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 Title:
 Bivariate symmetry tests with censored data
 Creator:
 Perkins, Laura Lynn, 1957
 Copyright Date:
 1984
 Language:
 English
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 University of Florida
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 University of Florida
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 Copyright Laura Lynn Perkins. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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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
to my parents, with love
ACKNOWLEDGE MENTS
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.
TABLE OF CONTENTS
PAGE
ACKNOWLEDGEMENTS .....................
iii
vi
ABSTRACT CHAPTER
INTRODUCTION ...... ..............
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
A CLASS OF STATISTICS FOR TESTING FOR
DIFFERENCES IN SCALE ... .......... 48
3.1 Introduction .... ........... 48
3.2 VI= P2 Known .... ........... 50
3.3 PI= P2 Unknown ... .......... 62
3.4 Asymptotic Properties ......... 73
3.5 Comments ..... ............. 86
A TEST FOR BIVARIATE SYMMETRY VERSUS
LOCATION/SCALE ALTERNATIVES ...
90
4.1 Introduction ....... 4.2 The Wn Statistic Using
Tnl,nc . . . .
4.3 The Wn Statistic Using
CD ... .........
4.4 Permutation Test . .
4.5 Estimating the Covariance. .
1
ONE
TWO
THREE
FOUR
. .. 90
. . .. 93
ill
121
123
F IVE
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
APPEND ICES
1 TABLES OF CRITICAL VALUES FOR TESTING
FOR DIFFERENCES IN SCALE............158
2 THE MONTE CARLO PROGRAM............171
BIBLIOGRAPHY..........................183
BIOGRAPHICAL SKETCH.........................185
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
August, 1984
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 N1, 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
to that of a Wilcoxon signed rank statistic. Thus, any
member in the class can be used to provide an exact test which is distributionfree for the null hypothesis. The statistic based on Kendall's tau is not distributionfree 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 distributionfree.
For the second and more general alternative, a small sample permutation test is proposed based on the quadratic form Wn = I t' Tn' where T is a 2vector of statistics composed of a statistic designed to detect location differences and a statistic designed to detect scale differences and is the variancecovariance matrix for Tn. For large samples, a distributionfree approximation f or T't T is recommended.
~nr
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
INTRO DUCTIO0N
Let Wand W2denote random variables; then the property of bivariate symmetry can be defined as the property such that (W1,W 2) has the same distribution as (W 2,1). 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 dental sensitivity is again assessed. if Xiand X2iare
the first and second sensitivity measurements, respectively,
of the ith 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 distribution of (Xli, X2i) is the same as that of (X2i,Xli), 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
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 1, 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 rth (r
experiment was fixed but random entry into the experiment
was allowed. It is this latter type of censoring which this
work addresses.
Now we statistically formulate the problem of
I I
interest. Let (X liX 2i) for i=1,2,...,n denote a random
sample of bivariate pairs which are independent and
identically distributed (i.i.d.) and C. i=1,2,...,n denote a
1
random sample of censoring times which are i.i.d., such that
Ci denotes the value of the censoring variables associated
I I
with pair (Xli,X 2i). In the case of random right censoring,
the observed sample consists of (X1iX2i,6i) where
9 9
Xli = min(Xli,Ci), X2i = min(X2i,Ci) and 6i is a random
variable which indicates what type of censoring occurred,
6i Description
I I
1 XIi
1 I
2 XIiCi
1 I
3 X1i>Ci,X2i
t I
4 Xli>Ci,X2i>Ci
Now we state a set of assumptions which are referred to
later.
Assumptions:
1 I
Al. (Xli,X2i) i=1,2,...,n are i.i.d. as the
I I
bivariate random variable (X11 ,X21).
! t
A2. (X11,X21) has an absolutely continuous bivariate x1 i x2 12 distribution function F( 1, ) 2
1 02
where F(u,v) = F(v,u) for every (u,v) in R2. The
parameters pl (2) and 01 (02) are location and
scale paratmeters, respectively. They are not
necessarily the mean and standard deviation of the
marginal distributions.
A3. C1,C2,...,Cn are i.i.d. continuous random
variables, with continuous distribution function
G(c).
A4. The censoring random variable Ci, is independent
9 T
of (X1i,X2i) i=1,2,...,n and the value of Ci is
the same for both members of a given pair.
I I
A5. P(X1i>Ci,X2i>Ci) < 1.
1
A6. G(Fx ( 1/2 )) < 1 where FX denotes the marginal i i
cumulative distribution function (c.d.f.) of Xil
i=1,2.
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 Ho: Pi=2, al=a2 versus the alternatives:
1. The case where Pi=P2=P with p known,
Ha: 1 0 G2
2. The case where Pi=P2=P with p unknown,
Ha: al #02
3. Ha: I # P2 and/or 01 # 02.
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 Hailer (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 distributionfree 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 following. Under H0, the pairs (X1i,X2i) i=1,2,...,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, RN
RN = 1 R 12 : : : 1n
( R 21 R 22 R 2n)
where Rji is the rank of Xji in the pooled sample j=1,2 i=1,2,...,n. Let S(RN) be the set of all rank matrices that can be obtained from RN by permuting within the same column of RN for one or more columns. Under H0, each of the 2n elements of S(RN) is equally likely and thus, if Tn is a statistic with a probability distribution (given S(RN) and Ho) which depends only on the 2n equally likely permutations of RN, Tn is conditionally distributionfree (conditional on the given RN and thus S(RN) observed). Sen's statistic Tn can be defined as
1 n
Tn n E NR
i= 'Ri
where EN,i is a score function based on N=2n and i alone.
For the test of location differences only, Sen suggests using the Wilcoxon scores (ENi I ) or the quantile F
N~i N+1
scores (EN,i = F() where F is an appropriately chosen absolutely continuous c.d.f.). The AnsariBradley scores (ENi N 2 i N ) or the Mood Scores
i~ 2 2
(ENi N f i ) 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 (Xli,X i) i=1,2,...,n 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 (YliY2i) where Y iC Xli X2i and Y2i= X1i + X2i. The
9
resulting test they suggest when dealing with a bivariate normal distribution is to reject Ho if fBl1 > t(01;n2) or IB21 > t(2;n1) where
/2l r(Yi,2 12Y
B = and B = 2 y 1
(1r 2 (YI'Y2 )2 )i S
and r(YI,Y2) is the sample correlation coefficient of the Ylii's and Y2's, Y Iand S2 are the sample mean and unbiased sample variance, respectively of the Yli's and t(3;n) represents the critical value for a t distribution with n degrees of freedom which cuts off a 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 = 261 + 262 4a162
so relatively small values for a, and 62 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. Ile suggested
10
Dn=ff {Fn(x,y) Fn(y,x)}2dFn(x,y)
where
n
F (x,y) I (X )I (X
nn i=1 ( ,x] li (= y] 2i
is the bivariate empirical c.d.f. He notes that nDn is not distributionfree nor asymptotically distributionfree when H is true, and thus proposed a conditional test in which the conditioning process is based on the 2n data points
(j )( )n
{((x11,x21 1 ,...,(xn'x2n n k= 0 or
for k = 1,2,...,n}
which are equally likely under H Here we let (s,t)(0) = (s,t) and (s,t)(1) = (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 (YliY2i) 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
statistic, n
T Tf( YI )(Y y2)l
n (n) i Ylj i Y2j 2i
2)
where
()= fi ift>O
T~ttO
S0 if t<0
which is Kendall's Tau applied to the transformed observations. He noted that 7n is neither distributionfree nor asymptotically distributionfree in this setting and thus recommended a permutation test which is conditionally distributionfree based on 7n for small samples. This permutation test was based on conditioning on what he called the collection matrix, Cnl
n Y 21 Y 22 2n
He noted that under Ho and conditional on Cn' there are 2n equally likely transformed samples (TiIYlily2i) possible, each being determined by a different collection of T where i= {I 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 wn' which Kepner notes, is that 7n is insensitive to unequal marginal
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
Vn = Tn tn Tn
where
/ {W+ n(n+l)/4}
(nl) nl
n T /2 IT 1/2)
W+is the Wilcoxon signed rank test statistic calculated on
n
the Yli's and nn 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 distributionfree 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., X1i X 20 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 Nand N2 where N, 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
T (N IN (1L ) 2 T (N ) + L/2 *
n 2 n In 1 n T2(N2)
where Tin is the standardized Wilcoxon signed rank statistic calculated on the N1 uncensored pairs, and T2n = N2 12 (N2RN2L) where N2R is the number of pairs for which Xli is censored and X2i is not, and N2L is the number of pairs for which X2i is censored and Xli is not (note N2R+N2L= N2). The weight Ln is a function of N1 and N2 only such that OLn l and Ln + L Note that Tln is a distributionfree statistic calculated only on the uncensored pairs (and is a common statistic used for testing for location in the uncensored case) while T n is a statistic based only on the type 2 and 3 pairs (as previously defined in this introduction). The statistic T2n is designed to detect whether type 2 pairs are occurring more often (or less often) than should be under the null hypothesis. Under Hog T2n is a standardized Binomial random variable with parameters N2=n2 and p= 1/2 and thus distributionfree. Popovich obtains asymptotic normality for the statistic Tn(NIN2) under the conditions (1) that N, 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 (TWL) (1980) based on logistic scores. The results show that these statistics perform as well as TWL (better in some cases) and that they
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 7rn 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 I represents a possible contour of an absolutely continuous distribution of this form. The alternative hypothesis for which this test statistic is developed is H a : al a2; 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.
17
x 21
x 11
Figure 1. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales.
x 21
xl
Figure 2. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales.
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 (Xli,X2i) for i=1,2,...,n denote independent
identically distributed (i.i.d.) bivariate random variables which are distributed as (XI1,X21). Consider the following orthogonal transformation of the random variables (Xli,X2i); let
Yi X + Xi and Y2i Xli X' for i=1,2,...,n.
Figure 3 illustrates what happens to the contour given in Figure 1 (i.e., the contour of an absolutely continuous distribution under Ho) when this transformation is
applied. Figure 4 shows what happens to the contour given in Figure 2 (i.e., under Ha) when this transformation is applied. Note, as can be seen in Figure 3, under this
I I
transformation and Ho, YII and Y21 are not correlated
I I
although XII and X21 possibly were. Similarly, as can be seen in Figure 4, under this transformation and Ha) YII and 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 between Y,, and Y21 Kepner (1979) suggested the use of Kendall's tau to test for correlation between YII and Y21. Kendall's tau was chosen, due to the fact it is a Ustatistic and, thus, the many established results for Ustatistics 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 of XI, or X21 (or both) is not known, and thus YlI or 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 introduced.
Recall, (Xli,X2i) denotes bivariate random variables
which are distributed as (Xil,X21). Let CIC2,..., Cn denote the censoring random variables which are independent and identically distributed (i.i.d.) with continuous
x 21 x 11
Figure 3. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Equal
Marginal Scales under the Transformation.
x 21 X 11
~21
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 Ci denotes the value of
I I
the censoring variable associated with pair (X1i,X2i). In the case of random right censoring, the observed sample
consists of Xli = min(Xli,Ci) and X2i = min(X2i,Ci). These pairs can be classified into four pair types which are
Pair Type Description
I T
1 X1i
T I
2 XiCi, X 2i>Ci
!2 !
3 Xli>Ci X2i
9 9
4 X1i>Ci, X2i>Ci
Consider the following orthogonal transformation applied to the observed sample:
Yli li + 2i and Y2i = X1i 2i for i=1,2,...,n.
Notice that, due to censoring in type 2,3, or 4 pairs, the
9 T
true values of Yli and Y2i (denoted Yli and Y2i i.e., the values had no censoring occurred) are not actually observed. The following table, Table 2.1, summarizes the
9 1
relationship of the true values of Yli and Y2i to the observed values.
Table 2.1 Summarizing the Relationship Between the True
I I
Values of Yli and Y2i to the Observed Values
Description XliCi XIi>Ci, X2i(Ci Xli>Ci', X2i>Ci
Relationship Between Yli and Yli
yl =
Yli li yli > Yli y1i >li Yli ) li
Relationship Between Y21 and Y2i
~2i 2i
T
Y2i = Y2i
I
Y2i < Y2i
I
Y2i > Y2i
uncertain
o
(i.e., Y2i > Y2i
or Y21 4 Y2i)
The modified Kendall's tau (denoted CD for 9oncordantdiscordant) can now be defined as
CD = 1 a.. b. where for i
(n) i
2
1 if Y Yj
li 1j
a ij= 1 if Yi YIj
ij li lj
0 if uncertain of
the relationship
I if Y Y
2i 2j
and b ij= 1 if Y 2 Y2j
ij 2i 2j
0 if uncertain of
the relationship
(2.2.1)
Pair Type
1
2 3
4
(here Yli Yj can be read as "Y'i is definitely smaller
than Y1j").
For example, if the ith pair is a type 1 and the jth pair is a type 2 and it was observed that Ylii < Y,,, then aij = 1 since Y =i = Y i and YIj < YIj (thus YIi < YIj. If Y ii> YIj had been observed, then aij = 0, since the relationship between Yli and YIj is uncertain. Similarly, b.,=0 ic 2 2 an
if Y2i < Y2j, then bij = 0, since Y21 = Y2i and Y2j < Y2j (thus, the relationship between Y2i and Y2j is uncertain). On the other hand, if Y2i > Y2j had been observed, then bij= 1 (by a similar argument).
Table 2.2 summarizes the necessary conditons for aij
and bij to take on the values of 1, 1 or 0. The product of aij and bij results in a value of 1 if the ith and jth 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 ith pair is a type 4 (i.e., both Xli and X21 were censored) then bij will always be 0 since the relationship between the ith and jth pair is always uncertain regardless of the jth pair's type. Thus, type 4 pairs always contribute O'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).
Table 2.2 a = 1 :
Summarizing the Values of aij and bij for i
ith pair type
jth pair type
aij = 1: if Yi > YIj and one of the following occurs,
aij = 0:
bij = 1:
ith pair type
1
2 3
jth pair type
1 1
1
for all other cases if Y2i < Y2j and one of the following occurs,
ith pair type
jth pair type
bij = 1: if Y2i > Y2j and one of the following occurs,
ith pair type
1 1 3 3
th pair type
1 2 1
2
bij = 0: for all other cases
27
Next, we establish some properties of the CD statistic.
Lemma 2.2.1: Under Ho,
E(CD) = 0
and
Var(CD) = a + 4(n 2) Y
n(n 1) n(n 1)
where
a = 4P(a ij= 1,b = 1) ij i.j
and
y =2P(aij= 1,bij= 1,aij= 1,bj= 1)
S2P(a ij iij= 1,aij'= 1,b i,= 1)
+ 2P(a ij= 1,bij= 1,aij= 1,b ,= 1) ij bi J bij,
 2P(a = 1,b = 1,aij'= 1,bij'= 1)
 2P(a ij= 1,bij= 1,aij, ,bij= 1)
 4P(aij= 1,bij= 1,a j, 1,b ,= 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 Ho,
S d C
(XliX2i Xlj,X2jCi,Cj) = (X2i,X11,X2j,1l ,Ci,Cj)
28
and therefore it follows that
(X1iX2iXij,X2j,6i,6j) (X2ixliX2j,X1j'f( i),f(6 ))
(2.2.3) where
X1i = min(Xli,Ci), X2i = min(X2i,Ci),
6i indicates what type of pair (X1i,X2i) is,
and
f(6i) indicates what type of pair (X2iX li1) is. Thus, f(.) is the function defined below.
f(6i
1
3
2
4
Let Yli = X1i + X2i and Y2i = li X2i; thus from (2.2.3) (YliY2i,Y1jY2j, i 6i j (Yli,Y2iYlj'Y 2j,f(6i) jf( i) Applying the definition of aij and bij in (2.2.1) (or using Table 2.2) to the above, it follows that (aijbij) = (aij,bij)
and thus
P(aij = 1,bij = 1) = P(aij = 1,bij = 1)
and
P(aij = 1,bij = 1) = P(aij = 1,bij = 1)
(2.2.4)
1
E(CD) = 1
n)
2
E(aij bij) = (1)P(aijbij
E(aij b ) i
= 1) + (1)P(aij bij = 1)
= P(aij = 1,bij = I) + P(aij = 1,bij = 1)
 P(aij = 1,bij = 1) P(aij = 1,bij = 1) .
Applying (2.2.4) to the above, it follows that E(a ijb ij) = 0
and thus E(CD) = 0.
Note, that under Ho,
and thus
d
(YIiY2i,Y jY2j,6i 6 (Y1jY2j YIiY2i,6j 6i)
(2.2.5)
Applying the definition of aij and bij as before, it follows
(aij,bij) = (aij,bij
and also
Now,
where
(X1i,X2iX jX2j, i,6j) (XijX2jXliiX2i,6j,6i)
P(a ij= 1,bij = 1) = P(aij = 1,bij = 1) and
P(ai = 1,bij = 1) = P(aij = 1,bij = 1)
Now,
2
Var(CD)  [1 ] a (..I
(2)] i
2
= [ I ] I Cov(a. .b.ij., a.,i bi, )
(2) i
The three possible cases to consider for the covariance are
1) ifi', j#j', i
2) i=i', j=j', i
3) where exactly two of the four subscripts
i
Case 1) i#i', j#j':
In this case, Cov(aijbij, ai'j'bi'j') = 0 since the bivariate pairs are i.i.d.
Case 2) i = i', j = ':
In this case,
Cov(aijbij, aijbi) = E[(a ijbij)2
= P(aij = 1,bij = 1) + P(aij = 1,bij = 1)
+ P(aij = 1,bij = 1) + P(aij = 1,bij = 1)
= 4 P(aij = 1,bij = 1) = 4a (by part a).
Case 3) Exactly two of the four subscripts i
the same.
Now,
Cov(aijbij, aikbik) = E(aijbijaikbik)
Define the following events:
A1,1: {aij = 1, aik = 1}
A1,1: {aij = 1, aik = 1} A1i: {aij = 1, aik = 1}
AI,: {ai = 1, aik = 11
and similarly define the events B1 ,1, B1,1, B11 and B1,i. Using this notation, E(a ijbijaikbik) can be written as
E(aiJ bijaikbik) =
1 1 1 1
(I)k++m+np(A
S(1) P(A
k=0 =0 m=0 n=0 (Table 2.3 describes the events A
k ,B
1)k ,(1)
in more detail and the restrictions placed on the 6's.
Now, to simplify the probabilities in (2.2.6). Note, under H
(Xli,X2iX 1j,X2j'X1kX2k,6i,'6j,6k)
d
= (X2i, X11i,X2j,Xlj' 2k^1lk f(i), j k *6x
)
(1)m,(1)n
(2.2.6)
and B
(_I)k' ( )( m ( I n
Applying the transformations
Yi = X li + X2i and Y2i = Xi X2i it follows that
(YIi'Y2i Y Ij Y2j' Y1k'Y2kI6i,6j,6k)
= (Y1i,Y2iY1j ,Y2j Ylk,Y2k,f(6 i ),jf ( ),f(6k))
Now, applying the definitions of aij and bij in (2.2.1) (or using Table 2.2), notice that if bij = 1 (i.e., Y2i Yij, f(6i)E(1,3) and f(6 )E(1,2) which would yield bij = 1.
Using similar arguments, it follows
(aijaik' ,bijbik) d (aij ,aik,bij,b ik and thus
P(A1, B ,1) = P(A1, ,BI,1) P(A1 l ,B11) = P(A1,1 ,B11)
P(A1,1 ,BI,1) = P(A__1 ,B_,)
P(A1,1 B1,1) = P(A1,1 'B 1,1)
P(A1,1 B1,1) = P(A1,1 B1,1) P(A1I, ,B1,1) = P(A1 ,1 'BI,1 P(A1,1 ,BI1,1) = P(A1,1 ,B1,I)
P(A1,I ,B1,1) = P(A1,_ ,BI,1)
Table 2.3 Describing the Events A and B and the Restrictions on the 6's
Event
Description
A1, 1B1 ,
A1 ,1B1 ,
A1, 1B1 1 A1,1B1, A_IB I
A_, A B
A1 ,1IB1 1 A1,IB1 I
A 1Bl ,I
A1,1B1, A1,1B1,I
Al,1B1,1 A1,1B1,1 A1 ,1B1 ,1
AI,1B1,1 A
A1 , 1B1,I A1,1B1 ,1
(YliYlj Yli
(YliYlk) (Yli
(YliYI ,Y1i>Ylk) (Yli>YIj YliYIj YiY1k)
(YliYlk)
(Yli
(Yli>YIj Yli>Ylk) (Yli>YIj Yli>Y 1k)
(Yli>Ylj Yli
(YIi Yl k) (Yli>Y1j Yli>Ylk)
Restrictions on the 6's
(Y21Y2j Y2iY2k) (Y2iY2j Y2iY2k) (Y2i> 2j Y2iY2k) (Y2i>Y2j Y21Y2k)
(Y2i>Y2j Y2i>Y2k) (Y21>Y2j' Y2i>Y2k) (Y2i>Y2j Y2i>Y2k)
1
1
1
1
I 1,2
1
1
1
1
1
1
1
1
1,3
6
1,3
I
1
1,3 1,2 1,3
1
1
1
1,2 1,3 1,2
1
1
1
1,2
1
6k
1,3 1,3
1
1,3
1,2
1
1,3 1,2
1
1
1,2
I
1
1,2
1
1
m 
Similarly, under Ho
(Xli'X2iXlj'X2j'Xlk'X2k'6i'6j'6k)
d
= (XliX21,X1k,X2kXIjX2j li,6k, j)
and applying the definition of aij and bij in (2.2.1) it follows that
(aij,aik,bij,bik) d (aik,aij ,bik,bij) This yields that
P(A1,1 ,B1,1) = P(A1, 1 ,B1,1) P(A 1,l B1,1) = P(A ,1 B1 1)
P(A1,1 'B 1 ) = P(AI,1 1 ,B 1)
P(A1,1 ,B1,1) = P(A_1,1 ,B1,1)
Thus, E(aijbijaikbik) can be reduced to a sum of six terms, instead of the original sixteen; i.e.
E(a ijb a ik b ik =
1 1 1 1 k+k+m+nB
I 1 (1) P(A ,Bz )
k=O k=0 m=O n=0 (1)k,(1) (1) ,(1)
= 2P(AI,1 ,B1,1) + 2P(A11 ,B 1) + 4P(A11 ,B 1)
 2P(A1,1 ,B1,1) 2 P(A_,1 ,B_1,1) 4P(A 1,1B,) = Y
Note, the subscripts are arbitrary; thus E(aijbijaikbik) = E(aijbijakjbkj) = E(a ijbijajkbjk)
and therefore combining the results from case 1, 2 and 3, it follows that
Var(CD) = 1 n) +n2 + n2
2 { 2) 2 1 (2 ) 21) y'
(n)
2 4(n 2)
n(n 1) (n )Y "
As seen in Lemma 2.2.1, the variance of CD depends on the underlying distribution of (X11,X21) and possibly C. Therefore, CD is not distribution free under Ho. 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 Ho
(xIi,x2i,Ci) = (x2i,xIi,Ci)
and thus
(Xli,X2i 6i) 4 (x2i,X1i,f(6i))
(2.3.1)
where
p 1
X1i = min(X1,Ci), X2i = min(X2i,Ci), 6i is the pair type (i.e. 6i = 1,2,3 or 4) and f(6i) is a function such that
1
3
2
4
Let k = 0 be an
i1 a
operator such that
(X1i,X2i,6i)
if k = 1
(Xii,x2 i)k6) =
(X2i' ,Xli,f(6 i)) if k = 0
and K = {k: k is a 1 x n vector of O's and 1's} (of which there are 2n different elements). Thus, applying this operator to (2.3.1), we see under Ho, P{(Xli,X2i,6i) = (Xli,X2i,6~i)0} = P{(Xli,X2i,'6i) = (XliX2,16i) 1}. Applying this idea to the entire sample (in which the observations are i.i.d), under Ho, it follows that
{(X11 ,X21,6i)k1,(X12,X22 62)k2,...,(X n, X 2n 6n)kn}
II I
d k k k
S{(X11,X21,61) kl,(X12,X22,62) 2,...,(X1nX2n 6n) n}
(2.3.2)
where k and k' are arbitrary elements of K. Therefore, under Ho, given
{(Xl,x21,l1),(x12,x22,62),...,(xln,x2n,6n)}, the 2n possible vectors
{(x11,x21,61)kl ,(x12,x22 l2)k2 **,(x1nx2n,6n)kn} are equally likely values for
{(X11,X21,6I),(XI2,X 22 62" ..., (X1nX2nn)}
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
{(x11 ,x21,61)kl,(x12,x2262)k2,...,(x1n,x2n,6n)kn}
Note, since the sample observed is censored, the 2n vectors {xll,x21 ,1)kl (x12 ,x22 2 )k2, ..(x nx2n 6n)kn} are not necessarily unique. If a pair is a type 4 (i.e., both Xlj and X2j were censored), then (xljx2j, j)1 = (xljx2j,6 )0. In fact, there are only 2(nn4) unique vectors (n4 = number of type 4 pairs), since P(XIli = X2i) =
0 if (XliX2i) is not a type 4 pair under assumption A2. As a result, the permutation test, in effect, discards the type
4 pairs (since a1j bij = 0 if the ith or jth pair is a type 4) and treats the sample as if it were of size nn4 with no type 4 pairs occurring.
With regards to the transformed variables (Yli 2i i = 1,2,...,n, the permutation test can be viewed in the following way. Consider the transformations Yli = Xli + X2i
and Y2i = X1i X21. Applying these to (2.3.1) and (2.3.2), we see that under H
d
(YliY2i,6i) 4 (Yli,Y2i,f(6i)) and similarly,
kIk kn
{(YIIY21,6i) ,(Y12,Y22,62) ,2..*,(YlnY2n,6n) n
I I I
k k k
d 1 2 n
={(Y11,Y21,61) (YI2' Y22,162) '"'(Y ln Y2n'6n) }
where
(YliY2il6i) if k = 1
k
(YliY2i,6i) =
(Yli,Y2if(6i)) if k = 0
and k and k' are arbitrary elements of K. That is, under Ho, given {(ylly21,61),(y12,Y22 62), ...,(YlnY2n,6n)}, the 2n possible vectors
k k k
1 2n
{(yllY21,161) ,(Y121Y22,62) (YlnY2n,6n) } are
equally likely values for
{(Yll,Y21,61),(Yl2'Y22,62), ""(YInY2n,6n)}
To perform the permutation test, the measurements (xli,x2i,6i) i = 1,2,...,n are observed and the corresponding value of CD is calculated. Under Ho, there are 2n equally likely transformed vectors for {(Y11,Y21'6),(Y12,Y22,62), ..**. (YInY2n,6n)}. The CD
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 Ho, 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 H0,
CD d
+ N(0,1) as n +
[Var(CD)]I2
where
2 4(n2)
Var(CD) = n(nl) c +n(n1) Y
Proof:
Note that CD is a Ustatistic with symmetric kernel h(XiX ) = (aijbij). Thus, by applying Theorem 3.3.13 of
Randles and Wolfe (1979), it follows that
d1 CD d+ N(0, ) as n +
n
where
S= E[h(X i Xj)h(Xi,Xk)] = E[a ijbij aikbik] = y
Note that
2 4(n2)
n(n1) a + n(n1)
n(nl)
+ 1 as n + ,
therefore after applying Slutsky's Theorem (Theorem 3.2.8, Randles and Wolfe, 1979)
CD d
+ N(0,1) as n + c.
[Var(CD)] 2
Corollary 2.4.2: If Var(CD) is any consistent estimator of Var(CD), then
CD d
+ N(0,1) as n + m. [Var(CD)] '2
Proof:
This follows directly from Theorem 2.4.1 and Slutsky's Theorem. O
Next, we consider the problem of finding a consistent
estimator for Var(CD). There are many consistent estimators
for a variance, but three which worked well in the Monte Carlo study are described in the following lemma. Lemma 2.4.3: Under Ho, the following are consistent estimators of Var(CD):
4 1
1) Varl(CD) = 4 { A B Ai},
S njk ijk
3 () 14i
where
AijkBijk = (a ibijaikbik + aikbikajkbjk + aijbijajkbjk),
2) Var2(CD)
4 2 2 2
 2 2[ I X I AijkBijk + {. (aij bij) ] (CD) },
n n(n1)
and
3)Var3(CD)
2 {[ 1 (a b )2] (CD)2}
n(n) (n2) 14i
+4(n2)n
+ 4(n2) Var2(CD)}
n(n1) 4
Proof:
First, it will be shown that nVarl(CD) + 4y.
Now,
1
nVar 1(CD) = 4{ [ A ijkB ijk}
3(3) 1~i
A kB
= 4{ I I
n
(3) 1i
* *
which shows that nVar1(CD) = 4U where U is a Ustatistic
n n
of degree 3 with symmetric kernel h = AijkBijk/3. Thus, it follows that nVar1(CD) + 4y since U + y by Hoeffding's
n
Theorem (Hoeffding, 1961).
Next, it will be shown that
n(Var2(CD) Varl(CD)) + 0 as n + .
First though, notice that Var2(CD) is equivalent to
(n2) 4 2 2 2
Varl(CD) + 2{[ (aij..b..) ] (CD) }
(n1) n n(n1)21
Thus,
n(Var2(CD) Varl(CD)) = n{(n2) 1} Var(CD) (ni)
+ 4{ 2n(n) 2 1 i
n(n1) 14i
1 (aiji)2
4{(n2) 1} U + 4{[ 1 (a b ] (CD) }
(n1) n (n1)(2) l
+ 0 as n + .
Therefore, Var2(CD) is a consistent estimator for Var(CD).
Lastly, it will be shown that
n(Var3(CD) Var2(CD)) ~+ 0 as n + .
Now,
n(Var3(CD) Var2(CD))
2 [ 1 ( i b )2 2
2 1 (a b )2] (CD)2
(n1) (2) 14i
+ {(n2) 1} nVar2(CD) + 0 as n + .
(ni)
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 Ustatistic which estimates y.
Sn 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 Ustatistic theory that y is the variance of a conditional expectation (Randles and Wolfe, 1979, p. 79) (i.e., y = Var[(albl)*] where (albl)* = E[a12b121(Y1IY21)]). Thus, in Var2(CD), for each (YliY2i), the conditional expectation is estimated using all the other (Y1jY2j)'s, j*i and then the variance of all these quantities is calculated. That is,
1 n 2
Var2(CD) {(a.b.) CD}
i=1
where
* 1
(a. bi) E[a .b I (y ,y21 )].
n i i j ij li 2i "
n1 j i
In contrast to Varl(CD) and Var2(CD), Var3(CD) is estimating the exact variance of CD (2.2.2) derived in
Section 2.2. It is using an estimator of y from Var2(CD) and estimating a with a difference of two Ustatistics which is estimating
)22
E(a ij b2 j [E(a ijb j)]2
Again, although under Ho, E(a ijbij) = 0, the sample estimate for E(aijb ij) (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 Comments
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 that CD be used as an approximation for
[Var(CD)]112
CD
Thus, for an a level test using the
[Va r(CD)]I/2
asymptotic distribution, the null hypothesis would be rejected if I CD Z where Za/2 is the value in
[Var(CD)] '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 I
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 n1+n2+n3. This is understandable since X = X2i = Ci and thus they supply no information about the scale of XI1 relative to X21.
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 = (ijb ij) 2 ), it
n(n) lbi
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=nl+n2+n3), the value for a and y would be larger than the value had type 4 pairs been included (since ty.pe 4 pairs only contribute O's and never I's or I'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 alevel
t e s t .
Comment 3
It is unclear how the CD statistic would be affected if the marginal distributions of Xii 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 ii and X 2i) or even if this is true, that CD does not perform well inthese 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 nonparametrically distributionfree (conditional on N, = n, and Nc = n2+n3) 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.
F( 1 2 ) where F(u,v) = F(v,u) for every (u,v)
01 2
in R2. Tests are developed for both of the following alternatives to the null hypothesis of bivariate symmetry:
Case 1. Pil = 2 known,
Ho: aI = G2 and Ha: (I < a2
That is, the marginal distributions have the same known location parameter but, under Ha' X21 has a larger scale parameter than XII. A possible contour of an absolutely continuous distribution of this form was given in Figure 2.
Case 2. PI = P2 unknown,
Ho: GI = a2 and Ha: aI < G2
Here, the marginal distributions have the same unknown location parameter but, under H a' X21 has a larger scale parameter than XI1.
(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 onesided 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 I and Case 2, respectively, will be presented which are nonparametrically distributionfree conditional on NI = n1 an N= n2+3
and Nc 2+n3 In both cases, the test statistics can be viewed as a linear combination of two independent test statistics Tnl and Tnc' where Tnl is a statistic based only on the nI uncensored observations, while Tnc will be a statistic based on the nc = n2+n3 type 2 and 3 censored observations. The conditioning of the random variables NI
and Nc on n1 and n2+n3 (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 n1, n2 and n3. Thus, the test statistics will be written as Tnl,nc and TMnl,n (for Section 3.2 and 3.3, respectively) which imply conditioning on N1 = n1 and Nc = nc = n2+n3. Section
3.5 will consider the asymptotic distribution of each test statistic.
3.2 1 = P2' Known
This section will begin by introducing the notation necessary for the statistic Tn, nc designed for the alternative in case 1. Recall, the sample consists of (X1i,X2i) i=1,2,...,n where Xli = min(Xli,Ci) and X2i = min(X2i,Ci). These pairs were classified into four pair types. They were the following:
Number of Pairs
Pair Type Description in the Sample
I I
1 Xlii
1 f
2 X1iCi n2
1 1
3 X1i>Ci, X2i
1 I
4 Xli>Ci' X2i>Ci n4
where n = n1 + n2 + n3 + n4.
For convenience and without loss of generality, let the type 1 pairs occupy positions i to nI in the sample (i.e., {(XI ,X21),I(XI2,X22),I ,(Xin X2n,)} ) in random order. Similarly, the type 2 and type 3 pairs will be assumed to occupy positions nl+l,nl+2,...,nl+nc in random order. Lastly, the type 4 pairs occupy positions nl+nc+l,nl+nc+2,...,n. What is meant by random order, is that the exchangeability property still holds within the nI type I pairs, within the n2+n3 type 2 or 3 pairs and within the n4 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 nI 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 nl!n4!(n2+n3)! equally likely arrangements when the positions and numbers of the pair types are fixed. Therefore, it follows, that each of the nj! arrangements of the n, uncensored pairs is equally
likely and that the exchangeability property still holds within the type 1 uncensored pairs. Similar arguements for the (n2+n3) type 2 or 3 pairs and the n4 type 4 pairs hold.
The following notation will be used in the statistic Tn1, a statistic which is based on the n1 type 1 pairs.
1
Define a variable Zi to be
Zi = IX2i I IX1i 1 for i=1,2,...,n1
where p is the known and common location parameter. Let Ri be the absolute rank of Zi for i=1,2,...,n1, that is, the rank of Zij among {IZII IZ21 ,... IZn1 } and let Pi be defined as
= ~(zi) 1 if Zi > 0
[i
0 if Zi < 0
Note, the variable Z is defined only for the uncensored pairs. The statistic Tnl is then n1
T = Y. R
n I Ri
1 i=1 i
the Wilcoxon signed rank statistic computed on the Zi'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 {Cnl +1,Cnl+2...,Cnl+n } and
I if the jth pair is a type 2 pair
yj 0 if the jth pair is a type 3 pair
for j = nl+l,nl+2,...,nl+nc. The statistic Tn is defined
C
as
n + n
1 c
T = y. Q.
Tn + i Q
c j=n1+1
 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 X1 (i.e., under Ha), then IX2i PI Xli Pl 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 Ho), then X21 XIli pi would be positive approximately
as many times as negative with no pattern present in the magnitudes of IX2i PI lxi il. Thus the test statistic would be comparatively less.
For the test statistic Tnc, if Ha is true, there should 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 Tnc would be large. In contrast, if Ho is true, the number of type 2 pairs should
___ ___I xl
Cl
Figure 5. Contour of an Absolutely Continuous Distribution
That Has Equal Marginal Locations and Unequal
Marginal Scales with Censoring Present.
not dominate nc and the test statistic Tn should not be unusually large.
Now we establish certain distributional properties for T and T
j Tnc
Lemma 3.2.1: Conditional on nl, Tnl has the same null distribution as the Wilcoxon signed rank statistic.
Proof:
First it will be shown that conditioning on the n1 type
1 pairs does not affect the exchangeability property (i.e.,
' d '
(X i,X2i,Ci) = (X2i,X i,C)) still holds. Let Wi (X Ii,X2i,C i) and W.i (X2i,'X liC) and
!11
Gw (t) =P(X 1 t1,X2i 4 t2,Ci < t ) = E[I(W < t)] where
1 if X 1 t ,X t ,C 4 t
I(W < t) = *
0 otherwise
Now, under Ho, for the entire sample, we have (X d
' d ( '
(Xli, 2i,Ci) = (X2i,XIi,Ci)
and applying an apropriate function (and Theorem 1.3.7 of Randles and Wolfe, 1979) thus d *
I(W. t)I(6 = 1) = I(W t)(f(6.) = 1)
i i i 1
Taking expectations, it follows that
E[I(W.i t)I(6 = 1)] = E[I(W t)I(f(5 ) = 1)] .
Now, recalling that 1 = 1 iff f(5i) = 1; thus
E[I(S6.= 1)] = E[I(f(6.) = 1)]
and it follows that and it follows that
E[I(W.< t)I(6 = 1)] E[I(W.4 t)I(f(6.) = 1)]
1 i 1 1
E[I(6i= 1)] E[I(f(6i) = 1)]
This shows that the c.d.f. of Wi 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
fl(a,b,c) = Imin(b,c) p min(a,c) 9 and applying Theorem 1.3.7 (Randles and Wolfe, 1979, page 16) it follows Z = IX21 I'l IX11 Il
= min(X21,C) I min(X11,C)
d .
= Imin(X1,C) lmin(X21 ,C) 1
= X1 PI 1x21 = Z
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). C1
Lemma 3.2.2: Under Ho, 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, Tnc has the same null distribution
as the Wilcoxon signed rank statistic.
Proof:
First, it will be shown that conditioning on the n type 2 and 3 pairs does not affect the exchangeability property. Define Wi' iGw(t) and I(Wi < t) as in Lemma
3.3.1. Now under Ho, for the entire sample, we have
, d ,
(Xi,X2i,Ci) = (Xi,XIi,Ci)
and applying an appropriate function (and Theorem 1.3.7 of Randles and Wolfe, 1979)
d
I(Wi 4 t)I(6ic(2,3)) = I(Wi < t)I(f(6i)c(2,3)). Taking expectations, it follows that
E{I(ji )l(6i(2,3))} =E{I( i< ) (f(6i)E(2,3))} Recalling that, 61e(2,3) iff f(6i)e(2,3), and thus
E[I(6ie(2,3))] = E[I(f(6i)e(2,3))]. It follows that
E[I(Wi t)I(6i E(2,3))] E[I(Wi t)I(f(6i)E(2,3))]
1 i 1 1
E[I(6i E(2,3))] E[I(f(6 )E(2,3))]
Therefore, conditional on the pair being a type 2 or 3, the exchangeability property still holds.
Thus, it follows that
I 9
P(yj = 1,C < c) = P(XIj cj ,X2j > cj ,Cj < c)
I I
= P(X2j < cXj > cjCj < c) = P(yj = 0,C c) .
Noting that,
P(yj = 1,Cj c) + P(yj = 0,Cj < c) = P(Cj < c)
and thus
2P(yj = 1 ,Cj c) = P(Cj < c) or that
P(y = 1,C < c) =1/2 P(C < c) = P(yj = 1)P(Cj < c) and thus we see that yj and Cj are independent.
To prove part b), let = (yn +1 'Yn +2 '"n'Yn + 1+ 1 1 nc
and
S= (Qn1 +1 'on1 +2 ''Qn1 +ne) By Theorem 2.3.3 (Randles
and Wolfe, 1979, page 37), 9 is uniformly distributed over Rn where
c
Rn = q : is a permutation of the integers 1,2,...,nc } nC C
Now, let q be any arbitrary element of Rn and let & be any
C
arbitrary nc vector of O's and l's. Thus,
P(y = ,Q = 9) = P( = q )P(Q = q ) (by part a) and
P( )P(Q % ) = n
 1
n
C n !
2 c
which proves part b). 0
By Lemmas 3.2.1 and 3.2.2, Tn1 and Tnc are
nonparametrically distributionfree conditional on n1 and nc, respectively.
59
Lemma 3.2.3: Under Ho, the following results hold.
a) Conditional on n1, E(Tn ) = nl(nl+1)/4 and
Var(Tnl) = nl(nl+1)(2nl+1)/24 .
b) Conditional on n E(Tnc) = nc(nc+l)/4 and
Var(Tn ) = n (n +1)(2n +1)/24.
c
c) Conditional on n1 and nc, Tn1 and Tnc are
independent.
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+1)/4 and variance of n(n+1)(2n+1)/24.
The proof of part c) is also trivial following from the fact Tnl and Tnc are based on sets of mutually independent observations.
With these preliminary results out of the way, the test statistic Tnl,n can now be defined by
Tnl,nc =L1 Tn1 + L2nc n1 nl+n
= L1 i iRi + L2 yjQj '
i=1 J=nl+1
where L1 and L2 are finite constants.
Theorem 3.2.4: Under Ho,
a) E(Tnn ) = LIE(Tn) + L2E(Tn )
1 nc 1c
= (L1nl(nl+1) + L2n c(n c+1))/4
b) Var(T ,nc CL
b) Var(Tn1,ne) = (Lnl(nl+1)(2nl+1)
+ L2n (nc+1)(2nc+1))/24
c) Tn1 ,n is symmetrically distributed about E(Tn1,nc and
d) for fixed constants L1 and L2, Tn1,n is
nonparametrically distributionfree.
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, Tnl and Tn are symmetric about E(Tnl) and E(Tnc), respectively. Since Tn1 and Tnc are independent (conditional on N1 = n1 and Nc = nc), the symmetry of Tnl,nc follows.
To prove part d), note that
P(T n= k) = P(Li Tn + L2Tn = k) =
P(L1 = kk L = k )P(L2 = kc) =
n c 2cnc c n
SP(LIT n = kk )P(L2Tn = k C)
{k } 1 c
where {kc} = set of all possible values of L2Tn
c
61
Now using the nonparametrically distributionfree property of Tni and Tn established in Lemmas 3.2.1 and 3.2.2, it follows that for fixed L1 and L2, L1Tnland L2Tn are also nonparametrically distributionfree.
The conditional null hypothesis distribution of Tnl,nc can be obtained using the fact it is a convolution of two Wilcoxon signed rank test statistics' null distributions. Thus, for fixed L1 and L2, the distribution can be tabled. Tables in the Appendix 1 give the critical values for Tnl,nc with L1 = 1 and L2 = I for n, = 1,2,...,15 and nc = 1,2,...,10 at the .01, .025, .05 and .10 levels of significance. The actual alevels are also reported for the cutoffs given. The decision rule for the test is to reject Ho 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: al G2) 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 Ho for larger n, and nc can be based on the
asymptotic distribution of Tn which will be presented in
Section 3.4.
62
3 .3 PI= 12' 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 Tnl,nc to be used in these settings. The modification will be to estimate the common location parameter using a "smoothed" median estimator based on the productlimit (Kaplan Meier) estimate of the survival distribution (Kaplan and Meier, 1958). This estimated location parameter M, replaces p in the previous definitions. That is, define the variable Zi to be
= lX2i Ml lXii Mn
The definitons of i, Ri, yj, Qj, T n, T and Tnl nc 1nc
remain unchanged. In this section, the statistic will be denoted by TMnl to identify the fact the location
n,nc
parameter was estimated with a "smoothed" median estimator based on the productlimit 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 Zi's, a condition which no longer exists. Also, in 3.2.3 c) Tnl and Tnc 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 productlimit estimate of the survival distribution. Let (Y(1),Y(2),...,Y(2nl+n2+n3)) represent the ordered uncensored observations. (This ignores the fact the original observations were bivariate pairs, and considers only the 2nl+n2+n3 uncensored observations, i.e., 2n, components belonging to type I pairs, the n2 uncensored components of type 2 pairs and the n3 uncensored components of type 3 pairs.) That is, Xij = Y(k) if Xij is uncensored and Xij has rank k when ranked among the set of all uncensored observations from either (both) components of the pairs for i=1,2 and j=1,2,...,n. Let n(i), i=1,2,...,2nl+n2+n3, be the number of censored and uncensored observations which are greater than or equal to Y(iW Thus,
2 n
(i)= I I(Xi Y(i), where
i=lj =I
I is the indicator function which takes on a value of one when the argument is true and zero otherwise.
The productlimit estimate of the survival distribution is defined as
I
1
S(t) = H (n(k) 1)In(k)
(k) (k)
k=1
0
if t < Y(1)
if Y) < t < Y +I) for
(j) (j+1)
j=1,2,...,2nl+n2+n31 if t > Y(2nl+n2+n3
(Note, that Y(1) is the smallest uncensored observation and Y(2nl+n2+n3) is the largest 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
S ) 0.5
m + (m2 m )
S(ml) S(m2
M =I
ml1
if ml m2 if mI = m2
A brief explanation of this estimator follows.
The productlimit estimate of the survival function,
9(t), is a right continuous step function which has jumps at
where
and
ml = min{Y(i): S(Y(i)) /2 }
m2 = max{Y(i): S(Y(i)) 2 1/2
the uncensored observations. An intuitive estimate for the common median is the value of Y(i) such that S(Y(i)) = 1/2, 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
and m2. If the Y(i) exists, such that S(Y(i)) = 1/2 then mI = m2 and M is that value of Y(i) by definition.
Lemma 3.3.1: The statistic M is a symmetric function of the sample observations.
Proof:
Let (Y( Y represent the ordered 2n
observations where Y(1 Y( Y(* This again is
'(2) (2.. n)'
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 productlimit estimator S(t) can be viewed as a function of the vectors (Y(1),Y(2), ...,(2n)) and (I(1),I(2), ...I(2n)) where
1 0 in the fact that
n(i)
if Y is censored
(i)
otherwise
2 n
SI I(xi> Y )
i=lj=1
= 2n + 1 (rank of Y(i) in (Y(1) (2),..Y(2n)).
In addition, S(t) can be expressed as
1 t < min{Y .i: (i)= 1}
^ t0 t > max{Y .): I )= 1}
S~t) =4 (1) (i)
~2n jj
Y 2n ) otherwise
* < (2n j + I
(j)
Thus, S(t) is a symmetric function with respect to the sample observations and therefore M, being a function of S(t), is also.
Lemma 3.3.2: Conditional on n1, Tn has the same null distribution as the Wilcoxon signed rank statistic.
Proof:
Let T = {1 2' *""'n }, where Ti = y(Z i) and
1i1
R = R1 R2 ,..., R } with Ri = absolute rank of Zi.
2 nII
1
Let T be any arbitrary element of
P = o : TO is a 1 x n, vector of O's and 1's},
67
(of which there are 21 different elements), and let r be any arbitrary element of R = {r : r is a permutation of the integers 1,2,...,nl}.
Now, under the null hypothesis,
' d '
(Xi,X2 i,Ci) = (X2i'X li,iCi)
9 I
and thus letting X1i = min(Xli,Ci) and X2i = min(X2i,Ci), it follows that
(Xli 2i (X2i, li
for i=1,2,...,n1 and these pairs are also exchangeable. Now, let k = 0 be an operator such that
(X i, X 2i) if k = 1 (xi' X2 k (X, X) if k = 0.
Thus, under Ho and using the exchangeability property, it follows
{(XI X2 ), (XI X2 ),..., (Xl X )}
11' X21 12' X22 In, 2n
1 1
d k k k
( {(X X ) 1 (X X ) 2 X I X ) nl} .
1 1 2 2 n n1
(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
f1 (1'Y2) = fY2 MI = Z ,
it follows from applying this function to (3.3.1) that
{Z1, Z2",..., n r {Z r rn
1 1 2 n
n1
d k k 2
{(zr ) (Z )
r k
1 2
ki where (Z ) =
r i
Z ri
r
Z
r
k
. (Zr nl}
n
if k = 1 if k = 0
Now defining a function f2(Z) = (T, R), where T and R are
1 x nI1 vectors such that
0
V J= 0
if Z. > 0
3
if Z. < 0
j
R. = absolute
for j=1,2,...,n1.
rank of Z., i.e., rank of IZI among { jZ1 iz 22 '' iZn I} Applying this function to (3.3.2) it
follows that
( l T2'" n R1, R2 ..., R n1
d
(T I,..., T R R ,..., R )
1r 2 n 1r 2 rn
1 n
k kk
d 1 2 kn
d ( r ) ( r ) ,.. ,( r ) 1 R rI R r2 '...,R }
r r r r
1 2 n 11 2n
(3.3.2)
and
69
k Trj if ki= 1
where ( ri )r. i
 if ki= 0
rr
Now since k and r were arbitrary vectors, it follows that
PQ R = = PQ To Z
I x
n
2 nI
Thus noting this produces the same null distribution for the Wilcoxon signed rank statistic, the proof is complete. 0
Lemma 3.3.3: Conditional on ni and nc, Tn and Tn are
i nc
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 yi be defined as before and let (xi,ci) denote the observed value of the ith type 2 or 3 pair i=nl+l,...,nl+nc. Note, one component was censored, and thus its observed value was ci while the other component was uncensored and its value is denoted by xi. This is not specifying which component (Xii or X2i) was censored.
Claim 1: Yi is independent of (xi,ci).
This follows by noting that under H and using the exchangeability property of type 2 and 3 pairs (as was shown in Lemma 3.2.2) that
P{Yi=1l (xi,ci)} = P{Xli=Xi,X2i=ci (xici)
= P{Xii=ci,X2i=xiI(xi,ci)} = P{yi=O(xi,ci)}
Since P{yi=ll(xi,ci)} + P{yi=OI(xi,ci)} = 1, Claim 1 follows. Now define y = (Ynl+l' Ynl+2."'Ynl+nc)
Claim 2: y is a vector of nc i.i.d. Bernoulli random
variables which are independent of
{(xnl+lcnl+l),(xnl+2cnl+2),c.,(xnl+nccnl+nc)}
This follows from Claim 1 and the fact that {(xnl+1,cnl+1),(xnl+2,cnl+2),...,(xnl+n ,cnl+nc)} are i.i.d.
Claim 3: y is independent of
{(xnl+1,cnl+1),(xnl+2'cnl+2),...,(xnl+n ,c l+n c)}
{(Xn +i i) ),(xx1nl~nic c1
xn and x where
n {(xllx12),(x12,x22),...,(xlnlx2nl)}
and
XIn4 {(cnl+n c+lcnl+nc+l)...,(Cncn)}
(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 xnl, Xn4, X(n) and C(nc)
where X(n ) denotes the observed ordered uncensored
members of type 2 and 3 pairs and (nc) denotes
the observed ordered censored members of type 2 and 3
pairs.
Note, this claim follows directly from Claim 3 and the fact that X(nc) and C(nc) are functions of
{( (nic ~1 ,(x c))( c) ny
{(xnl+1lcnl+l)'(xnl+2,cnl+2),...,(xnl+nccnl+nc)} only.
Claim 5: Yc is independent of xnl, Xn4' X(nc) and C(nc)
where Yc = {Yc(1)'Yc(2)''''Yc(n )}, c(i) is the ith element of c(nc) and yc(i) is the y which corresponds
c( ) (i)
to the pair of which c(i) 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 yc(i)'s.
72
Claim 6: Given xnl, xn4, x(nc) and c(n T is no longer
a random variable; that is, the value of Tnl is
observed.
This follows directly from the definition of Tn1
Claim 7: Note, that
n + n n + n
1 C 1 c
TYQj = JYc = W
n +I Jnl ~d +
c jn +1 j=n+1
which shows that Tnc is a function of yc and is
independent of Xn x n4 (n) and c(n ).
Thus, Tnc has a null distribution equivalent to the Wilcoxon signed rank null distribution and is independent of Tn n1
which is a function of xnl' Xn4' X(nc) and c(nc) only. E[]
With the proof of Lemma 3.3.3, Theorem 3.2.4 is valid for the modified test statistic TMul,n That is, under Ho Tnl~n has the same
and conditional on nI and nc, TMnnc has the same distributional properties stated in Theorem 3.2.4 for Tnlnc
n1and the tables in the appendix are valid.,n and the tables in the appendix are valid.
73
3.4 Asymptotic Properties
In this section, the asymptotic distribution of the test statistic Tn (and TMn ) under Ho will be
nl,nc nl,nc0
established. The asymptotic normality of the test statistic will be presented first, conditional on N1 = nI 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 + , Nl+Nc = (n number of type 4 pairs) + also. The asymptotics will be presented for the test Tn only. In the previous section, it was
n1,nc
shown that under Ho and conditional on N1 = nI and Nc = nc, Tn and TMn have the same null distribution; that is
d~n l,nc
d
T = TM Therefore, they have the same cumulative
nl,nc nl,nc
distribution function and thus their asymptotic distributions are the same. There is no need to prove them separately.
Theorem 3.4.1: Conditional on N1 = nI and Nc = nc, under Ho
T E(Tn )
n1'nc nc d
T n* N(0,1) as n + and n +
(T )n
where
E(Tn1,n ) = (L1nl(nl+1) + L2nc(nc+1))/4
and
a(Tn n c) = [Var(Tn )]/2
n,n nl,nc
2 2
= [(L1nl(nl+1)(2nl+1) + L2n (n +1)(2n +1))/24]'2 .
Proof:
First, it will be shown that Tnland Tnc have asymptotic normal distributions. Without loss of generality, it will be assumed that y = 0.
Note that
n
. iRi
n
11
Y(tX2il lxii) +
i1 1
where
T(IX2i IX1i + 1X2j1 1X1jl)
n
n i=1'(l X2i Xi )
and
U2,nl = 2' ( l Xlil + IX2j iX1j[)
12 1i
2
are two Ustatistics (Randles and Wolfe, 1979, page 83). It
follows
(n )1/2
1
(2') 1
(n )3/2
 nl(nl+ 1 )/4) = n (U
1( 2 1 1
 E(U in ))
1,n
+ (nl) 1/2(U 2,n
2, 1
 E(U 2 ))
2,1
ni
= nl(U1,n1) +21)(U2,n1)
U1,nl
Now notice, 0 < U < 1 and under Ho, E(Ul,n ) =
1,n1I 1 n
P{(IX2i IX1ii) > 0} = 1/2, so that IU1,n 1/21 <1 /2 Therefore,
(n )3/2 (ni)3/2
n U, 1/2) < n+(n ) + 0 as nI + .
(2
(n 12 1
Thus n1 n n 1(nl+ 1)/4) and (nl) U2,n1 1/2) have
(2
the same limiting distribution as nl co .
By Theorem 3.3.13 of Randles and Wolfe (1979), it is
seen that (n1)(U2,n 1 1/2) has a limiting normal distribution ,ni
with mean 0 and variance r2 1 (provided (I > 0) where
r2 = 22{E[T(lX2 X1l + X2j IX1j)
x ,( I2 lxi + IX2kI ix k] 1/4}
= 1/3
Thus, T nl(nl+ 1)/4 d
n ~I/
1 d
+ N(0,1) .
n
(2 3n
Note that,
n 1 )1/2
(2 ) 3n1 P
+ 1
nl (n + 1)(2nl + 1), 24
as n1 + 0 Therefore (after applying Slutsky's Theorem (Theorem 3.2.8, Randles and Wolfe, 1979)
 nl (nl + 1)/4
d
 N(0,1) .
o(T )
n
1
Similarly,
n +n
1 c
Tn = YJQ j
c j=n 1+1
n +n
I Yj + j=n +1
1
{yj(cj ck) + Yk(Ck cj)} n I +1
1 1 c
n
Snc(U3,nc) + (2c)(U4,nc
where
n +n 1 1 c
n Yj
c j=n +1
1
U4, nc n
(21 1 j ~n+
r c (2c n1+1
are two Ustatistics. It f
(n )1/2 c (T
n n
(c c
2
yjY(c. ck) + YkT(Ck c )}
ollows that
(n )3/2
 n (n + 1)/4) c (U
c c n 3,n
(2c c
+ (n )1/2(U
c 4,n
c
 E(U3n ))
c
 E(U4 ))
C
T n1
and
U3,nc
Note that 0 < U3,n
c
77
S1, so U3,
. ,He
 1/21 < 1/2 and thus
(n )3/2 (n )3/2
c U cI/
n (U 1< 0 as n + .
S 3,n n (n 1) + 0 asc
(2c c c c
Thus, C (T
n n
( c c
2
 n (n + 1)/4) and (n c)' U4,n 1/2) have
c c c 4,
the same limiting distribution as n + .
c
Again applying Theorem 3.3.13 of Randles and Wolfe (1979), it is seen that (n) 2(U 4,nc 1/2) has a limiting
C
normal distribution with mean 0 and variance r2I (provided r2 1>0 ), where
 Ck) + YkT(Ck C ))
x (jq(C ci) + yif(Ci C i))] 1/4}
= 22{E(y Y(C iCk k)(CkC ) ) Yj (Cj Ci) + YiP(CiC)) i 4}.
By the independence of yj and Cj (Lemma 3.2.2),
r 2 = 22 {P(y j= 1)P(C.> C C.> C.) + P(y J= l)P(yi= 1)P(C > Ck, Ci> C )
+ P(y k= 1)P(y j= 1)P(Ck> C C > C.)
r2 = 22{E[(yj (Cj
+ P(yk= 1)P(yi= 1)P(Ck> C., C i> C ) 1/4}
= 4 {1/2 P(C > Ck, C > Ci) +1/4 P(C > Ck, C i> C ) + 1/4 P(Ck> C C > Ci ) + 1/4 P(Ck> Cj C > C) 1/4 }
= 4{ + ( + + ) = 1
6 4 6 6 3 T 3
T n (n + 1)/4
n c c
n
2 c
d
+ N(0,1) .
Noting that
1 1/2 3n
c
n
2 c
(1 1/2
3n
c
p
+ 1 as n +
c
n (n + 1)(2n + 1) 1/2
24c c c /2
24
and applying Slutsky's Theorem, it follows that
T n (n + 1)/4 n c C
c
d
 + N(0,1) .
o(Tn
C
The conclusion of Theorem 3.4.1 then follows by writing
T E(T )
n,n n ,n
1 c 1' c
n ,n
(LT + LT ) (LIE(T) + L2E(T ))
i1n 2 n 1 n1 2 n
c c
(L2 o2(T ) + L2 02(T ))/2
1 n 2 nc
1 c
L o(T n) T E(T ) L 2o(T ) T E(T )
1 n n n1 2n n n
1 1 1 c c c
x + x
a(T )
nl,nc
a(Tn )
o(T )
n ,n
Thus,
o(T )
n
c
applying Slutsky's Theorem and utilizing the fact Tn and ni
Tn c are independent, conditional on N1 = n1 and N = nc. []
c
Next, and most importantly, the unconditional
asymptotic normality of Tnln will be established as n 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 {Tnl,nc } for nl=1,2,..., nc=1,2,..., be any array of random variables satisfying conditions (i) and
(ii).
Condition (i): There exists a real number y, an array of positive numbers [Wn1l,n } and a distribution function F(.) such that
lim P{T y < x W } = F(x)
min(n ,n )+ 1 c 1, c
at every continuity point of F(.).
Condition (ii): Given any 6 > 0 and n > 0, there exists v = v(C,n) and d d(e,n) such that whenever min(nl,ne) > v, then
 Tnln I < E nln
ni ,ni n
for all n',n' such that
1 c
In' nj < dnl, In' n < dnc} > 1 n
1 1 I c cl c
Let {nr} be an increasing sequence of positive integers
tending to infinity and let {N1r} and {Ncr} be random
variables taking on positive integer values such that
N. p
Nir
r + X. as r + , for some Xi such that 0
1 i 1
n
r
i=1,c. Then at every continuity point x of F(.)
lim P{TN N r+= ir, cr
 y < x [ n],[Xn]} = 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 Ustatistic Theorem (Hoeffding, 1948) but with the sample size as a random variable.
P{ IT ,
n ,n c
Lemma 3.4.3: Suppose that
U = (n)I f(X X ,... X ) n r BEB a1 82 ar SeB 1 2 r
where B is the set of all subsets of r integers chosen without replacement from the set of integers {1,2,...,n} and f(tl, t2,..., tr) is some function symmetric in its r arguments. This Un is a Ustatistic 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 values with probability one. If E{f(X1, X2,...,Xr) 2<
/ r2E N p
lim Var(n U ) > 0 and r + I then
n 1 n
n+ r lim P{(UN E(U )) N 12(r2 )2 } = (x)
Sr 1
r+ r r
where D(.) 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 C1 and C2 of Anscombe (1952) are valid and applying Theorem 1 of Anscombe (1952). Condition C1 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
C2 will be utilized in the proof of the major theorem of this section which follows.
Theorem 3.4.4: Under Hop
T E(T )
N N EN ,Nc)
1, c 1 c
o(T )
N ,N
1 c
d N(0,1) as n + .
 + 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
n1 ,nc
T E(T )
nl,n n1,nc
o(T )
n1,nc
, the standardized
T statistic. Theorem 3.4.1 shows that {T } for
n1,nc n1,nc
nl=1,2,..., nc=1,2,..., satisfies condition (i) of Lemma
3.4.2 with y = 0 amd an~,n = 1. Note that from assumption n1,nc
A5, it can be seen that Xi > 0 for at least one i=1,c. If xi = 0, for i=1 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 Xi > 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.
T E(T )
o(T )
n
1
, the standardized Tn
statistic. In the proof of Theorem 3.4.1, it was shown that
*
Let T
n
83
T has a limiting standard normal distribution by utilizing the Ustatistic representation of Tn1. As a result of Lemma
3.4.3 and this Ustatistic representation, it follows that Tn satisfies condition C2 of Anscombe (1952) (since T is equivalent to a Ustatistic 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 1> 0 and n > 0, there exists vI and dl> 0 such that for any n1> 1
1 1
P{iT Tn < for all n' such that In'l n < dn1 I n n'
1 1
1 n (3.4.1)
Similarly, as a result of the Ustatistic representation of T (as shown in the proof of Theorem 3.4.1) and from Lemma nc
T E(T )
n n
3.4.3, it follows that T c c satisfies
nc o(T )
n
c
condition C2 of Anscombe (1952). That is, for a given e 2> 0 and n > 0, there exists v2 and d2> 0 such that for any n > v2
c 2
1 I nI d
P{T T I E for all n' such that In' ncl < dn } In n 2 c c c c
c c
1r. 3.2
(3.4.2)
Consider T E(T )
* nl,nc n1nc
T =
nnc ~a(T )
nI ,nc
L 1a(T ) L2 o(T )
1 n1T* 2cn *
1 (T + x (T n
o(T ) 1 o(T ) c
n ,n n ,n
1 c 1 c
SL' T) + L' (T in n 2n n
1 c
Note that,
I 9
(1) Ln and L2n are functions only of N and N and the
in 2n 1 c
given L1 and L2 constants.
(2) (L n)2 + (L 2n2 = 1.
in 2n
(3) There exists constants L1 and L2 such that L in L1
, P ,
and L2n * L2 as n + .
First, it will be shown that condition (ii) is
I *
satisfied for LI(T n + L2(T n T n
1 c 1 c
Let E > 0 and n > 0 be given and let v1,' v2, dl, d2
satisify (3.4.1) and (3.4.2). Let v max(vl, v2) and
d min(d1, d2). Now,
P{ T n' Tn n < 2 for all n', n' such that
n1l c i c
In' n.I < dn ,i=1,c} n i i i '
' *
> P{(L Tn T *I + L2Tn, T ) < 2c for all n', n'
1 n n 1 2 n c
1i c c
such that n' ni < dni, i=l,c}
suchi Jhtn
> P{L Tn,
1
 T < E: and L2IT, T I < for all n, n'
n1 2 n n C C
1 c c
such that In! ni < dni, i=l,c}
= P{L1IT, n I < for all n such that n' n < dn1}
1 1
+ P{L2Tn, Tn < for all n' such that n' ne < dnc }
21 n n c Ic c
c C
 P{L T T I < e or L2 IT, T ( < for all n', n'
1n n C cn c
such that In' nil < dnI and In' ncl < dnc}
> P{L Tn,
1
+ P(L2 T ,
c
 T < e for all n' such that n n < dn
n 1 1 11 1
 T < E for all n' such that In' n < dn } 1.
n c c c c
CC
c
(3.4.3)
Now using inequalities (3.4.1) and (3.4.2) and applying them
' 12
to (3.4.3) with e = min{ E(L1) 12
9
P{ IT ,
n, ,n
1c
, 1/
C 2(L2) 2 } then
 T < 2E for all n' n' such that
n ,n 1 c
ni ni < dni, i=1,c} > (1 n) + (1 n) 1 = 1 2n *
86
Therefore Tnlnc satisifies conditon (ii) of Lemma 3.4.2 so
ni1,nc
I L I + LI T*
that Theorem 3.4.4 is valid for Tn,nc = Ln + L2Tnc
9 1
To see that the Theorem is valid if L1 and L2 are replaced
I I
by Lin and L2n, respectively, consider,
A'' *
T T = L I T + L T
nl,nc nnc In n 2n n
I I *
 L1Tn + L2Tnc
1
= (Ln L )T + (L2n L2 )T (3.4.4)
in 1 n1 2n 2 nc
Now, since Tn and Tn converge in distribution to standard
1 nc
normal random variables, Tn and Tn are 0 (1) (Serfling,
(1980), pg. 8). Also, since Lin + L1 and
I P I I I I I
L2n + L2 as n + m, (Lin Ll) and (L2n L2) are o (1).
Therefore (3.4.4) shows that
I I 5
(Ln L1 n1 + (L2n L2 n is o (1) and thus, Theorem
3.4.4 is valid.
3.5 Comments
From the results in Sections 3.2, 3.3,and 3.4, it is
clear that a distributionfree test of the null hypothesis
of bivariate symmetry versus the alternatives presented
could be based on Tn1n (or TMnnc ). For small samples,
an exact test utilizing the distribution of Tn1,n (and
TMn ,nc) conditional on N1 = n1 and Nc = nc could be
87
performed. For larger samples, the asymptotic normality of TnJ'nc (and TMnJ nc) could be used. In Chapter Five, a 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 I
In Section 3.2, the test statistic Tnl,nc, conditional on nc, 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 n3, the statistic would then have had a null distribution equivalent to the Wilcoxon rank sum statistic. Conditioning on nc and not on n2 and n3 was chosen because the observation of a particular n2 and n3 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 KaplanMeier 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 KaplanMeier 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 X1, and X2This 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 2vector, Wn9 of test statistics. The first statistic denoted TEnl is a
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 2vector 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)
n,nc
TMn but using separate location estimates for X and
X21 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
91
not distribution free, even when H is true. Thus, if n is the variancecovariance of Wn, the quadratric form W' n a tn _n
will not be distributionfree. A consistent estimator of tn' tn will be introduced in Section 4.5 and a test based on the asymptotic distributionfree statistic W' 1W will be n tn _n
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 = X X21 and R(IDiI) be the absolute rank of Di for i=1,2,...,n, that is, R(IDi is the rank of IDil among (ID1 ,jD2 ,..., Dn ). Define
1 if Z. > 0
1
0 if Z. < 0
1
Let TEn1 and TEnc be defined to be the following: n1
TE = W 'R(iD.I)
TEn i=1 (Dij)
and
TE = N N .
n c 3 2
c
Notice that TEn is the Wilcoxon signed rank statistic applied to the nI1 totally uncensored pairs. Popovich (1983) showed under H0, N3 is distributed as a Binomial random variable with parameters nc and p = 1/2 P2(0) = 1/2 P(type 2 or
3 pair). With a slight modification from Popovich, the statistic TEn1,n is
K (TE1) + Kn(TEc
inn 2n(E n
where
TE nl(nI + 1)/4
TE =
n (nl(n 1 + 1)(2n1 + 1)/24)/2
and TE
, n C
TE c
nc (n
and Kin and K2n are a sequence of random variables satisfying:
1) Kln and K2n are only functions of N1 and Nc,
2) there exists finite constants KI and K2 such
that KIn  K1 and K2n + K2 as n + o.
This is slightly different from the statistic Popovich introduced, the difference being that he required Kin = (1K2n) 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 TEnl,nc, TMnlnc and CD are not affected by their presence (other than in estimating the location parameter). It will be assumed that the sample is of size n = N1 + Nc.
4.2 The Wn Statistic Using Tnln
The first statistic to be considered for pairing with TEnn is similar to the statistic TMnl presentedd in
ln1 ,nc
Section 3.3. The difference being, that instead of using a common estimate for v as in Section 3.3, here we first consider using separate estimates which are denoted by M1 and M2 where M1 is the KaplanMeier estimate for v based on the X1i's alone and similarly, M2 is the KaplanMeier estimate for p based on the X2i 's. Define
n
Tn Ti R( l X2i M 21 l X i Mill
1 i=1
and
n +n
1 c
T = yj .Q.j
n 3+ 3
c j=n 1+1
(Note these are similar to statistics defined in Section
3.2, with a slight modification of using the separate estimators M1 and M12.) Similarly, define T nl(nl+1)/4
* n1
T
n1 n (nl+1)(2n +1)/24
and
andT n (n +1)/4
S n c c
T = c
n
c n (n +1)(2n +1)/24
c c c

Full Text 
PAGE 1
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
PAGE 2
to my parents, with love
PAGE 3
ACKNOWLEDGEMENTS I would like to thank Dr. Ronald Randies 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 ma king this all possible.
PAGE 4
TABLE OF CONTENTS PAGE ACKNOWLEDGEMENTS ABSTRACT CHAPTER ONE INTRODUCTION TWO A STATISTIC FOR TESTING FOR DIFFERENCES IN SCALE 2.1 Introduction 2.2 The CD Statistic 2.3 Permutation Test 2.4 Asymptotic Results 2.5 Comments THREE A CLASS OF STATISTICS FOR TESTING FOR DIFFERENCES IN SCALE 3.1 Introduction 3.2 pi= iio Known 3.3 Uj= v n Unknown 3.4 Asymptotic Properties 3.5 Comments FOUR A TEST FOR BIVARIATE SYMMETRY VERSUS LOCATION/SCALE ALTERNATIVES 4.1 Introduction 4.2 The W Statistic Using T n l' n c 4.3 The W n Statistic Using CD~ 4.4 Permutation Test 4.5 Estimating the Covariance 16 16 19 35 39 44 48 48 50 62 73 86 90 90 93 HI 121 123
PAGE 5
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 17 1 BIBLIOGRAPHY 183 BIOGRAPHICAL SKETCH 185
PAGE 6
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 August, 1984 Chairman: Dr. Ronald H. Randies 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 N~ 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
PAGE 7
to that of a Wilcoxon signed rank statistic. Thus, any member in the class can be used to provide an exact test which is distributionfree for the null hypothesis. The statistic based on Kendall's tau is not distributionfree 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 distributionfree. For the second and more general alternative, a small sample permutation test is proposed based on the quadratic form W n = T ^ E T where T ^ is a 2vector of statistics composed of a statistic designed to detect location differences and a statistic designed to detect scale differences and I is the va rianceco variance matrix for T For large samples, a distributionfree approximation for T" I 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 W 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.
PAGE 8
CHAPTER ONE INTRODUCTION Let W, and W2 denote random variables; then the property of bivariate symmetry can be defined as the property such that (W,,W,) has the same distribution as (W2Wj). 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 dental sensitivity is again assessed. If X,. and X~ are the first and second sensitivity measurements, respectively,
PAGE 9
of the i 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 distribution of (^jjX^^) "*" s t ie same as that of (Â•^i'^li^' 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
PAGE 10
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 (r
PAGE 11
was allowed. It is this latter type of censoring which this work addresses. Now we statistically formulate the problem of i i interest. Let (X. X^^ for i=l,2,...,n denote a random sample of bivariate pairs which are independent and identically distributed (i.i.d.) and C. 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 with pair (X, ^ ,X,. ). In the case of random right censoring, the observed sample consists of (X, i>^2'^^ where i i X,. = min(X,.,C.), X^j = min(X2 : ;,C i ) and 6is a random variable which indicates what type of censoring occurred, i Description 1 x Ii< c x 2i C i> X 2i C i' X 2i> C i Now we state a set of assumptions which are referred to later Assumptions : t 1 Al. ( x ii X 2i^ i=l,2,...,n are i.i.d. as the t 1 bivariate random variable (X..X,). A2 (X, 1 ,X, ) has an absolutely continuous bivariati 11 A 21 distribution function F( x, p xÂ„ p.
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where F(u,v) = F(v,u) for every (u,v) in R The parameters y, (pj) and :, (o^) a r ^ location and scale paratmeters, respectively. They are not necessarily the mean and standard deviation of the marginal distributions. A3. C, ,Ci C are i.i.d. continuous random variables, with continuous distribution function G(c). A4 The censoring random variable C. is independent 1 1 of (^ii>^2i^ i=l,2,...,n and the value of C. is the same for both members of a given pair. A5. P(xJ i >C.,X2 i >C i ) < 1. A6. G(F ( V2 )) < 1 where Fy denotes the marginal i i cumulative distribution function (c.d.f.) of X 11,2. 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 H Q : u^=V2> a l =a 2 versus tne alternatives:
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1. The case where y,=u2 = U with \i known, H : a i j* a , 2. The case where p,=p,=p with \s unknown, 1" M 2" H a : 0l + a 2 3. H : \i ^ ^2 and/or a, ^ On* 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
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conditionally distributionfree nonparamet ric 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 T following. Under H the pairs ( X ]i> X 2i) i=l,2,...,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, R., R ll R 12 'In 21 22 2n where R is the rank of X. in the pooled sample 1=1,2 i=l,2,...,n. Let S(R^) be the set of all rank matrices that can be obtained from R N by permuting within the same column of R^ for one or more columns. Under H each of the 2 n elements of S(R N ) is equally likely and thus, if T is a statistic with a probability distribution (given S(R N ) and H Q ) which depends only on the 2 n equally likely permutations of R N T n is conditionally distributionfree (conditional on the given R N and thus S(Rj observed). Sen's statistic T can be defined as t = y e n n Â• =i N > R li
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where E^ i 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., = Â— Â— ) or the quantile F w x N+l scores (E N ^ = F ( ) where F is an appropriately chosen absolutely continuous c.d.f.). The Ansar iB radley scores N+l I N+l (E (E N,i N,i } x ? 1^ or the Mood Scores ( TTTT ~ y ) ) 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 i ^ X li' X 2i^ i=lj2,...,n 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 nonparamet r i c tests for bivariate symmetry. In the normal case, they form the likelihood ratio test for the transformed observations (Y 1 ,Y 2i ) where Y 1 = Xj. X 2 and Y 2 = Xj. + X 2 The
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resulting test they suggest when dealing with a bivariate normal distribution is to reject H if b, > t(3,;n2) or b 2  > tC^J 11 1 ) where B l = (n2) l7 2 r(Y lS Y 2 ) (lr^Y^Y^)^ n^Y and B and r(Y^,Y2) ^ s c ^ e sam pl e correlation coefficient of the Â— 9 Y^'s and ^^'s, Y and S are the sample mean and unbiased sample variance, respectively of the Y, s and t(B;n) represents the critical value for a t distribution with n degrees of freedom which cuts off $ 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 = 20j + 23 2 40 1 & 2 so relatively small values for 0, and B 2 would need to be chosen The nonparame t r i c 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 ment ioned Hollander (1971) introduced a nonparame t ri c test for the null hypothesis of bivariate symmetry which is generally appealing and consistent against a wide class of alternatives. He suggested
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D = 10 F (x,y) F (y,x)} dF (x,y) where l n F (x,y) = I i = l I (,x] (X li )I (ro ,y]^2i ) is the bivariate empirical c d f He notes that nD is not r n distributionfree nor asymptotically distributionfree when H Q is true, and thus proposed a conditional test in which the conditioning process is based on the 2 n data points Cjj) (J n ) (x n ,x 21 ) ... .,(x ln ,x 2n ) for k = 1 ,2 ,n} j k = or 1 which are equally likely under H Here we let (s,t) (0) = (s,t) and (s,t) (l) = (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 nD It n becomes very prohibitive for even moderate n. Koziol (1979) developed the critical values for nD 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 ii> Y 2i^ of Bel1 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
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statistic, it 11 n\ r L f{(Y U" Y li )(Y 2j" Y 2i } i
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12 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 V n = T n tn T n where T = (W + n(n+l)/4 }/o n'2 U n V 2 ) W"T is the Wilcoxon signed rank test statistic calculated on the Y, 's and tt is as previously defined. Again, the conditioning of the test is on the collection matrix C He obtains the limiting distribution of the small sample permutation test and proposes a large sample distributionfree 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
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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., X,. X^) 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 N, 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
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14 T (N. ,N.) = (1L ) 7 2 T* (N. ) + L 7 2 T* (NÂ„) n 1 2 n lnl n2n2 where T. is the standardized Wilcoxon signed rank statistic calculated on the N, uncensored pairs, and 1/ = N, '2 (NopNjr ) where N~ R is the number of pairs for 2n which X, is censored and X^. is not, and N 2 t is the number of pairs for which X 2i is censored and X, is not (note ^2R +N 2L = N 2^* ^ e we ig nt L is a function of N, and N 2 only P such that 0 L Note that T, is a distributionfree statistic calculated only on the uncensored pairs (and is a common statistic used for testing for location in the uncensored case) while T~ is a statistic based only on the type 2 and 3 pairs (as previously defined in this introduction). The statistic T2 n is designed to detect whether type 2 pairs are occurring more often (or less often) than should be under the null hypothesis. Under H T 2 is a standardized Binomial random variable with parameters N 2 =n 2 and p= Vo and thus distributionfree. Popovich obtains asymptotic normality for the statistic T n ( N i> N 2) under the conditions (1) that N, and N 2 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 WL ) (1980) based on logistic scores. The results show that these statistics perform as well as T,,, (better in some cases) and that they
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15 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 t\ 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.
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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 H : a, t oo! 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
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17 21 Figure 1. Contour of an Absolutely Continuous Distribution That Has Equal Marginal Locations and Equal Marginal Scales.
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21 Figure 2. Contour of an Absolutely Continuous Distribution That Has Equal Marginal Locations and Unequal Marginal Scales.
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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 ( X M> X 2i) f r i = l>2,...,n denote independent identically distributed (i.i.d.) bivariate random variables i i which are distributed as (X,,,X 2 i). Consider the following i i orthogonal transformation of the random variables (X,.,X~.): let li = X. + X and Y = X, X li ~2i 21 'li for 11,2, n Figure 3 illustrates what happens to the contour given in Figure 1 (i.e., the contour of an absolutely continuous distribution under H ) when this transformation is
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2 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 i transformation and H Y, and Y are not correlated o 11 2 1 i i although X,, and X 2 i possibly were. Similarly, as can be i seen in Figure 4, under this transformation and H Y,, and Y 2] 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 between Y,, and Y 2 Â• Kepner (1979) suggested the use of Kendall's tau to test for correlation between Y, and Y 2 i Â• Kendall's tau was chosen, due to the fact it is a Ustatistic and, thus, the many established results for Ustatistics 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 ? it of X,. or Xj, (or both) is not known, and thus Y., or Y 2 (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, (X, ,X.) denotes bivariate random variables t i which are distributed as (X..,X 2 ,). Let C,,C 2 ,...,C denote the censoring random variables which are independent and identically distributed (i.i.d.) with continuous
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21 11 Figure 3. Contour of an Absolutely Continuous Distribution That Has Equal Marginal Locations and Equal Marginal Scales under the Transformation.
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22 21 Figure 4. Contour of an Absolutely Continuous Distribution That Has Equal Marginal Locations and Unequal Marginal Scales under the Transformation.
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23 distribution function G(c) where C^ denotes the value of the censoring variable associated with pair (X^X^^). In the case of random right censoring, the observed sample consists of X^ = minCX^.C^) and X 2i = min(X 2i C i ) These pairs can be classified into four pair types which are Pair Type 1 2 3 4 Description X ii< C i> X 2i X 2i> C i X ii> C i> X 2i C i> X 2i> C i Consider the following orthogonal transformation applied to the observed sample: 11 X li + X 2i and Y 2i = X li X 2i for i=1 > 2 Notice that, due to censoring in type 2,3, or 4 pairs, the T I true values of Y,. and Y 2i (denoted Y,. and Yj i.e., the values had no censoring occurred) are not actually observed. The following table, Table 2.1, summarizes the i i relationship of the true values of Y, and Y to the 1 1 2i observed values
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24 Table 2.1 Summarizing the Relationship Between the True Values of Y, and Y to the Observed Values li 21 Pair Type Description x li c. X li> C i X 2i< C i x li >c i x 2i >c i Relationship Between and Y li Y li = Y li Y li > Y li Y Ii > Y li li > Y li Rela t ionship ? Between Y and Y, 21 21 Y 2i = Y 2i Y 2i < Y 2i Y', > Y 2i uncertain (i.e., Y 2i > Y 2 or Y 21 Y 2i ) The modified Kendall's tau (denoted CD for concordantdiscordant) can now be defined as CD = 1 v ) a. b. where for i
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25 (here Y, K Y, can be read as "Y, is definitely smaller than Yj,"). .Â• th For example, if the i pair is a type 1 and the j th pair is a type 2 and it was observed that Y,. < Y, ., then a = 1 since Y u = Y^ and Y^ < yJj (thus y[ < y[ ) If Y,. > Y, Â• had been observed, then a. = 0, since the t t relationship between Y, and Y, is uncertain. Similarly, if Y 2i < Y 2 then b = 0, since Y 2 = Y 2i and Y^, < Y 2 Â• I T (thus, the relationship between Y 2 j and Y 2 ^ is uncertain). On the other hand, if Y 2 > Y 2 had been observed, then b. Â• = 1 (by a similar argument). Table 2.2 summarizes the necessary conditons for a. and b. to take on the values of 1, 1 or 0. The product of a^. and b^. results in a value of 1 if the i and j 1 pairs of the transformed data points are definitely concordant, a value of 1 if the pairs are definitely discordant and if it is uncertain. If the i pair is a type 4 (i.e., both X, and X 2 were censored) then b. will always be since the relationship between the i and j pair is always uncertain regardless of the j 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).
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26 Table 2.2 Summarizing the Values of a. Â• and b. for i Y, and one of the following occurs, i pair type 1 2 3 4 j 11 pair type a ii = ^ : ^ or a ^ otner cases b U = 1 if Y2J < Y2 ^ and one of the following occurs i pair type j fc pair type bjLj = 1 if ^2 Y 2i an ^ one ^ tne following occurs th i pair type j pair type 1 2 1 2 b. = 0: for all other cases
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27 Next, we establish some properties of the CD statistic. Lemma 2.2.1: Under H and E(CD) = 2) Var(CD) = a + Â— n(n 1) n(n 1) where and a = 4P(a. .= 1 ,b. .= 1) Y = 2P(a..= l,b..= l,a..,= l,b..,= 1) ij ij ij ij + 2P(a..= l,b..= l,a..,= l,b..,= 1) ij ij 3 ij + 4P( J l.b^l,a.. f = l.b Jf 1) 2P(a J l,b j l,a ij( = l,b..,= 1) 2P(. J l,b..= l,a..,= l,b..,= 1) 4P( ai .= l,b..= l,. lj( l,b ijt = 1) (2.2.2) Proof Throughout this proof, Theorem 1.3.7 in Randies and Wolfe (1979) will be used extensively and thus its use will not be explicitly indicated. Under HÂ„ I ^ X li > X 2i X l j X 2j ,C i C j ^ ( X 2i X li X 2j X l j C i C j ^
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28 and therefore it follows that (X li' X 2i' X lj X 2j 6 i' 6 j ) = (X 2i' X li' X 2j X 1 j f (5 i ) > f (6 j }) (2.2.3) where X u = minCxJ.C.), i X 2i = min(X 2i ,C ) 6 i indicates what type of pair (X. j.X^. ) is, and fCS^) indicates what type of pair ( x oi' X li^ is Thus, f() is the function defined below. h. i 2 3 4 f<6 ) Let Y l = X 1 + X 2i and Y 2i = X 1 X 2i ; thus from (2.2.3) (Y li ,Y 2i ,Y 1 .,Y 2j ,6 i ,6 j ) = (Y li ,Y 2 .,Y lj ,Y 2j ,f(6 i ),f(6 j )). Applying the definition of a., and b in (2.2.1) (or usinj Table 2.2) to the above, it follows that and thus (a, ,b. ) 2 (a. ,b. ) P(a ij = l.^Ij = 1> P < a ij l.bij = "I)
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29 and P(a i;j = l,b i:j = 1) = P( aij = l.b.j = 1) (2.2.4) Now (CD) = iI E(a. .b. ) where E(a. j b. j ) = (l)P(a..b.. = 1) + (l)P(a..b ij = 1 ) = PCa.j = l, bi = 1) + P(a j = l,b.. = 1) P (a j = l, bij = 1) P( j = l, bij = 1) Applying (2.2.4) to the above, it follows that E(a b.j.) = and thus E(CD) = 0. Note, that under H, (X 11 ,X 2 i x ij x 2j 6 l 6 j ) = (X lj X 2j x ii x 2i 6 j 6 i ) and thus < Y li' Y 2i> Y lj' Y 2j> 6 i>V = (Y lj' Y 2j' Y li' Y 2i 5 j' 6 i) (2.2.5) .pplying the definition of a, and t^. as before, it follows (a ij> b ij> ^ a ij'b ij) and also
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30 and Now Var(CD) = [P(a tJ = l.b^ = 1) = PCa.j = l, bij = 1) PCa.j l, blJ =1) = PCa.j = l, bij = 1) Â— ] Var( T a. b. ) 2 [ ~" ] I I Cov(a. .b. a. ,b. ) The three possible cases to consider for the covariance arc 1) 1*1 ', j/j', i
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31 Case 3) Exactly two of the four subscripts i A l,1 : < a ij = L > a ik = !> A l,l : U ij = _1 a ik = 1} A l,l : {a ij = "! a ik = H and similarly define the events B, B, _, B, and B_^ 1 Using this notation, E ( a. ^ b^ a .ub ik ) can be written E(a ij b ij a ik b ik ) = I I I I (D k+Â£+m+n P(A ,B m > k=0 Â£=0 m=0 n=0 (1) ,(1)* (1) ,(l) n (2.2.6) Table 2.3 describes the events A and B (1) ,(l) Â£ (D m ,(l) n in more detail and the restrictions placed on the 6's. Now, to simplify the probabilities in (2.2.6). Note, under H o (X li' X 2i ,X lj ,X 2j ,X lk' X 2k' 6 i' 6 j 6 k } = ^ 2 .,X li ,X 2 .,X 1 .,X 2k ,X lk ,f(6.),f(6 j ),f(6 k )
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32 Applying the transformations Y li = X li + X 2i and Y 2i X li X 2i it follows that (Y 1 .,Y 2i ,Y lj ,Y 2j ,Y llc ,Y 2k> 6 i ,6 j ,6 k ) (Y li ,Y 2i ,Y lr Y 2j ,Y lk ,Y 2k) f(5.),f(6 j ),f(5 k )) Now, applying the definitions of a. and b in (2.2.1) (or using Table 2.2), notice that if b = 1 (i.e., Y 2i Y f(5 i )e(l,3) and f(5.)e(l,2) which would yield b.. = 1. Using similar arguments, it follows (a ..,a.,,b..,b.,) = (a..,a.,,b..,b.,) ij ik' ij ik' v ij ik' ij ik' and thus PCA 1Â§1 ,B 1(1 ) P(A 1}1 .B.! x ) P(A P(A 1 ,1 B l ,1 1,1 '1, 1 P(A 1,1 B P(A 1,1 "l,l P ( A 1,1 B l, P(A P(A 1,1 B 1,1 1,1 D l, = P(A lfl .B^.p ) = P
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33 CO CM co CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCM >l>l><>H>4>l>l>l>l>l>>H> i >l>J>J VVVV/\VV/\.V/\/\VAA/\/\ CM CM CM CM CM CM CM CM CM CM CMCMCMCMCMCMCMCMCMCMCMCMCMCMCMCS >l>H>l>l>l>l>H>l><>(>H>l>J>l>l>H VVV/\VVAV/\VA/VV/\AA CMCMCMCMCMCMCMCM CM CM CM CM CM CM CM CM >H>H>I>H>I>1>I>H>I>I>H>I>I>J>JJM >"'>^>'>'><>'>lS><>l>>t>H>lt>'>l>. VVAVVAVVAAVAAVAA Â•rlilflH>Hi)iHHHHiliHHrtl>l>l>l>>H>l>l>l>l>l>lt)l>l>lfM ,r ) TÂ— J i~l **""> Â•Â—) *<~) Â•!") "i ) 1Â—) IÂ— ) Â•!Â— ) r) rn Â•!Â—) tÂ— j Â•?Â— j >I>I>1>I>H>I>H>I>I>H>I>I>I>H><>( VAVVVAAAVVV/\AAVA T(il'H>ilil>rliliHH Â— Â— I Â— I 1 I I H 1 1 I I I I H H . Â•> I I 11 ._, ^ m iHi^H^Moa i i q ~< CO CO l^n^HpapQCQqa I h h PQ C3 Â—i pa ^npopQ i^^^H^pa  i^^i Â— Â• 1 Â— Â— < Â— I I  ,H 4 _4  I T Â— T
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34 Similarly, under H (X li X 2i' X lj' X 2j' X lk' X 2k' 6 i' <5 j' 6 k ) = ( x ii X 2i' X lk' X 2k' X lj' X 2j' 6 i' 6 k' 5 j ) and applying the definition of a., and b.. in (2.2.1) it follows that (a lj ,a lk ,b ij ,b ik ) = (a ik ,a. j) b ik ,b. j ) This yields that P(A 1,1 > B 1,1> = P(A 1,1 B l,1> P(A 1,1 > B 1,1> P < A 1,1 B l,l> P(A 1,1 B 1 1> = P < A 1,1 B l,l> PC*!,! .B. 1(1 ) = HA. lfl .B.^.p Thus, ^ ( a Â• b^ a k b k ) can be reduced to a sum of six tern instead of the original sixteen; i.e. E(a i j b ij a ik b ik ) = I I I I (D k+A+m+n P(A ,B ) k=0 Â£=0 m=0 n=0 (1) ,(1)* (1) ,(1) = 2P(A lfl ,B lfl ) + 2P(A_ 1) 1 Bl)1 ) + 4P(A 1) 1 ,3.^) 2P(A 1,1 B l,l> 2 P(A 1,1 B 1.1> 4P(A 1.1 B 1.1^ = Y Note, the subscripts are arbitrary; thus E(a..b i;j a ik b ik ) = E(a..b. j a kj b k .) = E ( a b a k b fc )
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35 and therefore combining the results from case 1, 2 and 3, it follows that Var(CD) = "I n(n 1) 4(n 2) a + Â— Â— rr Y n2 21 Y) n(n 1) As seen in Lemma 2.2.1, the variance of CD depends on the underlying distribution of ( x ii X 21^ and possibly C. Therefore, CD is not distribution free under H 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 H (x li ,x 2i ,c 1 ) (x 2i ,x li ,c i )
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36 and thus where (X 11 ,X 2i ,5 i ) = (X 2i ,X li ,f(6 i )) (2.3.1) X^ = min( X^ ^ C^ ) X 2i = min( X 2i C.^ ) 6 ^ is the pair type (i.e. 6. = 1,2,3 or 4) and f ( X 2i > 5 i ) 1 y ( x i2 X 22 ^ 2 ^ 2 *" X l n X 2n 5 n ^ n ^ { (Xj j ,X 21 5j ) 1 (X. 2 ,X 22 6 2 ) 2,...,(X, n> X 2n ,6 n ) n} (2.3.2)
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37 where k and k' are arbitrary elements of K. Therefore, under H given {(x 11 ,x 21 ,6 1 ),(x 12 ,x 22 ,6 2 ),...,(x ln ,x 2n ,6 n )}, the 2 n possible vectors {(x 11 ,x 21 ,6 1 ) k l,(x 12 ,x 22 ,6 2 ) k 2,...,(x ln ,x 2n ,6 n ) k n} are equally likely values for {(X 11 ,X 21 ,6 1 ),(X 12 ,X 22 ,<5 2 ),...,(X ln ,X 2n ,6 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 2 n equally likely possible values of CD (not necessarily unique) calculated from {(x 11 ,x 21 ,6 1 ) k l,(x 12 ,x 22 ,6 2 ) k 2,...,(x ln ,x 2n ( 5 n ) k n} Note, since the sample observed is censored, the 2 n vectors {x 11 ,x 21 ,6 1 ) k l,(x 12 ,x 22 ,6 2 ) k 2,...,(x ln ,x 2n ,5 n ) k n} are not necessarily unique. If a pair is a type 4 (i.e., both X^ and X 2 Â• were censored), then (x Â• x 2 Â• 6 .. ) = (x 1 ,x 2 6 ) In fact, there are only 2^ nn 4' unique vectors (n^ = number of type 4 pairs), since P(X,. = X 2 ) = if ( x ii^2i^ *s not a c yP e ^ pair under assumption A2. As a result, the permutation test, in effect, discards the type 4 pairs (since a i jb 1 = if the i or j pair is a type 4) and treats the sample as if it were of size nn, with no type 4 pairs occurring. With regards to the transformed variables (Y,.,Y 2 ) i = 1,2, ...,n, the permutation test can be viewed in the following way. Consider the transformations Y, = X, + XÂ„.
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38 and Y 2i = X 1JL X 2i Applying these to (2.3.1) and (2.3.2), we see that under H ^ Y li' Y 2i' 6 i ) = ( Y liY 2i' f ( 5 i>> and similarly k k k {(Y n ,Y 21 ,5.) 1 ,(Y 12 ,Y 22 ,6 2 ) %...,(Y ln ,Y 2n ,6 n ) n } i i A 1 9 i = { ( Y, Y, ,6 ) ( Y, 9 Y 99 6 9 ) ,...,(Y, ,Y 9 11 x 21 ,u l 12' "22 u 2 ln l 2n ,0 n) } where (Y li ,Y 2 .,6 i ) *< (Y 1 .,Y 2 .,6 i ) if k = 1 (Y 11 ,Y 211 f(6 )) if k = and k and k' are arbitrary elements of K. That is, under H Q given { (y : 1 ,y 2 1 6 x ) ( y 1 2 y 2 2 6 2 ) (y x n y 2n 6 n ) } the 2 n possible vectors k l k 2 k n Hy n ,y 21 ,6 A ) ,(y 12 ,y 22 ,6 2 ) Â• Â• Â• (yi n y2n' 6 n ) } are equally likely values for {(Y u ,Y 21 1 ) ,(Y 12 ,Y 22 ,<5 2 ) ... .(Y ln ,Y 2n ,6 n )} To perform the permutation test, the measurements (x,.,x 2i ,5.) i = 1,2,. ..,n are observed and the corresponding value of CD is calculated. Under H there are 2 n equally likely transformed vectors for {(Y 11 ,Y 21 ,6 1 ),(Y 12 ,Y 22 ,6 2 ),...,(Y ln ,Y 2n ,6 n )} The CD
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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 H 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 H Q where CD VarCCD)]^ Â— > N(0,1) as n + Var(CD) = n(nl) 4(n2) n(nl) Proof Note that CD is a Ustatistic with symmetric kernel h ($i?j) = ^ a i i b li ^ Thus b y applying Theorem 3.3.13 of
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40 Randies and Wolfe (1979), it follows that d 4? 1 CD Â— N(0, ) as n where Note that ?1 = E[h(X i ,X j )h(X. ,X k )] E[a ij b ij a ik b ik ] Y Â• 4y 2 4(n2) a + Â—, r^r Y Â— 1 as n > oo n(nl) u n(nl) therefore after applying Slutsky's Theorem (Theorem 3.2.8, Randies and Wolfe, 1979) CD [Var(CD)] / 2 > N(0,1) as n D Corollary 2.4.2 : If Var(CD) is any consistent estimator of Var(CD), then CD 1/ Var(CD)l / 2 Â— >Â• N(0,1) as n > Proof : This follows directly from Theorem 2.4.1 and Slutsky's Theorem. *Â— Next, we consider the problem of finding a consistent .stimator for Var(CD). There are many consistent estimators
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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 H the following are consistent estimators of Var(CD): 1) Var l (CD) = n 1 { T7^ 1,1 I ^ijk*' 3[ 3 J Ki
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42 = 4 <77rr I I I A. .. B, .. ijk ilk 3 J Ki
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Now 43 n(Var 3 (CD) Var 2 (CD)) L ^T^T I I Ca b ) 2 ] (CD) : (n1) f" Ki (n1) D 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) = Â— U where U is a Ustatistic which estimates y 1 n n n Thus, Var,(CD) is estimating the asymptotic variance of CD. Var 2 (CD) is also estimating the asymptotic variance of CD, but in a slightly different manner. Recall, from basic Ustatistic theory that y is the variance of a conditional expectation (Randies and Wolfe, 1979, p. 79) (i.e., Y = VarUajbj)*] where (ajbj)* = E [ a j 2 b l 2  ( Y ] { Y 2 l ) ] ) Thus, in Var 2 (CD), for each ( Y iiY 2: j), the conditional expectation is estimated using all the other (Y, .,Y~.)'s, j*i and then the variance of all these quantities is calculated. That is, l Var,(CD) = V {(a .b ) CD} n i = I where (a iV* Â— .J. E[a i j b i j l^li^2i )] n1 j*i J J In contrast to Var^CD) and Var 2 (CD), VarJCD) is estimating the exact variance of CD (2.2.2) derived in
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44 Section 2.2. It is using an estimator of y from VarjCCD) and estimating a with a difference of two Us t a t is t ics which is estimating Again, although under H E(a. .b. .) = 0, the sample estimate for E(a..b.. ) (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 CD that A 1/ Var(CD)] 2 be used as an approximation for CD [Var(CD)] /2 Thus, for an a level test using the
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45 asymptotic distribution, the null hypothesis would be CD rejected if j [VarCCD)]^ > Z where Z /0 is the value in /Â•> a/2 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 n,+n+no. This is understandable since X, = X^.; = C. and thus they supply no information about the scale of X. relative to X2 1 In the asymptotic test, if one estimated the variance in (2.2.2) by estimating a and y with their sample 2 2 quantities (for example, a = Â£ Â£ ( a b ) ), it n(nl) Ki
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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+n,), the value for a and y would be larger than the value had type 4 pairs been included (since type 4 pairs only contribute O'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
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47 performed to achieve a legitimate distribution free alevel test Comment 3 It is unclear how the CD statistic would be affected if the marginal distributions of X,, and X~i 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 X ) or even if this is true, that CD does not perform well in these instances. Chapter 5 will investigate this problem in further detail.
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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 nonparame t ri cally distributionfree (conditional on N, = n, and N = n^+no) 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. F( xÂ„ p 1 ) where F(u,v) = F(v,u) for every (u,v) in R Tests are developed for both of the following alternatives to the null hypothesis of bivariate symmetry: 48
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49 Case 1 \i = y 2 known, H Q : a i = 02 and K a : o, < 02 That is, the marginal distributions have the same known location parameter but, under H X~ has a larger scale t parameter than X,,. A possible contour of an absolutely continuous distribution of this form was given in Figure 2. Case 2 y, = y ~ unknown, H Q : 0^ = o"2 and H : a ^ < 02 Here, the marginal distributions have the same unknown location parameter but, under H X~ 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 onesided 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 nonparamet rically distributionfree conditional on N. = n, and N = n2+n^ In both cases, the test statistics can be viewed as a linear combination of two independent test statistics T n and T n where T is a statistic based only 1 c 1 on the n, uncensored observations, while T will be a statistic based on the n = n^+n, type 2 and 3 censored observations. The conditioning of the random variables N,
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50 and N 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, r^ and n 7 Thus, the test statistics will be written as TÂ„ J n l n ( and TM n n (for Section 3.2 and 3.3, respectively) which 1 c imply conditioning on N, = n, and N = n = n 2 +n, Section 3.5 will consider the asymptotic distribution of each test statistic. 3.2 \i = y j > Known This section will begin by introducing the notation necessary for the statistic T n n designed for the 1 c alternative in case 1. Recall, the sample consists of ^ X li' X 2i^ i=l>2,...,n where X,. = min(X,.,C.) and i X 2i = min ( X^^ C^ ) These pairs were classified into foui pair types. They were the following: Pair Type 1 2 3 4 Pes crip t ion x li c i x li >c i x 2i c i x 2i >c i Number of Pairs in the Sample where n = n, + n + ti q + n,.
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51 For convenience and without loss of generality, let the type 1 pairs occupy positions 1 to ni in the sample (i.e., { (X n ,X 21 ) (X l2 ,X 22 ) (X ln ,X 2n )} ) in random order. Similarly, the type 2 and type 3 pairs will be assumed to occupy positions n, +1 n, +2 n,+n in random order. Lastly, the type 4 pairs occupy positions n,+n +l,n,+n +2 n What is meant by random order, is 1 c 1 c that the exchangeability property still holds within the n^ type 1 pairs, within the iin+n, 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^ ( n 2 +nj ) equally likely arrangements when the positions and numbers of the pair types are fixed. Therefore, it follows, that each of the ni! arrangements of the n, uncensored pairs is equally
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52 likely and that the exchangeability property still holds within the type 1 uncensored pairs. Similar arguements for the (n2+no) type 2 or 3 pairs and the n, type 4 pairs hold. The following notation will be used in the statistic T a statistic which is based on the ti, type 1 pairs. Define a variable Z. to be Z ,. = X 2i X 1 p for 11,2, where u is the known and common location parameter. Let R^ be the absolute rank of Z. for i=l 2 n, that is, the rank of Z among { Z, Zo defined as ZÂ„ } and let f. be n l x ? = V(Z ) = 1 if Z > if Z < Note, the variable Z. is defined only for the uncensored pairs. The statistic T is then t = v t. r, n l i=l x x the Wilcoxon signed rank statistic computed on the Z.'s. Notation will now be introduced for the statistic T n 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 Q, to be the rank of C= among {C n 1 + l' C n 1 +2""' C n 1 + n c > an<
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53 1 if the j pair is a type 2 pair if the j c pair is a type 3 pair for j = n, +1 ,n, +2 ,ni +n The statistic T is defined Â•r 1 *"! l TU c n, + n 1 c I Y. Q. c j=n 1 +l J J = l ranks of the C's for the type 2 p; A brief explanation of the logic behind the test statistic will be presented. For the test statistic T if X~ has a larger scale parameter than X, (i.e., under H ), then X 21 '1 "a X,. y should be positive and large. Thus, the test statistic T would be large. In contrast, if X2 and X^ have the same scale parameter (i.e., under H ), then [ X2 ^ u ~ j X y would be positive approximately as many times as negative with no pattern present in the magnitudes of j X2 ^ y   x ii ~ V\ Thus the test statistic would be comparatively less. For the test statistic T n if H is true, there should n c 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 T would be large. In c contrast, if H is true, the number of type 2 pairs should
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54 21 11 Figure 5. Contour of an Absolutely Continuous Distribution That Has Equal Marginal Locations and Unequal Marginal Scales with Censoring Present.
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55 not dominate n and the test statistic T n should not be c c unusually large. Now we establish certain distributional properties for T and T n n l n c Lemma 3.2.1 : Conditional on n 1 T n has the same null distribution as the Wilcoxon signed rank statistic. Proof : First it will be shown that conditioning on the n^ type 1 pairs does not affect the exchangeability property (i.e., (X li' X 2i' C i } (X 2i ,X li ,C i )) Stl11 holds Let W.= U^.X^.C.) and W*e (X 2i ,xj.,C.) and G tT (t) =P(X 14 < t.,X .< t ,C.< t,) = E[I(W < t)] where I(W < t) = i if x 1 < t 1 ,x 2 .< t 2 ,c.< t 3 otherwise Now, under H Q for the entire sample, we have it d (x li ,x 2i ,c i ) = (x 2 .,x li ,c i ) and applying an apropriate function (and Theorem 1.3.7 of Randies and Wolfe, 1979) thus I(W < t)I(5.= 1) = I(W* < t)I(f(6.) = 1) Taking expectations, it follows that E[I(W.< t)I(6.= 1)] = E[I(W*< t)I(f(5.) = 1)] Now, recalling that 5 i = 1 iff f(5 i ) = 1; thus E[I(5 i = 1)] = E[I(f(5.) = 1)] and it follows that
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56 E[I(W < t)I(5 = 1)] E[I(W.< t)I(f(6.) = 1)] E[I(5 1)] E[I(f(5.) = 1)] This shows that the c d f of V^ given it is a type 1 pair is equal to the c d f of W. given it is a type 1 pair and thus the exchangeability property holds within the type 1 pairs Now, by defining a function f,(a,b,c) = min(b,c) y min(a,c) u and applying Theorem 1.3.7 (Randies and Wolfe, 1979, page 16) it follows Z = l X 21 ~ v \ ~ l X ll ~ y I min(X2^,C) y min(Xi^,C) y  d ii' i = min(Xj,,C) y min(X2^ ,C) y = l X ll ~ v \ ~ l X 21 u \ = ~ Z and thus by Theorem 1.3.2 (Randies 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 (Randies and Wolfe, 1979, page so). a Lemma 3.2.2 : Under H the following results hold. a) Conditional on the fact the pair is type 2 or 3, the random variables yand C. are independent. b) Conditonal on n T has the same null distribution as the Wilcoxon signed rank statistic.
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57 Proof First, it will be shown that conditioning on the n type 2 and 3 pairs does not affect the exchangeability property. Define W. W.,G w (t) and I(W. < t) as in Lemma 3.3.1. Now under H for the entire sample, we have ii d i i (x li ,x 2i ,c i ) (x 2i ,x li ,c i ) and applying an appropriate function (and Theorem 1.3.7 of Randies and Wolfe, 1979) d I(W i < t)I(5 i e(2,3)) = I(W. < t)I(f (5 i )e(2,3)). Taking expectations, it follows that E{l(W i Cj.Cj < c) = PCX^ < c j x ij > c j c j < c) = P( Yj = O.Cj < c) Noting that, P(Yj = l.Cj < c) + P( Y j = 0,Cj < c) = P(Cj < c)
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58 and thus 2P( Y;j = l.Cj < c) = P(Cj < c) or that P(Yj = l.Cj < c) = 1 / 2 P(C :j < c) = P( Yj = OP(Cj < c) and thus we see that y a anc C. are independent. To prove part b), let y = (y ,y 2 Y n +n ) 11 1 c and Q = (Q ,, ,0 10 ,...,Q )Â• By Theorem 2.3.3 (Randies j n,+l n,+2 n. +n 11 1 c and Wolfe, 1979, page 37), Q is uniformly distributed over R n where c R = {q : q is a permutation of the integers 1,2, ...,n } Now, let q be any arbitrary element of R n and let g be any c arbitrary n vector of 0's and l's. Thus, P(Y = g,Q $) P(x Â§ )P(Q 3 ) (by part a) and P( Y = Â§ )P(Q ^ ) jjx Â— 9 C 2 c which proves part b). Q By Lemmas 3.2.1 and 3.2.2, T n and T n are 1 c nonparamet rically distributionfree conditional on n^ and n respectively.
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59 Lemma 3.2.3 : Under H Q the following results hold a) Conditional on n 1 E(T n ) = n^nj + D/4 and Var(T n ) = n 1 (n 1 +l)(2n 1 +l)/24 b) Conditional on n c E(T n ) = n c (n c +l)/4 and Var(T n ) = n c (n c +l)(2n c +l)/24. c c) Conditional on n, and n T and T n are 1 c independent 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 T and T_ are based on sets of mutually independent n l n c observations With these preliminary results out of the way, the test statistic TÂ„ Â„ can now be defined by n l' n c T = L,T + L 9 TÂ„ n l' n c l n l 2 n c n. n +n = L I f R + L I y Q 1 i1 X X L j=n 1 +l J J where L, and L, are finite constants
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60 Theorem 3.2.4 : Under H Q a > E(T n,,n ) = L l E < T n> + L 2 E < T n > l c 1 c = (L^Ci^ + l) + L 2 n c (n c + l))/4 b) Var(T n >n ) = (Lfn 1 (n 1 +l)(2n 1 +l) + Ln c (n c +l)(2n c +l))/24 c) T n n is symmetrically distributed about E(T ) and d) for fixed constants L, and L T 1 2' n lf n c nonparametrically distributionfree. is 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, T n and T n are symmetric about E(T ) and n l n c n i E(T ), respectively. Since T and TÂ„ are independent c n l n c (conditional on N, = n, and N = n ), the symmetry of T l 1 c c J n,,n follows To prove part d), note that P(T n lf n c = k ) = p (h T ni + L 2 T n c = k ) = I P^T^k kc  L 2 T = k c )P(L 2 T n = k ) = {k c } J, P(LlTn l = k k c )P(L 2 T n = k c> {k c } 1 c where {k } = set of all possible values of L T
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61 Now using the nonparamet r ically distributionfree property of T and T n established in Lemmas 3.2.1 and 3.2.2, it follows that for fixed L, and L L,TÂ„ and L TÂ„ are also I l 1 n i I n 1 c nonparamet rically distributionfree. D l' u c The conditional null hypothesis distribution of T 1 ( can be obtained using the fact it is a convolution of two Wilcoxon signed rank test statistics' null distributions. Thus, for fixed Li and L2 the distribution can be tabled. Tables in the Appendix 1 give the critical values for T with L, = 1 and L2 = 1 for n, = 1,2, ...,15 and n c = 1,2,. ..,10 at the .01, .025, .05 and .10 levels of significance. The actual alevels are also reported for the cutoffs given. The decision rule for the test is to reject H 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 H : o y Oj) 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 H for larger n, and n can be based on the asymptotic distribution of T which will be presented in n l c Section 3.4.
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62 3.3 u, = Uo> 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 n 1 ,n c in these settings. The modification will be to estimate the common location parameter using a "smoothed" median estimator based on the productlimit (Kaplan Meier) estimate of the survival distribution (Kaplan and Meier, 1958). This estimated location parameter M, replaces \i in the previous definitions. That is, define the variable Z i to be *2i M X 1:L M i = l ,2 ,n The definitons of T R y Q T T and T 1 i j j n^ n c ll ji u c remain unchanged. In this section, the statistic will be denoted by TM n to identify the fact the location parameter was estimated with a "smoothed" median estimator based on the productlimit 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 i 's, a condition which no longer exists. Also, in 3.2.3 c) T and T were based on sets of 1 c
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63 mutually independent observations. This is not the case in the current context. First, we introduce the "smoothed" median estimator and the productlimit estimate of the survival distribution. Let (*(!). *( 2 )i'" Y (2n 1 +n 2 +n 3 ) ) ^present the ordered uncensored observations. (This ignores the fact the original observations were bivariate pairs, and considers only the 2n,+n2+no uncensored observations, i.e., 2n, components belonging to type 1 pairs, the nÂ£ uncensored components of type 2 pairs and the n, uncensored components of type 3 pairs.) That is, X = Y/, \ if X. is uncensored and Xi ^ 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 (j)> i = l 2 2ni +ti2+rii be the number of censored and uncensored observations which are greater than or equal to Y^). Thus, 2 n n (i) = I I I(X ii Y (i) } Where K J i = lj=l J v I is the indicator function which takes on a value of one when the argument is true and zero otherwise. The productlimit estimate of the survival distribution is defined as
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64 S(t) = if t < Y (1) n <*,, ,D/n k = l (k) (k) if Y j=l ,2 2n 1 +n+n 1 if t > Y (2n 1 +n 2 +n ) (^te, that Y (1) i8 the smallest uncensored observatiQn ^ Y (2n 1 +n 2 +n 3 ) i the largest uncensored observation.) The definition given here assu.es 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 = < m 1 + SCm^ 0.5 if njj = m 2 where and m l = n>in{Y (i) : S(Y ()) > V 2 } '2 = raax ^ (i) : s(Y ()) < V 2 } A brief explanation of this estimator follows. The productlimit estimate of the survival function, SCt). is a right continuous step f unctlon which has jumps ^
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65 the uncensored observations. An intuitive estimate for the common median is the value of /.> such that S(Y/jO = V^. 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 n^ If t he Y/jn exists, such that S(Y/iJ = V2 then m, = n^ 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 : ft ft ft Let ( Y/ \ Y/ 2 \ Â• Â• Â• Y/ 2 n ) ) represent the ordered 2n ft ft ft observations where Y/ < Y/j) < Â•Â•Â• < ^(2 )' 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 Aft ft (Y (l)' Y (2)""' Y (2n) ) and (I (1)' I (2)'' I (2n) ) where
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66 (j) 1 if Y,., is censored (j ) otherwise in the fact that 2 n n, n = I I KX. .> Y, ..) = 2n + 1 (rank of Y (i) in ( Y ( { j Y ( 2 j Y ( 2n) ) In addition, S(t) can be expressed as S(t) = n Y, ,x max Y, : 1,.,= 1 L (i ) d ) 2n j "(j) 2n j + 1 J otherwise L (J) Thus, S(t) is a symmetric function with respect to the sample observations and therefore M, being a function of S(t), is also. Lemma 3.3.2: Conditional on n, T has the same null 1 x n distribution as the Wilcoxon signed rank, statistic. Proof : Let = (
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67 n, (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, i i d (x 11 ,x 21 ,c i ) = (x 2 .,x li ,c i ) and thus letting X,. = min(X, ,C, ) and X 2 = min( X 2i C ) it follows that (X li' X 2i> = (X 2i' X li ) Â• for i=l,2,...,ni and these pairs are also exchangeable. Now, let k = be an operator such that (x u x 2i ) (X li X 2i } if k = 1 (X 2i X li ) if k Thus, under H Q and using the exchangeability property, it follows (x n x 21 ), (x 12 x 22 ),..., (x ln ^, x 2n ^)} = l (X lr,' X 2r, ) '' (X lr X 2rJ 2 >'".^ lr X 2r ) n l} 1 1 2 2 n l n l (3.3.1) Recalling that M = the estimate of the location parameter, is a symmetric function of the components of the observation
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68 pairs from Lemma 3.3.1 and defining a function f i( y i> y 2 } = 1*1 m y 2 M l = z it follows from applying this function to (3.3.1) that Zj, z 2 ,..., z ( r r *"Â• j1 2 i where (Z ) i = {(z r )\ (z r Z 2 ,..., J if Z. < J R. = absolute rank of Z., i.e., rank of Z. amonj C Z 1 I Zo,..., z n 1} for j =1 2 n, Applying this function to (3.3.2) it follows that 1 2. n 1 l n = (f ,..., ,R ,R ,...,R ) r l r 2 r ni r l r 2 r n x {(% ) l ( ) 2 ,...,(^ r ) n l, R r R r ,...,R, 1 2 n, 1 2
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where ( ) 1= 69 if k.= 1 1
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70 Claim 1 : yis independent of (x.,c.). This follows by noting that under H and using the exchangeability property of type 2 and 3 pairs (as was shown in Lemma 3.2.2) that P{ Yi l(x i ,c 1 )} = P{X 1 .=x.,X 2i =c i  (x. Ci )} = P{X li =c i ,X 2i =x i  (x i ,c i )} = P{ Yi =0 (x i ,c i )} Since P{y i = l  (x^^)} + P{ Yi =  ( x c ) } = 1, Claim 1 follows. Now define y = (y n + 1 Y n +2' ,,, ^n +n )Â• 1 1 1 c Claim 2: y is a vector of n i.i.d. Bernoulli random ~ c variables which are independent of { (x ni + l c ni + l^ (x n 1 +2' c n 1 + 2 ) ( X n 1 +n ( C n 1 +n (; ) } This follows from Claim 1 and the fact that {(x n 1 +l> c n 1 +l ) (x n 1 +2> c n 1 +2 ) '' (x n 1 +n c > c n 1 +n c )} are d Claim 3 : y ^ s independent of {(x n x + l c nj + l' ,(x n 1 +2' c n 1 + 2 ) ( X n 1 +n (; C n 1 +n ( ) } xÂ„ and xÂ„ where ~n ] n 4 ~n : = {(x ll' x 12)'( x 12' x 22).".( x ln 1 x 2n 1 )} and x n, = ^ c n,+n +l' c n,+n +1 >Â•Â•Â•.< c n c n } } t 1 c 1 c (i.e., the observed totally uncensored type 1 pairs and
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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 x x X/ \ and C/ \ where x (n c ) denotes the observed ordered uncensored members of type 2 and 3 pairs and c/ \ denotes 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 X (nÂ„) and c CO are functions of ^ (x n 1 +l' c n 1 +l ) (x n 1 +2' c n 1 +2 ) "' (x n 1 +n c c n 1 +n c ) > onl y* Claim 5 : y is independent of x n > x n > x ( n ) an< i Â£( n \ where y Â„ = {yÂ„ ,yÂ„ Ic w c (1) Y c (2) Yc (n ) } C ^) 1S the 1 th element of c / \ and y is the y which corresponds to the pair of which c (i) was a member This claim follows from Theorem 1.3.5 of Randies and Wolfe (1979) and since y is a fixed permutation of y. Note that the i.i.d. property still holds for the y 's. (i)
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72 Claim 6: Given x_ xÂ„ x/ \ and c l l x (n ) and c (n )' T n, is no lon g er a random variable; that is, the value of T is n l observed This follows directly from the definition of T n l Claim 7 : Note, that n.+ n n, + n 1 c 1 c jttj+1 J J jt^+1 c (j) which shows that T_ is a function of y and is n c c independent of x_ xÂ„ x/ \ and c, \. ~ n \ ~ n 4 ^p ~kn c ; Thus, T n has a null distribution equivalent to the Wilcoxon c signed rank null distribution and is independent of T n l which is a function of x n x Q x/ n s and C/_ > only. ~1 ~4 ~ c c With the proof of Lemma 3.3.3, Theorem 3.2.4 is valid for the modified test statistic TMÂ„ Â„ That is, under H n p n c o and conditional on n, and n TM has the same 1 c n l n c distributional properties stated in Theorem 3.2.4 for T n l n c and the tables in the appendix are valid.
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73 3.4 Asymptotic Properties In this section, the asymptotic distribution of the test statistic T Â„ (and TM n ) under H will be n 1 ,n c n l' n c established. The asymptotic normality of the test statistic will be presented first, conditional on Nj = n 1 and N Â£ = n Q 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 > N ^ + N c = (n number of type 4 pairs) also. The asymptotics will be presented for the test T only. In the previous section, it was n 1 ,n c shown that under H and conditional on N^ = n^ and N c = n c T and TM have the same null distribution; that is n l' n c n l' n c a T = TM Therefore, they have the same cumulative n 1 ,n c n ls n c 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 1 = n^ and N c = n c under H Q T E(T ) n l' n c n l' n c _jl_ a(T n n > n l' n c N(0,1) as n, > and n + 1 c where
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74 E(T n ) = (L^.Cn.+l) + L,n fn +l))/4 l'"c '1 U P U 1 2 ll c^ u c and ''V'c' = IV(T niinc )]^ = [(Lin 1 (n 1 +l)(2n 1 + l) + L 2 n c ( n c + l ) ( 2n c +l ) ) /24 ] ly 2 Proof : First, it will be shown that T and T have asymptotic n l n c normal distributions. Without loss of generality, it will be assumed that y = Note that n l I V.R. = i = l X 1 I Hx 2 .  Xll ) + I I Hx 2i   Xli  + x 2i  x n ) i = l Kini ) + ( 2 )(U 2>ni ) where and U l, ni = n7"J 1 H \ X 2l\ ~ l X lil> U 2, ni = ~11 *(X 21  X U  + 1 1, Ki n l v 1 n l 1 n l > n n i 1 + (n l )1/2 ^ U 2, ni E(U 2, ni ))
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75 Now notice, < U. < 1 and under H. E(U, Â„ ) = 1 n o i n^ P{(x 2i  x 1 ) > 0} = V 2 so that u 1 n V 2  < V 2 Therefore (n L ) 3/2 U l,n,'^ (n 1 ) , rr> as n Â• n. (n 1 ) 1 Thus f T n ni (n 1+ l)/4) and U^O^ V 2 ) have the same limiting distribution as n By Theorem 3.3.13 of Randies and Wolfe (1979), it is seen that (n^^U V^ has a limiting normal distribution with mean and variance r E, (provided E, > 0) whe re _2 5j = 2 {E[ Thus Note that, x f(x 21  x lt  + x 2k  x lk )] i/ 4 } = 1/3 T n. (n.+ l)/4 n 11 1 lh N(0,1) 3n, 1 h 3n, n (n + l)(2n.+ 1) i Y'2 P Â— + 1 24
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76 as n Therefore (after applying Slutsky's Theorem (Theorem 3.2.8, Randies and Wolfe, 1979) T n. (n.+ l)/4 n l l 1 d_ o(T ) n l N(0,1) S imilarly n +n 1 c n +n 1 c ., ^ j=n +1 n +l + ( 2 )< D 4, > where and U 4,n, '3,n, Â— I I C\ n. +1 < j 1/2 < D 4 ; n E(U 4,Â„ Â• c c
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11 Note that < U < 1, so lu, V?! < V? and thu.< (n ) 3/2 (n ) 3/2 ^(U, 1/2) < C n ^ 3 n & n ( n 1 as n >co U jk Thus T n (n + l)/4) and (n ) '\ U Vol hav n cc ; c v 4 n z; the same limiting distribution as n + <*> c Again applying Theorem 3.3.13 of Randies and Wolfe (1979), it is seen that (n ) Zfu. Vol has a limiting c v h n *Â•' c 2 normal distribution with mean and variance r Â£i 2 (provided r Â£i>0 ), where r 2 C x = 2 2 {E[( Yj ?(C. C k ) + Y k T(C k C. ) x Y jMCj c.) + Yi nc i C.))] \ } 2 2 {E( Y .1'(C.C k ) + Y k nC k C j ))(Y j T(C j C i ) + y^lC.C.)) V4} By the independence of yand C(Lemma 3.2.2), rV = 2 2 {P( Y ,= DP(C > C, C > C ) + P(y.= i)p(y.= i)p(c> c, o c.) J 1 j k' 1 j + P(y k = l)P(Yj= l)P(C fc > C C > C.)
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p(Y k = i)p(y i)P(c k > c j} c.> c.) V 4 } = 4 {Vo P ( c > C. C > C ) + Va p ( c > c, c> C ) j k' j i J k' 1 j + V 4 p(c k > C Cj> c.) +V 4 p(c k > C., C.> Cj) V 4 1 ^ r 1 1 r 1 1 1 ^ 1, 1 4{ 6 + 4^ 6 + 6 + 3 J 4 } 3 Thus T n (n + 1 )/4 n c c 2 J l 3n J + N(0,1) Noting that 1 ,v, 3n n (n + l)(2n + 1) h c c c n x /2 Â— 1 as n v c 24 and applying Slutsky's Theorem, it follows that T n (n + 1 )/4 n c c Â£ * N(0,1) o(T ) n c The conclusion of Theorem 3.4.1 then follows by writing T E(T ) (L.T + LÂ„T ) (L.E(T ) + LÂ„E(T )) n, ,n n, ,n In, 2 n In, 2 n lclc 1 c 1 c o(T ) n l' n c (L2 a 2 (T ) + 12 o 2 (T )) 7 2 1 n, 2 n 1 c L.a(T ) T E(T ) LÂ„a(T ) T E(T ) In, n, n, zn n n 1 1 1 c c c n ,n n o(T ) a(T ) n n n 1 c c
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79 applying Slutsky's Theorem and utilizing the fact T n and T are independent, conditional on Nj = n^ and N c = n^. n c Next, and most importantly, the unconditional asymptotic normality of T will be established as n 1 Â• 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 Anscorabe (1952). Lemma 3.4.2: Let {T Â„ } for n,=l,2,..., n =1,2,..., be 1 c any array of random variables satisfying conditions (i) and (ii). Condition (i): There exists a real number y, an array of positive numbers {co } and a distribution function F ( Â• ) such that lim P{T y < x a } = F(x) n.,n n.,n min(n, ,n )+* 1 c 1 c 1 c at every continuity point of F(). Condition (ii): Given any e > and n > there exists v = v ( e n ) and d = d ( e n ) such that whenever min(n,,n ) > v, then
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80 P{ T ,T 1 n 1 II l'c c l c Let {n } be an increasing sequence of positive integers tending to infinity and let {N.} and {N cr > be random variables taking on positive integer values such that N. p lr r + X. as r for some X. such that 0
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81 Lemma 3.4.3: Suppose that n v r l i f(x x ,..., x ) 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,, t,..., t ) is some function symmetric in its r arguments. This U is a Ustatistic of degree r with a symmetric kernel f(). Let {n } be an increasing sequence of positive integers tending to infinity as r > and {N } be a sequence of random variables taking on positive integer 1' X 2' Â•Â• X r 2 values with probability one. If E{f(X X ,...,X )} < V. 2 N r P lira Var(n 2 U ) = r f,> 0, and Â— Â• 1 then n 1 n lim P{(U N E(U N ))  r r where ( Â• ) represents the c.d.f. of a standard normal random variable Proof : This is Lemma 3.3.3 in Popovich (1983). a One comment is needed about this result. The proof of this lemma follows as a result of verifying that conditions C, and C2 of Anscombe (1952) are valid and applying Theorem 1 of Anscombe (1952). Condition C, 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
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82 C ? will be utilized in the proof of the major theorem of this section which follows. Theorem 3.4.4 : Under H V.N E(T N,,N > i Â£ + N(0,1) as n > 1 c Proof : The proof which follows is very similar to the proof of Theorem 3.3.4 in Popovich (1983). T E(T ) n n n n Let T = Â— the standardized n l' n c o(T ) n l' n c T statistic. Theorem 3.4.1 shows that { T } for n l n c n l n c 11,1,2, .... n =1,2,..., satisfies condition (i) of Lemma 3.4.2 with y = amd co =1. Note that from assumption n l n c A5, it can be seen that X. > 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 i > 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. T E(T ) n l n l Let T = the standardized T n n l a(T ) 1 n l statistic. In the proof of Theorem 3.4.1, it was shown that
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83 T has a limiting standard normal distribution by utilizing n l the Ustatistic representation of TÂ„ As a result of Lemma n l 3.4.3 and this Ustatistic representation, it follows that T satisfies condition C, of Anscombe (1952) (since T is n l 2 n l equivalent to a Ustatistic which satisfies condition C 2 of Anscombe (1952) as proved by Sproule (1974)). This condition C 2 can be stated as follows. Condition C?: for a given e > and n > there exists v. and d > such that for any n > v. 1**1 1 P{ T T < e for all n.' such that n J 1 n n 1 1 '1 < d^j} > 1 n Â• (3.4.1) Similarly, as a result of the Ustatistic representation of T (as shown in the proof of Theorem 3.4.1) and from Lemma c 3.4.3, it follows that T T E(T ) o(T ) n satisfies condition C 2 of Anscombe (1952). That is, for a given eÂ„> and r\ > there exists Vo and d 2 > such that for any n > v Â„ c 2 P{ T T < eÂ„ for all n' such that n' n l 2 c 1 c < dÂ„n } > I c 1 n (3.4.2)
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84 Consider T E(T ) 1 c 1 c 1' c a(T ) n l' n c L l 3(I n' L 2 o(T n } L_ < ( T ) + r__ < f T a(T ) n l' a c 1 a(T ) n l' n c In 1 n ; 2n v n 1 c Note that, (1) L, and L Â„ are functions only of N. and N and the In zn l c given L, and Lj constants. (2) ( L ; n ) 2 + (L ; n ) 2 = i. (3) There exists constants L, and L 9 such that L In and L 2n L. as n First, it will be shown that condition (ii) is satisfied for L,fT 1 + LÂ„fT ) = T 1 ^ n, ; 2 v n ; n.,n 1 c 1 c Let e > and n > be given and let v^, v ^ ^i ^2 satisify (3.4.1) and (3.4.2). Let v = max(v 1 v 2 ) and d = min(di, dj ) Â• Now, P{ T T < 2e for all n' n' such that n! n. < dn. ,i=l,c} l l i l > p ul;i<. i i T + LÂ„ T T n l 2 l n i 1 c < 2e for all n n' 1 c
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such that n! n < dn., i = 1 c } I 1 1 I l > P{L,t T I < e and LÂ„ I T ,T I < e for all n' n 1 1 I n n,i 2 I n n I 1 c 1 1 c c such that n '. n. < dn., i = l,c} I i i I i ic it I = P{L, T T < e for all n. 1 such that n J n. < dn.} 1 I n n I 1 I 1 l 1 1 + P{LÂ„ T T < e for all n 1 such that n' n < dn } 2inni c ice' c P{L.T T I < e or LÂ„ I T ,T I < e for all n' n' 1 n n' 2'n' n < lc 11 c c such that n .' n, < dn, and n' n < dn } II II 1 I c c I c > P{L, T T < e for all n,' such that n ,' n, < dn,} lln' n I 1 11 II 1 + P{LÂ„ T T < e for all n' such that n n < dn } 1 2  nn l c 'cc 1 c c c (3.4.3) Now using inequalities (3.4.1) and (3.4.2) and applying them i 1/ t 1/ to (3.4.3) with e = min{e (L ) 2 f e (L ) '2} then T T < 2s for all n' n 1 such that 'n.'.n' n,,n' l'c 1 c 1 c n i n i I < d ni 11, c} > (1 n) + (1 n) 1 1 2n
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8 6 Therefore T satisifies conditon (ii) of Lemma 3.4.2 so ni n. 1 ll c that Theorem 3.4.4 is valid for T = L,TÂ„ + L T 2 \' n i> n c _1 n : To see that the Theorem is valid if L, and L^ are replaced T I ky L n anc ^2 n respectively, consider, T. + L 2n A n, L,L + L ? T 1 z n c 1 n = (L In L l)T + (L 2n Lo)T, (3.4.4) X X Now, since T and T converge in distribution to standard X X normal random variables, T and T are "0(1) (Serfling, P (1980), pg. 8). Also, since L, Â— Â• L, and i P i it ii LÂ„ > LÂ„ as n (L ln L l ) and (L 2n L 2 ) are (1). Therefore (3.4.4) shows that (Lj n l)T* + (L^ L 2) T n is p ^ l ^ and thus Theorera 3.4.4 is valid. ^ 3 5 Comment s From the results in Sections 3.2, 3. 3, and 3.4, it is clear that a distributionfree test of the null hypothesis of bivariate symmetry versus the alternatives presented could be based on T (or TM ). For small samples, 1 > u c l l li c an exact test utilizing the distribution of T Q (and 1 c TM ) conditional on N, = n, and N = n coul n 1 ,n (; / 11 c c d be
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87 performed. For larger samples, the asymptotic normality of T (and TMÂ„ Â„ ) could be used. In Chapter Five, a n 1 ,n c ni ,n c 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 T Â„ conditional n l' n c on n was presented which had a null distribution c r equivalent to the Wilcoxon signed rank statistic. If instead of conditioning on n the statistic had been presented (with some minor adjustments) conditional on n2 and no, the statistic would then have had a null distribution equivalent to the Wilcoxon rank sum statistic. Conditioning on n and not on n2 and n^ was chosen because the observation of a particular n2 and Ti2 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.
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88 Comment 2 In Section 3.3, the KaplanMeier 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 KaplanMeier 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
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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 X.. and X,. This idea will be further investigated in Chapter Four.
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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 2vector, W of test statistics. The first statistic denoted TEÂ„ is a n l n c 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 2vector 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 n (Chapter Three, Section 3.3), (2) n l n c TM_ but using separate location estimates for X and X 21 and (3) the CD statistic (Chapter Two). It will be shown in Sections 4.2 and 4.3 that the distribution of W is 90
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91 not distribution free, even when H is true. Thus, if E is o rn the va r iancecova riance of W the quadratric form W t~ W ~ n ^ ~nrn~n will not be distributionfree. A consistent estimator of rn' fn w iH be introduced in Section 4.5 and a test based on the asymptotic distributionfree statistic W* fcw will be ~nrn~n 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 D.^ = Xj^ X 2 and R(  D I ) be the absolute rank of D i for i = l,2,...,n, that is, R( j D  is the rank, of j D^ I among (  D,   D*  Â•  D  ) Define t = (D jL ) = 1 if Z. > if Z. < Let TE_ and TE n be defined to be the following n l n c and TE = V *_, R( I D. I ) "l 11 X l! TE n = N 3 N 2 Notice that TEÂ„ is the Wilcoxon signed rank statistic n l applied to the n^ totally uncensored pairs. Popovich (1983) showed under H Q Nj is distributed as a Binomial random variable with parameters n and p = V2 P 2 (0) = V2 P(type 2 or 3 pair). With a slight modification from Popovich, the statistic TEÂ„ Â„ is n 1 ,n c
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where and 92 K lT ,(TEÂ„ ) + K 9 (TE n ) In v n 1 2n ^ n c TE TE n. (n, + l)/4 n 1 1 1 (n 1 (n 1 + DC211J + l)/24)'2 TE TE Cn ft and Ki and Ko are a sequence of random variables satisfying : ^ ^ln anc ^2n are on 'y functions of N, and N 2) there exists finite constants K, and K 2 such P P that K,Â„ K, and K Â„ + K as n In "2n This is slightly different from the statistic Popovich introduced, the difference being that he required K, = (I'Koj,) 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 > TM and CD are not n l' n c 1 C affected by their presence (other than in estimating the location parameter). It will be assumed that the sample is of size n = N, + N
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93 4.2 The WÂ„ Statistic Using T n ~n u l c The first statistic to be considered for pairing with TE is similar to the statistic TM presented in n l> n c n l' n c Section 3.3. The difference being, that instead of using a common estimate for \i as in Section 3.3, here we first consider using separate estimates which are denoted by Mj and M 2 where Mj is the KaplanMeier estimate for y based on the Xji's alone and similarly, M 2 is the KaplanMeier estimate for p based on the X 2i 's. Define T = V ?.R( X..M_ X Â• M ) n .^. l Il2i zi ill i'i and n, +n 1 c I Y. Q. l c JS +1 J J (Note these are similar to statistics defined in Section 3.2, with a slight modification of using the separate estimators M, and M 2 Â• ) Similarly, define T = T n. (n,+l)/4 n 11 n l n 1 (n 1 +l)(2n 1 +l)/24 and T n (n +l)/4 n c c n c n (n +l)(2n +D/24 c c c
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94 which would be the standardized versions of T and T in n l n c Section 3.3, had not M, and M 9 been used. We now defined K, TE + KÂ„ TE In n, 2n n 1 c In 1 L. T + L T Inn, 2n n 1 c where L, and L are a sequence of random variables in zn ^ satisfying: 1) Li and L Â„ are functions only of N, and N In / n J 1 c, and 2) there exists finite constants L, and L^ such that 'In Â•+ L, and L 2n Lo as n = Note, the statistic for scale L T Â„ + L ~ T Â„ is slightly 1 n n z n n Â•> 1 c different than the forms presented in Chapter Three. The difference is that here, the two components are standardized before taking the linear combination. Appropriate weighting variables can be chosen though, which make this form of the statistic equivalent to that presented in Chapter Three. The test statistic for the alternative of differences in location and/or scale is W^ \\~ w ln wnere \\ is tne variancecovariance matrix for W^ n The derivation of the asymptotic distribution of W^ fl W ln' wil1 be accomplished in a series of proofs. Theorem 4.2.1 shows that under H Q if the common location parameter was known and used instead of the estimators M, and M 2 the vector T = (TE_ ,T_ (u),TE* ,T* )" has a limiting multivariate n l n l n c n c normal distribution. Here T (y) denotes the statistic T n l n l
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95 when the value of p is used in its calculation. T (y) is n l the standardized T n which was presented in Section 3.2. Next, Theorem 4.2.3 will prove that using the estimates, M, and M2 for the common location parameter, does not affect the asymptotic results in Theorem 4.2.1. This is achieved by applying results about Ustatistics with estimated parameters (Randies, 1982) which are stated in Theorem 4.2.2. Finally, the asymptotic distribution of W^ fi w i n will be presented in Theorem 4.2.5. Theorem 4.2.1 : When \i is known and used in calculating T under H and conditional on N,=n, and N =n o lice' \ TE T = N(0,f T ) where ^.j = ((o^ a ) is the 4x4 variancecovar iance matrix for T with 0t (1,D 0t (2,2) = ffT (3,3) 0t (4,4) 1 > a^ 1 2 = 12P (as defined on page 99) a T < 3 4 > = (3/4)V 2 ai (l,3) = ^(2,3) = ai (l,4) = 0t (2,4) = Proof : Recall in the proof of Theorem 3.4.1., it was shown that
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96 and T* = (n c )^(3)^(D 4i V 2 ) + o p (l) where U 2,n II f(x 21 W UK] and U. = Â— I I {y.VCc.c, ) + y. ^(c, c )} 4 n c f n c, n 1+ lni V 2 ) + o p (l) and where and TE n = < n c> U 3,n c t 1 n Â— I I ^ X n" X 0+ X 1_ X 9 > 1,n l ( n l) Ui
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97 Thus TE T n>> TE, (n^ (3)^ (U 1Â§ni V 2 ) Ul ) 1/2 (3) V 2 (U 2>ni V 2 ) (n c ) 1 /2(u 3>nc ) (n c )^(3)^(n 4f V 2 ) and therefore, if we can show the right hand side has the appropriate distribution, the proof will be complete. First, it will be shown that n / 2(U 3>tlc ) n V 2(U 4>n V 2 ) N(0,1 ) "u where t = ((o U b) )) and o U b) = I Â— T Â— U b) u L A 1 2 I i = l r (a) r (b) n A.= lim [ ) and j; is the covariance term described in n n i Theorem 3.6.9 of Randies and Wolfe (1979, pg 107). Note, conditional on N,=n, and N =n the problem can be considered as a two sample problem. By Theorem 3.6.9 (Randies and Wolfe, 1979, pg. 107) it follows that
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u = 98 ^Cu 2ini ^) n / 2(U 3)nc ) n /2 (U 4 ,n V 2 ) Â— "(o, f u ), where J u = ( ( a ( a b } ) ) and (a) (b) 2 r r cu\ v i i (a,b) a (a > b > = I i C i = l l for ^ = lim (Â— ) and (r^ a ,r. ) the degrees for Ustatistic U fl Here, Uj Q and U 2 are of degree (2,0) 3>n is of degree (0,1) and U 4 Q is of degree (0,2). We now evaluate the matrix I Now (2,2) u 2^2 (2,2) 0x0 (2,2) Xj c l X 2 ^2 2x2 (2,2) where ?1 (2 > 2) = Cov{4'(x 2i p X U i  + X 2J y x 1;j T ( I X 2~ M l X li A + l X ?VU X llr u) v 2k 'Ik = E{y(x 2i X M ii + X,,U 2j x>1 ( l X 2i" Â•*! l X li" A + l X 2k I l X lj" u) : lk y  ) } V 4 Notice that under H Q x 2 u x 1;L u is symmetrically distributed about 0. Thus,
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and S imi larly where 99 ?1 (2,2) = 1/3 l/4 = 1/12 (2 2) = l/Oxp fl U.l) = i><2 r (l.D ?1 (1 U = Cov{f(X ljL X 2 + K X ), T(X U X 2 + X lk X 2k ) = 1/12 Likewise o (1 l) = 1/(3^) a (3,3) = (i/x 2 )5 2 < 3 3 > = (l/X 2 )Cov{l2 Yi ,l2 Yi } = (l/X 2 )4xVar( Yi ) = 1/A 2 a (4 4) = (4/X 2 ) Cov{ Yi T(c.c.) + Yj ^(c j c i ), Yi n Ci c k ) + Y k *(c k c )} = 4/X 2 ( 1/12) = 1/(3X 2 ) (1,2) 2x2_ _(1,2) 0x0 (1,2) X l l X 2 2 = (4/X 1 )Cov{?(x 21 y X U M  + X 22 U  ~ X 12 U \) tCXjj x 21 + x 13 x 23 ) = (4/A 1 ){Pr [( x 21 h x n y + x 22 u x 12 ~ u) > 0, (x u x 21 + x 13 x 23 ) > 0] V 4 }
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100 i 4P*/X 1 (3 4) = (2/x 2 n 2 < 3 > 4 > = (2/X 2 )Cov{ Yi 4'(c i c k ) + Y k Y(c k c i ), 12y } = (4/X 2 ){E[ Y ?4'(c ;L c k )+Y :L Y k i'(c k c i )J V 4 } = (4/X 2 ){E[Y i *(c i c k )]+E[Y i Y k H'(c k c i )] V 4 } = ~(4/X 2 )( V 4 + \ x V 2 x V 2 \\ ( by Lemma 4.2.1) = 1/(2X 2 ) and a (l,3) = a (2,3) = Q> Thus we have n / 2(U 2>nr V 2 ) nV2(U 3,n c ) .* /2 (u 4>n V 2 ) Â— + N <9 tu^ where 1/(3X 1 ) (4/X^P* t u =  (4/xpP* l/OX^ 1/X 2 1/(2X 2 ) 2 1/(3X 2 Define a 4x4 matrix A to be, A = (3X 1 ) 1/ 2
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101 and applying Corollary 1.7.1 in Serfling (1980, pg 25), it follows (nA^ ( 3 )V 2 (U 1>ni V 2 ) (nX^ (3) V 2 (U 2>n V 2 ) ] d (nX 2 //2 (U 3)nc ) (nX 2 ) l7 2 (3) V 2 (U 4>n V 2 ) AU = N(0, T ) where TT A tu A 1
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102 Theorem 4.2.2: Given the following three conditions and 1) Assume there exists a B,>0 such that j h(x 1 ,x r ; Y)h(x L x r ; u )  < Bj for every Xj,..,,x and all y in some neighborhood of u, where h( Â• ;t) denotes the kernal of the Ustatistic U 2) Suppose there is a neighborhood of X, call it K(X) and a constant Bo>0 such that if yeK(X) and D(y,d) is a sphere centered at y with radius d satisfying D(y,d) c K(X) then E[ Sup h(X ,.. .,X ;y') h ( X X ; y )  ] (Condition 2.9A) n 7 2 [U (y) U (y)l Â— See Randies (1982). D
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103 Theorem 4.2.3 : Under H Q (n ) 7 2 [T* (y) T* (M.,M.)1 B+ 1 n i n 1 2 J where T_ (p) is the statistic T which used u, while n l n l T n (M^,M 2 ) is the statistic which use estimates of u, M, and Mo Proof : The proof of Theorem 4.2.3. follows, if the conditions of Theorem 4.2.2 hold. Although Theorem 4.2.2 has been stated here in terms of one parameter, the theorem is valid for a pvector parameter (i.e., u). Thus, the unknown parameter in this case is (u^,^) which is being estimated by (M^,M 2 ). Next we need to show that the necessary conditions hold. Note, Condition 1 follows directly from the fact that the kernel for T_ is an indicator function, that is n l no = 1 t>0 t<0 and thus HX 2 .Y 2  X 1JL Yl  + X 2J Y 2  x irYl ) v(x 21 u I x i i vj I + x 2 jy  x iju) < 1
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104 To prove Condition 2 holds, it needs to be shown that E[ sup h( X 2i Y2 X e D ( i d ) X li~Yll + l X 2jY 2 X ljYl) Hx 2i Y 2  X 1 Y1 I + X 2J Y2  x irYl )] < B 2 d. (4.2.1) First, consider the following change of variables, let Y li = X li" Y l and Y 2i = X 2i" Y 2* This simplifies (4.2.1) to showing Y lj E[ sup T( Y 2 + Y 2 . y YeD(Y.d) ZJ nY 2i (Y 2 Y2)+Y 2j (Y 2 Y 2 ) Y li(YlY;) Y lj^ 1 Y;)) < B d (4.2.2) Recalling that  f ( t ) If ( t ) j < 1, we need to show P' : M ; Y  + Y 2j 2i Y lil l Y lj ^Y 2i (Y 2 Y 2 )+Y 2j (Y 2 Y 2 ) Y li(YlY;) Y lj( Yl Y; = 1 1 < B d (4.2.3)
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105 To prove (4.2.3) we will utilize the fact that y 2 ~Y 2  < ^ and J Y 2 ~ Y 2 I ^ (i.e., y eD(y>d)) and first consider the region where  Y ^ ^ j > d and I Y 2] l > d for k=i,j. Without 1 loss of generality, it will be assumed that Y 2 >Y 2 an ^ Yi>Yi It will be argued that this region can be appropriately bounded, and similarly that the regions which have not been included here can be bounded also. For the first region we are considering (i.e., l Y lkl ^ anc l Y 2kl ^ ^ k=i J) notice that it can be 1 1 divided into 16 subregions determined by Yj,(yiYi) and I Y 2k~^ T 2 _Y 2M k=i J that is, determined by whether Y,^ > d or < d for k=i,j and whether Y~. is > d or < d for k = i,j. Consider the subregion where Y,.> d, Y 2 .> d, Y, .> d and Y 2 > d. In this region (4.2.3) simplifies to Pr[HY 2i + Y 2j Y u Y U ) ^ Y 2i + Y 2j" Y li" Y lj" 2 (Y 2 Y 2 ) + 2( Yl Y ;) = 1 (4.2.4) Letting Y = Y 2 + Y 2j Y 1 Yj,, (4.2.4) becomes P[*(Y*) ?((Y*2(y 2 "Y 2 ) + 2( Yi yJ)) = 1] which is equal to 2(y 2 "Y 2 ) 2(y 1 ~Y 1 ) f(y )dy if 2(y 2 "Y 2 ) 2(y 1 "Y 1 ) >
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106 f(y )dy if 2(y 2 Y 2 ) 2^Y^ < 2(Y 2 Y 2 ) ~ 2( Y 1 "Y 1 ) Â•k ft & where f(y ) denotes the density function of Y Now f(y ) is bounded, if X,, and X 21 have a bounded joint density. Letting this bound be denoted by B (finite), then 2(y 2 ~Y 2 ) 2( Y 1 "Y 1 ) f(y )dy X 2(y 2 ~Y 2 ) 2( Yl Yl ) d(y) and similarly, u / < 4Bd f (y*)dy* < 4Bd 2(Y 2 "Y 2 ) 2(y 1 "Y 1 ) Thus, this subregion is bounded. With similar arguments, the remaining 15 subregions can be shown to be bounded with the same type of expression. Similarly, the other regions, (i.e., {JYi^l < d and l Y 2kl < d k=i 'JK { l Y lkl > d and l Y 2kl < d k=i >J> etc > can also be bounded by K, d for some constant K, This completes the proof of Condition 2. For Condition 3, under the following conditions, J  ( f (us ,u)f (v+s v) ds du dv v X X
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107 and v = J f J f (us ,u)f (v + s v) ds du dv oo v J f f f ( vs v ) f (u+s u ) ds du dv oo U co V = / J f f (us ,s)f (v + s v) ds du dv oo u where f Â„( Â• Â• ) represents the density of (X,,,X2i), (4.2.5) it can be shown that the differential is zero. For the next requirement of Condition 3, under certain regularity conditions, it can be shown that n'2[M y] = (1) for 11,2 The regularity conditions which are required are the following : a) that F Y (Â•) is continuous, where F Y is the X, marginal c.d.f. for X., i=l,2 b) that G( ) is continuous, and c) G(F X ( l/ 2 )) < 1 Note, conditions a and b are satisfied by assumptions A.2 and A3 and that condition c requires the censoring distribution
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108 to have support which includes the location parameters y and \i Â„ which is satisfied by assumption A6 (See Sanders (1975).) Thus Condition 3 holds since n 7 2 [u ( u ) E(U (u)] Â— > N(0,o 2 ) n n was shown is Section 3.2. and therefore the proof of Theorem 4.2.3. is complete. Corollary 4.2.4 : Under H Q V p n 2 [T (u) TM ] where TMÂ„ denotes the T statistic which uses the combined n l n l sample estimate for u. Proof : This can be viewed as a special case of Theorem 4.2.3. Note, in this case it is easily shown that E [h(X. ,X ; y ) ] has a zero differential by noting that X2^~y ~ ^ii~Y has a symmetrical distribution about for any y* The extra conditions stated in (4.2.5) are not needed D It has been shown in Theorem 4.2.3 (or Corollary 4.2.4) that using the estimates M, and M2 (or M) does not affect the limiting multivariate normal distribution of the vector T of test statistics. The last major theorem of this section, Theorem 4.2.5., states the resulting asymptotic distribution of the quadratic form Wf l7 w l n In c his
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109 theorem T (M,,M 9 ) will denote the T statistic using th n, v "l ,ll 2 g the separate estimators M, and M 9 Theorem 4.2.5 ; Under H the following are true, K, TE + KÂ„ TE In n 2n n (1) ?ln = ( L la T n 1 (M l' M 2 ) + L 2n T n 1 c NCO.jlj) wh ere ^ = (a (a b) ) with a/ 1 1 = k 2 + k 2 a x ( 2 2 > = L 2 + L 2 and ai (1 2) = nKjLjP* K 2 L 2 (3/4) 1/ 2 < 2 > "in tl 1 Sin Â— ^ X( 2) (3) if iis any consistent estimator of I,, then Hn ?I lw ln Â— X( 2) Proof To prove part (1), note that from Theorem 4.2.3., w have T = T n (M 1 ,M 2 ) Â— > N(0, )
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110 where JÂ™ was defined in Theorem 4.2.1. Defining a matrix A to be Kj K 2 \ A =  ~ and applying Theorem A in Serflinj L l L 2 (1980, pg. 122) we see that W ln = AT Â— N(0, A T A') where AT = K,TE + K TE In, 2 n 1 c L T (M ,M ) + L T In, 12 2 n 1 ( fl A ti A Thus, part (1) follows by noting that K. TE + KÂ„ TE (K..TE + KÂ„TE Inn, 2nn 'In, 2n 1 c 1 c = f K. K, TE + KÂ„ K_ TE ^ In 1 ; n, *Â• 2n 2' n 1 ( and since TEÂ„ and TEÂ„ converge in distribution to standard n l n c normal random variables, TE n and TE fl are 0(1) (Serfling, 1 p c P (1980), pg.8). Also, since (K, Â— + K,) In 1 P and (K 2n >Â• K ) as n > thus (K ln ~ Kj) and (K 2n ~ K 2^ are o p (l). Therefore, (K ln K^TE* + (K 2n ~ K 2 )TE* is o (1). A similar arguement holds for L ln T n (M,,M 2 ) + L 2n T n and thus 1 c
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Ill the vector W, has the same distribution as AT. In Parts 2 and 3 follow directly from (1) and well known results The results in Theorem 4.2.5 also hold for !2n K, TE + K n TE In n, 2n n 1 c L. T (M) + L. T In n, 2n n 1 ( where T_ (M) denotes the T statistic using the combined n l n l location estimate and thus will not be stated separately. The quadratic form based on W Â„ is denoted WoÂ„ I W Â„ where n ~ Zn ~2nrzzn rl = t2" Note that each quadratic form mentioned in this chapter is not distributionfree, although each is asymptotically distributionfree. Section 4.5 will investigate consistent estimates for X 4.3 The W 3n Statistic Using CD The last statistic to be considered for pairing with TE. l' u c is the CD statistic presented in Chapter Two. Here, the W statistic will be denoted by H, to indicate this third type of scale statistic used. Theorem 4.3.1: Conditional on N, = n, and N = n and 1 i c c under H
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w, 3n 112 K. TE + K TE^ In n 2n n CD N(0,L) where io is the variancecovariance matrix for W 3n and CD is the standardized CD statistic of Chapter Two. Proof Recall from Chapter Two (ignoring type 4 pairs), that CD = Â— r I I a. .b. r n.+n v L .h ij ij n +n 1 c i
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113 1 pairs occupy positions 1,2,. ..,ni in the sample while the type 2 or 3 pairs occupy positions n^+1 n2+2 n i+ n c Â• Thus, CD can be written as CD = Â— Â— I y y 3j .b, + y y a, .b. n l +n c) l2c + n l X n c U 3c where and U, = Â— I I a..b.. lc n K l
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114 and TE (np^O^CU^^V 2 ) + o p (l) K = (n c ) 2 (U 3> ) and from Theorem 3.6.9. of Randies and Wolfe (1979, pj 107 ) it follows that nV2(U 3,n c ) n V 2 (U lc ) n 7 2 (U 2c ) n V 2 (U 3c ) Â— N(0,i ) where f u (a (a b) ) and 2 r U) r (b) (a,b) = y i i (a,b) 1 = 1 X. l Here U^ n and U. are Ustatistics of degree (2,0), Uo n is of degree (0,1)., U 2c is of degree (0,2) and U. is of degree (1,1). From the proof of Theorem 2.4.1, we have a (l,l) = i/OXj), o {2 2) = 1/X 2 and a (1 2) = 0. In addition, a (3 3) = (4/X 1 )Cov{a ij b. j a k bk  6 5 ., =6 k =l } f (4 4) = (4/X 2 )Cov{a i:J b i:j ,a ik b ik (6 1 ,6 j ,6 k ) Â£ (2,3)}
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115 a (5 5) = (l/X 1 )Cov{a. j b ij ,a lk b lk  6 l (6 j 6 k )e(2 ,3 )} + (l/X 2 )Cov{a. k b. k ,a jk b jk 6 i =6 j =l,6 ke (2,3)}, ,(3,4) = a (l,4) = ,(2.3) = 0> a (3,5) Â„ ( 2 /A 1 )Cov{a ij b. j a ik b ik  6 = 6 j = 1 6 k e ( 2 3 ) } a (4,5) ( 2 /X 2 )Cov{a i:j b i;j ,a ik b ik  ( 6 6 j )e ( 2 3 ) 6 fc l } a (1 5) = (2/X 1 )Cov{nx 1 .X 2i +X lj X 2:J ), a ik b ik  5 i =6 j =l,6 k e(2,3)} a (1.3> = (4/X 1 )Covfnx 1 .X 2i +X lj X 2j ), a. k b ik 6 i =6 j =5 k =l} a (2 5) = (l/X 2 )Cov{l2 Yj ,a 1J b 1 j6 1 l,je(2,3)} (,(2,4) = ( 2 /x 2 )Cov{l2 Y: j ,a i jb 1J (6 1 ,j)e(2,3)} Â• Next, we get the distribution of n (U l, ni ~ l k >Â• n (U 3,n c > n < CD > Note that, n /2 (CD) 1/ 1 n l 2 )2c + n l n c" 3 ct Â•
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116 lim = lim n (r^1 ) n(nl) = X lim = X. and lim n* = lim 2n, n 1 c n(nl) 2X X X 2 Thus, we need 10 1 A 2 ftnc ) n V 2 (U lc ) n V 2 (U 2c ) n V 2 (U 3c ) V2 CD lfni ^) n / 2(U 3jIlc ) ^ 2 (X l U lc + X 2 U 2c + 2 ^i^ 2 U 3c ) i'i ( u 3jnc ) A (CD) with variancecovariance matrix A  U A' = J!q D = (o^D where (1,1) = rt (lD U CD G (2,2) = G (2,2) a CD a CD (1 3) ^ia (1 3) + 2X 1 X 2 a< 1 5 > CD (2 3) = ^2(2 4) + 2X 1 X 2 a< 2 > 5 > and
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117 ;cD ( 3 3 > = aV 3 3 > + ZA^UX^)^ 3 5 ) + 2X2(2X 1 X 2 )c (4 5) + X 2 a (4 > 4) + (2X 1 X 2 ) 2 a (5 5) ( 3 3 ) Next, it will be argued that o en is as y n P tot i ca ^y the same as the variance for CD (i.e., 4y) derived in Section 2.4. Note that, 2 (35) 2 (45) 22 (55) 2X (2X^2)0^ ; + 2X2(2x^2) a '+ 4X L X 2 a } 8X. X.Covfa, b, ,a ., b .. I <5 5 1 6. e (2 3 ) } 12 l ij ij lk lk l x j k + 8X 2 X 1 Cov{a lj b ij ,a ik b. k (6 i ,6 j )e(2,3),6 k =l + 4X 1 X 2 Cov{a ij b ij ,a ik b. k  5 .=l,(6.,6 k ) e (2,3) + 4x'x 2 Cov{a ik b. k ,a. k b. k 6 i =6.=l,6 k e(2,3)} = 4X 1 X 2 Cov{ 2 + 4X X 2 Cov{ 2 + 4X 1 X 2 Cov{ + 4X X 2 Cov{ 2 + 4X X.Covf 2 i + 4X X 2 Cov{ 6 1 J l i a k e(2,3)}  1 k l,6 j e(2,3)} (6. ,6 ) Â£ (2,3),5 k l I (6. ,6, )e(2,3),6 =1 'IK J I S = 1 (6 ,6 k )e(2,3) 6.e(2,3),6 j =6 k =l} (3 3 ) Thus, combining all the terms in Op D we have D < 3 > 3 > = 4X 3 lC ov{ 3 + 4X 2 Cov { 2 + 4X 1 X 2 Cov{ 6 i =6.=6 k =l} I (6 i ,6 j ,5 k )Â£(2,3)} 6 i =6 j =l,6 k Â£(2,3)
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118 + 4X 1 X 2 Cov 2 + 4X i X oCov 2 + 4XJX2COV 2 + 4X 1 X 2 Cov 2 + 4X 1 X 2 Cov Recalling that 6 i =6 k =l,6 j e(2,3)} ( 6 6 j ) e (2 ,3) 6 k =l } (6 i ,6 k )c(2,3) .jl} 6 k l,(6j,6 k )e(2,3)} 6 i e(2,3),6 j =6 k =l} (4.3.1) Â— = (proportion of sample which are type l's) P + X. = (probability of being a type 1), and, thus, the X coefficients in front of each covariance term are the probabilities necessary to uncondition each covariance term. For example, LCov(a. _.b,. 6 6, =5.1) j b ij a ik b ikl 6 i =6 j =6 k = P(6 i =6 j =6 k =l)C v(a. j b.. a. k b ik 6 i =6 j =6 k =l) = Cov(a..b. j Â• lk b lk 1 J k l) Now note that the eight covariance terms correspond to the eight possibilities for the subscripts i, j and k (i.e., 3 subscripts with 2 possibilities for each, that is, each 3 subscript is either a 1 or (2,3) yields 2 =8 combinations) and thus acD (3,3) Cov(a. j b.. .a.,^,.) = 4 Y The last step of the proof, (i.e. showing Wo + N(0,I ) w 3 follows from observing that
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119 KjOXp^ K 2 (X 2 ) l7 2 ^^nf^)' (4 Y )~ L/ 2 J x j A (U 3jn ) A (CD) ^O^CnX^CU VjP + K 2 (nX 2 ) 1/ 2(U 3>nc ) CD by Theorem A in Serfling (1980, pg 122), where \ 3 = (a 3 < a > b) ) N(0,L) a 3 '3 n,n = K + 4 c,< 2 2 > 1 and n. 2 ) = ^OX^ 2 {xja (1 3) + 2X 1 X 2 a (1 5 M(4Y)l/ 2 + K 2 (X 2 ) V 2 {X 2 a (2 4) + 2X 1 X 2 a (2 5) }(4 Y )1/ 2 After some simplification, similar to that used to show that 'CD ( 3 3) was equivalent to 4 Y it follows that 03 is equal to K 1 (4 Y )l/ 2 (3^)^ 4Cov[^(X 1 .X 2 + Xlj X,, ) a^b^  6. =6 =1 ] } + K 2 (4 Y ) J/2 (X 2 ) 1/ 2 {2 Cov(l2 Yj a ij b. j S jÂ£ (2,3))}
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120 Recalling that 1 c Â— Â— Â— * A, and Â— Â— Â—* AÂ„ and using a similar N 1 N 2 arguement as in Theorem 4.2.5 (pgs. 109110), we get K, TE + KÂ„ TE In n 2n n 1 < CD N(0, ) 3 and the proof of 4.3.1 is complete Note that, similar to the case with W, and W^ W 3n T3 W 3n is not distributionfree, although it is asymptotically distributionfree The following corollary, states results which follows directly from Theorem 4.3.1. Corollary 4.3.2: Under H Jo and (1) ?3n E 3 2 ?3n ~+ *(2) (2) I, is any consistent estimator of h, then Id 2 ^3n Z 3 ?3 Â— *(2) Proof : The proof is omitted, since (1) and (2) follow directly from Theorem 4.3.1 and well known results. D
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121 This section has established the asymptotic distribution for the quadratic form W, Io W, which could be used for a large sample test for the general alternative of location and/or scale differences. The next section, will discuss a permutation test which could be performed for any of the quadratic forms (based on H W~ or W, ) mentioned in Sections 4.2 and 4.3. Section 4.5, will discuss consistent estimators for I,, I2 and L. 4 .4 Permutation Test In the situation where the sample size is small, there may not exist a good estimate for I. 1=1,2,3 or for the 2 limiting X(2) distribution to provide an adequate approximation for the distribution of W. Â£7 W. 1=1,2 or 3. In this case, a small sample permutation test is recommended Recall, in Section 2.3, a permutation test for CD was discussed. It was based on the 2 n possible samples [X 11' X 21'6 1 ) 1 ,(X 12 ,X 22 ,5 2 ) 2 >...,(X ln ,X 2n ,6 n ) n ] : k^ = or 1 for i=l,2, ,n: which are equally likely under H Here
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122 (X 1 .,X 2 .,6.) i = (X u ,X 2i ,6 1 ) if k (x ,x ,f(a )) if k. The permutation test in this section is based on the same 2 samples which are equally likely under H (A slight change is present though, since n = N, + N here, that is, no type 4 pairs are included in the sample for the calculations.) Without loss of generality, it will be assumed that W, ti Wi is the test statistic for which the permutation test is being done. The permutation tests for the statistics based on W and W, are performed similarly. It ~2n on r is also assumed that a particular K, and K2 n (L, and Lo n ) have been chosen by the researcher. Let w, (1) denote the first component in W, ; that is the location statistic which is w. ( 1) = K. TE + KTE In Inn 2nnc and let w, (2) denote the second component in W, ; that is the scale statistic used for W,. For each of the 2 n equally likely samples, the statistics w i n (l) w i n ^^ anc w, w, (2) are computed and their values tallied. From these In 1 n tallies, the relative frequency of each possible value of w, (1), w, (2) and w, (l)w, (2) is determined and these In' In In In relative frequencies are then the probabilities that w i n ^^' w, (2) and w i n (l)w, (2) assume the corresponding distinct values. Using these probabilities, the actual conditional variance of w.(l) and w, (2) can be calculated and the In 'In'
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123 actual conditional covariance of w i n O) an d w i2n^^ can ^ e calculated Let i denote the conditional variancecovariance matrix for W, Now we calculate the 2 n not necessarily distinct values for W ln j:" 1 W ln determined by the 2 n equally likely samples under H From these calculations compute the relative frequency for each distinct value of W, i _1 W, thus obtaining the conditional probability In Tc in' distribution of W, j:" 1 W ln The null hypothesis is rejected if W, t~ W, for the actual observed sample is too large according to this conditional distribution. 4.5 Estimating the Covariance In Section 4.3, the asymptotic distribution of WC tT 1 VI. for 11,2,3 was established. In each case, I. ~ in ri in ii depended on the underlying distributions F(,) and G() (the c.d.f. of (Xjj, X 2i ) and C respectively). Hence, we can not perform a large sample test based on WÂ£ Q j:^ W^ n unless I. i=l,2,3 is known. This section will discuss estimation for the components of t, i=l,2,3 which depend on the underlying distribution. These estimators, when substituted into the appropriate quantities they are estimating, provide asymptotically distributionfree statistics that can be used in the
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124 hypothesis testing situations considered in this dissertation. For the var iancecovariance matrix I, and, thus, I2 since it is identical, the term in I which depends on the underlying distribution is a^ 1 2 ^ = 12KjL.jP* K 2 L 2 (3/4) // 2 (page 109). The dependence due to P which was defined as P = Pt{( XÂ„u X, ,u + X Â•21 11 Â•22' 12 Â•u > 0, (Xjjx 21 + x 13 x 23 ) > } V 4 was a result of the asymptotic covariance between TE and T_ Lemma 4.5.1 defines a consistent estimator for the n l quantity 12P (the asymptotic covariance of TE and T ). n J J r n i n  First, though, we describe some notation which will be needed Define x i = (Xj.,x 2i ), h (1) (X ) = f(x 2i MXj i M ), (2) h h (3) (x ) = f (x 1 x 2i ) h (4) (Si.Sj) = ^ X li" X 2i + x ijX 2j>' h (1 3) (X.) = h (1) (X.)h (3) (X i ),
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125 h(1 4) (?i?j) = h(1) (?i) h(4) (?i?j> + h (1) (X j )h (4) (X i ,X j ) h(2,3) (? i ? j ) = h (2) (X.,X j )h (3) (X i ) + h (2) (X i ,X j )h (3) (X j ) and h 2 2 4) (?i'?j> h(2) (?i'?j) h(4) (?i?j) h [ 2 4) (? i .?j'?k ) h(2) (?i'?j) h(4) (?i + h (2) (x i ,x k )h (4) (x i + h (2) (X.,X j )h (4) (X j (2) (4) + h^ / (x j ,x k )h v w (x i + h (2) (X i ,X k )h (4) (X j + h (2) (x r x k )h (4) (x. ?k> 5k> ?k> The quantities h (1) (), h (2) (.,.), h (3) (.) and h (4) (.,) are actually the kernels of Ustatistics or kernels of Ustatistics with an estimated parameter which are used in the representation of T n and TE n (See page 74 for the exact Ustatistic representation of T A similar representation for TEÂ„ can be defined.) The quantities 1 h (*Â•), h^ '(,Â•), etc., are needed to calculate to covariance between the kernels of the Ustatistic representations of T and TE A consistent estimator for 11 1 n l Cov(T ,TEÂ„ ), which will be defined in Lemma 4.5.1, can be n l n l viewed as estimating the exact covariance between TE and n l T n using the sample covariance. In the proof of Lemma
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126 4.5.1, it will be argued that this estimator is a consistent estimator for the asymptotic covariance. Lemma 4.5.1: Under H Co^TE^.T^) = r 4 "(1,3) 4 ;(1,4) 4 ;(2,3) "" 2 C + (ii,!) c + (n,l) C (a 1 l) Â£ 1 1 1 a. 4U 1 2) C(2,4) __2 "(2,4) is a consistent estimator for 12P (the asymptotic covariance of TEÂ„ and TÂ„ ) where n l n l (1,3). JL I h U,3) h (l) h (3) 1 n i i=i JCi.*>._L. j j h (1 4) (x. ,x.) h (1) h (4) (2,3)_ 1 Â£I I h (2 > 3) (x ,X ) 1) i = _L M h (2 > 4) (X.,X.) h (2) h (4) 2 iI I h (2 4) (X.,X.) h' ( n il i
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127 (1) value of a Ustatistic with kernel h y ) and analogous definitions for ti t/ 3 ^ and la '. Proof : Note that r A (2 4 ) P Cov(TE ,T ) 12c 1 Â—+ 0. L V n l 1 Thus, the proof will be complete if it can be argued that A ( 2 4 ) X, is a consistent estimator of P This follows 1 directly since for a Ustatistic, U based on a kernel, h P U Â— h by Hoeffding's Theorem (Hoeffding, 1961). Q Note that this is just one of many possible consistent estimators for 12P This estimator is presented because it worked well in the Monte Carlo study presented in Chapter Five. Although other estimators may appear to be reasonable, all too often in practice their determinant will be less than or equal to zero. Now we consider estimators for the variancecovariance matrix for W~, the quadratic form using CD. In looking at the variancecovariance matrix L derived in Section 4.3, we notice immediately that the estimation needed here is more complicated than that of I, First, a consistent estimate for the asymptotic variance of CD (i.e., 4y) is needed. Secondly, we need to estimate two asymptotic ( 1 2 ) covariances which are used in the calculation of o"^
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128 (i.e., the asymptotic covariance between TE and CD ). 1 c The first estimation problem has already been taken care of in Section 2.4. The method of solving the second estimation problem is similar to that used for estimating 12P That is. the exact covariance between TE n and CD is derived n l n c and the sample quantities are then used in its calculation. Lemma 4.5.2 will present this estimator and argue that this is a consistent estimator for the asymptotic covariance. This estimator will be actually in the form of two estimators; one which is estimating the covariance of TE and CD and the other which is estimating the covariance n l of TE and CD. This is equivalent to estimating the c quant it ies Cov[f(X ir X 2 + X X 2j ), a. k b ik 6.=6.=l and Cov[ (1 2 Y .), a i:j b i:j I <5j e(2 ,3) in o 3 ^ 2) First though, we describe some notation which will be needed. Let h (3) (X i ) and h**'(X lf Xj) be defined as before. In addition, define (lc) = t.(2c) (3c) (X 1 ,X j ) = h^ c ^(X.,X.) = h^ c '(X 1 ,X J ) = a..b i:j h <5) (Xj) = 12YJ, h(3,1C) ( X i?j) = h< 3 >(X i )h (lc >(X 1 ,X j ) + h C3) (X j )h (lc) (X 1 ,X j )
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129 h l ( ,lc) = h(4)( ?i (4) + h + h< 4 > < X i < X i + h< 4) ( Xj Xj)h (lc) (X, X j )h (lc) (X j Xu)h (lc) < X i ?k )h(1C) (?j Xu)h (lc) < x i x k )h (lc) ( Xi x k> ?k) x k> x k> x j>> and h 2 (4,1C) h(4) (?i.?j> b(1C) (?i.?J>. h (3,3c) ( x. )X _.) = h (3) (X.)h (3c) (X i ,X j ), h^'^^X^Xj,^) = h (4) (x 1 ,x j )h C3c) (x 1 ,x k ) + h (4) (X i ,X j )h (3c) (X j ,X k ) h(5,2c) ( x i x j) h(5) (!i) h(2c) (!i!j) + h (5) (x j )h (2c) (x i ,x j ), h (5 3c) (x i ,x j ) = h (5) (x j )h (3c) (x i ,x. j ). Note, the quantities h^ lc) (,), h (2c) (,) and h (3c) (,) are actually the kernels in the Ustatistic representation of CD given on page 113. The quantities h^ 3 1 c ( Â• Â• ) 1^ c (Â•,*,Â•), et c are needed to calculate the covariance between the kernels of the Ustatistic representation of TE n and CD, and TE n and CD. In Lemma 4.5.2, the consistent estimator is now defined.
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130 Lemma 4.5.2 : Under H a consistent estimator for K 1 (4 Y )" l/2 (3X^2 [4Cov{T(X 11 X 2l +X lj X 2j ), a j b J  6. j =1 } ] + K 2 (4 Y ) 2 (X 2 ) /2 [2Cov{l2 Yj a ij b ij  6 e(2,3)} ] is the followinj K Uy)"^ (3 A X ) V 2 r 4 a (3 lG) + 4U 1" 2) J(4,lc) 4n a Â„ v 4n 2 M4,lc) c *(3,3c) c ( 4 3c) (n1) ^ 2 + (nlXnj1) S (n1) ^ + K (4v)" 1/ 2 (X ) V 2 r 2Uc r (5 2c) + 2ni ;< 5 3c >l k. 2 ^y; u 2 ; [ (n 1) ^ + (n 1} c;^ j where 4y is a consistent estimate for 4y, X 1 1 n,+ n 1 c Xo = 2 n .+ n 1 c and 5j 1C \ E[ 4,lc) l^ ,lc \ etc. are Ustatistics which are summarized in the following table.
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131 Table 4.1 Summarizing the Us tat is tics Used in Estimatinj the Covariance for lo UStatistic p(3,lc) (4,1c) p(4,lc) :(3,3c) g l :(4,3c) g l p(5,2c) g l p(5,3c) Kernal h (3,lc) hi (4,lc h2 (4,lc h (3,3c) h (4,3c) h (5,2c) h (5,3c) ,Â•) ) ) Conditions on the 6's 5i =6.=l 6 i = 6 j =6 k = l 6 i = l ,6je(2,3) 6 i =6 j =l ,6 k e(2,3) (6 1 ,6 J )e(2 1 3) 6 i l,5 j e(2,3) Proof : Let the estimator defined in Lemma 4.5.2 be denoted by Cov(TE n ,CD ). Notice that n l c ^ J /o A Vo r 4(n i" 2) ^(4 lc) {Cov(TE ,CD ) K l( 4 Y ) (3X^2 [^jC 4 lc) 1 c l 4n c M4.3C), (n1) S I J./ /s 1/ 2 ( n ~ 1) K 9 (4 Y ) / 2(X ? ) / 2[ ,;_Â„ 2n _c 11 M5,2c) 1 M5,3c) (n1) ? j + (n1) S Â£* 0. Thus, if we can show that
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132 4(n 2) A ,, N 1 (4 ,1c) (n1) S 4n c *(4,3c) (n1) 1 is a consistent estimator for 4Cov{(X. X..+ X..X..), a., b.. lfi.6.l}, 1 1 1 2i 1 j 2j lk lk i i j and that 2(n c l) ^ (5>2c) 2n l ^ (5>3c) (n1) S + (n1) ^ is a consistent estimator for 2Covfl2y., a. b. 6 e(2,3)} the proof will be complete. Recalling that n l P n c P Â— Â— > X, and Â— Â— X Â„ we observe that asymptotically, n In 2 J the coefficient in front of each z, term is the probability necessary to uncondition the term so that within each estimator the terms can be combined appropriately. (See page 118 for a similar argument.) Also, since each estimator is a Ustatistic, it follows that each is consistently estimating the appropriate covariance term and the proof of Lemma 4.5.2 follows. D
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CHAPTER FIVE MONTE CARLO RESULTS AND CONCLUSION 5 1 Int roduction The first three chapters of this dissertation have been devoted to developing tests statistics for the purpose of testing for scale differences in censored matched pairs. Chapter Four used the statistics proposed in Chapters Two and Three to develop a vector of statistics designed to test for the more general alternative of location and/or scale differences. This chapter will investigate the performance of some of the test statistics proposed. In Section 5.2, a simulation study will be presented to compare the intermediate sample size performance of some members of the proposed class of statistics presented in Chapter Three and the CD statistic of Chapter Two. Similarly, Section 5.3 will present a simulation study for selected W vectors and a test statistic proposed by Seigel and Podger (1982). In each of these simulation studies, the asymptotic distribution of each test statistic is being used 133
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134 5.2 Monte Carlo for the Scale Test In this section, nine statistics will be investigated to compare their performance under the null and alternative hypotheses. Three of the nine, are versions of the CD statistic, where each version uses a different estimator for the variance of CD (Section 2.4). The next three statistics are members of the class of statistics proposed in Section 3.2, where the common location parameter was known, while the following three are members of the class of statistics proposed in Section 3.3, where the common location parameter was unknown and thus estimated. Table 5.1 summarizes the nine statistics considered in this Monte Carlo. In this Monte Carlo three bivariate distributions (each with common location (0,0)) were considered for generating the bivariate samples. Although, generally the common location is not (0,0), it was used without loss of generality. The first distribution, the bivariate normal, was generated using the subroutine GGNSM of the International Mathematical and Statistical Library (IMSL). It allows specification of the var iancecovariance structure for the bivariate pairs. The remaining two distributions were generated using a technique for generating elliptically symmetric distributions proposed in Johnson and Ramberg (1977). The two distributions are both Pearson Type VII multivariate distributions which were generated in the following manner. For a sample of size n, 2n uniform [0,1]
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135 Table 5.1 Summary of the Test Statistics Considered in the Monte Carlo for Scale.* Test Section Statistic Description in Thesis 1 CD statistic using Var^CD) 2.4 2 CD statistic using Var 2 (CD) 2.4 3 CD statistic using Var 3 (CD) 2.4 4 T n p n c = W T n c ( e L 1 = L 2 = 1 > 3 2 5 W 2 X+ T n c (L. .^2.^1) 3.2 6 T npn c = T ni + 2T n c < i Â• e Â• L 1 1 L 2 = 2 } 7 Â™n 1 ,n c = %<*> + T n c Ci Â•Â•Â• .^1 ,L 2 1) 8 Â™ni. c 2 ^1 (M) + Tn c ^Â• e L l2 L 21 ) 9 TM n n = T n (M) + 2T n ( i e ,L 1 1 L 2 2 ) 3.3 3.2 3.3 3.3 Tn,(M) denotes the T n statistic of Section 3.3 which uses an estimate for the common location parameter. random variables (denoted U i = l 2 2n) were first generated and then the following transformations were applied : X li = CU 2 il 1/U V) 1)/2 cos ( 2TrU 2i ) X 2 = (U 2i1 l/(1 v) 1) /2 sin(2TrU 2i ) where the parameter v (for bivariate pairs, v > 1) specifies a particular Pearson Type VII distribution. To generate bivariate pairs with scale parameters a 1 and a 2 and correlation p, the following transformation was applied:
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136 X 1 o 1 X 11 and 2 V 2 ^2i = P2^1i + a 2^ ~ P ) ^2i The values of v which were chosen were v = 1.5 (which corresponds to a bivariate Cauchy distribution) and v = 3 (a distribution with moments and moderate tailweight). To generate the censoring random variable, the natural logarithm of the Uniform [0,B] distribution was used. The choice of B was made separately for each distribution with specific correlation p, so that under H approximately 25% of the total sample was censored in some manner. Three values for p were chosen: (1) p = .2 (weak correlation), (2) p = .5 (moderate correlation) and p = .8 (strong correlation). Note that the value of p affects the type of censoring occurring in the samples, that is, when p = .8, type 4 pairs dominate the observations which are censored, while when p = .2, type 2 and 3 pairs dominate the observations which are censored. Since the results presented in this section apply to the pattern of censoring described above, any conclusions drawn only apply to this form of censoring. In each case the null hypothesis was H : a, = Oo and the alternative was H : Oo > o, The tests were conducted at the .05 level of significance using the asymptotic distribution for each test statistic. The first Monte Carlo study consisted of generating 1000 independent censored
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137 samples of size 25, while the second utilized 1000 2 independent samples of size 40. In each, the value of Oi 2 was 1.0 while the value of Oo was 1.0 (under H ) or 2.0 or 3.0 (each value corresponding to a different run of the fortran program listed in Appendix 2). Tables 5.25.4 give the results of the Monte Carlo for each distribution type with entries corresponding to the number of times, a statistic rejected H The nine test statistics are J o numbered in accordance with the listing in Table 5.1. The standard deviation associated with each entry e can be estimated by ( e ( 1 000e ) / 1 000 ) ^. Inspecting Tables 5.25.4, we see that as the correlation increases between the components in the bivariate pairs, that the power increases for all the tests, regardless of the distribution considered. This exhibits the fact that the tests were designed to use the intrapair information or at least some of it, in the case of the distributionfree statistics which correspond to columns 49. The CD statistics (columns 13) which use more intrapair and interpair information than the distributionfree statistics are performing the best across all of the distributions considered. Recall, the only difference in the CD statistics (columns 13) is the method of estimating the variance. The CD statistic corresponding to column 2 which uses Var 2 (CD) (an estimate for the asymptotic variance (4y) which is the variance of a conditional expectation, page 41 in this
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138 Table 5.2 Approximate Powers of the Tests for the Bivariate Normal Distribution P n
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139 Table 5.3 Approximate Powers of the Tests for the Bivariate Cauchy Distribution (Pearson Type VII, v=1.5) P = 2 n
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140 Table 5.4 Approximate Powers of the Tests for the Bivariate Pearson Type VII, v=3 = .2 n
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141 thesis) has some problem maintaining the significance level under H and thus is not recommended. The CD statistics o corresponding to columns 1 and 3 are maintaining the .05 level under H with the statistic corresponding to column 3 performing the best over the alternatives. Recall, the CD statistic which uses Var^(CD) (column 1) is using a Ustatistic to estimate the asymptotic variance (4y), while the CD statistic which uses Var^CCD) (column 3) is 2 4(n2) estimating the exact variance that is Â— , rr a + Â— 1 TT Y Â• n(nl) n(nl; For the CD statistics, only once was there a negative estimate for the variance which occurred for the Pearson 2 Type VII, v=3 distribution with n=25 and o 2 = 2.0 One basic disadvantage of the CD statistics is the fact they require the use of a computer to perform the calculations for even moderate sample sizes. The CDSTAT subroutine of the fortran program listed in Appendix 2 could be used. If a computer is not available or the necessary knowledge to use it to program the calculations for the CD statistic, then a distributionfree test could be recommended. Of the distributionfree tests in columns 49, the tests which uses weights of L,=l and L 2 =2 (columns 6 and 9) appear to be performing the best. Column 6 corresponds to the statistic for the case when the common location parameter is known, while column 9 corresponds to the statistic for the case when the common location parameter is unknown. The corresponding test statistics using equal weights, that is, L,= Lo" 1 (column 4 corresponds to the
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142 case of the known location parameter, while column 7 corresponds to the case of the unknown location parameter) follow closely behind in terms of power. If the common location is known, in each case the power is improved by using that known value (column 6 for weights L,=l and L 2 = 2 or column 4 for equal weighting). When the correlation is high, note that all the statistics considered in this Monte Carlo are performing well. In summary, the best statistic to use is CD using Varo(CD) when the necessary computations which require a computer can be done. If it is not possible to calculate the CD statistic, the distributionfree test statistic usinj weights of L,= 1 and l>? = 2 (corresponding to column 6) is recommended when the location parameter is known. If the location parameter is unknown, then the distributionfree test statistic using weights of L,= 1 and L2= 2 (corresponding to column 9) is recommended. 5.3 Monte Carlo for the Location/Scale Test In this section, twelve statistics will be investigated to compare their performance under the null and alternative hypotheses considered in Chapter Four. The alternative hypotheses studied here include alternatives for location differences only, for scale differences only and for location and scale differences. The first nine statistics
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143 which will be considered are various versions of the quadratic forms presented in Chapter Four. The tenth statistic was introduced by Seigel and Podger (1982) and will be defined in this section. The last two statistics, which are from Chapters Two and Three, are included here to determine their performance under the alternatives considered in this section. The first is the distributionfree statistic TMÂ„ Â„ which uses weights of L,= 1 and L 9 = 2 n i n c x (corresponding to statistic 9 in Section 5.2). The second is the CD statistic which uses Var.j(CD) to estimate the variance of CD (corresponding to statistic 3 in Section 5.2). Of the nine quadratic forms from Chapter Four, the first three are versions of W~ o ^2n wnere ^2n """ s C he vector of statistics which uses TEÂ„ (which tests for n l n c location) 'and TM_ (which tests for scale) with the n l'c common location estimate. The three different versions correspond to different choices for the weights, K^ n K.2 n L, and L used in forming W The next three quadratic In Zn' ~/n forms are versions of W ln f, W ln where W ln is the vector of statistics which uses TE and TÂ„ (which tests for n l' n c n l' n c scale) with the separate location estimates. Similar to the first three quadratic forms, these three different versions correspond to different choices for the weights K^ n K.2 n L, and L 2n The choices of weights used here are identical to those used in W and will be defined shortly. The last three quadratic forms correspond to different versions of
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144 i 1 K, I, H, where W 1n is the vector of statistics which uses jn n in ~ jn TEÂ„ and CD (the standardized CD statistic) with n l n c Varo(CD) as an estimate of the variance of CD. Again, the three versions correspond to different choices for the weights K, and K For the weights K. K^, L^ n and L2n' the following choices were used: (1) K, = K~ = L, = L 2 =l (This amounts to just summing the standardized statistics to form the location statistic and the scale statistic.) n n (2) K, = L, = Â— and K 9 = L 9 = ~ (This In In n ,+ n 2n 2n n ,+ n l c l c weights each statistic proportionally to the sample size used in its calculation and will be denoted as SS weights.) and a(TE n ) (3) K ln= a(T E n i' n c K a(TE ) n c 2n" a(TE ) n i' n c L, = (T ) n i In a(T ) n l' n c and KÂ„ = o(T ) n c 2n o(T ) n l' n c where a(TE n ), a(TE n ) and a(TE_ ) n l n c n l' n c represent the standard deviations of TEÂ„ TEÂ„ and n l n c TE + TE respectively, under the null hypothesis. Similarly, o(T n ), o(T n ) and l c a(T ) represent the standard deviations of
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145 and T Â•Â„ Â„ > respectively, under the null "1 u c n l' n c hypothesis. (This weights each statistic proportionally to its null standard deviation and will be denoted as STD weights.) For the quadratic forms using CD (i.e., versions of Wq to Wo ) only the weights K, and K~ were used since jn Tj jn J In ^n the definition of CD did not include any weights. For the quadratic forms based on W and W,Â„. the three choices for the weights, K, K 2 L, and L correspond to the three versions of each statistic which will be presented. Table 5.5 summarizes the twelve statistics considered in this Monte Carlo The test which was proposed by Seigel and Podger (1982) can be viewed as a special case of the procedure commonly referred to as the log rank method proposed by Mantel (1966). This test assumes the null hypothesis that the survival curve for X,, is identical to that of Xjl at a ^ points. The alternative for this test is that differences exist between the two curves. Note, that this is a more general alternative than the alternatives specified in Chapter Two and Three but the test was included to determine how well it performed for the alternatives considered here. Now, to define the test statistic. Let n represent the total number of type 1, 2, and 3 pairs and let n be the number of pairs for which X, > ^n' Similarly, define n to be the number of pairs in which X. < Xji' Note that
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146 Table 5.5 Summary of the Test Statistics Considered in Monte Carlo for Location and/or Scale Alternatives Test Section Statistic Description in Thesi; 1 ?2n h 1 ?2n with K ln =K 2n = L ln = L 2n = 1 4 2 2 W^ ^2* W 2n with ss weights 4.2 3 W2 n fa 1 W 2n with STD weights 4.2 4 W[n JI 1 "la with K ln =K 2n =L ln= L 2n =1 4 2 5 W^ n Ij 1 W ln with SS weights 4.2 6 w[ n l^ 1 W ln with STD weights 4.2 7 ?3n l~3 l ?3n with K ln =K 2n = 1 4 3 8 W 3n li 1 W 3n with STD weights 4.3 9 W^ n $2^ w 3 n with SS weights 4.3 10 Siegel Podger Statistic 5.3 11 TM = T (M) + 2T 3.3 n 1 ,n c n x n c 12 CD statistic using Var 3 (CD) 2.4 n + n = n. The test proposed by Seigel and Podger (1982) is to compare the observed frequencies n and n against the expected values of n/2 (under H Q ) using the binomial distribution or when appropriate, an approximate large sample distribution. A suggested statistic which would be appropriate for the approximation is McNemar's statistic which could be defined here as T SP (n t n u )2/n 2 Under H Q T sp has a limiting Xfn distribution.
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147 For this Monte Carlo study, the uncensored bivariate samples were generated using the techniques described in Section 5.2. Those techniques generate bivariate pairs with scale parameters Oi and ao and correlation p. Without loss of generality, the location parameter for X 2 j was chosen to be 0, while location parameter for X.. was \i 2 > T ^ e censoring random variables were also generated in the same manner as in Section 5.2, so that under H approximately 25% of the total sample was censored in some manner. The values for p considered in this section are .2, .5 and .8 (as in Section 5.2). For each Monte Carlo run the null hypothesis was H jj = y~ and o, = On, and the alternative was H : y,
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and 148 (3) \ii" 0.5 and a 2 1.0, (4) Ml = 1.0 and a 2 = 1.0, (5) y : = 0.5 and a 2 = 2.0, (6) pj0.5 and a 2 = 3.0 (7) y,= 1.0 and a 2 = 3.0. Tables 5.65.8 give the results of the Monte Carlo for each distribution type with entries corresponding to the number of times a statistic rejected H The twelve statistics are " o numbered in accordance with the listing in Table 5.5. The column headings Null, Scale, Location and Location/Scale refer to the type of bivariate pairs being generated and the type of alternative that it reflects. The standard deviation associated with each entry e can be estimated by (e(500 e)/500)'2 Covariance estimates between certain entries were also estimated but are not reported here. The following discussion refers only to statistics 1 to 10 in Tables 5.65.8. The discussion of statistics 11 and 12 will follow. Inspecting Tables 5.65.8, we see that as the correlation increases between the components in the bivariate pairs that the power increases for all the tests. Similar results were observed in the Monte Carlo in Section 5.2. For the null hypothesis (column 1), all the tests are maintaining the significance level under H Q fairly well, although, the levels for the tests using CD show more
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149 Table 5.6 Number of Rejections in 500 Replications for the Bivariate Normal Distribution
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150 Table 5.6 continued.
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151 Table 5.7 continued
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152 Table 5.8 Number of Rejections in 500 Replications for the Bivariate Pearson VII, v=3 Null Scale Location Loca t ion/ S cale y = 0.0 0.0 0.0 0.5 1.0 0.5 0.5 1.0 Stat a 2 = 1.0 2.0 3.0 1.0 1.0 2.0 3.0 3.0 2 P.2 1
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153 Table 5.8 continued.
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154 statistic (rows 79) are performing better than the quadratic forms using the distributionfree scale statistic TM (rows 13) or its analog which uses the separate n l c location estimates (rows 46). No basic differences exist between the three quadratic forms which use CD for these two alt erna t i ves For the alternatives in which the bivariate pairs were generated with location differences only (y,= 0.5 and oj?= 1.0, or p ,= 1.0 and Oo = 1.0), in general, the Seigel Podger statistic (row 10) is performing the best because it specializes in this type of alternatives. If we look at the bivariate Pearson VII distribution with v=3 and the bivariate normal distribution only, the statistics corresponding to rows 1, 2, 4, 5, 7 and 8 are all performing equivalently to the Seigel Podger statistic. It is in the bivariate cauchy distribution where the Seigel Podger statistic seems to have a slight advantage. Note that the statistics corresponding to rows 1, 4 and 7 use the equal weighting scheme while the statistics corresponding to rows 2, 5 and 8 use the sample size weighting scheme in forming the corresponding W vector. A possible reason why the standard deviation weighting scheme seems to diminish a quadratic form's performance is that the variance for the term TE is relatively small compared to the variance of TEÂ„ Thus, in the linear combination K,Â„TEÂ„ + K Â„TE n Inn, znn' most of the weight is being given to TE n Popovich (1983
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155 ) observed in his Monte Carlo, that this, in some cases, reduces the performance of the location statistic. Now we consider the alternatives in which the bivariate pairs were generated with location and scale differences {\iy= 0.5 and a 2 = 2.0, or u i = 0.5 and a 2 = 3.0, or u^ 1.0 and a 2 = 3.0). For these alternatives, the statistics corresponding to rows 7 and 8 are performing the best overall. The statistics corresponding to rows 1 and 2 are performing equivalently to rows 7 and 8, except for the alternative where y,= 0.5 and a 2 = 3.0 for the bivariate Pearson VII distribution with v=3 and p=.2, and the bivariate normal distribution with p.2 or p=.5 This possibly is reflecting the fact that CD performs slightly better than the distributionfree scale statistic TM n 1 ,n c when scale differences exist and, that, for this alternative (i.e., y^= 0.5 and a 2 = 3.0) large scale differences exist. As the correlation increases within any distribution, we see that all the statistics are performing moderately well and that for the last alternative where y^= 1.0 and o 2 = 3.0 when p=.5 or p=.8 no differences exist between the ten statistics in general. In summary, the recommended statistic is the quadratic form which uses CD and the sample size (SS) weights for TE (row 8). This statistic provides the best power in general for the alternatives considered here. The statistic corresponding to the quadratic form which uses CD and equal weights for TE n (row 7) performs for the most part 1 c
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156 equivalently but is not recommended due to its higher fluctuation of levels under the null hypothesis. Note, if instead of the more general alternative hypothesis considered here (i.e., H fl : y 1 ^ y 2 and/or a ^ ^ o 2 ) we can restrict the alternative to be more specific (i.e., H : a 2 t Oo) then, in many cases, the power of the test can be improved by using a statistic which is specifically designed for that alternative. Finally, we turn our attention to the last two statistics (11 and 12) included in this Monte Carlo. These statistics were included for two reasons. The first reason was to investigate their performance when the bivariate pairs were generated with equal marginal scale parameters but unequal marginal locations. The second reason was to determine what effect unequal marginal locations had on the power of the tests when the bivariate pairs had unequal scale. Looking at the columns labeled Location, we see that both tests are fairly robust when the components of the bivariate pairs have equal scale but unequal marginal locations. From the last three columns of each table, we observe that slight differences in the locations parameters do not affect the power of the scale statistics appreciably but as the location differences become more pronounced the power of each test is dramatically reduced. In conclusion, if slight differences exist between the location parameters
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157 for the marginal distributions, the tests for scale still perform appropriately but as the differences increase the tests have definite drawbacks.
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APPENDIX 1 TABLES OF CRITICAL VALUES FOR TESTING FOR DIFFERENCES IN SCALE The tables in this appendix list the critical values of T = T + T (i.e., T with L, = L = 1) for .01, .025, .05 and .10 n l > n c 12 levels of significance for n =1 2 1 5 and n 2 = l 2 3 10 These tables are also appropriate for TM Â„ since TÂ„ TMÂ„ For larger values of n, or n 1 ,n c n l n c n l' n c l n the asymptotic normal distribution of T (and TM ) could be used. When n,= or n = 0, the critical values can be obtained from the critical values for the Wilcoxon signed rank, distribution based on n, or n observations (respectively). The critical values for this test statistic were derived for each n, and n by convoluting two Wilcoxon signed rank statistics (based on n, observations and n observations). Thus, it follows that the critical values for a test based on n,= a and n = b are the same as the 1 c critical values for a test based on n,= b and n = a T observations. Therefore, the tables can also be viewed as listing the critical values for n, = 1 2 1 and n =1 2 1 5 These critical values are tabled for the test H : a i = 2 versus H : aiOo or l /u 2 H a : o^o 2 the symmetry of T about 1 n c 158
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159 n, (n,+ 1) + n (n +1 ) 11 c c could be used to calculate the necessary cutoffs. Since the Wilcoxon signed rank, distribution is discrete, exact .01, .025, .05 and .10 level critical values do not always exist. These tables list the following: 1) The critical value (c) for a specific a level, such that P{T n >c}c} = pvalue). n l n c 3) The attained significance level of the next closest critical value is given in the square brackets (i.e., P{T n >(cl)}). 1 c For example, let n,= 10 and n = 5, the critical value for a .05 level test would be 53. The attained signficance level for the test would actually be .048. The next closest critical value would be 531 = 52, with an attained signficance level of .059. When n, and n are both very small, (generally less than 3), many times a critical value does not exist for a specific level of significance. Then, the value in the
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160 n (n +1) + n c (n c +l) bracket is P(TÂ„ Â„ >ra) where m = (i.e., n l' n c ? the largest value T could be). n l' n c
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161 n c l n,=l n 1= 2 n 1= 3 01
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162 n c = 2 01 025 05 10 n 1= 2 Â— ( )[.063] Â— ( )[.063] ( )[.063] 6 (.063)1.1881 ni 3 Â— ( )[.031] Â— ( M.031] 9 (.031)[.094] 8 (.094)1.188] n 1= 4 Â— ( )[.016] 13 (.016) [ .047] 12 (.047)1.094] 11 (.094)1.172] 01 025 05 .10
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163 n c = 3 a
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164 n = 4 a
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165 n c = 5 a
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166 n c = 6 a
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167 n =7 a
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168 a
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169 n c =9 01 025 ,05 10 01 025 05 10 01 025 05 10 n 1= 9 n, =10 1^ = 11 73
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170 n =10 _2L 01 025 ,05 10 1^ = 10 88 (.008)[.010] 83 (.023)[.027] 79 (.045M.053] 74 (.093)1.106] 96 (.008)[.010. 91 ( .021 ) [ .025 86 (.047)[.054. 81 ( .092) [ 104 1^ = 12 104( .010) [ .012; 99 (.023H.026. 94 (.047) [ .053 88 (.098)[.109 01 ,025 ,05 10 n 1 13 1 14 ( .009) [ .010] 108( .023) [ .026] 103(.045) [ .051] 96 (.099)[.110] n 1 = 14 124( .009) [ .010 117 ( .025) [ .028 112( .046) [ .052 105( .097) [ 107 n 1 = 1 5 135( .009) [ .010 128 ( .023) [ .026 122(.046) [ .051 1 1 5 C .092)[ .101
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APPENDIX 2 THE MONTE CARLO PROGRAM The Monte Carlo program listed in this appendix, was written for this research using fortran (FORTXCG, i.e., SYSTEM/370 fortran H extended (enhanced)). Computing was done utilizing the facilities of the Northeast Regional Data Center of the State University System of Florida, located on the campus of the University of Florida in Gainesville. It used available IMSL subroutines (e.g., GGUBS GGNSM, RANK, ..., etc.) whenever possible. The single precision version of this library was used. 171
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172 DCOBLE PRECISION DSESD,DSEED2 DIMENSION POWERS (45) X1 ( 40) X2 ( 40) SIGMA (3) X(4Q,2) d)IRKVÂ£C{40) ,81fVEC(40) ,WKVEC (2) DFTST(3) ,DFMST (3) ,C(40) SEAL L1 (3) ,L2{3) INTEGER EEPS,N,NS,NNS,NPROb, NCD, NOCENS (4 1, 4 1 ) ,0(40) C C THIS PEOGB&S BUNS A MONTE CAELO FOR A SAMPLE SIZE UP TO 40 C C OBTAIN PARAMETERS EOE THIS RUN Of TEE MONTE CARLO C CALL I NIT (EEPS,N,NS,NNS,XMU,L1,L2,SIGMA,DSÂ£ED,DSEFD2) DC 5 1=1, NNS 5 POWERS (I) =0, NPEOB=0 NCD=0 C NPR03 IS THE OF SAMPLES HITH NO UNCENSOEED PAIRS, WHILE C NCD IS THE # OF TIMES CDTST HAS A NEC VARIANCE ESTIMATE C DC 10 1=1,41 DC 10 J=1,41 10 NOCENS (I, J) =0 C C START THE REPLICATIONS C CO 100 IREPS=1,RÂ£PS C C GENERATE AN 3 RANDOM BIVAHIATS NORMALS C WITH COVARIANCE MAIRIXSIGMA C CALL SAM?LE(DSSED,DSEED2,N,SIGMA,WKVEC,IHEPS,X,X1, M2,C) C C NOW TO PREFORM THE CENSORING ON THE RANDOM VARIABLES C CALL CENSOR (X 1 X2,C, N,D) C C CALCULATE THE TEST STATISTICS C CALL DFSTAI (X1 X2, D, N, XMU, LI ,L2,NPR03, NCCE NS,DFTST, olDFMST) IF (DFTST(1) .EQ. 999.9) GO TO 100 C CALL CDS TAT (X 1 X2, D, N, CDTST, CDTST2 ,CDTST3, NCD) C C COLLECT SUMMARY AND POWER STATISTICS C CALL POWER (NS,NNS,DFTST, DFfcST, CDTST, CDTST2 ,CDTST3, 3POWERS) 100 CONTINUE C
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173 C PEIST GDI THE RESULTS C WHITE (6,500) NPSOB 500 FOBS AT {'' ,'THE NUMBER OF SAMPLES DISCARDED, DUB ', Â£'TO N1 = WAS 1 1X,I8) SEITE (6,502) NCD 502 FORMAT ('', 'THE NUMBER OF SAMPLES RHICU HAD A ', a> NEGATIVE VARIANCE ESTIMATE FOR CD .if AS ,lX f l8) WEITÂ£(6,505) 505 FORMAT (Â•Â• ,'TtIE DISTRIBUTION OF CENSORING: THI2 SOWS', 3 ARE FOR TIPS 2 OR 3 AND THE COLUMNS FOR TYPE U //) DO 510 1=1,41 510 *RITS(6,515) I, (MOCESS (I, J) J=1,26) 515 FORMAT(' Â• ,12 : Â• ,4X 2614 ) IF (N .LT. 26) GO TO 530 DO 520 1=1,41 520 WRITS(6,525) I, (NOCENS (I G) J = 27,41) 525 FCRMAT( Â• 12 Â• : Â• ,4 X, 2514) 530 CONTINUE NPRCP=Q NX=N+1 DO 600 1=1, NX DC 601 J=1,KX bO 1 NPROP=NPBOP+ (((I1) + (J1)) *NOCÂ£NS (I, J)) 600 CONTINUE AVG=(FLOAT (NPROP) )/ (FLOAT ( REPS) ) KRITE(6,610) AVG d10 FORMAT ('', 'THE AVERAGE NUMBER OF OBSERVATIONS', a)' CENSORED IS: ',F105) WHITE(6,550) 550 FORMAT ('', 15X, 'FINAL RESULTS', /, 20X, a, 'STAT *1: DFIST1', / 20X,'STAT *2 : DFTST2', & / 20X,'STAT #3: DFTST3', / 20X,'STAI #4; DFHST1*, a / 20X,'STAT #5: DFMSI2', / 20X,'STAT *6: DFKST3', i / 20X,'STAT #7: CDTST (WITHOUT MEAN)', & / 20X,STAT #3: CDTST2 (WITH MEAN)', a) / 20X,'STAT 49: CDTST3 (ASYKP WITH MEAN)*) C CALL USWSH (Â• POWER MAT RIX/REJECTS Â• ,20 POW ERS NS 1) STOP END SUBROUTINE IN IT (REPS, N, NS, KNS, XMU, L1 L2, SIGMA DSE ED, (HD3EZD2) C C THIS SUBROUTINE READS THE NUMBER OF REPLICATIONS (REPS), C THE SAMPLE SIZE PER RUN (S ) THE POPULATION COVASIANCE C MATRIX (SIGMA) AND THE VECTORS LI AND L2 FOE THE STATISTIC C DFTST. IT STORES SIGMA IN SYMMETRIC STORAGE MCDE (IXSL) C DOUBLE PRECISION DSEED, DSEED2 REAL L1 (3) ,L2(3) DIMENSION SIGMA (3) INTEGER REPS
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174 REPS=10G0 N=25 N3=9 NS=(NS*(NS + 1) )/2 L 1 ( 1 ) = 1 L2(1)= 10 LI (2) =2.0 L2(2) = 1.0 L1 (3) = 1.0 L2(3)=2. DSEED=335768.D0 DSEED2=672344. DO C SHITE (6, 100) REPS, K 100 FORMAT (Â•Â• ,15X,' ft HEPS = ', 14, 10X '3 AMPLE SIZE (N) = Â• di!2) C C NOW TO READ IN THE COMMON LOCATION PARAMETER (XMC) C XKU=0.0 c C NOW TO ENTER THE COVARIAHCE MATRIX (SIGMA) C SHC=. 2 VARX1=1 ,0 V A 2X2= 3. SIGMA (1)=VASX1 SIGHA(2)=8H0*(SQ2T(VARX1) ) (SQBT (VAEX2) ) SIGMA {3)=VARX2 C ECHO CHECK C MRITE(6, 105) 105 FOSKAT(0'// 10X, BIVARIATE N03MAL DISTRIBUTION ', ^'GENERATED') CALL OSBSH(COV. MATRIX SIGMA 1 17, SIGMA, 2, 2) RETURN END SUBROUTINE SAMPLE (DSEED, DSEBD2 H,S IGMS,iKV EC, IREPS, a)X,X1,X2,C) r C THIS SUBROUTINE GENERATES THE NX2 RANDOM VECTOR OF C OBSERVATIONS C D0U3LS PRECISION DSEED, DSEED2 DIMENSION X(N,2) ,WKVEC{2) ,SIGMA (3) X1 (N) ,X2 (N) diU (40) ,C (N) INTEGER N,IEH C C CALL THE IMSL NORMAL RANDOM VECTOR SUBROUTINE C m'KVÂ£C(1) = 1. IF (IS EPS ,EQ. 1) WK?EC(1)=0.0
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17 CALL GGNSH(DSEED,N,2,SIGMA,H,X,WKVEC,IÂ£R) IF (IER .NE. 0) 1RITE{6,500) IÂ£EPS # IE2 500 FORMAT (', 10X, 'GGNSH ERROR, ESPL IC AI ICK= Â• 14, (Â£ IEfi= ,I i *) DO 25 1=1, N X1{I)=X(I,1) X2(I)=X(I,2) 25 CCNTINOE C C SOU TO GENERATE THE CENSORING DISTRIBUTION C C FIRST GENERATE A SAMPLE OF N UNIFORM (0 1 ) [; V Â• S C CALL GGUBS(DSEED2,N,U) C C NOB TO GENERATE THE CENSORING EANDOII VAFIAELES C DO 28 1=1, N ^8 C(I) =ALGG (o,9975*n (I)) C28 C(I)=ALCG(6.8371*U (I) ) C28 C(I)=ALOG(6, 3369*0 (I) ) C RETURN END SUBROUTINE SANK (HO ,Z HZ, IEi VEC SHV EC) C C THIS SUBROUTINE CALCULATES THE VECTOR RANKS C DIMENSION Z(NU) ,RZ (NU) ,IRWVEC(NO) RWVEC (NU) EFS=0. 00000001 CALL NHRANK(Z,NU,EPS,IRMVEC,RVEC,RZ,S2,S3) RETURN END SUBROUTINE CEfiSOH (X1 ,X2,C N, D) C C THIS SUBROUTINE CENSORS THE DATA AND CREATES A VECTOR D OF C THE TYPE OF CENSORED PAIR, A PARTICULAR PAIR 15, C DIMENSION X1 (N) ,X2 (?I) ,C (N) INTEGER D(N) C DO 6 1=1, N IF (X1(I) ,NE. X2 (I) ) GO TO 100 IF (X2{I) .LE. C(I)) D(I)=1 IF (X2(I) .GT, C(I)) GO TO 102 GO TO 6 100 IF (X2(IJ LE. X1(I)) GO TO 4 IF (X2(I) .GT. C(IJ) GO TO 105 D(I)=1 GO TO 6 105 IF (X1(I) .LE, C(I)) GO TO 110 102 D(I)=4
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176 X1(I)=C(I) X2(I)=C(I) GO TO 6 1 1C D(I)=2 X2(I)=C(I) GO TO 6 4 IE (X2(I) .LE. C(I)) GO TO 115 D(I)=4 X1(I)=C(I) X2(I)=C(I) GO TO 6 115 IF (X1(I) ,LE. C(I)) GO TO 120 D(I)=3 X1(I)=C(I) GO TO 6 120 D(I) = 1 6 CONTINUE I5ETUBN END SUBROUTINE ESTHU (X1 X2,B ,K,EMU) C C TUIS SUBROUTINE CALCULATES THE COMBINED SAMPLE MEDIAN C USING THE KAPLAN MEIES ESTIMATOR KITE SMOOTHING (E3U) C AND WITHOUT SMOOTHING (SMU) C DIMENSION X1 (N) ,X2{N) ,BVEC(80) BY ( bO) Y ( 80) ,11(80) dIiVEC(30) r S(80) INTEGER D(N) ,DD(S0) ,DYY(80) K=1 DO 10 1=1, N IF (D(I)3) 20,20,11 20 J=2*K JJ=J1 Y(J)=X2(I) Y(JJ)=X1 (I) IF (D(I)2) 22,24,25 22 DD(J)=1 DD(J1)=1 GO TO 28 24 DD(J)=0 DD(J1)=1 GO TO 28 26 DD(J) = 1 DD(J1)=0 GO TO 28 11 J=2*K JJ=J1 Y(J)=X2(I) +0. 000000 1 Y(JJ)=X1(I) Â£D(J)=0 DD(JJ)=0 28 K=K+1
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177 10 CONTINUE C CALL HANK (J, Y,E Y,IWVÂ£C, SVEC) DC 780 1=1, J L=IPIX (aY(I) ) YY(L)=Y(I) 7&0 CYY(L)=DD(I) C C COMPUTING TOE KAPLAN MEIES ESTIMATE CF THE SURVIVAL C FUNCTION C XJ=FLOAT (J) S (1) = ((XJ1. 0)/XJ) **DYY{1) DC 500 1=2, JJ XI=FLGAT(I) 500 S(I)=S (11) *(( (XJXI)/(XJXI+1.0) ) **DYY(I) ) S (J) =0.0 C C NOa TO CALCULATE THE BSD I AH ESTIMATES C DO 550 1=1, J IF (S (I) .500000) 530,540,550 540 EMU = YY(I) GO TO 560 b30 E1 = YY(I) E2=YY(I1) EMU = E1((E1E2)*(.5 000 0GS(I))/(S (11) S (I) ) ) SMU = (E1+E2)/2.0 30 TO 560 550 CONTINUE 560 CONTINUE C EETUEN END S UB POUT I NÂ£ DFS TAT ( X1 X2 D N XS U L 1 L 2 N? B CB, NOC ENS oBDFTST,DFMST) C C THIS SUBROUTINE COMPUTES THE TEST STATISTIC CALLED DFTST C (DFMST), WHICH IS THE COMBINATION OF TWO SIGNED SANK C STATISTICS DFTST (I) = L 1 ( I) *i*ILCG XON + L2 (I) HI LCOXCN C SEAL L1 (3) ,L2(3) DIMENSION XI (N) ,X2 (N) ,Z(40) ,T2 3 (40) PHI (40 ) BH VEC (40) aDFHST(3) DFTST (3) ,3Z(40) ,T(3) ,VAET(3) ,SDT(3) ,RT23(40) d)KZZ(40) IE VEC (4 0) GAS (40) ZZ (40) ,ZPHI (40) ,TT (3) E I( 3) INTEGER D(N) NCCSNS ( 41 4 1 ) 81=0 NC=0 C C NCa TO ESTIMATE THE VALUE OF MO (EU) USING THE PEODUCTC LIMIT ESTIMATOB BASED ON THE ENTIEE SAMPLE C CALL ESTMU(X1,X2,D,N,EMU)
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178 C IKS FIRST PART OF THE SUBROUTINE, UTILIZES WHAT TYPE OF C PAIR (X1,X2) IS: TYPE 1 = UNCEKSGRÂ£D,TYPE 2= X2 CEHS03ED, C TYPE 3= XI CENSORED, TYPE 4=CCTH CENSORED C AND PLACES THE UNCEN3CE3D CALCULATION'S IN T1 WHILE TEE C TYPE 2 OR 3 CENSORED C VALUES GO INTO THE VECTOR T23 C DO 6 1=1 ,N IF (D(I) EQ. 4) GO TO 6 IF (D(I)2) 5,3,4 C C A PAIR HILL GO TO 3, IF IT IS A TYPE 2 PAIR C 3 NC=NC+1 GAM (NC) =1.0 T23 (HC)=X2(I) GO TO 6 C C A PAIR WILL GO TO 4, IF IT IS A TYPE 3 PAIS C 4 NC=NC+1 GA(NC)=0, T23 (NC)=X1 (I) GO TO 6 C C A PAIR WILL GO TO 5, I? IT IS A TYPE 1 PAIS C o N1= N1 + 1 Z(H1J=(A3S ((X2(I)) iaO) ) (ADS ( (X 1 (I) )XfiÂ£J) ) ZZ(N1) = (AB3((X2(I) )EMU))(ABS((Xl (I)) EMU)) PHI (N1) = (SIGN (1.0,Z (Nl))/2.0) +0. 5 ZPHI(NI) = (SIGN (1.0, ZZ(Nl))/2.0)+0, 5 Z(N1) = ABS IZ(N1)) ZZ(N1) =AES (ZZ(N1)) 6 CONTINUE IF (N1 .EQ. 0) GO TO 100 C C TO INSERT WHAT TYPE OF CENSORING OCCURRED INTO THE MATRIX C NOCENS (NN4=#TYPE 4 PAIRS + 1, NNC=#TYPE 2 OR 3 PAIRS + 1) NN4=1+NN1NC NNC=NC+1 NOCENS (NNC,NN4)= NOCENS (NNC,NN4) +1 C C CALCULATING THE ABSOLUTE RANKS FOR THE S1 UNCSNSCRED 003. C CALL RANK(Nl,Z,RZ,IRWVEC,EfcVÂ£C) CALL RANK (N 1 ZZ 3ZZ, IR VEC RWVEC) C C NOW TO CALCULATE THE RANKS OF THE C'S FOR TOE TYPE 2 AND 3 C IF (NC ,NE, 0) GO TO 25 WRITE (6,23) NC 2 3 FORMAT (Â•,Â• THERE ARE NO TYPE 2 OB 3 CENSORED Â•,
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179 S'OESEHVATIOSS, NC = Â• ,13) GC TO 23 25 CONTINUE CALL RANK (NC, T23,BT23, IRWVEC,BIVEC) C 28 CONTINUE C C NOW TO CALCULATE THE WILCOXON TYPE STATISTICS, SDMI AND C SUHC, AND THE CORRESPONDING EXPECTED VALUES AND VARIANCES C SUM1=0, S0H1 1=0. DO 30 1*1, Ml SUK1 = SUM1 + (PHI(I)*EZ (I)) sumii=sumiu (zphi (i) *bzz(i)) 30 CONTINUE VAR 1 = (FLOAT (N1* (N1 + 1 ) ( (2*S1) + 1} ) )/24. C E1 = (FLOAT (N1*(N1 + 1)) )/4.0 C SUMC=Q, IF (NC ,NE 0) GC TO 3 3 VAHC=0.0 EC=0.0 GC TO 35 i^ CONTINUE DO 34 1=1, NC SUMC=SUMC+ (GAM (I) *BT23 (I) ) 34 CONTINUE VASC= (FLOAT (NC* (NC + 1 ) ( (2*NC) + 1) ) ) /2 4 EC= (FLOAT (NC*(NC+1)))/4.0 C 35 CGKTINUE C C NOW TO CALCULATE THE DFTST C DO 38 1=1,3 T(I) = (L1 (I)*SUM1) + (L2 (I) *SUMC) TT (I) = (L1 (I)*SUM11) + (L2(I) *SUMC) ET(I) = (L1 (I)*E1) + (L2 (I)*EC) VAET(I)= ((L1 (I) **2)*VAR1) + ( (L2 (I) *2) *AEC) SDT (I)=SQRT(VAST(I)) EFTST(I) = (T(I)ET(I) ) /SDT (I) DFMST (I) = (TT(I) ET(I) )/SDT(I) 3e CONTINUE GO TO 47 100 CONTINUE C C THERE IS A PROBLEM, N1=0, THUS THE SAMPLE IS NOT GOING TO C BE USED IN THE POSER STUDY. THE TEST STATISTIC KILL 5E C SET TO 999.9 WHICH WILL BE USED AS AN INDICATGE. C DO 102 1=1,3 DFMST (I) =999.9
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180 102 DFTST (I) =999.9 KEITE(6,104) 104 FOBHAT( l *** P20BL2M, B1 = 0, THE SAMPLE KILL DOT Â•, 3BE USED IS THE POWER STUDY 1 ) NPfiGE=N?ROB+1 47 8ETUEN END SUBROUTINE CDSTAT {X1 ,X2, D, N, CDTST, CDTST2 CDTST3, NCR) C c C THIS SUBROUTINE CALCULATES THE CCNCOREANT 'JI5CC3DANT C TYPE STATISTIC (CDTST) C DIMENSION X1 (N) ,X2(N) Y1 (40) Y2{40) INTEGER D(N) ,A{4O,40),B (40,40) SO BCD, S U3 A SUM 3, SUM 1 a>SUK2,SUa3,SUMGG DO 1 1=1, N DC 2 J=1,N A(I,J)=Q 2 B(I,J)=0 1 CONTINUE DO 5 1=1, a Y1 (I) =X1 (I) +X2 (I) 5 Y2 (I)=X1 (I)X2(I) C aoa TO CALCULATE THE A (I, J) AND B(I,J) MATRICES C KN=N1 DO 10 1=1, N'N 11=1+1 DO 20 J=II,N IF (Y1 (IJY1 (J)) 200,230,220 IF (D(I) ,EQ, 1) A(I,J) = 1 GO TO 230 IF (D(J) ,EQ, 1) A(I,J)=1 IF (D(I) EQ. 4) GO TO 10 IF (D(J) EQ. 4) GO TO 20 IF (Y2 (I)Y2(J)) 240,20,260 IF (D(I)3) 242,20,20 IF (D(J) .20.. 1) B(I,J)=1 IF (D(J) ,Â£Q. 3) B(I,J)=1 GO TO 20 IF (D(J)3) 262,20,20 IF (D{I) .EQ. 1) B(I,J)=1 IF (D(I) .EQ, 3) 3(I,J)=1 20 CONTINUE 10 CONTINUE 200 220 230 240 242 2b0 2o2 C CALCULATING THE CONCOSDANIDISCOBDANT STATISTIC (CDTST) C SUMCD=0 SUA=0
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181 SOMG=0 S0H1=0 SOM2=0 SUH3=0 SUHGG=0 I N = iilv=NÂ— 1 DO 40 1=1, BH 11=1+1 DO 42 J=II,N SOKCD=SUMCD+(A(I r J)*B (I,J) ) SUHA = SUMA+IABS(A (I J) *E (I J) ) IF (J EQ. li) GO TO 45 JJ=J+1 DO 45 K=JJ,N SUM 1 = SUM 1+ (A (I, J) *A(I,K) *5 (I, J)*5 (I, K) ) SUM2=SUK2+(A(I,K)*A (J,K)*3 (I,K) *B(J,K) ) SOH3=SOH3* (A (I, J) *& { 3, K) *B {I, J) *li{ J, K) ) IN=IN+3 45 CONTINUE 42 CONTINUE 40 CONTINUE VN = FLOAT(N) VNN=VN* (VN1,000) CE=2. 000* (FLOAT (SUMCD) )/VKN SUMG=SUaUSUM2 + SUM3 S[JMGG=SUM1+S'JM2+SUr'3+SUMA AAA=(2.000* (FLOAT(SUMA))/VNS)(CD*CD) VIu=FLGAT(IN) VNNN=FLOAT(N* ( (N1)**2)) G= (FLOAT (SUaG) ) /VIN GGG=( (2.000* (FLOAT (SUHGG) j )/VN8N)(CD*CD) VA5CD=(4.000*G)/VN VAECD2=( (2,0Q0*AAA)(4,00 *(VN2, 0) *GGG) )/VNN VABCD3= (4. 000*GGG) /VN IF (VA2CD2 .61. 0,0) GO TO 55 CDIST=0. 00 CDTST2=0.00 CDTST3=0,00 NCD=NCD+1 GO TO 65 55 SDCD=SQET(VAKCD) SDCD2=SQET(VARCD2) SDCD3=SQET(VARCD3) CDTST=(CD)/ (SDCD) CDTST2= (CD)/(SDCD2) CDTST3=(CD)/(SDCD3) 6 5 CONTINUE EE1DKN END S OBROUTI HE PO Efi (NS, N US, DFT3T DFHS T CUTS! CDTST2 SCDTST3, POKERS)
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182 NOTE: DIMENSION POWERS (NN3) REJECT (9) ,DFT3T (3) ,DFM3I (3) NS=NUiiBEB OF STATISTICS CALCULATED SSS= NS(NS+1)/2 GIVE THE CRITICAL VALUES FOR THE TEST STAIISIICS ZCEIT=1.645 BZCRIT=1.b45 GE,
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BIBLIOGRAPHY Anscorabe, F.J. (1952). LargeSample Theory of Sequential Estimation. Proc. Cambridge Philos. Soc 48, 600607 Bell, C.B., and Haller, H.S. (1969). Bivariate Symmetry Tests: Parametric and Nonparametric. Annals Math. Stat., 40,259269. Hajek, J., and Sidak, Z. (1967). The Theory of Rank Test; New York: Academic Press. Hoeffding, W. (1948). A Class of Statistics with Asympt oticallyNormal Distribution. Ann. Math. Statist. 19, 293325. Hoeffding, W. (1961). The Strong Law of Large Numbers for US tat is tics Institute of Statistics Mimeo Series No. 302 University of North Carolina at Chapel Hill. Hollander, M. (1971). A Nonparametric Test for Bivariate Symmetry. Biometrika 58, 203212. Johnson, M.E., and Ramberg, J.S. (1977). Elliptically Symmetric Distributions: Characterizations and Random Variate Generation. Stat. Comp. Sect. Proc. of ASA 262265 Kalbf leisch, J.D., and Prentice, R.L. (1974). Marginal Likelihoods Based on Cox's Regression and Life Model. Biometrika 60, 267278. Kaplan, E.L., and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. J. Amer. Stat. Assoc. 53, 457481. Kepner, J.L. (1979). Tests Using the Null Hypothesis of Bivariate Symmetry. Ph.D. Dissertation in Statistics, University of Iowa. Koziol, J. A. (1979). A Test for Bivariate Symmetry Based in the Empirical Distribution Function. Comm in S t at A8, 207221. Mantel, N. (1966). Evaluation of Survival Data and Two New Rank Order Statistics Arising in its Consideration. Cancer Chemother. Report, 50, 163170. 183
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184 Miller, Jr., R.G. (1981). Survival Analysis New York: John Wiley and Sons. Oakes, D. (1982). A Condordance Test for Independence in the Presence of Censoring. Biometrics 38, 451457. Popovich, E.A. (1983). Nonparame t r i c Analysis of Bivariate Censored Data. Ph.D. Dissertation in Statistics, University of Florida. Randies, R.H. (1982). On the Asymptotic Normality of Statistics with Estimated Parameters. Annals of Stat. 10, 462474. Randies, R.H., and Wolfe, D.A. (1979). Introduction to the Theory of Nonparametric Statistics New York: John Wiley and Sons. Sander, J.M. (1975). The Weak Convergence of Quantiles of the ProductLimit Estimator. Stanford Technical Report #5, Stanford University. Seigel, D., and Podger, M. (1982). A Sign Test for Significance Differences in Survivorship Curves from Paired Truncated Data. Controlled Clinical Trials 3, 6971 Sen, P.K. (1967). Nonparametric Tests for Multivariate Interchangeabi lity Part 1: Problems of Location and Scale in Bivariate Distributions. Sankhya 29, 351372. Serfling, R.J. (1980). Approximation Theorems of Mathematical Statistics New York: John Wiley and Sons. Sproule, R.N. (1974). Asymptotic Properties of UStatistics Trans. Am. Mathematical Soc 199, 5564. Woolson, R.F., and Lachenbruch, P .A (1980). Rank Tests for Censored Matched Pairs. Biomet rika 67, 597606.
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BIOGRAPHICAL SKETCH Laura Lynn Perkins was born in Harvey, Illinois, on August 1, 1957. She moved to Lawndale, California, in 1961 and remained there until she moved to Titusville, Florida, in 1964. After graduating from Titusville High School in 1975, she enrolled at the University of Florida. Upon receiving her Bachelor of Science degree in mathematics in 1978, she entered the Graduate School and received her Master of Statistics degree in 1980. She expects to receive the degree of Doctor of Philosophy inAugust, 1984. She is a member of the American Statistical Association and the Biometric Society. Her professional career has included teaching various courses in the Statistics Department and consulting in the Biostatis tics Unit of the J. Hillis Miller Health Center at the University of Florida. She has been the recipient of Graduate School fellowships, graduate as s is tant ships the Statistics Faculty Award and nominated for a Graduate Student Teaching Award during her academic career at the University of Florida. 185
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ronald H. Randies, Chairman Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. >1 IvoA^Malay Gho; Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. John G. Saw Professor of Statistics I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Jonathan J. Shuster Professor of Statistics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Wayne Profes'sor of Environmental Engineering This dissertation was submitted to the Graduate Faculty of the Department of Statistics in the College of Liberal Arts and Sciences and to the Graduate School, and was accepted for partial fulfillment of the requirements of the degree of Doctor of Philosophy. August, 1984 Dean for Graduate Studies and Research
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UNIVERSITY OF FLORIDA 31262 08285 339
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