A PARAMETRIC APPROACH TO LIFE-TESTING
•
AND 'J.lHE THEORY OF COMPETING RISKS
by
MELVIN L. MOESCHBERGER
'1
•
Institute of Statistics
Mimeograph Series No. 706
Raleigh
1970
iv
TABLE OF CONTENTS
Page
LIST OF TABLES . .
viii
LIST OF FIGURES
1.
INTRODUCTION AND REVIEW OF LITERATURE
Introduction to the Problem . . . . . . • •
Development of Competing Risk Theory and Some Approaches
That Do Not Make Specific Distributional
1
Assumptions . . . . . . . . . . . . . . . . . . . . .
4
1.3
Competing Risk Theory and Life-Testing When a Specific
Class of Underlying Distributions is Assumed • • • •
Selection of the Underlying Life (or Failure)
Distribution • • . • •
Objectives of the Study • . • . • • • • •
1.5
8
10
11
MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM SOME CONTINUOUS
DISTRIBUTIONS AND SOME PROPERTIES OF THESE ESTIMATORS • • ••
14
2.1
2.2
Likelihood Equations for Continuous Distributions.
Underlying Exponential Populations
14
16
2.2.1
2.2.2
2.2.3
17
20
2.2.4
2.3
2.4
Properties of ~i • . • • • . •
Properties of e.*. . . . . ..
. ...
Estimation of H~zard Function and Properties of
This Estimator • .
. . . . . . .
Extension Allowing Failure Due to a Combination
of Causes
. . • . . • .
23
24
2.3.1
2.3.2
2.3.3
25
25
Populations with Known Equal Shape Constants
Populations with Unknown Equal Shape Constants •
Populations with Unequal Shape Constants .
Underlying Normal Populations .
2.4.2
Populations with Different Means, Common
Variance . • •
Populations with Different Means, Different
Variances
CENSORING
3.1
22
Underlying Weibu11 Populations
2.4.1
3.
1
1.1
1.2
1.4
2.
vii
30
30
35
40
Type I Censoring
3.1.1
28
Underlying Weibu11 Populations with Equal Shape
Constants
. . • . • • • . • • . • • ••
40
42
v
TABLE OF CONTENTS (continued)
Page
3.1.2
3.2
4.2
4.3
Underlying Weibu1l Populations with Equal Shape
Constants
. . • • . • ••
. . . . .
. . . .
68
....
68
General Likelihood Function When Some Observations are
Exactly Known, Grouped, and/or Censored
• ••
71
Weibu11 with Equal Shape Constants
73
Boag's Data
5.1.1
5.1.2
5.1.3
5.2
62
65
APPLICATIONS
5.1
7.
59
61
Weibull with Unequal Shape Constants •
Grouping in the Case of Censored Observations
4.5.1
4.6
55
56
59
Equidistant Group Limits for Finite h
Equidistant Group Limits for Infinite h
Estimates Based on the Theory of Non-Grouped
Samples . .
4.4
4.5
52
55
Likelihood Function for Complete Grouping of
Independent Observations • • •
Weibull with Equal Shape Constants
Exponential
. • • . • • .
4.3.1
4.3.2
4.3.3
6.
50
GROUPING
4.1
5.
47
Type II Censoring
3.2.1
4.
Underlying Weibul1 Populations with Unequal
Shape Constants .
73
Fitting Weibulls with Unequal Shape Constants •
Fitting Exponentials (Assuming Continuous
Data)
Fitting Exponentials (Assuming Grouped Data)
..................
..
Simulated Exponential Example
6. 1
Summary
Suggestions for Further Research
. . . . . . . . . . . .
LIST OF REFERENCES
.. .............
80
83
84
87
SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH
6.2
76
...
87
88
90
vi
TABLE OF CONTENTS (continued)
Page
8.
APPENDIX
8.1
8.2
Review of Methods Which Obtain the Inverse Moments of
the Positive Binomial Variate . . • • • • • •
Nelson's Method of Selecting an Underlying Life
Distribution by Hazard Plotting
8.2.1
8.2.2
8.2.3
Exponential Distribution •
Weibull Distribution . . •
Applying Nelson's Method to Boag's Data
94
95
96
97
97
vii
LIST OF TABLES
Page
5.1.1.
Boag's data.
75
5.1.2.
Comparison of the parametric method proposed with Kimball's
and Chiang's methods for equi-spaced intervals. • • ••
81
Comparison of the various methods when expected number of
deaths within each interval are approximately equal
82
5.1.3.
5.2.1.
Simulated examples with three underlying exponential
populations where 6 =1, 6 =1.5, 6 =2 (or·n =.46l5,
2
1
l
n =.3077, n =.2308) • • . • . • .3• • • • • • • •
2
3
86
APPENDIX
8.2.1.
Boag's data and hazard calculations
99
viii
LIST OF FIGURES
Page
8.2.1.
Exponential hazard plot of Boag's data (cancer present) •• 102
8.2.2.
Weibull hazard plot of Boag's data (cancer present)
• 103
8.2.3.
Exponential hazard plot of Boag's data (cancer not
present) • • • •
• • • • • • • • •
• • 104
8.2.4.
Weibull hazard plot of Boag's data (cancer not present) •• 105
CHAPTER 1.
INTRODUCTION AND REVIEW OF LITERATURE
1.1
Introduction to the Problem
It is often desirable to assess statistically the life characteristic of various types of individuals.
organisms,
1.~.,
These individuals may be living
interest may center upon making inferences regarding
the length of life after some treatment has been applied or some
operation has been performed on animals or human beings; or they may
be inanimate objects,
~.&.,
light bulbs, electrical components such
as fuses or vacuum tubes, and various kinds of physical equipment such
as ball bearings.
In either situation we speak of the lifetime of an
individual as meaning the time from some starting point to failure of
the individual under consideration.
It is assumed that this lifetime
is a random variable with a certain underlying distribution, usually
of the continuous type.
Generally, observations are obtained from individuals under their
natural operating conditions or from individuals that are subjects in a
planned laboratory experiment.
Some work has been done to determine
which family of distributions is appropriate for studying the life
characteristic of different individuals.
A considerable amount of
literature deals with estimating parameters from such underlying
distributions when the cause of failure is left unspecified or when
there is a single cause of failure.
Buckland (1964) and Barlow and
Proschan (1965) discuss these problems in addition to giving central
references.
A rather extensive bibliography on life-testing and
related topics is given by Mendenhall (1958) and Govindarajulu (1964).
2
Sometimes, however, the failure of an individual can be attributed
to one of k possible causes.
Since the failure of an individual from
a specific cause usually precludes failure from all other causes and
since the failure of an individual is usually not a repetitive event,
the ordinary life-test analysis is not applicable.
The basic information
available in this type of trial or life-test is the time to failure
of the individual under consideration together with the cause of failure.
Associated with each cause there is a characteristic life distribution,
sometimes referred to as a failure distribution, so that the observed
lifetime of an individual will be the minimum of k theoretical
observations, each theoretical observation being the time to failure
for the k respective causes if each one were acting alone.
The pro-
cedure used to estimate the parameters of the underlying life distributions
will be based upon the use of this minimum.
Perhaps it should be noted that problems may arise in coding causes
of failure when two or more causes seem to be responsible for failure.
The way in which such decisions are made can, of course, have important
effects on the results.
However, this paper is concerned only with
estimation procedures once such decisions have been made.
In the medical field, the type of situation which has been
described (viz., competing causes of failure) has been discussed in what
has come to be known as "competing risk" theory.
The immediate
objective of human mortality studies and medical follow-up is to
estimate life expectancy and survival rates for a defined population
at risk.
Other quantities of interest in the study of human survival
are (cf. Chiang, 1968)
3
i) the crude probability which is the probability of death
from a specific cause in the presence of all other risks
acting on a population,
ii) the net probability which is the probability of death if
a specific risk is the only risk affecting a population, and
iii) the partial crude probability which is the probability of
death from a specific cause when another risk (or risks)
is eliminated from the population.
A basic quantity useful in life testing and upon which the crude,
net, and partial crude probabilities may be based is the intensity
function (sometimes called hazard function, force
£f mortality,
perhaps more appropriately, conditional failure rate).
or,
It is the
probability that an individual alive at time t will fail during the
interval (t,t+&t).
The intensity function corresponding to the jth
cause of failure can be written as
= Pj (t)
(1.1.1)
l-P. (t)
J
where p.(t) and P.(t) are the probability density function and cumulative
J
J
distribution function, respectively, of the theoretical lifetime associated with the jth cause of failure, acting alone.
Often a distinction is made between "risk of death" and "cause
of death."
Basically, both terms refer to the same condition.
Prior
to death the condition referred to is called a risk and after death
the same condition is the cause.
4
Section 1.2 will elaborate further on the development of the study
of competing risks and will review some of the approaches that do not
make specific underlying distributional assumptions.
Section 1.3 will
review methods employed when a specific class of underlying distributions
is assumed, and hence the material of this section will be closely
related to work presented in this dissertation.
Procedures used to
select the appropriate underlying life distributions are discussed in
section 1.4 and, finally, objectives of the dissertation are outlined
in section 1.5.
1.2 Development of Competing Risk Theory and Some
Approaches That Do Not Make Specific Distributional Assumptions
The basic concept of competing risks was developed by Bernoulli
and D'Alembert during the 18th century in a controversy over the value
of vaccination for smallpox.
Each of these men considered the effect
of eliminating smallpox as a cause of death in a given population.
Todhunter (1865) relates the development of these early efforts in his
history of probability.
About 100 years later, W.
~.
Makeham (1874)
drew upon and simplified this earlier work to formulate a theory of
multiple sources of decrement which became a basis for future actuarial
methods.
More recently, Neyman (1950) presented a formal discussion of the
concepts of competing risk theory.
Fix and Neyman (1951) approached
the problem of competing risks in cancer patients by finding crude
and net probabilities of transfer from one state of health to another.
The four primary states of health considered were (1) initial state of
being under treatment for cancer, (2) death from cancer or operative
5
death attributable to cancer, (3) apparent recovery from cancer, and
(4) lost after "recovery," either through death from some cause other
than cancer or through difficulties in following a patient.
They
assume that changes in probabilities of the various risks during the
period of observation can be neglected and that the "period of observation
is divided into a large number of elementary time intervals
which are so short that a given patient exposed to various risks cannot
succumb to more than one during a single such elementary time interval."
This approach requires a fairly detailed knowledge of the situation in
addition to placing rather restrictive assumptions on the model.
Cornfield (1957) discussed competing risk considerations in a general
manner.
Two interesting approaches have since been developed that make use
of grouped data,
1.~.,
data which contain information indicating cause
of death of individuals but only pinpoint the time of death within a
certain interval.
Kimball (1957) in finding estimates of disease incidence in
populations subject to multiple causes of death obtains the crude
probability of death from cause i, say wi' within an interval (he refers
to it as simply a "conditional probability") by assuming a multinomial distribution for the observed number of deaths classified in a
two-way table according to cause of death and age-at-death.
If there
are k causes of death and the data are grouped into h intervals, then
this model depends on the assumption that the kh probabilities are
constant over the period of observation and that the observations are
mutually independent.
6
Kimball then introduces a
~ew
set of conditional probabilities
where a specific disease has been eliminated.
For example, the
probability of death from the ith cause within an interval given the
elimination of the first disease would simply be wi /(l-w ).
l
In other
words, the conditional probability of death from a particular disease
in a given interval is "increased i~ proportion to itself by an amount
equivalent to the probability assigned to the disease which has been
eliminated from consideration in the same interval."
Such conditional
probabilities are essentially partial crude probabilities.
Chiang (1960a, 1960b, 1961a, 1961b, 1968) has approached the
problem somewhat differently.
In theory, Chiang's basic approach is
perfectly general and applies to any set of lifetimes arising from
continuous distributions.
This is theoretically, as well as biologically,
pleasing.
However, although Chiang gives a formula for the crude probability
which is applicable to any underlying continuous distribution, he does
not use it in practice since a knowledge of the force of mortality
associated with each cause of failure is required.
In practice, Chiang
assumes an underlying multinomial distribution, in somewhat the same
spirit as Kimball, to obtain his estimate of the crude probability.
In fact, Chiang's estimate of the crude probability is identical to
that of Kimball's.
Differences in the two methods arise in their
estimation of the partial crude probability.
Upon assuming the relative forces of mortality, which are defined
to be rj(t)/r(t), j • l, ••• ,k, to be independent of time in the
intervals of interest
(!.~.,
each specific force of mortality may vary
7
in absolute magnitude but it must remain a constant proportion of the
total force of mortality in an interval), Chiang obtains relationships
among the crude, net, and partial crude probabilities.
From these
relationships, he uses the estimates of the crude probability to
estimate the net and partial crude probabilities.
Although this method of obtaining net and partial crude probabilities
may appear to be free of the form of the underlying distributions, the
assumption of constant relative forces of mortality holds only for
certain continuous distributions of interest.
David (1970) has shown
that this assumption will be satisfied whenever the underlying distributions of life have one of the three possible forms of the extremevalue distribution of the minimum.
Of the specific distributions
considered in this thesis, Chiang's assumption will be satisfied
whenever the underlying distributions are exponential or Weibull with
equal shape constants.
However, if the theoretical lifetimes associated
with each cause follow Weibull distributions with unequal shape constants
or are normally distributed,then the relative forces of mortality are
no longer independent of time within an interval.
assumes risks to be mutually independent,
(~.~.,
Chiang's model
risk of death from
one cause is independent of and unaffected by changes in risks of death
from other causes).
Asymptotic results for properties of Kimball's and Chiang's
estimators are given by both authors.
Atta and Kimball (1968) have done
Monte Carlo studies in an effort to learn more about the small sample
distribution of Kimball's estimators.
of the two models.
Kimball (1969) gives an appraisal
8
1.3 Competing Risk Theory and Life-Testing When a
Specific Class of Underlying Distributions is Assumed
Boag (1949) in an analysis of survivorship in cancer therapy uses
a model which assumes that a logarithmic transformation of the lifetimes of patients who died with cancer present is normally distributed.
Although he is primarily interested in estimating the proportion of
cancer patients cured by a particular treatment, he also estimates, by
the method of maximum likelihood, the mean and standard deviation of the
logarithm of the lifetimes of those patients who died with cancer present.
Kodlin (1961) gives a model closely related to Boag's.
Both
authors postulate that there is a fraction c cured of a particular
disease and subject only to the risk of unspecific death and the
remaining fraction l-c is uncured and subject to risk of death by cancer
as well as unspecific death.
Boag's model is slightly more complex in
that he uses more information, viz., he classifies his survivors as
being symptom-free or suffering from a persistence or recurrence of
cancer.
Kodlin associates known underlying distributions with survival time
to cancer death and survival time to unspecific death.
case for underlying exponential distributions.
He works out the
For c·o his likelihood
function corresponds to the one given by Berkson and E1veback (1960)
who deal with the problem of competing exponential risks, with particular
reference to the study of smoking and lung cancer.
The latter authors
obtain net probabilities of death as defined in section 1.1 by employing
the simplifying assumption of constant relative risks which obviously
holds for underlying exponential life distributions as has been discussed
in section 1.2.
9
The case c=o is the situation of interest in this dissertation,
~.~.,
we will only be concerned with the situation where the entire
population is subject to all risks present in the system.
Sampford (1952a, 1952b, 1954) deals with the general problem of
estimation of response-time distributions in a series of three papers.
In the second paper he discusses an "accidental model" which is most
relevant to work done in this thesis.
as follows.
The situation dealt with there is
Two causes of death are considered to be distinguishable,
their joint action is assumed independent, and the survival times are
uncorre1ated.
It is assumed that the distribution of treatment survival
times or some suitable function thereof is normally distributed.
Then
he estimates the mean and standard deviation of this normal distribution
by the method of maximum likelihood and shows that the actual numerical
estimates obtained will not depend upon the distribution of survival
times from accident or some other secondary cause, provided the two
distributions have no parameters in common.
This last point will be
elaborated upon in section 2.1.
Hoem (1968) has given some consideration to the estimation of forces
of decrement (also known as forces of mortality) when there exist k
causes of decrement.
The estimation technique is based upon the use of
aggregated lifetime as operational time.
His discussion is confined to
the negative exponential distribution.
Marshall and 01kin (1967) present a multivariate exponential model
within the context of reliability considerations.
This model is
relevant to competing risk situations where the failure of an individual
may be due to a combination of causes.
will be taken up in section 2.2.4.
Further discussion on this point
10
To the author's knowledge, literature dealing with life-testing in
the situation of competing causes is scant.
However, a note by David
(1957) dealing with the breakage of airplane wings for two normal
parents is relevant.
Buckland states that, although situations
involving competing causes of failure for individual units have been
largely confined to the medical field, "life-analysis in other fields
could become involved as knowledge of the fundamental mechanisms of
failure improves."
1.4
Selection of the Underlying Life (or Failure) Distribution
A major problem in life-testing is deciding upon the form of the
statistical distribution which mathematically describes the length of
life of the individual in question.
It is obvious that the different
modes of possible failure will affect the analytic form of the life
distribution.
Three major types of failure are early failure, random failure,
and wear-out failure.
The hazard function may be useful in determining
the forms of the underlying life distributions since a decreasing,
constant, or an increasing hazard function corresponds to the early,
random, or wear-out failure situation, respectively.
Nelson (1969) presents a method of plotting data on hazard paper
to obtain information on the distribution of time to failure, specifying what type of hazard paper to use for the exponential, Weibull,
normal, log normal, and extreme value distributions.
His method is
applicable even though the different individuals have different
censoring times,
i.~.,
the time period for which a given individual
is under observation varies from individual to individual.
Although this
11
procedure is primarily concerned with a single cause of failure, the
methods presented may be adapted to a life-testing situation involving
competing causes of failure, as indicated in sections 5.1 and 8.2.
Berman (1963) suggests a method to determine the form of the
underlying life distributions when there exist competing causes of
failure.
He shows that the joint distribution of the index signifying
cause of death and the observed lifetime uniquely determines the underlying parent life distribution function associated with each cause of
failure.
This joint distribution may be determined for human populations
from actuarial data.
He makes some standard assumptions, viz., that the ages at which
different causes strike are stochastically independent and that each
individual is exposed to all risks operating in a population.
The
constant relative force of mortality assumption which is necessary for
Chiang's approach need not be made in Berman's method.
1.5
Objectives of the Study
If the underlying life-distributions may be confidently restricted
by previous knowledge to a specific class of distributions,then one would
wish to use such knowledge and the procedures of this dissertation
become applicable.
Although work by authors mentioned in section 1.3 covers some
specific situations in which the life distributions are known, it seems
desirable to present a general approach to the estimation of the parameters
of the underlying life distributions from which one can estimate
the relevant probabilities in competing risk theory.
The general like-
lihood function given in section 2.1 dispenses with the usual assumption
12
of independence, although the only specific case of dependent causes for
which explicit expressions have been obtained is the multivariate
exponential model of section 2.2.4.
Some specific families of distributions whose parameters will be
estimated by the method of maximum likelihood are the negative exponential, Weibu11, and normal.
Properties of these estimators, both
asymptotic and for small samples whenever possible, will be discussed.
The estimation methods will, of necessity, often involve iterative
techniques which can only be carried out in practice by the use of a
computer.
Important ramifications of the general method to be considered are:
1) Type I censoring,
i.~.,
an individual failure is observed only if it
occurs within some specific time period, 2) T¥pe II censoring,
!.~.,
only the failure of the first m individuals is observed, where m is some
pre-chosen integer, and 3) grouped observations,
i.~.,
the observed
lifetimes are known only to fall within specified time intervals.
Censoring will be considered in Chapter III and Grouping in Chapter IV.
Since the chief interest in the theory of competing risks is
concerned with crude, net, and partial crude probabilities of death and
since these probabilities can be deduced once we have estimated the
forces of mortality associated with the respective causes of death,
another aim will be to estimate such relevant probabilities.
A worked
example will be included in section 5.1 to illustrate the point.
Appendix contains some additional details on certain points, viz.,
a review of methods used to obtain inverse moments of a positive
The
13
binomia~
variate and an explanation of Nelson's method of selecting an
underlying distribution by hazard plotting.
14
CHAPTER 2.
MAXIMUM LIKELIHOOD ESTIMATION OF PARAMETERS FROM SOME
CONTINUOUS DISTRIBUTIONS AND SOME PROPERTIES OF THESE ESTIMATORS
2.1
Likelihood Equations for Continuous Distributions
Suppose each individual is subject to k (finite) causes of failure,
labelled C , C , ••• , C •
l
2
k
Associate with each cause C a non-negative
t
random variable Y representing the age of an individual at failure if
t
Ct. is the only cause of death.
Suppose Y has an absolutely continuous
t
cumulative distribution function (c.d.f.), denoted by Pt(y) and a
probability density function (p.d.f.), denoted by Pt(Y)' t=1,2, ••. ,k.
A single trial of an experiment consists of observing the time to
failure of an individual, min (Y , ••• , Y ), and noting the cause of
l
k
failure.
Thus, for a single trial, the observable random variable,
min {Y t } will be equal to Y if C is the cause of failure.
i
i
t
this observed lifetime by Xi' ~.~., Xi = YilY i = min {Y t }.
Denote
t
Let p(Yl, ••• ,Y k ) be the joint p.d.f. of the Yt's.
Then the p.d. f.
of Xi is
£i (xi) = likelihood of [Y i = xi IY i = min(Y 1 ,··· ,Yk»
likelihood of [Y i = xiAY i = min(Y1, ••• ,Y k )]
=
Pr[Y i = min(Y1,···,Y k )]
k
co
!
Where
Y =
i
Note:
~i
xi
P(Yl' •.. 'Yk)
= Pr[Y i = min{Y t })
ITdYn
t=l ~
t;'l
is assumed to be non-zero and
t
xi in integrand.
~i
= Pr[failure due to cause Ci )·
(2.1.1)
15
Suppose that n i individuals have failed from cause
total sample size is n =
~ni'
Ci'!'~"
the
Let X denote the lifetime of the jth
ij
individual failing from cause C. (j=1,2, ••• ,n.; i=1,2, .•• ,k).
1
1
Then
the p.d.f. of X is
ij
k
II dyR,' (2.1.2)
1
fi(x i ·) = J
7T
t=l
t;i
Since the selection of the individuals was carried out randomly,
i
we have
(2.1.3)
k
dy t
II
•
t=l
Ml
It should be noted that the ni's are themselves random variables
with the multinomial probability function
k
II 7T ni.
. 1 i
1=
(2.1.4)
Hence the likelihood function of our sample is
k
II
t=l
t;l
dYt' (2.1.5)
If we further make the usual assumption that the causes of failure
act independently,
L
n!
= --.,.........:.:;.;:....-k
II (n !)
i
k
II
..!.. ~.,
n
i
II.
•••'., k) are independent, then
Y R. (R.=l,
Pi(X i ')
J
k
II
i=l j=l 1-P i (X ij ) R.=1
[l-P R. (X
ij
)] •
(2.1.6)
i=l
Under the independence assumption, (2.1.1) becmmes
fi(x i ) =
7T
k .
Pi(x )
i
[l-P (x )] R.~l[l-PR.(Xi)]·
i i
i
(2.1. 7)
16
Sampford (1952), using a slightly different argument, obtains
essentially the same likelihood function as in (2.1.6) for the cases
k = 2,3 by considering the simultaneous operation of two or more
stimuli producing different responses such that the occurrence of
one response precludes that of the others.
He points out that if the
joint action of the stimuli is independent,
!.~.,
the Y are independent,
t
and if the p.d.f. P1(Y1) does not contain any of the parameters in the
remaining Pi(Y )' i = 2, •.• ,k, then the parameters of P1(Y 1 )' the
i
density of a particular response of interest, may be estimated and
these numerical estimates will be the same regardless of the distributiona1
form of the remaining Pi(Yi)' i = 2,3, ••• ,k.
This can be
seen, in general, from (2.1.6) since
ni
k
L = g(x
ij
IT
IT
(1-P (X ))
1 ij
i';"l j=l
; j=l, ••• ,n ; i=l, ••• ,k)
i
where g does not contain any of the parameters in Pl(Y )'
l
Specific
illustrations of this can be seen in sections 2.2, 2.3, and 2.4.2.
It should be noted, however, that the distribution of such
estimators will depend upon the distributional form of the Pi(y ),
i
i:;;2,3, ••. ,k.
2.2
Let Pi (Y i ) =
Underlying Exponential Populations
~)
i
exp f-(yi/e i )} 'Y i 2:.. 0, e i > 0,
and assume Y to be independent, i=l, ••• ,k.
i
From (2.1.6) we have
k
n
i
log L = constant + L
L
i=l j=l
k
log [rh..) exp{ - ( L
'9 i
R,el
L)
e R,
x
ij
}]
17
-
or log L = constant
alogL
So
ae.l.
k
k
ni
k
En 10g8 - (E 1:..::.) E E x '
R,=l 8 R, i:l j=l ij
i=li
i
ni
t
= --+-8' i
8. 2
(2.2.1)
(2.2.2)
l.
where t =
k
ni
E
E
x
i=l j=l
ij
'
Thus the likelihood equations are
8
2.2.1
= tln
i
i
(i=l, •.• ,k) •
(2.2.3)
Properties of 8.
l.
Now (T, N , .•. ,Nk _ ) is minimal sufficient for (8 "" ,8 k) and so
l
l
1
"
A
,e k)
is minimal sufficient for (e l' ... , 8 ) •
k
1
k 1
Let ~ = E 8-' Then employing the easily established fact that
i=l i
(8 , ...
1
7T.
l.
= Me.l.
(2.2.4)
we see from (2.1.7) that
(2.2.5)
Thus, the observed lifetimes are identically distributed
of the cause of failure.
Also the density of Z = min{YR,} is identically
JI,
equal to those of the Xi in (2.2.5).
Sethuraman (1965) encounters this type of characterization - which
he calls "complete confounding" of random variables Xl, ••• ,Xk , and Z in a study of damage models, viz., situations whose physical conditions
are such that the experimenter, after observing a single trial of an
experiment, can only determine whether an observation is a "damaged"
one or an "undamaged" one.
For our situation an undamaged observation
18
would correspond to the lifetime of an individual failing from a
specific cause of interest, whereas, a damaged observation
would cor-
respond to the lifetime of an individual failing from some cause other
than the one of interest.
Sethuraman shows that such confounding, viz., that the densities
of X1 ' ••• 'X ' and Z are identica1,occurs if and only if there exists
k
constants ci>O such that
i=2,3, ... ,k.
In section 2.3, it will be seen that Weibull parent distributions with
equal shape constants also lead to complete confounding.
It should be noted that fi(x ) in (2,2.5) and
i
1 . - t n-1 exp(uc/ll), t
f(t) = - _
~ nr (ll)
are conditional on the n •
i
~
0
(2.2.6)
However, these conditional densities are
identical to the unconditional densities.
Hence X and N are
ij
i
independent, and T and N are independent.
i
Thus
~i
T'
=~
is the ratio of two independent random variables.
i
In general, if the random variable X is independent of random
.
-
variables Y (i=l,.,.,k), then, subject to the existence of the
i
expectations involved,
The asymptotic moments of the
~i
may be examined directly.
a result due to Derkson (1939) and Slutsky (1925) the asymptotic
By
19
111
----- (i~j) may be approximated by the first
expectations of --,
Ni N. 2 ' NiNj
few terms of the appr6priate Taylor series expansions.
Now, from (2.2.6), we find that
2
Var(T) = n6 .
E(T) = ri6,
(2.2.8)
Hence, following (2.2.7), we have that, asympotically,
= ri6[_1mT i
+ 0(1 )]
(2.2.9)
and
n2
2 2
n 6 [
2 +
1
(mT i)
1
2 +
1
3 ] + 0(2")
(nn )
n
i
(nn )
i
3
1
8 In6 + 0("2)'
i
Therefore Var(a.)
1
A
2(1-n i )
(2.2.10)
Also
n
A
1
Therefore, COV(8 ,8 .. ) = 0(2")
i i
(i~i"').
(Z.Z.11)
n
Hence, our estimates are consistent and asymptotically unbiased .
.
.
1
1
1
For finite samples, E(--N), E(---Z) and E(N N ) are infinite since
i
N
i
i
~
.
a finite probability is associated with the event Ni=O.
Thus, a slight
modification must be made for looking at finite sample properties.
A
natural alternative might be to consider the estimator 8 *, where 8 *
i
i
is 8
i
conditional on ni>O.
following section.
This estimator will be examined in the
20
2.2.2
P~operties
Let 8i *
of
T
=N
*'
i
~i*
where Ni * is a binomial (n,
~i)
variate truncated
at O.
We now proceed to obtain the density of 8
transformation 8.
J.
t
= -n
(holding n
i
i
fixed).
i
*.
In (2.2.6) make the
Thus we have
(2.2.12)
Furthermore, we know that
(2.2.13)
and upon substituting (2.2.12) and (2.2.13) into the above expression,
we obtain the
fee
*)
i
~.d.f.
e *n-1
= _.__;;;;.i
of 6 i *
_
~nr(n)[l-(l-~ )n]
i
(2.2.14)
..
* directly from (2.2.14) or
T and Ni * and use (2.2.7). We find
Now one can evaluate the moments of 8
take advantage of the independence of
i
(2.2.15)
Several authors, viz., Grab and Savage (1954), Mendenhall and
Lehman (1960), Stephan (1945), and Clark and Williams (1961), deal
with the problem of evaluating E(__l__) to various degrees of accuracy,
N
i
*r
21
where r is an integer.
David and Johnson (1956-57) also suggest an
1
approximation for E(---- ) and Govindarajulu (1963) gives some recurN.*r
1
rence relations involving inverse moments of the positive binomial
variable.
See section 8.1 in Appendix for a more detailed discussion
of some of these methods.
If the values of n and
~i
fall outside the range of tables given
by Grab and Savage or Mendenhall and Lehman, then one can obtain the
1
values of E(----) in the following manner. If one is content with
N *r
i
two-place accuracy, then the Mendenhall and Lehman approximation seems
to be more than satisfactory.
This method's chief advantage lies in its
If a computer is available, then, for
n~i>l5,
Stephan's
approximation yields any desired accuracy and, for
n~i<l5,
one can
simplicity.
evaluate the explicit expression
Thus estimates of expressiom in (2.2.15) may be based on the
estimates 6 •
i
Also, from (2.2.14),
"
pr[~i*~60]
and upon making the transformation
"
pr[r6
*~e
i -
0
]
n
L
00
=fe
2x
"
"
f ('6 *)de *
o
i
i
,1\
= ni6i*/~'
we obtain
~.
co
(n) (_1_) n i
n =1 n
1
l-~
iii
(2.2.16)
which is just a weighted sum of chi-squared integrals.
Bartholomew's (1963) estimate of the mean of an exponential
distribution when the sample is censored at a fixed time is also a
weighted sum of chi-squared integrals.
22
2.2.3
Estimation of Hazard Function and Properties of This Estimator
It may be desirable to estimate 0i =
1
ewhich
is the hazard or
i
intensity function (also called the conditional failure rate in 1ifetesting) •
Then, from (2.2.3) and the invariance property of maximum 1ikelihood estimators,
0i = ni/t (i=l, ... ,k) .
(2.2.17)
The sufficiency of these estimators is as before.
Now
Hence
(2.2.18)
is an unbiased estimate of 0i for all n.
Also
var(oi') =
1
[(n-l)oi
n(n-2)
l:,.
Cov(oi '',OJ'')
=
°i OJ
1'1(1'1-2)
and
(2.2~1~)
(i;'j) .
Here we can obtain unbiased, consistent estimators of 0i whose
variances and covariances are relatively simple quantities for all n.
It is of interest to note that, for discrete time models with a
"single cause of failure," estimators of conception rate (a type of
"
hazard function) are, of the same form as 0i in (2.2.17), viz.,
'.
number of conceptions divided by the number of months of exposure
(£f.
Sheps, 1965).
However, the two situations are in contrast in that
one cannot obtain a function of the number of conceptions/number of
months of exposure which is asymptotically unbiased, whereas, in the
exponential situation described above an unbiased estimate of the
hazard function is obtained in (2.2.18) for all sample sizes.
23
2.2.4
Extension Allowing Failure Due to a Combination of Causes
Practical considerations may lead one to study situations where
the failure of an individual is due to a combination of causes.
This
kind of situation may still be studied for the exponential case using
the framework of our model.
Marshall and 01kin (1967) propose a multivariate exponential
distribution which is motivated by reliability considerations.
Their
"fatal shock" model supposes that the components of a two-component
system die after receiving a shock which is always fatal.
Independent
Poisson processes Zl(t;A l ), Z2(t;A 2), Z12(t;A 12 ) with parameters
A ,A 2 ,A 12 govern the occurrence of the shocks.
l
Events in the process
Zi(t;A ), i=1,2, are shocks to component i, and events in the process
i
~12(t;A12)
are shocks to both components.
If X and Y denote the life
of the first and second component,then
Pr[X>s, Y>t] = Pr[Zl(s;A ) = 0, Z2(t;A ) = 0, Z12(max(s,t);A ) = 0]
l
2
12
= exp[-Als - A t - A max(s,t)].
2
12
(2.2.20)
Let Y (i=1,2) denote the time elapsed until component i has
i
received a fatal shock and Y denote the time elapsed until both
3
components receive a fatal shock simultaneously.
..
Now X
= min(Y l ,Y 3)
and Y=min(Y ,Y ) are bivariate exponential whose distribution is
2 3
characterized by (2.2.20).
It is an immediate consequence of the
model that Yl 'Y 'Y are independent exponential random variables.
2 3
Furthermore, it can be easily seen that min(X,Y) = min(Y ,Y ,Y ) has
l 2 3
the density
24
This model relates to our situation in the following manner.
Yl ' Y2 ' Y be
3
thetheo~tical
Let
lifespan of an individual failing from
cause Cl only, cause C only, and causes C and C jointly.
2
2
l
!.~.,
we
are regarding the failure of an individual due to causes C and C
l
2
jointly as a separate cause of failure, say C .
3
Thus X = min(Y ,Y ) is
l 3
the lifetime of an individual failing from cause C only, or in
l
combination with cause C •
2
Similarly, Y = min(Y ,Y ) is the lifetime
2 3
of an individual failing from cause C only, or in combination with
2
cause C .
l
The following correspondences hold in going from the Marshall
and Olkin model to our model:
Al ~ 6L
1
A
2
~
Pr [X<Y] =
A
- = TIl = Pr[death due to C ]
l
1
6
Pr [X>Y] =
2
A -+- L
12
6
3
-+-
~
A
A
2
-+- - = TI =- Pr[death due to C ]
A
6
2
2
2
A
12
A
Pr[X=Y] = -A--+- - = TI = Pr[death due to C ]
6
3
3
3
L
6
Al
A
1
-;;:-.
Thus, one can treat failures from more than one cause essentially
as before by introducing additional "causes" corresponding to any
combination of basic causes.
2.3
Underlying Weibull Populations
The Weibull p.d.f. is of the form
ci
c -1
.{ ( c,/ )}
= ~- ) Yi i
exp - Yi ~ 6i Yi::0 ,
i
where, as usual, we assume Yi to be independent (i=-l, ••. ,k).
(2.3.1)
25
2.3.1
Populations with Known Equal Shape Constants
Under the assumption that c
= c
l
2
= ••• = c
k
= c, the density of
an observed failure from cause C is from (2.1.1)
l
(2.3.2)
Here the observed lifetimes, as in the exponential case, are
identically distributed irrespective of the cause of failure.
If one transforms
the~data
c
by z = x , then the observations will be
identically exponentially distributed and hence the estimation procedure
and properties of the estimators hold as discussed in section 2.2.
2.3.2
Populations with Unknown Equal Shape Constants
From (2.3.1) and (2.1.6) we have, upon letting
log L
= constant
k
k
+ n log c -
E n i log Si + (c-1) E
i=l
i=l
n
-
E
Now
.
,
..
nR,
i
Cl10gL
=-+
ClSR,
S,Q,
S 2
,Q,
Cl10gL =.!!.+
E E log x
ij
dC
c
i j
i
k 1 k
c
[E - ] E
E x '
,Q,=1 S,Q,° i=l j=l iJ
(2.3.4)
(,Q,=1, • •• ,k)
k
(E
.!-)
,Q,=l S,Q,
E E
i j
(2.3.3)
(x ' Clog x )
ij
iJ
which gives rise to the likelihood equations
!
E E log x
+ : =[E E XiJ,clog xij]/(E Ex, .c)
ij
n i j
c
i j
i j 1J
(2.3.5)
26
Clearly, once we obtain c by iterative techniques from the first
e~'s
equation in (2.3.5) the
will be trivially determined.
From (2.3.4), we find
2
a logL
ae 2
~
2L LX .• C
i j 1J
n~
=~ ~
2
a logL
=
acae~
2
a logL
ae~ac
•.. ,k)
~
=(_1_)
e 2
(x
L L
i j
~
= -
(~=l,
e 3
ij
Clog x
ij
)
(2.3.6)
k 1
c
2
n 2 - (L -e) L LX.. (log x .. ) .
~=l
c
~
i j
1J
1J
Thus the inverse of the information matrix is
t::.
e
o
3
1
e
2
1
•
•
V
•
•
•
•
A
o
= n-1
~
k
k+1,k+1
A
A
e
1
where
A
~
=~'fo
c
("F,) x
2c-1
D=1 +~ , B
2
c
'w
••
=
2
-1
A
• • • e 2
D
k
c
log x exp(- x /t::.)dx
f:
(f)x
2c 1
- (10g x)2 exp(-xc/t::.)dx.
Expressions A and B may be simplified further as follows.
c
x
y = ~, and using the relationships
f
~
0
log y (l-y)e-Ydy = -1
and
f
~
0
(2.3.7)
(log y) 2 (l-y)e-Ydy = -2 f
00
0
log y e- y dy
Upon letting
27
we have
00
!J.
A = -("t) [1 + f
log y e-Ydy + log !J.]
o
and
. 00
f
(log y)
2
e-Ydy + 2(l+log ~)f
o
However,
-
00
00
f
exp{ -e
Z
-2
- z}dz
_00
0
and
log y e-Ydy
0
00
00
z
f
-z
exp{-e -z}dz
2
_00
0
are just the first two moments of the extreme value distribution waose
values are known to be ';
= 0.5772157 ...
(Euler's constant) and
2
(~_) + ,2, respectively (Gumbel, 1958).
Hence A
= -(%)
[1-Y+1og !J.]
2
B
= (~) [~
:c2
6
+ '1 2 - 2(l+log
~)y+
2log 6+ (log t,)2].
Also
D=.!:.-+~
2
6
c.
t
= ( \ ) [O.-y+log
<
•
..
c
6 ) 2+
~
2
]
Following a matrix inversion suggested by Rao (1965)
Gk,k
V: k+ 1 ,k+1 =
~1
,
n-1
(2.3.8)
!i:,k
II
e
where
1
28
o
3
+
6(l-y+10g ll)2
e e"
2
'IT
e
o
~
F
6c(l-y +10g ll)
=-It, 1
'IT
3
e
2
2
H ~
.§.£...
'IT
2.3.3
k
2
e
~
Populations with Unequal Shape Constants
From (2.3.1) and (2.1.6) we have
log L
~
constant +
k
L
i~l
ni
k
L
L
j~l
L
L n, log c, -
i~l
,Q,~1
1
i~l
1
ni
k
k
k
n,
log e, +
1
1
L
i-1
(c,-l) L log X
ij
1
j=l
x" c,Q,
(1 J
)•
(2.3.9)
e,Q,
Hence
(L LXi,C,Q,)
n,Q, + i j J
o,logL
= -(-)
ae,Q,
e,Q,
e 2
,Q,
n,Q,
n
dlogL
= (-.!) + L log x,Q,j
ac,Q,
c,Q,
j=l
(2.3.10)
[L
-
L(X, . c,Q,log x
i J.
.
1J
ij
)]
e,Q,
(2.3.11)
(,Q,=l, •••. ,k)
$
which leads to theA likelihood equations
[L
e,Q, =
~Xij c~]
i J
n,Q,
(,Q,=l, ••• ,k)
(2.3.12)
n
n,Q,
c,Q,l
]
nn [ L LXi'
og x,. = [(~) + L log X,Q,j] ~ ~ x ij c,Q, (,Q,=l, ••• ,k).
'" i j
J
1J
c,Q,
j=l
(2.3.13)
From (2.3.10) and (2.3.11) we see that
[2 L L xi.c,Q,]
2
n
J
i j
a 10gL
= (-,Q,-)
2
e 2
e 3
,Q,
,Q,
ae ,Q,
(2.3.14)
29
o210gL
=
oSQ.ocQ.
o210gL
oCQ.0SQ.
[L L (x
i j
=
CQ.
ij
CoQ.
2
o 10gL
ocQ.
2
[L L x. .
nQ.
- --2cQ.
1.J
i j
log
S 2
Q.
(log
X
X
ij
ij
)]
(2.3.15)
)
2
J
(2.3.16)
SQ.
If ci's are not equal, then the observed lifetimes are no longer
identically distributed,
cause C is
i
xi
f i (xi) =
c -1
i
the density of an observed failure from
i.~.,
k
exp
cQ.
{- L
xi
/SQ.}
.I!. =1
Joo
z
c -1
i
(2.3.17)
k
exp
{- L
cQ.
z
/S Q. }dz
Q.=l
0
Thus, asymptotically, the variance-covariance matrix of the
estimators is
-1
o
o
o
v= -n1
o
(2.3.18)
o
•
o
.
..
h
were
o
o
__2__)
AQ. = ( S 3
Q.
1
k
DQ. =-(-2) L
SQ.
i=l
no
Go = -"'-
'"
cQ. 2
1
+ (-)
SQ.
(2.3.19)
k
L
i=l
ci
(-S.)
Joo
0
co+ci-l
xi '"
(log
1.
(Q.=l, ••• ,k)
30
and
1T
R,
=
z
c R, -1
k
exp{-
E
i=l
ZC i
/ 8i }dZ.
Simplifying the inverse in (2.3.18) by a well known matrix inversion
we get
Gl
AlGl-D l
2
• G
k
0
AkGk-D k
V
= -n1
0
Dk
2
Dk -AkGk
0
2
• (2.3.20)
D
l
2
Dl -AlGI
Al
0
AlGl-D l
2
•
D
k
2
Dk -AkG k
0
Note:
D
l
2
Dl -AlGI •
0
•
A
k
~Gk-Dk
2
If the shape constants are known, then the estimators 8R, are
obtained from (2.3.12) where cR, are replaced by cR, and the asymptotic
variance-covariance matrix will simply be
L
Al
'.
0
0
0
1
Ak
·0
2.4
2.4.1
Underlying Normal Populations
Populations with Different Means, Common Variance
Suppose Y
i
n
N(~i,a
2
) i
= 1,2.
Let
of N(O,I) random variable, respectively.
~
and
~
be p.d.f. and c.d.f.
31
Then from (2.1.6) we have
log L
=
constant
n
+
i l l x1'-~1 2 ~
1
1 X2j-~2 2
log [ - exp{- - ( J
)}f
exp{- "2 (
) }dx 2j ]
j =1
-y2;cr
2
cr
x 1JV2rrcr
cr
I:
n
+
2
1
1 X2j-~2 2 ~
1
1 Xlj-~l 2
]
log [ - exp{- - (
)}f
exp{- "2 (
) }dx1j
-fi;r cr
2 cr
x 2f'.!2rr cr
cr
j=l
I:
n
=
n
1
2
122
constant - n log cr - - [I: u, + I: v ]
2
n
+
n
1
I:
j=l
. 1
J=
log
[l-~(u,')]
J
+
J
'1
J=
j
(2.4.1)
2
I:
log
[l-~(vj')]'
j=l
(2.4.2)
(2.4.3)
'.
(2.4.4)
32
So the likelihood equations are
n
2
~1 = xl + ~ E B.
n 1 j=l
J
(2.4.5)
n
=
~2
1
E A.
;Z. + ~
2
n 2 j=l
J
n
=
n
1
1
(-;;z) [ E
o
+
j=l
n2 "
E v. "'B.
j=l J J
n
i
where xi
=
(i=1,2)
xi' In.
E
j=l
J
1.
and u "', v "', A , B are the corresponding functions u "', v "', A , B
j
j
j
j
j
j
j
j
with
~.
1.
and
0
~.
replacing
and
1.
respectively (i=1,2).
0,
Now, to look at the information matrix we need the second
derivatives of terms in (2.4.3).
2
a 10gL =
,,2
O~l
2
a 10gL
a~la~2
'.
".
l....
0
=
n2
E
(V .... B.-B.
J
2 . 1
J=
2
a 10gL
a~2aU1
J
J
They are
2
)
= 0
n
1
2
n
a 10gL
1
2
= - E (u. "'A.-A. 2)
2
2
2
J J J
j=l
0
0
aU 2
n
2
2
2
a 10gL =_.
a 10gL
1
E [ v. ... 2B. -v. ... B 2 - Bj ]
=
2
aoaU
au 1 ao
J
J J j
j=l
1
0
2
2
a 10gL
aoa~2
2
a 10gL
2
a0
=
a 10gL
a~2ao
n
n
2
-2"
0
1
1
=-
E
0
j=l
2
(2.4.6)
[u .... 2A .-u . "'A. 2_A .]
J
J
J
J
J
-2"
n
n
n
2
1
1
~ u. 2 + ~ v 2] + l.... ~
t.
t.
j
,..2 J' =t.
j=l J
j=l
v
1
[ vj ... 3 Bj -vj ... 2Bj 2- 2v j "'B]
j .
2
0
1
E
u.
j=l
J
n2
E v.
j=l J
[u . . 3AJ' -u ' . . 2AJ' 2- 2u A]
j
j
j
J
33
Now from (2.1.2),
=
ep(u.)[l-~(u."')]
J
J
cr [l-HO]
(2.4.7)
= ep(vj)[l-~(vj"')]
where
cr~(O
~
=
).11-).12
\j2cr
Noting from (2.4.2) the relations
U
j
...
= uj + -Y2~
and
v
j
= vj -
-...{2f,
•
We have, upon transforming the variables in (2.4.7),
ep(u.)[l-~(u. +\f2~)]
f
1 (u j )
=
J
(1-~~~)]
_ ep(Vj)[l-~(vj -l/2~)]
~(f,)
f 2 (v j ) -
Now
and employing the densities in (2.4.8) we have
=_~
exp{- ~2/2}
2-y2:IT
exp{- ~2/2}
= ~(~) 2~
(2.4.9)
~ (0
1
Dl(~) = ~(f,)
2
00
f
-00
[ep(z)]
I-Hz)
ep(z+~'2~)dz
V~
(2.4.10)
Hence, realizing that E(n ) =
1
n[l-~(~)],
E(n ) =
2
n~(~)
(2.4.11)
we have
(2.4.12)
34
Similarly,
(Z.4.13)
where D is defined in (Z.4010).
1
Z
Z
Now
E[- 0 logL]
ocrO).ll
=
In 1
E{E[- 0 logL
'" '"
acra).ll
=
ZE(n 1 )E(u.)
Z J
cr
-
]}
' nZ
E(n Z)
Z
Z
---~Z- [E(v.~ B.) - E(v.~Bj ) - E(B )]
j
cr
J
J
J
and employing densities in (Z.4.8) and expectations in (Z.4.11), we
obtain
E[- o,Z10gL]
ocro).ll
n exp{- F;Z/Z}[DZ(F;)
cr 2 Z-{;
=
(Z.4.14)
-v;
(Z.4.15)
Similarly,
E[- oZ10gL]
ocro).lZ
n eXP{_F;Z/Z}[Dz(-F;)
crZz-y:;
=
(2.4.16)
-y-:;
where DZis defined in (Z.4.15).
Finally,
Z ..
E[- 0 lo gL]
ocr Z
=
3
Z
=
E{E[- 0 logL In n]}
ocrZ
1, 2
Z
Z
cr Z [E(n1 )E(u j ) + E(n 2 )E(v j )]
E(n 1 )
~3
~Z Z
cr 2 [E(U j Aj ) - E(u j Aj )
2E(U ~Aj)]
j
E(n Z)
~3
~z 2
cr Z [E(V j Bj ) - E(V j Bj ) - ZE(Vj~Bj)]
Upon employing densities in (Z.4.8) and expectations in (Z.4.11)
we have
35
(2.4.17)
where
2.4.2
00
D3(~)
=
1_ 00
[1!~(~)]
z2
exp{~~)2}dZ.
(2.4.18)
Populations with Different Means, Different Variances
2
Suppose Y. r'I N(J.l,.a. ) i=l, .•. ,k.
J.
J., J.
Let
~
and
c.d.f. of N(O,l) random variable, respectively.
k
log L = constant k
k
1
n 10ga, i=1 i
J.
L
k
2
ni
~
be the p.d.f. and
Then from (2.1.6) we have
2
u '
i=1 j=1 iJ
L
L
ni
(2.4.19)
+ L L L
t=l i=l j=l
i;t
k
1
Then
ni
L A,
]
U.+ L
tJ
i=l j=1 J.j, t
i;t
=-
n
t
a10gL
a = 1- [-n + L
at
crt
t
j=l
k
2
U
tj +
(2.4.20)
ni
L L
i=l j=1
i;t
(t=l, ••• ,k)
=
where Aij,t
HU ij ,t)
1-~ (u
ij , t) .
Hence the likelihood equations are
n i ...
n t '"
k
L u. + L L Ai'
J,t
j=l tJ
i=l j=l
i;t
n
t
L
j=l
U j
t
2
k
+
=0
n,
J. '"
u..
i=l j=1 J.J,t
i;t
L
(t=l, •.• ,k)
L
A
ij ,t
(2.4.21)
36
where Utj ' uij,t' and Aij,t are the corresponding functions Utj ' Uij,t'
A
A
at replacing ~t and at'
and Ai"
J,t with ~t and
It is of interest to note that the likelihood equations are k sets
of two equations in two unknowns.
For the case k=2,
1.~.
,the situation calls for two underlying
normal populations with different means and different variances, we
have from (2.4.20)
n
alogL
a~l
n
2
=- [ L u
+
L
A " 1]
1j
a1
j=l 2J,
j=l
1
1
n
n
1
2
a10gL
1
2
= - [...n + E ulj + L u ,1 A , 1 ]
1
2j
a0
a1
2j
j=l
1
j=l
n
alogL
a~2
(2.4.22)
n
2
1
1
= - [L u
+
L
Alj 2]
2j
a
2 j=l
j=l
'
n
alogL
1
= - [-n +
a
2
aa
2
n
2
2
1
L u 1j 2 A1j 2]'
2j + j =1
'
,
j=l
u
L
2
So the likelihood equations are two sets of two equations in two
unknowns, vi z . ,
a
".
..
n
1
xl + -
2
F_
L
A , 1
~1
n 1 j=l 2J ,
n
n1
2
2
L u ' 1 A , 1
n = L U
+
2J ,
2J ,
1
1j
j=l
j=l
nl
a
2
~2 = x·2 +n- E A
2 j=l 1j,2
A
A
A
n
n
2
=
2
L
j=l
n
A
u:aj
2
+
1
L
j=1
A
u 1J" , 2 A1j ,2
(2.4.23)
37
(i=1,2)
where xi
and u ij ' uij,t' Aij,t are the corresponding functions
u ij , uij,t' Aij,t with ~i and cri replacing ~i and cr i , respectiwely.
To determine the information matrix, we need the second derivatives
of terms in (2.4.22).
They are
c2 10gL
c~1
- n ]
1
2
c2 10gL
ccr
2
l
- n ]
2
..
c2 10gL
ccr
2
2
o for
i;&j, (i,j=1,2).
(2.4.24)
38
Now from (2.1.2), we have, analogous to (2.4.7) that
=
~(ulj)[1-~(ulj,2)]
(2.4.25)
(Jl[1-~(O]
=
~(U2j)[l-~(U2j,1)]
(J2~(O
where
Then upon transforming variables in (2.4.25) we can obtain the
densities of u .. and u ..
1J
n
1J ,)(,
(Q,~i).
These densities can be used to
evaluate the expected values of terms in (2.4.24) just as the densities
in (2.4.8) were used to evaluate the expected values of terms in
(2.4.6).
Thus
(2.4.26)
00
where Dl
=
f
Hz)
[ l-~(z)]
exp{-
_00
..
(2.4.28)
39
2
Now E[- d 10~L],
d1J
2
EI-
2
d lo g L], and
d1J d0
2 2
E[-
d
2
g L]
lo 2
may be obtained,
d0
2
from (2.4.26), (2.4.27), and (2.4.28), respectively, by symmetry, by
replacing
",
"
.
~1
by
1J
2 and 01 by
°2 ,
40
CHAPTER 3
CENSORING
3.1, Type I Censoring
It may be convenient, or perhaps even necessary due to the
physical limitations of one's experiment, to observe an individual's
failure only if it occurs within some specified time period.
The
length of such periods may vary from individual to individual; however,
each period will be assumed to begin at time zero.
There are two
obvious examples of situations which might fit this type of censoring.
First, in some life-testing work, units are put on test at various
times but are all taken off together, either because the experimenter
has run out of money or perhaps he has a time schedule to meet.
Second, in medical follow-up studies patients usually enter a study
(!.~.,
receive some treatment or have an operation performed) at
different times but the terminal point of the study is usually the
same for all patients.
Also, if contact with some patients is lost
(as often is the case), then this type of censoring becomes applicable.
Suppose that, for n (fixed) individuals, each individual is under
..
observation until time
Yi
(i=l, .•. ,n).
individuals whose lifetimes are
..
only if z.<y .•
1- 1
!.~.,
~l""'z
n
,
~.
1
for a sample of n
will be known if and
Thus an individual whose failure is not realized
before his corresponding censoring time has been reached,
Zi>Y i'
will be considered a survivor.
!.~.,
Estimation of the mean life
of an exponential distribution (for a single cause of failure) in
the context of such censoring is well known, Bartholomew (1963).
41
In this section we consider maximum likelihood estimation, where
each individual has a different censoring
~ime
as described above, in
the presence of competing causes of failure as in section 2.1.
Thus,
in addition to the terminology of section 2.1, the following notation
will be used.
Let m denote the total number of failures and r the
total number of survivors, so that n=r+m.
failed from cause C
i.~.,
(i=l, .. o,k),
i
Suppose m individuals have
i
m=Em .
i
Let x
ij
denote the time
to failure of the jth individual whose failure was due to cause C
i
( i =1 , . . . , k ;
J= l, . . . , mi)
0
Then the censored likelihood function is
likelihood of obtaining a sample of Size~
the likelihood Of]
m as described above the m failures and •
the
m failures
,
1
[ and r i survivors
[ r surV1vors
I
o
or
k
L
I
a:
k
[ IT
i=l
k
m1
IT dy (IT
IT rr i . ) n"
2=1
i=l j=l J
00
f x,.
1J
i
t~i
(3.1.1)
(X ij <y ij)
where rr •• = Pr [jth failure due to ith cause has occurred before]·
1J
the corresponding censoring time has elapsed
n" = Pr [r=n-m individuals have not failed from any cause beforel
J
their corresponding censoring times have elapsed
Y = censoring time of jth individual failing from ith cause.
ij
~
.
Note:
If all xij>Y ij for some i, then mi=O and
m
i
IT in (3.1.1) may
j=l
be defined as 1.
Thus
L
I
a:
m 00
k
i
IT
IT f
x ij
i=l j=1
00
o •
•
1X
k
IT
dY n
P(Y1'''·'Yk)1 yi-x
ij £=1 t
1J
o ·,
..
(3.1.2)
t~i
Upon making the usual assumption that the causes of failure act
independently,
.!.~.,
y 1,0 .• ,Y
k are independent,
42
(l-Pn(x . .
)f {~£=1
~J ~
'"
(3.1.3)
(X ij <y ij)
where y(£) (£=l, ... ,r) denotes the censoring
times of the r survivors.
If the censoring times are all equal, say y, then from (3.1.3)
(X
3;1.1
ij
<y)
(3.1.4)
Underlying Weibull Populations with Equal Shape Constants
From (3.1.3) we wish to maximize
k
10gL
= constant + m log c - .E
I
~=1
k
k
( E 1/8£) ( E
-
£=1
a8
E
i=l j=l
i
x
c
+
ij
~
E
£=1
c
Y(£) ] •
(3.1.5)
(3.1.6)
i
k
mi
E
E
k
log x .. ~J
i=l j=l
where
r
[t+ Eye ]
t=l (£)
8 2
m.
_-2:. +
8.
a~ogLI
Hence
m
i
k
mi
m 10g8 + (c-l) E
E log x
ij
i
i
i=l j=l
(t) [Ei=l
mi
E
(xijClOgXij)
j=l
t =
Thus the like1ihoodnequations are
[t+
8
i
£r Y C]
= _-,-£:...-.;:;;.1~(£~)_
m
i
(i=l, ••• ,k)
(3.1.7)
43
A
[ .t+ rL Yo,)c ]
,Q,=1
[(~) +
m
c
r
m
k
L
i
log x, ,] =
L
1J
i=l j=l
k
L
m
i
L
c
(x
i=l j=l
ij
log x
ij
)
c
+ L Y<'0 logy (,Q,)
,Q,=1
c
where t = L LXi'
i j
J
From (3.1.6) above, the second derivatives are
r
2 [t+ L y (£) c l
,Q,=1
e
(3.1.8)
3
i
(i=l, ••• ,k)
m
i
~
j=l
c
1J
r
(x, ,logx, .) +
1J
c
L Y(,Q,)logY(,Q,)l
£=1
(3.1.9)
a210gL r
de 2
m
=- 2
c
(3.1.10)
The following notation and random variables will be useful in
obtaining the information matrix.
Let zl"'" zn be the lifetimes of the n individuals in our sample
whose corresponding censoring times are Y1 ""'Y n '
Let a,Q, = 1 if ,Q,th individual in the sample dies before his
his corresponding censoring time, Y,Q,' has elapsed,
=
a
otherwise.
Let b,Q, = 1 if ,Q,th individual dies from ith cause before his
corresponding censoring time, Y,Q,' has elapsed,
= 0 otherwise.
k
mi
c
L x
expressed in previous notation
Then t = L
ij
i=l j=l
44
expressed in new notation.
=
=
Now E(t)
n
=
=
E E[a1E(z1CJa1)]
1=1
n
E {1'E(z1 c1z 1<Y1) Pr[z1<Y 1 ] + Q'E(z1 C1z 1>Y 1 ) Pr[z1>Y 1 ]}
1=1
n
E
=
1=1
Pr[ z1<Y ] E(z1 Clz <y ).
1
1 1
But from (2.3.2) we know the density of the z1 (1=1, ••• ,n) and
(3.1.11)
n
E(t) =
E {~[1-exP(-Y1c/~)] - Y1
1=1
c
exp(-Y1C/~)}.
(3.1.12)
Also
r.
C
n
Thus E[ E Y(1) ] = E Y c E(l-a )
1=1 1
1
R.=1
..
=
(3.1.13)
n
Furthermore,
mi
E h leads to
x.=1 1
n
L Pr[1th individual died from ith cause before
1=1
time Y1]
=
=
n
E Pr[death due to ith cause],pr(Z1<Y1Ideath due to ith cause)
1=1
~ (~i) [1-exP(-Y1c/~)].
1=1
Therefore,
(3. L 14)
45
Note that
k
n
k
E(m) = E( E mi ) = (E !-) E [1-exp(-y c /6)]
t
i=l
i=l ei t=l
n
c
= L [l-exp(-Y /6)].
t
t=l
0.1.15)
Thus, from (3.1.12), (3.1.13), and (3.1.14), we have
a2 logL r
(_6_)
E [- -----",..::;.]
ae
i
2
e.
3
].
n
L [l-exp(-Y t c /6)].
t=l
Similarly, we obtain
2
a
E[-
logL
r
aeiac]
1
= - (-2)
ei
(3.1.16)
(i=l"."k)
n
L
t=l
c
c
{[l-exp(-Y t /6)]E[Zt logz t "IZ t <Y t ]
+ Yt c logy t exp(-Y t c /6) }
(3.1.17)
and, using (3.1.15),
n
L [1-exp(-y t c /6)]
t=l
(3.1.18)
The inverse of the information matrix can be formed in the usual
.
,
way from (3.1.16), (3.1.17), and (3.1.18) following a matrix inversion
used to obtain (2.3.8) from (2.3.7).
,
.
The following special cases are of interest:
1) All the censoring times are equal,
2) c=l,
i.~.,
i.~.,
Yt=Y
for t=l, ••• ,n,
the underlying populations are exponential,and
3) Yt=Y (t=l,.",n) and c=l.
46
The estimators and the quantities leading to the information
matrix for case 1) may be obtained by setting y£=y in (3.1.7)
and
in (3.1.16), (3.1.17), (3.1.18), respectively.
For case 2), we have, from the first equation in (3.1. 7),
the
maximum likelihood estimators
r
[t+ L Y(£)]
£=1
m.
(i=l, .•• ,k)
(3.1.19)
1
where t = L L x ...
1J
i j
E(t+ ~ Yet»~
Note that
£=1
E(8.)~
E(m. )
1
1
= 8.
1
.!.~.,
, using (3.1.12), (3.1.13), and (3.1.14),
8. are asymptotically unbiased, as one might expect .
1
From (3.1.16), we have
8 3
1
o
•
•
Var(~) ~
0
where 8
{~
8 3
n
L (1-exp(-y£/~)]}
£=1
-1
(3.1.20)
k
8
1
•
..
..
•
•
For case 3), we have upon setting y£=y for £=l, ••• ,n in (3.1.19) and
(3.1.20),
8
i
= [t+ry]
m
i
(i=l, ••• ,k)
(3.1.21)
47
and
8 3
1
o
A
Var(~)
{nA[l-exp(-y/A)]}
-1
(3.1.22).
8 3
k
o
Since Halperin (1960) has shown the density of
Z
=L
L x
ij
j
i
+ ry
to be
g(z)
=
exp(-z/A)
n
E
s=l
=
E
s-l
j>n-z/y
for z<ny
0 for z>ny, and
Pr[z=ny] = exp(-ny/A),
there seems to be little hope of obtaining any meaningful expression
for the density of 6ilmi>0.
3.1.2
Underlying Weibull Populations with Unequal Shape Constants
From (3.1.3) we need to maximize
k
10gL
r
= constant +
m
i
E E
i=l j=l
k
-
Hence
..
dlogL
aa t
r
k
k
m.
J
E milogc. - E m,log8 i + E (ci-l) E log. x ij
i=l
~
i=l ~
i=l
j=l
c
c
yi
k
r
k
xi' ~
E ( J ') - E ,E ( (t) ,)
(3.1. 23)
8
8
t=l
t
t=l 1=1
i
m
c
r
1
c
+ (~) [E E x ij t +
= - (...!.)
a
Y(t) t]
i
j
R::l
t
at
(t=l, .•• ,k)
a10gL
m
m
t
r = (~)
E log
+
C
aCt
j=l
t
-
X
tj
ct
1
(-)[E E x
log x
ij
ij
8t i j
r
c
+ E y(t) t logy (t)] •
t=l
(3.1.24)
48
Therefore, the likelihood equ~tions are
r
c
[L L
Ct + L Yet) t]
=
i j x ij
t=l
at
( t=1, ••• ,k)
m
(3.1.25)
t
m
t
log
L
X
j=l
tj =
From (3.1.24), the second derivatives are
a210gL r
aa 2
m
=~
-
(_2_) [
t
3
at a t
t
L L xij
i j
C
t
r
+ L
C
Y(~) t]
(3.1.26)
t=l
r
= (_1_) [l: L x .. Ctlog x .. + L Y(t)Ct10gY(t)]
a 2
i j 1J
1J
tal
=
t
.Now following the argument used to obtain (3.1.14) and (3.1.15)
we find
(3.1.27)
..
~
k
C
i
c·-1 exp{- L z CnJ<,/a }dz
since Pr[death due to ith cause] = (-)
f Z 1
0
tal
t
ai
and the density of an observation failing from the ith cause is given
by (2.3.11), and
r
E [ L Y (t)
tal
C
n
t]
=
l: y~Ct
tal
exp{-
k
L y~Ci/ai}.
i=l
(3.1.28)
49
Define
a~i
= 1 if
tthihd~idua1
dies from ith
caus~
by time Y
t
= 0 otherwise.
Then
So
n
k
C
E(I: I: x
t) = I: . I: E(a Zt Ct)
ij
u
t=l ~=1
i j
n
k
E E(a
= I:
u E(Zt Ctl au
t=l i=l
n
k
I: E(ztCtldeath from ith cause
= I:
• Pr(death from ith
t=l i=l
by time Y )
cause by time Y )
t
t
»
•
J
[
.C i
Y't c -1
k C
("8) f
z i exp (- I: z t / et ) d Z •
i
t=l
0
n
k ci
YQ,
E(I: I: XijCt) = I:
L (-) f
i j
tal i=l ei
0
Therefore,
C +C
Z
t
i
-1
k
exp(- L zCR./S)dz],
~ -1
t
(3.1.29)
Thus, using (3.1.27), (3.1.28), and (3.1.29) we get
E[-
a210gL1
as 2
]
t
..
+
n
L Y Ct exp (_
~=1 t
n
C
Yt
t
() I: f 0
e 3 t=1
t
(3. L 30)
c -1
z t
k.
C
exp(- I: z t/eQ,)dz .
t =1
50
Similarly,
n
=-
8 2
i=l i=l
t
+
k
E
(_1_) [E
c
(f)
YRf0
i
n
k
R-=1
~=l
E YR-CtlogYi exp(- .E y Ci /8 )]
i
i
(3.1.31)
E[-
+
l8t
Y
k
f R- zCt+c~-l(log
~
z) 2exp(- E z c t/8R-)dz
i=l i=l 8 i 0
R-=l
n
[~6
k
~
6
C
-!
(3.1.32)
3.2
Tyee II Censoring
It may be desirable to only observe the failure
ot
the first m
individuals, where m is some predetermined integer (m<n).
the total number of individuals that have not failed,
(known).
k
Let r be
1.~.,
r=n-m
Suppose mi individuals have failed from cause C ,
i
!.~.,
m= E mi' Here ml, ••• ,mk are random variables. It should be noted
i=l
that in this section we assume all individuals are under observation
'.
~
.
for the same length of time, as opposed to the more general situation
in sec tion 3.1.
Let xi(j) denote the time to failure of the individual with the
jth longest lifetime among those individuals whose failure is
attributed to cause Ci .
Alternatively, it will be useful to let
z(R-) be the time to failure of the indiVidual with the R-th longest
lifetime, irrespective of the cause of death.
Thus our sample
51
consists of the observations xi(j),(i=l, ••• ,k; j=l, ••• ,m i ) or,
alternatively, z(t) (t=l, ••• ,m).
Note that z( )=max{x.( )} denotes the mth individual to fail
m i
1 m
i
and hence, z(m) (a random variable) will be the common censoring time
for all individuals.
Now the censored likelihood function is
L11
=
[likelihood of first m fai1ureslm1, ••. ,~][1~ke1ihood of m1""'~]
k
oc
[
IT
mi
__1____
IT
i=l ~.mi j=l
1
k
• (IT
mi
~i
)n
r
i=l
J
Xi(j)
P(Yl""'Yk)
IY =xi(j)
(3.2,1)
i
where n=Pt[not failing from any cause by time Z(m)]
~i=Pr[failure
due to ith cause by time Z(m)]'
Assuming the causes of failure act independently,
(Xi(l)<· •• <xi(m ); i=l, .•• ,k)
i
From (3.1.4) and (3.2.2) we note that the maXimization of the
two likelihood functions L1 and L yields the same estimators if one
11
takes y=z(m)'
Thus, when all individuals are under observation for the
same period of time, only one computation is necessary to find maximum
52
likelihood estimators for Type I or Type II censoring, viz., finding
parametric values which maximize
m
k
L'"
i Pi(x ij )
k
k
r
(
IT {l-P (x )}] [ IT {l-P~(y)}].
i=l j=l I-Pi x ij ) t=l
t ij
i=l
~
= [IT
IT
(X
ij
<Y)
(3.2.3)
3.2.1 Underlying Weibu1l Populations with Equal Shape Constants
From (3.1.7), upon replacing x
ij
(i=l, .•• ,k; j=l, ••. ,m ) and
i
Y(i) (t=l, ••• ,r) by z(t) (t=l, •.• ,m) and z(m)' respectively, as
suggested in section 3.2, we have the likelihood equations
"
(i=l, .•• ,k)
"
(3.2.4)
c
t+rz( )
m
[
m ] [(~) + L log
m
c
t=l
"
m
"
L Z(~)'
where t ...
t=l
From (3.1.8), (3.1.9), and (3.1.10), the second derivatives are
a 210gL
ae 2
i
.-
a 2log1
aeidc
I
m
(_i_)
e 2
i
.
=
2[t+rz(:)}
~ 3
i
c
a210gL .. (~)[ m
L Z(~)log z(1) + rZ(m)log z(m)]
acae ·
e
i
i
(3.2.5)
(3.2.6)
t=l
-.
and E(Z(:»
i<J
... ~
t
L
1
j=l n-j+l
(3,2.8)
53
since if a statistic X has a p.d.f. given by (2.3.2). which is how the
unordered z~'s (~;l, ••• ,n) are distributed, then ~c is distributed as
an exponential variate and the lower moments of
an underlying exponential population are well
orde~
statistics from
~nown.
Now using (3.2.8) we can find the expected values of terms in
Hence
(3.2.5).
a210gLI
m6
E[- ---=-...;...] = -
(3.2.9)
S 3
dS 2
i
i
2
d 10gL 1
E[- -----,..-...::;. ]
aSide
1
= -(~)
S 2
i
+ (r+l) E[Z(~)lOg Z(m)]}
2
d 10gL
E[ -
dC
2
1]
= (!!L)
c
2
+
1 {m-l
(~)
(3.2.10)
c
I: E[Z(~)(log
~=l
c
+ (r+l) E[z(m)(log z(m»
2
]}
o<z(~)<co
(3.2.11)
(3.2.12)
If c=l, then the estimators are, from the first equation of (3.2.4).
(i=l, ••• ,k)
(3.2.13)
m
where t
=
I: z(~)'
R,=l
Directly, one can see that
a.1.
are asymptotically unbiased since
54
From (3.2.9) we have
a
1
3
o
'"
Var (~)
(3.2.14)
tV
o
a 3
k
55
CHAPTER 4.
GROUPING
4.1
Likelihood Function for Complete Grouping
of Independent Observations
In most statistical work the random sample of interest consists of
individually known observations.
However, situations arise in practice
where individual values of the observations are not recorded, but rather
the data are grouped into intervals.
In the context of .competing causes of failure, the observations
are individual lifetimes which are only known to fall within some time
interval.
As before, it is assumed that the cause of each failure has
been recorded.
An additional assumption, unique to grouped data,
made is that the observed number of deaths classified in a two-way
table according to cause of death and time interval of death follows
a multinomial distribution.
Suppose that the range of variation (zo,zh) of the lifetimes is
part~tioned
into h intervals with the endpoints of the intervals being
Zj·(j=O,l, ••• ,h) such that zo<zl<",<zh'
Let n ij denote the number
1.~.,
of individuals failing from ith cause in the jth interval,
the interval
(Zj_~j)'
Then n i .= E nij individuals have failed from
j
cause C (i=l, ••• ,k) and
i
n~=~nij
causes in the jth interval.
individuals have failed from all
The total sample size is
n=~
En
i j
ij
•
The distinguishing feature of such grouped data is that the
n
ij
in
(i=l, .•• ,k; j=l, ••• ,h) and the Zj (j-O,l, ••• ,h) contain all
the information in the sample.
56
Since the n
ij
are mu1tinomia11y distributed, the likelihood
function is
(4.1.1)
where n
ij
is the probability of an individual failing from the ith
cause in the jth interval.
NQw n
ij
= n i Pr[individua1 fails in jth interva11fai1ure due to ith cause)
(4.1.2)
= Pr{failure
where n.],
and F
i
due to ith cause] as in section 2.1,
is c.d.f. of Xi whose density is given by (2.1.7).
4.2
From (4.1.2), n
where tJ. and 6
i
Weibu11 with Equal Shape Constants
ij
tJ.
6
= (--
)[exp(-~
c
j-1
i
/tJ.)
are as in section 2.3.
For notational convenience, let a
j
= eXP(-zj c /tJ.).
Then from
(4.1.1)
or
k
Log L = constant + n log tJ.
~
h
E n log 8i + E n. j 10g (a j _ 1-aj ). (4.2.1)
i=l i •
. j=l
Hence
(i-1, ••• , k)
(4.2.2)
57
Therefore, the likelihood equations are
"
(i=1, ••• ,k)
(4.2.3)
and
h
I: n
j=l·j
(4.2.4)
=
where a.
J
1
1-1
8
6
(~)+ ••• ~)
1
•
k
The obvious simp1fications Z =0, Z =00 a =1 a = 0
o
h' 0 ' h
can be made at one's convenience.
From the k equations in (4.2.3) we obtain the relationship
h
I:
=0
n.
j=l .J
(4.2.5)
Now (4.2.4) and (4.2.5) may be rewritten as
"
"
c
. c
n'(Zj 10 g z j -z. 1 10gz . 1)
-J _
-;;. J Jc
c /"
eXP[zj -Zj_1) ~]-1
h-1
I:
j=l
-
h
I: n
j=2 .j
h
c
~ n. zJ'-l log Zj_1 = 0
j=2 .J
Zj_1
c
= o.
(4.2.4a)
(4.2.5a)
1
Equation (4.2.5a) is of the form of Ku11dorff's (1961) equation
(2.36).
Solving equations (4.2.4) and (4.2.5) or, alternatively, (4.2.4a)
and (4.2.5a) by some iterative technique will give us
~
and c.
lIt should be noted that Ku1ldorff (1961) has shown that there
exists a root to equation (4.2.5a) iff n <n and n <no He points out,
however, that as n+OO the probability for'!he distutRing cases n l=n
and n h=n tends to zero, and the probability for the existence'
of a solution to equation (4.2.5a) tends to unity.
58
Upon substituting (4.2.5) into (4.2.3) yields
Thus
a"" i
= --- (i=l, ••• ,k).
nl1
ai
are trivially determined after l1 has been obtained.
(4.2.6)
ni.
Now, towards the end of obtaining the asymptotic variancecovariance matrix, we find
a210gL
aa 2
_
-
nl1(l1-26 )
i
a 4
i
i
1
h
(4.2.7)
- --- E n
a 4 j=l .j
1
(4.2.8)
2
a 10gL
aaiac
h
= (_1_) L n
M 2 j-1 .j
i
+
a210gL =
ac 2
Note:
h
(_1_) E n
a 2 j=l .j
i
(4.2.9)
h
(.L) E n
l1 2 j=1 .j
c
= [exp(Zj /l1) -
(4.2.10)
c-1
ex p (Zj_/l1)] •
59
Upon using the facts that
E(n, ) = nA/S, and E(n ,) = n(a, I-a,)
~.
.J
~
J-
(4.2.11)
J
we have
E[-
2
a logL] = nA +
as 2
S 3
i
i
Z
2c
(~) {
1
c
S. 4
exp(zl /A)-l
+
j=2
~
2
zl c
L
c 2
c
h-1
[
.
(Zj -Z, -1 )
-
J
2
] -A }
eXP(Zj c/A)-exP(zj~l/A)
2
(4.2.12)
(z,c- z c )
{
c
+ L [
J
j-l
]_A 2 }
exp(zl /A)-l
j=2 exp(zj c/A)-exP(zj~l/A)
h-1
(4.2.13)
2c
c
c
c
c
~2l L
zl log zl
h-l (z. -z'_l)(Zj logzj-zj_llogzj_l)
E[- 0 og]= _(_n_){
+ L [J
J
_
_ _
_}
as.ac
2
c
c
c
~
AS i
exp(zl /A)-l
j=2
exp(zj /A)-exp(zj_1/A)
(4.2.14)
and
E[-
~2l
o
2c
L
2
. zl (logzl)
h-l
o~ ]=(n ){
+ L
c
2
ac
A exp(zl /A)-l
j=2
4.3
Exponential
For underlying exponential populations, the likelihood equations
"
are given by (4.2.5a) where c is replaced by 1 and (4.2.6).
Also the quantities leading to the information matrix are given by
(4.2.12) and (4.2.13) where c = 1 wherever it appears.
4.3.1
..
Equidistant Group Limits for Finite h
If the group limits
finite), i.e.,
z'=~21
J
zo~o,zl,z2, .•.
,zh_l are equidistant (h
for j=O,l, ... ,h-l, then the likelihood equations
are, upon simplifying (4.2.5a) and substituting into (4.2.6),
S
i
=
i
h
Si = ni.10g[1+(n-n.h)/.L2(j-1)n.j]
J=
(4.3.1)
60
Also
2
a
E[-
= n~ +
logL]
ae
i
a
2
(4.3.2)
3
i
and
2
a lo gL] ( n ){
aa i aa i" = a 2 a 2..
i i
E[ -
(4.3.3)
(i;&i .. )
Hence, the asymptotic variance of
n~
a
1
+ nA
a
3
1
nA
a 2a 2
1 2
4
~
is
a 2a 2
1 k
nA
e 2a 2
1 2
Vk,k
•
•
=
-1
nA
•
•
•
•
•
nA
nA
a 2a _2
1 k 1
where A
=
nA
nA
a 2a 2
1 k
a 2a 2
2 k
zl
nA
~+ nA
e3
k
a4
k
(4.3.4)
2
exp(zl/~)[l-exp(-(h-1)zl/~)]
[exp(zl/~)-l]
2
-~
2
.
v- 1 = n B C
But
B
k,k k,k k,k
where B is a diagonal matrix with diagonal element l/a i 2
and
C
k,k
=
D + AJ
where D is a diagonal matrix having diagonal
k,k
k,k
element ~a. and J is a matrix whose elements
1.
k ,k
are all unity.
61
Hence
But (£i.
Graybill, 1969)
-1
C
= D-1 +
y
A
Y=---7
where
provided A
a£~
1+A/6
and
2
2
~
-6 ,
111
e ).
k
a = T. (S;:-'.'"
Therefore, upon simplification,
(e 3 /6)+ y~e12
1
.
y~ele2
y~ele2
.
1
v=n
•
..
.
•
y~elek_l
•
y ~e2ek
y"e e
1 It
2
2
6 [exp(zl/ 6 )-l]
A
where y ~=--- =
A+6
4.3.2
.
•
2
-~2-----=--------
- 1.
zl exp(zl/6) [1-exp(-(h-1)zl/6)]
Equidistant Group Limits for Infinite h
If Zj=jz1 for j=O,l, •.. ,
~,
then the likelihood equations are
nZ
1
.
ei = -----.;;;;....~-----
'.
(4.3.5)
n i • 10g[1+n/ L (j-l)n j]
j=2
.
Also
2
2 g
zl exp(zl/ 6 )
a
10
L]
_ 62 }
(-E-)
.
=
n6
+
E[{
2
3
e
4
ae 2
e.1[exp(zl/6)-l]
i
i
and
2
2
zl exp(zl/6)
2
a
10gL
]
6 }..
= ( ~ 2) {
E[2
aeiaei~
[exp(zl/ 6 )-1)
e i ei~
(i~i ~)
(4.3.6)
(4.3.7)
62
Here the asymptotic variance of 6 is as in (4.3.4a)
where A =
2
zl exp(zl/ A)
[exp (z / A) -1 ]
4.3.3
2
A and hence y
2
2
2
A [exp(zl/A)-l]
=
-l.
2
zl exp(Zl/A)
Estimates Based on the Theory of Non-Grouped Samples
A natural estimate of 6 , and one which is widely used in practice,
i
from a grouped sample may be formed by replacing the unknown individual
observations within an interval by the central value of that interval,
i.e., all observations in the jth interval are given the value
(Zj_l + zj)/2, and the estimator for ungrouped data
as given by
(2.2.3) is then used.
Now this procedure leads to an estimator
(4.3.8)
For equi-spaced intervals,
~.~.,
Zj = jZl for j=l,2, •••
,~,
(4.3.8a)
Kul1dorff (1961, p. 24) obtains a similar estimator for a single
cause of failure.
When the number of groups is infinite,
of (4.3.8) is well defined.
e.
i.~..
'~
i
would be infinite since
the probability of the event n.h=O tends to 0 as
this event may occur.
~,
the estimator
When the number of groups is finite,
however, then particular care must be exercised.
be zero for otherwise 8
, h=
First,
zh=~'
nt~
D.
h
must
Of course,
but in practice
Second, assuming n.h>O, some arbitrary decision
has to be made concerning the values the observations falling in the
last group should be given.
For a single cause of failure, several
63
alternative values have been assigned to such observations.
We shall
confine our attention to the case of an infinite number of groups.
When
h=oo, we have, from (4.3.8),
1
E(81.') = -2
But E(n ./n
i
..J .
(4.3.9)
L (z'_l+ Zj) E(n j/n. ).
j=l J
•
l..
) = E[E(n ./n. In .. , j=l, ... ,h)]
.J
=E [
1. •. 1.J
n (TI . -TI i . )+n i .
.J
J
J ].
n1.
Hence
(4.3.10)
which will be infinite for finite n, where TI.
j
= LTI ..
i
1.J
However, asymptotically,
E(_l_) =(_1_) + O(l.-)
n1.
·nTI1.
n2
(4.3.11)
where TI
= LTI .
i.
j iJ
Now, for any two random variables X and Y,
E(X)
E(!) = E(Y)
Y
Cov (X, Y) +
[E(y)]2
Var(Y)
+
[E(y)]3
n ..
TI i ·
Thus E(-21.) =-21.+ 0(1.)
n
TI.
n.1..
1..
(4.3.12)
Therefore, substituting (4.3.11) and (4.3.12) into (4.3.10) we
"0
have
(4.3.13)
which substituted into (4.3.9) yields, asymptotically,
hL (z. l+z,) TI ..
TIL j=l
JJ.J
"E (8.)
= -2-1
1.
+
0(1.)
n
64
and realizing
TI
i. =
~i from (2.2.4) and
TI.
p
j = eXP (-zj_ 1 /6)-ex (-z j /A)
from (4.1.2) we have
a.
E(8i ) =
(2~)
1
00
L (zJ'_l+Zj) [exP(-zj_1/6)-exP(-zJ./6)]+ O(n) (4.3.14)
j=l
Hence, ""
ai is, asymptotically, not an unbiased estimator for
thus ""
ai is not a consistent estimator.
For equi-spaced intervals,
E(8.)
'V
1
a,zl
(.2:....:...) {
6
i.~.,
Zj=jZl for j=1,2, •..
1
1
- -}
l-exp(-zl/6)
2
ai
and
,00,
(4.3.15)
The usual method used to obtain a consistent estimator in such
a situation is to set the left hand side of (4.3.15) equal to '"
ai and
solve for a .
i
However, the problem is further complicated by the
presence of 6, so we need to find an asymptotically unbiased estimator
of 6.
From (4.3.15)
(4.3.16)
Setting the left hand side of (4.3.16) equal to
..
(4.3.17)
•
00
log[l+n! L (j-l)noj]
j=2
•
It is interesting to note that, for a single cause of failure,
6 is the parameter of interest and that
A is
simply the asymptotically
unbiased maximum likelihood estimator of 6 as given by Kulldorff (1961).
Now replacing 6 by
A in
(4.3.15) and solving for
asymptotically unbiased estimator
ai
yields the
65
ei
nZ
=
l
-----=.-----
(4.3.18)
~
n
i.
log[l+n/ E (j-l)n j]
j=2
.
which is just the maximum likelihood estimator found in (4.3.5).
4.4
Weibull with Unequal Shape Constants
Employing (4.1.2) where Fi is the c.d.f. of a r.v. whose p.d.f.
~ij=~i,pr
is in (2.3.17), we obtain
[individual fails in jth interval
Ifailure due to ith cause]
Z
= f
c
= (a-)
i
o
k
c -1,
x i
i
f
c
x 1
exp{- E (-e-) }dx
1=1 1
j
Zj_1
c -1
x i
k
c
'x 1
exp{- E (-e-) }dx
t=l 1
(4.4.1)
Thus, from (4.1.1),
or
k
k
h
10gL = constant + i=E ni .[lo g c i - .loge i ] + E E nijlo g b ij (4.4.2)
l
i=l j=l
..
where b ..
~J
~1
CJ
og
L
oe 1
=
f
Zj
C
Zj_1
x i
-1
k x Cl
exp{- E -e-} dx.
1=1 1
nn
1
k
h n
Zj
c +c -1
k x Cn
_ _ (_i<._.) + _
E E ij [f
x i 1 exp{- E -i<.}dx]
e1
e 2 i=l j=1 b ij . Zj_1
1=1 e 1
1
+
(4.4.3)
h n
Z
C
k c
1
i
E'.=!i [f j
(1- ~)
x c 1-1 log x exp(- E T)dx]
(4.4.4)
j=l ~1j
Zj_1
e1
i=l i
k
h n
1
- (--)
E E b ij
e1 i=1 j=l ij
i;l
for 1=1, ••. ,k.
f
+
1
k xC~
xCi c1- log x exp(- E -e-~)dx
Zj_1
i=1 i
Zj
66
Setting the terms in (4.4.3) and (4.4.4) equal to zero yields the
likelihood equations.
If
are known, then equations may be oPtained
c~
by setting (4.4.3) equal to zero.
k
Now, letting w = exp(- L x
C
~/S~),
~=l
2
E[- a logL]
as 2
(4.4.5)
R,
= -(
k
c.
h
Z
n
) { L (..2) L [J j XCR,+c~,,+ci-l w dx
SR,SR, .. 2
i=l Si j=l Zj_l
- b l (JZj xCi+c£-lw dX)(JZj xCi+cR, ..-lw dx)]}
ij Zj_l
Zj_l
E[-
k
C
h
(4.4.6)
xc~ C +C -1
(1- -S-)x R, i log x w dx
Zj_l
~
z.
_(--E..-) { L (..2:.) L [I J
S 2
~
i=l Si j=l
(4.4.7)
••
(4.4.8)
67
k ci h
z.
1
2
+(~) E (--)
E {f J xCi+Ct- (log x) w
8 t i=l 8 i j=l ~j-1
i#t
+ (1-) [fZj x Ci+ Ct- 1 log x w dx]2}
8t
Zj_1
1
zJ'
- (---)[f
b tj
Zj_1
xCt
C -1
][ zJ'
C ~+c -1
]'
(1- ---) x t log x w dx f
x t
t log x w dx :
8t
Zj_1
Z
1
j
C
- (-b--)[f
x t
t ~j
Zj_1
- (
k
(4.4.9)
c.
~+c -1
t
h
z.
log x w dx]
+ +
1
n ) E (~) E {f J xCi c t Ct~- (log x)
8 x,n 8 x,n ~ 1=
• 1 e.. 1
z.J- 1
1 J=
2
w dx
i#t
i#t'
- (f
-.
If c
••
t
ZJ'
C +C 1
zJ'
C +C .-1
x i t- log x w dx)(f
x i t- log x w dx)}.(4.4.10)
Zj_1
Zj_l
are known then quantities in (4.4.5) and (4.4.6) are the
only relevant ones .
68
4.5.
Grouping in the Case of Censored Observations
An important situation occurring in practice is one in which the
lifetimes of some individuals, whose cause of failure is known, are
grouped into intervals up to some point in time and the remaining
observations are censored.
If the group intervals are zo,zl, ••• ,zh
and the common point of censoring is zh' then the likelihood function
may be written as
(4.5.1)
where n
ij
and n
are as in section 4.1, and- r denotes the number of
ij
censored observations.
4.5.1
Weibu1l with Equal Shape Constants
h
Let n • = L n , denote the number of individuals who have failed
i
j=l i J
k
from the ith cause by time zh and n., = L n
be the number of
J
i-1 ij
individuals who have failed from all causes in the jth interval.
Here the total sample size is n = L L n
j
i
ij
+ r, n1j and r being
random variables.
Now employing the density of (2.3.1) with c
..
••
i
= c for all i, we
have, from (4.5.1),
L ~
k
IT
h
IT
{~ (aj_1-aj)}nij exp(- rZhc/~)
(4.5.2)
i=l j=l 6 i
where a
j
= exp(- Zj
logL = constant + (n-r) log
~
c
/~),
-
or
k
L n
i=l i .
k
log6
+ j:1n.j10g(aj_l-aj) -rz h c /~ •
i
(4.5.3)
69
Hence
dlogL _ (n~r)!.1 _ ni. +
ae. - e 2
e.
1.
i
(2...-)
1.
h
L:
(c
c)
Zj-t j - l - Zj ) a j
a. I-a.
n..
e. 2 J'=l J
J-
1.
c
(4.5.4)
+ rZ h2
e'
i
J
Therefore, the likelihood equations are
C
h
L:
c
A
A
(Zj la. l-Z' a.)
- JJ J }
a = (_l_){(n-r)~ +rz C+ n .
h j=l oJ
i
ni .
(a. I-a.)
JJ
and
A
A
A
(4.5.6)
(i=l, ... ,k)
h
L: n
(4.5.7)
j=l .j
Now, taking derivatives of (4.5.4) and (4.5.5) and using the facts
n~
that E(n. ) = (e) [l-exp(- zh
i
1.0
E(n .)
.J
2
c
= n(a.J- I-a.),
J
/~)],
E(r) = n exp(- zh
c
/~:),
and
we have that
Zl
A
2c
h
C
C
(z. -z. 1)
2
E[- a 10gL] = ~ + (~) {
+ L: [
J
J]_~
3
4
c
C
c
ae 2
ai
ei
exp(zl /~)-l
j=2 eXP(zj /~)-exP(zj_l/~)
i
n exp(-zh
+
'.
f.
= (
ai
n
2
e.1.
2
ei~
c
2
}
/~)
(4.5.8)
3
2c
__
z~l~
_
){
C
exp(zl /~)-l
+ ~'-' [
C
C
(Zj -Zj_l)
__
c
2
c
j=2 eXP(zj /~)-exP(zj_l/~)
(4.5.9)
(4.5.10)
70
and
(.!!-) {
~2
nZ
+
c
h
zl
2c
(log zl)
eXP(zlc/~)-l
2
+
h
c
c
2
(z. log z.-z. 110g Zj_1)
L [J
J
J-
_
j=2 exp(zj c/~)-exP(zj:1/~)
]}
2
c
(log zh) exp(-zh /~)
~
(4.5.11)
Letting c=l in (4.5.6) yields the likelihood equations when the
underlying populations are exponential and setting c=l in (4.5.8) and
(4.5.9) leads to the associated information matrix.
Applying a matrix
inversion as suggested in section 4.3.1 yields the asymptotic variancecovariance matrix
e
3
1
"e 2
(~+B) + Y 1
.
,
71
4.6
General Likelihood Function When Some Observations
are Exactly Known, Grouped, and/or Censored
In practice, samples may be obtained as a mixture of grouped and
censored observations as well as observations which are exactly
specified.
In essence, this involves a combination of situations
discussed in sections 2.1, 3.1, and 4.1.
In place of censoring in the sense as described in section 3.1,
we shall introduce a more general form used by Swan (1969).
The most
general form in which an observation may be censored is to limit it
between a pair of values (L , U ).
i
i
as being confined.
Swan speaks of such an observation
For a single cause of failure the concept of
confined observations is general enough to include both grouped and
censored observations.
However, for the multiple causes of failure
situation, grouped observations deserve separate treatment because
of the added knowledge of cause of failure.
To be specific, suppose m individual lifetimes are observed
exactly within some specified time period (01' 02) with m individuals
i
failing from the ith cause.
Further assume that n
ij
(i a 1, ••• ,k;
j=l, ••• ,h) individuals have failed from the ith cause in the jth
interval (the intervals may be adjacent to one another but need not
k
~ mi-E En
individuals are known only to be
ij
i=l
i j
confined between the r pairs (L , U ), t=l, ••. ,r.
t
t
be), and that r=n
I.
If the causes act independently, then the general likelihood
function is
72
mi
k
L
cr
[II
II
i=l j=l
k
r
.[ II
II {P.(U t ) - P.(L )}]
t
1
1
t=l i=l
where x
ij
(4.6.1)
is the jth individual failing from ith cause in
specified time period (01' 02)'
and
TI ••
1J
is as defined in (4,1.2).
If there are no observations falling into the categories of
being exactly known, grouped, and/or confined, then the first, second,
and/or third terms in brackets, respectively, of (4.6.1) are each taken
to be unity.
Thus, most special cases of interest, including all those
considered in this thesis, may be deduced from (4.6.1).
-,
73
CHAPTER 5.
APPLICATIONS
5.1
Boag's Data
Consider the numerical example due to Boag (1949).
He cites a study
involving 121 patients treated for cancer of the breast by surgery
and/or x-ray therapy in one particular hospital.
Each of the patients
entered the study during the years 1929-1938 and remained under
subsequent observation until February 28, 1948 or until their death.
All surviving patients were in the study for at least 110.5 months.
The stage of advancement of the cancer was estimated by the clinician
when the patient first presented herself for treatment and cases in
which the spread of the disease had become so extensive as to prohibit
any cure by treatment were excluded from the study.
The patients were classified into the following four distinct
groups:
1) those who died with cancer present, the growth being either
a persistence or a recurrence of the original cancer,
2) those who responded favorably to treatment and were free
from signs or symptoms of cancer at their death from other
causes,
I.
3) those who responded favorably to treatment and were still alive
and symptom-free at the conclusion of the study, and
4) those who were alive at the conclusion of the study but were
suffering from a persistence or recurrence of cancer.
S~rvival
times of the patients in groups 1 and 2 and follow-up
times of those in groups 3 and 4 originated at the commencement
74
of treatment.
Up to 30 months the survival times are given to the
nearest tenth of a month and after 30 months the survival and follow-up
times are given to the nearest month.
Consequently, any analysis
that assumes the survival and follow-up times to be continuouslY
distributed will obviously be a slight approximation.
The data are given in Table 5.1.1, except that we have combined
Boag's last two categories (3 and 4) above into a single group which
will be designated as
3')those who were still alive at the conclusion of the study.
In section 5.1.1 the data are analyzed under the assumption that
the failure distribution of those patients who died with cancer present
is a Weibull (equation 2.3.1) and the failure distribution of those
individuals who died from other causes is also Weibull (different
parameters).
From the estimates of the parameters it is shown how to
obtain the relevant probabilities in competing risk theory.
For illustrative purposes, in sections 5.1.2 and 5.1.3 the
failure distributions are assumed to be exponential for the continuous
case and for a grouped situation, respectively.
This assumption
should not be far from reality, however, since Boag states that the
negative exponential does a fine job of explaining deaths due to
cancer of the breast (although he assumes a log normal distribution).
It may be questioned whether or not the failures due to other causes
follow an exponential, although the estimated shape constant of this
failure distribution as seen in (5.1.1) is remarkably close to 1
(~.~.,
the exponential case).
Following a method of selecting an
underlying life distribution put forth by Nelson (1969), it is shown
75
Table 5.1.1.
Boag's data
a
Survival times (in months) of those
Who died from
Who died with cancer present
other causes
Length of time (in
months) survivors
were in the study
0.3
12.2
17.5
28.2
41
78
0.3
110
III
136
5.0
12.3
17.9
29.1
42
80
4.0
III
112
141
5.6
13.5
19.8
30.0
44
84
7.4
ll2
113
143
6.2
14.4
20.4
31
46
87
15.5
132
ll4
167
6.3
14.4
20.9
31
48
89
23.4
162
114
177
6.6
14.8
21.0
32
48
90
46
ll7
179
6.8
15.7
21.0
35
51
97
46
121
189
7.5
16.2
21.1
35
51
98
51
123
201
8.4
16.3
23.0
38
52 100
65
129
203
8.4
16.5
23.6
39
54 114
68
131
203
10.3
16.8
24.0
40
56 126
83
133
213
11.0
17.2
24.0
40
60 131
88
134
228
11.8
17.3
27.9
41
78 174
96
134
aup to 30 months the survival times are given to the nearest
tenth of a month and after 30 months the survival and follow-up times
are given to the nearest month.
,.
in the Appendix, section 8.2.3, that one has some additional reassurance
that the exponential does a fair job of describing both failure
distributions.
As one might expect, the more general Weibu11 distribu-
tion seems to fit each failure distribution slightly better; however,
the fit is still not as good as one might hope.
76
5.1.1
Fitting Weibulls with Unequal Shape Constants
From (3.1.25) we obtain the maximum likelihood estimates of the
parameters of the failure distributions associated with those individuals
who died with cancer present and those who died from other causes as
c
•
c
= .8356,
l
2
=
8
1
1.0578,
= 48.9730
(5.1.1)
8
2
=
581.004,
respectively, where the month is the time unit used in solving the
equations.
The estimates were obtained by a standard Newton-Raphson
iterative technique.
Using the parametric estimates as given in
(8.2.8) the method achieved convergence after five iterations.
Now, from (2.3.1)"
(5.1. 2)
and
median(Yi) = (6 log 2)(1/c i ).
i
Thus, from (5.1.1) and (5.1.2), the estimated expected survival
times for those individuals who died with cancer present and for those
who responded favorably to treatment but died from other causes are 9.65
and 33.46 years, respectively.
II
..
The estimated median times for the
respective populations are 5,66 and 47.47 years.
It is not possible to
obtain estimates of the asymptotic var,iances of the estimators (when
all sample values are utilized) since the differenc censoring times
for each individual must be known, as can be seen from (3.1.30),
(3.1.31), and (3.1.32).
However, we know that all individuals were under study for at
least 110.5 months so if we use that figure as a common censoring time
77
it will enable us to obtain estimated asymptotic variances (although
we will not utilize information on the eight individuals whose
surviving time is greater than 110.5 months).
Assuming a censoring
time of 110.5 months for each individual, the estimates are
c
l
= .9092,
8
1
= 62.6128
(5.1.3)
c
= .9861,
2
8
2
= 460.636.
Here the estimates given in (5.1.1) were used as starting values and
by the Newton-Raphson method convergence was achieved in eight
iterations.
The estimated expected survival times, employing the estimates in
(5.1.3) for those who died with cancer present and for those who
recovered from cancer but died from other causes are 8.22 and 42.10
years, respectively.
The estimated median times for the respective
populations are 5.27 and 28.86 years.
The estimated asymptotic standard deviations of the estimates,
as obtained from the information matrix whose elements are given by
(3.1.30), (3.1.31), and (3.1.32) (where Y = 110.5 for all t=l, ••• ,n),
t
are found, by numerical integration using Simpson's Rule, to be
s.d.(8 ) = 2.14 years
l
..
A
A
s.d. (c ) = .0915
l
A
A
s.d.(8 ) = 38.99 years,
2
A
A
s.d.(c ) = .2262.
2
(5.1.4)
78
The estimates of the parameters from the second failure distribution, viz., c
2
and 8 , in (5.1.3) appear to be somewhat unreliable due
2
to the small number of deaths occurring with cancer absent.
The cor-
responding estimates obtained in (5.1.1) should be more reliable;
however, the extent to which this is true is not known.
The definitions of the crude and net probabilities as given in
section 1.1 are formulated as (cf.
b
Chiang, 1968, pp. 245-246)
t
Qi = fa exp{-f a
r(~)dT}ri(t)dt
(i=1,2)
(5.1.5)
which is the probability of death from cause i in the interval (a,b)
in the presence of all other risks acting on a population and
b
qi = l-exp{- f
a
r.(t)dt}
l.
(i=1,2)
(5.1.6)
which is the probability of death in the interval (a,b) if risk i is
the only risk affecting a population, where r (t)=P.(t)/[l-Pi(t)] and
i
l.
r(t)=r (t)+r (t).
1
2
For an underlying p.d.f. as in (2.3.1)
c -1
r.(t) = c.t i
l.
l.
/8
i
(5.1. 7)
and hence, from (5.1.5) and (5.1.6),
.
•
(5.1.8)
and
q. = l-exp{-(b c·l.-a ci )/8.}.
l.
l.
(5.1.9)
A numerical integration is necessary to evaluate (5.1.8). Simpson's
Rule was used in the calculations of Tables 5.1.2 and 5.1.3.
79
Kimball's estimate of the crude probability is identical to the one
Chiang uses in practice as pointed out in section 1.2.
Their crude
probability of death within an interval from ith cause is the ratio
W.
1
=
number of deaths due to ith cause in the interval
•
number of survivors entering the interval
(5.1.10)
(i=1,2)
The net probability of death from the first cause within an interval
(a,b) as given by Kimball is the ratio
(5.1.11)
where wi is simply the crude probability as calculated in (5.1.10).
Similarly, the net probability of death from the second cause within
the interval (a,b) is
(5.1.12)
Chiang, upon assuming constant relative forces of mortality as
discussed in section 1.2, gives the net probability of death within
an interval from the ith cause to be
(5.1.13)
where wi is the crude probability as calculated from (5.1.10),
q
= wl +w 2
(the probability that an individual who enters the
interval alive will die in the interval from some cause),
and
p
= l-q
(the probability that an individual who enters the
interval alive will not die in the interval).
80
Table 5.1.2 and Table 5.1.3 compare the crude and net probabilities
of death in various intervals with those obtained by Kimball and Chiang.
The intervals in Table 5.1.2 are equi-spaced and those in Table 5.1.3
were selected such that the expected number of total deaths within each
interval should be about equal (based on an exponential fit).
By way of appraising the parametric approach presented here with
the methods already in use, three points become apparent.
First, the
major advantage of the parametric method is that one need not be
restricted to grouped data and, in fact, is encouraged to use ungrouped
data which should tend to increase the value of one's information.
Second, Chiang's and Kimball's crude and net probabilities fluctuate
from interval to interval, as can be seen in Tables 5.1.2 and 5.1.3,
depending upon how the intervals were chosen, whereas the parametric
method tends to "smooth out" these probabilities over the lifespan
of the individuals (this difference should become less marked as the
sample size is increased).
Third, the selection of the class of
underlying failure distributions is an important consideration in the
parametric method (although one may select a fairly general family,
~.~.,
the Weibull), whereas Kimball's method is entirely nonparametric
and Chiang's method, as discussed in section 1.2, is partially parametric and partially nonparametric.
5.1.2
Fitting Exponentials (Assuming Continuous Data)
Assuming that both failure distributions follow the negative
exponential law and utilizing all the data in Table 5.1.1, we obtain
the maximum likelihood estimates from (3.1.19)
..
e
Table 5.1.2.
e
"
Intervals
47.357163.1428
15.785731. 5714
31.571447.3571
e
20
24
12
8
4
1
2
121
97
Cancer present
Kimball & Chiang
(5.1.10)
Parametric (5.1.8)
.1653
.1818
Other causes
Kimball & Chiang
(5.1.10)
Parametric (5.1.8)
63.142878.9285
78.928594.7142
94,7142110.4999
2
5
3
1
2
2
2
72
58
49
45
38
.2474
.1460
.1667
.1347
.1379
.1278
.0408
.1229
.1111
.1191
.0789
.1161
.0331
.0280
.0103
.0312
.0278
.0324
.0172
.0332
.0408
.0338
.0444
.0342
.0526
.0346
Cancer present
Kimball (5.1.11)
Chiang (5.1.13)
Parametric (5.1.9)
.1709
.1683
.1852
.2500
.2488
.1484
.1714
.1692
.1370
.1404
.1392
.1301
.0426
.0417
.1251
.1163
.1138
.1213
.0833
.0812
.1182
Other causes
Kimball (5.1.12)
Chiang (5.1.13)
Parametric (5.1.9)
.0396
.0362
.0314
.0137
.0119
.0339
.0333
.0304
.0349
.0200
.0186
.0356
.0426
.0417
.0361
.0500
.0472
.0365
.0571
.0549
.0369
No. of deaths with cancer
present
No. of deaths from other causes
No. of survivors entering
interval
Net
Probabilities
.
Comparison of the parametric method proposed with Kimball's and Chiang's methods for
equi-spaced intervals
015.78"57
Crude
Probabil..:.
ities
.'
~
-
-
_. __. _ - - _ . _ -
._--
-
--
--
---
00
I-'
e
Table 5.1.3.
e
(
.
>
e
•
Comparison of the various methods when expected number of deaths within each interval
are approximately equal
07.75
7.7516.35
16.3526.03
Intervals
26.03- 37.0837.08
49.97
49.9765.42
65.4284.73
84.73110.50
No. of deaths with cancer present
8
14
16
9
11
6
4
6
No. of deaths from other causes
3
1
1
0
2
2
2
3
121
110
95
78
69
56
48
42
.0661
.1057
.1273
.0924
.1684
.0943
.1154
.1003
.1594
.1100
.1071
.1245
.0833
.1464
.1429
.1817
.0248
.0140
.0091
.0170
.0105 0.00
.0198 .0230
.0290
.0271
.0357
.0327
.0417
.0407
.0714
.0536
.0678
.0670
.1069
.1284
.1279
.0932
.1702
.1694
.0953
.1154
.1154
.1015
.1642
.1619
.1116
.1111
.1092
.1266
.0870
.0852
.1496
.1538
.1485
.1871
.0265
.0257
.0149
.0104
.0097
.0179
.0127 0.00
.0115 0.00
.0208 .0243
.0345
.0316
.0288
.0400
.0378
.0349
.0455
.0435
.0441
.0833
.0772
.0593
No. of survivors entering interval
Cancer present
Kimball & Chiang (5.1.10)
Crude
Pro b a b'l't'
l. l. l.es Parametric (5.1.8)
Other causes
Kimball & Chiang (5.1.10)
Parametric (5.1.8)
Cancer present
Kimball (5.1.11)
Chiang (5.1.13)
Net
Parametric (5.1.9)
Probabilities Oth er causes
Kimball (5.1.12)
Chiang (5.1.13)
Parametric (5.1.9)
00
N
83
6
6
1
2
= 102.9333 (8.58 years)
(5.1.14)
= 446.0444 (37.17 years).
If we assume a common censoring time of 110.5 months then the estimates
become, from (3.1.21),
6
6
1
2
= 92.5311 (7.71 years)
(5.1.15)
= 489.093 (40.76 years)
and the corresponding estimated asymptotic standard deviations are,
from (3.1. 22) ,
"
s.d. (6 )
1
"
.8778 years
(5.1.16)
"
s.d.(6 ) = 10.6677 years.
2
Of course, the estimated expected survival times for each
population are given by the estimates of 6
i
(i=1,2).
The estimates of
the median times associated with the 6. in (5.1.14) are 5.95 years,
1
25.76 years and in (5.1.15) 5.34 years, 28.25 years.
Crude and net probabilities for the various causes could be
obtained easily as in Tables 5.1.2 and 5.1.3.
5.1.3
Fitting Exponentials (Assuming Grouped Data)
It is assumed that the failure distributions of interest are as
in section 5.1.2, however, the data are assumed to be grouped.
logical grouping might be to select equi-spaced intervals.
A
For
illustrative purposes, the interval endpoints given in Table 5.1.2 are
used.
Of course, the choice of the group limits may vary and such
choices may influence the result to some extent.
In Boag's data the
censoring point common to all individuals is 110.5 months as stated
earlier in section 5.1.
84
From equation
(4.5.6) where c=l, we obtain the maximum likelihood
estimates as
6
6
= 92.6910 (7.72 years)
1
(5.1.17)
= 489.938 (40.83 years).
2
Using the crude estimates obtained by setting all observations in
an interval equal to the midpoint of the interval, viz., 6
6
1
= 93,
2 = 492, the Newton-Raphson iterative method converges to the estimates
(5.1.17) after only two iterations.
The corresponding estimated asymptotic standard deviations obtained
from (4.5.12) are
s.d.(6 ) = .8884 years
l
A
A
s.d.(6 )
2
(5.1.18)
10.7859 years
Using the interval endpoints given in Table 5.1.3 we obtain the
estimates
6
= 92.6023 (7.72 years)
1
(5.1.19)
A
8
= 489.469 (40.79 years)
2
and the corresponding estimated asymptotic standard deviations are
s.d.(8 ) = .8868 years
l
A
A
(5.1.20)
s.d.(8 ) = 10.7671 years.
2
5.2
Simulated Exponential Example
In the absence of experimental data that contains the exact value
of all lifetimes and associated causes of failure, some samples were
simulated where the theoretical lifetimes were assumed to be independently
exponentially distributed.
These samples are used to illustrate the
calculation of the estimates from (2.2.3) as well as the estimated
85
asymptotic variances from (2.2.10) and the estimated finite sample
variances from (2.2.15).
The samples are simulated from underlying exponential populations
whose parametric values were 6 =1, 6 =1.5, and 6 =2, corresponding
2
1
3
to
TI
l
=.46l5,
TI
=.3077, and
2
TI
=.2308, respectively.
3
The estimates of 6., along with the estimated variances are given
1
in Table 5.2.1 for samples of size n=20,40,60,80,lOO.
The number of
individual observations coming from the respective underlying populations,
viz., n , n , n , are also given.
l
2
3
Note that the estimate 6
than 6 ,
3
1
is better than 6
2
This is as it should be since n >n >n .
l 2 3
which in turn is better
Another trivial
observation is that the estimated finite sample variances of the
estimates approach the estimated asymptotic variances as the sample
size is increased.
e
e
Table 5.2.1.
•
e
Simulated examples with three underlying exponential populations where 8 =1,
1
8 =1.5, 8 =2 (or TI =.4615, TI =.3077, TI =.2308)
2
3
2
1
3
Estimates
Estimated asymptotic
variances of the est.
"
"
82
81
83
Estimated finite sample
variances of the est.
"*
Ell *
82
83 *
Sample size
n
8"
20
0.8807
2.8624
3.8165
0.0597
2.0484
4.8553
0.0709
6.2692
40
1.0122
1. 3236
1. 7207
0.0603
0.1348
0.2961
0.0752
0.1972
0.5638
60
0.8422
1. 5947
2.4645
0.0224
0.1496
0.5521
0.0244
0.1994
0.9883
80
1. 3356
1.4692
2.5927
0.0541
0.0720
0.3954
0.0602
0.0817
0.5457
100
1. 0433
1.5323
2.3350
0.0232
0.0734
0.2596
0.0247
0,0836
0.3337
Sample size
n
n
n
n
20
13
4
3
40
17
13
10
60
32
17
11
80
33
30
17
100
47
32
21
62
1
1
2
8"
3
A
A
10.379
3
00
0'\
87
CHAPTER 6
SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH
6.1
Summary
This dissertation presents a method for assessing statistically
the life characteristic of various types of individuals when the
failure of each individual can be attributed to one of many possible
causes.
Some knowledge of the underlying life distributions is assumed either
from past experience or by employing procedures designed to aid one
in choosing an appropriate class of underlying distributions,
~.~.,
Nelson's method.
In Chapter 2, a general likelihood function is obtained which
allows for dependence of the causes, although the multivariate
exponential model of section 2.2.4 is the only specific case involving
dependent causes for which explicit expressions have been obtained.
When the causes of failure are assumed to act independently, the
method of maximum likelihood is used to estimate the parameters of
some specific families of distributions, viz., the negative exponential,
Weibull, and normal.
.discussed.
Asymptotic properties of these estimators are
In addition, for underlying exponential distributions, we
may obtain small sample properties of conditional maximum likelihood
estimators (conditional upon at least one individual failing from
each cause), including the exact distribution of such estimators, and
we may also obtain the maximum likelihood estimator for the hazard
function and small sample properties of this estimator.
88
Situations involving censoring are considered in Chapter 3, both
Type I and Type II as defined in section 1.5.
The cases of completely
grouped observations and grouped observations with censoring are
investigated in Chapter 4.
General likelihood functions are obtained
in each case, along with the maximum likelihood estimators of parameters
for specific distributions.
are given.
Asymptotic properties of such estimators
For the simplest case, viz., underlying exponential distribu-
tions, the results are quite tractable.
Chapter 5 deals with applications.
The main example illustrates
the estimation procedure for Type I censoring.
In this study, two
basic causes of death were recorded and it is assumed that the underlying life distributions are two independent Weibu11 distributions with
different parameters.
From the estimates of these parameters, the
relevant probabilities in competing risk theory are obtained.
The Appendix contains a review of methods used to obtain inverse
moments of a positive binomial variate and an explanation of Nelson's
method of selecting an underlying distribution by hazard plotting.
6.2
Suggestions for Further Research
First, it would be interesting to note how this parametric
approach might compare with a nonparametric approach or semi-nonparametric approach, say when the underlying failure distributions have
monotone conditional failure rate.
Since one often has dependence of causes in actual physical
situations, it might be helpful to work out the maximum likelihood
estimation procedure for some specific multivariate distributions.
89
Such work would probably not lead to clean expressions,
i.~.,
math-
ematical approximation and/or numerical analyses on the electronic
computer would likely be necessary.
The integrals in the information matrix for the normal case are
fairly complicated and thus some simplification might be accomplished
with the aid of mathematical approximations.
Finally, it would be desirable to know more about the small
sample properties of the estimators in the Weibull and normal case.
Hence, a simulation study for these cases might be helpful.
90
CHAPTER 7
LIST OF REFERENCES
Atta, G. J. and A. W. Kimball. 1968. Monte Carlo investigation of a
model for competing risks. Journal of the National Cancer
Institute 40:525-534.
Barlow, Richard E. and Frank Proschan. 1965. Mathematical Theory of
Reliability. John Wiley and Sons, New York.
Bartholomew, D. J. 1963. The sampling distribution of an estimate
arising in life testing. Technometrics 5:361-374.
Berkson, Joseph and Lila E1veback. 1960. Competing exponential risks,
with particular reference to the study of smoking and lung cancer.
Journal of the American Statistical Association 55:415-428.
Berman, Simeon M. 1963. Note on extreme values, competing risks,
and semi-markov processes. Annals of Mathematical Statistics
34:1104-1106.
Boag, J. W. 1949. Maximum likelihood estimates of the proportion of
patients cured by cancer therapy. Journal of the Royal
Statistical Society Series B 11:15-53.
Buckland, W. R. 1964. Statistical Assessment of the Life Characteristic.
Hafner Publishing Company, New York.
Chiang, C. L. 1960a. A stochastic study of the life table and its
applications: I. Probability distributions of the biometrics
functions. Biometrics 16:618-635.
Chiang, C. L. 1960b. A stochastic study of the life table and its
applications: II. Sample variance of the observed expectation
of life and other biometric functions. Human Biology 32:221-238.
Chiang, C. L. 1961a. A stochastic study of the life table and its
applications: III. The follow-up study with the consideration
of competing risks. Biometrics 17:57-78.
Chiang, C. L. 1961b. On the probability of death from specific
causes in the presence of competing risks. Proceedings of the
Fourth Berkeley Symposium 169-180.
Chiang, C. L. 1968. Introduction to Stochastic Processes in
Biostatistics. John Wiley and Sons, New York.
Clark, Charles E. and G. Trevor Williams. 1961. Estimates from
censored samples. Institute of Statistical Mathematics
12:209-226.
91
Cornfield, Jerome. 1957. The estimation of the probability of
developing a disease in the presence of competing risks.
Journal of the American Public Health Association 47:601-607.
David, F. N. and N. L. Johnson. 1956-57.
Poisson variables. Metron 18:77-81.
Reciprocal Bernoulli and
David, H. A. 1957. Estimation of means of normal populations from
observed minima. Biometrika 44:282-286.
David, H. A. 1970. On Chiang's proportionality assumption in the
theory of competing risks. Biometrics 26:336-339.
Derkson, J. B. D. 1939. On some infinite series introduced by
Tschuprow. Annals of Mathematical Statistics 10:380-383.
Fix, Evelyn and Jerzy Neyman. 1951. A simple stochastic model of
recovery, relapse, death and loss of patients. Human Biology
23:205-241.
Govindarajulu, Zakkula. 1963. Recurrence relations for the inverse
moments of the positive binomial variable. Journal of the
American Statistical Association 58:468-473.
Govindarajulu, Zakkula. 1964. A supplement to Mendenhall's bibliography on life testing and related topics. Journal of American
Statistical Association 59:1231-1291.
Grab, Edwin L. and I. Richard Savage. 1954. Tables of the expected
value of l/X for positive Bernoulli and Poisson variables.
Journal of the American Statistical Association 49:169-177.
Graybill, Franklin A. 1969. Introduction to Matrices with Applications
in Statistics. Wadsworth Publishing Company, Belmont, California.
Gumbel, E. J. 1958. Statistics of Extremes.
Press, New York.
Columbia University
Halperin, Max. 1960. Extension of Wilcoxon-Mann-Whitney test to
samples censored at the same fixed point. Journal of the
American Statistical Association 55:125-138.
Hoem, Jan M. 1968. Estimation of forces of decrement with aggregated
lifetime as operational time. Unpublished. University of Oslo,
Oslo, Norway.
Kimball, A. W. 1957. Disease incidence estimation in populations
subject to multiple causes of death. Bulletin of the International Statistical Institute 36:193-204.
92
Kimball, A. w. 1969. Models for the estimation of competing risks
from grouped data. Biometrics 25:329-337.
Kodlin, D. 1961. Survival time analysis for treatment evaluation
in cancer therapy. Cancer Research 21:1103-1107.
Kulldorff, Gunnar. 1961. Estimation from Grouped and Bartially
Grouped Samples. John Wiley and Sons, New York.
Makeham, W. M. 1874. On an application of the theory of the
composition of decremental forces. Journal of the Institute
of Actuaries 18:317-322.
Marshall, Albert W. and Ingram Olkin. 1967. A multivariate exponential distribution. Journal of the American Statistical
Association 62:30-44.
Mendenhall, W. 1958. A bibliography on life testing and related
topics. Biometrika 45:521-543.
Mendenhall, W. and E. H. Lehman, Jr. 1960. An approximation to the
negative moments of the positive binomial useful in life testing.
Technometrics 2:227-242.
Nelson, Wayne. 1969. Hazard plotting for incomplete failure data.
Journal of Quality Technology 1:27-52.
Neyman, J. 1950. First Course in Probability and Statistics.
Henry Holt and Company, New York.
Rao, C. R. 1965. Linear Statistical Inference and Its Applications.
John Wiley and Sons, New York.
Sampford, M. R. 1952a. The estimation of response-time distributions.
I. Fundamental concepts and general methods. Biometrics
8:13-32.
Sampford, M. R. 1952b. The estimation of response-time distributions.
II. Multi-stimulus distributions. Biometrics 8:307-369.
Sampford, M. R. 1954. The estimation of response-time distributions.
III. Truncation and survival. Biometrics 10:531-561.
Sethuraman, J. 1965. On a characterization of the three limiting
types of the extreme. Sankhya Series A 27:357-364.
Sheps, Mindel C. 1965. Applications of Probability Models to the
Study of Patterns of Human Reproduction. Public Health and
Population Change, University of Pittsburgh Press, Pittsburgh.
93
Slutsky, E. 1925. Uber stochastische Asymptoten und Grenzwerte.
Metron 4:1-89.
Stephan, F. F. 1946. The expected value and variance of the
reciprocal and other negative powers of a positive Bernou11ion
variate. Annals of Mathematical Statistics 16:50-61.
Swan, A. V. 1969. Computing maximum likelihood estimates for
parameters of the normal d'istribution from grouped and censored
data. Applied Statistics 18:65-69.
Todhunter, I. 1865. A History of the Mathematical Theory of
Probability. Chelsea, New York.
94
CHAPTER 8
APPENDIX
8.1
Review of Methods Which Obtain the Inverse Moments
of the Positive Binomial Variate
Grab and Savage (1954) have exact values (to five decimal places)
of E(l/N *), where N * is positive binomial variate, for
i
i
n = 2(1)20
'IT 1 = .01, .05(.05) .95, .99
n = 21(1)30
'IT ,
0
,01, .05 ( •05) .50
1
They also suggest several approximations for E(l/N.*) which
1
usually give at least two place accuracy for large values of n'IT •
i
Mendenhall and Lehman (1960) give exact values (to five decimal
r
places) of E(l/N * ) for
i
r = 1,2,3,4
n = 5,10,15,20,30,40
'IT. = .05(.05).95 .
1
They also give an approximation which is claimed to be slightly
more accurate than Grab and Savage's for n'IT >5.
o
1
Their method is
based upon approximating the probability distribution function of N
i
with a Beta distribution by equating the first two moments of the
two distributions.
This technique yields
E(l/N *)
o
1
::<
n-2
n(a-l)
(n-2) (n-3)
2
, where a=(n-l)'IT
n (a-l)(a-2)
i
which is correct to two decimal places for n'IT.>5
and in several
1instances is correct to six decimal places.
Stephan (1945) gives methods for calculating E(l/N.* r ) to any
1
degree of accuracy by expanding l/N
i
*
in a series of inverse factorials.
His approximations are advantageous for large values of n'IT .
i
95
Clark and Williams (1961) give a method for obtaining E(l/Ni*r),
r=1,2.
If
TIi~O.8,
then they give expansions in powers of l-TI .
i
TI <O.8, their expansions are in terms of x=nTI , W(x), and
i
i
W(x) and
~(x)
are tabulated for x=O(.l)lOO.
~(x),
If
where
Although these tables are
accurate to six significant figures, the computed value of E(l/N
i
*r )
may be considerably less accurate.
8.2
Nelson's Method of Selecting an Underlying Life
Distribution by Hazard Plotting
The intensity (hazard) function of a distribution is defined in
(1.1.1).
Now the cumulative hazard function (c.h.f.) for cause j is
x
Uj(x)
=f_~
(8.2.1)
rj(t)dt
or
(8.2.2)
H.(x)= -log[l-P.(x)]
J
J
This c.h.f.plays the same role in hazard plotting as the c.d.f.
does in probability plotting in that the grid on which one plots the
c.d.f. or c.h.f., respectively, versus the random variable of interest
is chosen so that if the random variable is from a particular class
of distributions, then the plotted points will be a straight line.
In
probability plotting, for a sample of n failures, the increase in the
sample c.d.f. at a particular failure time equals the probability of
the failure, lIn, and the sample c.d.f. based on the sum of these
increments approximates the theoretical c.d.f.
For hazard plotting,
the increase in the sample c.h.f.at a failure time equals the
conditional probability of the failure,
l/n~,
where
n~
is the number of
individuals in the study at the time of the failure, including the
individual that failed.
Then the sample c.h.f., based on the sum
96
of the calculated increments associated with each failure time,
approximates the theoretical coh.f.
The grid upon which one plots the
c.h.f. versus times to failure depends upon the different classes of
distributions.
Three points make this method particularly appropriate to the
problem considered in this thesis.
l.~.,
First, the data need not be complete,
some of the observations may be censored.
Furthermore, the
censoring times may be different for each individual.
Second, if some
individuals fail from some cause other than the one of interest, then
the times to failure of such individuals may be regarded as censoring
times for the particular cause of interest and the sample c.h.f. corresponding to this primary cause may be obtained.
This method is based
on the assumption that the censoring time associated with an individual
is statistically independent of the time to failure.
Thus one is only
justified in employing Nelson's procedure in the competing causes of
failure situation for statistically independent times to failure.
Third, in addition to providing a means of judging the worth of a
particular class of distributions for a situation, it is possible to
obtain crude estimates of the parameters of the distributions.
This
point will be taken up further when looking at specific distributions.
8.2.1
Exponential Distribution
If the time to failure from cause j is exponential, then, from
(8.2.2) we have
H,(x)
J
=
x/e,
J
(8.2.3)
97
Thus the time to failure x as a function of the c.h.f. H. is
J
x(H. ) = ejH
j
J
(8.2.4)
Thus time to failure x plots as straight line function of H. on
J
square grid graph paper.
Furthermore, e
which H =l, hence a crude estimate of 8
j
j
j
is the value of the time for
may be obtained from the
straight line graph.
8.2.2
Weibu11 Distribution
If the time to failure from cause j has the p.d.f. (2.3.1), then
from {8.2.2) we obtain
(8.2.5)
or
x(H. ) = (8 j Hj )(l/c.)
J
J
1
log x = (-)log H. + (1:...) log e.
cj
c.
J
J
J
or
(8.2.6)
Thus x, the time to failure, plots as a straight line function
of H on log-log graph paper.
j
l/c
j
The slope of the straight line is
and for Hj=l, the corresponding time x equals 8 (l/c j ).
j
These
facts are used to obtain, first, a crude estimate of c. and then an
J
estimate of e .•
J
8.2.3
Applying Nelson's Method to Boag's Data
In the example of section 5.1, Nelson's method is applied with
respect to underlying exponential and Weibull distributions for each
cause of failure.
Table 8.2.1 restates Boag's Data and gives the
sample cumulative hazard function corresponding to each time of
failure for each cause.
From these calculations, Figures 8.2.1, 8.2.2,
98
8.2.3, and 8.2.4 have been obtained by plotting the times to failure
for a specified cause versus the sample c.h.f.
FollQwing the suggestions as stated in sections 8.2.1 and 8.2.2,
crude estimates of the underlying population parameters may be obtained.
From Figures 8.2.1 and 8.2.3, respectively, we find
6
1
62
=
85.5
(8.2.7)
= 439
and, from Figures 8.2.2 and 8.2.4, respectively, we obtain
C
c
l
=
2
= 1.07
.97,
6
1
, 62
= 81
= 560
(8.2.8)
These estimates may be used as starting values for an iterative
technique which is needed to obtain the maximum likelihood estimates
put forth in this thesis •
•
99
Table 802.1.
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
(121)
(120)
(119)
(118)
(117)
(116)
(115)
(114)
(113)
(112)
(111)
(110)
(109)
(108)
(107)
(106)
(105)
(104)
(103)
(102)
(101)
(100)
(99)
(98)
(97)
(96)
(95)
(94)
(93)
(92)
(91)
(90)
(89)
(88)
(87)
(86)
(85)
(84)
(83)
(82)
(81)
(80)
(79)
(78)
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 (77)
46 (76)
Boag's data and hazard calculations
Ordered survival
and follow-up
a
times in months
0.3
b
0.3
b
4.0'
5.0
5.6
6.2
6.3
6.6
6.8
b
7.4
7.5
8.4
8.4
10.3
11.0
11.8
12.2
12.3
13.5
14.4
14.4
14.8
b
15.5
15.7
16.2
16.3
16.5
16.8
17.2
17.3
17.5
17.9
19.8
20.4
20.9
21.0
21.0
21.1
23.0
b
23.4
23.6
24.0
24.0
27.9
28.2
29.1
Hazard
.8264
.8333
.8403
.8475
.8547
.8621
.8696
.8772
.8850
.8929
.9009
.9091
.9174
.9259
.9346
.9434
.9524
.9615
.9709
.9804
.9901
1.000
1.010
1.020
1. 031
1.042
1.053
1.064
1.075
1.087
1.099
1.111
1.124
1.136
1.150
1.163
1.177
1.191
1.205
1.220
1.235
1.250
1.266
1.282
1.299
1.316
Cumulative
Cumulative
hazard (cancer'
hazard
(cancer present) not present)
.826
.833
1.67
1.67
2.53
3.39
4.26
5.14
6.02
2.57
6.92
7.83
8.75
9.68
10.6
11.6
12.5
13.5
14.4
15.4
16.4
17.4
3.58
18.4
19.5
20.5
21.6
22.6
23.7
24.8
25.9
27.0
28.1
29.3
30.4
31. 6
32.7
33.9
35.1
4.80
36.4
37.6
38.9
40.2
41.5
42.8
100
Table 8.2.1 (continued)
Number
•
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
(75)
(74)
(73)
(72)
(71)
(70)
(69)
(68)
(67)
(66)
(65)
(64)
(63)
(62)
(61)
(60)
(59)
(58)
(57)
(56)
(55)
(54)
(53)
(52)
(51)
(50)
(49)
(48)
(47)
(46)
(45)
(44)
(43)
(42)
(41)
(40)
(39)
(38)
(37)
(36)
(35)
(34)
(33)
(32)
(31)
Ordered survival
and follow-up
times in months
30.0
31.0
31
32
35
35
38
39
40
40
41
41
42
44
46 b
46
b
46
48
48
51
51 b
51
52
54
56
60
b
65
b
68
78
78
80
b
83
84
87
b
88
89
90b
96
97
98
100
110b
c
Hl
b
lll
b
ll2
Hazard
1.333
1.351
1.370
1.389
1.409
1.429
1.449
1.471
1.493
1.515
1.539
1.563
1.587
1.613
1.639
1.667
1.695
1. 724
1. 754
1. 786
1.818
1.852
1. 887
1.923
1.961
2.000
2.041
2.083
2.128
2.174
2.222
2.273
2.326
2.381
2.439
2.500
2.564
2.632
2.703
2.778
2.857
2.941
3.030
3.125
3.226
Cumulative
Cumulative
hazard
hazard (cancer
(cancer present) not present)
44.1
45.5
46.8
48.2
49.6
51.1
52.5
54.0
55.5
57.0
58.5
60.1
61.7
63.3
6.44
65.0
8.13
66.7
68.4
70.2
9.95
72.1
74.0
75.9
77 .9
79.9
12.0
14.1
82.0
84.2
86.4
16.3
88.7
91.1
18.8
93.6
96.1
21.4
98.8
102
104
24.4
27.5
30.7
101
Table 8.2.1 (continued)
Number
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
(30)
(29)
(28)
(27)
(26)
(25)
(24)
(23)
(22)
(21)
(20)
(19)
(18)
(17)
(16)
(15)
(14)
(13)
(12)
(11)
(10)
(9)
(8)
(7)
(6)
(5)
(4)
(3)
(2)
(1)
Ordered survival
and follow-up
times in months
112 c
113 c
c
114
114
c
114
117 c
121 c
123 c
126
c
129
131
c
131
b
132
c
133
c
134
134 c
136 c
c
141
c
143
162b
c
167
174
177 c
c
179
189 c
201 c
203 c
c
203
c
213
228 c
Hazard
3.333
3.448
3.571
3.704
3.846
4.000
4.167
4.348
4.546
4.762
5.000
5.263
5.556
5.882
6.250
6.667
7.143
7.692
8.333
9.091
10.00
11.11
12.50
14.29
16.67
20.00
25.00
33.33
50.00
100.00
Cumulative
Cumulative
hazard
hazard (cancer
(cancer present) not present)
108
113
118
36.3
45.4
129
aUnfootnoted numbers denote survival times of patients who died
with cancer present.
bSurviva1 times of patients who died from other causes.
cFo11ow-up times of patients who were alive at the conclusions of
the study.
102
140
•
130
•
120
•
110
.-..
.- •
til
.c: 100
+J
s::0
s
'-'
90
Q)
l-l
::J
.-l
•.-1
CIl
4-l
•• •
80
•
• ••
70
0
+J
til
Q)
60
.~
E-l
50
.
'
..
'
, ,-
40
30
•
..
'
.
••
• ••
•• •
••
•
20
10
0
Figure 8.2.1.
10
20
30
40
50
60
80 90
70
(Cumulative hazard (%))
100
110
120
Exponential hazard plot of Boag's data (cancer present)
103
300
200
••
100
70
"'"'
to
..c:
50
s::
40
+J
o
-8
.(1)
.."
30
~
::l
OM
...-l
III
.......
20
4-1
o
r
....
•
J
•
...' ••
I
.,•
+J
to
••
(1)
o~
10
E-l
••
7
• • •
• •
5
4
3
2
•
1
2
Figure 8.2.2.
3
4
5
7
10
20
30 40 50
(Cumulative hazard (%»
70
100
Weibu11 hazard plot of Boag's data (cancer present)
150
104
"
500
400
Q)
~
..-l
300
'M
ell
4-l
a
+J
{/)
Q)
~
E-l
200
180
160
140
120
100
80
60
40
20
o
Figure 8.2.3.
10
20
30
40
50
60 70
80
(Cumulative hazard (%))
90
100
Exponential hazard plot of Boag's data (cancer not present)
105
400
300
200
100
""'
Ul
.c
.j.I
70
l:l
50
s
40
0
'-"
III
,...
::J
30
.-l
OM
til
4-l
20
0
.j.I
Ul
III
o~
E-t
10
7
5
4
3
A
2
1
2
Figure 8.2.4.
3
4 5
7
10
20
30 40
60
(Cumulative hazard (%»
100
150
Weibull hazard plot of Boag's data (cancer not present)