ESTIMl\TION IN THE GENERAL MULTIPLICATIVE rvDDEL FOR SURVIVAL
TECHNICAL
NORMAN L, JOHNSON
AND
REGINA C. ELANDT-JOHNSON
JULY 1978
U, S. ARJ\iY RESEARCH OFFICE
RESEARCH TRIANGLE PARK NORTH CAROLINA
J
DAAG29-77-C-0035
DEPARTfVENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA AT CHL\PEL HILL
CHL\PEL HILL NORTH CAROLINA 275111
J
APPROVED FOR PUBLICRELEASE
DISTRIBUTION UNLIMITED
ESTIMATION IN THE GENERAL MULTIPLICATIVE
MODEL FOR SURVIVAL
Norman L. Johnson*, Department of Statistics
and
Regina C. Elandt-Johnson**, Department of Biostatistics
University of North Carolina at Chapel Hill
SUMMARY
The general multiplicative model (formula (2)) represents the
hazard function as the product of an "underlying" hazard rate,
of unspecified form and a certain function of known form,
where
~
is a vector of concomitant variables, and
of unknown parameters.
Assuming that
A(t)
f
A(t),
g(~;~),
is a vector
can be approximated by a
constant between any two consecutive failures, the general forms of
likelihood function are derived (formulae (6) and (7), or (9) and (11)).
The likelihood utilizes the available information on the time of
exposure to risk of each individual (until failure or withdrawal) .
Special cases, when the z's do not depend on
some detail (Section 7).
t
are discussed in
Multiple failures are handled in a simple
manner - no ordering of failures is required (Section 8).
Estimation
of empirical survival function when there are no covariates is
discussed in Section 9.
An example using heart transplant data, is
given (for illustrative purpose only).
* Research supported by the Army Office of Research, under Contract
DAAG29-77-C-0035.
**
Research supported by US. National Heart and Lung Institute Contract
NIH-NHLI-7l-2243 from the National Institutes of Health, and by
Public Health Research Grant 1 ROI CA 17107 from the National
Cancer Institute.
-21.
INTRODUCTION
Of recent years there has been considerable interest in the use
of multiplicative models for hazard rates, as a means whereby the
influence of concomitant variables on survival can be expressed, and
so allowed for in the analysis of survival data.
Methods of analysis
have been developed for estimation of parameters, reflecting
dependence of survival on concomitant variables, which are robust
with respect to the form of hazard rate function (provided the
multiplicative model is appropriate).
Use of these methods does
involve sacrifice of some of the information typically available in
survival data.
In this paper the nature and likely effects of the omitted information are studied, mainly by comparison of the "critical functions"
of the parameters reflecting dependence on concomitant variables which
have to be maximized to obtain maximum likelihood estimators, according
as to what information is included and what further assumptions about
the model are made.
2.
We suppose that
NOTATION
N individuals in all are under observation at
some time or other during the study.
We also suppose that observation
continues over a single interval of time - that is, that no individual
withdraws and then reenters the study later.
be denoted by
(i);
for each individual
of entry and of withdrawal or death
concomitant variables
(zl.'" . ,z .)
1
Sl
(i),
The i-th individual will
the data include times
(failure) and also values of
= "'1
z!.
s
-3-
Time
t
= 0 may correspond to date of initiation of a treatment
(as in many clinical studies) or to date of birth (as in some
occupational studies).
In the first type of study it will very often
be the case that the time of entry is 0 for all individuals, but this
need not be so in general.
If, among the
N individuals,
n
are observed to fail during
the course of the study, we denote the ordered failure times by
We denote the individual failing at time
(Multiple failures, when
(iU))·
the same time,
t.,
J
m. (>1)
J
by
t.
J
failures are recorded at
are discussed in Section 8.)
The set of individuals in the study at a time
t - those "exposed
to risk" of being observed to fail - is called the risk set at time
t.
In particular the risk set at ("just before") the j-th failure - at
time
t. - is denoted by
J
R.. We also use the notation R!
J
J
to denote
the set of all individuals who are under observation for at Zeast part
of the interval
I.:: (t. I' t.]
J
3.
J-
J
U
=
I, ... , n) .
FORMULATION OF THE PROBLEM
We are interested in estimating the survival distribution
function (SDF)
S(tl~)
where
=
T denotes failure time.
Pr[T<tl~],
(1)
Usually it is impracticable to have
sufficient numbers of individuals with common (or even approximately
common) sets of values of the
z's,
assumptions about the way the
z's
so it is necessary to make some
might combine to affect the SDF.
It might also appear necessary to make assumptions about the form of
-4-
dependence on
t,
specifying at least a parametric model for this.
While this is so, it is possible to estimate parameters reflecting
dependence on the
zls,
without making detailed assumptions about
the form of SDP as a function of
4.
t.
THE GENERAL MULTIPLICATIVE MODEL
These methods are based on use of the general multiplicative
model for the hazard rate function
- d 10gS(t I~)/dt = A(t I~) = A(t)g(!;~) ,
where
A(t)
is an "underlying" hazard rate of unspecified form
(except that
of
l'
to
s,
(2)
A(t)
unknown parameters
and
g(' )
g (.)
while
~O),
(31' ... , Sl' '
is a known function of
Usually
is assumed to be a function of
this is not essential.
If
z
'"
is taken equal
l'
L~=l(3uZu ,
but
varies with time, we can write
z
'"
and
!(t)
on the right hand side of (2).
5.
MAXIMUM PARTIAL LIKELIHOOD ESTIMATORS
A commonly used likelihood function for estimating the
utilizes information only on individuals in the risk sets
that a failure does occur at
supposing
z
=
z(t),
t.,
among the risk set
J
R.,
J
the probability that the individual
(3's
R..
J
Given
and
(i(j))
is
the one which fails is
g(z.(.)(t.);
"'1
where
£.
E
R.
J
not depend on
J,
J
~)[ L g(!o(t.); ~)]-l,
.- £.ER.
J
means that individual
-t-
J
~
belongs to
A(t). Cox (1972) used the product
R.•
J
(3)
This does
-5-
n
TT
j=l
(4)
as a "partial likelihood" from which estimation of the
found by maximizing this statistic.
estimators of the
(i)
SIS
can be
While this approach leads to
uninfluenced by
A(t)
Only the information t., R.
J
SIS
and
J
information on events inside the intervals
it should be noted that:
(i(j))
I.
is used, that is,
is ignored.
J
(This is,
why the term "partial likelihood" is applied to (4);
(ii)
In order to estimate
S(tl~)
it is still necessary to
introduce some assumptions about the form of
6,
~~XIMUM
LIKELIHOOD ESTIMATION
In order to approximate the form of
that it remains constant over each
from interval to interval.
A(t) = A.
A(t)
interval~
it may be supposed
but can change
I, ,
J
Under this assumption
for
J
A(t).
t. 1 <t
J-
~t.
J
,
(5)
and the likelihood function for the data described in Section 1, with
the model (2) is
n
=
11
L. (A.;
j =1 J J
S)
~
(6)
where
L.eA.; S) =A.g(z.(,)(t,); S)exp[-A.
J
J
~
J
1.
J
J
~
J
L J
f€Rj I j f
g(~o(t); ~)dt],
-l-
(7)
-6-
and
I
jt
denotes that part of I.
over which
J
the difference between
R!
J
and
R.-R!
J
(t)
is observed.
consists of
J
R.,
J
Note
plus all
those individuals who withdraw.
We first maximize (6) with respect to the
the
B's
A,' s
J
Csupposing
fixed) and then maximize the resultant value,
with respect to the
B's.
Clearly
n
n
max
L(A;~) =
[max L,(A,; B)] .
Al' ... , An '"
j =1
Aj J J '"
(8)
It is easy to show that
-AW
-1-1
= w e
. Hence
LC~; ~)
max Ae
A
corresponding to
-1
A= w
is maximized by
~. (B)
(9)
J '"
and the maximized value is
A
LCA(B); B)
n
__ e -nr-rI I
j=l
I
g(z·C')(t.);
B)
"'1 J
J
'"
I
OVI
-{..E'''j
J
I jt
-{..
The maximum likelihood estimators of the
which maximize the critical function
(10)
g(~o(t); S)dt
'"
S's
are those values
-7-
n
nj=l
n '"
eL(;\((3);
(3)
~
~
~
g(z.(,)(t.);
(3)
~l
J
J
~
(ll)
Comparing (11) with (4) we see that the numerators are identical but
the denominators differ,
being replaced by
\fuereas the quantity in (4) depends only on the risk set
t
values of the concomitant variables "at
the risk set
individuals
during
(R! )
J
(l)
in
7.
In practice
1.
J
j'
t!
(R. )
in (11) it depends on
and the whole periods of exposure of
1..
J
SOME IMPORTANT SPECIAL CASES
!(t)
will not be known for all
t.
In many cases,
only fixed values of concomitant variables are used - that is
does not depend on
and
r
t,
z.
and so can be represented as
~
!(t)
If this is
so ,then
J
I
where
g(!l; ~)dt
=
h'og(zo;
(3),
J-C
,..,
~-c
(12)
jl
is the length of time ("person x time units exposed to
risk") for which
(l)
is under observation in
I ..
J
For individuals
-8-
under observation for the whole of
I.,
J
(13)
If precise information on times of entry and withdrawal are not
available, more or less arbitrary approximations must be used.
(l)
in
If
either enters or withdraws from the study (but does not do both)
1.
J
one might take
withdraws in
I.,
h
jl
-;-
t
one might take
J
if
h ;
j
h
jl
-:-
both enters and
(l)
+h
j
.
Sometimes there
may be specific information on withdrawal and/or entry "habits" which
might lead to modification of these formulae.
From (9) and (12) we see that
~. (~)
J
G
L h· g(~o;
l€.R ! J -t-t-
=
o
S)]-l
(14)
~
J
and the critical function (11) takes the form
(15)
This is a constant multiple
n
r-r
r-r -1]
n h.
[ j =1 J
g(z.(.);
~l
J
of
S)J
~
(16)
j =1
where
8'0 = h'o/h.
J-tJ-t- J
under observation.
is the proportion of
1.
J
for which
(l)
is
-9-
The critical function (partial likelihood) (4) is now
n
nj=l
(17)
Comparison of (16) and (17) shows that they differ in their
denominators: - (a) the summation in (16) is over
in (17, is over
R. ,
J
while that
thus omitting individuals withdrawing in
J
(b) the factors . 8
R!,
jl
I.;
in (16) reflect different proportions of
J
1.
J
during which the corresponding individuals were under observations.
To sum up, (17) could be obtained from (16) by (a) ignoring individuals
who withdrew during
I.,
J
and
(b) supposing all individuals who
enter (and do not withdraw) during
I.
J
to have been under observation
for the whole of the interval.
Situations
when there are no new entries deserve, perhaps,
special attention.
These are typical in clinical trials, where
t
is
the follow up time since initiation of the treatment.
For computational purposes, it might be useful to represent (16)
and (17) in more convenient forms.
Let
Ll denote the time of departure (by failure or withdrawal)
(l).
for individual
Then
8
jl
=
0
for
Ll
(Ll - t.J- 1) Ih J.
for
t. 1 < Ll < t j
J-
1
for
Ll
:0;
~
t.J - 1
t. ,
J
(18)
-10-
so that (16) takes the form
n
g(z.(.); B)
"'1 J
'"
nj=l
(19)
If we were to define
'[ > t.
J
'[ : : : t.
J ,
if
if
(20)
formula (19) would correspond to (17).
8.
MULTIPLE FAILURES
If several failures - m.,
J
say - are recorded at the same time,
then, in practice, it means that the time unit of record is not
sufficiently small to distinguish them.
However, it is possible to
establish a likelihood function, treating the failures as if they
really did occur at the same time, and so to obtain maximum likelihood
estimators.
We denote the
(i (j, 1)),
m.
J
individuals failing at time
(i (j, 2)), ... , (i (j, m.))
J
we must have
m
+
m
+ .•• + m
~
n
The likelihood function is
1
n
L (A; B)
=
nn=l
2
t.
by
J
(j = 1, 2, " ' , n).
Of course,
N.
m. mj
~.J{n
g(~(.J, k) (t.);
.§)}exP{.-A. 2 J
k=l
J
J tsR!
rJ
J
l
g(.et(t);
.§)dt-f}·
-
jt
(21)
-11-
1
Maximizing with respect to
~jCiP
:: mj [
gives
f gC~
I
lER! 1.
J
0
J-L
f)dt~-l
Ct);
l
(22)
and the maximized value of the likelihood is
m.
J
A
LC;\C@);
f)
:: e-
n
r-r
gC~·C·
k::l
n
n m.m.)J r-r
[r-r
j=l J j::1
1.
I
J
J
Ij~
The 6's
=I j
k)Ct.); f)
J
1---------1
lER!
If
J,
I.
gC~lCt); f)dt
(23)
0
J-L
in (22), we obtain Breslow's (1972) formula.
are obtained by maximizing the critical function
m.
J
Tlg C~. C· k) Ct.);
k=l
1. J,
J
f)
(24)
which differs from Cll) only by substitution of
m.
J
~ gC~·C· k) Ct.);~)
k=l
'-1.
J,
J'-
for
gCz·C·)Ct.); ~),
"'1. J
J
.-
in the numerator.
If the z's
do not depend on
t,
then in place of (24) we can
use the critical function
m.
J
n
r-r=1
j
r-r
gC~. (.
k=l
1.
J,
k); ~)
(25)
-12-
9.
ESTIMATION OF SURVIVAL FUNCTION WHEN THERE ARE NO COVARIATES
When the covariates are absent so that
g(o)
~l~
formula (22)
gives
~.
=
L
m. {
J
J
tERl
J
h . o}-l
J-t.-
The corresponding estimate of the SDP (for
~
= exp -
j-l
I
mf
f=l
(J~l
I. 8ft
tER
t >t
For
we have
n
J-
n
relatively few data for these
t
is
I . e.J-t.-J~
t. 1::; t < t .
Set) = SCt)
t. l::;t <t.)
JJ
- m. (t - t. -1) h -l(
.
J
J .
J
tERj
f
for
(26)
J
0
(27)
(j = 1, •.. , n).
formally, but there may be
(they correspond to individuals
under observation for longer periods than that to the last failure).
If
if
f =Rf
R
8
ft
=1
in (16) - that is, there are no new entries, and
- that is, there are no withdrawals - then (27) gives
Set) =
l
expr.}~l
mfR - - m.R~l(t - t'_l)h~ll,
f
&~ 1
J J
J
J J
By comparison, the Kaplan-Meier estimator of
r.
S* (t) =
nj-l
f=l
where
R
f
(1 - mR -1 )
f
t. l::;t <to . (28)
JJ
Set) is
(t. 1 <t<t.),
J- J
is the number of individuals in
(29)
R ,
f
Another approximation, given by Thompson (1977), leads to a
formula like (29), but with
R
f
increased by one-half of the number
-13-
of withdrawals in
If'
Of course, when there are no multiple failures,
formulae (22) through (29) are valid with
10.
m 's
f
replaced by 1.
EXAMPLE
The data in Table 1 are adopted from Crowley and Hsu (1974)
("Covariance Analysis of Heart Transplant Survival Data:' Technical
Report No.2, Division of Biostatistics, Stanford University).
are survival data for
N =64
These
patients who had heart transplants.
The date of entry corresponds here to the date of operation, so that
the time at entry is equal to zero for each patient in the follow-up.
The investigation covers the period January 6, 1968 to April 1, 1974 ,
that is 2276 days.
individual
withdrawal).
(i)
The time
(in days) is the time beyond which
was no longer under observation (after death or
The data are arranged in increasing order of
(column 3 of Table 1).
Note that for patient (1)
assigned arbitrarily the value
0.5 days.
L
l
= 0+;
Li
we have
The numbers in parentheses
in column 1 are the ID numbers of the patients, assigned to them at
entry.
Column 2 gives the values of the indicator variable
equal to 1 if individual
There were
Li(j)=t j .)
(i)
is observed to die, and
n = 39 deaths, at times
The values of
t .•
J
when
0
0i'
otherwise.
(Note that when
are distinguished by
asterisk.
Out of 8 concomitant variables given in the original paper, only
two are shown here;
"mismatch score",
patient.
age at entry,
z2'
zl
(in years), and the so called
based on tissue and blood typing of each
-14-
We use these data for illustrating the methods of estimation.
There is no claim to evaluate the study as a whole.
The model used here is simply
•
The parameters
(a)
sets
and
Rj
are estimated by two approaches.
J
We notice a multiplicity at
ejl
t
17
(m
17
defined in (18),
:::: 2).
The estimates
and their estimated standard errors are:
8's
8 :::: 0 .0705,
1
"'-
82 :::: 0.6644,
e
A.'s
The likelihood function (19), with
A
#
the
Using the information on withdrawal times, with the risk
is used.
of
8 , 8 ,
1
2
(b)
8.0, (8 ) ::::0.0233;
1
8.0. (8'" 2 ) ::::0.2876
Ignoring the information on withdrawal time, that is, using
The likelihood function is still given by
only the risk sets,
R..
(19), but with
defined by (20).
8
J
jl
The results are:
A
8 :::: 0.0699,
1
82 ::::0.6817,
A
8.0.(8 ) ::::0.2883
2
We also evaluate the estimated survival functions ignoring covariates,
that is when
g(.):::: 1,
in three different ways.
A
(a)
sets
8l(t)
~
utilizing information on withdrawal times with
R'j ,.
(b)
S2(t) - assuming that the withdrawal time is at the beginning
of the interval for all those who withdraw in the interval, with
sets
R..
J
-15-
Since there were no new entries,
A. (t) ,
J
is estimated from the
formula
~. (t)
J
with
ej £
(b).
Then
64
m.
[I e. (Jh . ] -1
J £=1
(30)
J-t.. J
defined by (18) for appraoch (a), and by (20) for approach
Sl(t)
and
r-
S2(t)
are calculated from (27) and (28) ,
respectively.
(c)
S*(t) - the product-limit estimates, with sets
R.•
J
The resulting SDF's are given in the three last columns of
Table 1.
There are not many withdrawals between any two deaths (this
is especially true for early deaths).
In these circumstances is to
be expected that these three estimates of SDF do not differ greatly
as is, indeed, the case.
ACKNOWLEDGEMENT
We would like to thank Paul Wan for computation in the Example.
REFERENCES
Breslow, N.E. (1972). Discussion of D.R. Cox (1972) J.B. Statist.
Soc. Ser. B ~~, 216-217.
Cox, D.R. (1972).
Soc. Ser. B
Regression models and life tables.
187-202.
J.B. Statist.
~~,
Thompson, W.A. (1977). On the treatment of grouped observations in
life studies. Biometrics ~~, 463-470.
-16-
·e
TABLE 1
Estimation of Survival Funct ion for IIcart Transplant Data
•
,
e
£
(ID)
Of
T .'
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
(38)
(28)
(l00)
(4)
(20)
(74)
(98)
(3)
(18)
(70)
(90)
(79)
(92)
(22)
(45)
(10)
(37)
(83)
(87)
(47)
(55)
(36)
(32)
(73)
(13)
(68)
(65)
(89)
(97)
(ll)
(24)
(94 )
(96)
(67)
(93)
(51)
(16)
(84)
(78)
(58)
(88)
(86)
(81)
(80)
(76)
(64)
1
1
0
1
1
1
0
1
1
1
1
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
1
1
1
0
1
0
1
1
1
0
1
0
0
0
0
0
1
0
0
1
0
1
0
0.5
1
13
10
(71)
(72)
(7)
(69)
(23)
(63)
(30)
(59)
(56)
(46)
(21)
(49)
(41)
(l4)
(40)
(33)
(34)
(25)
t.
J
l
12
1315
23
25
26
29
3039
44
46
47
48
50
51}
51
54
60
63
64
65
66
68
109127
136
161
166228
236253
280
297
304322
338*
388438455*
498*
551
588591624
659730
8141
836
8370
8740
994
1
1 1024
0 11050 1263*
1 1350
0 13660 15360 15480 1774-
1
2
0.5
1
h.
zIt
Zu
R:J
0.5
0.5
41.5
54.3
35.2
40.4
45.3
29.2
28.9
54.3
46.9
53.0
52.5
54.0
45.8
42.8
36.3
42.5
61.7
53.3
46.4
47.2
52.5
49.1
64.4
56.4
54.6
45.3
51.3
51.4
23.6
48.0
52.0
43.9
26.3
19.8
47.8
48.8
49.5
42.8
49.3
48.1
55.4
48.9
52.9
46.5
52.2
48.9
46.5
26.7
51.0
48.0
58.4
32.7
45.0
41.6
38.1
48.5
43.4
36.8
45.5
54.1
48.6
49.0
40.5
33.2
0.87
0.47
0.67
1.66
2.76
0.61
0.77
1.11
2.05
1.68
0.82
1.08
0.16
1.38
0.00
0.61
0.87
3.05
2.25
1.38
1.51
2.09
0.69
2.16
1.89
1.68
1.12
1.33
1.78
0.36
1.62
1.20
0.46
1.02
0.33
1.08
1.12
0.60
0.81
1.82
0.68
1.44
1.94
1.41
1. 70
0.12
0.97
1.46
1.32
1.20
0.96
1.93
1.58
0.19
0.98
0.81
1.13
1.35
0.98
0.87
0.75
0.91
0.38
1.06
J
3
4
5
3
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2
7
2
6
7
8
9
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15
23
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26
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2
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11
12
13
14
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16
39
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20
21
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32
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106
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S2(T(.)
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.9844
.9690
.9690
.9532
.9375
.9217
.9163
.9057
.8897
.8736
.8576
.8416
.8399
.8252
.8089
.7926
.7762
.7599
.7435
.9844
.9688
51
50
49
48
47
46
.9844
.9690
.9690
.9532
.9375
.9217
.9164
.9058
.8898
.8737
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.7113}
.7ll2}
.7087}
43
42
41
40
39
38
37
43
43
41
40
39
38
37
35
34
33
32
31
30
28
27
29
28
27
26
25
.6949
.6785
.6622
.6458
.6295
.6131
.5968
.5850
.5800
.5632
.5463
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.5290
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.4931
.4752
.4699
.4566
.4549
.4497
.4446
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.4385
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.4197
.4186
.4069
.3974
.3789
.3547
.3486
.3483
.3394
.3119
.2753
.2619
.2377
.2254
.6922
.6757
.6592
.6428
.6293
.6098
.5933
36
34
33
.6950
.6786
.6623
.6459
.6296
.6132
.5969
.5853
.5804
.5635
.5467
.5454
.5294
.5237
.5117
.4937
.4758
.4705
.4573
.4558
.4511
.4465
.4450
.4410
.4362
.4235
.4225
.4114
.4020
.3837
.3606
.3548
.3546
.3457
.3184
.2810
.2696
.2486
.2378
Rj
Sl (T
64
63
64
62
60
59
61
60
59
58
56
55
54
53
57
56
55
54
53
52
50
49
48
47
46
63
24
19
18
16
15
14
13
12
11
8
9
8
5
}
.9531
.9372
.9214
}
.9052
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}
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j
.4280
+
.4013
}
.3726
+
.3416
}
.3036
.2657
}
.2205
1
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Data entered)
READ INSTRUCTIONS
BEFORE COMPLETING FORM
REPORT DOCUMENTAJION PAGE
I. REPORT NUMBER
2. GOVT ACCESSION NO.3.
4. TI TL E (lJtld Subtitle)
RECIPIENT'S CATALOG NUMBER
S.
TYPE OF REPORT &. PERIOD COVERED
Estimation in the General MUltiplicative Model
for Survival
TECHNICAL
6.
,
PERFORMING
O~G.
REPORT NUMBER
Mimeo Series No. 1187
7,
AUTHOR(.)
8.
Norman L. Johnson and
Regina C. Elandt-Johnson
9,
CONTRACT OR GRANT NUMBER(s)
DAAG29-77-C-0035
10,
PROGRAM ELEMENT. PROJECi, T/ISK
AREA a WORK UNIT NUMBERS
CONTROLLING OFFICE NAME AND ADDRESS
12.
REPORT DATE
ARGD
Research Triangle Park, NC
13.
NUMBER OF PAGES
PERFORMING ORGANIZATION NAME /lND ADDRESS
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
11,
~, MONITORING~GENCY
NAME
JUly 1978
16
a~A-D-D-R~ES~S~(I~I~d~lf~~-N-n-t~fft-m-,
~CO-,-,tr-o~II~~-~~O~I~N-re~)~~175,~S~E~C~U=R~IT~Y~C~L~AS~S~.~0~'~t~hl~·~-r-~-)I-.'~tJ~--~
UNCLASSIFIED
~:
16.
DECLASSIFICATION/DOWNGRADING
SCHEDULE
DISTRIBUTION STATEMENT (of thl. Report)
Approved for Public Release:
Distribution Unlimited
17,
DISTRIBUTION STATEMENT (01 the ..bslr..ct ent"'red In Block 20, If dlflerenl from Report)
18.
SUPPLEMENTARY NOTES
19.
KEY WORDS (Continue on rever.e side 1/ nece ....ry IJtId Identify by blocl< number)
Estimation, Multiplicative mortality model, Maximum likelihood, Partial
likelihood, Ties, Multiple deaths
,
20,
I
ABSTRACT (Contln"e on rever,'e ~Ide If n .. c .. •. ...rr and IdJlntlfy b~ bloc) r,.ulfber)
h
'd
The general multiplicative moael (tormUla l2j) represents the azar function
as the product of an "underlying hazard rate, A(t), of unspecified form and a
certain function of known form, g(z; 8), where z is a vector of concomitant
variables, and 8 is a vector of unKnown parameters. Assuming that A(t) can be
approximated by~a constant between any two consecutive failures, the general form~
of likelihood function are derived (formulae (6) and (7), or (9) and (11)). The
likelihood utilizes the available information on the time of exposure to risk of
each individual (until failure or withdrawal). Special cases, when the zls do
DO
1
~~~~3 1473
EDITION OF 1 NOV 6S IS OaSOL-ETE
UNCLASSIFIED
SECURITY CLASSII'ICATION 0' THIS P"OE
(Wt".,. fl.l.
l!n/.,.d)
SECURITy CLASSIFICATION OF THIS PAGE(Wh"" DIOI" Enl.. red)
20.
not depend on t are discussed in some detail (Section 7). Multiple failure
are handled in a simple manner - no ordering of failures is required
(Section 8). Estimation of empirical survival function when there are no
covariates is discussed in Section 9. An example using heart transplant
data, is given (for illustrative purpose only).
"
,
- - - - - - - - - - - - ._------