LINEAR RANK TESTS FOR THE NONRESPONDERS PROBLEM
WITH CENSORED DATA
By
VASILIS BILL PIKOUNIS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1991
Copyright @ 1991
by
Vasilis Bill Pikounis
To All Those Who Keep Their Dreams Alive
ACKNOWLEDGEMENTS
I make no apologies for my indulgence in the following sentences, since my
completion of this dissertation was helped in so many ways by so many people. First
and foremost, I thank my parents Stamatios and Evangelia for their everlasting love
and their teachings of the tremendous values to be gained from working hard and
earning things in life. The rest of my family and relatives also deserve thanks.
I truly appreciate the wisdom of Dr. P.V. Rao in his guidance of my Ph.D. pro-
gram. I am fortunate for the enthusiasm of statistics shared with a handful of my
fellow students who have become valued colleagues and friends, especially J. Ben
Lang and Carolyn Hansen. I readily acknowledge all the Statistics Department
faculty in the main division that have imparted their vast knowledge of statistics
to me since I was a eager freshman at the University of Florida. I feel very lucky
with the invaluable experience gained in my four years as a data analyst and sta-
tistical consultant in the Division of Biostatistics, due in large part to the superb
professionals that serve as faculty there. In particular, I thank Dr. Michael Con-
Ion and Mr. Phil Padgett for their vision and thankless support that has led to the
workstation computing environment so crucial to this manuscript. In particular, the
wonderful document preparation system ILTEX (Lamport, 1986) and its genesis TEX
(Knuth, 1984) were used to typeset this manuscript; S-plus (Becker, Chambers, and
Wilks, 1989; Statistical Sciences 1990) produced the graphics and simulations, and
Mathematica (Wolfram, 1988) provided numerical integration. Dr. Scott Emerson
deserves thanks for constant help (and never once complaining) in my early days of
trying to learn the workstation environment.
I am grateful for my many longtime friends that have always supported me, and
inspired me to keep going in those times of inevitable valleys.
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS................................................. iv
LIST OF TABLES ................................. ................. viii
LIST OF FIGURES .................................. ...................... x
ABSTRACT ................ .................................... ........... xi
CHAPTER
1 INTRODUCTION ............................................ 1
1.1 The Nonresponders Problem ............................ 1
1.2 Overview of this Manuscript .................... ...... 4
2 STATISTICAL FORMULATION OF THE NONRESPONDERS
PROBLEM ................................................. 6
2.1 Introduction ......................................... 6
2.2 The Mixture Model .................................... 6
2.3 Literature Review ..................................... 20
3 METHODS FOR CENSORED DATA ........................ 33
3.1 Introduction ....................................... 33
3.2 The Linear Rank Statistic v............................. 34
3.3 Conditions for Asymptotic Properties Under Ho ........ 49
3.4 Asymptotic Normality of v ............................ 60
3.5 Evaluation of Expected Scores ......................... 75
3.6 Properties Under Ha .................................. 81
4 FORMS OF THE TEST STATISTIC ........................... 99
4.1 Introduction ........................................... 99
4.2 Score Functions for Inference about r ................. 100
4.3 Uncensored Data Expected Scores .................... 106
4.4 Censored Data Expected Scores ...................... 110
5 COMPARATIVE STUDIES ................................. 117
5.1 Pitman Asymptotic Relative Efficiencies ............... 117
5.2 Simulation Study .................................... 128
5.3 Real Data Examples ................................. 157
6 SUMMARY AND CONCLUSIONS .......................... 165
REFERENCES ..................................... ...................... 168
BIOGRAPHICAL SKETCH............................................... 171
LIST OF TABLES
Table page
2.1 Pitman ARE for misspecified A, mixed normal distribution ........ 30
2.2 Pitman ARE for location-model vs. mixed-model linear rank test, mixed
normal distribution ................................................ 31
5.1 Pitman ARE for misspecified A under a mixed normal distribution with
exponential and uniform (in parentheses) censoring patterns ....... 121
5.2 Pitman ARE for location-model vs. mixed-model linear rank test, mixed
normal distribution with exponential and uniform (in parentheses)
censoring patterns ................................................ 123
5.3 Pitman ARE for misspecified A under a mixed logistic distribution with
no censoring ....................................................... 125
5.4 Pitman ARE for misspecified A under a mixed logistic distribution with
exponential and uniform (in parentheses) censoring patterns ........ 125
5.5 Pitman ARE for location-model vs. mixed-model linear rank test, mixed
logistic distribution with absence or presence of exponential and uniform
(in parentheses) censoring patterns .............................. 126
5.6 Pitman ARE for misspecified A under a mixed extreme-value
distribution with exponential and uniform (in parentheses) censoring
patterns ................................................... 127
5.7 Pitman ARE for location-model vs. mixed model linear rank test, mixed
extreme-value distribution with absence or presence of exponential and
uniform (in parentheses) censoring patterns ....................... 129
5.8 Empirical powers under a mixed normal distribution and no censoring,
with group sample sizes of 20 ..................................... 140
5.9 Empirical powers under a mixed normal distribution and no censoring,
with group sample sizes of 50 (40) ............................... 142
5.10 Empirical powers under a mixed extreme-value distribution and no
censoring, with group sample sizes of 50 ........................ 144
5.11 Empirical powers under a mixed extreme-value distribution and no
censoring, with group sample sizes of 20 ........................ 147
5.12 Empirical powers under a mixed extreme-value distribution and 10%
exponential and uniform (in parentheses) censoring rates, with group
sample sizes of 50 ............................................. 150
5.13 Empirical powers under a mixed extreme-value distribution and 25%
exponential and uniform (in parentheses) censoring rates, with group
sample sizes of 50 ................................................ 152
5.14 Empirical powers under a mixed normal distribution and 10%
exponential and uniform (in parentheses) censoring rates, with group
sam ple sizes of 20 .................... ........................ ..... 154
5.15 Empirical powers under a mixed normal distribution and 40%
exponential and uniform (in parentheses) censoring rates, with group
sample sizes of 50 .................. ............................. 156
5.16 Change in pain measures from diabetic neuropathy study .......... 158
5.17 Comparison of test statistic values, uncensored data example ...... 159
5.18 Test statistic values and computed p-values,
uncensored data example ........................................ 160
5.19 Survival times in days from Veteran's Administration lung cancer trial,
patients receiving test therapy .................................... 163
5.20 P-values for VA lung cancer censored data example .............. 164
LIST OF FIGURES
Table page
2.1 Graph of extreme-value mixture model; A = 0.5. ................... 13
2.2 Graph of extreme-value mixture model; A = 1.0. ................... 14
2.3 Graph of extreme-value mixture model; A = 2.0. .................. 15
2.4 Graph of extreme-value mixture model; A = 3.0. ................. 16
5.1 1000 random observations from a normal distribution ............. 134
5.2 1000 random observations from an extreme-value distribution ...... 135
5.3 Graph of test statistic behavior, uncensored data example ......... 162
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
LINEAR RANK TESTS FOR THE NONRESPONDERS PROBLEM
WITH CENSORED DATA
By
Vasilis Bill Pikounis
December 1991
Chairman: Pejaver V. Rao
Major Department: Statistics
In a two-sample situation, a two-parameter mixture model is postulated for the
treatment group where a proportion 7r of the subjects are affected favorably by the
treatment, and the remaining proportion of nonresponders behave like the control
patients. The response of interest is subject to random censoring.
A linear rank statistic is developed that has scores that are derived from the
probability of the underlying rank vector arising from data governed by the mixture
model. The technique used to generate the scores is one that emphasizes optimality
properties for values of the parameter 7r in the neighborhood of zero. Writing the
linear rank statistic as a stochastic integral leads to a large sample test procedure.
The performance of the test procedure is compared to standard rank-based tests
in terms of empirical power for a variety of underlying distributions, censoring
xi
patterns, and moderate to large sample sizes. Comparisons via asymptotic relative
efficiency and real data examples are presented as well.
Simulation studies indicate that the mixture model-based test procedures are
more efficient than the standard tests in detecting improvement due to treatment
when the proportion of responders is small and the improvement is substantial.
Because the mixed model test statistic depends on an unknown parameter, the
simulations also provide some guidance in choosing a parameter value that provides
a test that is both sensitive to detect a true treatment effect and is valid in terms
of the chosen significance level.
CHAPTER 1
INTRODUCTION
1.1 The Nonresponders Problem
The "nonresponders" problem may arise in studies designed to evaluate the
efficacy and/or toxicity of new drugs. Such studies generate data from two groups-
the Treatment group consisting of responses to the drug and the Control group
consisting of responses to a placebo. In cases where the drug is designed to combat
one of several different factors that can be the chief cause of the disease, a portion
of the treated subjects will respond like the control subjects. In other words, this
portion of subjects will not respond to the treatment, and the response of interest
can be regarded as an observation that would occur had the subject been a member
of the control group. The subjects are defined to be nonresponders to treatment.
Salsburg (1986) presents an excellent discussion of the nonresponders problem.
He notes that many prospective drug compounds are designed to combat a particular
cause of disease that can be successfully identified and targeted. One example
cited by Salsburg (1986) is an enzyme-inhibitor compound for emphysema. Such
a compound shuts down the activity of a particular proteolytic enzyme that is
known to be a major factor in the presence of the disease. Other patients with
emphysema may not have the enzyme as a primary agent, so that the compound
will consequently not affect the emphysema.
In a similar vein, serum activity of certain enzymes increase when the disease
of hepatitis is present. Medical researchers may try to isolate the enzyme(s) and
investigate whether a lowering of the serum enzyme levels) will result in amelio-
ration of the hepatitis for patients. We may be able only to determine a specific
enzyme as a major risk factor by treating a patient with a compound designed to
specifically act on serum enzyme levels. In cases where the enzyme is the cause of
the disease, the patient will respond, in other cases he/she will not. There is no
external mechanism to determine whether or not a patient is a responder to the
treatment.
Besides enzyme-inhibitors, Salsburg (1986) mentions receptor site agonizers and
hormone-mimickers as prospective compounds to combat conditions such as con-
gestive heart failure and arthritis. Clinical investigations of such compounds will
encounter the same nonresponders problem. Boos and Brownie (1986) mention
the areas of nutritional supplementation and behavioral toxicology where the as-
sumption of the existence of nonresponders is reasonable. The use of antibiotics on
rats infected with the Herpes virus served as an illustration of the nonresponders
phenomenon for Good (1979).
Johnson, Verill, and Moore (1987) reported on a study by Carano and Moore
(1982) in which subject-to-subject variation of complex body mechanisms for deal-
ing with medications resulted in only a subset of subjects reacting to treatment.
Carano and Moore (1982) were interested in studying Sister Chromatid Exchanges
(SCE) as an indicator of carcinogenicity in human smokers. A larger frequency of
high SCE values per cell appeared in many subjects who were smokers, but subject-
to-subject variation was highly prevalent and made it difficult to detect a difference
in SCE distributions for smokers and nonsmokers. The damage in cells was de-
scribed as persistent although only occurring in a small fraction of them. Data for
a similar situation where chemotherapy is a toxic substance that is hypothesized
to increase SCE counts in some cells is given in the Johnson et al. (1987) article.
The counts for a sample of cells from a patient are measured before undergoing the
therapy and again for a second sample taken after administration.
The examples of the nonresponders phenomenon described thus far, particularly
the last one of chemotherapy, involve data that are completely observable. A nat-
ural extension is the scenario in which some of the observations are incomplete, or
censored. This is a feature for responses of the time-to-event nature, such as time-
to-relapse, time-to-death, time-to-recurrence, etc. A subject's observation may only
be known to exceed some value. In medical studies this aspect of censoring produces
what is called survival data.
A clinical trial of cancer patients who receive one of two therapies will involve
data of the time-to-event nature with the feature of censoring. Patients receiving
conventional radiation therapy are considered the "controls," while other patients
receiving a combination of radiation and chemotherapy make up the treatment
group. Conceivably, there may be patients where the chemotherapy has no effect,
and those treated nonresponder patients will have survival times like the control
patients. Clinical trials invariably produce patients surviving at time of analysis,
or perhaps patients "lost to follow-up" by reason of moving away, or death due to
a cause unrelated to treatment. Such patients will have censored times.
Other medical studies attempt to assess the effects of transplanted organs on
survival. The human body's tendency to reject foreign tissue may render the trans-
plantation ineffective, and the individual dies just as if no transplant was received.
1.2 Overview of this Manuscript
The presence of nonresponders in the treatment group is considered in this
dissertation. The potential of censoring on the collected data is taken into account.
Standard techniques as well as newly developed methods for detecting a treatment
effect in the presence of nonresponders and censored data are investigated.
Chapter Two presents statistical formulation of the nonresponder problem. A
model to describe this situation is given. Techniques put forth in the statistical
literature for the nonresponder problem are reviewed. These techniques consider
uncensored data. Chapter Three contains the development of methods for censored
data. Specific forms for these methods under chosen error distributions are given
in Chapter Four. Efficiencies, simulations, and illustrations on real data in order to
compare new methods with the standard methods make up Chapter Five. Summary
and conclusions are in Chapter Six.
The referencing system in this manuscript will follow the convention of number-
ing equations within section. Tables and figures will be numbered within chapter.
Theorems, Corollaries, and Lemmas are counted within section and are numbered
5
independently of one another. For example, the first theorem in Section 2.3 is Theo-
rem 2.3.1, and the first lemma in Section 2.3 is Lemma 2.3.1. The second referenced
equation in that section is (2.3.2).
CHAPTER 2
STATISTICAL FORMULATION OF THE NONRESPONDERS PROBLEM
2.1 Introduction
Statistical models for the nonresponders problem have been consistently written
in terms of cumulative distribution functions in the literature. To accommodate the
prevailing convention in the treatment of censored data, we shall present our models
in terms of survival functions.
In Section 2.2, models suitable for statistical treatment of the nonresponders
problem are formally presented along with notations defined for right-censored data.
A review of statistical research literature pertinent to the modelling for the nonre-
sponders problem is contained in Section 2.3.
2.2 The Mixture Model
Let Xo denote the response variable associated with members in the control
group, and let X1 denote the response variable for those subjects in the treatment
group. Call R the indicator variable which delineates whether or not a treated
subject is a responder, that is,
R 1, if the treated subject is a responder;
0, if the treated subject is a nonresponder.
Then the true proportion of responders in the treated population is given by
r = Pr(R = 1). (2.2.1)
The distribution of Xi, the response for a treated subject, can be represented as
a mixture of two conditional distributions--one the distribution of the response for a
responder subject and the other the distribution of the response for a nonresponder
subject. Let the associated survival functions for the two conditional distributions
be denoted by
FR(x) = Pr(Xi > x I R = 1); FNR(x) = Pr(Xl > x I R = 0),
respectively. Then with 7r as in (2.2.1),
G(x) = Pr(Xi > x)
= Pr(XI > x R = 1)Pr(R = 1) + Pr(Xi > x R = 0) Pr(R = 0)
= 7rFR(x) + (1 rT)FNR(X) (2.2.2)
is the survival function associated with the distribution of X1.
The following assumptions are made throughout the remainder of this disserta-
tion.
Al. The distribution of the measured value for a subject in the control group
is the same as the distribution of a measured value for a nonresponding
subject in the treatment group. That is,
F(x) = Pr(Xo > x) = FNR(X).
A2. The measured value for a responding subject in the treated group is
stochastically larger than the measured value for a treated nonresponder.
That is,
FR(z) > FNR(X) for all x with strict inequality for at least one z.
A3. Both FR and FNR are members of a family of survival functions indexed
by a real parameter A:
F = {F(x : ): 0 < A < +oo}, (2.2.3)
where F(x :A) is continuous with density f(x : A) and is such that
F(x : 0) = FNR(X) = F(x).
Before proceeding further, some comments concerning the three assumptions
are in order. As noted before, Al is a reasonable assumption in many practical
settings. It implies that nonresponding subjects in the treated group behave like
the subjects in the control group. The second assumption, A2, can be interpreted
as implying that the treatment effect, if present, will increase the probability that
the measured value for a responding subject will be larger than a given value. Of
course, the methods that are developed under A2 can be modified to the case where
A2 is replaced by the assumption
A2'. the measured value for a responding subject in the treated group is
stochastically smaller than the measured value for a treated nonrespon-
der. That is,
FR(x) < FNR(X) for all x with strict inequality for at least one x.
Assumption A3, and particularly (2.2.3), when used in the mixture model at
(2.2.2), implies that the survival function for an experimental subject has the form
G(x : r, A) = rF( : A) + (1 7r)F(x),
0 < < 1, O
where F(x) = F(x : 0) is the survival function for the control subjects. The
corresponding density function is given by
g(x : r, A) = rf(x : A) +(l 7r)f(x),
O< r < 1, < A < 00.
The survival functions at (2.2.4) represent a two-parameter family of models for
the two-sample nonresponders problem. In this family, the claim that "there is a
treatment effect" can be interpreted as the claim "a positive proportion of treated
subjects are responders." Thus, a test of the null hypothesis
Ho : No treatment effect,
against the research hypothesis
Ha : There is a treatment effect,
can be performed by testing
Ho : = 0 vs. Ha: R > 0.
(2.2.6)
An examination of the nonresponders model at (2.2.4) shows that the hypotheses
at (2.2.6) are equivalent to the hypotheses
Ho: G(x : 7r, A)= F(x) vs. Ha: G(x: 7r, A) > F(x). (2.2.7)
(2.2.5)
The research hypothesis at (2.2.7) means that X1 is stochastically larger than Xo.
Thus, any two-sample rank test for stochastic ordering, such as the Wilcoxon rank
sum test, will provide a nonparametric test for testing H0 vs. H,. However, since
these tests do not use the specific form of G(x) under Ha, it is natural to search
for tests that are more efficient for testing for treatment effects under the model at
(2.2.4).
For the case where there is no censoring of the data, several authors (Good,
1979; Boos and Brownie, 1986; Johnson et al., 1987; Conover and Salsburg, 1988)
have developed tests of Ho assuming one of the following forms for F(x : A):
(i) Location model: F(x A), -oo < x < oo, A > 0;
(ii) Scale model: F(x/eA), x > 0, A > 0;
(iii) Lehmann model: [F(x)]" -oo < x < oo, A > 0;
(iv) Lehmann model: 1 1 F(x)] -oo < x < oo, A > 0.
Note that F(x : A) is an explicit function of F(x) in the last two cases.
In the context of survival data, the model for scale alternatives appears more
appropriate because survival times are nonnegative random variables whose dis-
tributions are often skewed. However, since the natural logarithm transformation
T(x) = log(x) transforms the scale alternatives to location alternatives, rank-based
tests for the case of a scale model are identical to those for the location-shift model
under the log transformation. Accordingly, in the remainder of this work attention
is confined to location models only.
The models (iii) and (iv) have been explored by Conover and Salsburg (1988).
Their models are expressed in terms of cumulative distribution functions. Indeed, if
we assume our model (iii), then it is straightforward to show that the corresponding
cumulative distribution function is
1 G(x: r, A) = r [1 [F(Z)]] + (1 r)[1 F(x)],
which is their "Model 2." It should also be noted that their parameter a is related
to A through the relationship
a=ea, for a > 1.
Similarly, under model (iv),
1 G(z : r, A) = r [1 F(x)]^ + (1 r) [1 F(x)]
corresponds to the Conover and Salsburg (1988) "Model 1."
A Graphical Example
Consider the location-shift model
G(x : r, A) = rF(x A) + (1 7r)F(x)
for (2.2.4). Values of 7r, the true proportion of responders in the treated population,
will influence the shape of the density in (2.2.5):
g(x : A) = 7rf(x A) + (1 7r)f(x).
The shape of g(x : r, A) will match the shape of the density f(x) for Xo when 7 = 0
or 7r = 1. The degree of modification to the density shape increases as the value of
A tends away from zero and/or the value of Ir tends away from either zero or one.
For the purpose of illustration, let us consider the case where the random variable
X1 IR = 1 has the extreme-value distribution with density
f(x : A) = f(x A) = exp (-(x A) e-(-) .
The series of graphs comprising Figures 2.1-2.4 indicate how the values for r
and A dictate right-tail behavior for the distribution of X1. The location shift
parameter A is labelled Delta in the following graphs. The parameter r is labelled
Pi. When A = 0.5, the shapes of the distributions for X1 (solid line density for
the treatment population) and Xo (dotted line density for the control population)
remain quite similar for the range of values of ir. Differences in the densities only
become apparent for values of 7r > 0.4. In comparison, a distinction between the
densities of Xo and X1 when A = 1 can be seen as early as for the value 7r = 0.2.
The shape for the density associated with X1 "flattens out" at around 7r = 0.5 and
"recovers" as 7r moves away from 0.5. Figures 2.3-2.4 pronounce this phenomenon
more clearly, with suggestion of a bimodal shape.
Figures 2.1-2.4 give us an example of how a location shift parameter A and 7r
interact in terms of their influence on the density shape for the treatment group.
Random Censoring
The presence of censoring in survival data directly affects the information in
samples from the governing model. Several kinds of censoring mechanisms have
been investigated in the statistical literature, each different in its influence on con-
struction of a likelihood for the data. (For example, refer to Kalbfleisch and Prentice
Extreme-Value, Delta-0.5
PI-02
60I3
PI-O..
I 0 *
P0028
aI a a a a
PI-0.1
Pi-0.4
Figure 2.1: Graph of extreme-value mixture model; A = 0.5.
- Treatment density g(x : 7, 0.5)
--- Control density f(z)
PI-0.9
P140.9
I
4 0 a 4 4
P-0.7
-a 4 2 4 I
d
PI-0.1
* 2 4 *
PI-OA
4 P 0. 4
l\
PI.0.7
4 4
Extreme-Value, Delta-1
Pi602
4 2 4 4 a
PI-05
-t o |
P)-O4
2 0 2 4 I i
P10.3
4 8 2 a a
P10.6
I .
P0.9
!,
Figure 2.2: Graph of extreme-value mixture model; A = 1.0.
- Treatment density g(x : 7, 1.0)
-- Control density f(x)
Extreme-Value, Delta-2
PI-02
P1.0.4
P-W7
P6O
P6M
.2 2 2 4 S I
-I I 4 8 1
PI0.5
a 2 4 a a
Fb4~
P.3
-2 I 4 S 8
6!"
i
1 4
Pl6".
P5-o
- I 2
Figure 2.3: Graph of extreme-value mixture model; A = 2.0.
- Treatment density g(x : 7r, 2.0)
--- Control density f(x)
Pi0.1
P60A
Pi-0.7
- 5 2 4 aI i1a
Rsemme
Extreme-Value, Delta-3
PI-02
-a S 2 4 4 a 'S
PI-0.5
2 A
2 a 2 4 10
Pi i0
b: A A
* i i 1
*3 2 1
P 1-0.6
P6.O.6
S a 4 a l
PI.0.9
4 a a 4 $ 10a
I =ere=1
Figure 2.4: Graph of extreme-value mixture model; A = 3.0.
- Treatment density g(x : 7r, 3.0)
--- Control density f(x)
(1980, Ch. 5) for a survey and discussion of the different censoring types and their
appropriateness.)
There are N subjects under study, of which No are in the control group and
V1 are in the treatment group. Let r = 0 for the control group and r = 1 for the
treatment group. Associated with the ith subject in the rth group are two random
variables:
Xri = the potential failure time for the ith subject in group r,
Cr, = the potential censoring time for the ith subject in group r.
For the ith subject in group r, we observe the values of the random variable
X,i = min(Xi, Ci)
and
1,r if Xri = Xi;
0i if X7i = C. .
Thus 6,; takes on the value 1 if the observation for subject i in group r is not
censored and 0 otherwise.
Under the random censorship model, we assume
(1) (Xoi,Co0),..., (XoNo,CoNo), (XI1, C1),..., (X1N, CiN), are mutually in-
dependent pairs.
(2) X,i and C,i are independent and continuous, with survival functions
G(x : 7r, A) and L(x), respectively.
(3) The form of L(x) does not depend upon r or A.
The three assumptions of random censorship implies a noninformative structure
about the censoring mechanism. Most of the results in the statistical literature are
developed under these assumptions. One feature of the noninformative structure is
an opportunity to construct a partial likelihood for the observed data.
Other Notations
When it is necessary to use the random variable associated with the survival
function F(x), the notation of either F or F(X) will be employed in this dissertation.
This convention applies to other random variables as well. A (fixed or random)
function that takes on the values F(x) at the point x may be denoted by either
F(.) or F.
The notation
F,(x) = G(x) L(x) = Pr(Xi > x)
is for the survival function of Xi. Also,
(x) = g( )
G(x)
denotes the hazard function at time x, and
A(x) = jA(s)ds
denotes the cumulative hazard function at time x. The integral of a function such
as A(s) with respect to the variation of s over some interval in R (the real line) will
be denoted as in
0 A(s)ds.
This last integral could also be represented as a Lebesgue-Stieltjes integral with
respect to the total variation of A(x), such as in
fO,x) dA(s).
Estimates from the data of such quantities will be denoted using the "hat" notation;
e.g. F(x) is some reasonable estimate of F(x).
Realizations of Xri are denoted by x,i whether censoring occurs or not. Now sup-
pose that we wish to refer to all the observations irrespective of group membership.
Dropping the subscript r from x,i produces
X1, ... XN
as the observed times for the N subjects in the combined sample. Let ko (kl) denote
the number of distinct uncensored times in the control (treatment) sample, and set
k = ko + ki. Let
0 x(0) < X(1) < X(2) < ... < X(k) < X(k+l) oO
represent the ordered uncensored times in the combined sample. Within the interval
[x(i), x(+l)),i = 0,...,k, suppose that there are mi total censored observations
X(i)1, ..., (i)m,. Define
1, if x(i) corresponds to a treated subject;
0, otherwise.
Thus z(i) is the group membership indicator corresponding to x(i). The censored
values x(i)g, with ( = 1,...,mi, analogously have the covariates z(i)t to indicate
their group memberships. Let
mi
Mi = E z(i
(=1
Then Mi is the number of censoring times belonging to the treatment group in the
interval [X(,), X(;+i)). So mi Mi is the number of censored observations in the
control group that reside in [x(i), x(i+i)). The quantity
k
Ri= (mi + 1)
j=i
will be called "the size of the risk set at time x(i)" and is the number of subjects
whose values are known to be at least x(). The quantities mi, Mi, and Ri do not
have indices in parentheses since they are examples of functions of the x(i). Note
that (i,) does not follow this convention.
2.3 Literature Review
Subrahmanian, Subrahmanian, and Messeri (1975) noted that the use of the
independent sample Student's t-test will suffer from reduced power when the alter-
native hypothesis specifies a mixture of normal distributions. Good (1979) appears
to be the first to introduce the mixture model to account for the existence of non-
responders in the treatment group. To account for the fact that the presence of
nonresponders will decrease the difference between the means of the two groups
and increase the variance of the treatment group, Good (1979) proposed a new
randomization test based on the statistic
X Xo]2
v(0.67) = 0.67 1 I+ + (1 0.67)(N 1)S, (2.3.1)
0+- Vj
where Xo (XI) is the sample mean for the control (treatment) group, and S2 is the
usual sample variance for the treatment group. Notice that the test statistic is a
weighted sum of functions involving the difference in group means and the variance
for the treatment group. The choice of 0.67 by Good (1979) as the weight was made
in hopes of wide applicability of the test over the unknown range of values for ir
and A. Other weights within the interval [0,1] could be used-a weight of 1.0 gives
Fisher's randomization t-test.
The empirical power investigations of Good (1979, Table 1) determined that
the test based on v(0.67) in (2.3.1) performed markedly better over the range of
values 7r E (0, 0.8] than Student's t-test for sample sizes under ten when F(x : A)
represented the standard normal distribution and A = 1,2, or 4. It also beat out
Fisher's randomization t-test over the entire range r E (0, 1] when the underlying
distributions were log-normal. Applications of the test based on v(0.67) in (2.3.1)
showed its evidence of increased "sensitivity" (Good, 1979) relative to Student's t
for real data that exhibit the nonresponders situation. Presumably, the existence of
greater variability in addition to a shift in location for the treatment group responses
made the test based on (2.3.1) better suited than the t-test for rejecting the null
hypothesis of "no treatment effect."
Boos and Brownie (1986) addressed some interesting questions raised by Good's
(1979) paper. The interpretation of a significant p-value was one issue. Boos and
Brownie (1986) emphasized that acceptance of the alternative hypothesis does not
necessarily imply a shift in mean. An increase in the variability could be solely
22
responsible. The graphs in Figure 2.4 indicate how a small r value and a large A
value produces more variability in the treatment group distribution although the
means of the two groups are similar. In addition to correcting the procedure for
implementing a one-sided test with (2.3.1), Boos and Brownie (1986) used Monte
Carlo study results to argue that Student's t-test and the Wilcoxon rank sum test
are at least as powerful as the test based on (2.3.1) when the underlying distributions
were normal or extreme-value and 7r > 0.6. The advantage of (2.3.1) appeared as A
increased and 7r was less than 0.6. Group sample sizes of eight were used. A study
with group sample sizes of twenty did not include (2.3.1); Boos and Brownie (1986)
justified its omission with a statement by Good (1979) of diminished effectiveness
of (2.3.1) as No and N1 grew larger.
The non-technical article by Salsburg (1986), which discussed several applica-
tions of the nonresponder phenomenon, indicates that an interpretation of some
change in the treatment group whether it be primarily due to variability or shift in
mean is useful in preliminary investigations of new clinical compounds. The sample
sizes in such studies are usually small, so that the presence of nonresponders will
decrease power of tests such as Student's t and the Wilcoxon.
The use of Lehmann alternatives in the mixture model for the treatment group
was proposed in a subsequent article authored by Conover and Salsburg (1988). Un-
like the articles of Good (1979) and Boos and Brownie (1986), a formal mechanism
to derive tests sensitive to the alternative hypothesis was used. The mechanism was
Conover's (1973) technique for deriving a locally most powerful rank test (LMPRT)
under more general models than the usual location-shift or scale change settings.
The LMPRT is a linear rank test that is a sum of expected or approximate scores
specifically generated from the given model and assigned to the observations.
Recall that the mixture model (2.2.4) has two parameters, 7r and A. Conover and
Salsburg (1988) took the approach of fixing one of the parameters and deriving the
LMPRT scores for the other. In both cases, the score functions were independent
of 7r. Since both r and A indicate effect due to treatment in the model at (2.2.4),
Conover and Salsburg (1988) were interested in a loss of power that would arise
in using one score function when the other score function was more appropriate.
A comparison of the two score functions in terms of Pitman asymptotic relative
efficiency (ARE) lent creedence to a compromise score function based on a value of
A that maximized the Pitman ARE.
Test statistics based on the scores developed by Conover and Salsburg (1988)
were shown to be asymptotically normal in distribution. For smaller samples,
Conover and Salsburg (1988) suggest the use of the two-sample Student's t-test
computed on the scores. Monte Carlo studies indicated that the empirical signif-
icance levels for the tests were close to the chosen nominal significance level for
sample sizes as few as five in each of the groups when the t-test approximation was
employed. Applying the tests on two examples of real data indicates better perfor-
mance than Student's t, the Wilcoxon, and Good's test based on (2.3.1) in the sense
of calculated p-values. One example, which compares pain values for control and
treated patients undergoing acute painful diabetic neuropathy had sample sizes of
No = N1 = 10. The other example of SGOT liver function values for heart patients
had samples sizes of No = 28 and N1 = 30.
A similar formal approach of ranks was taken by Johnson et al. (1987), where a
location-shift was postulated for F(x : A). Johnson et al. (1987) clearly stated the
testing problem to be the one considered in this manuscript at (2.2.6):
Ho: r = 0 vs. Ha: r >0.
The linear rank tests derived by Johnson et al. (1987) were noted to be based on
score functions that accentuated large responses, which is contrary to the usual
aspect of score functions for rank tests such as the Wilcoxon. The example in
Chapter One of this manuscript that referred to higher SCE counts for patients
receiving chemotherapy served as an illustration of the usefulness of the tests derived
by Johnson et al. (1987). In asymptotic relative efficiencies and Monte Carlo studies,
the derived tests performed better than standard tests such as the Wilcoxon for a
broad range of configurations that reflect the nonresponders problem.
Two underlying forms for F(x : A) were considered-uniform and normal. The
normal distribution with a location shift generated a score function that was de-
pendent on A. The resulting linear rank test could then be used to get an exact
conditional distribution under the null hypothesis. As the sample sizes No and
N1 increase, computations become prohibitive and an asymptotic distribution is
convenient. Also, the score functions may not have tractable expectations so that
approximate scores that are asymptotically equivalent are useful. Results for dealing
with these problems and others are systematically dealt with in the Johnson et
al. (1987) article. The following subsection summarizes their findings.
Properties of Uncensored Data Linear Rank Tests
Under the mixture model at (2.2.4), the score function assigned to z(i) is q(ui),
where
ui = 1 F(x() : 0) = 1 F(x(i)),
and
f(F-l(1-- u) A)
O(u) = f(F u) 1. (2.3.2)
f(F-1(1 u))
The function (2.3.2) will be subsequently designated the "mixed-model" score func-
tion.
The form of the statistic for the LMPRT for testing H0 : r = 0 vs. H. : 7 > 0 is
N
T = z(i) ()(ui)), (2.3.3)
i=1
with the expectation taken over the joint density of the order statistics of a
random sample of size N from a uniform (0,1) distribution. The fact that the ith
order statistic from a uniform (0,1) distribution has a beta distribution with density
N-
N! 1 u -1(1 u1)(N+-)-1 0 < ui < 1,
(i 1)! (N i)!u 0
can be utilized to evaluate T under specific forms for 0 such as when the underlying
distributions F(x A), A > 0 are normal.
Exact significance levels associated with T at (2.3.3) can then be calculated via
the permutation principle. The score function (2.3.2) can also be used with the
approximation
(u.)) (u.) = [ iN+ 1-
to get a statistic
N
T*= Ez(i) (2.3.4)
that is asymptotically equivalent to T at (2.3.3). The approximate scores not only
have easier computational forms but also possess closed-form expressions for most
of the common distributions.
Since computations with all possible permutations can quickly become too in-
tensive as N increases, asymptotic results are employed for assessing significance
levels when N is large. Under mild regularity conditions, standardized versions of
both (2.3.3) and (2.3.4) will be asymptotically normally distributed under the null
hypothesis.
THEOREM 2.3.1 Assume that
(i) 0(u) as defined by (2.3.2) is monotonic.
1
0
(ii) f 2(u)du < oo.
If v is either (2.3.3) or (2.3.4), then under the null hypothesis Ho : r = 0,
ar(v)
where
Var(v) = NN1 (u)du
No + N, J
converges in distribution to a standard normal random variable as min(No, NI) --
oo.
Proof. See Johnson et al. (1987), Theorem 3.1. 0
Behavior of the statistics T at (2.3.3) and T* at (2.3.4) under alternative hy-
potheses has been handled in the context of sequences of local parameter values.
Johnson et al. (1987) provide formal details in the appendix of their article. The
distributions of T and T* under sequences of local alternative hypotheses are asymp-
totically normal and can be used to obtain efficiency expressions. As mentioned
before in review of the Conover and Salsburg (1988) article, the Pitman ARE can
be used to help determine a value for A in order to conduct the test. Let a > 0 be
some constant. Under a limiting sequence of local alternatives
a
r* = 7, min(No, Ni) -- oo, (2.3.5)
the Pitman ARE can be computed to explore loss of efficiency for a particular LM-
PRT due to inaccurate specifications such as the value of A. Since the test statistics
depend on A, a choice of some value to substitute for a unknown parameter is an
important issue. This section introduces some evidence of the problem. Further
study in hopes of finding adequate solutions is discussed in later chapters, after the
censored data case is considered.
Suppose ch(u) denotes the chosen score function to be used, while Ot(u) denotes
the score function which contains the true value of A. That is, let
/ch (u) = -1
'Oh(u) = f(F-1(1 u) Ah) 1,
f(F-'(1 u))
and
Sf(F-(l u) At)
f (F-1(1 u)) -
where Ach is the chosen value of A, and At is the true value of A.
THEOREM 2.3.2 Under the conditions of Theorem 2.3.1 and (2.3.5), the Pitman
ARE of the test statistic v based on Och(u) relative to #t(u) is
J [ch(u)t(u)du
eff(Ah, At) )= 10
f (udu J(u)du
Proof. See Johnson et. al (1987), Theorem 3.2. 0
The computation of a Pitman ARE may be simplified with the next result.
LEMMA 2.3.1 For the mixed-model uncensored score function 0(u),
1
/ (u)du = 0,
0
or equivalently,
/f(F-1(1-u A)du = 1
f (F-(1 u))
Proof. The integration substitution x = F-1(1 u) produces
1 00f :A
I(u)du = f ) f(x)dx 1
0 --oo
= 1-1=0.0
Cross product terms that arise when components of the Pitman ARE are ex-
panded out will involve the same type of integral as given in Lemma 2.3.1, thereby
leading to simplified evaluations.
An Example
Theorem 2.3.2 can be used to evaluate a (Pitman) ARE for some particular
distributions when the unknown parameter is misspecified. Any mention of ARE
in this dissertation implicitly refers to Pitman asymptotic relative efficiency.
The ARE expression in the theorem can also be employed to compare the mixed-
model scores test with the linear rank test derived under the usual location-shift
model. Score functions such as those for the Wilcoxon and log-rank take the place
of Och(u). The mixed-model scores are substituted into 4t(u).
An illustration using the normal distribution for F(x A) is now presented.
Recall that Johnson et al. (1987) chose the normal distribution to derive a mixed-
model score function. First we note that because 0(u) does not have a closed-form,
the expected score corresponding to x(i) must be numerically evaluated. A general
expression for the expected score is
CN! 1
= (i -1)! (N i)! (i)(Ui)i1 u)
Now refer to Theorem 2.3.1. Since all components of the first derivative
AeF-(1-u)
( f(F-1(1- u))
exceed 0, condition (i) holds. Also, condition (ii) holds for any A > 0, since by
Table 2.1: Pitman ARE for misspecified A, mixed normal distribution
At
Ach 0.5 1.0 1.5 2.0 2.5 3.0
0.5 1.00 .860 .518 .194 .042 .009
1.0 .860 1.00 .831 .443 .141 .026
1.5 .518 .831 1.00 .801 .393 .115
2.0 .194 .443 .801 1.00 .784 .732
2.5 .042 .141 .393 .784 1.00 .780
3.0 .009 .026 .115 .372 .780 1.00
straightforward integration,
1
J0(u)du = e2 -1.
o
This provides the asymptotic properties of v when the underlying survival distribu-
tion is mixed normal.
For studying misspecification of A, Table 2.1 is an extension of Table 3 in John-
son et. al (1987). Again by straightforward integration, the algebraic form of the
ARE for misspecified A is
Le-1 +A2-At+&,h)2) 2]
ef[e A ] [ei i]
Now we turn to model misspecification. Table 2.2 is an extension of Table 3 in
Johnson et al. (1987). In comparison to the normal scores tests of Fisher & Yates
Table 2.2: Pitman ARE for location-model vs. mixed-model linear rank test,
mixed normal distribution
A for mixed-model score
0.5 1.0 1.5 2.0 2.5 3.0
0.880 0.582 0.265 0.075 0.012 0.001
(1963) and van der Waerden (1953), Table 2.2 indicates that the mixed-model scores
have a distinct advantage for A > 1.0. This was pointed out in Table 1 of Johnson et
al. (1987). The SCE data example in Johnson et al. (1987) makes the only mention
by the authors on a choice for A in practice, which was A = 1. A look at the results
of their Monte Carlo study suggests that the mixed-model test when the underlying
distributions are normal has better power than the standard location-shift normal
scores test or the Wilcoxon test. This was particularly true for true values of A = 2
or A = 3 and samples sizes of 10, 20, and 40 per group. Choosing A = 0.5 or 1.0 or
1.5 or 2.0 to calculate the test statistic gave very similar empirical powers in most
cases.
Discussion
The articles that we have reviewed in the literature have dealt with rank-based
methods for the nonresponders problem with complete observations. A believed
advantage of deriving scores based on ranks is their robustness to departures from
the assumed underlying distributions F(z : A). Additionally, the invariance to
monotone transformations is a worthwhile feature. The property of locally most
powerful rank tests is appealing as well, since it is usually the case that such rank
tests do well for nonlocal alternatives. Extensions of these procedures constitute
the thrust of the remainder of this manuscript. One topic is the use of other dis-
tributional forms for F(x : A) other than the normal. The main interest, however,
is to develop scores and linear rank tests that allow for the potential of censored
observations. One way to derive such scores is from the so-called rank likelihood.
Once such scores are generated, distributional properties need to be considered for
inferential purposes.
The issue of how to choose A to conduct the test in practice demands more
attention than what has been given in the literature. To this end, more study is
warranted on the behavior of the tests as different values of A are chosen.
CHAPTER 3
METHODS FOR CENSORED DATA
3.1 Introduction
Investigation of the properties associated with a linear rank test statistic for
censored data requires a different approach than that for the case where no censoring
occurs. A useful technique in similar and common inference situations is to cast the
test statistic as a stochastic integral. This enables the use of the theory of counting
processes and martingales to derive asymptotic properties. Aalen (1978) and Gill
(1980) were responsible for much of the foundational work of this approach. Their
findings have been applied numerous times by researchers in the area of censored
data linear rank statistics.
In Section Two of this chapter, a rank-based likelihood is derived for censored
data under the mixture model (2.2.4). The likelihood is used to determine the form
of the linear rank statistic, v, for testing Ho : 7 = 0. The equivalent expression
of v as a stochastic integral is also derived. Section Three explores asymptotic
equivalence of linear rank statistics when v employs certain scores that approximate
the expected scores. Also introduced are some preliminary results that are needed
to establish the asymptotic normality of a properly standardized version of v. The
asymptotic properties under the null hypothesis make up the content of Section
Four. The results and discussion in Section Five focus upon score expressions for
33
I
v that may be useful in practical applications. Properties of v under alternative
hypotheses are presented in Section Six.
3.2 The Linear Rank Test Statistic v
Rank Likelihood for the Data
Kalbfleisch and Prentice (1973) defined a censored data rank statistic as the set
of all rank vectors of uncensored survival times that can give rise to the observed
ranks of the data. The probability of the observed censored data rank statistic,
P(r), can be calculated by summing up the probabilities of all possible underlying
rank vectors given the censoring pattern of the data. All possible rankings of the
sample that might be observed if we could measure the actual survival times for
each subject are added together to get P(r).
The Kalbfleisch and Prentice (1973) formulation does not take into account the
order of the censored observations inside an interval defined by two adjacent ordered
uncensored values. Sacrifice of this ordering information should be negligible in the
relative presence of a decent number of uncensored observations (Prentice, 1978).
Recall the mixture model at (2.2.4) as
G(x : r, A) = G(x) = rF(x : A) + (1 r)F(x),
with A > 0 and 7r E [0, 1]. A representation of P(r) for data sampled under model
(2.2.4) and the random censorship structure is
k
P(r) = /. J f(X,)1)-'l [Wf(7() : A) + (1 7r)f(x() : 0)]6)
X(I)
x [F(x() : 0)]m"-M'[rF((,) : A) + (1 r)F(x() : 0)]M dx(). (3.2.1)
Because the censoring distributions are assumed to not depend on Ir and A, the
censoring distributions are not included in the likelihood expression at (3.2.1). This
"rank likelihood" allows generation of scores in linear rank statistics for various
choices of F(x : A), A > 0. A mechanism for deriving linear rank tests from
P(r) parallels the technique for constructing locally most powerful rank tests with
uncensored data for the usual types of alternatives-location, scale, and regression
(e.g. Hajek and Sidik, 1967).
A General Form
There are three conditions of Hajek and Sidak (1967, p. 70) that are sufficient for
deriving locally most powerful rank tests. A version of these conditions appropriate
for the purposes of this dissertation is:
(i) For some e > 0, the survival function G(x : r, A) and the density
g(x : 7r, A) are continuous in r : :r E [0, e] for every x.
(ii) The limits
dG(x: r, A) .iG(x : 7, A) G( : 0, A) l G(x : r,A)-F(x)
dir r=O -.,o 7 .-O -7
and
dg(x : r,A) g(X :r,A)--g(x :O,A) im g(x :r, A) f(x)
=li : lim g (m
dTr ,=O r--0 7" r-o
exist for every x.
(iii)
( : 7d da < oo for every A
0 dir
-00
and
SdG(x:rA) f() d < oo for every A.
dir F(x)
-00
We now introduce the general form of v in the following theorem.
THEOREM 3.2.1 Consider the two-sample data situation given by (2.2.4):
G(x) = G(x : r, A) = 7rF(x : A) + (1 r)F(x).
Suppose the support of F(x) is contained within that of F(x : A), and there exists
a positive constant B < oo such that
SF(x) f() dx = B. (3.2.2)
-oo
Then a rank test for testing
Ho: = 0 vs. H,: > 0
in the model (2.2.4) can be based on the statistic
k
v = z(zci + MC) (3.2.3)
i=1
where
k
ci = ..]. (u,) {Rj(1 ui)m du,}, (3.2.4)
l<
and
C,= /.. (u) {R(1 u)m dua}, (3.2.5)
u <...
with
= f(F-1(1 u): A)
f(F-1(1 u))
and
Iu) F(F-1(1 u) : A) F(F-(1 u): A)
F(F-1(1 u)) (1 u)
Proof. Following Johnson et al. (1987, p. 654), a proof of Theorem 3.2.1 can be
given by verifying the three conditions on page 35 of this manuscript. For some
e > 0, (i) the survival function G(x : r, A) = (1 r)F(x) + rF(x : A) and the
density g(x : 7r,A) = (1 7r)f(x) + f(x : A) will be continuous in r : r E [0,e]
for every x. The one-sided nature of the interval [0, e] does not affect the validity of
Hajek and Sidak's conditions according to the argument of Johnson et al. (1987).
Next, (ii) the limits
lim G(: 7r, A) G(x: 0, A) = im G(x) F(x)
r-O 7 r-*O r7
= lim (1 r)F(x) + rF(: A) F(z)
= F( : A) F(z)
and
g( : r,A) g(x: 0,A) limg() f()
lim = im
7r-O r-.O
lim (1 r)f(x) + Irf(x: A) f(x)
r-+O 7
= f(x: A) f(x)
exist for every x; and (iii)
+00
SJIG(x : -r,A)- G(: 0,A)(x : A
S G(x : 0, A): OA)(dx
-00
+ F( : A) F(x)j
=- F(x) f(x)dx
-00
+00 +00
< F(x:A)f( + f F (x))dx
-00 -00
+00
J F(x : A)
f(x) dx + 1
-00
=B+1< oo
when the supposition at (3.2.2) holds, and
+00
I g(x: r,A) -g(x : O,A)dx
-00
+00
= Jf (fx:A)-f(x)\dx
-00
+00 +00
f If(x: A)( dx+ If(x)ldx
-00 -00
= 1+1= 2 < o.
To derive the linear rank test statistic based on (3.2.1), refer to Prentice (1978) as
a basis for the following representations. Conditions (i)-(iii) being satisfied ensures
that differentiation and integration interchange is permissible. The rank probability
(3.2.1) can be expressed as
P(r)= ... (2-dx(<),
X(1)
where Pi represents the contribution to the likelihood of failure time xz( and the
censored observations x(o)i,... (m, contained in the interval [x(),x(i+l)):
Pi = f(x(i))-z' [rf (x() : A) + (1 7r)f(x(i : 0)](')
x [F((,) : O)]m,-M'[7F(x(.) : A) + (1 r)F(() : 0)]M. dx().
Then
Slog P(r) P(r) /P()
P(r) ) "'"II f1dx(
p(r) [J ... Ik Pax(]
4 I 1 dX*]
(r) / i= /
[I '" log Pj) Pi dx(o ]
P(r)= ilog PI Pidx(i) (3.2.6)
1r) [=[1 i=1
Evaluation of the right-hand side of (3.2.6) at the null value 7r = 0 provides the
form of the linear rank statistic given by (3.2.3). To illustrate this, first note that
P(r) = II f (x())F(x(,))m' dx(),
r=O X(i)<...
which can be directly integrated (Prentice, 1978) to arrive at the result
P(r) = H -. (3.2.7)
r=o i=1
Next, observe that
a log Pi
i=1 r=O
= -{(1 z(j))log f(x()) + (mi Mi)logF(x(1))
i=1
+z() log[7rf(x() : A) + (1 r)f (x())]
+ Milog[7rF(x(:) A) + (1 7r)F(x())]}
Ir o
= (x() A) + ( r)f(x(i))]
= [ z,)[rf(x(3) A) + (1 :)f(xy)]
+ F((i : A) F(zx())
[rF(x() : A) + (1 r)F(x(.))] --o
k f f(x() : A) f(x(,)) F(x() : A) F(x())
f(xzi) F(x() f
= z f( (i) :) 1 + M [ F(x()) -: (3.2.8)
Putting (3.2.6)-(3.2.8) together and then using the substitution uj = 1 F(x(j))
yields the linear rank test statistic given by (3.2.3) with scores (3.2.4) and (3.2.5).
0
The score functions (3.2.3) were reported in Johnson et al. (1987) for the situa-
tion of uncensored data and location-shift form for F(x : A). Conover and Salsburg
(1988) derived specific scores corresponding to a Lehmann-alternative choice of
F(x : A). The next corollary summarizes the notion that the resulting linear rank
statistics provide locally most powerful rank tests in the uncensored data case.
COROLLARY 3.2.1 Under the conditions of Theorem 3.2.1, suppose that there are
no censored observations. The locally most powerful rank test (LMPRT) is given by
the form (3.2.3) with the scores (3.2.4).
Proof. Set Mi = 0 in (3.2.3). This gives (2.3.3), the LMPRT as discussed in Section
2.3. o
The following lemma contains a sufficient condition for (3.2.2) in Theorem 3.2.1
when F(x : A) = F(x A).
41
LEMMA 3.2.1 Let B be a positive constant such that B < oo, and let (.-) be defined
as in (2.3.2) or (3.2.4). If F(x : A) = F(x A), then
2(u)du = /(F-[(1 u) A)du < B/2
0 0
implies that
oo F(X-A)
I F (x)) f(x)dx < B.
Proof. Make the substitution u = 1 F(x) and integrate-by-parts to show
7 F(x A) F(F-'(1 u) A)d
I F(x) 1 u
-oo 0
= [-log(1 u)F(F-(1 u) )
/ ^f (F-'(1 u) A)
log( ) f(F (1 u) A)du. (3.2.9)
0 f(F-1(1 u))
Now the first term on the right-hand side of (3.2.9) contains the form oo 0 when
evaluated at u = 1 and thus requires some extra handling to show that its value is
zero. Let I[A] denote the indicator function which takes on the value 1 when the
event A occurs and 0 otherwise. Let X be the random variable with the survival
function F. The substitution x = F-l(1 u) leads to
lim log(1 u)F(F-(1 u) A)] = lim log(F(x))F(x A)
u-"*1 X"-oo
= lim- log(F(x))E[x>,_a]
< lim E[-log(F(X))]r[x>z-a]
(for log(F) is monotone increasing)
= lim [- log(F(X))]
X--00
limn [- log(F(X))]ZTx<;_-Al
= 1-1=0
by the Monotone Convergence Theorem and the fact that
1
S[- log(F(X))] = log(1 u)du = 1.
0
The second term on the right-hand side of (3.2.9) is also finite since
2
/ g f(F-i(1 u) A)) 2
log(1 f(F1(--u)- du
<([log(,1- u)]2 du) ([f (F-1(1 u) A)
< 2. (B/2) < oo
by the Cauchy-Schwarz inequality and by the sufficient condition of the lemma. 0
The condition in Lemma 3.2.1 is closely related to large-sample properties for
the linear rank statistic v. As can be seen from Theorem 2.3.1, it is the primary
component of an expression for the asymptotic variance of v with uncensored data,
which is required to be finite.
Alternate Summation Form for v
It has been common and convenient in the literature to display linear rank statis-
tics for testing equality of survival distributions in the form of a sum of weighted
differences between the number of observed deaths and the number of conditionally
"expected" deaths. Conditioning is done on the prior failure and censoring history
observed. Tarone and Ware (1977) and Prentice and Marek (1979) formally discuss
this form for some of the more well-known test statistics. Authors such as Harring-
ton and Fleming (1982) consider whole classes of statistics that are distinguished
by the choice of the weights on the differences.
Recall the definition R, of the size of the risk set in Section 2.2. Let Rli be
the number of treated subjects in the risk set at failure time x(i) and Roi be the
corresponding number in the control group. Note that Roi + Rui = Ri. The present
goal is to represent v (defined in (3.2.3)-(3.2.5)) in the following form:
E w z(O) (3.2.10)
where
wi = c C (3.2.11)
is the weight function associated with the failure time x(i).
A sufficient condition for v to have the form at (3.2.10) and (3.2.11) is that the
scores ci and C, satisfy
RCi-C = ci + (Ri 1)C; i = 1,...,k, (3.2.12)
where Co = 0. The condition at (3.2.12) was first derived by Prentice and Marek
(1979), who used it to represent the log-rank (Mantel 1966; Cox 1972), Gehan
(1965), and Peto & Peto (1972) statistics in the form (3.2.10). Since v can be
rewritten (Prentice and Marek, 1979) as the sum of the form (3.2.10) and a quantity:
v = wi (z() -R + I z(i) (ci Ci Ri(Ci-I Ci)), (3.2.13)
1=1 i =1
where the second term on the right-hand side of (3.2.13) equals zero when (3.2.12)
holds, we concentrate on showing that the relation (3.2.12) indeed holds for the
mixed-model scores c, and Ci.
In full notation, (3.2.12) is expressed as
F(F-'(1 u-) : A) k
F1 Rj(1 uj)m' dui
u <.
ff (F-1(1 u) : A) k
J..J R(F-'(l1 ui)) -1 k Rj(1 uj)"i dui
ul <.
+((I F(F-l(1 u,) : A)
< ui<'-__
k__
x iI Rj(1 uj)j' duj.
j=1
Expanding the above full version of (3.2.12) yields
/../ (F(F- t(1 ~i-) :F R I R(1 u,) du,
ul <...__
( f (F-'(1 u,) : A) k__
J (f (F-1(1 u1)) j R '(1 us)'m du,
Ul<...
A 1) F(F-1(1 uj) RA) +
U1<...__
k__
x J< Rj(1 uj)m, duj.
j=1
Cancelling common terms on both sides and simplifying gives for (3.2.12):
F(F-1(1 unj-) : A) k
-- i F(F-(1- ui) Rj(1 -uj)' du
u<...__
f (I / f(F-'(1 u) :A)\ )__
=( 1f(F1I(l )) } I R,(1 u,)m' duj
f(F-1(1 ui)) ,.=l
) j F(F-1(1 u) : A)
+ < F(F-1(1 u,))
k
x iI R,(1 uj)' duj. (3.2.14)
j=1
Assume that Ri > 1. We adapt a technique employed by Mehrotra, Michalek,
and Mihalko (1982). Starting with the second term on the right-hand side of
(3.2.14), first integrate out ui+1,... k to get
(R 1)1 1 F(F-'(1 u) : A)( uRi
JuiI 0 a F(F-1(1 uj))
i-1
x I {R(1 uj)'m' du,}du,
j=1
=(R-)- J1) n{R(1l-uj)"' du,}
U-_2 j=
x F(F- R(1 ui)R-.du. (3.2.15)
Ju,_- (1 ui)
Next, apply the integration-by-parts technique to the integral over u,. Let
U = F(F-(1 ui) : A), and
dV = Ri(1 ui)R-dui.
By the sequence of relations
y =F-l(1-u),
F(y) = 1- u,
f(y)dy = dui,
dy 1
dui f(y)'
it follows that
dl dF--(1 a )
d f(F-1(1 u) : A)dF-(1 ui)
dui dui
= -f(F-i(1 u) :A) d
du,
f(F-1(1 u) : A)
f(F-(1 u)) '
and
V = R- ( i
Use of these relations in the last integral of (3.2.15) gives
F1 F(F-(l ui): A)Ri(1- ui)R- du
= j F(F-1(1 u) : A)d [- 1( u)-1
= R- F(F-( u,) A)(1 u)1] 1
+ f (F1(1 uj) A) R .-1
(R,1 u) d
I,_i f (F-(1-u)) R4 1
S f(F(1 u ): A u)R- (3.2.16)
R 1 Ju.- f (F-I (1 -)) "u))
[= {F(F-(1I- ui-) : A)(1 us) '-]1
l 1 f(- (1-: ) : )
+ u ) d(F- 1 ) (3.2.16)
Then with the additional relation implied by the definition of Rj in Section 2.2:
m, = R1 Rj-1 1, j =1,..., k,
substituted into (3.2.16), we get
(R1 1)(C, + 1)
R-1
x ." F(F-(1 ui-1 ) : A)(1 ui-) 'l-
Ri 1
-(Ri 1) {Rf(1 u)m du}
wi-2 j=
Ri l f(F-(1 ui) : A) (1
XRi,- 1 f(F-1(1 ui))
=^ 1 1* /'i-2}
= R, Ii 1 i {Ri(1 us)mJdu1}
(1 u.-1)
1 1 i-1
si- j=1
f (F-1( i): A)(1 ) du
f (F-I(1 ui))
= Ri(Ci-I + 1) (ci + 1),
which is just a re-expression of (3.2.12) or (3.2.14) since ci and Ci are of the same
form after integration over ui+1,..., uk.
Now if Ri = 1, we have k = i, and mk = 0. Here, the condition (3.2.12) reduces
to
ck = Ck-. (3.2.17)
The relation (3.2.17) can be verified as follows:
c 1 f(F-1(1 uk) : A)
x f (H-{R (1- u ) dui }duk
lk-2 1 =
-2 1 (1 Uk-1) 1
k-2
xR(k-1)(l U(k-1))1+Mk-I I{R(1 uj)m'duj}duk
j=1
1 .0 [F(F-(1 uk ) : A) -
U-2 J(1 -U k-1)
k-2
xRk-I(1 Uk-1)R-1-1I R( uj)"du}duk
j=1
= Ck-1
Stochastic Integral Form for v
The statistic at (3.2.10) can be readily translated into the form of a stochastic
integral. Recall that the control group sample is labelled sample 0; the treatment
group sample is labelled sample 1.
Recall the notation defined in Section 2.2, and recall the definition of indicator
function I(A) = 1 or 0, according to whether the event A occurs or does not occur.
For the groups r = 0, 1, with survival time random variables
Xr, j = 1,..., N,
and censoring indicators (as defined in Section 2.2)
6,j, j = ,...,
let
Nr
N,(x) = fI(x,j < X6, = 1).
j=1
Note AfN(x) is the number of failures for group r that occur no later than time x.
Also, let
N,
Y.(x) = ZI(Xx, > x),
j=l
the size of the risk set for group r at time x. Note that Y,(x(i)) = Rri in our earlier
notation for risk set sizes. Let
YI() + Yo(x)= Y(x),
n (x) + Ao(x) = A(x),
and
dN,(x) = NX(x) N,(x-).
The assumptions of continuous survival distributions implies that the probability
of ties among the x(i) is zero. Thus with probability 1,
d (x) = Z(i), if X = (i);
0, otherwise,
and
f1 (i), if X = (i);
0, otherwise.
So a stochastic integral expression for v can be written as
= (i) + RoR _Ri\
=1li
= wz(i) wi (1 zA)
k w(x) d (x) (w) d (x)
= ,(z) Yo(X)YY(x) 0Y(x) dYo(z)
-oo Yo(X)) + ( YW) (x) Yo(x) '
(3.2.18)
where
w(x) = ci Ci, X(i) < x < Xi+1.
3.3 Conditions for Asymptotic Properties Under Ho
The formulation of v as a stochastic integral allows us to apply the martingale
theory of Gill (1980) with suitable adaptations. His work includes the statement
of certain conditions that are sufficient to derive an asymptotic distribution for a
50
properly standardized version of v. Recall the definitions of 0(u) and O(u) at (3.2.4)
and (3.2.5), respectively. The notation of 0(u) and 4(u) as score functions that cor-
respond to uncensored and censored observations, respectively, was introduced by
Prentice (1978) in the framework of regression models. Subsequently, the notation
has been used in the area of censored data linear rank statistics (Harrington and
Fleming, 1982; Cuzick, 1985). Thus we adopt the notation for the remainder of this
dissertation.
In this section we lead up to sufficient conditions that permit investigation of the
asymptotic properties of the censored data test based on v. In the process we will
show that for large samples, the expected scores ci and Ci in v can be respectively
replaced with approximate scores that are easy to compute.
Approximate Scores
The computational complexities associated with linear rank procedures that
use expected value scores ci = E[4(ui)] has long been recognized in the literature.
For uncensored data linear rank procedures designated to detect a shift in a dis-
tribution, for example, the normal (van der Waerden, 1953) approximate scores
c, = F-'(1 j ), where 1 F is the cumulative distribution function of the
standard normal distribution, are much easier to evaluate than the expected scores.
In fact, the normal expected scores are the E(Z()), i = 1,... N, where Z(i) is the
ith order statistic in a random sample of size N from the standard normal distri-
bution. These scores involve integrals that do not have closed-form expressions.
More so in the censored data case, the expected scores involve integrals that do
51
not have streamlined closed forms and require numerical integration methods to be
evaluated.
Prentice (1978) conjectured that the expected scores ci and Ci from the rank
likelihood he derived could be approximated by the respective scores 0(1 F(x(i)))
and 4(1 F(x())). A subsequent article by Cuzick (1985) confirms the truth of the
conjecture with the requirement that some fairly mild conditions on the uncensored
score function q(u) be satisfied. The resulting asymptotic equivalence of the linear
rank tests based on the two types of scores is one in the sense that the distance
between the two test statistics converges to zero in absolute mean as N oo.
Before stating Cuzick's (1985) main result, we state two lemmas.
LEMMA 3.3.1 The following relationship holds for i = 1,..., k:
1 1
(u) = i (v)dv.
U Ju
Proof. Utilize the transformation v = 1 F(u) to get
0 (v)dv
1 (f(F-l(1 -v) : A) 1) d
Ju f (F-'(1 v))
=- [F(F-'(1 v): A)] (1 u)
= -F(F-1(0) : A) + F(F-'(1 u) : A) (1 u)
= F(F-(1 u): A) (1 u),
so that dividing both sides by 1 u and substituting with the relation
F(F-'(1 u)) = 1 u completes the proof. O
Lemma 3.3.1 provides a well-defined functional relationship between uncensored
and censored score functions. Mathematical conditions on 0(u) such as continuity
will imply similar ones for (I(u).
Consider the following three sample quantities to estimate the survival function
F(x):
KM(x) = (R-1)(3.3.1)
X(,)< R j
() = (3.3.2)
X(i)) (Ri + 1)
A(x) = exp(- ), (3.3.3)
where Rj is the size of the risk set at time x(j).
The estimator FKM is the so-called product-limit estimator (Kaplan and Meier,
1958). The estimator FP is discussed in Prentice (1978), and fA was proposed by
Altshuler (1970). For generating censored data linear rank test statistics in the case
of location alternatives, a judicious choice from these three estimators has led to
nice closed-form expressions for expected and approximate scores (Prentice, 1978;
Harrington and Fleming, 1982). In numerical computations for large-samples, any
one of the three estimators is reasonable. The actual relationship (Cuzick, 1985)
among the three is
fKM(x) < fA(x)
Unless specified otherwise, F will be the general representation for the three
estimators. One danger in the use of (3.3.1) is when it takes on the value 0 and causes
an infinite-value approximate score. This situation may arise for the normal case
53
for example, since both the uncensored and censored score functions are unbounded
as u -- 1. One remedy in practice is to replace an occurrence of 0 with a value of
0.001 for instance, but this is a very subjective device that can cause considerable
difference in values of the approximate scores. Harrington and Fleming (1982)
employ
=KM (R 1)
X(j)
which takes on values in the interval (0, 1]. Both (3.3.2) and (3.3.3) strictly take
on values in the interval (0, 1). Also, (3.3.2) can be regarded as a generalization for
the value
N+1
that is always used in the literature for approximate scores on uncensored data. So
FP is the estimator of choice in practice for this manuscript.
Theorem 3.3.1 below is a modified statement of Theorems 1 and 2 of Cuzick
(1985). The theorem provides us with a set of sufficient conditions for the difference
between v based on the approximate function scores and v based on the expected
scores to be negligible in the limit as N -- oo. The sufficient conditions are fairly
mild; the scores that make up some of the well known linear rank statistics such as
the log-rank, Peto & Peto, and Harrington & Fleming's (1982) GP, 0 < p < 1 class
satisfy them (Cuzick, 1985).
THEOREM 3.3.1 Assume the following conditions on the score function f(u).
C-I The uncensored score function q5(u) is twice continuously differen-
tiable on (0, 1) with first and second derivatives 0'(u) and 0"(u).
C-2
lu'(u)l + lu2q"(u)I < for some a < 1/2 and B < oo.
C-3
lim NVar(-v) > 0.
N-oo N
Then the scores ci and Ci in (3.2.8) & (3.2.9) converge in probability to Q(1 F(x))
and 4(1 F(x)) as N -- oo, respectively. F(x) can be either (3.3.1), (3.3.2), or
(3.3.3).
Proof. Refer to Theorems 1 and 2 of Cuzick (1985). The fact that q(u) can be
differentiated twice with continuity maintained will imply the same properties for
Q(u) from Lemma 3.3.1. 0
The general forms of the derivatives of 0(u) are messy, so that the score functions
for specific distributional choices must be checked to see if C-1 and C-2 are satisfied.
The non-vanishing Condition C-3 on the variance will follow from the structure
obtained when asymptotic properties of v are established in Section 3.4. Cuzick
(1985, Theorem 3) indicates that C-3 is satisfied for such a structure of the variance.
Gill's Conditions
Theorem 3.3.1 brings us to the point where the martingale-based results of Gill
(1980) can be used to establish an asymptotic variance for v as well as normality
(under the null hypothesis of equal survival functions) for its asymptotic distribu-
tion. The usual random censorship model (cf. Section 2.2) is assumed.
The stochastic integral form for v in (3.2.18) can also be considered in the form
J(o ( dA/,(x)) /o(x) (3.3.4)
-co Ytl() Yo(x)
where
K(x) = w(x) Yo()+Y () (3.3.5)
Yo(z) + YI(x)
is a weight function in the class CK of Gill (1980). Gill (1980) defines members K E K
as a function of the observations that equals zero whenever min(Yo(x), Yi(x)) = 0.
The paper of Andersen, Borgan, Gill, and Keiding (1982) shows that the class
of statistics from Prentice (1978) that uses the FP estimator can be written as
a stochastic integral of the form (3.3.4) with K(x) as in (3.3.5). Moreover, the
Prentice (1978) class is a subset of the Prentice and Marek (1979) class based on
the "preservation of scores" condition at our (3.2.12). The Prentice and Marek
(1979) class can likewise be written in the framework of (3.3.4) and (3.3.5).
Gill (1980, p. 72) provides three conditions that are sufficient for the asymptotic
normality of test statistics of the stochastic integral form (3.3.4). These sufficient
conditions can be applied with many distributional forms for the survival and censor-
ing variables, and any test statistics with weight functions K(x) that are members
of the general class )C. We state Gill's (1980) conditions as they are relevant to our
case under the null hypothesis.
Let 1* be the set with membership
{ x I min(Fo(x) > 0, F(x) > 0) },
(3.3.6)
where
Fo(x) = F(x).Lo(x),
PF(x) = G(x) LI(x).
Let A,(x), r = 0, 1 be the cumulative hazard functions associated with the control
and treatment groups, respectively.
1. For r = 0,1:
a. A,(x) is finite on 2*.
b. K2(x)/Y,(x) converges uniformly in probability to h,(x) as N -- oo on each
closed subinterval of Z*. The function h,(x) is left-continuous, bounded, and has
right-hand limits on each closed subinterval of Z*.
c. Yr(x) --+ oo in probability as N --+ oo for each x E Z*.
II. Let S = sup 2'. If S i Z*, then for r = 0, 1:
a.
h,(x)dA,(x) < oo.
lim lim sup Pr(
xTS N--oo J[,S]
for every e > 0.
II. If S < oo, then for r = 0,1:
J(,oo K2(x)/Yr(x)dA,(x)
converges in probability to 0 as N -+ oo.
K2(t)/Y((t)dA,(t) > e) = 0
When conditions I, II, and III hold, two consistent estimators for the variance of v
are
-1 _-o [K2(z)] d(N'r(x)
V = X) (3.3.7)
Y, -o L(x) JY.(X)
and
=1 K (x) d(Nx (x) +ANo(x))
V2o =- 0 yo ()J Yi(x)+Yo(x)
= 2(x)o d(AN (x) + Ko(x)). (3.3.8)
[-oo Yr(x)Yo(x)J
The variance estimators at (3.3.7) and (3.3.8) are suggested by Gill (1980, p. 47).
Lemma 4.3.1 of Gill (1980) establishes that both (3.3.7) and (3.3.8) are consistent
estimators under Ho for the true limiting variance of v based on the form (3.3.4).
Gill (1980) notes that there is no true ordinal relationship between V, and V2, and
we would expect them to be fairly equal in practice for large samples. For the null
hypothesis case, there is some creedence to choosing V2 on the basis that it utilizes
both samples to get a pooled-type of estimator of the common cumulative hazard
function A(x).
In order to verify the three conditions of Gill (1980) and thus obtain variance
expressions and asymptotic normality of v, we consider some mathematical results.
LEMMA 3.3.2 (A continuous function preserves uniform convergence.)
Let {fN(x)} be a sequence of random functions that converges uniformly in prob-
ability to f(x) as N -+ oo for x E some set E. If g(.) is a continuous function
on the range of {fN(x)}, then g(fN(x)) is uniformly convergent in probability to
g(f(x)) as N -, oo for x E E.
Proof We know that for arbitrary 6 > 0,
Pr(sup IfNv(x) f(x)I > 6) -, 0 as N -- oo.
xEE
Let e > 0 be given. Since g(-) is continuous, we can find 6 > 0 such that Ig(fN(x))-
g(f(x))| < e if IfN(x) f(x)| < 6. Thus for arbitrary e,
Pr(sup Ig(fN(x)) g(f(x))| > e) = 1 Pr(sup Ig(fN(z)) g(f(x))l < e)
xEE zEE
S1 Pr(sup fN(x) f(x)j 6)
xEE
= Pr(sup fN(x) f(x) > 6)
zEE
--- 0
as N -- oo. O
LEMMA 3.3.3 Let {AN(x)} be a sequence of random functions that is uniformly
convergent to A(x) in probability as N -- oo. Let {BN(x)} also be a sequence
of random functions that uniformly converges to B(z) in probability as N -- oo.
Assume that A(z) and B(x) are bounded for all x E E. Then AN(x)BN(z) converges
uniformly in probability to A(x)B(x) as N oo.
Proof Let e, c > 0. Then
Pr(sup I(AN(x) A(x))(BN(x) B(x))I > e)
zEE
= Pr(sup I(AN(x) A(x))(BN(x) B(x))\ > e,sup IBN(z) B(x) <5 e/c)
zEE zEE
+ Pr(sup I(AN(x) A(x))(BN(x) B(x))l > e,sup IBN(x) B(x)I > c/c)
zEE zEE
< Pr(sup I(AN(x) A(x))I > c)
zEE
+ Pr(sup IBN(x) B(x)I > e/c)
xEE
-- 0+0=0.
(3.3.9)
The boundedness condition implies that IA(x)l < M for some positive number
M < oo. It is also assumed that
Pr(sup IBN(x) B(x)| > -) -- 0.
xEE M
Combining these last two facts with the relation
Pr(sup IA(x)[BN(x) B(x)]j > e) < Pr(sup IBN(x) B(x)I >
xEE xEE M
implies the result
Pr(sup IA(x)[BN(x) B(x)]l > e) --- 0
xEE
(3.3.10)
as N -* oo.
Commuting the roles of the two sequences {AN(x)} and {BN(x)}
similarly gives
Pr(sup IB(x)[AN(x) A(x)]| > e) -- 0.
xEE
(3.3.11)
Next employ the identity
AN(x)BN(x) A(x)B(x)
= (AN(x) A(x))(BN(x) B(x)) + A(x)(BN(x) B(x))
+B(x)(AN(x) A(x))
along with (3.3.9)-(3.3.11) to give
Pr(sup IAN(x)BN(x) A(x)B(x)I > e)
xEE
< Pr(sup I[AN(x) A(x)][BN(x) B(x)]l > )
xEE 3
+ Pr(sup I[AN(x) A(x)]B(x) > -)
zEE
+ Pr(sup IA(x)[BN(x) B(x)] > )
xEE 3
--40+0+0=0
as N oo, which is the desired result. C
3.4 Asymptotic Normality of v
Recall that Theorem 3.3.1 provides a set of sufficient conditions whereby the
expected scores in the censored data rank statistic v can be replaced by approx-
imate scores for large samples. The test statistics considered by Gill (1980) in
establishing asymptotic properties can all be regarded as censored data linear rank
statistics based on approximate scores. Therefore, in the development of asymptotic
properties for v that follows, we will consider the version of v that is based on the
approximate scores, i.e.
k
V = z(qw(1 F(x())) + MOi(1 F(xi)).
i=1
Under the conditions of Theorem 3.3.1, the same properties for v based on the
expected scores (as in (3.2.2)) will follow.
Verifying the Conditions
Gill (1980) used the log-rank statistic as a prominent member of the class K:.
Among the many properties he explored for this member, he showed the meeting of
the three conditions listed on page 56 of Section 3.3 that are sufficient for a limiting
normal distribution of the properly standardized version of the statistic under the
null hypothesis of equal survival distributions. In the form (3.2.18) for v, setting
w(x) = 1 gives the log-rank statistic. The corresponding weight function in the
class KC is
Yo(x)Yi(x)
KC(z) = Y( Y() (3.4.1)
Yo(x) + y,(x)
If we multiply Kc(x) by the weight function
Sf(F-1((x)) : A) F(F-((x)):A) (3.4.2)
f(F-1(F(x)) F(F-1((x)))
for our model at (2.2.4), we get a weight function K(x) that is a member of Gill's
(1980) class KC. The corresponding function tb is random since it is a function of
the approximate scores 0(1 F) and 4(1 F). Note that w(x) at (3.3.5) is not
random as it is a function of the ci and Ci. Convergence properties associated with
tb(x) at (3.4.2) in combination with results already well-established (Gill, 1980) for
the log-rank statistic will provide the course for verifying the three conditions that
are sufficient for asymptotic normality of v.
LEMMA 3.4.1 Suppose that f(t : A),A > 0 and f(t) = f(t : 0) are continuous
densities with support over the interval (-oo, oo). If A1 < oo and A2 < oo are
positive constants such that
f (t : A) f (t : A)
lim f A1, and lim = A,
t-oo f(t) t--oo f(i)
then ti(x) defined at (3.4.2) is bounded on the interval (-oo, oo).
Proof. By L'Hopital's rule,
im F(t : A) l f(t : A)
km = hm = Ai.
t-oo F(t) t-foo f(t)
Therefore,
lim f(t : A) F(t: A) limf(x : A) limf(t: A)
lim = lim lim
t-oo f(t) F(t:) t-oo f(x) t-oo f(t)
= A -A =O0.
Also,
lim f(t:A) F(t : A)} = f(t: A) li F(t: A)
hlm- = lim hm
t---oo f(t) F(t) t.--,o f(t) t---oo F(t)
SA2-1 < oo.
Now since
f(t : A) F(t : A)
f(t) F(t)
is continuous, then for any finite values a, b, the function (3.4.3) is bounded on every
finite interval [a, b]. Since (3.4.3) is finite at the points -oo and oo, the substitution
t = F-'(F(x))
shows that ib(x) at (3.4.2) is bounded on (-oo,oo). 0
The value of the Lemma 3.4.1 will be seen when we choose specific forms for
F(x : A), but it should be noted that it does not cover the case where both the
censored and uncensored score functions tend to infinity as x tends to infinity. For
this case, the "indeterminate" form oo oo for the "limit" of tb(x) requires special
handling. An example of this occurs in Chapter Four.
THEOREM 3.4.1 (Uniform convergence in probability of the Kaplan-Meier esti-
mator.) Assume the model outlined in Section 2.2, with the support of G(x) to be
(-_0, oo). Let s E (-oo, oo) be such that the risk set size
Y(s) = Yo(s) + Yi(s) c0. (3.4.4)
Then under the null hypothesis of F(x) = G(x) for all x, the Kaplan-Meier estima-
tor FKM converges uniformly on x E [-oo, s] in probability to F as N -- oo. That
is,
sup FKM(x) F(x)j ) 0.
Proof. See Theorem 4.1.1 of Gill (1980), who proves a more general case. His
notation uses F(x) for the cumulative distribution function. The relation (3.4.4),
which is given as Condition Ic of Gill (1980), has the interpretation that the size of
the risk set at any point x grows to infinity as N -+ oo. O
If the support of the censoring distributions is also (-oo, oo), then s = oo.
Sometimes, however, censoring distributions are assumed to have support of the
form (-oo, s), where s < oo, and examples can be constructed to show that the
uniform convergence of FKM fails for x > s.
COROLLARY 3.4.1 Theorem 3.4.1 holds for FKM(x) replaced by either FP(x) or
FA(x).
Proof. The case for FA is also covered in Gill's (1980) Theorem 4.1.1. For the
estimator FP, first consider the following relation from the lone lemma of Cuzick
(1985):
IPP(x) FKM(X) PP(X) 2 (3.4.5)
Y()" (3.4.5)
Then
sup ) P(x)- F(x) = sup IP(x)- FKM()+ FKM(x) F(x)[
-o
< sup PP(x) FKM(x)I
+ sup FKM(x)-F(x)
-oo
P
-+0+0=0
since
sup IFKM(x) F(x)j I 0
-oo
by Theorem 3.4.1 and
sup FP() FKM(x) < sup (x).--2
-oo<~~
~~
< 2/ inf Y(x)
-oo<0~~
P~~
-+ 0
by (3.4.4) and (3.4.5). 0
Using these last two results, we now consider a key property for the weight
function tb(x) at (3.4.2).
THEOREM 3.4.2 Assume the conditions of Theorem 3.4.1 and Lemma 3.4.1.
Then the weight function i(x) represented by (3.4.2) is bounded and tZ converges
uniformly in probability to w* on (-oo, oo) as N -- oo, where
f(x : A) F(x : A)
w'(x) = (3.4.6)
f(x) F(x) (3.4.6)
Proof Since F(x) is assumed continuous, its inverse function F-'(x) is also con-
tinuous. The function
f(F-'(.) : A) F(F-'(.) : A)
f= (F-'(.)) F(F-'(.))
will also be continuous since it is a composite function of continuous functions. The
survival function estimator F takes on values in the interval [0, 1], so Lemma 3.4.1
insures that tb(x) is bounded when it is defined on the extended line [-oo, oo].
Applying Lemma 3.3.4 with (3.4.7) in the role of the continuous function and F(z)
as the argument, where P has the uniform convergence properties in Theorem 3.4.1
and Corollary 3.4.1 proves the uniform convergence of tb(-). 0
As mentioned previously, the three conditions of Gill (1980) have been discussed
and verified for the member Kc(x) which corresponds to the log-rank statistic.
Actually, Gill (1980) attaches a standardization factor on to the weight function at
(3.4.2) to get
NoN, Yo(x) Yi(x) No + N
S No +N, No N, Yo(x)+ Y1(x)
The multiplicative factor is attached to ensure that the variance of the test statistic
corresponding to K.(x) is bounded away from 0 and oo as N -+ oo. Note that this
provides a test statistic
No + N,
v = -N V
NoV N
where v remains defined as in (3.2.10) and (3.2.18). The variance estimators V1 and
V2 respectively given by (3.3.7) and (3.3.8) can be adjusted accordingly, and their
consistency for the variance of v* is established in Lemma 4.3.1 of Gill (1980). That
lemma is a precursor to Gill's (1980) Corollary 4.3.1 and Proposition 4.3.3, which
specify the asymptotic distribution for the test statistic based on 1Kf(x).
The standardization factor of Gill (1980) is specifically
No + N,
The previous paragraph defined the relationship between v and v*. Suppose there
exists a variance of v*, Var(v*), such that
v*
Var(v*)
is a random variable with some asymptotic distribution. Also suppose that the
variance estimators Vi and V2 are such that
S No + N1
V N NV,
No N1
S=No + NV
2 NoNx
are consistent estimators of Var(v*). By Slutsky-type arguments,
V* V* V*
S and and-
Var(v*) I VA
will have the same asymptotic distribution. But note that
V V
Var(v*) Var(v*)
V* V
1 and
vW~7
We V-V2
so there are numerous expressions of random variables that are asymptotically equal
in distribution.
The proofs and surrounding discussions of Gill's (1980) Corollary 4.3.1 and
Proposition 4.3.3 note that
sup Y() x) 0; r = 0, 1, (3.4.9)
-oo<;
where F,(x) = Pr(X,i > x) as introduced in Section 2.2, and
N,
min(No, N1) oo; N---o -' pr E (0,1) (3.4.10)
No + N1
are sufficient to establish conditions I-III for various censorship models and thus
the asymptotic distribution of the log-rank statistic. The Glivenko-Cantelli theorem
can be used to show that (3.4.9) is satisfied for the mixture model at (2.2.4) under
the random censorship model. (Gill, 1980; Harrington and Fleming, 1982).
THEOREM 3.4.3 Assume the random censorship model described in Section 2.2,
and let (3.4.10) obtain. Suppose zb(x) is defined by (3.4.2), Kc(x) is defined by
(3.4.1), and KI(x) is defined by (3.4.8). Let Z* be defined as in (3.3.6), and suppose
tb(x) is bounded on 1*. Then the test statistic
= No + N No + N, kr R1i
0 =R1 NNV= N1
will satisfy Conditions I-III on page 56 as they are required. Define
f( :' ( A) F(x1: w) 0
Var(') f(x : A) F(x : ) o(x)() i dA(x). (3.4.11)
-oo f() F(x) Po oW(x) + ph i (x)
Recall the mixture model at (2.2.4). Under the null hypothesis 7r = 0,
V* V
Var(v*) N Var(v*)
Vr~r(V) =7-N&+-N,=lV
converges in distribution to a standard normal random variable as N oo. Fur-
thermore, both estimators
V11 2 (X) Y_,(x) d,( ()
r=o = Y0 (x) + Y (x). d,(
and
V2 = 1_ tb2(x) Y d(Ao(x) + AN(x))
-(Yo(x) + YI(x))2
will be such that
No + N, No + NV
NoN, NON,
converge in probability to Var(v*) as N -- oo.
Proof Under the null hypothesis r = 0 and the random censorship model outlined
in Section 2.2, Conditions (3.4.9) and (3.4.10) are sufficient for conditions I-III to
hold for Kc(x). We start with verification of Condition Ia. For those values of x
in I* such that min(Fo(x),Fi(x)) > 0, both F(x) > 0 and G(x) > 0. Thus the
corresponding cumulative hazard functions will be less than infinity, i.e. finite on I.
In Condition Ib, let
K ,(X) = l (x) "W
Theorem 3.4.2 and Lemma 3.3.2 imply that
p f(x : A) F(x : A)
S f(x) F(x)
uniformly on [-oo, oo) as N -+ oo. The uniform convergence will hold on closed
subintervals of 1* as well. Applying Lemma 3.3.2 with
AN(X) = -(x)
and
BN(X) = K; (x)
gives
K2(x)/Y,(x) h,(x); r = 0,1
where
,r) { f(x : A) F(x : A) 1p2 P Fo(x)F(x) 120
X fF (x) F(x) ',() poLo(x) + pix) ( r
(3.4.12)
uniformly on closed subintervals of Z* as N -- oo. Let
h( ,() poFo(x) + pF(x)
be the corresponding function for KI(x). Then hr is the product of hc and the
square of expression (3.4.3) from Theorem 3.4.2. It is the product of two bounded
left-continuous functions with right hand limits, and thus is itself a bounded left-
continuous function with right hand limits.
Condition Ic is covered automatically by the random censorship model (Gill,
1980).
Theorem 3.4.2 implies there is a positive number M1 such that
f(x : Z) F(x: ) < M for all x E [-oo, oo],
and therefore
h,(x)dA,(x) = H(.{ -2 h} ()dA,()
< M, hc(x)dA,(x) < oo,
so that Condition IIa would be satisfied if S Z*, since
j h(x)dA,(x) < o0
was established for the log-rank statistic in Gill (1980). Analogously, there is a
positive number M such that zi2(x) < M, so Condition IIb follows from
liimlimsupPr( I K(x)/Yr(x)dAr(x) > e)
xTS N-.oo Jx
< limlimsup Pr( MK2(x)/Yr(x)dA,(x) > e)
-zS N-oo Jx
= limlimsupPr(M / K2(x)/Yr(z)dAr(x) > e)
xTS N-*oo J
= limlimsup Pr( K[ 2(x)/Y,(x)dA(x) > --)= 0,
xTS N-oo Jx M
the last step verified by Gill (1980).
The bound M used in the verification of Condition II also is central to a veri-
fication of Condition III if S < oo. Finally, the expression for Var(v*) comes from
substitution of (3.4.12) into Gill's (1980) expression
f_ (hi(x) + ho(x))dA(x)
in his Corollary 4.3.1. The expressions for the estimators V1 and V2 are determined
after substitution of
b(x) Kc(x)
for K(x) in (3.3.7) and (3.3.8). The convergence associated with V1 and V2 follows
from Gill's (1980) Lemma 4.3.1. 0
COROLLARY 3.4.2 Under the conditions of Theorem 3.4.3,
v v
and
converge in distribution to a standard normal variable as N -+ oo.
Proof. Recall that
V* =N Vt N ; =1,2,
N, No
in the discussion before Theorem 3.4.3, and apply a Slutsky-type argument. 0
The expressions given in Corollary 3.4.2 are most convenient for calculating
the statistic on data. If the support of the survival and censoring variables is
(-oo, 00), then S = oo, and Condition III will then be empty. This is also true if
no censoring occurs. On the other hand, it is often tenable in practice to assume
uniformly distributed censoring times with finite support. In this case Condition
III is required, as S < oo.
Use of the Test
Back in Section 2.2, the mixture model (2.2.4) was introduced with the pre-
sumption that A > 0, i.e., that larger values of the response were associated with
the treatment group (See Assumption A2). The asymptotic normal distribution of
the test statistic developed in Theorem 3.4.3 suggests that the opposite situation
of smaller values (A < 0) can be handled as well, as long as the required conditions
hold. Note that the special case of Theorem 2.3.1 is included here.
If a test against the alternative
with strict inequality for at least one x,
G(x) F(x),
is desired, the rejection region will be in the lower tail of the standard normal
distribution. For A > 0, the upper tail is appropriate. Absence of any knowledge
about A should have a rejection region involving both tails.
It is not true, however, that v or the test statistic that is a standardized version
of v is an odd or even function of A. The fact that v is not a symmetric function
of A gives another aspect in the consideration of the choice of A for conducting the
test. However,
k
(ci + miC.) = 0 (3.4.13)
i=1
holds, since the sum (3.4.13) is the average of sums of scores that occur from all
possible underlying rank vectors given the data (Prentice, 1978). Note that (3.4.13)
implies
k k
v = z(ici + MC) = ((1 z))c + (mi- Mi)C) (3.4.14)
i=1 i=1
Hence summation of the scores assigned to the control group observations (when
A > 0 is employed) produces a small value for an appropriately standardized test
statistic that could be compared with the lower tail of the standard normal distri-
bution.
If larger values for the control group are present, the assigned scores for the
control group will likewise tend to be larger than those scores for the treatment
group. Then v (cf. 3.4.14) based on the treatment group scores with A > 0 will
tend to be small, and the test statistic in Theorem 3.4.3 should be compared with
the lower tail of the standard normal distribution. Summing the negatives of the
scores in (3.4.14) would produce large values of the test statistic based on v, and
the rejection region should then be based on the upper tail of the standard normal
distribution. This approach was suggested by Conover and Salsburg (1988) for
their scores derived from Lehmann alternatives and uncensored data. Therefore, if
smaller values of the response for the treatment group is of interest, Theorem 3.4.3
permits us to either
1. Choose a value of A < 0 to calculate the scores, and compare v with the lower
tail.
2. Take the negatives of the scores based on a A > 0, and compare v with the
upper tail.
3. Take the scores based on A > 0 and compare with the lower tail.
Choices (2) and (3) will be equivalent since the rejection regions are based on the
standard normal distribution.
Dealing with Ties
Those who are practitioners of statistics know that ties in the observed data are
often prevalent. Continuity assumptions on the survival and censoring distributions
imply that the probability of ties is zero, but measurement error forces us to group
the data at plausible times. Let dl(,) denote the number of observed failure times
in the treatment group at time z(i), and let d(i) be the total number of observed
failure times at x(;). The test statistic v as in (3.2.10) can be modified by the usual
convention (e.g. Prentice and Marek, 1979):
v = ( (( d( i)-L). (3.4.15)
i=1 R.
The stochastic integral form (3.2.17) already covers the modification as does the
expression (3.4.11) for the limiting variance. Now define
AAV.(x) = X.((x) X.(x-)
to be the number of failure times at x that belong to treatment group r, r = 0, 1.
(Note that A is not related to the parameter A.) The variance estimators are
modified as
= J W() Yo(x) + Y(x) ~ Y(x) --1
r f00 .^OW + Yr(X)-
and
00o Yo(x)Y&(x) 1W) + &V W 1()
V2= t 2(x) YOx)Y() 1 Wo(x)+ A'(- 1) W d(Afo(x)+Ni(x)).
=J-o (Yo(z + Yi())2 Yo(x) + Yi(x) 1
It is also mentioned that if all the ties at a particular failure time belong to the same
group, the modifications are not needed, since v can be computed with a breaking
of the ties in some arbitrary order. Ties among censoring values are not a concern
since they are all assigned the same scoring value within the interval defined by
adjacent uncensored values.
By the same arguments in establishing (3.2.18) for v, the variance expression V2
is equal to
i ,? 1- R i Ri }d(i) (3.4.16)
i=l \i RR Ri 1
75
with 0/0 = 0 and tb = (x(;)) 4(x(i)). When there are no ties in the observed
data, (3.4.16) reduces to
wtR, 1 (3.4.17)
Expression (3.4.16) has often been interpreted in the literature as a weighted sum
of hypergeometric variances because of the argument of Mantel (1966) in deriving
the log-rank test from a series of independent 2 x 2 contingency tables.
3.5 Evaluation of Expected Scores
The forms of the expected scores in (3.2.4) and (3.2.5) produce very involved
multiple integration when some of the mj are nonzero if we approach the evaluation
in a brute-force manner. But suppose we assume that 0((1 ui)) = 1 in (3.2.4).
Then the integration is straightforward (Prentice, 1978) and equals the value of 1.
This suggests that
k
g*(ul,...,uk)= II Rj(1 uj)m", 0 < u < -- < uk < 1
j=1
constitutes a joint density in the uis. Since the expected scores involve score func-
tions that only depend on a particular ui, the scores c, and C, can be represented
as
ci = (ui)g;(ui) du,; (3.5.1)
C, = 4 (u)g*(ui) du,, (3.5.2)
where
k
g;(u,) = J.* J I Rj(1 uj)'mdu, ... d. ddui+ ... duk (3.5.3)
ul<-.-__
__
is the marginal density in ui corresponding to the joint density g*(ul,..., uk). It
turns out that for the score functions considered in this dissertation, the integrals
in (3.5.1) and (3.5.2) do not have nice closed-form expressions. But numerical
evaluation of the integrals can be carried out in a straightforward manner.
The Form of gf(ui)
To gain some insight, consider the case i = 1. Integration over u2,...,uk in
g*(u,. ,uk) yields
g (ui) = ... {R,(1- uj)m-duj} = R1(1 -u1)R-1,
uk-1 u1 j-'
To obtain g2(U2), we can integrate out u3, ,uk and then ul:
1 1 u2 k
g(u2)= 1 ..jJ1{RA(1-u,)mdu,}
Uk-1 u2 0 j=l
u2
= RR2(1 u2)R2-1 (1 u)m(')du1
0
O < ul < 1.
U2
= RR2(1 u2)R2-1 J(1 u1)R-R2du
0
= R1R2(1 u2)R2-1 1 1 (1 u2)R-R2]
= RR2 (1 u2)R2-1 (1 u2)R-1
SR1 R2 R+ -- R "
For i = 3, the housekeeping becomes more extensive. We get
S 11 u3 u2 k
g;(u3)= -J J JI {Rj(1 u.)mrdu.}
Uk-1 U3 0 0 j=1
u3 u2
= RIR2R(3)(1 u3)3)- //(1 u2)m((1 u1)m()duldu2
0 0
= RlR2R(3)(1 u3)R3-1
u3
xI(31 /R1
= RRR3(1 u3)R3-1 (R R2) (1 u2)-R3- du2
0
-R1R2R3(1 u)R3-1 (R- ) (1 U2)R-R-Id
= R1R2R3(1 -u -
S R2)(R2- R3)
-RIR2R3(1 u3)n3-1
SIf--(I--u3)R-R3}
S(R R)(R R3)
= R1R2R3(1 U3)R3-
[ 1 1 (1
x(R, R2)(R2 R) R2) R3)
+R1R2R3(1 u3)R2-1 1
(R (R, R)(R2 R)
= RR2R3(1 3 )R-1 1
(R- )R)(R3 R3)]
= (Rx,( R2))"(R- (RI R2)(Rs R3 )
+R R2R3(l U3 R2-11
(RI R2)(R R2)]
+R1R2R3(1 u3)R1 -(R1 R)R R)
1R23(1 U3)R- (R2 R)(R3 R1)
A reasonable conjecture then for arbitrary i in (3.5.3) is
1=1
The validity of (3.5.4) can be proven by mathematical induction. We start with a
result that will be used in the proof.
78
LEMMA 3.5.1 The risk set sizes
R, i =l ,...,k+ (Rk+ 0),
satisfy
1 1
= i+1, i= ,...,k.
S(RI -=) (R, R,)
I=1 =11
Proof. The product term in the denominator of the left hand side can be regarded
as a polynomial in Ri+, with the simple roots R1, R2,..., Ri. Then the whole
expression on the left hand side is a ratio of two rational functions. Since the
denominator is a polynomial, we can decompose it into more elementary functions
that involve the factored terms. A discussion of the technique can be found in
Gradshteyn & Rhyzik (1980, p. 56-57). The following notations of f and 0 mimics
theirs and is to be assumed only in the local context of this proof. Denote
f(R) = H(R- Ri);
1=1
(R) =
as the two rational functions. Then
f(Ri+x) A1 A2 Ai
= + +-..+
f/(R+l) Ri+l R Ri+l R2 Ri+ Ri'
where
A- (R1) A (R2) = (Ri)
f'(R1)' f'(R2)' f'()
Now
f(Ri+,) = (Ri+1 RI)(R.+i R2) ... (R?+l -R),
i
f'(Ri+l) = 1(Ri+e R,) +
I=1
which implies that
which implies that
f'(R1)
f'(R2)
f'(Ri)
i i
j(Ri+ Ri) + + JQ(Ri+ Ri),
1=1 I11
1 2 1ii
= l(R RI),
1=1
1#1
i
= ll(R2- R),
1=1
1;2
-- II(R, RI,)
as all other terms drop out. Since O(Ri) = 1, 1 = 1, -, i, we have
L=1
f(Ri+i) 1
f (A+I) (-+i Ri)(I (Ri Ri)
11
1
ft
( +l R4)( H (Ri Ri)
i+1 i+1
I (Rt R1) 1 (li
1=1 I=1
19d1 10i
1 1
(1)i-1 +1
j=l n(RI Rj)
1=1
Coupled with the fact that
f(R+1)
1(Ri+1)
1
(-1)i
we get the desired equality. 0
As a first step in the induction argument, note that (3.5.4) holds for i = 1. Next
assume that (3.5.4) holds for i. From this assumption it must be proven that a
version of (3.5.4) with i + 1 replacing i everywhere is also true.
From the joint density in the u1,... U, we need to integrate out ul,..., ui (call
this "subrange 1") and u(i+2),..., uk ("subrange 2") in order to obtain the marginal
density g,'1 (ui+1). Integration over subrange 2 yields
Ri+ (1 u+)R+,1-1 /** f Rj(1 uj)dui. (3.5.5)
subrange 1 j=1
To see what happens over subrange 1, write (3.5.4) as
R,(1 ui)-1 R ,) E i. (3.5.6)
1= / i (Rt Ri)
Note the similarity with (3.5.5). Because we are assuming the case for i is true, the
expression
('= l =1 \ J (R1 Rj)
1;1
in (3.5.6) is the value of the integral of g*(u, ..., uk) over u1, ..., ui-1. The subse-
quent integration of (3.5.7) over ui produces
Rj,' (1 u,)R-Ri+-'du,
i=l = I (R i- Rj) o
1t$
= R- (R) -(1 U+)R-'+] (3.5.8)
= x (R, R -) R- R+i+
'= 1
The substitution of (3.5.8) into (3.5.5) gives
1=1
( ,) (1 -R u +l)+Z- I (i) l R (1 ui+i)R 4 J+1 1
\' =1 (R Rj) (R,) R) R Ri+-
1 i+ 1R-1 i+ Ri+I-1
= Ii+ i+
\'=+1 j=' n (RI Ri) i=' n (RI Ri)
= R(j,) i+1 + i+ I (3.5.9)
'= i= H (Ri Rj) l (Ri Ri+i)
1=1 1=1
10i IJi+l
with the last step arising from substitution of the equation in Lemma 3.5.1 to get
the last term on the right hand side of (3.5.9). Collecting the i + 1 terms in (3.5.9)
into one summation gives
i+1 i+1 ui+R-l
'= j= (RI Rj)
which is the density in ui+1 that matches the form (3.5.4) for u,. This completes
the induction argument that (3.5.4) holds for all i > 1.
Although the form gf(ui) in (3.5.4) is not simple, analytical or numerical eval-
uations of ci and Ci will involve single integrals.
3.6 Properties Under H,
The behavior of the linear rank test statistics derived in previous sections can be
studied in terms of (Pitman) asymptotic relative efficiency (ARE). The structure of
local alternatives of parameter values {7rN} converging to the null hypothesis value
ro = 0 will allow us to compare the local powers of v for different scores.
Gill's (1980) foundational work will again be applied. In particular, his Chapter
Five provides sufficient conditions for asymptotic (N -- oo) normality for stan-
dardized v under a sequence of local alternatives in the location-shift model. Gill's
(1980) results can be used to evaluated the "efficacy" of the linear rank test statistic
based on v and the sequence of local parameter values. The Pitman ARE of one
test with a competitor can be established by evaluating the ratio of their efficacies.
The efficacies in general depend on the censoring distributions for the two sam-
ples. Thus we can study the influence of censoring distributions on behavior of the
tests. For the exponential distribution (extreme-value in the logarithmic scale) it is
known that fixed (not dependent on N) and equal censoring distributions for the
control and treatment populations are sufficient for the log-rank to be fully efficient
(Crowley and Thomas, 1975). Gill (1980) extended this result to other distribu-
tional forms including the logistic, double-exponential, and normal distributions.
The Structure of Local Alternatives
Let { r)} denote a sequence of parameters belonging to (0,1] where
a
for some constant a > 0. This choice has been used by authors in the uncensored
data case (Johnson et al., 1987). When convenient, the value of a = 1 can be
assumed without loss of generality. The sequence of corresponding control and
treatment population survival functions is
FN(x) = F(x) (unchanged from Ho)
GN(x) = 7rNF(x: A) + (1 rN)F(x).
Consistent with previous definitions, the respective cumulative hazard functions are
AN(x) = -log(F(x)) and A(x) = -log(GN(x)).
Sufficient Conditions
In sections 3.3 and 3.4 of this manuscript, Gill's (1980; p. 72) conditions sufficient
for a limiting null distribution were stated and verified for v in the mixed model
at (2.2.4). Under a sequence of local alternative parameter values, Gill's (1980)
conditions on his page 106 in addition to those given on his page 72 (Gill, 1980)
will have to be verified in order to establish asymptotic properties.
We adopt the notational convention of Gill (1980) with suitable modifications
for stating these conditions. To deal with local alternatives, certain parts of the
conditions in our Section 3.3 must be replaced with more general statements. The
following is a list of Gill's (1980) conditions stated in terms of our mixed model
setup (2.2.4).
I'. There exist 7,(x) such that
7,(X) = lim i 1 r = d0, 1
N-o No + N1 dAo =/
uniformly on each closed subinterval of {x : F(x) > 0}. Also, let
7(x) = 7o(x)- -i(x).
Let k(x) and
K(x) = t(x)K(x) = t(x) YO Y ()
Yo(x) + Yi(x)
be such that
No + NM
lim N + K(z) = k(x)
N-.oo NoNi
uniformly in probability on closed subintervals of
T = {x | min(Fo(x) > 0, F (x) > 0)}.
The function k(x), called a "limiting weight function" by Gill (1980), is left contin-
uous with right hand limits such that
k+(x) = lim k(t)
t-*x+
is of bounded variation on closed subintervals of Z*. The function k(x) is defined
to be zero in regions outside of 2*.
a. FN(x) converges uniformly on Z* to F(x) as N --+ oo; GN(x) converges uniformly
on I" to G(x) as N -- oo. Also,
A,(x) = lim AN(x)
N-oo
is finite on Z*.
b. K2(x)/Y,(x) converges uniformly in probability to hr(x) as N -+ oo on each
closed subinterval of 1*, where h,(x) is left-continuous and has right-hand limits on
each closed subinterval of 2*. Also,
lim h,(t)
t-r+
must be of bounded variation on each closed subinterval of Z*.
c. Y,(x) oo in probability as N -- oo for each x E Z*.
II'. Let S = sup Z*. If S I Z, then for r = 0, 1:
a.
/. h,(x)dA,(x) < co.
b.
limlimsup Pr(/ K(t)/Y(t)dAN(t) > e) = 0
xTS N oo J[x,S]
for every e > 0.
III*. If S < oo, then for r = 0,1:
S,oo) K'(x)/Y)(x)dA(x)
converges in probability to 0 as N -+ oo. From page 106 of Gill (1980) the extra
conditions are
IV'. If S I', then for r = 0, 1:
a.
Ik(x)y,(x)l dAo(x) < oo
lim lim sup Pr(
zTS N-oo J[z,S]
IK(t)l IdA'(t) dAo(t) > )= 0
for every e > 0.
V'. If S < oo, then for r = 0,1:
JS,oo) K(x)I IdA(x) dAo(x)I
converges in probability to 0 as N -+ oo.
Some of the components of II* and III* are essentially no more than a substitu-
tion of A (x) for Ar(x) in conditions II and III. Also, note that
AN((x) = Ao(x) = A(x).
In order to verify I*- V* for the mixed model setup, we follow our strategy in the
null hypothesis situation, where each condition was verified in two steps:
1. Establish that such conditions have been verified by Gill (1980) for the case
K(x) = Kc(x) (log-rank); and
2. Use (1) along with the fact that our weight function K(x) can be expressed
as a product of a bounded weight function tb(x) and Kc(x).
This approach will be referred to as the "two-point" mechanism in the remainder
of this section.
In the verification of I*-III*, the case r = 0 brings us back to the conditions
already verified for the null hypothesis case of Section 3.3. Only the case r = 1
remains. We start with condition I*a. The functions GN(x), N > 1, and G(x) are
continuous survival functions and
lim GN(x) = G(x); lim GN(x) = G(x) for all x.
N- -oo N-.oo
Due to a well-known result due to Polya (see e.g. Serfling (1980)), the uniform
convergence is proved. Condition I*b follows just as condition Ib did because K(x) =
d(x)K(x) where zt(x) is bounded.
To evaluate y(x) and other expressions, we need
dA~'(z)
gN(x)
GN(x)
rN(x)f(x : A) + (1 rN(x))f(x)
rN(x)F(x: A) + (1 rN(X))F(x)'
and
AN( dAo(x)
Aox) dx
f(x)
F(x)
= Ao(x).
Then
dhN(X)
dAo(x)
and
dAo(x) o(x)
7rN(x)f(: A) +(1 rN(x))f(x) F(x)
rN(x)F(x : A) + (1 rN(x))F(x) f (x)
rN(X) ( 1) + 1
((F(x:A 1) + 1"
Therefore -o(x) = 0, and 7(x) = -71(x). Now,
Y1(x) = lim NN -( 1)
N-.oo VNTo + N dAo(x)
lim ONNo + N (1
N-oo N N No + N dAo(x)
= No o1 7N *N /()-- F(S)_._]
= m 0+ oN 1 +
N--oo No + N No+N F+1)
Upon substituting 7N = -, we get
N0 (X) = hAm a F(x)
N-oo No +N N N1a VTrN ( ) i) +
S f(xA) F(x:)
= \/Pa ( f(X) F( x)j
= "Plpaw*(x)
as N -* oo, where w*(x) was defined at (3.4.6) in Theorem 3.4.2. This last con-
vergence is uniform on each closed subinterval of {x : F(x) > 0} by the same
reasoning (cf. (3.4.12), two-point mechanism) employed in Theorem 3.4.3. Likewise
as in (3.4.12),
i No + N I (x f(x : A) F(z : A) ( 'o(x)fl(x)
N--oo V-No f() F(x) I poo(x) + p- i(x) k(x
uniformly in probability on closed subintervals of 2*, with k(x) being left continuous
with right hand limits and k+(x) of bounded variation, again since 1b(x) is bounded
and the version of these properties for the log-rank statistic have been verified in
Gill (1980).
Condition II*a is the same as condition II in the null hypothesis case and thus is
already verified. Verification of conditions II*b and III* again makes use of the two-
point mechanism, along with Lemma 4.3.2 of Gill (1980) and his proof of Proposition
4.3.3. A pair of inequalities
K2 (x) Yo(x) Y(x) No + N, No + N, Yo(x)
Yi(x) No N Yo(x)+Y (x) N No
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