e·
.
CONDITIONAL RATE DERIVATIOIJ IN THE PRESENCE OF
INTERVENING VARIABLES USING A STOCHASTIC MODEL
by
Richard H. Shachtman
~ohn R.
Schoenfelder
Carol ~. Hogue
Department of Biostatistics
University of North Carolina'at Chapel Hill
Institute of Statistics Mimeo S.ries No.
,June 1900
1292
.CONDITIONAL RATE DERIVATION IN THE PRESENCE OF
INTERVENING VARIABLES USING A STOCHASTIC MODEL
Richard H. Shachtman
(1),
,John R.
Schoenfelder
(1),
Carol ,J.
Hogue
(2)
ABSTRACT
When conducting inferential and epidemiologic studies, researchers
are often interested in the distribution of time unti) the occurrence of
some specified event, a form of incidence calculation.
.
Furthermore, their
interest will often extend to the effects of intervening factors on this
length.
In this paper we impose the assumption that the phenomena being
investigated are governed by a stationary Markov chain and review how one
may estimate the above distribution.
We then introduce and relate two
interpretatively different methods of investigating the effects of
intervening factors.
In particular, we show how an investigator may
evaluate the effect of potential intervention programs.
Finally, we
demonstrate the proposed methodology using data from a population study.
(1)
University of North Carolina, Chapel
(2)
University of AT'kansas -For Medical Sciences, l.ittle Rock, Arkansas
~ill,
North Carolina
1
CONDITIONAL RATE DERIVATION IN THE PRESENCE OF
INTERVENING VARIABLES USING A STOCHASTIC MODEL
,.
Richard H.
Shachtman, John R. Schoenfelder, Carol J.
Hogue
Biological and epidemiological outcome investigations often assess the
relationship between chaT'acteristic
variable~
(e.
g.,
personal or demographic
variables, often called T'isk factors) and condition vaT'iables
(e.
g.,
development of disease, often called response factors) to determine whetheT'
the condition occurs more frequently among those individuals exhibiting the
chaT'acteristic.
For example.
in a study of the
relation~hip
between smoking
and lung cancer, the characteristic variable is smoking status and the
condition variable denotes whether .lung cancer develops.
..
A reseaT'cher would
compare the incidence (new case) rate of lung cancer for smokers to that of
non-smo kers.
Frequently, research will investigate more than Just the characteristic
and condition; that is, one will want to consider intervening variables.
the above example, an intervening
vari~ble
with epidemiological implications
might be occupational exposure to asbestos powder.
investigate two ways of incorporating such variables
prior factors.
In
After noting that traditional
In this paper,
we
as posterior or
epidemiolog~cal
approaches
to both methods of intervening variable analysis often exert an unreasonably
lc.H'ge data demand, we shol'J how a Markov chain 01Cl may be used to
significantly reduce the data requiT'ements.
First, howeveT', we
conceptually define two methodologies for incoT'porating intervening
variables.
S~bsequently,
we will provide mathematical definitions and
derive relationships for the two techniques.
2
Consider a cohort (homogeneous group) of smokers demarcated at time
zero.
We may consider
~sbestos
powder (the intervening variable) either
as a POSTERIOR 'actor (we consider all smckers at time zero and later
compute the lung cancer rate among those who. as an additional constraint.
have not been exposed to asbestos powder as time proceeds) or as a PRIOR
factor (we consider only the subset of smokers who will not be exposed to
asbestos powder and compute their lung cancer rate),
For a specific individual.
treating asbestos powder exposure as a
posterior factor considers the probability that
s/~e
will develop lung
c~ncer
without being occupationally exposed to asbestos powder GIVEN that s/he was
initially smoking.
Treating this same intervening variable as a prior factor
considers the probability that a specific individual will develop lung
cancer GIVEN that s/he was initially smoking and will not experience asbestos
powder exposure.
•
These approaches differ in the way they handle the
intervening variable constraints.
.The posterior conditioning analysis
treats the intervening variable as a condition which mayor may not occur
during the period of observation whereas the prior conditioning controls
for it before observation begins.
The resulting incidence rates may
differ significantly and. as will be seen.
re~uire
distinct formUlae.
Researchers may analyze intervening constraints in a direct
epidemiological analysis by using characteristic and intervening variables
to partition the data into distinct cohorts;
specific rates.
In our example.
they then compute cohort
treating asbestos powder
~xposure
as a
POSTERIOR factor is exemplified by the comparison o' workers in an' industrial
setting which may.
shipbuilding).
but need not.
involve asbestos
pow~er
exposure (e.g.,
At the termination of the study, one computes lung cancer
rates within each cohort (smoker. non-smoker> separately for those
individuals who do (do not) experience asbestos powder exposure.
3
Considering asbestos powder exposure as a PRIOR 'actor requires that
..
.
the subcohorts of smokers, for example,
the study.
be demarcated at the initiation of
One subcohort consists of those smokers who will be exposed to
asbestos powder' during the period of observation and the other of those who
will not.
When the study terminates, one computes the appropriate lung
cancer rate within each subcohort; a comparison
0'
those rates serves to
assess the effect of exposure to asbestos powder on the incidence of lung
can~er
within a cohort of smokers.
By defining
simil~r
subcohorts of
non-smokers we may complete the analysis of smoking and lung cancer while
controlling,
in a PRIOR sense,
for the effect of Clsbestos powder exposure.
From this example it is clear that both of these traditional approaches
necessitate a large amount of data in order to obtain accurate estimates of
the rates under consideration.
The data demand becomes
a major
concern
when the condition andlor intervening variable is rare or when a number of
•
intervening variables are considered simultaneously; the latter situation
req,uires the formation of many subcohorts,
each
of which must be "reasonably"
large.
Efficiency criteria, or simply the problem of obtaining sufficient
information,
indicate the advantage of using analytic models rather than
following up multiple cohorts;
the time-dependent,
probabilistic nature of
many empirical processes implies that such analytic models be stochastic
processes.
~nvestigatoTs may then use these models to estimate ~he
probability distributions of
predetermined state
the process in
(e.
c~rtain
(i) the time until the
g., a disease condition),
states, and
proce~s
arrives at a
(ii) the length of stay of
(iii) the number of visits to key states.
The scientist may condition all of these distributions by intervening
variables as previously stipulated,
thereby utiliZing the madel to account
for various confounding or concomitant variables.
4
In
...
we assume that the underlying process is a stationary
Section 2
(time-homogeneous)
MC,
present formal analytic definitions of both
types (prior and posterior) of intervening variable analyses, and derive
a relationship
betw~en
them.
We then show how a re.earcher may perform
both analyses, as well as the ordinary (non-intervening variable) analysis,
using functions of the
MC
transition matrix.
Thus all analyses demand
only sufficient data to estimate the transition matrix.
these
~oncepts,
To illustrate
we will present a numerical example based on an
for a stUdy of induced abortion.
We
employ a
state,
79
MC
model
time-homogeneous
model described in Shachtman and Hogue (2).
1.
CONDITIONING THE RATES
TRANSITION FUNCTIONS WITHOUT INTERVENING VARIABLES
A.
•
Assuming that the phenomena under investigation are representable
by a stationary MC,
let
=
P(X
=
P
n
Jk
k I x
n-1
that the underlying process is in state
state
J
at time
do not depend on
be the probability
J)
at time
k
n
given that it was in
note that because of stationarity these probabilities
n-1;
n.
=
The chain's transition matrix is then
P
=:
».
«p
We
. Jk
may obtain the
n-step
transition probabilities,
(n)
p
n
by the expression
P
=:
(n) »;
«p
that is,
p
n-th
power of the transition matrix.
passage time probabilities,
(n)
f
P(X
x
o
= k
=:
= P(X =
ki
n
Jk
(n) is the (J' k)-th element
Finally, we express the first
X
m
/= k,
1
<= m <= n-1 : X = J)'
o
by
p
(n)
Jk
= SUM/m(l,n)
J) I
Jk .
Jk
of the
=:
n
Jk
f
(n-m)
(m) p
Jk
kk
n
>=
1
5
and hence may obtain them iteratively by the
f
(1)
Jk
Cn)
f
=
P
=
p
Jk
Jk
(The symbol
m=l
to
~ormula
Cn) - SUM/m(l,n-l)
Jk
means unequal and
/=
m=n.
(m)p
f
Jk
SUM/m(!,n)
(n-m)
n :>= 2.
kk
represents the sum from
Using these probabilities, a researcher may employ known
techniques to compute such quantities as mean time necessary to "travel"
between any two states, mean number of visits to any state, and mean
length of stay in a given state.
To be definite let us now suppose the researcher is interested in the
distribution of the time required to reach state
k
from state
J.
As
expressed in Shachtman, Schoenfelder, and Hogue (3) this distribution is
given by
«
F
(n ):
= 0, 1, . ..
n
»
where
state
k
f
Jk
the probability that the process visits state
it was in state
= SUM/m(l,n)
(n)
F
Jk
at time
J
by time
n
O.
(m)
is
Jk
k
by time
n
given that
The unconditioned probability of Visiting
is then expressible as
G (n)
= SUM/J
(0) F
a
k
J
(n),
Jk
where the summation is over all states of the chain and
« a
(0)
= P(X
"J
distribution.
all states
o
Thus, for examp Ie,
represent diagnosis
task respectively,
o~
J
constitutes the initial probability
»
if
n
indexes months and states.
lung cancer and performing a "pulmonary
thEm
F
(n)
k, J
~unction"
will be the probability that a person is
Jk
diagnosed to have lung cancer by month
a "pulmonary function" task at time
O.
n
given that he was performing
The quantity
G (n) probability
k
probability that a person develops lung cancer by month
of his status at time
O.
n
irrespective
6
These distributions are of particular interest when state
«
In this situation
en):
F
=
n
»
0,1, ...
k
is. absorb ing.
is the distribution for time
Jk
to absorption from state
J
«
and
G (n):
n=O·l, ...
»
the time
h
k
to absorption distribution.
Moreover, F
Throughout the remainder of this paper
treat
k
= p
(n)
(n) for
Jk
we will,
for ease of expression,
as if it were an absorbing state; we ulill,
explic~t use of the assumption.
however,
Hence it could be dropped,
we would be computing "time to visit"
absorbing.
k
Jk
make no
in which case
rather than "time to absorption"
distributions.
B.
TRANSITION FUNCTIONS WITH INTERVENING VARIABLES AS POSTERIOR FACTORS
As previously mentioned,
interest will often extend beyond an analysis
of initial and absorbing states;
that is, a researcher will want to know the
probability of absorption without visiting one or more prespecified states.
Letting
h
represent such a taboo state, an informative conditional
probability,
p
called a post-conditioned taboo probabi.lity (POSTA13) is given by
(n )
= P (X =
k;
n
h Jk
X
1= h,
the probability of being in state
state
h
1
m
k
<= m <=
at time
n-1
n
without having entered
given that the process started in state
post-conditioned first passage probability (FOSTAB)
f
h Jk
Note that foT'
(n):::
=
P(X
1=
k;
n
h ::: k,
f
h Jk
h, k
1
J.
~s
<= m <=
m
(n)
=
first passage time probability.
f
(n)
Jk
The corresponding
th€ ordinClT'y,
n-1
X
o
unconditioned
7
Analogous to the non-taboo situation,
«
F
h Jk
(n):
where
» ,
n :.:: 0, 1, . "
F
the probability distribution
(n)
= SUM/me
post-conditioned taboo
Likewise
n)
«
G (n):
to absorption from state
ti~e
n = 0,1, ...
»,
where
POSTABs
and
FOSTAEs
(m),
is the
J
distribution.
G (n) = SUM/J a (0) F (n),
h k
J
h Jk
is the post-conditioned taboo time to absorption
The
f
h Jk
h k
LEMMA:
1,
h Jk
~istribution.
follow the same functional relation-
ships as unconditioned probabilities;
in particul.:.r, assuming
h
k,
/=
we have
p (1) =
h Jk
(I)
e
P
d
(i I)
PROOF:
of'
= p
(1) = of'
h Jk
Jk
(n) = SUM/me 1, n)
Jk
of'
h Jk
h Jk
p (n) = SUM/rer /=
h Jk
h)
(m) p (n-m)
h kk
p (n-1) p
h Jr
rk
n
>=
2.
c;.
n :>= ,.,
Obvious by inspection.
0.
By virtue of part
POSTABs
of this lemma one may easily derive
from the unconditioned transition probabilities whenever
(Recall that for
probabilities.)
h-th
(ii)
row of
P,
h = k ,
Define
the
T'OW
the
POSTABs
are the usual
h
first-p~ssage
/~
time
P * to be the matrix formed by replacing the
corresponding to the taboo state,
k.
by zeroes.
B
1
The sequence of matrices defined by
LEMMA:
n
iS5uch that
PROOF:
R
= «
p
h
(n)
R
n
"" P,
R
*
n-1
= R
P,
n
:>= 2,
».
Jk
Obvious by inspection.
o
n
R is this the matrix of
Note that
C.
n-step
POSTABS.
TRANSITION FUNCTIONS WITH INTERVENING VARIABLES AS PRIOR FACTORS
We have previously suggested another way of incorporating intervening
Rather than being interested in time to
variables into an analysis.
absorption for a "subcohort" which does not visit a taboo state during the
time of observation, the researcher may want to know the time to absorption
distribution for a cohort which is explicity restricted from visiting a taboo
state during the period of observation.
In this case the taboo restriction
is placed behind rather than in front of the conditioning bar.
These probabilities are expressed as
r
(n)
= P(X
n
h Jk
=k
x
/= h,
1 'C== m
<=
0-1
x
m
o
the pre-conditioned taboo probability (PRETAB), and
s
(n) ==
h Jk
P(X
==
k
n
X /=
m
h, k ,
1 <== m <:= n .... 1;
the pre-conditioned first passage taboo probability (FRETAB).
r
h Jk
(1)
==
s
h Jk
(1)
=P
Jk
He
define
e9
D.
RELATING THE PRE- AND POST-CONDITIONED RATES
To this point we have introduced two methods of.
n-step
de~ining
transition functions for intervening variable analysis.
As discussed
in the introduction each has a distinct interpretation involving either
The following theorem,
prior or posterior conditioning.
relating pre-
conditioned and post-conditioned first passage probabilities,
provides a
direct comparison of these types of taboo conditioning.
Furthermore, by expressing
FOSTABs,
FRETABs
t~e
as a function of the
the latter being iteratively obtainable from the unconditioned
transition probabilities, this theorem also provides a convenient method of
computing the former.
relates
PRETABs
to
Although not explicitly stated, a similar theorem
POSTABsi
since the latter are obtainable from the
unconditioned transition probabilities, so are the former.
Hence all
taboo probabilities and distributions discussed aT'e computable as
Me.
functions of the initial transition matrix goverflingthe
(FRETABs
THEOREI'l:
(i)
s
h
F
where
r
(ii )
(n)
=
Jk
(n)
from
f
FOSTABs)
(n)
/
= SUM/m(l,n)
f
.1'
)0=
it follows that
..
(n-l)
k Jh
(10),
s
h
h /= k, we have
n
)0=
f
h Jk
(m)
2.
Ji
we have
1
F
(n-1)-
h Jk
Ji
n
F
1 -
(
h Jk
IT for all
Assuming that
(n)
Jk
SlJl'1/ m( L n )
)0=
f
h Jk
(n
>.
+
f
k
(rn)
Jh
) (
1,
10
PROOF:
Define th e following sets:
(i)
A
B
- «
=
X
n
(e X
k
) )
h, k
1=
1
m
(e X
o.
C ==
=
=
<=
m
<= n-1
) )
»
J
Note that by definition we have
f
h J.k
(m)
= pex =
ki
1= h, k,
X
m
1 <.= u <.= m-l
u
= P(X =
k
for the first time and no
h
has occurred
= P(X = h
for the first time and no
k
has occurred
m
Similarly,
f
(m)
k Jh
x
o
m
Now, define
as the union of states
d
follows that
X
= k
--)
m
1= h
X
and
= h
m
pex
= d
k.
--).
Then, since
pex
m
P(X
m
h /= k,
it
X /= k, we see
m
for the first time;
X
= k
X
0
= J)
=
X
0
= J)
m
m
+
=
and
X
m
(m) + f
(m) =
f
h Jk
k Jh
h
= J >.
=d
for the first time;
=d
for the first time
X
m
h
X = J ).
0
Hence,
F
h Jk
(n)
+
F
(n)
= SUM/m(l,n)
(
= SUM/me 1, n)
P(X
k Jh
= P(X = d
..
m
and
(m) + f
(m)
f
k Jh
h Jk
= d
for the first time
m
for some
m,
1 <= m <:= n
X
0
0"
0
X = J)
0
= J>'
11
1
-
-
( rd
F
h Jk
(n) == 1
F
k Jh
=
-
P(X
==
1 <:= m <:== n
m,
for some
d
X
m
0
1= d,
P(X
1 ..,::: m
<== n
X
m
0
1= h, k,
== P(X
<:== m
1
(=
=
J)
J>
X
n
m
=
0
=
J ).
Now,
P(ElIC)
= P(X
x =
1 <:= m <:= n-1
1= h, k,
o
m
=1
F
J)
(n-1)-
h Jk
Then
s
(n)
h Jk
= P(AIBC) = P(ABC)/P(BC) = P(ABlC)P(C)/P(BIC)P(C) =
=
f
(ii )
F
1( 1 -
(n)
Obvious as
h
nor
be absorbing,
k
s
(n)
h Jk
or,
"waiting time" before absorption.
corresponding analyses,
the single state
COROLU\RY:
If
s
(n)
k Jk
PROOF:
(n-1)
>.
h Jk
F
(n-1)F
(n-1 )
k Jh
h Jk
0<:1 -
The condition necessary to have
state
F
(n-1)-
k Jh
h Jk
:>=
f
h
(n)
<:=
1.
is that neither
Jk
if they are,
that there be an infinite
One may extend this theorem, and the
H of taboo states
to a collection
rat~er
h.
h == k,
=
of
then
(n)
/
(i)
(1 -
Jk
P(ABiC)/P(BIC)
reduces to
F
(n)
).
Jk
Obviou; by inspection.
n
than
12
E.
TRANSITION FUNCTIONS WITH J.NTERVENING VARIABLES AS DUAL PRIOR AND
POSTERIOR FACTORS
In many epidemiological
inve~tigation5
a researcher is interested
in particular intervention strategies which may be time dependent.
For instance, one may ask what is the probability.
without visiting state
this
~uestion
h
(or
k)
'or the first
we consider a combination
probabilities described above.
0'
0'
t
absorption in itate
time units?
k
To answer
the taboo conditioned
These probabilities, merging
FRE- and
FOSTABs, are called dual-conditioned taboo probabilities and,
for tile case
of the first passage times, are written as
(nIt) = P(X
fs
hk
n
Jk
= ki
X 1=
u
h, k ,
X 1=
v
h, k
t+1 <:= u
<=
n-1
, 1 <:= v <:= t
X
= J ).
n
>=
t+1.
0
Using the following corollary to the previous theorem, we relate this
dual-conditioned probability to the FOSTABs,
thereby obtaining an iterative
method for computing the former.
COROLLARY:
fs
hk
Cnlt)
Jk
=
f
h Jk
(n)
I
( 1
F
r-
(t)-
k Jh
(t)
n
),
>=
t+1.
h Jk
D
Thus the distribution formed by summing the
hk
over
n
>=
from state
t+1
J
'$ Jk'(nIt)
probabilities
gives the desired dual-conditioned taboo time
distributions,
FS
hk
(NIt)
Jk
= SUM/n(t+1,N)
~o
fs
hk Jk
absorption
(n
It).
By computing this distribution for all initial states and averaging over
the initial distribution, we arrive at the general (irrespective of
initial state ) dual-conditioned taboo time to absorption distribution,
13
hk
GS (NIt) :: SUt11 J
k
a (0)
J
(Nit).
FS
By varying the value of
Jk
hk
the researcher mey assess the efficacy of a proposed intervention strategy
of explicitly preventing a visit to the taboo state as a function of time.
By altering the conditioning statements. a researcher may obtain
interesting variations of the above dual-conditiqned taboo probabilities.
e. g. , on e ma y c on sid e r on e
( i>
= ki
P (X
n
X 1= k,
v
0
f the foIl ow i n g e x pre s s i on s :
1 <:= v <:= n-1
1= k,
X
1 <:= u <= ti
X
=
J)
I
0
U
n ::.-== t
(i i )
P( X
n
=
ki
X 1= h, k,
v
1 (= v <:= n-1
1= k,
X
1 <:= u <:= t
X
i
n
(iii>
= ki
P(X
1= h, k,
X
n
1 <.= v <:= n-1
X
1= h,
1 <:= u <.= t
J)
>=
t
I
X = J)'
0
n >= t
I
u
v
=
0
U
One may express each of these taboo probabilities in terms of the initial
transition probabilities and post-conditioned taboo probabilities by deriving
an appropriate corollary to the previous theorem.
the numerator of the theorem by
f
(n)
To obtain
(i)
and the denominator by
one replaces
1-F
Jk
to obtain
(t)
1-F
Jk
2.
(ii)
and
and
(iii)
(t) i
Jk
one replaces the denominator of the theorem by
<t), respectively.
I-F
Jh
AN APPLICATION TO POPULATION EPIDEMIOLOGY.
To demonstrate these techniques we consider an
Me
model originally
designed for an inferential study of the effect of induced abortion on
subsequent pregnancy outcome (2).
Data used in this example come from a
historical prospective' study of "'lacedonian women residing in SkOPJe,
14
Yugoslavia who aborted their first pregnancy during 1968-69, Hogue (1).
Interviews conducted in 1972 produced,
for each woman, a chronological
record of her reproductive behavior between the beginning of menstruation and
Quality checks verified that the women interviewed
the interview date.
were indeed representative of the target population of Yugoslavian l#omen
(1)'
The interval between the initial abortion and the first delivery is an
important consideration to both population epidemiologists and demographers.
Let. _
represent an induced abortion,
d
a delivery, and assume that the
total mass of the initial distribution is concentrated on
«
basic time to delivery distribution is
F
(n»)
Then the
a.
where
ad
F
(n)
ad
= SUM/m(l,n)
f
(m)
and
n
indexes months since the initial
ad
This distribution, which appears in
abortion.
column 2
necessarily reflects diverse contraceptive use patterns;
further investigation is warranted.
rhythm or coitus interruptus), m
condom), and
e
=
effective method
(e.
each of these states as a taboo state,
i
= moderately
g.,
Table 1,
consequently.
The state space of this
all types of contraception into one of three states:
(e. g.,
of
=
Me
partitions
ineffective method
effective method (e. g.,
birth control pills).
By treating
the corresponding post-conditioned
taboo time to delivery distribution assesses the probability of a delivery
by month
n
without having resorted to the taboo type of contraception.
These distributions are listed in
columns 3-5
of
Table 1.
The dual-conditioning technique provides an assessment of at-month
limited intervention.
For example,
suppose a researcher wishes to evaluate
a program which would @xplicitly pr'event the use of ineffective
•
contraception for
t
months.
(NOTE:
of
.Consequently, women who desire to employ a
mothod of contraception must use either an effective or
method.
method~
moderat~ly
effective
This is not to imply that the women who avoid ineffective
contraceptive methods .will continualllJ'
or even temporarily,
employ other,
15
more effective methods.)
the dual-conditioned taboo
By treating
tim~
i
as a taboo state and evaluating
to delivery distribution
<N: t),
FS
id
may assess the effect of such a policy on those wompn who do not
during the first
t
We list these distributions in
months.
we
ad
delive~
Table 2.
Tables 1,2
This application addresses epidemiologic questions about the effect of
different patterns of contraceptive use on the distribution of time to
delivery.
By computing the post-conditioned time to delivery distributions
<Table 1) we are able to obtain insight into the population dynamics of
cohorts of women electing to use different levels of contraception.
Note
the large magnitude of the relative decrease in the unconditioned time to
delivery distribution <181.
= <1001.)
x <. 5344 - .4366)/.5344
which results when ineffective contraceptive methods,
taboo.
Thus the women who avoid
i
by month
60)
i, al'e regarded as
are less likely to deliver during
a set period of time than the group as a whole;
possibly this is because
the members of this subgrolJp who elect to use contraceptive methods
emp.loy a method which is at least moderately effective.
Even more
surprising is the similarity of the distributions which result when
moderately effective and effective contraceptive practices are treate~
as taboo,
/
avoiders of
As expected, however,
m
both of the associated subgroups,
and the avoiders of
the subgroup which avoid
i
e, are more likely to deliver than
1.
Finally, by computing the
distribution using
the
du~l-conditioned
taboo time to delivery
as the taboo state <Table 2)
we are able to assess
the efficacy of preventing ineffective contraceptive practices.
Note that
the taboo time to delivery distribution follOWing the termination of this
16
implementation policy appears to be invariant across the duration Or that
policy.
For
ex~mple,
the probability of a delivery less than
following termination of policy is approximately
of
t
studied.
Also note that these
in columns
2,3,4
a
delay.
t-month
t-month
0.29
24
months
for all three values
limited distributions
are not the unconditioned (column 2) distribution with
Clearly, states other than
i
may be of interest as the taboo state
in the dual-conditioning investigation.
the state of susceptibility to pregnancy;
Or particular concern might be
similar distribution calculations
would give insight into the efficacy of an implementation program encouraging
all women to employ some type Or contraceptive
practi~e.
These distributions
would provide benchmarks or total acceptance against which family planning
administrators could compare actual practice.
dual-condition representations,
Furthermore,
by modifying
one may calculate similar distributions ror
more complicated patterns Or contraceptive, switching within cohorts or users.
3.
SUMMARY
We have
made
use of a typical epidemiologic: problem ·to illus.trate the use
of conditional rate information.
Using a traditional cohort rJrmulation such
an investigation would have required large amounts Or data -- an expensive
proposition,
even ir data were available.
Alternatively, we are able 'to
employ the conc:epts of pre- and post-conditioned probabilities and the theorem
relating them.
In the example presented above,
the
Me
was
s~bmitted
of-fit tests in order to verify the Markov property (3).
application,
validation,
to formal goodness-
Depending on the
the researc:her may prefer to (i) employ structural assumption
1. e.,
verify the t'iarkov property by comparing probabilities
for all pairs of states visited over appropriate time lengths,
(ii) accept
17
the model as a reasonable approximation to the empirical process under
stud~,
or (iii) loosely validate the property by examining functions of the
transition matrix
(e.
g.,
those leading to the time to absorption in a state)
and seeing if these functions replicate eqUivalent empirical distribution
functions derived directly from the underlying data.
We caution
th~t
the
latter is not a strong validation since, as seen in (3), distinctly
different populations may produce similar functions from their respective
trans~tion
matrices.
Making use of an
Me
offers the additional advantage of employing partial
path information which is unavailable in cohort studies.
follow an individual up to a certain point,
That is, we may
lose that person to observation (or
follow-up)· and not have use of that "path" Tor a cohort study.
However,
all
transitions made up to the point of loss are valid for computing maximum
likelihood estimators of the chain's transition probabilities.
Hence the
behavior, for example in visiting various combinations of contraceptive
states, of a "lost" individual may be reflected in the transition probability
estimators.
When applied to the dual-conditioned taboo probability estimators,
this information is especially advantageous in maximizing the use of
available data.
In addition,
by using either a parametric analysis varying particular
transition probabilities or simulation,
one may investigate the effects of
alternative assumptions about the magnitude of certain probabilities ~n
final cumulative distributions.
In our example, for instance,
one may wish
to raise or lower estimates of particular contraceptive uses, and/or
probabilities of Jumping to "susceptible" states, and test the effect on
functions of the transition matrix, while simultaneously taking into account
one or more intervening variables.
In summary, we have shown how a stochastic model allows researchers
to assess the effect oj implementing intervention in.situations where
18
financial and/or ethical considerations prohibit cohort studies.
Furthermore,
we have established new analvtic versions of conditional rates for use in
biological and
epidemiologic~l
studies and
h~ve
generated various
produce such l'ates in the face of fntervening variables.
presented relationships among them,
Finally,
thereby further illustrating
a Markov chain as a tool of (statistical) inference.
WdY
to
we have
th~
use of
19
REFERENCES
1.
C. .J.
Hogue, "Low Birth Weight Subsequent to Induced Abortion:
Historical Prospective Study of 948 Wumen in SkOPJe,
Yugoslavia," AM . .J. OBSTET. GYNECOL. 123, 675-81 (1975>.
2.
R. H.
Shachtman and .C . .J. Hogue, "Markov Chain Model for Events
Following Induced Abortion," OPNS. RES. 24, 916-32 (1976>.
3.
R. H.
Shachtman, .J. R. Schoenfelder and C. .J. Hogue, "Using a Stochastic
Model to Investigate Time to Absorption Distributions,"
Submitted
to OPNS. RES. (1979 >.
A
20
TABLE 1
Time to Delivery Distributions*
(initial abortion to delivery)
Mo.nth
n
Unconditioned
(n)
F
ad
12
24
36
48
60
•
*
States:
a =
d
i =
m=
e =
0.0392
0.2047
0.3382
0.4465
O. 5344
Post-Conditioned
F
(n)
i ad
0.0356
0.1797
O. 2892
0.3727
0.4366
induced abortion
= delivery
ineffective contraceptive met~ods
moderately effective contraceptive methods
effective contraceptive methods
F
(n)
m id
O. 0384
O. 1991
0.3278
O. 4314
O. 5149
F
(n)
e id
O. 0390
0.2025
0.3334
0.4386
O. 5230
21
..
Tab Ie 2
Split-Conditioned Time to Delivery Distributions following an
induced abortion
Taboo state is ineffective contraceptive methods*
Month
N
Unconditioned
Split-Conditioned
(N)
F
12
24
36
48
60
e
.0392
.2047
.3382
.4465
.5344
id
ad
t=12
t=24
t=36
. 1651
.3027
.4067
.4878
. 1594
.2936
.3962
. 1541
.2840
•
*
(Nlt>
FS
ad
States:
a = induced abortion
d = delivery
i = ineffective contraceptive methods