BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
B Y
LAURA LYNN PERKINS
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
1984
BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
BY
LAURA LYNN PERKINS
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
1984
to my parents, with
love
ACKNOWLEDGEMENTS
I would like to thank Dr. Ronald Randles for originally
proposing the problem. Without his enormous patience,
encouragement and guidence, it would not have been
possible. I would also like to thank Dr. Jim Kepner for his
help in its original conception. To my family, especially
my parents, I am grateful for the mental and financial
support they provided when I needed it the most. I would
like to thank Robert Bell for his patience and
understanding. More than once, when I could not see the
end, he was there to reassure me and give me confidence. To
my typist, Brenda Prine, I express my gratitude for many
hours spent with no complaints. Last, but not least, I
would like to say thank you to the Department of Statistics
for making this all possible.
iii
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS iii
ABSTRACT vi
CHAPTER
ONE INTRODUCTION 1
TWO A STATISTIC FOR TESTING FOR DIFFERENCES
IN SCALE 16
2.1 Introduction 16
2.2 The CD Statistic 19
2.3 Permutation Test 35
2.4 Asymptotic Results 39
2 .5 Comments 44
THREE A CLASS OF STATISTICS FOR TESTING FOR
DIFFERENCES IN SCALE 48
3.1 Introduction 48
3.2 u 2 Known 50
3.3 p^ = P 2 Unknown 62
3.4 Asymptotic Properties 73
3 .5 Comment s 86
FOUR A TEST FOR BIVARIATE SYMMETRY VERSUS
LOCATION/SCALE ALTERNATIVES 90
4.1 Introduction 90
4.2 The W Statistic Using
4.3 The Wn Statistic Using
CD Ill
4.4Permutation Test 121
4.5Estimating the Covariance 123
IV
FIVE MONTE CARLO RESULTS AND CONCLUSION. . 133
5.1 Introduction 133
5.2 Monte Carlo for the Scale Test . 134
5.3 Monte Carlo for the Location/
Scale Test 142
APPENDICES
1 TABLES OF CRITICAL VALUES FOR TESTING
FOR DIFFERENCES IN SCALE 158
2 THE MONTE CARLO PROGRAM 171
BIBLIOGRAPHY 183
BIOGRAPHICAL SKETCH 185
v
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
BIVARIATE SYMMETRY TESTS WITH CENSORED DATA
By
Laura Lynn Perkins
Augus t, 19 8 4
Chairman: Dr. Ronald H. Randles
Major Department: Statistics
Statistics are proposed for testing the null hypothesis
of bivariate symmetry with censored matched pairs. The two
types of alternatives considered are (1) the marginal
/
distributions have a common location parameter (either known
or unknown) and differ only in their scale parameters and
(2) the marginal distributions differ in their locations
and/or scales. For the first alternative, two types of
statistics are proposed. The first is a statistic based on
Kendall's tau modified for censored data, while the second
type is a class of statistics consisting of linear
combinations of two statistics. Conditional on N^, the
number of pairs in which both members are uncensored, and
N2 the number of pairs in which exactly one member is
censored, the two statistics used in the linear combination
are independent and each has a null distribution equivalent
vi
to that of a Wilcoxon signed rank statistic. Thus, any
member in the class can be used to provide an exact test
which is distribution-free for the null hypothesis. The
statistic based on Kendall's tau is not distribution-free
for small sample sizes and thus, a permutation test based on
the statistic is recommended in these cases. For large
samples, a modified version of the Kendall's tau statistic
is shown to be asymptotically distri bution-free.
For the second and more general alternative, a small
sample permutation test is proposed based on the quadratic
form Wn = T^ j; ^ T where T' is a 2-vector of statistics
composed of a statistic designed to detect location
differences and a statistic designed to detect scale
differences and | is the variance-covariance matrix for
T For large samples, a distribution-free approximation
for T' 1 T is recommended.
~ n T ~ n
Monte Carlo results are presented which compare the two
types of statistics for detecting alternative (1), for
sample sizes of 25 and 40. Quadratic form statistics Wn
using different scale statistic components are also compared
in a simulation study for samples of size 35. For the
alternative involving scale differences only, the statistic
based on Kendall's tau performed best overall but requires a
computer to do the calculations for moderate sample sizes.
For the more general alternative of location and/or scale
differences, the quadratic form using the scale statistic
based on Kendall's tau performed the best overall.
vii
CHAPTER ONE
INTRODUCTION
Let Wj and denote random variables; then the
property of bivariate symmetry can be defined as the
property such that (W ^ W 2 ) has the same distribution as
(W2*W^). This property of bivariate symmetry is also
referred to as exchangeability (or bivariate
exchangeability). Commonly, this property arises as the
null hypothesis in settings in which a researcher has paired
observations, such as, when the subjects or sampling units
function both as the treatment group and the control group
or possibly the researcher has matched the subjects
according to some criteria such as age and sex.
For example, a dentist may want to assess the
effectiveness of a dentifrice in reducing dental
sensitivity. The dentist randomly selects n patients and
schedules two appointments for each patient at three month
intervals. During the first visit, a hygienist assesses the
patient's dental sensitivity after which the patient is
given the dentifrice by the dentist. At the end of the
three month usage period, the patient returns and his or her
f
dental sensitivity is again assessed. If X^ and X2^ are
the first and second sensitivity measurements, respectively,
1
2
t" Vi
of the i 1 patient, the dentist has n bivariate pairs in the
sample. If there is no treatment effect, then effectively
the two observations of dental sensitivity are two
measurements of exactly the same characteristic at two
randomly chosen points of time. In which case, the
ft t f
distribution of is t^ie same as that of (X2i,X^i),
and so a test using the null hypothesis of bivariate
symmetry would be appropriate.
The possible alternatives for a test which uses a null
hypothesis of bivariate symmetry are numerous. The three
types of alternatives which will be considered in this work
are the following:
1) The marginal distributions have a common
known location parameter and differ only in
their scale parameters.
2) The marginal distributions have a common
unknown location parameter and differ only in
their scale parameters.
3) The marginal distributions differ in their
location and/or scale parameters.
The situation under consideration in this work is
further complicated by the possibility of censoring.
Censoring occurs whenever the measurement of interest is not
3
observable due to a variety of possible reasons. The most
common situation is when the measurement is the time to
"failure" (i.e., death, the time until a drug becomes
effective, the length of time a drug remains effective,
etc.) for an experimental unit subjected to a specific
treatment. If at the end of the experiment, the
experimental unit still has not "failed," then the
corresponding time to "failure" (referred to as survival
time) is censored. All that is known, is that the survival
time is longer than the observation time for that unit and
thus has been right censored. An example of censoring in
bivariate pairs could be the times to failure of the left
and right kidneys or the times to cancer detection in the
left and right breasts (Miller, 1981).
Many different types of right censoring exist (Type I,
Type II and random right censoring), each determined by
restrictions placed on the experiment. Type I censoring
occurs if the observation time for each experimental unit is
preassigned some fixed length T. Thus, if the survival time
for a unit is larger than T, it is right censored. Type II
censoring occurs when the experiment is designed to be
terminated as soon as the r*"^1 (r
occurs. Random right censoring is a generalization of Type
I censoring, in which the experimental units each have their
own length of observation (which are not necessarily the
same). This would occur, for example, if the length of the
experiment was fixed but random entry into the experiment
4
was allowed. It is this latter type of censoring which this
work addresses.
Now we statistically formulate the problem of
I
interest. Let (X^,X2^) fr i=1>2,...,n denote a random
sample of bivariate pairs which are independent and
identically distributed (i.i.d.) and Ci i=l,2,...,n denote a
random sample of censoring times which are i.i.d., such that
C ^ denotes the value of the censoring variables associated
t I
with pair (X^,X2i). -*-n t^e case random right censoring,
the observed sample consists of (^ii>^2i^i) w^ere
I T
X j ^ = min(X,Â£,C^), X2j_ = min(X2i,Ci) and 6i is a random
variable which indicates what type of censoring occurred,
r
0i Description
1
2
3
4
Xli
x;ici
Xli>Ci> x2i
x;i>ci,x;i>ci
Now we state a set of assumptions which are referred to
later .
Assumptions:
I I
Al. (^ii>^2i) i=l2,...,n are i.i.d. as the
t
bivariate random variable
A 2
f t
(X^j,X2j) has an absolutely continuous bivariate
distribution function F(
X1
X2 U2
)
5
O
where F(u,v) = F(v,u) for every (u,v) in R The
parameters y ^ (p 2) and (a 2) ate location and
scale paratmeters, respectively. They are not
necessarily the mean and standard deviation of the
marginal distributions.
A3. Cj,C2*..,C are i.i.d. continuous random
variables, with continuous distribution function
G(c) .
A4. The censoring random variable is independent
f I
of i=l2 n and the value of is
the same for both members of a given pair.
A 5. P(xJi>C.,X2i>Ci) < 1.
A6. G (F ^ )) < 1 where F^ denotes the marginal
i i
T
cumulative distribution function (c.d.f.) of X^^
i=l,2.
Note that under A5, the probability is positive that the
sample will contain observations that are not doubly
censored.
With this notation, the null and alternative hypotheses
can now be formally stated. The null hypothesis is
Hq : y^=P2> 1=ct2 versus the alternatives:
6
1. The case where p^=y2=M with p known,
Ha: 1 a2
2. The case where pj=p2=U with p unknown,
Ha: i a2
3. Ha : p^ \i 2 and/or ^ a 2*
Chapter Two and Three will present test statistics for
alternatives 1) and 2). Chapter Four will present a test
for the more general alternative stated in 3). Monte Carlo
results and conclusions will be presented in Chapter Five.
First though, we describe related work in the literature.
Since this dissertation combines two areas of previous
development, that is, bivariate symmetry and censoring, the
first part of the review will deal with related works in
bivariate symmetry without a censoring random variable
considered. The second part of the review will mention
related works for censored matched pairs.
The first four articles to be considered, Sen (1967),
Bell and Haller (1969), Hollander (1971) and Kepner (1979),
all suggest tests directed towards specific alternatives to
the null hypothesis of bivariate symmetry. The work of
Kepner (1979) more directly influenced the development of
this thesis than the others, but they were direct influences
on the work of Kepner and thus will be mentioned.
Sen's article (1967) dealt with the construction of
7
conditionally distribution-free nonparametric tests for the
null hypothesis of bivariate symmetry versus alternatives
that the marginal distributions differed only in location,
or that the marginal distribution differed only in scale, or
that the marginal distributions differed in both location
and scale. The basic idea behind his tests is the
I I
following. Under Hq, the pairs (xii>X2i^ i=l2,...,n are a
random sample from an exchangeable continuous
distribution. He pools all the elements into one sample (of
size N=2n), ignoring the fact the original observations were
bivariate pairs and then ranks this combined sample. From
this, Sen obtains what he refers to as the rank matrix,
n / R11
R 1 2
R1 n
RN "
\ R 2 1
R 2 2 *
R2n
where R^^ is the rank of X^^ in the pooled sample j=l,2
i=l,2,...,n. Let S(R^) be the set of all rank matrices that
can be obtained from R^ by permuting within the same column
of R^ for one or more columns. Under HQ, each of the 2n
elements of S(R^) is equally likely and thus, if Tn is a
statistic with a probability distribution (given S(RN) and
H ) which depends only on the 2n equally likely permutations
of Rn, Tn is conditionally distri bution-free (conditional on
the given R^ and thus S(R^) observed). Sen's statistic Tn
can be defined as
T
n
n
n
l
i = l
,R
1 i
8
where ^ is a score function based on N=2n and i alone.
For the test of location differences only, Sen suggests
using the Wilcoxon scores (E^ ^ ) or the quantile F
scores (E^ ^ = F ^) where F is an appropriately chosen
absolutely continuous c.d.f.). The Ansari-Brad1ey scores
(En ^ ^ ~ |i ~ j|) or the Mood Scores
(Ew = ( 4 )^) are suggested for use when the
alternative is that the marginal distributions differ only
in their scale parameters. For the more general
alternative, that the marginal distributions differ in
location and scale, he recommends making a vector (of size
2) of his statistics where one component is one of the
statistics for differences in location and the other for
scale.
One basic weakness of Sen's proposals, as mentioned by
Kepner (1979), is that the procedure basically ignores the
correlation structure within the original observations
t
^Xli,X2i^ and, thus, suggests that a better test
could possibly be constructed by exploiting the natural
pairing of the observations.
The test proposed by Bell and Haller (1969) does
exploit this natural pairing of the observations. They
suggest both parametric and nonparametric tests for
bivariate symmetry. In the normal case, they form the
likelihood ratio test for the transformed observations
(^i i, Y2i) where Y1;L= X2i and Y2i = X1;L + X2i* The
9
resulting test they suggest when dealing with a bivariate
normal distribution is to reject Hq if |B ^ j > t(3^;n-2) or
j B 2| > t(g25n-l) where
(n-2)1/2 r(Y ,Y.) n^ Y
B. = and B =
1 2 l/ 2 c
(l-r'(Y1,Y2))/2 b
and r(Y^,Y2) is the sample correlation coefficient of the
2
Yj^'s and ^2i's ^ j and S are the sample mean and unbiased
sample variance, respectively of the Y^^'s and t(3;n)
represents the critical value for a t distribution with n
degrees of freedom which cuts off 3 area in the right
tail. The main problem with this test, as Kepner (1979)
also states, is that the overall level of the test, a, is
a = 23 1 + 2 3 2 43^
so relatively small values for 3^ and 32 would need to be
chosen .
The nonparametric tests they suggest are either
complicated, due to many estimation problems involved, or
have low power or are just unappealing due to the fact the
test is somewhat researcher dependent. (That is different
researchers working independently with the same data could
reach different conclusions.) Thus, they will not be
mentioned.
Hollander (1971) introduced a nonparametric test for
the null hypothesis of bivariate symmetry which is generally
appealing and consistent against a wide class of
alternatives. He suggested
10
D
n
/[ fFu-y)
- Fn(y,x)}2dFn(x,y)
where
is the bivariate empirical c.d.f. He notes that nDn is not
distribution-free nor asymptotically distribution-free when
Hq is true, and thus proposed a conditional test in which
the conditioning process is based on the 2n data points
(J i )
{((xHx2i)
V 0 or 1
for k = 1,2,...,n}
>
which are equally likely under HQ Here we let
(s,t)^) = (s,t) and (s,t)^^ = (t,s). This statistic
performs well even for extremely small sample sizes (n=5)
with one major drawback as mentioned by Hollander which is
the computer time which it takes to evaluate nDn. It
becomes very prohibitive for even moderate n. Koziol (1979)
developed the critical values for nDn for large sample
sizes, which work much better than the large sample critical
value approximations originally suggested by Hollander.
Kepner (1979) proposed tests based on the transformed
observations (Y^,Y2) of Bell and Haller for the null
hypothesis of bivariate symmetry versus the alternatives
that the marginal distributions differ in scale or that the
marginal distributions differ in location and/or scale. For
the alternative of differences in scale, he proposed a test
where
1 if t>0
'i'(t)
0 if t < 0 ,
which is Kendall's Tau applied to the transformed
observations. He noted that tt is neither distribution-free
nor asymptotically distribution-free in this setting and
thus recommended a permutation test which is conditionally
distribution-free based on tt for small samples. This
permutation test was based on conditioning on what he called
/
the collection matrix, Cn,
He noted that under HQ and conditional on Cn, there are 2n
equally likely transformed samples possible,
each being determined by a different collection of T* 's
j|f
where Â¥^ = {1 or -1}. For larger samples, he obtains the
asymptotic distribution which can be used to approximate the
permutation test.
One nice property of the statistic tt which Kepner
notes, is that tt is insensitive to unequal marginal
1 2
locations and thus location differences do not influence the
performance of the test.
For the more general alternative of location and/or
scale differences, a small sample permutation test for
bivariate symmetry was proposed based on the quadratic form
whe r e
T
n
W + is the Wilcoxon signed rank test statistic calculated on
the Yj^'s and tt n is as previously defined. Again, the
conditioning of the test is on the collection matrix Cn. He
/
obtains the limiting distribution of the small sample
permutation test and proposes a large sample
distribution-free approximation which is computationally
efficient.
The second collection of articles which will be
mentioned deals with the topic of censored matched pairs.
Much work has been done recently in the area of censored
data, but the work of Woolson and Lachenbruch (1980) and
Popovich (1983) most directly influence the results in this
thesis and thus will be described here.
Woolson and Lachenbruch (1980) considered the problem
of testing for differences in location using censored
13
matched pair data. The situation they considered is
identical to the situation developed in this thesis if one
assumes equality of the scale parameters. They utilized the
concept of the generalized rank, vector introduced by
Kalbfleisch and Prentice (1973) to develop tests by
imitating the derivation of the locally most powerful (LMP)
rank test in the uncensored case. Although they imitate the
development of LMP rank tests for the uncensored case, it is
unclear whether these tests are LMP in the censored case.
Scores for the test are derived for (1) if the underlying
distribution the differences (i.e., X2^) is logistic
and (2) if the underlying distribution for the differences
is double exponential. In each case the statistic developed
reduces to usual statistic (Wilcoxon signed rank statistic
and sign test statistic for an underlying logistic density
or double exponential density, respectively) when no
censoring is present. Asymptotic results for the tests are
derived based on the number of censored and uncensored
observations tending to infinity simultaneously.
Popovich (1983) proposed a class of statistics for the
problem of testing for differences in location using
censored matched pair data. The class consists of linear
combinations of two statistics which are independent given
Nj and N2 where is the number of pairs in which both
members are uncensored and N2, the number of pairs in which
exactly one member is censored. The class of statistics can
be expressed in the general form of
14
T
n
(N1 ,N2 )
(l-L )V2
T, (N. )
In 1
Vo *
I/2 T (N.)
n 2 n 2
where T^n is the standardized Wilcoxon signed rank statistic
calculated on the uncensored pairs, and
T2n = N2_ /2
^N2R N2L^
where
N2R
i s
the number of pairs for
which Xis
censored
and X2^
is
not
, and N2L is the number
of pairs for which X2^ is censored and X^ is not (note
^2R+^2L= ^2^* The weight Ln is a function of and ^ only
P *
such that 0
distri bution-free statistic calculated only on the
uncensored pairs (and is a common statistic used for testing
Â£
for location in the uncensored case) while T2n is a
statistic based only on the type 2 and 3 pairs (as
previously defined in this introduction). The statistic
k
T2n is designed to detect whether type 2 pairs are occurring
more often (or less often) than should be under the null
hypothesis. Under Hq T2n is a standardized Binomial random
variable with parameters N2=n2 and p= V2 and thus
distribution-free. Popovich obtains asymptotic normality
for the statistic under the conditions (1) that
and N2 tend to infinity simultaneously and (2) under a more
general condition as n tends to infinity. In a Monte Carlo
study, he compares five statistics from this class to the
test statistic of Woolson and Lachenbruch (T^) (1980) based
on logistic scores. The results show that these statistics
perform as well as T WL (better in some cases) and that they
1 5
are computationally much easier to calculate. Furthermore,
exact tables can be generated for any member of the class
proposed by Popovich.
With the background established for the research in
this thesis, the attention will now be focused toward the
development of the test statistics to be investigated
here. Chapter Two will present a statistic for testing for
differences in scale which can be viewed as an extension of
Kepner's tt n for censored data. In Chapter Three, another
statistic will be presented for the same alternative but
more in the spirit of the work proposed by Popovich, that
is, the linear combination of two statistics which are
conditionally independent (conditioned on the number of type
1 and (type 2 + type 3) pairs observed). For the more
general alternative (i.e., differences in location and/or
scale), Chapter Four will present a statistic(s) which is a
vector of two statistics (one for scale and one for
location) following the work of Kepner. Lastly, Chapter
Five will present a Monte Carlo study of the statistics
developed in this dissertation.
CHAPTER TWO
A STATISTIC FOR TESTING FOR DIFFERENCES IN SCALE
2.1 Introduction
In this chapter a statistic will be presented for
testing the null hypothesis of bivariate symmetry in the
presence of random right censoring. Figure 1 represents a
possible contour of an absolutely continuous distribution of
this form. The alternative hypothesis for which this test
statistic is developed is : o^ t i.e., the marginal
distributions differ in their scale parameters. The
marginal distributions are assumed to have the same location
parameter. Figure 2 represents a possible contour of an
absolutely continuous distribution of this form.
The basic idea for this statistic was introduced in a
dissertation by Kepner (1979). He suggested the use of
Kendall's tau on an orthogonal transformation of the
original random variables to test for differences in scale
in the marginal distributions. The presence of a censoring
random variable was not included. To extend this idea to
include the presence of random right censoring, the concept
of concordance and discordance in the presence of censoring
which was used by Oakes (1982) was applied.
16
17
Figure 1. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales.
18
Figure 2. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales.
19
Section 2.2 will present the test statistic and the
notation necessary for its presentation. A small sample
test will be discussed in Section 2.3. Section 2.4 will
investigate the asymptotic properties of the test statistic,
with comments on the statistic following in Section 2.5.
2.2 The CD Statistic
In this section, the test statistic will be presented
which is designed to test whether the marginal distributions
differ in their scale parameters. First, since the work, is
so related, the test statistic which Kepner (1979) proposed
to test for unequal marginal scales will be presented. This
will give the reader an understanding of the motivation for
the test statistic.
Let for i=l,2,...,n denote independent
identically distributed (i.i.d.) bivariate random variables
which are distributed as (X,j,X2^)* Consider the following
t I
orthogonal transformation of the random variables (X^,X2^);
let
Yli = Xli + X2i and Y2i = Xli X2i for
Figure 3 illustrates what happens to the contour given in
Figure 1 (i.e., the contour of an absolutely continuous
distribution under HQ) when this transformation is
20
applied. Figure 4 shows what happens to the contour given
in Figure 2 (i.e., under H ) when this transformation is
applied. Note, as can be seen in Figure 3, under this
I
transformation and H Y, and Y01 are not correlated
o 11 11
t
although Xjj and X2 ^ possibly were. Similarly, as can be
seen in Figure 4, under this transformation and and
f
Y21 are correlated (negatively in this case). Thus, the
original problem of testing for unequal marginal scales has
been transformed into the problem of testing for correlation
I
between Y^ and Y2^ Kepner (1979) suggested the use of
T
Kendall's tau to test for correlation between Y^ and Y9y .
Kendall's tau was chosen, due to the fact it is a
U-statistic and, thus, the many established results for
U-statistics could be applied.
The test statistic which will be presented in this
section is very similar to the above mentioned statistic.
However, when censoring is present, the true observed value
If
of X ^ ^ 0^ X 21
(or
both) is not known,
a nd
thus
Y11 Y21
(or both) are
also
affected. To take
this
into
account a
modified Kendall's tau will be used which was presented by
Oakes (1982) to test for independence in the presence of
censoring. First though, some additional notation must be
int roduced.
I I
Recall, (Xjj,X2i) denotes bivariate random variables
t 1
which are distributed as (xii>^21^* Let 0^,02, Cn denote
the censoring random variables which are independent and
identically distributed (i.i.d.) with continuous
21
Figure 3.
Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales under the Transformation.
22
Figure 4. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales under the Transformation.
23
distribution function G(c) where denotes the value of
! t
the censoring variable associated with pair In
the case of random right censoring, the observed sample
f I
consists of X1 = min(Xj^,C^) and X2^ = mi n ( XÂ£ ^ C ^ ) These
pairs can be classified into four pair types which are
Pair Type Description
f
1
Xli
x2i
*
2
Xii
X2i> Ci
3
XÂ¡i>Ci>
X2i
4
Xii>Ci>
X2i>Ci
Consider the following orthogonal transformation applied to
the observed sample:
/
Yli = Xli + X2i an(* Y2i = Xli X2i ^or i = 1 ^ . n.
Notice that, due to censoring in type 2,3, or 4 pairs, the
T
true values of and Y2^ (denoted Y^ and Y2i i.e. the
values had no censoring occurred) are not actually
observed. The following table, Table 2.1, summarizes the
t I
relationship of the true values of Y ^ and Y2^ to the
observed values
Table 2.1
24
Summarizing the Relationship Between the True
t
Values of and Y2^ to the Observed Values
Pair
Type
Description
Relationship
Between
Yli and Yli
Relationship
tBetween
y21 and Y2i
1 i
X2i
Yli
=
Yli
Y2 i
= Y2 i
li
x2i>ci
YIi
>
Yli
Y2 i
< Y2 i
li>Ci
x2i
1
Yli
>
Yli
Y2i
> Y 2 i
1 i>Ci
X2i>Ci
Yli
>
Yli
uncertain
or Y2 ^ < Y2 i )
The modified Kendall's tau (denoted CD for
c^oncordant-discordant) can now be defined as
CD
a., b.. where for i < i ,
ij ij
r
r
1
if
Y
! Y, .
1
if
Y
v Y
l i
Ij
2 i
2j
<
-1
if
Yli '
Y. and
1 J
bu- <
-1
if
Y
2i
\ y
2j
0
if
uncertain of
0
if
unce r t ain of
V
the
relationship
V
the
relationship
(2.2.1)
25
(here Y11
I
can be read as
i
Y is definitely smaller
than
*- l_ *_ >
For example, if the iL pair is a type 1 and the j
pair is a type 2 and it was observed that Yli < Ylj. then
i t i i
aij = 1 since Y^ = Y^ and Y^j < Y^ (thus Y^ < Y^). If
Yli > Ylj had been observed, then a^^ = 0, since the
t f
relationship between Y^ and Y^j is uncertain. Similarly,
if Y2 < Y2j then f'ij = 0 since Y2 = Y2i and Y2j < Y2j
I
(thus, the relationship between Y2i and Y2j is uncertain).
On the other hand, if Y2i > Y2 j had been observed, then
b^j = -1 (by a similar argument).
Table 2.2 summarizes the necessary conditons for a^^
and b^j to take on the values of -1, 1 or 0. The product of
a^j and b^j results in a value of 1 if the it'1 and j*"*1 pairs
of the transformed data points are definitely concordant, a
value of -1 if the pairs are definitely discordant and 0 if
it is uncertain. If the iC^ pair is a type 4 (i.e., both
I
Xu and ^2^ were censored) then b^j will always be 0 since
the relationship between the i n and jcn pair is always
uncertain regardless of the pair's type. Thus, type 4
pairs always contribute 0's in the sum for CD. Notice, also
in the case of no censoring this modified Kendall's tau
reduces to the Kendall's tau applied to the transformed
data, the statistic investigated by Kepner (1979).
26
Table 2.2 Summarizing the Values of a. and b.. for i
= 1: if and one of the following occurs,
ith pair type
Jth
pair type
1
1
1
2
1
3
1
4
a .
ij
= -1 :
if Y^ > Y^j and one
of
t he
following occurs,
t" Vi
i pair type
3th
pair type
1
1
2
1
3
1
4
1
a .
ij
= 0:
for all other cases
bij
= 1 :
if Y2^ < Y2j and one
of
the following occurs,
ith pair type
3th
pair type
1
1
2
1
1
3
2
3
bij
= -1 :
if Y2^ > Y2j and one
of
t he
following occurs,
ith pair type
3 th
pair type
1
1
1
2
3
1
3
2
bU
= 0:
for all other cases
27
Next, we establish some properties of the CD statistic.
Lemma 2.2.1: Under HQ ,
E ( C D ) = 0
and
Var(CD) = a +
n ( n 1)
n ( n 1 )
where
a nd
a = 4P(a .= 1,b. = 1 )
ij iJ
2p
bi j =
1 a . =
ij
f N
*H
II
*-)
*H
xÂ¡
+ ZPia.j.
= 1 a . ,
ij
= -1,b . =
ij
1)
+ 4P(a1.=
= -1,b . =
ij
1)
- 2P(a1j =
l.bpj-
= 1,b. =
ij
1)
- 2PUlj-
-l.bij
= -1 a .
ij
-i,bir
= 1)
- 4P(a1;j-
'bij"
1 "
1)
(2.2.2)
Proof:
Throughout this proof, Theorem 1.3.7 in Randles and
Wolfe (1979) will be used extensively and thus its use will
not be explicitly indicated.
Under H
>
(Xli,X2i,X1j ,X2j ,Ci ,Cj ) = ^X2i,X1iX2j X1j Ci Cj^
28
and therefore it follows that
(XliX2iXlj X2j 0i>6j} (X2i,Xli,X2j ,X1 ,f(6 ) ,f(6j ) )
(2.2.3)
where
Xu = min(xJi,C1),
X2i = rain(X2i ,C),
^ indicates what type of pair (X^.X^)
and
f(6i) indicates what type of pair (X2i,Xli^ is
Thus, f(*) is the function defined below.
5i
f (6)
1
3
2
4
Let Yli Xli + X2i a nd Y2i Xli X2i; thus from (2.2.3)
(YliY2iYljY2j6i>6;j> = (Yli,-Y2.,Ylj,-Y2j,f(6i),f(6j))
Applying the definition of a^ and b ^ j in (2.2.1) (or using
Table 2.2) to the above, it follows that
and thus
p
29
and
p (ai j -1> bij Pbij -1J
(2.2.4)
Now,
E ( C D ) = I E ( a . b . )
(?) i
where
ECa^bjj) (DF^jbjj 1) + (-I)P(aijblj -1)
p
- P(atj l.bjj -1) P(ai:j -l,btj 1) .
Applying (2.2.4) to the above, it follows that E(a^^.b^j) = 0
and thus E(CD) = 0.
Note, that under H Q ,
(Xli,X2i*Xlj ,X2j <5i,<5j) ^Xlj X2j xiix2i5j >
and thus
(YlfI2i.'ilj-y2J-Si.5j>
(2.2.5)
Applying the definition of and b^j as before, it follows
(au>bij) (-aij-'bij)
and also
30
and
P(a13 l,b13 1) P(a^j -l.bij -1)
P p
Now,
Var(CD) = [] Var( Â£ a b..)
' ^ 1 i !J 1 J
(J) ^<3
, 2
= [ ] J y Cov ( a . b . a , b )
r?i i*
The three possible cases to consider for the covariance are*
1 ) ii j i
2) i-i' j=j i< j i < j '
and
3) where exactly two of the four subscripts
i
Case 1) i/i j tj' :
In this case, Cov(a^jb^j, a^ij,b^iji) = 0 since the
bivariate pairs are i.i.d.
Case 2) i=i', j =j':
In this case ,
Cov(aijbij at j bjj ) EU.jjby) ]
P(aij >bij > + P(aij 1 bij -*>
+ P> + P
= 4 p(aij = 1 t> i j = 1) = 4a (by part a).
31
Case 3) Exactly two of the four subscripts i
the same.
Now,
Cov(aij bj
aikbik)
- E(atJb
i j aik
the following
events:
Al,i:
a i j "
1 aik =
1}
Al,-1:
{aij =
aik =
-1}
A-1 1:
{aij =
_1 aik
= 1}
A-1,-l
: {aJ =
-1 aik
= -1}
and similarly define the events B_^ ^ and
B_j Using this notation, E ( a ^ ^ b ^ j a ^ ) can be written
as
E(aijbijaikbik) =
l I l I (-l)k+I+"+"p(A
k = 0 A-0 m = 0 n =0 (-1) ,(-1)*
,B )
(-l)m,(-1)n
(2.2.6)
Table 2.3 describes the events A and B
<-Dk,(-i)t <-i)m,<-nn
in more detail and the restrictions placed on the 6's.
Now, to simplify the probabilities in (2.2.6). Note,
under Hq
d
X2j ,XlkX2k,,Si5j ^ k ^
X2k>Xlk>f(6i)>f
32
Applying the transformations
Yli = Xli
X2i and Y2i Xli X2i
it f ollows that
6i.6j'Sk>
Now, applying the definitions of a^j and in (2.2.1)
using Table 2.2), notice that if b^j = 1 (i.e.,
6 e(1,2) and 6jÂ£(l,3), then ~Y. > -Y. f(6 ) e( 1 3)
f(6j)e(l,2) which would yield b^j = -1.
Using similar arguments, it follows
and
(a..,a.,,b..,b.,) = (a..,a..,-b..,-b.,)
ij ik ij lk ij ik ij lk
and thus
P(A,
1 9
1
,B 1,1 ) =
p(Ai,i
-1
P(A,
* 9
1
,B-1,1)
- P (A.
1 9
1 '
B1
P(A_1
9 ~~
1 ,B 1 1 )
= P(A_
1-1
9
P(A_j
9 ~~
1 *B-1,1
) = P (A
-1 ,
-1
B 1 1
P(A_X
, 1
,B 1 1 )
= P(A_1
, 1
P(A_j
, 1
B-l,1}
= P (A_
1 1
)
P ( A ,
1 9
-1
B 1 1 ^
= P(A,
1 9
-1
> B
-i.-i
P ( A,
L 9
-1
B-l,1)
= P(Aj
9
Bi,-i>
(or
Table 2.3 Describing the Events A and B and
Event Description
A1,1B1,1
CYll
(Y21
A-1, 1B1,1
(Y11>Y1jYli
A1,-1B1,1
Yik)
A1, 1B-1,1
Y2j
A1,1B1,-1
A-1,-1B1,1
(Yjl>Ylj,Yli>Yjk)
A-1,1B-1,1
Y1j *Yli
(Y2i>Y2j
A-1,1B1,-1
(Y1i>Y1J,Yii
A1,-1B-1,1
Ylk>
Y2j
A1,-lBl,-l
Ylk>
A1,1B-1,-l
^ Yli
Y2j
A-1,-1B-1,1
(YU>YljYli>Ylk>
Y2j
A-1,-1B1,-l
(Yl1>Yjj,Y,1>Yik)
(Y2i
A-1, 1B-1,-1
Ylj-Yli
(Y2i>l2j
A1,-1B-1 ,-l
Ylk)
Y2j
>
1
1
cc
1
1
H-
Ylj-YU>Ylk>
Y2j
the Restrictions on the
6 s
Restrictions
on the
6 s
6 j
Y2i
1
1,3
1,3
Y2i
1
1
1,3
Y2i
1
1,3
1
Y2i
1
1,2
1,3
Y2i>Y2k>
1
1,3
1,2
Y21
1,2
1
1
Y2i
1
1
1,3
Y2i> Y2 k >
1
1
1,2
Y2i
1
1,2
1
Y2i>Y2k>
1
1,3
1
Y2i>Y2k)
1
1,2
1,2
Y2i
1
1
1
Y21>Y2k>
1
1
1
Y2i>Y2k>
1
1
1,2
Y2i>Y2k>
1
1,2
1
Y2i>Y2k>
1,3
1
1
34
Similarly, under HQ
(Xli,X2i,Xlj,X2j,Xlk,X2k,<5i5jlSk^
= (xiiX2i>xlk>X2kXljX2j6i>6k6j)
and applying the definition of a^ and b^j in (2.2.1) it
follows that
(aij >aikbij >bik) (aikaij )bikbij}
This yields that
P
B-1, i^ = p(Ai,-i
B 1 -1
*-d
>
1
*
Bi ,-P = p(Ai ,-i
B 1 1
p
,B i i ) = P(A_1 ^l
B1 1 )
p
B -1 ,-P = P(A-1 ,
1 B-1,
Thus, E(ajb^jaikbik^ can re^uced to a sum of six terms
instead of the original sixteen; i.e.
E(ab..ab ) =
ij ij ik ik
1111
l l l l (-D
k = 0 Â£ =0 m = 0 n =0
k+Â£+m+n
p (A. . ,B )
(-i)k,(-i)* (-n,(-Dn
2P(A1,1 + 2P(A-1,-1 Vl> + 4P(A1,-1
' 2P 4P(A1,-1B1,1> 4
Note, the subscripts are arbitrary; thus
E(aijbijaikbik) = E(aijbijakjbkj) = E(aijbijajkbjk)
35
and therefore combining the results from case 1, 2 and 3, it
follows that
!-l ) Y}
Var(CD)
As seen in Lemma 2.2.1, the variance of CD depends on
the underlying distribution of and possibly C.
Therefore, CD is not distribution free under HQ. Section
2.3 will discuss a permutation test based on CD that is
conditionally distribution free. This test is recommended
for small samples. For larger samples, Section 2.4 presents
the asymptotic normal distribution of CD using a consistent
estimator of the variance. This result can be used to
/
construct a distribution free large sample test based on CD.
2.3 Permutation Test
In the situation where the sample size is small, a
permutation test based on CD is recommended. What is
considered a small sample size will be discussed in Chapter
Five when the Monte Carlo results are presented. Now, we
will develop the motivation for the permutation test.
Recall, under HQ
f
(X
2i,Xli>Ci>
36
and thus
x2i
, <5 i) = (X2i,Xli,f(6i))
(2.3
where
Xli =
min(Xji,), X2i
= min(X2i,Ci), 6^ is the
pair
type
( i e .
6 ^ = 1,2,3 or 4)
and f(<5^) is a function
such
that
1
2
3
4
f (5)
1
3
2
4
0
Let k = | j be an operator such that
r
(XliX2i6i)k = \
(X1iX2i6i)
if k = 1
(X2iXlif(6i)) if k = 0
and K = {k: k is a 1 x n vector of 0's and l's} (of which
there are 2n different elements). Thus, applying this
operator to (2.3.1), we see under HQ, P{(X^,X2^,6^) =
(Xu,X2i,6i)0} = P((Xli,X2i,6i) = (X1.,X2i,6.)1}. Applying
this idea to the entire sample (in which the observations
are i.i.d), under H it follows that
{(Xll*X21^i^ ^(^X2X22^2^ ^ ^XlnX2n,<^n^t1^
t I
^91 *6}) 1 ,(X.2,X22,62) 2,...,(X.,X2n,6n) n}
(2.3.2)
37
where k and k' are arbitrary elements of K. Therefore,
unde r HQ, given
{ (x
11 x21^1^ >(xi2,x22^2^* * ^ x1nx2 n 6n)>,
the 2n possible vectors
{ (xiix21^l) l(xi2x2 2^2^ ^^xlnx2n^n^ n}
are equally likely values for
{ ( X j i >XÂ£i <5 ^ ) ,(Xi2>X22><52^ * ^ln^2n^n^ *
The idea of the permutation test is to compare the
observed value of CD, for the sample witnessed to the
conditional distribution of CD derived from the 2n equally
likely possible values of CD (not necessarily unique)
calculated from
{ (xii ,x21 1>(x12x22^2^ 2,...,(xin>X2n(5n^ n ^
Note, since the sample observed is censored, the 2n
vectors {xii>x21^l^ l,(x^2x22^2^ ^''^xlnx2n^n^ n }
are not necessarily unique. If a pair is a type 4 (i.e.,
both and X2j were censored), then (xij >x2j >^j ) ^ =
(xjj,x2j 6j In fact, there are only 2^n-n4^ unique
vectors (n^ = number of type 4 pairs), since P(Xj^ = ^2^) =
0 if ^xii^2i^ is not a cYPe ^ pair under assumption A2. As
a result, the permutation test, in effect, discards the type
4 pairs (since a j b j = 0 if the i c or j C ^ pair is a type
4) and treats the sample as if it were of size n-n^ with no
type 4 pairs occurring.
With regards to the transformed variables (^ii*^2i^
i = 1,2 the permutation test can be viewed in the
following way. Consider the transformations = X^ + X2^
38
and Y2^ = ^2i* Applying these to (2.3.1) and (2.3.2),
we see that under Hq
(Yli,Y2i,6i) = (Yli,-Y2i,f(6i))
and similarly,
{ ( Y
11 L21
Si)
( Y
1 2 12 2
s2)
( Y
In 2 n
5n>
n-,
!
k
whe re
k.
if k = 1
= ^
(YliY2if(6i}) if
k = 0
and k and k' are arbitrary elements of K. That is, under
H0, given { (y:2 ,y21,Sx),(y12,y22 s2),..., (yln,y2n,sn)}, the
2n possible vectors
kl k2 kn
^yl1 y2161) >(yi2y22^2^ *^ylny2n^n^ } are
equally likely values for
{ (Yii Y2 i ^ i ) ,(Y12>y22,S2) . >(Y^n,Y2n,6ri)}
To perform the permutation test, the measurements
(x1i*x2i^i) ^ = l>2,...,n are observed and the
corresponding value of CD is calculated. Under Hq, there
are 2n equally likely transformed vectors for
{(YllY216l)(Y12Y2262)*(YlnY2n
39
statistic is computed for each of these possible vectors and
from this the relative frequency of each possible CD value
is determined. The null hypothesis is rejected if the
original observed CD value is too large or too small when
compared to the appropriate critical value of this
conditional distribution.
2.4 Asymptotic Results
In Section 2.3, a permutation test was presented to
test Hq, when the sample size was small. In larger sample
sizes, the permutation test becomes impractical and time
consuming. In these situations, the asymptotic results
which will be presented in this section could be employed.
Theorem 2.4.1: Under Hq,
* N(0,1) as n * ?
[ Var(CD) ]X/2
where
Var(CD)
2 4(n-2)
n(n-l) a n(n-1) Y
Proof;
Note that CD is a U-statistic with symmetric kernel
h(X^,Xj) = Thus, by applying Theorem 3.3.13 of
40
Randles and Wolfe (1979), it follows that
CD
+ N(0 ,
as n > oo
whe re
?1 = E[h(X.,Xj)h(Xi,Xk)]
Note that
= E[aijbij aikbik]
il
n
2 4(n-2)
n(n-l) a n(n-l) ^
Y
as n + <*>,
therefore after applying Slutsky's Theorem (Theorem 3.2.8,
Randles and Wolfe, 1979)
CD
[Var(CD)J^2
+ N(0,1) as n + .
Corollary 2.4.2: If Var(CD) is any consistent estimator of
Var(CD), then
+ N(0,1) as n -* .
[ Var (CD) ]^2
Proof;
This follows directly from Theorem 2.4.1 and Slutsky's
Theorem. ^
Next, we consider the problem of finding a consistent
estimator for Var(CD). There are many consistent estimators
41
for a variance, but three which worked well in the Monte
Carlo study are described in the following lemma.
Lemma 2.4.3: Under Hq the following are consistent
estimators of Var(CD):
1) Va r x(CD)
4
n
lllA
1 *
ijk*
y
where
AijkBijk = (aijbijaikbik + aikbikajkbjk + aijbijajkbjk)>
2) Var2(CD)
- 4 (
l l l + l I 1 > >.
n n(n-l)2 l*
and*
3)Var3(CD)
<[tj- l l 21
n(n-l) ( 2 ) 1*
4(n-2) rn .N.*
+ {- Var2(CD)}
n(n-1) 4
Proof:
First, it will be shown that nVar^(CD) 4y.
nVa r ^(CD)
lllA
1 < i < j
i jkBij k
}
Now,
42
, A B .
- Hr I l l
1*
which shows that nVar,(CD) = 4U where U is a U-statistic*
J- n n
of degree 3 with symmetric kernel h* = A^^^3^^^/3 Thus, it
follows that nVar^(CD) 4y since y by Hoeffding's
Theorem (Hoeffding, 1961).
Next, it will be shown that
n(Var2(CD) Var^CCD)) 0 as n .
First though, notice that Var2(CD) is equivalent to
2)Var1(CD) + -{ [ l l (a b )2] (CD)2}
(n-1) n n(n-l) l*
Thus ,*
(n-2 )
n(Var2(CD) Var^CD)) = n{ f 1} Var^CD)
(n- 1 )
+ 4{[
l l (a b )2] (CD)2}
n ( n-1 ) 2 1 < i < j < n 1 ^
(n-2 )
(n-1)
- U B* + {[ L l J (a b )2] (CD)2}
(n-1)(2) 1 *
0 as n > oo.*
Therefore, Var2(CD) is a consistent estimator for Var(CD).
Lastly, it will be shown that
n(Var^(CD) Var2(CD) ) * 0 as n + .
43
Now,
n(Var3(CD) Var2(CD))
-L- [tst I l (aiibii)2]
(n-1) (") 1*
+ {(n 2) 1} nVar2(CD)*
(n-1)
-> 0 as n -*
Next, we provide a brief explanation of each of these
estimators. As was shown in the proof of Lemma 2.4.3,
^ 4 *
Var,(CD) = U where U is a U-statistic which estimates y.
i n n n
Thus, Var^(CD) is estimating the asymptotic variance of
CD. Var2(CD) is also estimating the asymptotic variance of
CD, but in a slightly different manner. Recall, from basic
U-statistic theory that y is the variance of a conditional
expectation (Randles and Wolfe, 1979, p. 79) (i.e.,
y = Var[(ajbj)*] where (ajbj)* = E[a12b^2Â¡ (Y11,Y21)]) .
Thus, in Var2(CD), for each (^j^,Y2^), the conditional
expectation is estimated using all the other (Y^,Y2j)'s,
j*i and then the variance of all these quantities is
calculated. That is,
where
Var-(CD) = Y {(a.b.)* CD}2
4 n t l l
n-1 j*i J
In contrast to Var^(CD) and Var2(CD), Var3(CD) is
estimating the exact variance of CD (2.2.2) derived in
44
Section 2.2. It is using an estimator of y from Va^CCD)
and estimating a with a difference of two U-statistics which
is estimating
Again, although under HQ ECa^b^j) = 0, the sample
estimate for Eia^b^j) (i*e., CD) was left in to possibly
increase the power of the test under the alternative.
Each of these variance estimators will be considered in
the Monte Carlo study in Chapter Five. Although the
calculations look overwhelming if performed by hand, they
are all easily programmed on the computer. (See the CDSTAT
subroutine in the Monte Carlo program listed in Appendix 2.)
2.5 Comment s
This chapter has presented a statistic to test the null
hypothesis of bivariate symmetry versus the alternative that
the marginal distributions differ in their scale
parameters. For small samples, a permutation test is
recommended. A basic disadvantage of this is that it
generally requires the use of a computer for moderate sizes
(otherwise it is very time consuming to derive the null
distribution). For larger sample sizes, it is recommended
C D
that be used as an approximation for
[Var(CD)f^2
CD
Thus, for an a level test using the
[Var(CD)]1/2
45
asymptotic
rej ected if
distribution,
i CD
^ Vo
[Var(CD)] 2
the null hypothesis would be
> Z
a/2
where Z is the value in
a / 2
a standard normal distribution such that the area to the
right of the value is a/2.
Chapter Five will present a Monte Carlo study which
uses the asymptotic normal distribution of CD (with a
consistent variance estimator) to investigate how well the
test performs under the null and alternative hypotheses.
First though, some comments on this chapter.
Comment 1
One possible advantage of the CD statistic is the fact
it utilizes information between censored and uncensored
pairs whenever possible. In the permutation test, type 4
pairs have no effect on the outcome of the test. That is,
they can be ignored, treating the sample as if it were of
size ni+n2+n3* This is understandable since Xli = X2i = Ci
and thus they supply no information about the scale of
relative to
In the asymptotic test, if one estimated the variance
in (2.2.2) by estimating a and y with their sample
quantities (for example, a =
n(n-l) 1 < i < j < n
l l (aijbij) ic
46
is easily shown that the type 4 pairs have no effect on the
value of the test statistic. That is, the value of the test
statistic remains the same whether the type 4 pairs are
discarded or not. If a different estimate for the variance
is used, there is a slight change in the test statistic's
value if type 4 pairs are discarded, due to the different
variance estimator. Asymptotically, this difference goes to
zero, due to the fact the variance estimates are all
estimating the same quantity. Thus, in some sense, the
asymptotic test behaves similarly to the permutation test
with regards to type 4 pairs.
If a and y are known, they are a function of whether
type 4 pairs are included or not. That is, if type 4 pairs
were not included in calculating the test statistic (thus
n=n^+n2+n3), the value for a and y would be larger than the
value had type 4 pairs been included (since type 4 pairs
only contribute 0's and never l's or -l's). The effect of
type 4 pairs on a and y is such that the test statistic's
value would be the same (or at least asymptotically the
same) whether type 4 pairs were discarded or not.
Comment 2
A disadvantage of the test is that for small samples CD
is not distribution free. Thus, the permutation test,
conditioning on the observed sample pairings, must be
47
performed to achieve a legitimate distribution free crlevel
test .
Comment 3
It is unclear how the CD statistic would be affected if
the marginal distributions of and X21 have different
locations. It is possible that the assumptions made on the
censoring distribution might not be valid (in particular
assumption A4, which assumed the same censoring cutoff for
X^ and ^2i^ or even if this is true, that CD does not
perform well in these instances. Chapter 5 will investigate
this problem in further detail
CHAPTER THREE
A CLASS OF TESTS FOR TESTING FOR DIFFERENCES IN SCALE
3.1 Introduction
In the previous chapter, a test statistic was presented
to test the null hypothesis of bivariate symmetry against
the alternative that the marginal distributions differ only
in their scale parameters. A shortcoming of the statistic
was the fact the variance of CD depended on the underlying
distribution and, thus, for a small samples a permutation
test had to be done or for large samples the variance had to
/
be estimated. In this chapter, two test statistics will be
presented which are nonparametrica1ly distribution-free
(conditional on N^ = n^ and Nc = n2+n^) for all sample sizes
to test the null hypothesis of bivariate symmetry. The
alternative hypotheses are structured by assuming the
samples come from a bivariate distribution with c.d.f.
*1 U x2 p
F( ) where F(u,v) = F(v,u) for every (u,v)
a 1 2
2
in R Tests are developed for both of the following
alternatives to the null hypothesis of bivariate symmetry:
48
49
Case 1 = ^2 known,
Ho: ai = a2 an<^ Ka: o< a 2
That is, the marginal distributions have the same known
f
location parameter but, under Hfl, X2^ has a larger scale
I
parameter than X^ A possible contour of an absolutely
continuous distribution of this form was given in Figure 2.
Case 2 p j = pi 2 unknown,
a 1 = a 2 and Ha
a L < a 2
Here, the marginal distributions have the same unknown
location parameter but, under Ha, X2 ^ has a larger scale
parameter than X^
(Note, for both cases, the alternative has been stated in
the form for a one sided test. The procedure which will be
presented can easily be adapted for the other one-sided or a
two sided alternative. The latter is discussed at the end
of this chapter )
In Sections 3.2 and 3.3, tests statistics for Case 1
and Case 2, respectively, will be presented which are
nonparametrically distribution-free conditional on N ^ = n ^
and Nc = n9+n^. In both cases, the test statistics can be
viewed as a linear combination of two independent test
statistics T and T where T is a statistic based only
nl nc nl
on the n^ uncensored observations, while Tn will be a
statistic based on the nÂ£ = n2 + n3 type 2 and 3 censored
observations. The conditioning of the random variables
50
and Nc on n^ and n2+n^ (respectively) is used throughout
Section 3.2 and 3.3 and, thus, this condition will not
always be stated but will be assumed with the use of n^, n2
and n7. Thus, the test statistics will be written as T
3 ni n(
and TMn n (for Section 3.2 and 3.3, respectively) which
1 c
imply conditioning on = n^ and Nc = nc = t^ + n^. Section
3.5 will consider the asymptotic distribution of each test
statistic .
3.2 p ^ = p 2 > Known
This section will begin by introducing the notation
necessary for the statistic Tn n designed for the
1 c
alternative in case 1. Recall, the sample consists of
(Xli,X2i) i = l>2,...,n where = min(X^,C^) and
I
X2i = n(^2 iCi). These pairs were classified into four
types .
They
were the
following
Pair
Type
Description
Number of Pairs
in the Sample
1
X1i
X2i
nl
2
Xli
X2i>Ci
n2
3
Xii>Ci
X2i
n3
4
xU>ci.
X2i>Ci
n4
n^ + 02 +
where n
n3 + n4
51
For convenience and without loss of generality, let the
type 1 pairs occupy positions 1 to n^ in the sample (i.e.,
{(XjJ,X21),(Xj2,X22(Xln ,X2ni)} ) in random order.
Similarly, the type 2 and type 3 pairs will be assumed to
occupy positions n^+1,n^+2,...,n^+nc in random order.
Lastly, the type 4 pairs occupy positions
n^+nc+l,n^+nc+2,...,n. What is meant by random order, is
that the exchangeability property still holds within the n^
type 1 pairs, within the n2 + n2 type 2 or 3 pairs and within
the n^ type 4 pairs. This could be accomplished, if the
pairs were placed into their respective grouping (type 1, 2
or 3, or 4) arbitrarily, with no regard to their original
position in the sample. Much easier, from a researchers
point of view, would be to place the pairs into their
respective groupings in the same order they occurred in the
sample (i.e., the first uncensored pair is placed into the
first position among the n^ uncensored pairs, the second
uncensored pair into the second position, etc.) This
procedure would not affect the desired exchangeability
property, as deduced from the following argument. In using
the second method, the reseacher is actually fixing the
position of the type 1 pairs, type 4 pairs and type 2 or 3
pairs. Thus, instead of n! equally likely arrangements of
the original sample, there are n^!n^!(n9+n2)! equally likely
arrangements when the positions and numbers of the pair
types are fixed. Therefore, it follows, that each of the
n^! arrangements of the n^ uncensored pairs is equally
52
likely and that the exchangeability property still holds
within the type 1 uncensored pairs. Similar argueraents for
the ^2 + 1*2) type 2 or 3 pairs and the n^ type 4 pairs
hold .
The following notation will be used in the statistic
Tn^, a statistic which is based on the n^ type 1 pairs.
Define a variable to be
Zi I X 2 i y I I X1 i 11 I for i_12...n1
where p is the known and common location parameter. Let
be the absolute rank of for i = 1 2 . n ^ that is, the
rank of j Z ^| among {|z^|,|Z2|,...,|zn j} and let be
defined as
H'i = 'R C Z ) =
1 if Z > 0
0 if zi < 0
Note, the variable Z ^ is defined only for the uncensored
pairs
The statistic T is then
nl
ll
: = y i.
n. L 1
R
i = l
the Wilcoxon signed rank statistic computed on the Z^'s.
Notation will now be introduced for the statistic Tn ,
c
a statistic based only on the type 2 and type 3 censored
pairs. (The pairs in which only one member has been
censored.) Define Qj to be the rank of Cj among
^ Cn1 + 1Cn1+2 ,Cni + nc* and
53
YJ 1
h
1 if the j pair is a type 2 pair
0 if the j11*1 pair is a type 3 pair
for j = n^+1,n^+2,...,n,+n^. The statistic Tn is defined
1 c
a s
n, + n
1 c
C = l y. Q.
nc j=n1+l J J
= y ranks of the C's for the type 2 pairs
A brief explanation of the logic behind the test
statistic will be presented. For the test statistic Tn if
X2 has a larger scale parameter than Xj (i.e., under ) ,
then 1^2^ pJ |x^ p| should be positive and large.
Thus, the test statistic Tn would be large. In contrast,
if X2 and X: have the same scale parameter (i.e., under HQ ) ,
then I X 2 ^ ~ pI ~ |x^ p| would be positive approximately
as many times as negative with no pattern present in the
magnitudes of 1x2^ p| |x^ pj. Thus the test
statistic would be comparatively less.
For the test statistic T if H is true, there should
nc a
be a preponderance of type 2 censored pairs (relative to the
number of type 3 censored pairs) and these pairs should have
the more extreme censoring values. Figure 5 illustrates
this idea. Thus, the test statistic Tn would be large. In
contrast, if H is true, the number of type 2 pairs should
54
Figure 5 .
Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales with Censoring Present.
55
not dominate n and the test statistic T should not be
c nc
unusually large.
Now we establish certain distributional properties for
T_ and T_ .
n. n
Lemma 3.2.1:
distribution
Conditional on n^ T has
as the Wilcoxon signed rank
the same null
statistic.
Proof:
First it will be shown that conditioning on the
1 pairs does not affect the exchangeability property
(XliX2iCi) (X2iXjiCi)) still holds. Let
(XiXÂ¡i.Ci) and W* = (X2i,x|i,Ci) and
G (t) =P(xJt< t x X2i^ t2Ci< t3) = E[I(W < Â£)] where
r 1
i(w < t)
i
0
if X..< t
11
otherwise
X. <
2i
t2Ci<
nl type
( i e ,
Now, under HQ for the entire sample, we have
(Xii>X21>Ci> Cl>
and applying an apropriate function (and Theorem 1.3.7 of
Randles and Wolfe, 1979) thus
I(W. < t)I(5i= 1) = I(W* < t)I(f(6.) = 1) .
Taking expectations, it follows that
E[I(W.< t)I(6.= 1)] = El I(W < t)I(f(5.) = 1)] .
Now, recalling that 5^= 1 iff f(5^) = 1; thus
E[ I ( 5 i = 1)] = E[ I ( f ( <5 i ) = 1)] ,
and it follows that
56
E [ I( W < t)I(6jL= 1)] E [ I (W* < t)I(f(6i) = 1)]
' ~" 1
E[I(5.= 1)] E[I(f(5) = 1)J
This shows that the c.d.f. of given it is a type 1 pair
is equal to the c.d.f. of Wi given it is a type 1 pair and
thus the exchangeability property holds within the type 1
pairs.
Now, by defining a function
fj(a,b,c) = |min(b,c) y| |min(a,c) y| and applying
Theorem 1.3.7 (Randles and Wolfe, 1979, page 16) it follows
Z | X 21 ~ y |
- 1X11 l
= |min(X21>C) yj
j
- min ( X C )
d | ,
i
- |min(X2^ ,C)
= |min(X1j C) y 1
= 1 X1 l n| -
1X21 ~ =
and thus by Theorem 1.3.2 (Randles and Wolfe, 1979, page
14), the random variable Z has a distribution that is
symmetric about 0. The proof of Lemma 3.2.1 follows
directly from Theorem 2.4.6 (Randles and Wolfe, 1979, page
50) D
Lemma 3.2.2: Under HQ, the following results hold.
a) Conditional on the fact the pair is type 2 or 3, the
random variables Yj and Cj are independent.
b) Conditonal on nc, Tn has the same null distribution
as the Wilcoxon signed rank statistic.
57
Proof :
First, it will be shown that conditioning on the n c
type 2 and 3 pairs does not affect the exchangeability
property. Define W^, GW^ Â£ ^ and I(W^ < t) as in Lemma
3.3.1. Now under HQ for the entire sample, we have
(^1 i,X2i ^i) = (^2iXli>^i^
and applying an appropriate function (and Theorem 1.3.7 of
Randles and Wolfe, 1979)
d *
I(Wt < t)I(5ie(2,3)) = I(W. < t)I(f(5i)e(2,3)).
Taking expectations, it follows that
E{l(Wi
Recalling that, 6^e(2,3) iff f(6^)e(2,3), and thus
E[I(5ie(2,3))J = E[I(f(6i)e(2,3))].
It follows that
E[I(W.< t)I(ie(2,3))] E[I(W*< t)I(f(6 ) e(2,3 ) ) ]
E[I(5ie(2,3))] E[I ( f ( 5 ) e ( 2 3 ) ) ]
Therefore, conditional on the pair being a type 2 or 3, the
exchangeability property still holds.
Thus, it follows that
P(Yj = l.Cj < c) = P(xJj < > Cj.Cj < c)
= Cj^lj ^ ^ c) = P ( Y j = 0>Cj c)
Noting that,
P(Yj = l.Cj < c) + P(Yj = O.Cj < c) = P(Cj < c)
58
and thus
2P ( yj = l.Cj < c) = P(Cj < c)
or that
P ( Y j = l.Cj < c) = V2 P (Cj < c) = P ( Y j = 1 ) P (Cj < c)
and thus we see that Yj and Cj are independent.
To prove part b), let y = (Yn +1 >Yn +2 Yn +n
1 1 1 c
and
Q = (,0n +2 ,...,Qn +n ). By Theorem 2.3.3 (Randles
11 1 c
and Wolfe, 1979, page 37), Q is uniformly distributed over
Rn where
c
Rn = {q : q is a permutation of the integers 1,2, ....n^ }.
Now, let q be any arbitrary element of Rn and let g be any
c
arbitrary nc vector of 0's and l's. Thus,
P (Y = g >0 = 5) = P(X = g ) P (Q = <{ ) (by part a)
and
P(Y = g ) P (Q = <{ ) = x
- C n !
2 c
which proves part b). Â£3
By Lemmas 3.2.1 and 3.2.2, TR and Tn are
1 11 c
nonparametrica1ly distribution-free conditional on n^ and
nc, respectively.
59
Lemma 3 2 .3 : Under HQ, the following results hold,
a) Conditional on n^ E(Tn^) = n^(n^+l)/4 and
Var(Tn^) = nj(nj+1)(2nj+l)/24
b) Conditional on n E(T ) = n (n + l)/4 and
c n c c c
Var(Tn ) = nc(nc+l)(2nc+l)/24.
c)
Conditional on n, and n .
1 c
independent.
T and T are
nl nc
Proof:
The proof of parts a) and b) follow directly from
Lemmas 3.2.1 and 3.2.2 and the fact that the Wilcoxon signed
rank statistic based on a sample of size n has a mean of
n(n+l)/4 and variance of n(n+1)(2n+l)/24.
The proof of part c) is also trivial following from the
fact Tn and Tn are based on sets of mutually independent
observations. ^
With these preliminary results out of the way, the test
statistic T can now be defined by
1 c
T
nlnc
L1 T
1 nl
+ L2Tn
nl
n l + nc
y 'v. r .
i=i 1 1
+ L2
+
ii
II
)
where Lj and L2 are finite constants.
60
Theorem 3.2.4: Under HQ ,
> E L1E + E2E(Tn >
i c i c
= (LjiijCnj + l) + L2nc(nc + 1 ) )/4
b) Var(Tn n ) = (L^(nj + 1)(2ni+l)
1 c
+ L|nc(nc+l)(2nc+l))/24
c) Tn n is symmetrically distributed about E(Tn n )
and
d) for fixed constants Lj and L2 Tn n is
nonpararaetrically distribution-free .
Proof :
The proof of parts a) and b) follow directly from
Lemmas 3.2.2 and 3.2.3. To prove part c), it is known that
the Wilcoxon signed rank statistic is symmetric about its
mean. Thus, Tn and Tn are symmetric about E(T ) and
1 c 111
E(Tn ), respectively. Since Tn and Tn are independent
c 1 c
(conditional on Nj = n^ and Nc = nc), the symmetry of Tn n
follows.
To prove part d), note that
P (T
nl nc
= k) = P(L1Tn + L9 Tn = k) =
2 n,
l P(LlT1- | L2T kc)P(L2Tn k ) ,
{kc> C
where {kc}
l P(LlTn, k-kc>p
{kc} 1 c
set of all possible values of E2Tn .
61
Now using the nonparametrica1ly distribution-free property
of Tn and Tn established in Lemmas 3.2.1 and 3.2.2, it
follows that for fixed L, and L0, L,T and L0T are also
1 Z in, Z n
c
nonparametrically distribution-free
can be obtained using the fact it is a convolution of two
Wilcoxon signed rank test statistics' null distributions.
Thus, for fixed and L2 the distribution can be tabled.
Tables in the Appendix 1 give the critical values for Tn n
with Lj = 1 and L2 = 1 for n^ = 1,2,.. .,15 and
nÂ£ = 1,2,...,10 at the .01 .025 .05 and .10 levels of
significance. The actual a-levels are also reported for the
cut-offs given. The decision rule for the test is to reject
Hq if the calculated test statistic is greater than or equal
to the critical value given in the table at the desired
level of significance. A two tailed test (i.e., for Ha:
a
Oj a2) could be performed by using the symmetrical
property of the null hypothesis distribution and the table
to determine the lower critical value for the test
statistic.
A test of Hq for larger n^ and nÂ£ can be based on the
asymptotic distribution of Tn
which will be presented in
Section 3.4
62
3.3 y^ = U2> Unknown
In the previous section, the common location parameter
was assumed to be known. Generally, this is not the case.
More often we may assume a common location parameter, but
this parameter is unknown. This section will present a
slight modification to the test statistic T_ n to be used
nl nc
in these settings. The modification will be to estimate the
common location parameter using a "smoothed" median
estimator based on the product-limit (Kaplan Meier) estimate
of the survival distribution (Kaplan and Meier, 1958). This
estimated location parameter M, replaces y in the previous
definitions. That is, define the variable to be
i = l ,2
M
M
X
1 i
The definitons of T R, y Q. T T and T
i iii n. n n , n
J J 1 c 1 c
remain unchanged. In this section, the statistic will be
denoted by TMn n to identify the fact the location
parameter was estimated with a "smoothed" median estimator
based on the product-limit estimate of the survival
distribution. This estimation does not affect the results
in Section 3.2, but Lemmas 3.2.1 and 3.2.3 c) must be
reproved, since in the proof of 3.2.1, we utilized the
independence of the Z^'s, a condition which no longer
n
c
were based on sets of
63
mutually independent observations. This is not the case in
the current context.
First, we introduce the "smoothed" median estimator and
the product-limit estimate of the survival distribution.
Let (Y(l)Y(2)-**Y(2n1+n2+n3)) represent the ordered
uncensored observations. (This ignores the fact the
original observations were bivariate pairs, and considers
only the 2n^+n2+n3 uncensored observations, i.e., 2n^
components belonging to type 1 pairs, the n2 uncensored
components of type 2 pairs and the n^ uncensored components
of type 3 pairs.) That is, X^j = if is uncensored
and X^j has rank k when ranked among the set of all
uncensored observations from either (both) components of the
pairs for i=l,2 and j=l,2,...,n. Let n(^)>
i = l 2,...,2n^+n2 + n3, be the number of censored and
/
uncensored observations which are greater than or equal to
Y(i). Thus,
n
U)
2
= I
n
l
I(X
i = lj=l
ij
Y(i)}
whe r e
I is the indicator function which takes on a value of one
when the argument is true and zero otherwise.
The product-limit estimate of the survival distribution
is defined as
64
r
s(t) =
i
j
n (n
k = l
if t < Y
(1)
ur 1)/n(k> lf Y(j) c < Y(j+n f0,r
j=l, 2 ,...,2n1+n2+n3-l
if t > Y
(2n +n2+n )
(Note, that Y^^ is the smallest uncensored observation and
^(2nj+n2+n^) t^ie larSest uncensored observation.)
The definition given here assumes no ties in the uncensored
observations. This is valid under assumptions A2 and A3.
Using the above definition, the "smoothed" median estimator
M is
M =
where
and
^ S(m ) 0.5
+ T x (m m )
S(mx) S(m2) Z 1
<
if m^ t m2
if ml = m2 ,
m1 = minCY^^: S (Y ( )) > V2 }
m2 = max{Y^. j: S(Yq^ < V2 }
A brief explanation of this estimator follows.
The product-limit estimate of the survival function,
S(t), is a right continuous step function which has jumps at
65
the uncensored observations. An intuitive estimate for the
common median is the value of Y ^ ^ such that S(Y^j) =
which often does not exist due to the nature of S(t). Thus,
the "smoothed" estimator was suggested by Miller (1981, pg .
75), which can be viewed as a linear interpolation between
m^ and If the Y ^ ^ exists, such that S(Y^)) = V2 then
m^ = m2 and M is that value of Y^.^ by definition.
Lemma 3.3.1: The statistic M is a symmetric function of the
sample observations.
Proof :
"/c "fc ^|p
Let (Y^j,Y^2)Y/2n)) represent the ordered 2n
observations where Y^^ < ^(2) ** ^(2n) This again is
ignoring the fact that the original observations consisted
of n bivariate pairs and treats the sample as if it
consisted of 2n observations (some of which are censored).
Under assumption A2, there are no ties among the uncensored
observations. Similarly, by assumptions A2 and A3, there
are no ties between an uncensored and a censored
observation, although there may be ties (of size two) among
the censored observations because type 4 pairs contribute
two components with the same value. The product-limit
estimator S(t) can be viewed as a function of the vectors
^|p ^|p ^|p
(Y(1),Y(2)* * *>Y(2n)) and (r(1)I(2) * X(2n)) where
66
(j)
1 if Y,.. is censored
(j )
0 otherwise
in the fact that
2 n
/, \ = y y i(x..> y,. )
(1) lilj-l ^
X
= 2n + 1 (rank of Y^^ in (Y(i)>^(2)> ' >Y(2n)^
In addition, S(t) can be expressed as
S(t) =
1
0
t < min Y , : I, =
1 (i) (i)
1}
t > ma x Y x : I/.N
1 (i) (i )
= 1
n
2n j
X(j)
* ^2n j + 1
V (j)
otherwise
Thus, S(t) is a symmetric function with respect to the
sample observations and therefore M, being a function of
S(t), is also. Cl
Lemma 3.3
. 2 : Conditional on n^ Tn has the same null
distribution as the Wilcoxon signed rank statistic
Proof :
Let V = {Tj, 'P 2 where = and
R = R, R R I with R. =
1 1 2 n ^ i
Let V be any arbitrary element of
P = i V is a 1 x n, vecto
1 -o -o 1
absolute rank of Z ^
67
(of which there are 2 different elements), and let r be
any arbitrary element of
R = {r : r is a permutation of the integers l,2,...,n^}.
Now, under the null hypothesis,
(X
liA2i
,C.) 2 (x
f
2 i
iCi>
t f
and thus letting = min(X^,C^) and X2^ = mi n ( X2 ^ C ^ ) it
follows that
(X2i>Xll)
for i=1,2,.
Now, let k
n^ and these pairs are also exchangeable.
}. be an operator such that
(X
li
X2 i )
if
k = 1
(X2f Xll>
if
O
II
Thus, under Hq and using the exchangeability property, it
f o 1lows
l(Xll- X21> (X12 X22>>-"'>
d k1 k2 k
= {(xlr x2r ) (xlr x2r ) ,...,(xlr x2r ) ni} .
1 1
(3.3.1)
Recalling that M = the estimate of the location parameter,
is a symmetric function of the components of the observation
68
pairs from Lemma 3.3.1 and defining a function
fl(yly2) = lyl Ml ly2 Ml = Z
it follows from applying this function to (3.3.1) that
{ Z, Z , Z | Z y Z y . y Z
1 1 2 n J 1 r r r 1
1 12 n^
k k k
= {(zr ) \ (zr ) ,..., (zr ) ni}
(3.3.2)
where (Z ) =
r .
i
if k = 1
-Z if k = 0
r .
Now defining a function 2(Z) = (Y R) where 'f and R are
1 x n^ vectors such that
and
V
II
r-)
1
if Z.
J
V
o
0
k
if Z .
J
< 0
= absolute rank of Z^ ,
i e ,
rank of j Zj | among
{ | Z i | y
1 Z 2 1
> j j
Zn,l>
for j=1,2,...,n,. Applying this function to (3.3.2) it
follows that
(^ 1 ^ o > f j R > R )
12 n ^ 1 2 n ^
= (T V V R R . R )
r, r r r r r
12 n^ 1 2 n^
H kl k? k
= {(Yr ) (V r ) >...,('Fr ) "i, R r R r ,...,Rr }
69
where (f ) 1 =
i
if k.= 1
i
1
r. i
i
Now since k and r were arbitrary vectors, it follows that
P(Y = T*, R = R*) = P(T = X R = r) =
1 1
x
1
1
Thus noting this produces the same null distribution for the
Wilcoxon signed rank statistic, the proof is complete. Q
Lemma 3.3.3: Conditional on n^ and n Tn and Tn are
1 c
independent.
Proof:
This proof is done in a series of steps which are
stated as Claim 1 to Claim 7 in an attempt to avoid
confusion.
Let y Â£ be defined as before and let (x^,c^) denote the
. v
observed value of the iz type 2 or 3 pair
i=n^+l,...,nj+n Note, one component was censored, and
thus its observed value was c^ while the other component was
uncensored and its value is denoted by x^. This is not
specifying which component (x^ or X2i^ was censore<3.
70
Claim 1: y^ is independent of (x^,c^).
This follows by noting that under Hq and using the
exchangeability property of type 2 and 3 pairs (as was
shown in Lemma 3.2.2) that
P{yi = l | (xi,ci)} = P{X1i=xi,X2i = ci| (x ct) }
= P{ X1 i = ci X21 = xi | (xi c ) } = P{ Y^O | (xj. c )}
Since P{ Yi = l | ci ) } + P { y = 0 | ( x c ) } = 1, Claim 1
follows. Now define y = (Yn + i Yn +2>**>Yn +n
1 1 1 c
Claim 2: y is a vector of n i.i.d. Bernoulli random
variables which are independent of
^xnL + l cn1 + l ^ ^xnL + 2 cn1 + 2 ^ * (xn1 + nc cni+nc^
This follows from Claim 1 and the fact that
{ (xnj + l cn1 + l ^ ^xn:+2 cni+2 ^ * ('Xn1+nc Cn1 + n(;
)}
are i.i.d.
Claim 3: y is independent of
* ^xnj + l cn1 + l ) (xni+2 cni+2 ^ * (xn1 + n(, Cn1+n(.)
x and x where
~nl ~n4
~nl {Ull,X12)(xl2x22)>---.
and
-n< = {(cn,+n +lcni+n +1) * (cncn^}
4 I C 1C
(i.e., the observed totally uncensored type 1 pairs and
71
the observed totally censored type 4 pairs,
respectively ) .
This follows from Claim 2 and the fact y is a function of
the type 2 and 3 pairs only.
Claim 4: y is independent of xn
in,
5
where x
( nc )
1 ~ 4 "c' "*c
denotes the observed ordered uncensored
members of type 2 and 3 pairs and c/n \ denotes
~ n c '
the observed ordered censored members of type 2 and 3
pairs .
Note, this claim follows directly from Claim 3 and the fact
that X/n \ and c/n \ are functions of
c' ~ c '
^^n^l cni + l ) > (xn1+2 cn1+2 ^ (xn1+nÂ£ > cn1+nc) ^ only *
Claim 5:
yc is independent of x xQ^, x^n ^
Y
and c
( nc )
where y = {y ,y
~c c(l) c(2)
element of c^n ^ and yc
th
(i)
to the pair of which c
(i)
C(n )) c(i) ls the r
is the y which corresponds
was a member.
This claim follows from Theorem 1.3.5 of Randles and Wolfe
(1979) and since yc is a fixed permutation of y. Note that
the i.i.d. property still holds for the y 's.
(i)
72
Claim 6: Given x x X/ \
-nj ~ n4 ~(nc)
a random variable; that is
observed .
and S(nc)
Tn is no longer
the value of T.
i s
This follows directly from the definition of Tn .
Claim 7: Note, that
n + n n, + n
1 v c 1 v c d +
T I Y, Qj I J Yc W
c j=n1+l J J j=n1+l (j)
which shows that T is a function of y and is
nc 1 c
independent of x x x, \ and C/ \.
~nl -n4 ~(n ^) ~Cnc)
Thus, Tn has a null distribution equivalent to the Wilcoxon
c
signed rank null distribution and is independent of T
nl
which is a function of xn > xn > x(n ) and cjn \ only. Q
'*1 ~ 4 ~v c ^ ~ v c'
With the proof of Lemma 3.3.3, Theorem 3.2.4 is valid
for
the modified test
statistic TM .
nl ,nc
That
is, under HQ
and
conditional on n^
and n TM has
the
same
c nl > nc
distributional properties stated in Theorem 3.2.4 for T
nl nc
and the tables in the appendix are valid.
73
3.4 Asymptotic Properties
In this section, the asymptotic distribution of the
test statistic T
_ (and TM
n ^ nr ni ,n,
) under H will be
-c Jl*uc
established. The asymptotic normality of the test statistic
will be presented first, conditional on = n^ and Nc = nc
both tending to infinity and second, conditional on n
tending to infinity. In the second case, this is the
unconditional asymptotic distribution since it only requires
that the sample size go to infinity. Note that, under
assumption A.5 (A.5 stated that the probability of a type 4
pair is less than one), as n -v , N^ + Nc = (n number of
type 4 pairs) - also. The asymptotics will be presented
for the test Tn n only. In the previous section, it was
shown that under H and conditional on N, = n, and N = n
o lice
T and TM have the same null distribution; that is
nlnc nlnc
T = TM
nlnc
n n Therefore, they have the same cumulative
distribution function and thus their asymptotic
distributions are the same. There is no need to prove them
separately.
Theorem 3.4.1: Conditional on N, = n, and N = n under H
11 c c o
nlnc
- E ( T
niV
(Tn n )
nlnc
N(0,1) as n^ + and nc +
where
74
E ( T
nlnc) = (Llnl(nl+1) + L2nc(nc+1))/4
and
a(Tn n ) = [Va r(T
nlnc nln,
)}lf2
= [(L^n1(n1 + l)(2n1 + l) + L2nc ( nc + l ) ( 2 nc+l ) ) / 2 4 ]l/2
Proof:
First, it will be shown that T and T have asymptotic
nl nc
normal distributions. Without loss of generality, it will
be assumed that p = 0.
Note that
y ?. r =
. i i
i = l
I n|x2i| |xldL I) + l l V( IX211 |x1| + |x2i
i = l
where
and
2 n
1 < i < j <; n
2jI lXlj
1
n,
Dini' ~Vi-iT(|X21^'|Xi1^
! ST J. .1 T(lX2i
(,)
l', 1 < i < j < n
X1i I + lX2j I lXlj I }
1
are two U-statistics (Randles and Wolfe, 1979, page 83). It
f o 1lows
, a/2
(T
(/)
3/2
(n )
V"l+ 1)/4) E(V>)
(2) 1
+ (
"l)1/2(U2,E(U2,n>)
75
Now notice, 0 < U. <1 and under H, E(Ui ) =
i n i 1 >n ^
P { ( I X2 i I |xlih > > = 14* so that lUl,ni l,l\ < \
Theref ore ,
(n, )
3/2
(n )
3/2
(U V2) < 7 r-r *
n j v 1 n 1 z; n (n 1 )
0 as n .
(/)
(n,)V2
Thus, (T n (n.+ 1 ) / 4) and (n.)^n V2) have
(2) 1 1
the same limiting distribution as n - .
By Theorem 3.3.13 of Randles and Wolfe (1979), it is
seen that (n )^n l^j has a limiting normal distribution
1 n
o
with mean 0 and variance r E, ^ (provided Â£ ^ > 0)
where
r2?1 = 2 2{E[Â¥( |x2i
Xlil + IX2 j I lXljP
x ?(|X2| | x i i | + | X2 k I lXlklP ^4 >
= 1/3
Thus ,
T n.(n.+ 1)/4
nl 1 1 d
U)
+ N(0,1) .
Note that,
i ,v2
U1) W
1
n (n + 1)(2n + 1 ) ,
_ 1 j fz
P
- 1
24
76
as n1 > oo Therefore (after applying Slutsky's Theorem
(Theorem 3.2.8, Randles and Wolfe, 1979)
T n (n + 1 ) /4
1 1 1 d_
a (T )
nl
N ( 0,1 ) .
Similarly,
n, +n
1 c
n. +n
1 c
l Yj +
V
j-n,+l n ^ +1 < j
T = y Y.Q.
n J J
c J =n ^ +1
l { Y j f ( c j ck) + Yk(ck
= j)}
nc(D3,n > + (2C)(U4, >
where
and
U4,n
1
n, +n
1 c
3 ri = YJ
c J =n^ +1
1
l l Y i ^ (c - c, )
fc\ n.+1
l 9 J 1 1 c
k) + Yk'i'(ck Cj)}
are two U-statistics. It follows that
"2
E_ rT ^ j. \ ^ c
n ^ n
(2C)
c(c+ DM) (o3 ECO ))
(,C)
+ )
77
Note
that 0 < n < 1 so | n V2I < V2 and thus
* r* 9 C
, ,3/2 ,3/2
(n ) (n )
c t 1 / >, c
(u, V2) < C'
n ^ 3 n v n (n
+ 0 as n *
w)
c c
1 ) c
u> 1,
Thus, (T^ n^(n^+ 1 ) / 4) and (n^'^U^ ^ V2) have
n v n c c
Uc)
the same limiting distribution as n^> <*> .
Again applying Theorem 3.3.13 of Randles and Wolfe
(1979), it is seen that (n ) U. Vo] has a limiting
c v 4 n "
c 0
normal distribution with mean 0 and variance r Â£ ^
2
(provided r (j>0 ), where
r2^ = 22{E[(YjVCCj Ck) + ykf(ck c.))
X (Yj'KC. Ci) + YiT(Ci Cj ) ) ] V4 }
= 22{E(Yj'i'(Cj-Ck) + YkKCk-C. )](Y.T(C.-C1) + Y^(C.-C. ) ) V4} .
By the independence of
Yj and Cj (Lemma
r2^ = 22{P(Yj =
+ p(Yj- i)p(y1=
+ P(Yk= 1 )P(Yj =
1 ) P(Cj > ck,
1)P(C > C
J <
l)P(Ck> c ,
3.2.2),
C.> C. )
J 1
c.> c.)
1 J
c.> c.)
J 1
78
+ P(yk= 1)P(Y.= l)P(Ck> Cj C.> C.) -I/4}
= 4 {V2 P ( C > C, C > C ) + Va P ( C > C. C > C )
z J k j 1 H j k 1 j
X p(ck> c., C> C.) +V4P(Ck> c., c.> C.) -1/4}
. A/I.If 1*1*11 -L .1
Thus ,
T n (n + 1 )/4
n c c ,
c d
+ N(0,1) Noting that
n t 1 }/2
(,c)
n (n + 1 ) ( 2n + 1 ) i7
c c C nVo
+ 1 as n *
c
24
and applying Slutsky's Theorem, it follows that
T -n(n+l)/4
n c c ,
-* N(0,1 ) .
o(T )
n
c
The conclusion of Theorem 3.4.1 then follows by writing
T E(T ) (L.T + L T ) (L E(T ) + LE(T ))
n, ,n n, ,n In, 2 n In, 2 n
l cl c 1 c 1 c
o(T )
nlnc
(L2 a2 ( T ) + I2 a2 ( T )),2
1 n, 2 n
1 c
L. o(T ) T E(T )
1 n ^ n n ^
La(T ) T E(T )
2 n n n
c c c
a(T n )
nlnc
o ( T )
nl
a(Tn )
nlnc
a ( T )
n
c
x
X
79
applying Slutsky's Theorem and utilizing the fact T and
Tn are independent, conditional on = n^ and NÂ£ ~c
= n
Next, and most importantly, the unconditional
asymptotic normality of T will be established as n
n 1 n c
tends to infinity in Theorem 3.4.4. Prior to proving this,
several preliminary results will be stated which are
necessary. These preliminary results which are stated in
Lemmas 3.4.2 and 3.4.3, were proved by Popovich (1983) and
thus will be stated without proof. Minor notational changes
are made in the restatement of his results to accommodate
the notation in this dissertation.
The first preliminary result, Lemma 3.4.2, is a
generalization of Theorem 1 of Anscombe (1952).
Lemma 3.4.2: Let {T } for n,=l,2,..., n =1,2,..., be
n 1 >n c 1
any array of random variables satisfying conditions (i) and
( ii ) .
Condition (i): There exists a real number y, an array
of positive numbers {cun n } and a distribution function
F ( ) such that
lim P { T y < x a) } = F (x)
min(n^ ,n )-* 1 c 1 c
at every continuity point of F().
Condition (ii): Given any e > 0 and n > 0, there
exists v = v(e,n) and d = d(e n) such that whenever
min(n^,nc) > v, then
80
p{ T ,-T < e m for all n ,n' such that
I n n n,,n' n,,n 1 c
1 c 1 c 1 c
In' n,| < dn., |n n | 1 n .
I 1 1 I 1 1 c c' c
Let (nr} be an increasing sequence of positive integers
tending to infinity and let {N^r} and {N c r} be random
variables taking on positive integer values such that
N. p
> X as r > for some X. such that 0
l i i
n
r
i=l,c. Then at every continuity point x of ?()
lim P{Tn
r >oo 1 r
N
cr
y < xto , r , }
[Xinr],[Xcnc]
F (x)
where [a] denotes the greatest integer less than or equal to
a .
Proof :
This is Lemma 3.3.1 in Popovich (1983).
The last preliminary result necessary is a result of
Sproule (1974) which is also stated in Popovich (1983) as
Lemma 3.3.3. It can be viewed as the extension of the well
known one sample U-statistic Theorem (Hoeffding, 1948) but
with the sample size as a random variable.
81
Lemma 3.4.3: Suppose that
U
n
where B is the set of all subsets of r integers chosen
without replacement from the set of integers {l,2,...,n} and
f(t ^ t2>..., t ) is some function symmetric in its r
arguments. This Un is a U-statistic of degree r with a
symmetric kernel f(). Let {nr} be an increasing sequence
of positive integers tending to infinity as r > and (Nr)
be a sequence of random variables taking on positive integer
2
values with probability one. If E{f(X^, X2.,X )} <
1/ o Nr P
lim VarCn^ U ) = r ? > 0, and 1 then
n 1 n
n+ r
r \ r2^)^ } = Mx) ,
lim P{(UN E(Un )) < N
r-)- r r
where $ ( ) represents the c.d.f. of a standard normal random
variable.
Proof: This is Lemma 3.3.3 in Popovich (1983). ^
One comment is needed about this result. The proof of this
lemma follows as a result of verifying that conditions
and C2 of Anscombe (1952) are valid and applying Theorem 1
of Anscombe (1952). Condition is valid under the null
hypothesis and the verification of condition C2 is contained
in the proof of Theorem 6 by Sproule (1974). This condition
82
C 2 will be utilized in the proof of the major theorem of
this section which follows.
Theorem 3.4.4: Under Hq ,
\ ,Nc E(TN, ,N
a(TN,,N }
1 c
N(0,1) as n .
Proof:
The proof which follows is very similar to the proof of
Theorem 3.3.4 in Popovich (1983).
Let T
nlnc
T E (T )
nl ,nc nl ,nc
o(T )
nlnc
, the standardized
T statistic. Theorem 3.4.1 shows that {T } for
n1nc nlnc
n^=l,2,..., n =1,2,..., satisfies condition (i) of Lemma
3.4.2 with y = 0 amd
A5, it can be seen that X^ > 0 for at least one i=l,c. If
X^ = 0, for i=l or i=c, then Theorem 3.4.4 follows directly
from Theorem 1 of Anscombe (1952) and Lemma 3.4.3. Thus, it
will be assumed that X ^ > 0 for i=l,c. The proof of
Theorem 3.4.4 follows if it can be shown that condition (ii)
of Theorem 3.4.2 is satisfied.
Let T
: E (T )
nl ni
a (T )
nl
the standardized T
n,
statistic
In the proof of Theorem 3.4.1, it was shown that
83
T has a limiting standard normal distribution by utilizing
nl
the U-statistic representation of T
1
As a result of Lemma
3.4.3 and this U-statistic representation, it follows that
Â£
T satisfies condition C., of Anscombe ( 1 952) (since T is
n 1 L nl
equivalent to a U-statistic which satisfies condition C2 of
Anscombe (1952) as proved by Sproule (1974)). This
condition C2 can be stated as follows.
Condition C2: for a given e^> 0 and n > 0, there
exists Vj and d^> 0 such that for any n^>
* *
T T ,
n 1 nl
< e for all nj such that |n|
- n,
< d1n1 } >
1 n
(3.4.1)
Similarly, as a result of the U-statistic representation of
Tn (as shown in the proof of Theorem 3.4.1) and from Lemma
c
3.4.3, it follows that T
T E (T )
n n
c a (T )
n
c
satisfies
condition C2 of Anscombe (1952). That is, for a
given z0 and n > 0, there exists V2 and d2> 0 such that
for any n^>
P{ T T < e for all n such that n
in n I 2 c I c
c c
n < dn } >
cl 2 c
1 n
(3.4.2)
84
Consider
T E(T
nlnc
nlnc
nlnc
a(Tn n }
nlnc
L.o(T )
1 nl
a ( T )
nlnc
* V(Tn >
) + *
1
a (T )
nlnc
f >
K )
Note that,
(1) L^n and L2n ate functions only of N^ and Nc and the
given and constants.
(2) (Lln)2 + (L2n)2 = 1
I ! P I
(3) There exists constants and 1^ such that * L^
. P .
and L * L as n + .
7 n 7
First, it will be shown that condition (ii) is
f Â£ f -Jc
satisfied for L.T 1 + L_T ) = T
1 ^ n, ; 2 ^ n -1 n , n
1 c 1 c
Let e > 0 and n > 0 be given and let v^, v2, ^2
satisify (3.4.1) and (3.4.2). Let v = raax(vp v2) and
d = min(d^, d2). Now,
P { T , T < 2 e for all n1, n such that
In' n' n, ,n I 1 c
1 c 1 c
< dn. ,i=l,c)
i 1 i
1 *
* i
' i *
* 1 "V
T
- T 1
+ L J T ,
- t h
1 n'
n, 1
2 1 n '
n 1 1
< 2 e for all n', n'
1 c
35
such
that | n
i
n.
i 1
< dn
i 1 =
1 ,c}
*
V -
1
* i
T i
nl
< t and
t .
l2I
*
V
c
* i
" T j
n i
c
< Â£
for
all
S
such
that | n
t _
i
n .
i i
< dn
i 1 =
1 c}
*
* 1
T Â¡
nl
< e for
all
ni
such
that
lni
' nl
<
*
' n "
c
* i
T
n i
c
< e for
all
n
c
such
that
In
i c
- n
c
| <
i *
lTn:
*
- T
n,
Â¡ < e or
I
l2I
*
T ,
n
*
- T
n
1 ^ E
for
all
S
c c
such that n' n, < dn, and n' n < dn }
11 II 1 I c cl c
f Â£ Â£ i i i
> P{L. T T < e for all n' such that n' n. <
II n j n^l 1 II II
1 I * I I I
+ P{L_ T T < e for all n' such that n' n <
2 I n 1 n
c c
c c
(3
Now using inequalities (3.4.1) and (3.4.2) and applying
to (3.4.3) with e = min{e^(L^) ^ ^ } then
P T . T < 2e for all n,, n such that
1 I n, ,n' n,,nl 1 c
1 c 1 c
ni ~ nil < dni> 1=1>c} > (1 n) + (1 n) -1 = 1 2n
n*
c
dn l }
dn }
c
n'
c
dn l }
dn } 1 .
c
.4.3)
them
86
Therefore T satisifies condition (ii) of Lemma 3.4.2 so
nl *nc
that Theorem 3.4.4 is valid for T_ = L,T + L0T.
nl nc
In. 2 n '
i c
To see that the Theorem is valid if and L2 are replaced
t I
^ and respectively, consider,
nl >nc
Tninc Lln Tnl
c
' *
- LlTn
' *
2 n n,
+
' *
L2Tn,
= (L
1 n
- +
(3.4.4)
Now, since Tn and TQ converge in distribution to standard
normal random variables, Tn and Tn are 0p(l) (Serfling,
1 n c
I ^ T
( 1980 ), pg. 8). Also, since L^n > L ^ and
t P 11 11
L 2 n + L2 as n > , (Lln L ^) and (L2n L2) are op(l).
Therefore (3.4.4) shows that
(L^n L^)T* + (^2n ^2^n p ( 1) and thus, Theorem
3.4.4 is valid.
3.5 Comme n t s
From the results in Sections 3.2, 3.3,and 3.4, it is
clear that a distribution-free test of the null hypothesis
of bivariate symmetry versus the alternatives presented
could be based on T (or TM ). For small samples,
nlnc nl>nc
an exact test utilizing the distribution of T (and
n 1 ,nc
TM
) conditional on N, = n,
,nQ' 1 1
and N_ = n could be
87
performed. For larger samples, the asymptotic normality of
T (and TM_ ) could be used. In Chapter Five, a
n1nc nlnc
Monte Carlo study will be presented which compares the CD
test with the two tests presented in this chapter. For
each, the asymptotic distribution will be used for samples
of size 25 and 40 to investigate how the statistics compare
under the null and alternative hypotheses for various
distributions. First though, we make some comments on this
chapter.
Comment 1
In Section 3.2, the test statistic Tn n conditional
on nÂ£, was presented which had a null distribution
equivalent to the Wilcoxon signed rank statistic. If
instead of conditioning on nc, the statistic had been
presented (with some minor adjustments) conditional on n2
and n^, the statistic would then have had a null
distribution equivalent to the Wilcoxon rank sum
statistic. Conditioning on nÂ£ and not on n2 and n^ was
chosen because the observation of a particular n2 and n^ in
itself, seemed important. That is, if only type 3 pairs had
occurred (ignoring the number of type 4 pairs) that was
significant, since under the null hypothesis, the
probability a bivariate pair is type 3 is equal to the
probabiltiy the pair is type 2. The signed rank statistic
incorporates this idea and thus was used.
88
Comment 2
In Section 3.3, the Kaplan-Meier estimate of the
survival distribution was used in estimating the common
location parameter. The usual median estimator (the sample
median) could not be used, because in the presence of right
censoring this estimator is negatively biased. Thus, the
"smoothed" estimator based on the Kaplan-Meier estimate of
the survival distribution was the logical choice.
Comment 3
The tests presented in this chapter are not recommended
for situations in which heavy censoring occurs early on,
that is, a lot of censoring in the smaller measurements. If
this heavy censoring was to occur, many type 4 pairs would
be present in the sample which are not used in the
calculation of the test statistic other than to estimate the
common location parameter. This test was more designed for
situations when the extreme values (i.e., the larger values)
tended to get censored.
Comment 4
In this chapter, statistics were presented to test for
differences in scale when (1) the common location parameter
89
was known or (2) the common location parameter was
unknown. The next natural extension would be to test the
null hypothesis of bivariate symmetry versus the alternative
that differences in scale existed with unknown location
parameters which could be potentially different. This idea
could be incorporated into the test statistic by using
separate "smoothed" estimators for and This idea
will be further investigated in Chapter Four.
CHAPTER FOUR
A TEST FOR BIVARIATE SYMMETRY VERSUS
LOCATION/SCALE ALTERNATIVES
4.1 Introduction
In Chapters Two and Three test statistics were
presented to test the null hypothesis of bivariate symmetry
versus the alternative hypothesis that the marginal
distributions differed in their scale parameter. This
chapter will consider a test for the more general
alternative, that is, that the marginal distributions differ
in location and/or scale. To do this, two statistics will
be made the components of a 2-vector, W of test
statistics. The first statistic denoted TE_ is a
n 1 nc
statistic which is used to detect location differences. It
was introduced by Popovich (1983) and is somewhat similar to
the statistic introduced in Chapter Three. The second
component of the 2-vector will be a statistic(s) which is
designed to test for scale differences. Three different
statistics will be considered for this second component.
They are (1) TM (Chapter Three, Section 3.3), (2)
nlnc
TM but using separate location estimates for X and
n 1 y n c ^
X2^ and (3) the CD statistic (Chapter Two). It will be
shown in Sections 4.2 and 4.3 that the distribution of Wn is
90
on
not distribution free, even when H is true. Thus, if f is
o rn
the variance-covariance of W the quadratric form W'
~ n n ~nrn~n
will not be distribution-free. A consistent estimator of
A
j] n j-n t>e introduced in Section 4.5 and a test based
the asymptotic distribution-free statistic U' i W will be
recommended for large sample sizes. For small sample sizes
a permutation test will be recommended. First though, we
introduce the TE statistic by Popovich (1983) with a slight
change in notation to accommodate this thesis.
Let
Di
= Xli
- x2i
a nd
R( |
Di|)
be
the
absolute
Di
for i
= 1,2
> >
n, that
is ,
R(
1 Di 1
is
the
rank of |
among (|
Dll>
1 ^2 1
|Dn
| ).
Define
1
if
Z .
^ 0
T .
1
= Â¥(D1)
= <
1
0
if
Z .
1
O
V
Let
TE
nl
a nd
TEn
c
be defined
to
be
t he
following:
nl
TE
nl
i
II 0^1
H-*
4'1 R (
lDi
)
and
TEn = N N2 .
c
Notice that TE
is the Wilcoxon signed rank statistic
applied to the n^ totally uncensored pairs. Popovich (1983)
showed under Hq, N^ is distributed as a Binomial random
variable with parameters nc and p = V2 ?2(0) = V2 P(type 2 or
3 pair). With a slight modification from Popovich, the
n 1 n<
statistic TE
is
92
where
+ K2n
TE
TE n. (n, + 1 )/4
n ^ 1 1
1 (r^Ct^ + 1 ) ( 2n L + l)/24)1/2
and
TE
TE
"c (
(n ) z
c
and K^n and K2n are a sequence of random variables
satisfying:
1) and I<2n are only functions of and Nc,
2) there exists finite constants and such
P P
that Kjn + Kj and ^2n K2 as n .
This is slightly different from the statistic Popovich
introduced, the difference being that he required
Kjn = (l~K2n) which is not being required here.
One comment before proceeding to Section 4.2. In this
Chapter, type 4 pairs will be ignored (except in estimating
the location parameter for the scale statistics). This has
no real affect since TE_ n > TMn and CD are not
n1 nc nl c
affected by their presence (other than in estimating the
location parameter). It will be assumed that the sample is
of size n
*
* |