Exact Comparison of Hazard Rate Functions of Log-Logistic
Survival Distributions
Except where reference is made to the work of others, the work described in this
thesis is my own or was done in collaboration with my advisory committee. This
thesis does not include proprietary or classifled information.
Asha Dixit
Certiflcate of Approval:
Nedret Billor
Associate Professor
Mathematics and Statistics
Asheber Abebe, Chair
Associate Professor
Mathematics and Statistics
Hyejin Shin
Assistant Professor
Mathematics and Statistics
Joe F. Pittman
Interim Dean
Graduate School
Exact Comparison of Hazard Rate Functions of Log-Logistic
Survival Distributions
Asha Dixit
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulflllment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
Aug 09, 2008
Exact Comparison of Hazard Rate Functions of Log-Logistic
Survival Distributions
Asha Dixit
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at
their expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Vita
Asha Dixit was born on July 17, 1980 in Sringeri, India. She graduated with
Bachelor of Science from Kuvempu University, Shimoga, Karnataka, India in 2000.
ShealsocompletedherBachelorofEducationfromKuvempuUniversityin2001where
she got training for teaching physics and mathematics. In 2002, she joined Bangalore
University for MS program in Applied Mathematics. She was awarded Master of
Science in 2003. Following her graduation, she taught mathematics to undergraduate
students. She joined Auburn University in fall 2006 in the department of mathematics
and statistics.
iv
Thesis Abstract
Exact Comparison of Hazard Rate Functions of Log-Logistic
Survival Distributions
Asha Dixit
Master of Science, Aug 09, 2008
(M.S., Bangalore University, 2003)
(B.Ed., Kuvempu University, 2001)
(B.S., Kuvempu University, 2000)
69 Typed Pages
Directed by Asheber Abebe
A comparison of hazard rates of multiple treatments are compared under the
assumption that survival times follow the log-logistic distribution. Exact test proce-
dures are developed for ordered comparisons of the worst case hazard rates of several
log-logistic survival functions. In particular, critical constants are computed for test-
ing the null hypothesis that all dose levels give the same maximum hazard rates versus
the alternative that the maximum hazard rates are decreasing with increasing dose
level. In addition, critical constants are given for comparing equal maximum hazard
rates against the alternative of valley ordered hazard rates. A procedure for build-
ing simultaneous confldence intervals for certain contrasts is provided. The procedure
proposed in this thesis is then compared to two nonparametric simultaneous inference
procedures compare it to two nonparametric procedures: the Jonckheere-Terpstra test
and the Mack-Wolfe test.
v
Acknowledgments
This dissertation could not have been written without Dr. Abebe who not only
served as my supervisor but also encouraged and challenged me throughout my aca-
demic program. I thank Dr.Abebe and my committee members, Dr. Billor and
Dr.Shin, for their guidance.
Thanks to all the faculty members and GTA?s in Department of Mathematics
and Statistics. I thank my Father Markandeya Dixit, my mother Nagalaxmi Dixit,
my brother Vishwanath Dixit for their love and continuous support.
I dedicate this work to my belloved husband Madhu Kirugulige, for his love and
support.
vi
StylemanualorjournalusedJournalofApproximationTheory(togetherwiththe
style known as \aums"). Bibliograpy follows van Leunen?s A Handbook for Scholars.
Computer software used The document preparation package TEX (speciflcally
LATEX) together with the departmental style-flle aums.sty.
vii
Table of Contents
List of Figures ix
List of Tables x
1 Introduction 1
2 The Log-Logistic Distribution and its Properties 5
2.1 Density and Distribution Functions . . . . . . . . . . . . . . . . . . . 5
2.2 Relationship with the Logistic Distribution . . . . . . . . . . . . . . . 8
2.3 Moments, Mode and Median . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Survival and Hazard Functions . . . . . . . . . . . . . . . . . . . . . 9
2.5 Comparison of Log-Normal and Log-Logistic Distributions . . . . . . 12
2.6 Sampling Distribution of the Logistic Sample Median . . . . . . . . . 14
3 Simple Ordering of Hazard Rates 19
3.1 Test Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Computing the Critical Constants . . . . . . . . . . . . . . . . . . . . 25
3.3 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.1 Nominal Level Simulation . . . . . . . . . . . . . . . . . . . . 29
3.3.2 Power Simulation versus the Jonckheere-Terpstra Test . . . . 31
4 Valley Ordering of Hazard Functions 34
4.1 Dose Response experiment . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Test Statistic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.3 Computing the Critical Constants . . . . . . . . . . . . . . . . . . . . 37
4.4 Monte Carlo Simulation Studies . . . . . . . . . . . . . . . . . . . . . 39
4.4.1 Nominal FWER Simulation . . . . . . . . . . . . . . . . . . . 39
4.4.2 Power Simulation versus the Mack-Wolfe Test . . . . . . . . . 42
5 Simultaneous Confidence Intervals 48
6 Summary 51
Bibliography 53
A Appendix 58
viii
List of Figures
2.1 The pdf of LL(1;fl) for difierent values of fl . . . . . . . . . . . . . . 6
2.2 The cdf of LL(1;fl) for difierent values of fl . . . . . . . . . . . . . . . 7
2.3 The pdf of LL(1;fl) for difierent values of fl . . . . . . . . . . . . . . 11
2.4 The cdf of LL(1;fl) for difierent values of fl . . . . . . . . . . . . . . . 12
2.5 The pdf of log-logistic and log-normal distributions . . . . . . . . . . 14
2.6 The cdf of log-logistic and log-normal distributions . . . . . . . . . . 15
2.7 The pdf of logistic distributions with median as location parameter. 17
2.8 The cdf of logistic distributions with median as location parameter. . 18
ix
List of Tables
2.1 comparison of log-logistic and log-normal distribution . . . . . . . . . 16
3.1 Table of critical values ck;m;fi for simple ordered alternatives . . . . . 30
3.2 Table of signiflcance level fi for simple ordered alternatives . . . . . . 31
3.3 Power comparison against JT for k = 3 . . . . . . . . . . . . . . . . . 33
4.1 Values of qk;h;m;:05 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 Values of qk;h;m;:01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3 Values of fi from simultaneous confldence intervals for fi = 0:05 . . . 43
4.4 Values of fi from simultaneous confldence intervals for fi = 0:01 . . . 44
4.5 Simulated Power of the Mach-Wolfe Test against our Test . . . . . . . 46
5.1 Simultaneous confldence interval for m = 5 and fi = 0:05 . . . . . . . 50
x
Chapter 1
Introduction
In survival analysis, the survival and hazard functions are very functions used to
characterize the important characteristics of survival up to a certain specifled time
and instantaneous death or break-down at a speciflc time. Although the Weibull
distribution is frequently used in modeling survival data, its use is restricted since
its hazard function is either monotonically increasing or decreasing. The log-logistic
distribution is similar in appearance to the log-normal distribution but its hazard
and survival functions can be computed e?ciently. Thus it is a good choice when
the hazard rate function is desired to have increasing and then decreasing shapes in
addition to monotone increasing/decreasing shapes. Moreover, the log-logistic distri-
bution can be easily employed in the presence of censored data which is very common
in survival or reliability analysis. It is very di?cult to use log-normal distribution
in such cases. Consequently, the log-logistic distribution has seen increasing use re-
cently. For example, Diekmann [12] used the log-logistic distribution as a model for
event history analysis, Bennett [8] used it to model survival data, and Singh, Lee and
George [41] used it to model censored survival data.
The log-logistic distribution is a derivative of the very popular logistic distrib-
ution. The logistic distribution was initially developed to model population growth
by Verhulst [47, 48]. Verhulst [47] noticed that exponential distribution was used
1
in studies involving growth of biological populations such as cancer cells and bacte-
ria. But when there is a limitation of food and space for a large population then its
growth will not follow the exponential curve but rather the logistic curve. The use
of the logistic distribution for economic and demographic purposes was very popular
in the nineteenth century. The logistic distribution is also known by names such as
growth function, autocatalytic curve and so on depending on its application. The
name ?logistic ?was coined by Reed and Berkson [32]. Berkson [10] noted that under
some circumstances if the dosage of a drug is expressed in proportion to its logarithm,
the efiect, as a percentage, follows the form of a more or less symmetric sigmoidal
curve, the integral of a normal curve has been employed for the estimation of the
potency of a drug. The logistic distribution has been used on human population by
Pearl and Read [31], on flsh by Jensen [21], on animals by Miller and Botkin [11], on
bacteria and cells by Tan [43] and on tumor cells by Eisen [15], and on breast tumor
by Moolgavkar [30].
In many practical situations theories and previous evidences or conditions suggest
an expected ordering among the treatment efiects. Example of such situations include
severity of disease, drug dosage level etc. In our study, we consider a dose-response
relationship where increasing dose levels lead to certain order relationships among
the hazard rates. In particular we consider the situation where increasing dose leads
to decreasing hazard rate and the situation where increasing dose leads to decreasing
hazard rate up to a certain level and any more increase in the dose level results in an
increase in hazard rate.
2
Such type of ordered alternatives have been considered in the past. Robert-
son [33] considers umbrella ordering to flt multinomial distributions to cell counts.
Bartholomew [5, 6, 7] proposed a likelihood ratio test (LRT) for umbrella alternatives.
Simpson and Margolin [40] considered umbrella ordering in dose-response relation-
ships. Hayter and Liu [18] considered umbrella alternative for the normal distribution
and Singh et al.[42] for the logistic distribution. More recently, the test of a null of
no difierence against that of a u-shaped alternative was developed for the exponential
distribution location parameters by Abebe and Singh [1]. A nonparametric test for
umbrella alternatives was given by Mack and Wolfe [28].
When it comes to simple (increasing or decreasing) ordering of parameters,
Hayter and Liu [17] developed tests for the normal distribution location parame-
ters while Tebbs and Bilder [44] developed such tests for comparing proportions. A
nonparametric test for the simple order was given by Jonckheere [22] and Terpstra
[45].
In Chapter 2 the log logistic distribution and its properties are discussed along
with its comparison with the log-normal distribution. In Chapter 3, we develop
an exact simultaneous testing procedure to compare the maximum hazard rates of k
treatments (or doses) under a simple ordering restriction. From the union-intersection
test statistic, the required critical constants are computed using a recursive algorithm
and tables of critical constants are provided. A Monte Carlo simulation is performed
to verify the results and compare the new test procedure with the nonparametric test
due to Jonckheere and Terpstra. The fourth chapter deals with the valley ordering
3
restriction of the maximum hazard rates. A test statistic is introduced for testing for a
valley ordering and a recursive algorithm is given for computing the critical constants.
The power of this test is then compared to that of the Mack-Wolfe procedure for
umbrella alternative. Chapter 5 presents a discussion of simultaneous confldence
intervals for a certain set of contrasts.
4
Chapter 2
The Log-Logistic Distribution and its Properties
The log-logistic distribution is used in survival and reliability analysis as a model
for survival times and is similar in shape to the log-normal distribution (see for
example Kalb eish and Prentice [24] ). Its use is appealing because, like the lognormal
distribution, its hazard rate function takes several difierent shapes depending on
a value of a shape parameter. Recently it has seen increased use in hydrology to
model stream ow and precipitation and also in economics as a simple model of the
distribution of wealth or income due to its relationship to the generalized Pareto
distribution. For more on the use of the log-logistic distribution in hydrology, the
reader is referred to Shoukri et al.[39], Robson and Reed [35], or Ahmad et al.[3].
2.1 Density and Distribution Functions
A random variable T is said to follow the log-logistic distribution with scale
parameter and shape parameter fl, henceforth denoted by T ? LL( ;fl), if its
probability density function (pdf) is given by
f(t; ;fl) = (fl= )(t= )
fl?1
[1+(t= )fl]2 ; t > 0 ; (2.1)
where > 0 and fl > 0. The corresponding cumulative function (cdf) is given by
5
F(t; ;fl) = t
fl
fl +tfl : (2.2)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
t
pdf
?=1/2
?=1?=2
?=4?=8
Figure 2.1: The pdf of LL(1;fl) for difierent values of fl
In Figure 2:1, the pdf of log-logistic distribution is shown for various values of fl.
This is a non negative distribution of random variables which takes various shapes.
For fl ? 1 the pdf is a decreasing function, where as for fl > 1, the pdf is an increasing
decreasing function and there by has a peak. As value of fl increases, the peak of the
pdf shifts towards one and becomes more symmetric.
6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
cdf
?=1/2?=1
?=2?=4
?=8
Figure 2.2: The cdf of LL(1;fl) for difierent values of fl
In Figure 2:2 the cdf of the log-logistic distribution is shown for various values
of fl.
7
2.2 Relationship with the Logistic Distribution
A random variable X is said to follow the logistic distribution with location ?
and scale , written X ? L(?; ), if its probability density function (pdf) is given by
g(x;?; ) =
exp
?
?(x??)
?
h
1+exp
?
?(x??)
?i2 ; jxj < 1 ;
where ?1 < ? < 1 and > 0. The cdf is given by
G(x;?; ) =
h
1+e?(x??)
i?1
:
The relationship between the logistic and the log-logistic distributions is anal-
ogous to that between the normal and the log-normal distributions. In particular,
using a simple change-of-variable technique, one can show that X ? L(?; ) if and
only if T ? exp(X) ? LL(exp(?);1= ) distribution.
In later sections, we will exploit this relationship between the two random vari-
ables when constructing simultaneous tests and confldence intervals.
2.3 Moments, Mode and Median
Let T ? LL( ;fl). The kth moment of T for k < fl can be shown to be
ETk = kB(1?k=fl ; 1+k=fl) = k(k?=fl)csc(k?=fl) ;
8
where B(? ; ?) is the beta function given by B(a;b) = R10 sa?1(1?s)b?1ds and csc(?) is
the cosecant function. The kth moment is undeflned if k ? fl. In particular, if fl > 1,
we can show that the mean and variance of T are (see Tadikamalla and Johnson [2])
ET = (?=fl)csc(?=fl)
and
ET2 ?E2T = 2[(2?=fl)csc(2?=fl)?(?=fl)2 csc2(?=fl)] :
The LL( ;fl) distribution is unimodal with
fl ?1
fl +1
?1=fl
:
for fl > 1. The mode is zero for fl ? 1. The median m of LL( ;fl) is found by solving
F(m; ;fl) = 0:5 which gives m = .
2.4 Survival and Hazard Functions
The survival function, also known as the reliability function in engineering, is
the characteristic of an explanatory variable that maps a set of events, usually as-
sociated with mortality or failure of some system onto time. It is the probability
that the system will survive beyond a specifled time. The term reliability function is
common in engineering while the term survival function is used commonly in many
flelds, including human mortality. Lately, the log-logistic survival model is being used
9
increasingly. Conkin [14] used the log-logistic survival model as a model for hypobaric
decompression sickness as a consequence of ying high in the atmosphere. According
to Jones [23], even when the underlying survival function was best described by a
negative power curve, a log-logistic model flts the data well and provides more ver-
satility for fltting individual populations. In summarizing survival data, there are
two important functions, namely survival function and hazard function. The actual
survival time of an individual, represented by t is regarded as the realized value of
the survival time T, which is a random variable that can take any non negative value.
The survival function of T ? LL( ;fl) is
S(t) = P(T > t) = ?1+(t= )fl??1
.
This survival function is plotted in Figure 2:3. These decreasing survival func-
tions cross each other at t = 1 as S(t) becomes independent of fl for this particular
value of t.
The corresponding hazard function is the probability that an individual dies or
an equipment fails at instantaneously at time t, provided that the individual survived
up to time t. So the hazard function represents the death rate at a given point of
time t. The hazard function at t is given by
?(t) = f(t)S(t) = (fl= )(t= )fl?1?1+(t= )fl??1 :
10
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
t
Survival function
?=1/2
?=1?=2
Figure 2.3: The pdf of LL(1;fl) for difierent values of fl
Figure 2:4 explains the hazard rate function for difierent values of fl. Later on
we assume fl > 1 so that hazard rate function increases and decreases. We compare
the worst case scenarios.
Thus besides the similarity of the log-logistic distribution to the log-normal dis-
tribution, one using the log-logistic distribution to model survival data has the ad-
vantage of knowing explicit forms of the hazard and survival functions. This is not
the case with the log-normal distribution.
11
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
2
2.5
t
Hazard function
?=1/2
?=1?=2
Figure 2.4: The cdf of LL(1;fl) for difierent values of fl
2.5 Comparison of Log-Normal and Log-Logistic Distributions
In this section we will compare several characteristics of the log-logistic and log-
normal distributions.
Considering the shape of the distributions, the log-normal distribution, like the
Weibull distribution, is a very exible model that can empirically flt many types of
failure time data. The log-normal distribution has two parameters, shape parameter
12
and a scale parameter. The log-logistic distribution is also deflned using a scale and a
shape parameter and can also take a variety of shapes like the log-normal distribution.
Since scale parameter will not afiect the tails of either the log-normal or the log-
logistic distributions, we can take the scale parameter to be unity without any loss of
generality. The log-logistic distribution has heavier tail than log-normal distribution
when the shape parameter of the log-logistic distribution is less than or equal to 4p2?
times the shape parameter of the log-normal distribution Yanagimoto [50].
According to Bennett [8], the log-logistic distribution is very similar in shape
of the log-normal distribution but the log-logistic distribution is more suitable for
survival analysis than the log-normal distribution when the data contain censored
observations. Censored observations are quite common in survival analysis and the
log-normal distribution cannot be used directly in the presence of censored data. With
the Weibull distribution being monotonic increasing or decreasing, the log-logistic
distribution is a popular choice.
The difierences between the probability density functions of the log-normal and
log-logistic distributions is illustrated in Fig. 2.5 for the same grid values. Similarly,
the cdf of the log-normal and log-logistic distributions are shown in Fig. 2.6.
Some properties of the log-logistic and log-normal distributions are listed in Table
2.1.
13
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
pdf
Log normal
Log logistic
Figure 2.5: The pdf of log-logistic and log-normal distributions
2.6 Sampling Distribution of the Logistic Sample Median
We will be using the sampling distribution of the median of a random sample
from the logistic distribution to construct tests and intervals relating to the scale
parameter of the log-logistic distribution. The median is considered to be a good
estimator of the location parameter of the logistic distribution since
1. the logistic distribution is symmetric,
2. the logistic distribution is long-tailed,
3. the median is easy to compute, and
4. the closed form expressions of the pdf and cdf of the sampling distribution of
the logistic median are known.
14
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x
cdf
Log Normal
Log logistic
Figure 2.6: The cdf of log-logistic and log-normal distributions
Consider a random sample of size n from the logistic distribution with location
parameter ? and scale parameter . Without loss of generality, let m = (2n ? 1).
The probability density function (pdf) of logistic distribution L(?; ) in variable X
is given by Eq. 2.3
f(x;?; ) = a e
?a(x??)
1+e?a(x??)
(2.3)
and corresponding cumulative function (cdf) is given by
F(x;?; ) = 1
1+e?a(x??)
(2.4)
where a = ?=p3.
15
Log-normal Distribution Log-logistic Distribution
Support [0;1) [0;1)
Probability Density Function 1x p2? exp
?
?(ln(x)??)22 2
?
(fl= )(t= )fl?1
[1+(t= )fl]2
Cumulative Density Function 12 + 12'
?
(ln(x)??)
p2
?
tfl
fl+tfl
Mean e?+ 2 fi?=flsin(?=fl) if fl > 1, else not deflned
Median e?
Mode e(?? 2)
?
fl?1
fl+1
?1=fl
if fl > 1;0 otherwise
Table 2.1: comparison of log-logistic and log-normal distribution
Given a random sample X1;:::;Xn from the L(0;1) distribution, by ordering
X(1) ? X(2) ????X(n)
we can get the median as X(m). Using the formula for the distribution of order
statistics we can write the pdf of the sample median as
?m(x) = ?(2m)?2(m)F(m?1)(x)[1?F(x)]m?1f(x)
that, after some simpliflcation, can be written as
?m(x) = ?(2m)?2(m)a(e?ax)m(1+e?ax)?2m (2.5)
16
The cdf is given by
?m(x) = ?(2m)?2(m)
m?1X
j=0
m?1
j
?
(2m?j ?1)?1(?1)m?1?j(1+e?ax)j+1?2m : (2.6)
The probability density functions and cumulative distribution function of the
logistic distribution with median as location parameter is shown in Fig. 2.7 and Fig.
2.8. ?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
x
pdf
m=3
m=10
Figure 2.7: The pdf of logistic distributions with median as location parameter.
The asymptotic distribution of the standardized sample median is normal with
mean zero and variance [4nf2(0)]?1 since f2(0) = a2=16 this variance reduces to
12=n?2 Ref. [27]
17
?2 ?1.5 ?1 ?0.5 0 0.5 1 1.5 20
0.5
1
1.5
2
2.5
x
m=3
m=10
Figure 2.8: The cdf of logistic distributions with median as location parameter.
18
Chapter 3
Simple Ordering of Hazard Rates
Consider the situation where we have k dose levels of a certain treatment. Sup-
pose that n subjects have been randomly allocated to each of the k dose levels. Let
the survival times be given according to the log-linear model
logTij = ?i +?ij ; 1 ? j ? n ; 1 ? i ? k ; (3.1)
where ?ij are independent L(0; ) random variables and ?i is the unknown center of
the ith treatment and the scale parameter is assumed to be known. As we have
shown earlier, this condition is equivalent to assuming that Tij ? LL(exp(?i);1= )
for 1 ? j ? n ; 1 ? i ? k.
Let ?i(?) be the hazard function of the ith dose level and let
?maxi = sup
t
?i(t) ;
1 ? i ? k. We are interested in flnding out if increasing dose leads to a decrease of
the worst case hazard rate of individuals. In particular, we would like to perform the
simultaneous test of the null hypothesis
H0 : ?max1 = ??? = ?maxk
19
versus the simple ordering alternative
Hs : ?max1 ????? ?maxk
with at least one strict inequality.
As the following theorems show, for the log-logistic distribution, performing this
test is equivalent to making multiple comparisons of the scale parameters of the log-
logistic distribution.
Theorem 3.1. Let ?(?; ;fl?) be the hazard function of LL( ;fl?), for a given value
of fl?. Then ?(?; ;fl?) is a monotone decreasing function of .
Proof. The proof follows since for any given t, we can write
?(t; ;fl?) = fl
?tfl??1
fl? +tfl? :
Theorem 3.2. Let ?max( ;fl?) be worst case of the hazard function of LL( ;fl?), for
a given value of fl?. Then ?max( ;fl?) is a monotone decreasing function of .
Proof. For any given t, we can write log-logistic distribution as
?(t; ;fl) =
fl
?
t
?fl?1
1+
t
?fl :
20
If fl > 1, we can flnd the value of t that maximizes ?(t; ;fl) by solving
0 = @?(t)@t =
fl
?
(fl ?1)
t
?fl?2?
1+
t
?fl?
?fl
t
fi
?fl?1
fl
?
t
?fl?1
?
1+
t
?fl?2 :
This gives
0 = (fl ?1)+(fl ?1)
t
fi
?fl
?fl
t
?fl
and solving for t gives
tmax =
fl ?1
?1
fl
Substituting in ?(t; ;fl) we get
?(tmax; ;fl) =
fl
?
tmax
?fl?1
1+
tmax
?fl
=
fl
?"
fl?1
?1fl
#fl?1
1+
"
fl?1
?1fl
#fl
=
?
fl ?1
!fl?1=fl
21
Thus for a flxed fl = fl?
?max = k
where k is a constant.
One may consider a Cox proportional hazards model by imposing the restriction
?i(t) = ?0i(t)exp(?i)
where ?i is an unknown constant. This restriction produces a models that are in the
so-called Lehmann class (Lehmann [25] ). Although this is used often in the literature,
it imposes a proportionality assumption that is not necessary in our case. Thus, this
model will not be considered in this thesis.
In a similar manner to the proof of Theorem 3.1, one can show that the survival
function of the LL( ;fl) distribution is a monotone increasing function of for a
flxed value of fl. So, a similar analysis may be performed for survival functions. In
our case, it is true that decreasing ?(?) is equivalent to increasing S(?) and thus the
inferences that we develop for ?(?) may be used directly for S(?).
Let us now consider the test of H0 versus the simple ordering Hs. By Theo-
rem 3.2, the hypotheses can be written down as
H0 : 1 = ??? = k
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versus
Hs : 1 ????? k
with at least one strict inequality. Here i = exp(?i) and ?i is given in Equation 3.1
for 1 ? i ? k. Since exp(?) is a monotone increasing function, this is equivalent to
testing
H0 : ?1 = ??? = ?k
versus
Hs : ?1 ????? ?k
with at least one strict inequality. Recall that this corresponds to the case where
increasing dose levels are expected to give decreasing worst case hazard rates.
Thus we have reframed the problem as a multiple comparison of logistic location
parameters. This has been considered in Singh et al.[42]. Similar comparisons for the
case where the underlying distribution is normal has been considered in the literature
(see Robertson et al.[34] ). Williams [49] gave a test based on the maximum likelihood
estimators of the normal means ?1;:::;?k under the restriction given by Hs : ?1 ?
???? ?k. Hayter [16] gave a studentized range test for the normal case while Hayter
and Liu [17] gave a recursive computational procedure for computing the exact critical
values of this studentized range test.
We reiterate that if we know any information speciflcally about the data from
theory or from the previous experience that the hazard rates follow a certain order
prior to collecting the data, then it is very important to incorporate this information
23
into our analysis. If one neglects these factors then the potential consequence will be
improper interpretation of results.
3.1 Test Statistic
Consider the model given by Equation 3.1. Let Yi = medianflogTi1;:::;logTing
for 1 ? i ? k. For H0 versus Hs given above, the statistic for the union-intersection
test (Roy [36]; Sen [38] ) is given by
Wk;m = min
1?i a3;:0497)
1:0 1:0 1:0 1:0 0.0495 0.0423
(1:7)?1 (1:7)?1 (4:5)?1 (1:0)?1 0.6932 0.5800
(1:0)?1 (2:7)?1 (4:5)?1 (2:7)?1 0.7327 0.5895
(2:7)?1 (2:7)?1 (4:5)?1 (2:7)?1 0.5715 0.1956
(2:7)?1 (2:7)?1 (7:4)?1 (1:0)?1 0.8912 0.6424
(2:7)?1 (7:4)?1 (7:4)?1 (1:0)?1 0.9096 0.6375
Table 4.5: Simulated Power of the Mach-Wolfe Test against our Test
Based on the simulated power analysis, the following observations are noted:
46
? our test maintains the level of the test fi = 0:0497 when all the hazard functions
are the same level of accuracy
? our test is more powerful in detecting valley patterns in the hazard functions
compared to the Mack-Wolfe test.
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Chapter 5
Simultaneous Confidence Intervals
When the null hypothesis is rejected in favor of the alternative hypothesis, the
investigator is usually interested in determining exactly which hazard functions are
difierent from each other. If the null is not rejected, then no further action is needed.
Let us focus on the valley shaped alternative that was discussed in Chapter 4.
This case is chosen since it is more general than the simple ordering alternative
considered in Chapter 3. If one is interested in the simple ordering case, then one
can get the simultaneous confldence intervals by simply replacing h by 1(k) for the
decreasing (increasing) arrangement of hazard functions.
The simultaneous confldence intervals with simultaneous confldence coe?cient
of 1?fi for the pairwise difierences (?maxj ??maxi ), 1 ? i < j ? h and h ? j < i ? k,
may be constructed by inverting the fi-level test PH0(Vk;h;m < ?qk;h;m;fi) = fi. The
100(1?fi)% simultaneous confldence interval may be obtained as
1?fi = PH0(Vk;h;m ??qk;h;m;fi)
= P
?maxj ??maxi ? (Yj ?Yi)?qk;h;m;fi ; 1 ? i < j ? h; h ? j < i ? k
?
The investigator may also be interested in more general contrasts of the hazard
functions rather than just simple pairwise difierences. Let Y = (Y1;:::;Yk)0. Letting
?max = (?max1 ;:::;?maxk )0, one may want to construct a confldence interval for c0?max
48
for a given constant c 2