 1
1
1-_-_-_-_-_-_-_-_-I
I
1
I
I
1
,
I
I
1
I
I
I
I
I
I
I
1
I
I
1
I
f
I
I
I
I
1
I
I
I
I
I
I
I
I
I
I
I
I
I
f
I
I
I
,
I
I
I
I
I
I
I
1
I
I
I
I
I
I
1
I
I
I
I
I
I
I
t---------~-_-_-_-_-_-_-_-
I
1
1
I
I
,
1
I
I
I
I
f
II
I
I.
I
1
I
I
I
I
I
1*
1
I
I
,
I
I
I
I
I
1
I
I
I
•
I
I
1
·1
f
1
t
f
I
I
I1
I
I
f
I
I
I
f
I
I
I
I
I
,
I
I
1
I
I
I
I
I
,
I
I
I
I
t
I------------------~-----------------------~----------1
I
•
1 I I I f I I 1 1 I I 1 I 1 1 1 I I I 1 I I I I 1
I
I I I
I I I 1* 1 I I I I I I I I I 1 I 1 I I I I : I
I--------------------------~--------------------------II I1 I' II 1 I I 1*
II
I II II 1
I1 II II II II I.
I1 II I I II II II I1 II
,
I
I
"I-------------------------------------~---------------II II II I, I 1*
II I I1 I II II II I1 I1 II II I, 1
I1 II 1
I1 II I1 II II
I
1
I
I
1 1
I
I
I
,
I
I
I
I
I
1
I
1
1
1
I
1 I I II ,I II
I
II 1*
I
I
1
I
I
I
I
.1
,
I
1
I
1
I
I
I
1 1 1
I
I
1
I
I
I
1
I
I
I
I
1
1
I
I
f II I, II fI 1
I
1
1-----------------------------------------------------II I.
1*
,* 1*
1
I 1
I 1
I 1
I II 1
II II I1 I1 I1 II II I1 1*
1*
II II 1
,*
1*
II
I
I
1
I
f 1
I
I
I
I
,
1
I
I I I I I I 1 I I , I I I I , I 1 1 j I I I I I I I I
1*
* * 1*
1* 1*
1*
f*
1*
II
II
I1
II
II
II
1*
If
II
'*
f*
I1
,
I * 11* 1*
I
I*
I
1
I
1
1f*
f
I
If* 1
I
I
t---_------_-
-
-
~_~'-------------------- --
1-----------------------------------------------------1-----------------------------------------------------1* 1
1* 1* 1* 1* 1* 1
1* 1* 1* 1* 1* 1* 1*
I 1 I '1* 1* 1 1 1 1* 1*1
1
I.
I
I
I
I
til
I
I
I
I
I
I
I
I
I
I
I
I'
I
,------------------------------------------------------
1* 1* 1* 1* 1* 1* 1* 1* I~ 1* 1* 1* 1* 1*
! 1* 1*
! !* 1* !
,-----------------------------------------------------1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* 1* I 1 1 I I I 1* 1* I I I 1* 1* I
I
,
I
I
I
I
I
I
,
I
I
1
I
I
I
I
I
I
,
I
1
I
I
I
I
I
I
I
1
I
,
I
I
t
I
I
I
I
1
I
I
I
I
I
I
f
I
I
I
,
I
I
I
,
I
1
I
.-----------------------------------------------------:* I I 1
I I1 1 If I I I 1 .1 If 1* I : I I I I I If : I I I I I
-----------------------------------------------------I
:;
u
';:;,.,
~o.
~6
... ...
1'\
[)
P
~
~,
~,
,¥,
D
7
b
2
2
P
P
'1
.'
1
?
2
2
2
2
:'I
~
J
1\
~
4:'
n
?
P
P
?
P
2
2
2
2
2
'1
~
R
P
S
S
V
s
v
222222222
A
V
?
S
V
S
S
1
2
2
2
2
2
1
2
3
4
~
S
3
S
4
S
5
5
6
5
7
S
8
I
9
2
_
~9
M02 I
1
I
I
1
P02 I • I
1
1
1
1
I
1
I
1
I
1
117P2 1
I * I • 1
'-__ '
1
1
118P2 I
I· I * I
'
119P2 I
'
1I1P2 1
'
'
1
1
I
'
'
1
* , • 1
1
1
'
1
I
'
I
'
I
1
1
1
I· 1
1
1
1
1
1
I
1
I
I
1
* 11
I
1
1
1
I
I
I
I
1
1
I
I
,
I
1
1
'
1
1
I
f
1
I
1
1
1
1
1
1
I'
1
1
I
1
1
1
I
I_~_I
I
1
1
1
I
1
I
I
I
I
1
I
1
*
I
1
1
1
I
I
1
I
I
1
1
113P2 I
1
I
I
1
1
1
I
1
1
I
I
114 P2 I
I
1
1
I __ ~I
1
I
I
I
I
I
115P2 I
'
11
1
1
1
1
11
11
11
'1
,
1
116P2 1
1
R5I2 I
1,
'1
11
1I
I
1I
1
1I '1
'
I
I
1I
'1 * 11
,
1I
1
1I
,
I'
1
1
'
'
1
1'1
I
1
'
1
1
I
I
I'
'
'
1
1
1
I
I
1
1
R5V21
R5V22
R5V23
R5V24
1
1
1
1
1I
,
'
,
1
I
11
1
1
1
1
1
I
I
1
I
1,
*
I
'
1
I
1
1
I
*
1
1
I
,
1
I
I
1
I
1*1
1
1
1*1
'
1
'
I
1
II
1
1
*
*
'I
'
1
1
528 ,
1
529 ,
1
Il2 I
1
5A2 I
1
1
1
I
1
I
'
1I
I
'
I
1
I
1
1"
1
,
1
I
II'
'*1
1'1
1
1
1
1
1
'
1
1
,
I
I
I·'
I
I
I
1
1
1
1
1
1
1
'
I
I
,
II'
II
1
1
1
1
1
1
1
1
11
·'
11
1I I
1 '1
1I
1I
1
,
1
'
1
'
'I
I
I
1
1
1
I
I
I
I
1
,
1
I
I
I
"
1
1
1
f
1
1
I'
1
1
I
,
I
1
1
1
11
11
1I
1
*
1
1
1
1
1
*
1
1
1
1
1*1*
'
1
1·'*
1
*
I
1
1
I
1
I
I
,
I
I
1
I
I
I
1
1
'
1
I
1
I
1
I
I
I
1
I
1
I
1
1
I
1
1
I
1
1
I
I
1
I
1
I
'
"
1
1
I
I
1
I
1
I
1
I
1
1
I
,
1
I
1
I
1
I
1
I
I
I.
1
I
I· I •
I'
1
I
1
I
1
I
'
'
1
1
'
I
1
I
1
I
1
1
1
1
1
1
I
1
1
I
1
1
1
I
1
I
1
I • 1
1
1
1
,. I • 1
1
1
1
1
1
'
1
I
I
'1
1
I
1
I
1
I
'
I
I
I~
11
I1
I1
1I
1I
I1
1I
1I
I'
1I
1I
1I
11
11
11
11
I
· 11 • 11
1
1, . 1I • 11
I
1I
I
1I
I
1
1
1
1
I
1
1
1
'
*1·1*1
1
1
1
*1·'·1
1
1
1
·1*'.'
1
1
1
·1*'·1
1
1
I
I
1
*
1
1
1
1I
'
1
1
1
1
1
I
1
1
I
1
'*1*
1
'
1
1'11*1·
1
1
'
1
1
1
1
1
1
1
1
1
---'-r-'---I
I'
I
1
1
1
I
I
I
1
1
1
I
1
1
1
1
521 I I '
III
'*'
,
III
1
1
1
1
1
'
1
'
1
'
1
'
522 I
I
1
I
1
,
1*'
,
1'1
'
1
'
'
'
'
1
1
1
1
'
'
523 I "
1
1
,
'*'
I
II'
'_~'
I
I
I
I
I
I
I
t
'
I
524 I I
I
1
1
I
'*1
I
III
1-_-1_--1-__ 1__-1-__ '_-_1_-_1-__ 1-__ 1
1_-_1 __525 I I I
I
1
I
'*'
III
I
1
1
'
1
1
1
1
1
1
'
1
1
526 1
1'1
I
,
1·1
,
'I
1
1
1
1
1
1
1
'
'
'
'
527 1 1 1 1
I
I
1
*
'
I
I
I
'
1
1
'
1
1
'
1
1
1
1
I
1
'
1
'
I
'
1-__ 1
1>«'
'
,
1
I
1
'
1
III
1
1
I
1
1
1
1
I
I
1*'
1
1*1
1
'
1
'
I
'
1
'I
'
,
1
1
1
,
I
1
I
I
1
I
I
1
1
'
I
1
I
I
I
1
I
I
1
1
1
I
I
'
1
1
*
I
'
1
1
I
1
1
I
'
1
1
1
I
1
I
1
1
'
I
1
1
I
1
1
,
1
1
1
1
1
'
I
1
!'i2P2 1
1
...
1
1
'
'
1
'
I
I
1
1
,
I
1
1
1
1
1
~I
I
1
I
1
I
1
I1
I
1
I
1
I
1
1I
1
'
1
1
I
'
1I
I
1
I
1
I
1
I1
I
1
1
* , *
1
__ I
11
I
1
1I
1I
1I
1I
III
I
I
1
1
1
1
1
1
1
1
I
I
II
II
I
1
1
1
1
1
1
1
1
1
1
III
I
I
III
I
1
1
1
1
1
1
1
1
1
I'll
I
I
I
II
'·1*'·'*11.1
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
1·1*'·'*1
I
'·1
III
I
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1.1.1.1.1
I
11.11
I
I
I
1
1
1
1
1
1
'
'
'
'
1
1
1
'*1*1*1·1
I
I
11.11
I
I
:
1
'
1
'
1
1
'
1
1
1
1
1
'.1.1.1*1
I
,
I
11.1
I
I
'
1
1
1
'
1
1
1
1
1
1
1
1
1·'·1·1*11
I
I
I
1'·11
1
1
1
1
'
1
1
1
1
1
1
1
1
1
·
'
*
1
·
'
·
1
'
I
I
1
I
1
1
·
1
I
I
'
I
I
I
I
I__
I
I
I
I
11·1*1.'*'
I
1
1
1
1
1
1
1
'*'·1·1·1·1
I
1
1
1
1
1
1
1
I
'·1*1·'.1·1
'
1
1
'
1
1
'
1
' 11* 1
1 · 11 · 11* ' *
' 11
1
'
1
I
I
1
1
I
1
1
1
1
1
I
1
II
I
1
I
1
1
1
I
1
I
1
II
I_~I
I
·11
,
1
,
1
I
1
I
1
_
_
_
_
_
_
_
111·11
I
1
1
1
1
1
1
I
I
I
II
I
1
1
1
1
1
1
111111
'
1
1
1
1
1
1I
1I
1I
11
1I
II
,
·
I'D1
'
pn1
111 P1
!'Iap1
1!9P1
IA2
Pl1 P1
112 P1
~
~
(Jl
(Jl
g
::""::N
•..-t
0,.-4
ton p.,
C:l
'" I-<
0
I-<
(J]
D
p
[I
1
1
N
8
M
7
p
P
1
1
p..
C:l
'" I-<
0
I-<
1
A
i'I
9
P
2
1
M
M
M
S
5
p
3
p
~.P
[,
2
11
4
P
Ii
1
P
I
V
1
1
1
1
1
1
1
1
,
,
1
1
Pl3 P1' 1
I
1
1
1
I
I
I
1
I
I
1
I~
1'5 P1,,
116 P1
I
'
li5 I1 1
1
f
'
I1
'
__ I
,
I
1 '1
11
'I • 1
1
1
'
1,
I
1
*'
*'
1
1
1
1
'" 1
1'
I
1
• 1I
• 1
1
,1
*
1' I1
1
1
1
1
., •
'I
I
'
11
1
1
11
1
1
,'
1
1
1
1
R5V12 1
1
R5V1] 11
1
11
1
11
R5V14 1
1
511 1
1
1
1
1
'
1
I'
'
1
512 ,1
513 1
1
11
I
1
1I
I
1
11
1
1
11
1
'
1
1
11
1
1
11
"1
*'1
I
'
1'
*
*
1'
1
1
1
1
1
'
1
'
1
1
I
1
1
1
1
1
,
'
1
'
1
1
1
'
1
'
,
1
,
'
,
1
I
1
I
I
1
I
1
1
I
,
1
1
,
1
,
'I
1
1
*
1
1
1
'
*'
1
1
11
1
1I
1
I
1
'
1
1
1' I1 • '1 •
'1
11
1, .
*
'
1
'
11
1,
I
1
1
1
11 • 1,
'I • 11
'
*
*
1
'
'
1
'
1
1
1
1
1
1
1
1
1
1
,
'
I'
'
'
lit
1
1
'
11
1
'
11
1'_'
1
1
11
'
' I1
1
1
1
1
11
*
*
11
*
I • 1 * 1 • 1
'
1
1
1
1
•
1
•
1
•
1
1
'
1
11
1 • 1 * 1 * 1 • 1
1
1
1
'
1
I • 1 • 1 * I
,.
*
*
1
1
,
1
1
1
1
1
1
'
1
1
1
1
1
1
I
'
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
. 1
1
1
1
1
1
1
1
1
1
•
1
1
1
1
1
1
1
1
1
1
1__'__ 1
1
•
1
1
1
1
1
1
1
1
1
'
1
1
1
1
11
1
1
11
11
1
1
1I
11
,
1
11
1
1
1
1
11
1
'
·11
,
1
1
11
1'
11
1'
11
1
11
11
11
'
1
1
1
'
1
1
1
'
1
1'
11
11
11
,
1
11
1
1
,1
1
1
11
1
'
1
1
1
1
11
1
1
1,
1
1
11
1
1
,1
,
'
1
1
'
1
I
1
1
1
1
1
,
1
1
1
1
,
1
1
,
1
,
1
1
1
11
'
I· 1
1
1
11
1
'
11
1
1
11
,
1
11
1
'
11
1
1
1' • 'I • 11 • 11
• 1 * 1 * 1 * 1
1
1
1
1
1I
1
1
11
1
1
1
1
1
1
1I
1I
1I
,
. I1
'
'
'
1
1I
1I
1
1
'
'
1
1
'I
I
1
• 1 __
• II * 1I • 1I
• 11 • I1 • 11 .... 11
* ,' * ,1 * 1I • 1I
,1
I
1
11
1
1
1'
1
1
'1
1
1
1
,
1
1
,.
1
11
1
'
11
1
1
1,
,
1
1
1
1I
1
1
1
'
11
1
1
1
1
1I
1
1
1
1
11
1
1
1
1
I·'
11
1
1
,
1
*
515 ,,
516 1
'
,
1
11
1
1
'1
1
1
1I
I
1
'1
,
1
1I
I
1
1I
,
1
1, . ,1
I· 1
1
'
11
1
1
517 I
1
519'
,
1I
I
'
1
1
11
1
1
,
1
11
I
'
,
1
1I
1
'
I
'
11
,
11
1
1
1
1
'
518,
11
I
1
1
1
1
'
1
'
1I
1
1
,
'
1
'
'
1
1
1
'
1
'
,
'
•
1
1
1
,
*
'1__I
1
*
,
1
'
1
1
1
1
,
1
1
'
1
1
,
'
1
1I
I
1
11
1
1
1
,
1
'
,
'
I
1
,
1
1
'
1
'
,
'
,
1
I
'
I
'
1
'
1
*
-1
,
1
1
1
1
1
1
1
,
1
1
1
1
'
,
'
,
'
,
1
1
1
,
1
,
1
1
11
1
1
1I ·
I·
'
1
1
4
1I
1
1
'1
,
1
1
'
1
*
1
'
1
3
6
I
1
'
1
2
.~
8
9
S
1
1
1
I
'
1
3
S
I
'
'*
'*'
2
5
1
5
1
1
1
'
1
5
1
4
1
'
I
5
1
1
1
1
1
1
1
'
,
1
1
'
1
1
1
'
1
'
,
1
,
~
I·'
I
5
1
1
I
1
'
1
5
5
V
1
1
'
1
R
I
1
I'
1
1
R
S
V
I
1
514 ,,
1
I
1
1
1
1
1
1
1
1
1
I
'
1
1
11
1I
1
I
1
1
1
1
1
'
1
1
I
1
I
1 '1
1
1
'
'
'I
I' "
*
I
'I __
I
I
1
'
I
1
1
1
'I
1
'
I
1
1
1
1
I
1
1
'I'
1
1I
R5V11 ,1
1
1
R
S
V
------------------------------------------------------------------------1
I
1
I
,
,
1
,
1
1
1
1
1
1
1
1
,
,
'I
1
p
1
I~~-~-~-------------------------------'
1
I
I
1
1
I
1
I
1
1
1
1
1
1
'
1
'
1
I
,
I
I
I
I
1
I •
1
1
1
1
1
'
1
1
1
1
1 • , • 1
I· I
,
I
1
1
1
1
1
1
'
1
1
1
• , * ,
1 • 1
I
1
'
1
1
1
1
' __._1
1
* 11 • '1
1
I
I
1
1
'
1
1
1
I'
11
I
I
I
I
1
'
1
1
1
1
'
1
I
I'
I
I
1 *
1
'
1
'
1
1
1
1
I
1
1
1
1
1 *'
1
Pl4 P1'
..
11
g
I ' I·'
1
'I
'I
I
',
1
'
I
1
*
'
1
1
I'
1
1
1
1
1
1
1
1
'
I~
lit
'
1
1
I
'
1
• 1
'
*
*
* , *
*
1
'
1 • I
1
'
I • , • 1
1
I
1
I
'
*
*
'
1
1 • I
'
1
I • 1
'
1
1
1
,
'
,
'
1
1
1
1
1
1
1
*'
1
1
1
1
I·'
1
1
,
'
1
1
1
I.
1
1
1