*This work was supported by u.s. National Heart and Lung Institute
contract NIH-NHLI-7l-2243 from National Institutes of Health.
CONDITIONAL FAILURE TIME DISTRIBUTIONS UNDER COMPETING RISK
THEORY WITH DEPENDENT FAILURE TIMES AND
PROPORTIONAL HAZARD RATES*
by
Regina C. Elandt-Johnson
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1015
June 1975
CONDITIONAL FAILURE TIME DISTRIBUTIONS UNDER COMPETING RISK
THEORY WITH DEPENDENT FAILURE TIMES
AND PROPORTIONAL Iv\ZARD RATES
by
Regina C. Elandt-Johnson
Department of Biostatistics
School of Public Health
University of North Carolina
Chapel Hill, North Carolina 27514
U.S.A.
ABSTRACT
Suppose that death (or any non repetitive event) can occur due to
various causes, each having its own failure time.
Assuming independence
of failure times and proportional hazard rates over the whole range of
time, some authors shown that the single cause failure time distributions
conditional on the cause of death, each in presence of the remaining
causes, are the same as the distribution of observable failure time,
regardless of the cause.
It has been shown in the present article, that
this result is also valid without the assumption of independence (Section 3).
It has also been suggested (Section 5), that in the case of depen-
dent failure time, a conditional limiting distribution (as
could represent the failure time distribution when cause
nated.
T
ex
C
ex
is elimi-
Three examples (trivariate exponential, bivariate Compertz, and
U.S. Life Table 1959-61 data) are given as illustrations.
-2-
1.
INTRODUCTION
Consider a population in which
are operating.
k
causes of death,
C l' C , ... , C '
2
k
Each individual in this population is exposed to risk of
dying of anyone of these causes.
Methods of analyzing mortality experi-
ences in such a population are known as competing rick analysis.
A funda-
mental problem of competing risk analysis is the estimation of prolongation
of life expectancy where one (or more) cause(s) of death is (are) eliminated.
A useful and uptodate critical review on models which have been employed in
this field, has been recently given by Gail (1975).
We shall refer to his
article throughout this paper.
There are at least two kinds of distribution associated with failure
due to cause
C ,
a
which are important in competing risk analysis:
The failure-time distribution due to cause
(i)
that
(ii)
C
a
is the cause of death, in
p~esence
the failure-time distribution due to
alone.
C
a
C
a
conditionally
of other causes;
if
C
a
is acting
Of course, this latter distribution cannot be observed.
A central assumption in the analysis of competing causes is that the
force of mortality due to cause
same as in their
of
fo~ces
of
a
in the absence of other causes
1S
the
It is sometimes called an assumption of identity
p~esence.
mo~tality
C
(see (2.18)).
To be able to treat the problem in a more formal (mathematical) way,
the idea of a "due time",
duced.
T
for death from cause
a
C
a
is often intro-
It is supposed that each individual, presumably at birth, is
endowed with a set of such times
J
Tl' T'") ,
The actual time of death is the minimum of
assumed that the random variables,
T , T'") ,
l
... , Tk
one for each cause.
,
... , Tk
T , T ,
2
l
... ,
\
,
It is often
are mutually inde-
-3-
pendent.
Under this assumption, stochastic models in competing risk
an~dy-
sis become much simpler.
The purpose of this article is to discuss some mathematical and
practical consequences when neither the assumption of "identity of forces"
nor the assumption of independence hold.
In particular in Section 3, we
shall prove that in the class of distributions with rroportional forces
of
mortalit~
the failure distribution due to
death occur due to
C),
ex.
C
ex.
(conditional that the
in presence of other causes is identical with
that due to other (unspecified) cause(s), even though the assumption of independence
may not be valid.
For a joint fai lure time distribution of
T I' T2' ... , Tk '
when it
is expressed in a parametric form, we will also suggest (in Section 5) a
method of deriving failure time distributions when
and when
C
ex.
co.
is acting alone,
is eliminated.
To illustrate the results and the techniques, three examples:
joint
trivariate exponential distribution, joint bivariate C;ompertz distribution
and a mUltiple decrement life table model will be discussed in some detail.
2.
BASIC DEFINITIONS AND NOTATION
To make the paper selfcontained, we shall introduce some basic concepts of competing risk analysis.
We will use an approach essentially
similar to that used by Gail (1975) but with a somewhat different notation
which resembles (but is not identical with) the actuarial notation used in
multiple decrement tables.
We will be mainly concerned with the conse-
quences of certain assumptions on the mathematical models of failure-time
distributions rather than on the comparison of existing models under different assumptions.
-4-
If
T
a
is the failure time due to cause
F (t) = Pr{T
a
is the cumulative
failu~e
time
a
~
dist~ibution
C
a
alone, then
t}
(2. 1 )
due to
and
C
(1
5 (t) = Pr{T >t} = I-F (t)
a
1S
the
su~vival
a
function from
where
C
a
(2.1a)
a
is the only cause of death.
C
a
In most derivations in this paper, where it is convenient,the concept
of the survival function rather than of the failure time distribution will
be used.
The function
dlogS (t)
- - -a- -
dt
is known as the force of mortality or the
haza~d ~ate
(2.2)
(also the intensity
function) of the failure time distribution.
When
causes,
k
T , T ' ... , T
Z
k
l
failure times
failu~c
time
C , C , ... , C , are operating, hrith 'hypothetical'
1
2
k
respectively, one may introduce the joint
dist~ibution
(
:: . ::;)
and the corresponding joint sUI'vival function
k
51 2. . . k (t 1 ' t ~') , ... , t k )
Pdn
(T >t )}
0.=1
a
(when no ambiguity arises, we use simply the notation
We notice that
5(0,0, ... ,0)
=
1
and
5(00,00, ... ,00)
(2.4)
ex
S(tl,t::, ... ,t k )).
=0 .
Methods of analyzing mortality experiences of populations when many
causes
of death are operating usually imply a common assumption, which
-5-
we will call the fundamental assumption of survivorship analysis, and
which is
EACH DEATH IS DUE TO A SINGLE CAUSE.
This assumption has important consequences on the underlying models of
failure.
2.1.
First, we cannot simultaneously observe the failure times,
T , T , ... , T ,
l
2
k
so that
S(t ,t , ... ,t )
k
l 2
cannot be observed nor
can its form tested, without introducing further assumptions.
What, in
fact, can be observed is
(2.5)
The corresponding survival function is
k
SW(t) = Pr{W>t} = pr{n(T >t)} = S(t,t, ... ,t) ,
0.=1 a
where
t
is the observed failure time.
defined in (2.6), which we denote by
(2.6)
The hazard rate of the distribution
a~(t)
,
and which is the hazard rate
due to any (unspecified) cause when all causes are operating, is
aj..l(t) =
(2.7)
=
(The prefix
~
dS(td~··,t)
in the symbol
For a mathematical function
k
dS(t,t, ... ,t) =
I
0.=1
dlogS(t, ... ,t)
dt
/S(t, ... ,t)
a~(t)
can be regarded as standing for "all".)
S(t ,t , ... ,t )
k
l 2
as (t l' t 2' ... , t k )
at a
we have
I{t. =t}
J
dt .
-6-
It follows that
dlogS(t,t, ... ,t) =
dt
a~
Let
cause
C
a
a
(t)
k
I
0'.=1
alogS(tl,t z'" .,t k )
I
ato'.
(2.8)
{t.=t}·
J
denote the rate of decrement (i.e. hazard rate) due to
in presence of all causes operating.
Clearly, from the mathe-
matical relation (2.8), we have
a~
a
(2.9)
(t) =
and
(2.9)
It should be appreciated that (2.9) is a consequence of the fundamental assumption that each death is due to a single cause; when
this assump-
tion does not hold, (2.9) is not neceesarily true.
2.2
Integrating (2.9) over the interval
(O,t),
we obtain
(2.10)
Hence
and so
Sw (t ) = S (t , t , ... , t) = G1 (t) . G2 (t) ... Gk (t) ,
(2.11)
where
G (t)
a
(2.12)
-7-
for
a = 1,2, ... ,k.
(See also Gail (1975)).
Imagine a Ihypotheticall
with force of mortality
with
C
a
a~
a
population in which cause
(t)
.
Let
T'
a
C
a
is acting al.one
be the failure time associated
in the "imaginary" population.
We can see from (2.11) that
Sw(t) can be calculated as if the event
k
I
(W>t) were equivalent to the event
(T~>T)] , where T T2 , ... , TkI
0,=1
I
are mutually independent and T
has the survival function G (t)
Of
a
a
course, this does not mean that T , T , ... , T
are necessarily mutually
k
I
2
[n
independent)or that
2.3.
T
a
necessarily has the survival function
G (t)
a
.
Suppose that we are interested in the failure time distribution
due to all causes operating, except cause
take
i,
C
a
Without loss of generality
a = 1 .
One way of handling this problem, would be to ignore deaths due to
C
This does not mean that there are no such deaths; it only means that
a
we are not interested in them and omit them in the analysis.
We need to modify the notation, which may become rather awkward for
some functions.
In fact, the random variable
k
variates,
T , T , ... , T ,
2
k
I
W defined in (2.5) is the minimum of
and it can be denoted by
W
k
i.e.
(2.4a)
When only
k-l
causes are operating, (ignoring
C ), then we may have
I
(2.13)
From (2.6), we have
Sw (t) = S12 ... k(t,t, ... ,t) .
k
(2.6a)
-8-
When
C
l
is ignored, the survival function from the remaining causes
is the marginaZ survival function
Sw
k-l
(t) = S23
...
k(t, ... ,t)
(2.14)
S12 ... k(O,t, ... ,t)
(t), we obtain a new set of
k-l
causes together) and due to each cause separately
Applying a similar argument to
hazard rates due to
so that
S12 ... k(O,t, ... ,t)
k-l
Sw
which, in general, would not be the same as in the case of the original
causes.
k
Finally,
(2.15)
and, in general,
(2.16)
(See also Example 4.1).
Ignoring two causes
etc.
(k-l)-causes
Sw 1 (t)
where
C
k
Sk (t) = Pr{T
k
(e.g.
(e.g.
C , C ), three causes
l
2
Cl , C ' ... , C _ ),
2
k l
(e.g.
we obtain ultimately
(2.17)
= S12 ... k (0 , a, ... , t) = Sk (t) ,
> t}
is the marginal survival function due to cause
alone when other causes are ignored, with hazard rate
in (2.2).
C , C , C ),
3
l
2
Of course, the results are valid for any cause
~k(t)
C ,
a
defined
since the
subscript is immaterial.
2.4.
As we have mentioned before,s(t1, •.. ,t ) cannot be observed nor
k
its form tested, so that the results discussed above have only theoretical
meaning.
In applications, and in particular, in actuarial work a further
assumption is made, namely,
-9-
a1-1 (t) :: 1-1 (t) ,
a
a
a
=
1, 2, ... , k
known as the assumption on identity of forces of mortality.
(2.18)
This does imply
(2.19)
Sw ( t ) = S 12 ... k (t , t , ... , t) = S1 (t) • S2 (t ) ••• Sk (t) )
that is, that the events
(T
a
> t),
a = 1, 2, ... , k, are independent.
course, is does not necessarily mean that the random variables,
are independent (see also Gail (1975)).
Of
T ,T ,·· .,T '
k
1 2
However, to the best of my knowledge
no counter-examples, i.e. such a situation that (2.19) is true, and
does not hold, has been produced.
If assumption (2.18) is true, then we have
G (t) = S (t)
a
3.
a
,
(2.20)
a = 1, 2, ... , k .
SURVIVAL FUNCTIONS FROM DIFFERENT CAUSES WHEN THE
HAZARD RATES ARE PROPORTIONAL
3.1.
Of special interest is the situation when
the ratio of hazard
rates for any two causes is constant over the whole range
(0,00)
.
This
implies that
a1-1 (t)
a
with
I
a
c
a
1 ,
since
= c a • a1-1(t) , a =
I
a
G (t)
a
a1-1 (t)
a
= a1-1(t)
1, 2, ... , k
(3.1)
Hence
ex{caf: a~(U)dj
= exp C
J: a1-1(U)dj ca = [S ] ca
=
W
(3.2)
-10-
for
a
=
1, 2, ... , k .
Therefore, when the hazard rates,
proportionaZ the evaluation of
G (t)
a
the survival function from all causes,
a]J (t) ,
a
(a = 1, 2, ... , k)
are
,
present no difficulty, provided that
Sw(t) , is known.
It should be
appreciated that the assumption of proportionality over the whole range
(0,00)
does note require, in the general case, parametric form of
SW(t)
to be known.
We shall now prove another useful result under the proportion-
3.2.
ality assumption.
C
Q (t) denote the probability of dying from
a
a
presence of other causes, in the interval (O,t)
(Qo.(t)
Let
called the 'crude' probability).
aZone, but in
is sometimes
Thus
Q (t) = Ita]J (u) SW(u)du .
a
0 a
(3.3)
In particular, the proportion of all deaths which are due to cause
C
a
is
(3.4)
Q (00) = [a]J (u) Sw(u)du
a
0 a
Let
C
( -a)
denote all causes excluding
the (conditional) failure time distribution
presence of
C(_a) .
Co.
and
Fa; (-a) (t)
for deaths due to
C
a
denote
in the
Then we have
(3.5)
The corresponding survival function is
S
a; ( -a )(t) = 1 - Fa; ( -a )(t) .
(3.6)
Under the assumption that the hazard rates are proportional, we have
-11-
a~
a
(t) = c
a~(t)
a
.
It follows that
Qa (t)
rt
c
J
a 0
a~(u) Sw(u)du
(3.7)
and
Q
a
(00) = c
(3.7a)
a
Hence
Sa; (-a) (t) = 1 - Qa(t)/Qa(-oo)
=
1 -
J:av(u)
Sw(u)du
=
SwCt) ,
(3.8)
since
(3.9)
is the cumulative failure time distribution from any (unspecified) cause.
We have thus obtained the following result:
If the ratio of the hazard rate due to cause C (a=1~2~ . .. ~kJ in the
a
presence of the remaining causes c (-aJ to the hazard rate due to all
causes is
constant~
then the failure time distributions among failures due
in presence of
C
(-aJ
are identical with the distri-
bution of the failure time due to all causes.
Some authors (Sethuraman (1965), David (1970), Moeschberger and David
(1971)) have proved this result under the assumption of independence of
failure times,
T , T , ... , T
l
2
k
not necessary to ensure (3.8).
As we now can see, this assumption is
-12-
4.
EXAMPLES
Trivariate standard exponential distribution.
EXAMPLE 4.1.
To illustrate some of the results and techniques discussed in Section 2
and
3,
we will consider a multivariate generalized Farlie-Gumbel-Morgenstern
family of distribtuion. (Butkiewicz(1974), Johnson and Kotz (1975).) For this
family, the joint survival function (and, of course, the cumulative distribution function) can be simply expressed in terms of marginal survival functions of the individual variates.
In trivariate case with random variables
in the example instead of
vival functions
T , T , T
3
l
2
Sl (x), S2(y), S3(z) ,
S123(x,y,z)
x
= Sl (x)
[1
+ 8
+ 8
23
12
X, Y, Z (which will be used
respectively), having marginal surwe have
• S2(y) • S3(z)
Fl(x) F (y)
2
+ 8
13
Fl(x) F 3 (z)
Fl(x) F (y) F 3 (z)] ,
F (y) F (z) - 8
123
2
3
2
with the restriction
and two similar conditions, and
(See Johnson and Kotz (1975).)
The joint density for (4.1) is
(4.1)
-13-
x {I + 8
l2
[1-2F (x)][1-2F (y)] + 8 [1-2F (X)][1-2F (Z)]
l3
2
l
l
3
+ 8
where
F (x)
Y and
Z .
l
4.1.1.
= 1 - Sl (x)
l23
[1-2F (X)][1-2F (y)][1-2F (Z)]}
2
3
l
dS (x)
l
f (x) =- dx
l
and
(4.2)
and similarly for variates
We will consider here the case when the marginal distributions
are standard exponential, that is
= e -x
and similarly for
Y and
(In fact we must then have
F (x) = l-e
l
Z.
a
-x ,
We also take
< 8 <
e
8
12
-x
= 8 13 = 823 = 8 123 = 8
>
~).
Then the trivariate joint survival function is
(4.3)
and the joint density is
f
123
(x,y,z) = f(x,y,z) = e
-(x+y+z)
[1+8(2e
-x
-1)(2e
-y
-1)+8(2e
-x
-1)(2e
-z
-1)
(4.4)
The survival function
Sw (t)
from all causes at the observed time
t,
is
3
Sw (t)
=
S(t,t,t)
=
e
-3t
[1+8(2+e
-t
3
The hazard rate due to all causes,
a~(t)
,
is
)(l-e
-t 2
) ] .
(4.5)
a .
-14-
all(t) =
dlogS w (t)
_ _ _3_ _
=
~
t
3 1-8
dt
e- (1_e-t
1+8(2+e
C
The hazard rate for cause
2t
)(l-e
(i.e. associated
l
~
)
•
(4.6)
-t 2
)
X),
with failure time
is
dlogS(x,y,z)
dX
x=y=z=t
1
= 1 -
1+8(2+e
-t
"3
-t 2
)(l-e )
and exactly the same expression holds for
(4.7)
all(t)
a 1l (t)
2
and
a 1l (t) .
3
We then have
that is, we have a situation in which the hazard rates, all (t) ,
ex
portionaZ over the whole range
1:
1
are pro-
with proportionality coefficients
(0,00) ,
1
= c = c =3
2
3
From (3.2) , we obtain
1
G (t)
ex
for
e
-t
[1+8(2+e
-t
-t 2 3
)(l-e ) ]
(4.9)
ex = 1, 2, 3
From (3.8), the survival function from
is identical with
C
aZone in presence of
l
Sw (t)
C
2
given by (4.5), that is
3
S1 ; ( -1) (t) = Sw 3 (t) = e
-3t
[ 1- 8 (2 +e
-t
which also can be checked by direct calculations.
for
S2;(_2)(t)
and
) (1- e
-t 2
) ] ,
(4.10)
The same expression holds
-15-
Suppose that we now ignore cause
4.1.2.
vival function from
of
S123(x,y,z) ,
C
l
and
C
2
C .
3
S12(x,y),
Then the joint sur-
is a bivariate marginal
that is
S12(x,y) = S123(x,y,O) = e
-(x+y)
[1+8(l-e
-x-y
)(l-e )] ,
(4.11)
and the bivariate density function is
(4.12)
Using the same techniques as in the case of trivariate distribution,
Sw (t) ,
we obtain the survival function,
from any of the two causes,
2
Sw (t) = e
-2t
-t 2
[1+8(1-e)].
(4.13)
2
and
Let
a~(t;3)
C )
2
together, when
from cause
C
1
denote the hazard rate from the remaining causes
C
3
is ignored, and
in presence of
C when
2
C
3
a~l
(t;3)
(e
l
the hazard rate
is ignored, and similarly
We have
a~(t;3)
(4.14)
=
and
as 12 (x,y)
dX
Ix=y=t
1
= 2
and the same expression for
a~2(t;3)
.
all(t;3)
,
(4.15)
-16-
Thus
"21
=
and
a]J(t;3)
H (t) - which can be denoted here by
a
G (t;3)
a
=
[Sw (t)]
2
~
2
(4.16)
G (t;3) - is
a
= e -t [1+80-e -t ) 2 ] ~
2
(4.17)
,
a = 1,2 .
Clearly, from (4.9) and (4.17)
G (t;3).,tG (t)
a
a
4.1.3.
a
for
=
1,2.
Finally, if we ignore the mortality from
observe only deaths from
C ,
l
C
2
and
C ,
3
and
the survival function,
univariate standard exponential, that is
Sw (t)
=
Sw (t)
is the
l_t
Sl (t) = e
with the
1
hazard rate
a]Jl (t;2,3)
EXAMPLE 4.2.
Multiple
= ]Jl(t) = 1, and the density function
life tables, distinguishing different causes of
death are special examples of competing risk analysis.
include various 'standard' columns.
alx/ala)
a .
The
al
x
column
the total number of deaths between age
x
causes.
x
out of
t = x .
x
ala
al
x
who
(or more precisely, the pro-
corresponds to the survival function
in (2.6) and evaluated at time points
ad (a)
These tables usually
Among these, the column headed
represents the number of survivors at the exact age
portions
= e -t
Life tabZes by causes of death.
dec~ement
started life at age
f (t)
l
and
The
x
+
ad
1,
x
SW(t)
defined
column gives
and different
columns give the corresponding numbers of deaths from different
-17-
We may notice that
00
al(a) =
x
for
X l
a
1, 2, ... , k
=
explicitly given in the table).
cause
at
Ca
t
=
(4.18)
gives the number of individuals among
who eventually die from cuase
corresponds to the
a=x
ad(a)
x
(The
C
a
al (a)
Thus
x
al (a)
x
al
now aged
x
values are usually not
(or more precisely, al~a) /a16a))
survival function for individuals
in presence of other causes, i.e. to
dying
Sa;(-a) (t) ,
from
evaluated
x
If the hazard rates due to various causes were proportional, then the
ratios
alx/al
o
al~a)/a16a) should be approximately the same (in
and
view of the result (3.8)), for each
the ratios
al (a)/al
x
x
all values of
x
x
(0 < x < 00) •
should be approximately equal to
al (a)/al
In fact,
individuals of present age
x
x
This also means that
al (a) / al
o
for
0
represents the proportion of
x
who eventually will die from cause
C
a
these figures are sometimes given in multiple decrement tables.
Table 4.1 in this paper, is an extract from U.S. Life Tables by
Causes of Death: 1959-61, p. 44, (1968).
These are abridged tables, with
most ages at quinquennial intervals.
Table 4.1 gives the proportions
(C ) ,
l
al (a) /al
x
x
for:
malignant neoplasms
and major cardiovascular-renal diseases (C ) .
2
We notice that for malignant neoplasms, in the range of age
0-60
this ratio is fairly constant, but it decreases somewhat for older ages.
On the other hand, for cardiovascular diseases, this ratio is an increasing function of age so that the proportionality assumption for this cause
does not hold.
-18-
TABLE 4.1
The ratios
a1 (a) fa1
x
x
for malignant neoplasms
and eardio-vaseu1ar-rena1 diseases.
(U .5. Life
Table 1959-61, Total population)
Card.-vase.
-renal
diseases
Age
(x)
Malignant
neo 1asms
(x)
Malignant
neo 1asms
Card. -vase.
-renal
diseases
0
.15154
.61075
45
.15557
.65404
1
.15551
.62688
50
.15332
.66061
5
.15574
.62942
55
.14921
.66917
10
.15573
.63084
60
.14285
.67977
15
.15576
.63208
65
.13362
.69269
20
.15608
.63466
70
.12154
.70830
25
.15659
.63805
75
.10719
.72539
30
.15688
.64127
80
.09113
.74260
35
.15693
.64478
85
.07349
.75728
40
.15658
.64892
Age
-19-
5.
IS IGNORING THE DEATHS FROM A GIVEN CAUSE EQUIVALENT
TO THE ELIMINATION OF THIS CAUSE?
As was mentioned in Section 3 (see also Gail (1975)), one way of
evaluating the survival function when cause
is to ignore the deaths from this cause.
is actually eliminated,
C
a
In other words, just to consider
the marginal survival function associated with the remaining causes
We now ask:
is the effect of
The deaths due to
C
a
C
a
C(_o.) .
actually eliminated by this method?
are still occurring but we just do not take notice
of them.
It seems more reasonable to think that if we can achieve such a level
of health care that a disease associated with
in practice nobody dies from it, then
cause of death.
is under control and
C
a
we effectively eliminate
T
a
T
a
becomes
To present this problem in a formal way, we say that if the
limiting (conditional on
given
as a
Small pox can serve as an example.
The way in which this comes about is that the competing
very large.
C
a
T ) ,
a
distribution of surviving from
C(_o.) ,
exists, then this might be the survival function from
~ 00
is eliminated.
C
a
other causes, when
We then should have
k-l
Pdn (T.>t)!T
J
a = t}
a = S( -a )1 (t\oo) .
lim
a
(5.1)
00
j=l
t~
We may extend this concept to the elimination of two, three, ... etc.
(k-l)
causes, so that finally the survival function from the cause
s'
C
say, given the effects of the remaining causes was eliminated could be
defined as
I
lim pdT S>t To. = t
t~
a
a
a
= 1,
2, ...
(5.2)
-20-
This is to say, that (5.2) represents the survival function from
C
s
alone when other causes were eliminated (i.e. their failure times were "postponed" to infinity).
Of course, if the assumption of independence of failure times,
T , T ' ... , T ,
l
Z
k
is valid then
we have
pdT >t IT
S S a. = t a.'. a. = 1,2, ... , a.
~ S} = SS(t)
,
(5.3)
so that the methods described in Section 3 and in this Section produce
equivalent results.
We now present two examples, in which the method of conditional vs.
marginal distribution, when
a cause of death is eliminated, will be dis-
cussed.
EXAMPLE 5.1.
We consider the trivariate exponential distribution
from Examples 4.1.
5.1.1.
Suppose that cause
C
3
conditional limiting density function,
f
12joo
(X,y!OO) ,
f
f
Z
is eliminated, i.e.
(X,y!OO) = lim f 12 !3(x,y,lz=z)
12100
z-+oo
lim
+
00
The
is
lZ3
(x,y,z)
f 3 (z)
(5.4)
where
is given by (4.4) and
e
-x
The (conditional) bivariate 'survival function',
S121 00 (x,y\00)
is
oo 00
xfye-(x+Y)[1-8(2e-x-l)-8(2e-y-l)]dYdX
f
= e
and the survival function
-(X+Y)
-x
-y
[1+8(l-e) + 8(l-e .)] .
S(_3) 100(t!OO) ,
is
(5.5)
-2]-
SC_3) 100Ct I00) = s12looCt,t I00) = e
-2t
-t
[1+26Cl-e)]
C5.6)
This is different from that given by C4.l3).
5.1.2.
Suppose that
t~o
C
2
causes,
and
C ,
3
are eliminated, i.e.
y-+oo, x-+oo.
The conditional limiting density function,
flloo,ooCxloo,oo),
f 123 Cx,y,Z)
-x
6
-x
= lim--=';f=---Co---)--= e [1 - 1+6 C2e -1)] ,
x-xx>
12 x,y
y-xx>
where
f
x,y,z)
123 C
is given by C4.4) and
Thus the survival function,
when
C
2
and
C
3
x,y)
12 C
Slloo ooCtloo,oo)
C5.7)
by C4.l2).
from cause
C
l
alone,
are eliminated, is
6
-x
dx
1+6 (2e -1)]
Sl!oo,ooCt 100 ,00) = Iooe-X[l
t
= e
-t
[1 +
which is different from the marginal
EXAMPLE 5.2.
f
is
e
6
-t
(I-e ) ]
1+6
(5.8)
-t
Bivariate Gompertz distribution,
We wish to consider here some modification of bivariate extreme-value
Gumbel distribution.
Gumbel (1965) (see also Johnson and Kotz (1971), p. 251)
suggested a general bivariate form
(5.9)
for
where
F (x) , F (y)
2
l
F Cx) ,
2
we use
e
< 1 ,
are univariate extreme-value distributions.
gest here some modification of (5.9).
and
0 <
S12(x,y) , Sl (x)
CNote that this produces a different
First, instead of
and
S2(y)
F
12
We sug-
(x,y) , FICx)
respectively.
distribution that (5.9).)
-ZZ-
Second, we shall use in this example, univariate Gompertz distributions,
each being a truncated (from below at time
x
= 0,
or
y
= 0)
extreme-
value Type 1 distribution.
The algebra is rather lengthy, and we confine ourselves to presenting
only the final results.
The hazard rates and survival functions for the two marginal Gompertz
distributions are:
alx
].II (x)
~1l [l-e a x1lJ~
(5.10)
exp ~2a Z [1-: 2Y1J .
(5.l0a)
Sl (x) = exp -a
Rle
aZy
SZ(y) =
].IZ(Y) = RZe
The joint survival function, SlZ(x,y)
1
is,
-e
Putting
x
= y = t,
(5.11)
we obtain the 'observable' survival function
Sw (t) = SlZ(t,t)
Z
After rather lengthy algebra, we find the limiting survival function,
(5.12)
with the hazard rate,
].Izioo (t) ,
say,
(5.13)
which is again a Gompertz distribution with parameters
a
2
and
,
RZ
-23-
6.
CONCLUDING REMARKS
The results presented in this paper are intended as contributions to
both theoretical and practical aspects of competing risk analysis.
(i)
Various kinds of survival functions from single or combined
causes, in presence or absence of other causes were introduced and defined
using formal distribution theory.
Example 4.2 illustrates their relationship
to the multiple decrement life table functions.
(ii)
Clearly, such random variables as "times due to die" from dif-
ferent causes cannot be observed directly, and so the survival functions,
S (t)
a
or
Saloo(tI OO )
,
estimated without additional assumptions.
a =
1, 2, ... , k ,
cannot be
Nevertheless the analysis en-
hances the investigator's awareness of the possible consequences of relationships among survival functions under special assumptions (for example,
the assumption
(iii)
alJ (t) = lJ (t)).
a
a
In constructing multiple decrement life tables, various assump-
tions on the behavior of the
The most common are:
al
x
function in small intervals are made.
uniform distribution of deaths, constant hazard rate,
and proportional hazard rates from different causes (e.g. Chiang (1961)).
It would be probably possible to design some Monte Carlo studies to assess
the effects of these assumptions.
Especially some parametric models such
as exponential of Gompertz (see Examples 4.1, 5.1 and 5.2) would be useful to consider.
(iv)
Finally it seems feasible to conduct some experiments on labor-
atory animals in which diseases are genetically controlled, or on mechanical devices in which time to failure of various parts depend on the strength
-24-
of material of which they are made.
It would then be possible to estimate
directly, under some circumstances, the survival function,
possibly
setl, ... ,t )
k
for small number of causes
(e.g.
s0. I (tl oo )
and
00
k = 2
or
k
=
3)
-25-
REFERENCES
Butkiewicz, J. (1974). On a class of bi- and multivariate distributions
generated by marginal Weibull distributions. A paper presented at
the 7th Prague Conference, August 1974.
Chiang, C.L. (1961). On the probability of death from specific causes in
the presences of competing risks. The Fourth Berkeley Symp. Vol. IV,
169-180.
David, H.A. (1970). On Chiang's proportionality assumption in the theory
of competing risks. Biometrics~, 336-339.
Gail, M. (1975). A review and critique of some models used in competing
risk analysis. Biometrics 21, 209-222.
Gumbel, E.J. (1965). Two systems of bivariate extremal distributions.
The 35th Session of the Int. Statist. Inst., Beograd, No. 69.
Johnson, N.L. and Kotz, S. (1971). Distributions in Statistics: Continuous Multivariate Distributions, Wiley, New York, Chapter 41.
Johnson, N.L. and Kotz, S. (1975). On some generalized Farlie-GumbelMorgenstern distributions. To appear in: Communications in
Statistics, i.
Moeschberger, M.L. and David, H.A. (1971). Life test under competing risk
causes of failure and the theory of competing risks. Biometrics~,
909-933.
Sethuraman, J. (1965). On characterization of the three limiting types of
extreme. Sankhya Ser. A ~, 357-364.
United States Life Tables by Causes of Death: 1959-61. (1968). U.S. Dept.
Health~ Education and Welfare, Public Health Service, Vol. I, No.6.