Statistical Methods for Survival Data Analysis 3rd phần 4 pot
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Figure 6.11 Lognormal probability plot of the survival time of 234 male patients with
chronic lymphocytic leukemia. (From Feinleib and MacMahon, 1960. Reproduced by
permission of the publisher.)
Example 6.3 In a study of chronic lymphocytic and myelocytic leukemia,
Feinleib and MacMahon (1960) applied the lognormal distribution to analyze
survival data of 649 white residents of Brooklyn diagnosed from 1943 to 1952.
The analysis of several subgroups of patients follows. The survival time of each
patient is computed from the date of diagnosis in months. Analytical method
is used to fit the lognormal distribution to the data. The method is discussed
in Chapters 7 and 8.
Figure 6.11 gives the probability plot of the survival time of 234 male
patients with chronic lymphocytic leukemia, in which the horizontal axis for
the survival time is in logarithmic scale and the vertical axis is in normal
probability scale. When plotting 1 9 S(t) on this graph paper, a straight line is
obtained when the data follow a two-parameter lognormal distribution. An
inspection of the graph shows that the distribution is concave. Gaddum
(1945a, b) has pointed out that such a deviation can be corrected by subtracting
an appropriate constant from the survival times. In other words, the three-
parameter lognormal distribution can be used. Figure 6.12 gives a similar plot
in which the survival time of every patient plus 4 is plotted. The configuration
is linear and hence empirically it seems valid to assume that the lognormal
distribution is appropriate.
Similar graphs for male patients with chronic myelocytic leukemia and for
female patients with chronic lymphocytic or myelocytic leukemia are given in
Figures 6.13 and 6.14. Parameters of the lognormal distribution are estimated.
Feinleib and MacMahon report that the agreement between the observed and
calculated distributions is striking for each group except for women with
chronic lymphocytic leukemia. The corresponding p values for the chi-square
147
Figure 6.12 Lognormal probability plot of the survival time in months plus 4 of 234
male patients with chronic lymphocytic leukemia. (From Feinleib and MacMahon,
1960. Reproduced by permission of the publisher.)
goodness-of-fit test are as follows:
Chronic Myelocytic Chronic Lymphocytic
Male 0.86 0.73
Female 0.57 0.016
Since a large p value indicates close agreement, it is concluded that the
three-parameter lognormal distribution adequately describes the distribution
of survival times for each subgroup except women with chronic lymphocytic
leukemia. The shape of the observed distribution for the latter group suggests
that it might actually be composed of two dissimilar groups, each of whose
survival times might fit a lognormal distribution.
6.4 GAMMA AND GENERALIZED GAMMA DISTRIBUTIONS
The gamma distribution, which includes the exponential and chi-square
distribution, was used a long time ago by Brown and Flood (1947) to describe
the life of glass tumblers circulating in a cafeteria and by Birnbaum and
Saunders (1958) as a statistical model for life length of materials. Since then,
this distribution has been used frequently as a model for industrial reliability
problems and human survival.
148 -
Figure 6.13 Lognormal probability plot of the survival time in months plus 4 of 162
male patients with chronic myelocytic leukemia. (From Feinleib and MacMahon, 1960.
Reproduced by permission of the publisher.)
Figure 6.14 Lognormal probability plot of the survival time in months plus 4 of female
patients with two types of leukemia. (From Feinleib and MacMahon, 1960. Reproduced
by permission of the publisher.)
Suppose that failure or death takes place in n stages or as soon as n
subfailures have happened. At the end of the first stage, after time T
, the first
subfailure occurs; after that the second stage begins and the second subfailure
occurs after time T
; and so on. Total failure or death occurs at the end of the
nth stage, when the nth subfailure happens. The survival time, T, is then
T
; T
; % ; T
L
. The times T
, T
, , T
L
spent in each stage are assumed to
149
Figure 6.15 Gamma hazard functions with : 1.
be independently exponentially distributed with probability density function
exp(9t
G
), i : 1, , n. That is, the subfailures occur independently at a
constant rate . The distribution of T is then called the Erlangian distribution.
There is no need for the stages to have physical significance since we can
always assume that death occurs in the n-stage process just described. This
idea, introduced by A. K. Erlang in his study of congestion in telephone
systems, has been used widely in queuing theory and life processes.
A natural generalization of the Erlangian distribution is to replace the
parameter n restricted to the integers 1, 2, . . . by a parameter taking any real
positive value. We then obtain the gamma distribution.
The gamma distribution is characterized by two parameters, and . When
0 ::1, there is negative aging and the hazard rate decreases monotonically
from infinity to as time increases from 0 to infinity. When 91, there is
positive aging and the hazard rate increases monotonically from 0 to as time
increases from 0 to infinity. When : 1, the hazard rate equals , a constant,
as in the exponential case. Figure 6.15 illustrates the gamma hazard function
for : 1 and :1, : 1, 2, 4. Thus, the gamma distribution describes a
different type of survival pattern where the hazard rate is decreasing or
increasing to a constant value as time approaches infinity.
The probability density function of a gamma distribution is
f (t) :
()
(t)A\e\HR t 9 0, 90, 90 (6.4.1)
where () is defined as in (6.2.9). Figures 6.16 and 6.17 show the gamma
density function with various values of and . It is seen that varying changes
the shape of the distribution while varying changes only the scaling.
Consequently, and are shape and scale parameters, respectively. When
91, there is a single peak at t : ( 9 1)/.
150 -
Figure 6.16 Gamma density functions with : 1.
Figure 6.17 Gamma density functions with : 3.
The cumulative distribution function F(t) has a more complex form:
F(t) :
R
()
(x)A\e\HV dx (6.4.2)
:
1
()
HR
uA\e\S du
: I(t, )(6.4.3)
where
I(s, ) :
1
()
Q
uA\e\S du (6.4.4)
known as the incomplete gamma function, is tabulated in Pearson (1922, 1957).
151
For the Erlangian distribution, it can be shown that
F(t) : 1 9
L\
I
e\HR(t)I
k!
(6.4.5)
Thus, the survivorship function 1 9 F(t)is
S(t) :
R
()
(x)A\e\HV dx (6.4.6)
for the gamma distribution or
S(t) : e\R
L\
I
(t)I
k!
(6.4.7)
for the Erlangian distribution.
Since the hazard function is the ratio of f (t)toS(t), it can be calculated from
(6.4.1) and (6.4.7). When is an integer n,
h(t) :
(t)L\
(n 9 1)!
L\
I
(1/k!)(t)I
(6.4.8)
When : 1, the distribution is exponential. When :
and :
, where
is an integer, the distribution is chi-square with degrees of freedom. The mean
and variance of the standard gamma distribution are, respectively, / and /,
so that the coefficient of variation is 1/(.
Many survival distributions can be represented, at least roughly, by suitable
choice of the parameters and . In many cases, there is an advantage in using
the Erlangian distribution, that is, in taking integer.
The exponential, Weibull, lognormal, and gamma distributions are special
cases of a generalized gamma distribution with three parameters, , , and ,
whose density function is defined as
f (t) :
?A
()
t?A\ exp [9(t)?] t 9 0, 90, 90, 90(6.4.9)
It is easily seen that this generalized gamma distribution is the exponential
distribution if : : 1, the Weibull distribution if : 1; the lognormal
distribution if ; -, and the gamma distribution if : 1.
In later chapters (e.g., Chapters 7 and 9), we discuss several parametric
procedures for estimation and hypothesis testing. To use available computer
software such as SAS to carry out the computation, we use the distributions
adopted by the software. One of the very few software packages that include
the gamma or generalized gamma distribution is SAS. In SAS, the generalized
152 -
Table 6.4 Lifetimes of 101 Strips of Aluminum Coupon
370
706
716
746
785
797
844
855
858
886
886
930
960
988
990
1000
1010
1016
1018
1020
1055
1085
1102
1102
1108
1115
1120
1134
1140
1199
1200
1200
1203
1222
1235
1238
1252
1258
1262
1269
1270
1290
1293
1300
1310
1313
1315
1330
1355
1390
1416
1419
1420
1420
1450
1452
1475
1478
1481
1485
1502
1505
1513
1522
1522
1530
1540
1560
1567
1578
1594
1602
1604
1608
1630
1642
1674
1730
1750
1750
1763
1768
1781
1782
1792
1820
1868
1881
1890
1893
1895
1910
1923
1940
1945
2023
2100
2130
2215
2268
2440
Source: Birnbaum and Saunders (1958).
gamma distribution is defined as having the following density function:
f (t) :
""A?A
()
t?A\ exp [9(t)?], t 9 0, 90, 90(6.4.10)
To differentiate this form of the generalized gamma distribution from the
generalized gamma in (6.4.9), we refer to this distribution as the extended
generalized gamma distribution. It can be shown that the extended generalized
gamma distribution reduces to the Weibull distribution when 90 and : 1,
the lognormal distribution when ; -, the gamma distribution when : 1,
and the exponential distribution when : : 1.
Example 6.4 Birnbaum and Saunders (1958) report an application of the
gamma distribution to the lifetime of aluminum coupon. In their study, 17 sets
of six strips were placed in a specially designed machine. Periodic loading was
applied to the strips with a frequency of 18 hertz and a maximum stress of
21,000 pounds per square inch. The 102 strips were run until all of them failed.
One of the 102 strips tested had to be discarded for an extraneous reason,
yielding 101 observations. The lifetime data are given in Table 6.4 in ascending
order. From the data the two parameters of the gamma distribution were
153
Figure 6.18 Graphical comparison of observed and fitted cumulative distribution
functions of data in Example 6.4. (From Birnbaum and Saunders, 1958.)
estimated (estimation methods are discussed in Chapter 7). They obtained
: 11.8 and : 1/(118.76;10).
A graphical comparison of the observed and fitted cumulative distribution
function is given in Figure 6.18, which shows very good agreement. A
chi-square goodness-of-fit test (discussed in Chapter 9) yielded a value of
4.49 for 6 degrees of freedom, corresponding to a significance level between 0.5
and 0.6. Thus, it was concluded that the gamma distribution was an adequate
model for the life length of some materials.
6.5 LOG-LOGISTIC DISTRIBUTION
The survival time T has a log-logistic distribution if log(T ) has a logistic
distribution. The density, survivorship, hazard, and cumulative hazard func-
tions of the log-logistic distribution are, respectively,
f (t) :
tA\
(1 ; tA)
(6.5.1)
S(t) :
1
1 ; tA
(6.5.2)
154 -
h(t) :
tA\
1 ; tA
(6.5.3)
H(t) : log(1 ; tA)(6.5.4)
t . 0, 90, 90
The log-logistic distribution is characterized by two parameters , and . The
median of the log-logistic distribution is \A. Figure 6.19(a) to (c) show the
log-logistic hazard, density, and survivorship functions with : 1 and various
values of : 2.0, 1, and 0.67.
When 91, the log-logistic hazard has the value 0 at time 0, increases to a
peak at t : ( 9 1)A/A, and then declines, which is similar to the lognormal
hazard. When : 1, the hazard starts at A and then declines monotonically.
When :1, the hazard starts at infinity and then declines, which is similar to
the Weibull distribution. The hazard function declines toward 0 as t ap-
proaches infinity. Thus, the log-logistic distribution may be used to describe a
first increasing and then decreasing hazard or a monotonically decreasing
hazard.
Example 6.5 Byers et al. (1988) used the log-logistic distribution to
describe the rate of spread of HIV between 1978 and 1986. Between 1978 and
1980, over 6700 homosexual and bisexual men in San Francisco were enrolled
in studies of the prevalence and incidence of sexually transmitted hepatitis B
virus (HBV) infections. Blood specimens were collected from the participants.
Four hundred and eighty-eight men who were HBV-seronegative were ran-
domly selected to participate in a study of HIV infection later. These men
agreed to allow the investigators to test the specimens collected previously
together with a current specimen. For those who convert to positive, the
infection time is only known to have occurred between the previous negative
test and the time of the first positive one. The exact time is unknown. The time
to infection is therefore interval censored. The investigators tried to fit several
distributions to the interval-censored data, including the Weibull and log-
logistic by maximum likelihood methods (discussed in Chapter 7). Based on
the Akaike information criterion (discussed in Chapter 9), the log-logistic
distribution was found to provide the best fit to the data. The maximum
likelihood estimates of the two parameters are : 0.003757 and : 1.424328.
Based on the log-logistic model, the median infection time is estimated to be
50.4 months, and the hazard function approaches its peak at 27.6 months.
6.6 OTHER SURVIVAL DISTRIBUTIONS
Many other distributions can be used as models of survival time, three of which
we discuss briefly in this section: the linear exponential, the Gompertz (1825),
155
(a)
(b)
(c)
Figure 6.19 (a) Hazard function of the log-logistic distribution; (b) density function of
the log-logistic distribution; (c) Survivorship function of the log-logistic distribution.
156
Figure 6.20 Hazard function of linear-exponential model.
and a distribution whose hazard rate is a step function. The linear-exponential
model and the Gompertz distribution are extensions of the exponential
distribution. Both describe survival patterns that have a constant initial hazard
rate. The hazard rate varies as a linear function of time or age in the
linear-exponential model and as an exponential function of time or age in the
Gompertz distribution.
In demonstrating the use of the linear-exponential model, Broadbent (1958),
uses as an example the service of milk bottles that are filled in a dairy,
circulated to customers, and returned empty to the dairy. The model was also
used by Carbone et al. (1967) to describe the survival pattern of patients with
plasmacytic myeloma. The hazard function of the linear-exponential distribu-
tion is
h(t) : ; t (6.6.1)
where and can be values such that h(t) is nonnegative. The hazard rate
increases from with time if 90, decreases if :0, and remains constant (an
exponential case) if : 0, as depicted in Figure 6.20.
The probability density function and the survivorship function are, respec-
tively,
f (t) : ( ; t) exp[9(t ;
t)] (6.6.2)
and
S(t) : exp[9(t ;
t)] (6.6.3)
The mean of the linear-exponential distribution is 9(/) ; (/2)\L (/2),
where
L (x) : eV
V
y e\W dy
157
Table 6.5 Values of L(x) and G(x)
xL(x) G(x)
00.886 -
0.10.951 2.015
0.21.012 1.493
0.31.067 1.223
0.41.119 1.048
0.51.168 0.923
0.61.214 0.828
0.71.258 0.753
0.81.300 0.691
0.91.341 0.640
11.381 0.596
21.712 0.361
31.987 0.262
Source: Broadbent (1958).
Figure 6.21 Gompertz hazard function.
is tabulated in Table 6.5. A special case of the linear-exponential distribution,
the Rayleigh distribution, is obtained by replacing by
(Kodlin, 1967). That
is, the hazard function of the Rayleigh distribution is h(t) : ;
t.
The Gompertz distribution is also characterized by two parameters, and
. The hazard function,
h(t) : exp( ; t)(6.6.4)
is plotted in Figure 6.21. When 90, there is positive aging starting from eH;
when :0, there is negative aging; and when : 0, h(t) reduces to a constant,
eH. The survivorship function of the Gompertz distribution is
S(t) : exp
9
eH
(eAR 9 1)
(6.6.5)
158 -
Figure 6.22 Step hazard function.
and the probability density function, from (6.6.4) and (2.2.5), is then
f (t) : exp
( ; t) 9
1
(eH>AR 9 eH)
(6.6.6)
The mean of the Gompertz distribution is G(eH/)/eH, where
G(x) : eV
V
y\e\W dy
is tabulated in Table 6.5.
Finally, we consider a distribution where the hazard rate is a step function:
h(t) :
a
a
$
a
I\
a
I
0 - t : t
t
- t : t
t
I\
- t : t
I\
t . t
I\
(6.6.7)
where t
, t
, , t
I
are different time points. Figure 6.22 shows a typical hazard
function of this nature for k : 5. Using (2.2.4), the survivorship function can
be derived:
S(t) :
exp(9a
t)0- t : t
exp[9a
t
9 a
(t 9 t
)] t
- t : t
$
exp[9a
t
9 a
(t
9 t
) 9 %9 a
I
(t 9 t
I\
) t . t
I\
(6.6.8)
159
The probability density function f (t) can then be obtained from (6.6.7) and
(6.6.8) using (2.2.5):
f (t) :
a
exp(9a
t)0- t : t
a
exp[9a
t
9 a
(t 9 t
)] t
- t : t
$
a
I
exp[9a
t
9 a
(t
9 t
) 9 %9 a
I
(t 9 t
I\
) t . t
I\
(6.6.9)
One application of this distribution is the life-table analysis discussed in
Chapter 4. In a life-table analysis, time is divided into intervals and the hazard
rate is assumed to be constant in each interval. However, the overall hazard
rate is not necessarily constant.
The nine distributions described above are, among others, reasonable
models for survival time distribution. All have been designed by considering a
biological failure, a death process, or an aging property. They may or may not
be appropriate for many practical situations, but the objective here is to
illustrate the various possible techniques, assumptions, and arguments that can
be used to choose the most appropriate model. If none of these distributions
fits the data, investigators might have to derive an original model to suit the
particular data, perhaps by using some of the ideas presented here.
Bibliographical Remarks
In addition to the papers on the distributions cited in this chapter, Mann et al.
(1974), Hahn and Shapiro (1967), Johnson and Kotz (1970a, b), Elandt-
Johnson and Johnson (1980), Lawless (1982), Nelson (1982), Cox and Oakes
(1984), Gertsbakh (1989), and Klein and Moeschberger (1997) also discuss
statistical failure models, including the exponential, Weibull, gamma, lognor-
mal, generalized gamma, and log-logistic distributions. Applications of survival
distributions can be found easily in medical and epidemiological journals. The
following are a few examples: Dharmalingam et al. (2000), Riffenburgh and
Johnstone (2001), and Mafart et al. (2002).
EXERCISES
6.1 Summarize the distributions discussed in this chapter, answering the
following questions.
(a) What distributions describe constant hazard rates? Give the range of
parameter values.
(b) What distributions describe increasing hazard rates? If there are more
than one, discuss the differences between them.
(c) What distributions describe decreasing hazard rates? If there are more
than one, discuss the differences between them.
160 -
6.2 Suppose that the survival distribution of a group of patients follows the
exponential distribution with G : 0 (year), : 0.65. Plot the survivor-
ship function and find:
(a) The mean survival time
(b) The median survival time
(c) The probability of surviving 1.5 years or more
6.3 Suppose that the survival distribution of a group of patients follows the
exponential distribution with G : 5 (years) and : 0.25. Plot the surviv-
orship function and find:
(a) The mean survival time
(b) The median survival time
(c) The probability of surviving 6 years or more
6.4 Consider the following two Weibull distributions as survival models:
(i) G : 0, : 1, : 0.5
(ii) G : 0, : 0.5, : 2
For each distribution, plot the survivorship function and the hazard
function and find:
(a) The mean
(b) The variance
(c) The coefficient of variation
Which distribution gives the larger probability of surviving at least 3 units
of time?
6.5 Suppose that the survival time follows the lognormal distribution with
: 1 and : 0.5. Find:
(a) The mean survival time
(b) The variance
(c) The coefficient of variation
(d) The median
(e) The mode
6.6 Suppose that pain relief time follows the gamma distribution with : 1,
: 0.5. Find:
(a) The mean
(b) The variance
(c) The coefficient of variation
6.7 Suppose that the survival distribution is (1) Gompertz and (2) linear-
exponential, and : 1, : 2.0. Plot the hazard function and find:
(a) The mean
(b) The probability of surviving longer than 1 unit of time
6.8 Consider the survival times of hypernephroma patients given in Exercise
Table 3.1. From the plot you obtained in Exercise 4.5, suggest a
distribution that might fit the data.
161
CHAPTER 7
Estimation Procedures for
Parametric Survival Distributions
without Covariates
In this chapter we discuss some analytical procedures for estimating the most
commonly used survival distributions discussed in Chapter 6. We introduce the
maximum likelihood estimates (MLEs) of the parameters of these distributions.
The general asymptotic likelihood inference results that are most widely used
for these distributions are given in Section 7.1. We begin to used the general
symbol b : (b
, b
, , b
N
) to denote a set of parameters. For example, in
discussing the Weibull distribution, b
could be and b
could be , and p : 2.
b is called a vector in linear algebra. Readers who are not familiar with linear
algebra or are not interested in the mathematical details may skip this section
and proceed to Section 7.2 without loss of continuity. In Sections 7.2 to 7.7 we
introduce the MLEs for the parameters of the exponential, Weibull, lognormal,
gamma, log-logistic, and Gompertz distributions for data with and without
censored observations. The related BMDP or SAS programming codes that
may be used to obtain the MLE are given in the respective sections.
7.1 GENERAL MAXIMUM LIKELIHOOD ESTIMATION
PROCEDURE
7.1.1 Estimation Procedures for Data with Right-Censored Observations
Suppose that persons were followed to death or censored in a study. Let t
,
t
, , t
P
, t
>
P>
, , t
>
L
be the survival times observed from the n individuals,
with r exact times and (n 9 r) right-censored times. Assume that the survival
times follow a distribution with the density function f (t, b) and survivorship
function S(t, b), where b : (b
, , b
N
) denotes unknown p parameters
b
, , b
N
in the distribution. As shown in Chapter 6, an exponential distribu-
tion has one (p : 1) parameter , the Weibull distribution has two (p : 2)
162
parameters and , and so on. If the survival time is discrete (i.e., it is observed
at discrete time only), f (t, b) represents the probability of observing t and S(t, b)
represents the probability that the survival or event time is greater than t.In
other words, f (t, b) and S(t, b) represent the information that can be obtained
from an observed uncensored survival time and an observed right-censored
survival time, respectively. Therefore, the product
L
G
f (t
G
, b) represents
the joint probability of observing the uncensored survival times, and
L
GP>
S(t
>
G
, b) represents the joint probability of those right-censored survival
times. The product of these two probabilities, denoted by L (b),
L (b) :
P
G
f (t
G
, b)
L
GP>
S(t
>
G
, b)
represents the joint probability of observing t
, t
, , t
P
, t
>
P>
, , t
>
L
. A similar
interpretation applies to continuous survival. L (b) is called the likelihood
function of b, which can also be interpreted as a measure of the likelihood of
observing a specific set of survival times t
, t
, , t
P
, t
>
P>
, , t
>
L
, given a
specific set of parameters b. The method of the MLE is to find an estimator of
b that maximizes L (b), or in other words, which is ‘‘most likely’’ to have
produced the observed data t
, t
, , t
P
, t
>
P>
, , t
>
L
. Take the logarithm of
L (b) and denote it by l(b),
l(b) : log L (b) :
P
G
log[ f (t
G
, b)] ;
L
GP>
log[S(t
>
G
, b)] (7.1.1)
Then the MLE b of b is the set of b
, , b
N
that maximizes l(b):
l(b ) : max
b
(l(b)).
It is clear that b is a solution of the following simultaneous equations, which
are obtained by taking the derivative of l(b) with respect to each b
H
:
*l(b)
*b
H
: 0 j : 1, 2, , p (7.1.2)
The exact forms of (7.1.2) for the parametric survival distributions discussed in
Chapter 6 are given in Sections 7.2 to 7.7. Often, there is no closed solution for
the MLE b from (7.1.2). To obtain the MLE b , one can use a numerical
method. A commonly used numerical method is the Newton—Raphson iter-
ative procedure, which can be summarized as follows.
1. Let the initial values of b
, , b
N
be zero; that is, let
b : 0
163
2. The changes for b at each subsequent step, denoted by H, is obtained by
taking the second derivative of the log-likelihood function:
H:
9
*l(bH\)
*b *b
\ *l(bH\)
*b
(7.1.3)
3. Using H, the value of bH at jth step is
bH:bH\ ; H j:1, 2, . . .
The iteration terminates at, say, the mth step if #K#:, where is a given
precision, usually a very small value, 10\ or 10\. Then the MLE b is defined
as
b : bK\ (7.1.4)
The estimated covariance matrix of the MLE b is given by
V
(b ) : Co v(b ) :
9
*l(b )
*b *b
\
(7.1.5)
One of the good properties of a MLE is that if b is the MLE of b, then g(b ) is
the MLE of g(b) if g(b) is a finite function and need not be one-to-one. The
concept of the Newton—Raphson method for p : 1 is illustrated in detail in
Appendix A.
The estimated 100(1 9 )% confidence interval for any parameter b
G
is
(b
G
9 Z
?
(v
GG
, b
G
; Z
?
(v
GG
)(7.1.6)
where v
GG
is the ith diagonal element of V
(b ) and Z
?
is the 100(1 9 /2)
percentile point of the standard normal distribution [P(Z 9 Z
?
) : /2]. For
a finite function g(b
G
)ofb
G
, the estimated 100(1 9 )% confidence interval for
g(b
G
) is its respective range R on the confidence interval (7.1.6), that is,
R : +g(b
G
):b
G
+ (b
G
9 Z
?
(v
GG
, b
G
; Z
?
(v
GG
), (7.1.7)
In case g(b
G
) is monotone in b
G
, the estimated 100(19 )% confidence interval
for g(b
G
)is
[g(b
G
9 Z
?
(v
GG
), g(b
G
; Z
?
(v
GG
)] (7.1.8)
164
7.1.2 Estimation Procedures for Data with Right-, Left-, and
Interval-Censored Observations
If the survival times t
, t
, , t
L
observed for the n persons consist of uncen-
sored, left-, right-, and interval-censored observations, the estimation pro-
cedures are similar. Assume that the survival times follow a distribution with the
density function f (t, b) and the survivorship function S(t, b), where b denotes all
unknown parameters of the distribution. Then the log-likelihood function is
l(b) : log L (b) : log[ f (t
G
, b)] ; log[S(t
G
, b)]
; log[1 9 S(t
G
, b)] ; log[S(v
G
, b) 9 S(t
G
, b)] (7.1.9)
where the first sum is over the uncensored observations, the second sum over
the right-censored observations, the third sum over the left-censored observa-
tions, and the last sum over the interval-censored observations, with v
G
as the
lower end of a censoring interval. The other steps for obtaining the MLE b of
b are similar to the steps shown in Section 7.1.1 by substituting the log-
likelihood function defined in (7.1.1) with the log-likelihood function in (7.1.9).
The computation of the MLE b and its estimated covariance matrix is
tedious. The following example gives the general procedure for using SAS to
carry out the computation.
Example 7.1 If the survival time observed contains uncensored, right-,
left-, and interval-censored observations, one needs to create a new data set
from the observed data to use SAS to obtain the estimates of the parameters
in the distribution. For an observed survival time t (uncensored, right-, or
left-censored), we define two variables LB and UB as follows: If t is uncensored,
take LB: UB: t;ift is left-censored, LB : . and UB : t; and if t is
right-censored, then LB : t and UB:., where ‘‘.’’ means ‘‘missing’’ in SAS. If
a survival time is interval-censored, [i.e., one observed two numbers t
and t
,
t
: t
and the survival time is in the interval (t
, t
)], let LB : t
and UB : t
.
Assume that the new data set (in terms of LB and UB) has been saved in
‘‘C:!EXAMPLEA.DAT’’ as a text file, which contains two columns (LB in the
first column and UB the second column) separated by a space.
As an example, the following SAS code can be used to obtain the estimated
covariance matrix defined in (7.1.5) and the MLE of the parameters of the
Weibull distribution for the survival data observed in the text file ‘‘C:!EXAM-
PLEA.DAT’’. One can replace d : weibull in the following code with the
respective distribution in Sections 7.2 to 7.6 (see the SAS code in these sections
for details) to obtain the estimate.
data w1;
infile ‘c:!examplea.dat’ missover;
input lb ub;
run;
proc lifereg;
model (lb,ub):/covb d : weibull;
run;
165
7.2 EXPONENTIAL DISTRIBUTION
7.2.1 One-Parameter Exponential Distribution
The one-parameter exponential distribution has the following density function;
f (t) : e\HR (7.2.1)
survivorship function;
S(t) : e\HR (7.2.2)
and hazard function;
h(t) : (7.2.3)
where t . 0, 90. Obviously, the exponential distribution is characterized by
one parameter, . The estimation of by maximum likelihood methods for
data without censored observations will be given first followed by the case with
censored observations.
Estimation of for Data without Censored Observations
Suppose that there are n persons in the study and everyone is followed to death
or failure. Let t
, t
, , t
L
be the exact survival times of the n people. The
likelihood function, using (7.2.1) and (7.1.1),is
L :
L
G
e\HR
G
and the log-likelihood function is
l() : n log 9
L
G
t
G
(7.2.4)
From (7.1.2), the MLE of is
:
n
L
G
t
G
(7.2.5)
Since the mean of the exponential distribution is 1/ and a MLE is invariant
under an one-to-one transformation, the MLE of is
:
1
:
L
G
t
G
n
: t (7.2.6)
166
It can be shown 2n / has an exact chi-square distribution with 2n degrees of
freedom (Epstein and Sobel, 1953). Since : 1/ and : 1/ , an exact
100(1 9 )% confidence interval for is
L\?
2n
::
L?
2n
(7.2.7)
where
L?
is the 100 percentage point of the chi-square distribution with 2n
degrees of freedom, that is, P(
L
9
L?
) : (Table B-2). When n is large
(n . 25, say), is approximately normally distributed with mean and
variance /n. Thus, an approximate 100(1 9 )% confidence interval for is
9
Z
?
(n
:: ;
Z
?
(n
(7.2.8)
where Z
?
is the 100/2 percentage point, P(Z 9Z
?
) : /2, of the standard
normal distribution (Table B-1).
Since 2n / has an exact chi-square distribution with 2n degrees of freedom,
an exact 100(1 9 a)% confidence interval for the mean survival time is
2n
L?
::
2n
L\?
(7.2.9)
The following example illustrates the procedures.
Example 7.2 Consider the following remission times in weeks for 21
patients with acute leukemia: 1, 1, 2, 2, 3, 4, 4, 5, 5, 6, 8, 8, 9, 10, 10, 12, 14, 16,
20, 24, and 34. Assume that remission duration follows the exponential
distribution. Let us estimate the parameter by using the formulas given
above.
According to (7.2.5), the MLE of the relapse rate, ,is
:
21
198
: 0.106 per week
The mean remission time is then 198/21: 9.429 weeks. Using the analytical
procedures given above, confidence intervals for and can also be obtained.
A 95% confidence interval for the relapse rate , following (7.2.7),is
approximately
(0.106)(24.433)
42
::
(0.106)(59.342)
42
or (0.062, 0.150). A 95% confidence interval for the mean remission time,
167
following (7.2.9),is
(42)(9.429)
59.342
::
(42)(9.429)
24.433
or (6.673, 16.208).
Once the parameter is estimated, other estimates can be obtained. For
example, the probability of staying in remission for at least 20 weeks, estimated
from (7.2.2),isS (20) : exp[90.106(20)] : 0.120. Any percentile of survival
time t
N
may be estimated by equating S(t)top and solving for t
N
, that is,
t
N
:9logp/ . For example, the median (50th percentile) survival time can be
estimated by t
:9log0.5/ : 6.539 weeks.
Estimation of
for Data with Censored Observations
We first consider singly censored and then progressively censored data.
Suppose that without loss of generality, the study or experiment begins at time
0 with a total of n subjects. Survival times are recorded and the data become
available when the subjects die one after the other in such a way that the
shortest survival time comes first, the second shortest second, and so on.
Suppose that the investigator has decided to terminate the study after r out of
the n subjects have died and to sacrifice the remaining n 9 r subjects at that
time. Then the survival times for the n subjects are
t
- t
- %- t
P
: t
>
P>
: %: t
>
L
where a superscript plus indicates a sacrificed subject, and thus t
>
G
is a censored
observation. In this case, n and r are fixed values and all of the n 9 r censored
observations are equal.
The likelihood function, using (7.1.1), (7.2.1), and (7.2.2),is
L :
n!
(n 9 r)!
P
G
e\HRG(e\HRP)L\P
and from (7.1.2), the MLE of is
:
r
P
G
t
G
;
L
GP>
t
>
G
(7.2.10)
The mean survival time : 1/ can then be estimated by
:
1
:
P
G
t
G
;
L
GP>
t
>
G
r
(7.2.11)
It is shown by Halperin (1952) that 2r/ has a chi-square distribution with 2r
168
degrees of freedom. The mean and variance of are r/(r 9 1) and /(r 9 1),
respectively. The 100(1 9 )% confidence interval for is
P\?
2r
::
P?
2r
(7.2.12)
When n is large, the distribution of is approximately normal with mean and
variance /(r 9 1). An approximate 100/(1 9 )% confidence interval for is
then
9
Z
?
(r 9 1
:: ;
Z
?
(r 9 1
(7.2.13)
Epstein and Sobel (1953) show that 2r / has a chi-square distribution with
2r degrees of freedom. Thus a 100/(1 9 )% confidence interval for (see also
Epstein, 1960b) is
2r
P?
::
2r
P\?
(7.2.14)
They also develop test procedures for the hypothesis H
: :
against the
alternative H
: :
. One of their rules of action is to accept H
if 9c and
reject H
if :c, where c : (
P?
)/2r and is the significance level. Or if the
estimated mean survival time calculated from (7.2.11) is greater then c, the
hypothesis H
is rejected at the level. The following example illustrates the
procedure.
Example 7.3 Suppose that in a laboratory experiment 10 mice are exposed
to carcinogens. The experimenter decides to terminate the study after half of
the mice are dead and to sacrifice the other half at that time. The survival times
of the five dead mice are 4, 5, 8, 9, and 10 weeks. The survival data of the 10
mice are 4, 5, 8, 9, 10, 10;,10;,10;,10;, and 10;. Assuming that the
failure of these mice follows an exponential distribution, the survival rate and
mean survival time are estimated, respectively, according to (7.2.10) and
(7.2.11) by
:
5
36 ; 50
: 0.058 per week
and : 1/0.058 : 17.241 weeks. A 95% confidence interval for by (7.2.12) is
(0.058)(3.247)
(2)(5)
::
(0.058)(20.483)
(2)(5)
169
or (0.019, 0.119). A 95% confidence interval for following (7.2.13) is
2(5)(17.241)
20.483
::
2(5)(17.241)
3.247
or (8.417, 53.098).
The probability of surviving a given time for the mice can be estimated from
(7.2.2). For example, the probability that a mouse exposed to the same
carcinogen will survive longer than 8 weeks is
S (8) : exp[90.058(8)]: 0.629
The probability of dying in 8 weeks is then 1 9 0.629 : 0.371.
A slightly different situation may arise in laboratory experiments. Instead of
terminating the study after the rth death, the experimenter may stop after a
period of time T, which may be six months or a year. If we denote the number
of deaths between 0 and T as r, the survival data may look as follows:
t
- t
- % - t
P
-t
>
P>
: % : t
>
L
: T
Mathematical derivations of the MLE of and are exactly the same and
(7.2.10) can still be used. The sampling distribution of for singly censored
data is also discussed by Bartholomew (1963).
Progressively censored data come more frequently from clinical studies
where patients are entered at different times and the study lasts a predeter-
mined period of time. Suppose that the study begins at time 0 and terminates
at time T and there are a total of n people entered. Let r be the number of
patients who die before or at time T and n 9 r the number of patients who are
lost to follow-up during the study period or remain alive at time T. The data
look as follows: t
, t
, , t
P
, t
>
P>
, t
>
P>
, , t
>
L
. Ordering the r uncensored
observations according to their magnitude, we have
t
- t
- % - t
P
, t
>
P>
, t
>
P>
, , t
>
L
The likelihood function, using (7.1.1), (7.2.1), and (7.2.2),is
L :
P
G
e\HRG
L
GP>
e\HRG
>
and the log-likelihood function is
l() : n 9
P
G
t
G
9
L
GP>
t
>
G
(7.2.15)
170
and from (7.1.2), the MLE of the parameter is
:
r
P
G
t
G
;
L
GP>
t
>
G
(7.2.16)
Consequently,
:
1
:
P
G
t
G
;
L
GP>
t
>
G
r
(7.2.17)
is the MLE of the mean survival time. The sum of all of the observations,
censored and uncensored, divided by the number of uncensored observations,
gives the MLE of the mean survival time. To overcome the mathematical
difficulties arising when all of the observations are censored (r : 0), Bar-
tholomew (1957) defines
:
L
G
t
>
G
(7.2.18)
In practice, this estimate has little value.
Distributions of the estimators are discussed by Bartholomew (1957). The
distribution of for large n is approximately normal with mean and variance:
Var( ) :
L
G
(1 9 e\H2G)
(7.2.19)
where T
G
is the time that the ith person is under observation. In other words,
T
G
is computed from the time the ith person enters the study to the end of the
study. If the observation times T
G
are not known, the following quick estimate
of Var( ) can be used:
Var
( ) :
r
(7.2.20)
Thus an approximate 100(1 9 )% confidence interval for is, by (7.1.6),
9 Z
?
(Var
( ) :: ; Z
?
(Var
( ) (7.2.21)
The distribution of is approximately normal with mean and variance:
Var( ) :
L
G
(1 9 e\H2
G )
(7.2.22)
171
