RANK ORDER TESTS FOR It.'TERCHANGEABILITY IN SOME
RESTRICTED A.."ID INCOMPLETE MODElS
by
George R. Jerdack
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No.1826T
July 1987
RANK ORDER TESTS FOR INTERCHANGEABILITY IN SOME
RESTRICTED AND INCOMPLETE MODELS
by
George R. Jeroack
A Dissertation submitted to the faculty of The University
of North Carolina at Chapel Hill in partial fulfillment of
the requirements for the degree of Doctor of Philosophy
in the Department of Biostatistics
Chapel Hill
1987
Approved by:
:Adviser
,/
,/
....' /;>-:-: -'- '--
Reader
ABSTRACT
GEORGE R. JERDACK. Rank Order Tests for Interchangeability in
Some Restricted and Incomplete Models (Under the direction of Pranab
K. Sen.)
To study the effects of different treatments, or the effects of
different levels of some drugs, one may wish to test whether several
treatments or levels of a drug, may be used interchangeably, Le.,
within each block of observations the responses are equivalent. In
some cases we may have several sets of treatments, and within each
set of treatments we have several drugs. One may be interested in
testing whether the drugs wi thin each set of treatments may be used
interchangeably, for all sets simultaneously, this will be termed the
hypothesis of restricted interchangeability.
In the nonparametric case, Sen (1967) proposed a distribution-free
rank order test to test the hypothesis of interchangeability against shift
in location alternative. In this research, we extend this test to test the
folloWing three hypotheses:
a) HOh the hypothesis of restricted interchangeability against shift in
location alternative within each treatment set.
b) H02 ' the hypothesis of interchangeability against shift in location
alternative when some of the blocks are incomplete.
c) H03 ' the hypothesis of interchangeability against the alternative
hypotheSiS that the scale parameters are not equal and/or association
parametes are not equal.
Exact and large sample distributions are studied for each hypothesis.
For a) and c), when the distribution is normal, the parametric tests
(Wilks 1946, Votaw 1948) and the nonparametric tests (using normal
scores) showed an asymptotic relative efficiencies (A.R.E.) of values
close to one. While if the distribution is not normal the parametric
tests are less efficient for obvious reasons. In case b), we extend the
distribution-free rank order test using two ranking methods, Method 1
the intra-block ranking method (as in Friedman's chi-square procedure),
and Method 2 the ranking after alignment method. Under certain
assumptions, Method 2 showed better A.R.E., but under weaker
assumptions this may not be the case.
Applications to real life data are prOVided. All parametric and
nonparametric tests gave close results. Also, Methods 1 and 2 showed
same results when the data is incomplete.
It is a pleasure to express my appreciation to my adviser Dr. P. K.
Sen, for providing me with the topic and for his gUidance throughout my
graduate studies at the Department of Biostatistics. Special thanks are
due to Dr. Quade, for his valuable comments on my dissertation work,
and his gUidance during my work with him as a researcn assistant.
Thanks are also due to the other members of my advisory committee,
Drs. E. Davis (who provided me with the data for my dissertation),
S. Bangdiwala, K. Habib, and V. Kulkarni.
There are so many people who offered me a great deal of support
and encouragement throughout my graduate studies in Chapel Hill.
Among these are my former roommate Victor Peres-Abreu, my friends
in Biostatistics Dennis Cosmatos, Ali Barakat, Jerry Schindler, Nancy
Lucas, Cicilia Wada, Susan Reade, Janella Pantula, Marie Matteson,
Kathleen McCarroll and Muhammed Habib. I also wish to thank all the
friends that I have met in Chapel Hill especially my Lebanese friends
for their extraordinary support.
Finally, I would like to thank my family and relatives in Lebanon
and in the States, and especially my father Rafic Jerdak, for all the
encouragement and support they have given me throughout the years.
TABLE OF CONTENTS
Introduction and Literature Review
1
1.1
Introduction
1
1.2
Literature Review for the Parametric Case
4
1.3
Literature Review for the Nonparametric Case
13
1.3.1
Rank Order Test
13
1.3.2
Rank Order Test in Two Way Layouts
16
1.3.3
Incomplete Paired Observations
21
1.4
Proposed Research
23
Nonparametric Test of Restricted Interchangeability
26
2.1
Introduction
26
2.2
Preliminary Notions
28
2.3
Rank Permutation Test for the Case of Restricted
Chapter I
Chapter II
InterchangeabiIi tY
29
2.3.1
Permutational Distribution
29
2.3.2
Proposed Test
32
2.4
Asymptotic Permutation Distribution of W
N
Asymptotic Multinormality of the Standardized
33
2.5
38
Form of TN
2.6
Asymptotic Relative Efficiency
49
2.7
Remarks
52
Chapter III
Distribution-Free Rank Order Tests of
Interchangeability with Incomplete Data
53
3.1
Introduction
53
3.2
Preliminary Notions and the Exact Distribution
55
3.3
Proposed Test
61
3.4
Ranking After Alignment Method
66
3.4.1
AsymptOtic Permutation Distribution of the
Proposed Test
3.4.2
Asymptotic Multinormality of the Standardized
3.5
Form of T(2)
N
Intra-Block Rankings Method
3.5.1
AsymptOtic Permutation Distribution of the
Proposed Test
66
69
77
77
3.5.2
Asymptotic Multinormality of the Standardized
3.6
Form of TO)
N
Asymptotic Relative Efficiency
81
3.7
Remarks
90
Chapter IV
78
Rank Order Tests For Interchangeability Against
Non-Location Al ternati ve
91
4.1
Introduction
91
4.2
Preliminary Notions
93
4.3
Permutation Tests for H
O
Asymptotic Permutation Distribution of L
95
4.4
4.5
N
AsymptOtic Normality of SN' for Arbitrary F
101
103
e
Numerical Examples of the Test Procedures
122
5.1
Introduction
122
5.2
Examples of the restricted Interchangeability Test
123
5.3
Example on the Test of Interchangeability Against
Chapter V
a Non-Location Alternative
5.4
5.5
131
Example on Rank Order Tests of
Interchangeability With Incomplete Data
140
Recommendations for Future Research
149
References
150
Appendix
155
CHAPTER I
INlRCXJUC'llON AND LITERAnJRE REVIEW
1.1 Introc:hrlion
To study the effects of different treatments, or the effects of
different levels of some drug, one may wish to test whether several
treatments or level of drugs may be used interchangeably. Consider
the case of three treatments, and assume that the responses of
indi viduals on the three treatments have a normal three-variate
distribution (parametric case). Then the hypothesis of interchangeability is equivalent to the hypothesis that in a normal distribution
the means are equal, the variances are equal, and the covariances
are equal. When this hypothesis is true the normal distribution is
invariant over all permutations of variables and is said to have
complete symmetry. It is frequently more important, however, not
onl y to test that the treatments have completely symmetric
relations with each other but also they are interchangeable with
regard to their relation to some outside criterion measure (e.g.,
the criterion might be weight before treatment or time spent at the
clinic during treatment). Assuming that the responses of
individuals on the three treatments and the criterion have a normal
four-variate distribution, the hypothesis of interchangeability is
eqUivalent to the hypothesis of equality of means, equality of
variances, equality of covariances among the three treatments and
equality of covariances between treatments and criterion. When this
hypothesis is true, the four-variate normal distribution is invariant
over all permutations of the three variates (associated with
treatments) among themsel ves and so the variance-covariance
matrix has the following form :
A ICC C
--1------C
BOD
C lOB
C I 0 0
0
B
1
J
where the quantity A represents the variance of the criterion
measure, C represents the covariance between the criterion and
each treatment, B represent the variance of each treatment, and 0
represents the covariance between any two treatments. A normal
distribution for which this hypothesis is true is said to have
compound symmetry of Type 1. A more general case of compound
symmetry of Type I arises when there are several groups of
treatments (no two of which need to have the same number of
treatments) and several outside criteria. The hypothesis of
complete symmetry may also arise in certain medical research.
For example, suppose a measurement (e.g., % CO 2 in blood) is
made at each of three times (say T h T 2, T 3) on an experimental
animal, and assume that the three quantities have a normal threevariate distribution. One may then be interested in testing the
2
hypothesis of complete symmetry on the basis of measurements
(considered as a random sample) made on several experimental
animals. More generally, let there be two characteristics, say U
and W (e.g., % CO 2 in blood and % O 2 in blood), which are both
measured at each of two times, T 1 and T 2. Let it be assumed that
the four quantities, which we represent as UT 1, UT 2, WT 1, and WT 2,
have a normal four-variate distribution. One may then be interested
in testing the hypothesis that the means of the first two variates are
equal, that the means of the second two are equal, and that the
variance-covariance matrix has the form :
I
E
F
K
L
F
ElL
K
------1-----K
L
L
K
I
I
G
J
J
G
When this hypothesis is true, the four-variate distribution is said to
have compound symmetry of Type II which arises when there are h
characteristics and n times (h,n =2, 3, ... ).
The above case was the parametric case where we assumed that
the distribution was normal (more about the parametric case comes
in the next two sections). However, in the proposed research we
will concentrate on nonparametric structure in testing for interchangeability. In this case there is no need for the assumption of
normality. More specifically, suppose that we have n random
3
vectors Y l' Y 2' ... , Y n' where
Ya
= ( Ya 1 '
a
Ya2 , ... ,Yar )
= 1,
... ,n
r~2
has an unknown distribution F ( Y
ci ' and
i
j
= 1,
= 1,
,n
,r
Pj~l
where
{Yij }
j=l, ... ,r, are mutually exclusive subsets of Y for
i
i=l, ... ,no
Then our main interest is to test the hypothesis of interchangeability within {Y } for j=l, ... ,r simultaneously and for
ij
each i=1, ... ,no
As we will see in Section 1.3, nonparametric conditionally
distribution-free rank order tests have been developed for each of
the above parametric cases. Also in Section 1.2, some of what has
been done in the parametric case is explained.
1.2 Literature Review for the Parametric Case
Statistical test criteria for testing equality of means, equality
of variances and equality of covariances were introduced by Wilks
(1946). In his paper, a random sample of size n, ( Le., Y it i= 1,
... ,n) from a population which has a normal multi variate distribution with k=P1 variables was taken. Then he considered the
4
following three hypotheses :
(i) Hmvc ' the hypothesis that the means are equal, the variances
are equal, and the covariances are equal.
(ii)
Hvc ,the hypothesis that the variances are equal and covariances
are equal, irrespective of the values of the means.
(iii) H ,the hypothesis of equal means assuming that the variances
m
and covariances are equal. As in most applied problems, the
Neyman-Pearson method of likelihood ratios is used to develop the
following test criteria:
to test H ,
m
(b) L
(c) L
I s 1..1 I
vc
mvc
( s2 )k ( 1 - r )k-l ( 1 + ( k - 1 ) r ]
=L
where Is ..
lJ
matrix,
1
s ij =
vc
. L k- 1
m
to test Hvc
to test Hmvc ,
is the determinant of the sample variance-covariance
n1 2 na = 1
-
-
(yi a - yi ) (y j a - Yj )
5
The exact moments of each of the three criteria when the three
corresponding hypotheses are true were determined for any number
k of variables and for any size n of the sample. Asymptotically he
showed that -n In ( L
), -n In (L ), and -n ( k - 1 ) In (L )
mvc
vc
m
for large values of n are approximately distributed as chi-squared
wi th
~
k ( k + 3 ) - 3, ~ k ( k + 1 ) - 2, and ( k - 1 ) degrees of
freedom, respectively.
Also tables for 5 % and 1 % level of
significance were provided for certain values of k and for both the
exact and the asymptotic cases.
Votaw (1948) extended the above results to test the hypothesis
of "compound symmetry" in a normal multivariate population of
= 2:J=1Pj
k
variates ( k~3 ) on the basis of samples. After parti-
tioning k variates into r mutually exclusive subsets of variates with
p. variates in each subset, he formulated twelve hypotheses to test
J
interchangeability as follows:
(1) H1 (mvc) : The hypothesis that within each subset of variates the
means are equal, the variances are equal, and the
covariances are equal and between any two distinct
subsets of variates the covariances are equal.
(2) H1 (vc)
The hypothesis that within each subset of variates the
variances are equal and the covariances are equal and
that between any two distinct subsets of variates the
covariances are equal.
(3) Hi (m)
The hypothesis that wi thin each subset of variates the
6
means are egual given that H1 (vc) is true.
(4) Hk (MVC/mvc): The hypothesis that k normal t-variate distributions are the same gi yen that they all satisfy
H1 (mvc) for a particular division of variates
into subsets ( k
~
2).
(5) Hk (VC/mvc) : The hypothesis that k normal t-variate distributions have the same variance-covariance matrix,
gi yen that they all satisfy H1 (mvc) for a particular
di vision of the variates into subsets ( k
~
2 ).
(6) H (M/mvc) : The hypothesis that k normal t-variate distribk
utions are the same, given that they all satisfy
H1 (mvc) for a particular division of the variates
into subsets and that they all have the same
variance-covariance matrix ( k
~
2) .
-
The other six hypotheses, H1 (mvc) , H1 (vc), ... ,H (M/mvc), are
k
modifications of H1 (mvc), H1 (vc), ... , H (M/mvc), respectively.
k
In regard to any of these six H hypotheses, it is assumed that there
is a partition (h) (n = 2,3, ... ) of t variates ( t = nh ) and that in
each subset the variates are in a given order; thus each subset has
n variates and between any two distinct subsets of variates there
are n2 covariances, which form an n x n "block" in the variancecovariance matrix of the distribution. The hypotheses may now be
stated as follows :
7
-
H1 (mvc) : the hypothesis that within each subset of variates the
means are equal, the variances are equal, and the covariances are
equal and that between any two distinct subsets of variates the
diagonal covariances are equal and the off-diagonal covariances are
equal.
H1 (vc):
The hypothesis that within each subset of variates the
variances are equal and the covariances are equal and that between
any two distinct subsets of variates the diagonal covariances are
equal and the off-diagonal covariances are equal.
The statement of any of the hypotheses H1 (m) , Hk (MVC/mvc) ,
H (VC/mv c) , and H (M/m VC) is obtained from the statement of
k
k
the corresponding H hypothesis by simply substituting H for H .
As in Wilks' paper, Votaw used the Neyman-Pearson likelihood
ratio method to develop the twelve sample criteria (In this paper,
we will report the results that corresponds to H1 (mvc); the rest of
the results are similar and can be found in Votaw (1948) ). The
sample criterion for H1 (mvc) is :
L 1 ( m vc ) --
h
I vi j I [ITa=
1 ( va '
where,
,
va =
~
- 2
n1a Loa
i (y i a - y r )
' a
a
a
8
- (n -1)
- w')
a
a
"-1
]
I vrr' I
;
He also found the exact moments, the exact distribution, and the
approximate distribution for each criterion. In particular he showed
that -N In L 1 (mvc) is approximately distributed as chi -squared
wi th
1[k
( k + 3 ) - b ( b + 3 ) - r ( r + s ) ] - rb degrees of
freedom, where b is the number of subsets that contain one variate
each.
So far we have reviewed what is called the intraclass correlation
model; however, in the literature there are other types of models to
test for symmetric structure in the covariance matrix, for example :
H1 : }:
= Diag ( ~ ,
... ,
~
which is called sphericity.
)
which is called intraclass
correlation (special case of what
what was explained previously).
whIch is called circular
symmetry.
which is called a general
matrix form.
where
p .=p.=k
J
1
( dimension of each vector Y i )
9
}:e
= II cov
(Y.,Y.+
l l m
) II
cov (y.,Y.+
=}:
and
11m
)
=}:
m
p-m
m=O,1, ... , p .
For example, if p=4 and p=5 we obtain
~ }:. }:z }:.
}:
c
=
}:.
~ }:. }:z
}:z }:. ~ }:.
}:. }:z }:. ~
}:o }:. }:z }:2
}:=
C
}:.
}:.
~ }:. }:z }:z
}:2
}:.
}:o
}:.
}:2
}:2 }:2 }:. ~ }:.
}:. }:2 }:2 }:. }:o
It is clear that sphericity is a special case of intraclass
correlation, while the general matrix model is self explained. Olkin
(1969 and 1973) extended the circular symmetry model to test for
symmetric structure in the covariance matrix, by reducing it to a
canonical form; then he used the likelihood ratio test to test the
following:
a) H1 vs. H Le.,
3
b) H vs. H Le.,
3
2
c) H vs. H Le.,
3
4
Since each of the
sphericity versus circular symmetry,
intraclass correlation versus circular symmetry,
circular symmetry versus general form.
above involves circularity, we have p covariance
10
e
matrices which can be estimated by VII' ... , Vpp . Let,
Casel : p = 2m, Le. even;
V
Case2 : p
= 2m+ 1,
mm+ V m+2,m+2' V m+l,m+l )
Le. odd:
and define,
V.J
= VP-J.+2
j=2, ... ,p
then the test criteria for a), b), and c) are:
a)
2/N
A13
with
2
-2
P { -pIn L13 ~ z } ~ p { Xf:S: z } + a (n )
where
f =
!
mk ( k+ 1 )
_ 1 _ J P ( 3m+3:p ) -2 ] (2k 2 +3k-l )
p12nmp ( k+ 1 )
11
\
b)
23 -
( p-l ) (p-l )k
TIP_I
V. I
]-2
1
- L
22k(p-m-1) I ~=2 Vii ,p-1 - 23
with
p { -pIn L
23
~
z}
~
p { Xf2 ~ z } + o(n -2 )
where
f = ~ mk ( k+ 1 )
p = 1 _ (3mp - p2 - 3m + p - 3 ) ( 2k
2 + 3k - 1 )
12nm ( p-1 ) (k+ 1 )
2/N
\34
c)
=
22k(p-m-l) I V I
TI!=l
I Vj I
with
p { -pIn \34 ~ z }
= p { Xf2 ~ z } + o(N 2)
where
f = ~ k2 p ( p-l ) + ~ ( p-m-l ) k ( k+ 1 )
k
2
p = 1 - 24fN{ 2kp ( p-l ) ( 2kp+2k+q ) + ( p-m-l ) (2k +3k-1 )}
In the literature, the intraclass correlation model has now been
studied in detail in various contexts. Similarly, tests for sphericity
also have been studied in detail. The circular symmetry model can
be extended to test that the mean vectors are equal in blocks.
Similarly, one may simultaneously test for equality of the mean
12
vectors and circular symmetry. The key point in development of such
tests is to start with the canonical form from which the likelihood
ratio test may be readily obtained.
1.3 Literature Review for the Norparametric Case
1.3.1 Rank Order Test
Sen (1967a and 1967b) introduced distribution-free rank order
tests to test the hypothesis of interchangeability of P=Pl variates
from a multivariate population that has a continuous cumulative
distribution function. Further, the well known theorem of ChernoffSavage (1958) on the asymptotic normality of a celebrated class of
rank order statistics (in the case of two independent samples) was
extended to the multivariate sample case. Consider a sample
" to y(1)
(p) in Section 1. 1); t hen
X l' ... , Xp (correspondmg
i l' ... , Y i1
the null hypothesis is :
H : F (X l' ... , X ) is interchangeable for all (X l' ... , X ) E RP
O
p
p
Pool the n p-variate observations X a = ( X 1a' ... , X pa ), a= 1,
... ,n into a combined sample of size N ( = np ). Denote the order
statistic that is associated with the combined sample by { ZN, l' ...
,ZN,N }; then conSider a sequence of real valued rank functions
where B " is an explicit function of i and N, for i=1, .... , N (= p,
N ,1
2p, 3p, ... ). Let us also define
13
1
C (i)
N,a
ifZ N ,a is an X'f3
f3=1, .... ,n
1
=
{a
otherwise
and
rJ~L
~
-
(C
(i)
N,1"'"
C (i))
'-4... , ••• ,
1-
N,N
p
then Sen's test is based on statistics
TN ,1
=~ .. ~ =_1n £.a=
~ N 1 B N,a CNO),a
Let
R 11 ········R 1n
R
~=
21
R p1
R
R
2n
pn
be the matrix of ranks that corresponds to the n p-variate
observations. Denote by S(RN) = S(Tn) the set of all permutations
within each of the columns. Then the conditional distribution is
p {t / S(T ) , H } = (p!(n for all t E S(T )
a
n
n
n
n
Now, if we define
BN,a = I N ( N~ 1 ) , a = 1, ... ,N
14
BN,R = -l-~'p
B
P ~=1 N,R .
.a
la
-
BN=E(TN,i) , i=1, ... ,p
Then the test statistic is the following quadratic form :
Theorem : Under certain regularity conditions and the above
conditional permutational distribution, Wn has asymptotically, in
probability, a chi-squared distribution with p-1 degrees of freedom.
The efficiency of thiS test under Pitman's translation alternative
was compared to Wilks' test. SpeCifically, two important rank
order tests, namely, the rank-sum test and the normal scores test,
were considered. In his paper Sen (1967) showed that the ARE
(asymptotic relative efficiency) is greater or equal to 1 in the case
of the normal scores test, while for the rank sum test, ARE was
shown to be
.
3
-1
-1
ARE = - [{ 1-(6/rr) sin (pI2)} 1 (l-p))
•
rr
Also Sen argued that if the distribution is not normal the
nonparametric test will fare better.
15
1.3.2 Rank Order Tests in Two Way Layouts
1) ANaVA
Sen (1968a) constructed a class of aligned rank order tests for
the analysis of variance ( ANOVA ) problem relating to two-way
layouts.
Consider a two factor experiment comprised of n blocks, each
block containing p (
~
2 ) plots receiving p different treatments. In
accordance with the two way ANOVA model, we express the yield
X .. of the plot receiving the j-th treatment in the i-th block as,
IJ
X.. =
IJ
+ a. + to + E ..
1
J
IJ
}..l
i= 1, ... , nand j = 1, ... , p
where
is the mean effect
}..l
a 's
0
1
are the block effects
i= 1, ... , n
t.'s are the treatment effects
j=l, ... ,p
J
E .. 's
are the residual error components i= 1,
IJ
j= 1,
, n
, P
Assume that E. = (E. l' ... , Eo), i= 1, ... , n are independent and
1
1
Ip
identically distributed stochaStiC vectors having a continuous ( joint)
cumulative distribution function ( cdf ) G(x l' ... , x ) which is
p
symmetric in its p arguments. Then, without any loss of generality
we may set
2
t'
0
J
= 0, and frame the null hypothesis of no treatment
effects as,
•.• =t'
P
=0
16
If we define the aligned yields and errors by,
for i=1, ... , n , j,l = 1, ... , P , where c5
is the Kronecker delta,
jl
the test of the null hypothesis reduces to testing the interchangeability of (Y. , ... , Y. ), i=1, ... , n against shift alternatives,
11
Ip
which is similar to Section 1.3.1.
As is well known the usual ANOVA test based on the variance
ratio criterion is valid only when G is a p-variate (totally
symmetric) multinormal distribution. For arbitrary continuous cdf
G(x l' ... ,x )' intra-block rank tests are due to Friedman (1937)
p
and Sen (1968c). Hodges and Lehmann (1962) pointed out that intrablock tests do not utilize the information contained in the interblock
comparisons, and hence, are comparatively less efficient (than the
aligned rank order test). They suggested the use of ranking after
alignment. However, all nonparametric tests showed the same
power and efficiency compared to the parametric ANOVA using
normal scores, while it shows better efficiency when the distribution
is not normal, that is, classical ANOVA is not robust when the
normality assumption is not valid. Gerig (1969 and 1978) extended
2
the Friedman X test to the multivariate MANOVA case utilizing the
rank permutation distribution of Sen (1969) . He also did the ranking
after alignment. Again, his extension showed better efficiency than
the parametric tests.
One other method to recover the inter-block information (other
17
than ranking after alignment) was suggested by Quade (1979). In his
paper Quade suggested what he called "weighted ranking tests" to test
for treatment effects assuming that the block effects are additive.
His statistics is based on the weighted average rank correlation after
weighting the within-block ranks according to their credibility with
respect to treatment order. Silva and Quade (1980 and 1983)
studied the weighted ranking tests in more detail; in their (1980)
paper they computed the expected significance levels of the weighted
ranking, ordinary ANOVA, Friedman, and ranking after alignment
methods for normal, uniform, and laplace distributions. In their
(1983) paper Silva and Quade calculated the asymptotic relative
efficiency of the weighted ranking method in comparison with the
above three methods for the three given distributions.
The resul ts
in both papers varied from one distribution to another with the
weighted ranking statistics showing a strong efficiency in the unifom
distribution case (for more details about the weights in the suggested
statistics see Silva and Quade 1980 and 1983). Salama and Quade
(1981) studied the statistics and their distributions using different
correlation measures in both small and large samples.
2)MANOVA
Sen ( 1968b ) generalizes the above concepts ( including Section
1.3.1) ) to test for multivariate interchangeability.
Consider n independent and identically distributed stochastic
matrices,
18
e
Y' .. = (y~~1 ... , y~~)) .
Y.1 = ( Y.!,
... ,Yolr )
1
IJ
lJ
lJ
Then the null hypothesis is to test the interchangeability of the
stochastic vector Y i 1' ... , Y ir for each i=l, ... ,n. The null
hypothesis implies that the joint distribution of the elements of Y.
1
remains invariant under all possible (r!) permutations of the
columns of Y
r
This is termed the hypothesis of multidimensional
interchangeability. The problem of MANOVA in two way layouts can
be reduced to this problem as follows. The response ( p-vector )
X.. of the plot in the i-th block that receives the j-th treatment is
IJ
expressed in accordance with the usual MANOVA model as,
X .. = Jl + a. + t". + e.. , (~f=l
IJ
1
J
IJ
..J
t"J.
= 0) ,j = 1, ... ,r
i
i, ... ,n
=
where Jl is the mean effect ( vector), a's are the block effects,
t"' s
are the treatment effects and €i/s are the residual error vectors.
The null hypothesis of no treatment effects states that,
*
H 1: not all 0
The only assumption needed is that the joint distribution of e il ,
,
e. is continuous and symmetric in these r vectors, for all i= 1,
lr
n. Thus if we define
,
Y.. = X.. -r
IJ
IJ
-1
r
~!
iX, ,u.. = e .. -r
~=
15
IJ
IJ
-1
r
~1=1el.5' j=1, ... ,r
~
i= 1, ... ,n
the c.d.f. of uu' ... ,uir will also remain symmetric in the argument
19
vectors. Hence the test for H* reduces to that of testing the
O
interchangeability of Y 1' ... , Y ir for all i=l, ... ,n against the
i
shift alternative.
Also the proposed test is based on the Chernoff-Savage type of
statistic as follows. Let
Ri~) be the rank of yi~) among the N( = nr )
observations Y lkl, ... ,Y ~~ for j= 1, ... ,r; i= 1, ... ,n; k= 1, ... ,p;
thus, separate rankings are made for each of the p variates. Now,
define a class of rank scores as follows :
J (k)N
(_i_)
B (k)
N, i=N
+1
. 1 , ..., N and k=,...,
1
p
,1=
where the function JZ)satisfies the regularity conditions explicitly
stated in Section 4 of Sen (1 967a) . If we let
(k)
_
1 ~n
TN . - -~'-1
,J
n
1-
B (k)
._
(k) for J-1, ... ,r and k=1, ... ,p
N,R.
1.
then the conditional permutational distribution which we will denote
by Pn implies that there are ( r! (n equally likely realizations of
TN'
Then the proposed quadratic form test statistic is
S
N
= n ~p
"5"P
vkq(R. J ~~
(T(k) . 4= 1 ""1:]= 1
- N ~J= 1 N,J
BN(k))
(T(q). N,J
BN(q))
where
(k). / P } =
E { T N,J
n
BN(k) = N-1~!'J
B (k).
~l= 1 N,l
and
20
f
k 1
or = ,
j=1,
,p
,r
Then under the regularity conditions, the permutational ( conditional)
distribution of SN converges asymptotically in probability, to a chisquared di stribution with px (r-l) degrees of freedom. As a remark
in the paper it was mentioned that when using normal scores the
above test is as powerful as the likelihood ratio test for a normal
distribution function.
1.3.3 lra>mplete Paired Observations
Wei (1983) studied the case of bivariate interchangeability when
there are some incomplete pairs. More specifically, let (Xi ,Y 1)'
, (Xn ,Yn)' (X n+ l' . ), .. ,(Xn+ m ,
,Yn+ 1), ..
(0 ,Yn+l) be a random sample from a bivariate joint continuous
0
0
),
(
0
0
,
distribution function F with marginals F 1 and F2 respectively, where
, denotes a missing observation. Our null hypothesis then is
H : F(x,y) = F(y,x) for all (x,y) E R 2
O
If R., i= 1,
1
000
,m+n and S., j= 1,
j
000
,l+n are the ranks of X's and
Y's respecti vel y in the combined sample, then the test statistic has
the following form:
21
where
Then under certain regularity conditions, the conditional permutation
* /
of the random variable (TN-}-IN)
standard normal as N --+
*
}-IN =
.1
2
0:> ,
<1
* converges in distribution to a
N
where
""n
m
""n+m
"" n+1
L.i= 1 (Ai +B i ) + m+l ( L.i=n+ 1 Ai + L.j=n+ 1 Bj )
<7~2= *2:r=l (A(Bl + :;1 f t
](u) -
j J2
du
o
and
Wei also showed that the proposed test is more efficient ( when 1
is different from m ) than the test TLS proposed by Lin and Sti vers
(1974) and the test Se by Ekbohm (1976). Also, he showed that the
Wilcoxon version of the proposed rank test is more robust than the
parametric tests TLS and Se in the sense that it is less sensi ti ve to
a small deviation from the assumption . That is, TN is more
efficient than the parametric test when the shape of the true
under! ying distribution (the contaminated normal ) deviates slightly
from the assumed normal distribution.
22
1.3 Proposed Research
As explained in Section 1.2, tests of interchangeability in the
parametric cases may require the assumption of normality so that
the problem can be reduced to the test of (compound) symmetry. In
other parametric models, the situation becomes more complicated.
In comparison with the nonparametric tests, the parametric ones
showed less efficiency when the distribution was not normal. When
the distribution was normal the nonparametric (using normal scores)
tests were as efficient as the parametric ones. Thus, there is a
lack of robustness in the parametric case. Referring to Section
1.3.2, it was explained that some nonparametric tests are more
efficient than others, that is, the test is more efficient when the
ranking is done after alignment.
As before, let Y l' ...... ,Yn be n random vectors with an unknown
distribution function F (Y i) where
Y.1
= ( Y.1 1, ....
,Y.Jr )
(p)
(1)
= ( Y.1J. , ...
IJ
Y ..
, Y..
1J
)
P .~1; j=1, ... ,r;
J
. 1, ... ,no
1=
In the next three chapters, we extend Sen's (1967) test of interchangeability in three directions as we explain below. First a
general form of the exact distribution will be given. That distribution
will be used in developing our theory in later chapters.
(1)
(p.)
If Yij , ... ,Yij J are interchangeable for each j= 1, ... ,r
23
then the joint distribution of YN = (Y l' ... , Y
under the finite group
r n of transformations
ri
remains invariant
{g } (which maps the
n
* = (Yl'
* ••• 'Yn)
* , Yi* = (Yu*
sample into itself), where gn(YN)=YN
* and Y..
* is a column permutation of the matrix Y.. for
... ,Y.)
l~
~
~
each j= 1, ... ,r and i= 1, ... ,n.
Thus
rn
contains a set of
* which are permutationally equivalent to YN ;
[ Ilj=l Pj! ]n points Y N
this set will be denoted by S(Y~.
It follows from the above
discussion that under the null hypothesis of interchangeability, the
conditional distribution of YN' given a set S (Y~, is uniform over
the [ Ilj= 1 Pj! ]n possible realizations, which implies that
Let us represent thiS conditional distribution by the measure Pn'
The restricted interchangeability problem, Le., the hypothesis that
the variates within {Yo " i=l, ... ,n} are interchangeable for each j=l,
lJ
... ,r simultaneously, is studied in Chapter II. The test and its
rejection rule under the exact distribution are given. The asymptotic
permutation distribution of the proposed test under the null hypothesis
and its asymptotic distribution for an arbitrary distribution Fare
studied separately. Also, in that chapter the distribution under
Pitman's shift in location alternative is given. In Section 2.6 we
compare the proposed test to Votaw's Lim (Section 1.2) using the
asymtotic relative efficiency (A.R.E.).
Nonparametric tests for data with incomplete variates (for one set
24
of variates, i.e., r=1) are given in Chapter III. Namely, two methods
are given, the ranking after alignment method and the intra-block
rankings method. The proposed tests for both methods and their
rejection rules under the exact distribution are given in Section 3.3.
The asymptotic permutation distributions of the proposed tests and the
asymptotic distributions for arbitrary distribution F are studied for
each method separately. Theoretical and analytical comparisons
between the two methods are given in Section 2.6. The A.R.E. under
a given location al ternati ve is used in these comparisons.
So far, we have only considered the location alternative. In Chapter
IV, for r= 1, we study interchangeability within one set of variates for
a non-location alternative. The proposed test and its rejection rule
under the exact distribution are explained. The asymptotic permutaion
distribution is studied in Section 4.4. Also, the asymptotic normality
of the proposed test statistics for an arbitrary distribution is studied.
The comparison between the proposed test and Wilks' Lvc (Section
1.2) using the A.R.E. is given at the end of Section 4.5.
Finally, numerical examples for each of the above proposed tests are
gi ven in Chapter V.
25
CHAPTER II
NONPARAMElRIC TEST OF RESTRICTED INTERrnANGEABIUlY
2.1 Introdtrtion
Let Xl' ... ,Xn be n random vectors, each with an unknown
distribution function F (X.), where
1
(2.1.1)
X.1
= (X.1 1'
... ,X.lr')
i= 1, ... ,n ,
and
j= 1, ... ,r; p.
(2.1.2)
J
We are interested in testing the null hypothesis {H
~
a} of
1.
inter-
changeability within {X..} for j=1, ... ,r simultaneously, and for each
IJ
i= 1, ... ,no This hypothesis was first considered by Votaw (1948)
under the normality assumption. In his case, testing the null
hypothesis was reduced to testing for "compound symmetry" in a normal
mul ti variate population. Sen (1967) introduced distribution-free rank
order tests to test the hypothesis of interchangeability of one set of
variates from a multivariate population that has a continuous
cumulative distribution function.
In this chapter, we will extend the distribution-free rank order test
to test the hypothesis of restricted interchangeabi li ty as defined above.
Rank scores for each set are defined in Section 2.2; these scores are
very much dependent on the alternative hypothesis. Section 2.3.1 deals
with the exact permutational distribution and its first and second
moments. In Section 2.3, we define the test statistics and their
quadratic forms. Also, the rejection rule under the exact permutational distribution is given. In actual practice, however, n is usually
large. In this case, the labor involved in finding the rejection region
under the exact distribution increases tremendously. In Section 2.4,
we give a solution for the case when n is large by considering the
asymptotic permutational distribution under some five regularity
conditions (given as (C.!) to (C. 5)). As a result we will see that the
permutational distribution of the quadratic form converges in
probability to a chi-squared distribution, as occurs in all other cases of
distribution-free rank order tests. The decision rule for the asymptotic
case is given at the end of Section 2.4. However, the standardized
form of the test is studied asymptotically in Section 2.5, and we are
able to show that the distribution of the proposed quadratic form under
Pittman's shift alternative is chi-squared with noncentrality parameter
given in (2.5.50). Section 2.6 is devoted to studying the asymptotic
relative efficiency (A.R.E) of the proposed test in comparison with
Votaw's L 1m which is derived later for our case under the shift
alternative. We also show that, for normally distributed data, the
A.R.E. is close to unity when using normal scores in the rank order
test.
27
2.2 Preliminary Notions
Let R.(~) be the rank of X(~) among the N. (=np.) observations X~~),
p 1J
1J
J
J
1J
(
j)
c
k-1
.
'-1
Th
t
k'
,
..., X ij
lor - , ... , Pj ,1- , ••• , n.
us, a separa e ran mg 1S
made for each subset j, j=1, ... ,r. Then, the collection (rank)
matrix is defined as
(2.2.1)
R
(P1)
11 ... R 11 I
(1)
I
R (1) ... R (p 1) I
21
_
(1)
(r) _
RN - [R
, ... , R ]Nr
N1
21
I
I
I
I
R (1) ... R (p 1) I
n1
n1
J
···
· · ·
···
and has dimensions n x p, where N=2: =1 Nj
(Pr)
(1)
I R 1r
... R 1r
I
(Pr)
(1)
I R 2r
... R 2r
I
.
I
I .
I
(1) ... R (Pr)
I R nr
nr
J
and P=2: =1 Pj.
Define a class of rank scores as
(2.2.2)
. needs to be defined only at
13 for 13=1, ••• , N.. Also
NJ
fr.IT
J
define
J
where I
28
e
(2.2.3)
( 0)
1
_
T~ o,k J
(2.2.4)
n
( .)
_
, k-l, ... , PJ' ,
n ~=1 B~ .,R~~)
J
IJ
(j)
( ")
(")
0- (TN l'T \J 2'"'' T\J
) , J-l, ... ,r ,
"'
N.,
N.,p"
J
J
J J
and
(2.2.5)
_
(1)
(2)
(r)
1
2
r
- (TN ' TN ,. .. , TN ).
From the above definition,
.
column in the j -th set.
T~). ,k is the average score of the k-th
J
2.3 Rari< Permutation Test for the Case of Restricted
Interchargeabili ty
2.3.1 Permutational Distribution
Each row of the matrix RH)., defined in (2.2.1), is a permutation
of p. numbers, and thus ~) i~ a random matrix which has (p o!)n
J
j
J
possible realizations. But R N consists of r such matrices that
are permutational!y independent. This implies that
~
is a
(TIj= 1Pj!) n possible realizations.
unconditional distribution of ~ over the set of (TIj=l Pj!)n
stochastic matrix which has
The
realizations will eVidently depend on the unknown parent c.d.f. even
under the hypothesis of interchangeability. Nevertheless, the
conditional distribution in Section 3.2 of Chapter I generates the
conditional distribution of
~
on the set S(RN), obtained by applying
the, same group of transformations {gn} on~. Let us denote this
uniform probability distribution over the
29
(TIj= 1Pj!) n equally likely
realizations of
~
on S (~) by Pn' For each realization RH). or
there corresponds a
T~~
or TN' respectively. Then,
~
J
J
This implies that
(2.3.1)
p{ R{j) = R:*.
N.
.'"N.
J
J
/
o
° .),
S(R )}=(p.!)-n for any RN*.E S(R
N.
J
N
J
J
J
(2.3.2)
Since Pn is a completely known distribution, it is possible to find a
test function A(R ) (0(A(1), such that E{A(R ) / P }=a; 0(a(1,
N
n
N
where a is the level of significance of the test. Since TN depends
solely on
~,
we can equivalently write A(R ) as A(T ) and conclude
N
N
that there exists such a conditional test of exact size a, valid for all
c.d.f. 'so
Now in practice, we prefer to use a real valued function of TN as a
test statistic on which A(T ) may be based. For this purpose we
N
first find the permutational (conditional) expected value and variancecovariance matrix of TN'
Theorem 2.2.1 : Let
(2.3.3)
30
e
(2.3.4)
B~~,R~·)
=
J lJ
_1
~~~
BU)
(k)
p. "'1<-1 N .,R ..
J
J lJ
and
(2.3.5)
Then
(2.3.6)
E(TU) / p )
N .,k
n
J
= aU)
N.
J
and
(2.3.7)
(")
(h)
0 'h
2
(")
cov(T~J k,T
1) = ~ (ok1 p. - 1)CT (R.\~ ), k=1, ... ,p.;
N
N
Nj ,
h'
npj
J
j - 'N j
J
1= 1, ... , Ph; j,h= 1, ... , r,
where 0jh is the usual Kronecker delta.
Proof:
Since all subsets are permutationa11y independent, the expected
value of
T~~,k is
calculated Within the j-th subset. Sen (1967b)
J
showed that, for each subset j U=l, ... , r),
E(TU)
N .,k
J
/
p ) =
n
aU)
N.
J
and
(j)
(1)
_ 1
2
(j)
cov(TN .,k,TN .,l) -1if)."(Ok1 Pj - 1)CTN .(RN .)
j
j
J
j
J
k,l=l, ... , p. and j=1, ... , r.
j
Then what Is left to show Is that for j
31
*h, COV(T~~,k'T~~,l) = O.
But
this is true since, under the permutational distribution Pn' the two
subsets j and hare permutationally independent, which implies that
the two random values
TH~.k and TI:;~.1
are permutationally
independent hence the theorem.
2.3.2 Proposed Test
From (2.3.5),
(J~. (R~))
depends on the collection matrix, but
remains invariant uncrer S r~)
.
Thus if we use the inverse of the
(permutational) covariance matrix of TN' the proposed test statistic
is the following quadratic form:
(2.3.8)
Now, under HO' TN has the following location vector
- (r)
- (r))
B
, BN ' ... 'N
.
(2.3.9)
r
Hence it can be shown that, if
each j
U= 1,
r
(J~. (~)) is finite and non-zero for
J
J
... ,r), then under the permutational probability measure
P , WN will have (Il:- 1P.!)n possible realizations, which are all
n
J- J .
conditionally equally likely. On the other hand, if HO does not hold
and the p variates have locations which are not all equal, then at least
one of
T~~,k will
be different from
B~). for
j=l, ... ,r, and hence
J
J
WN , being a positive semi-definite quadratic form in TN' will be
stochastically larger. Thus it appears reasonable to base our permutation
32
e
test on the following rejection rule:
(2.3.10)
Thus, if in actual practice n is not large, we can consider the set
TN[S(~)] of
(Ilj=1 Pj!)n
values of TN (and hence of WN ), which will
provide us with the permutational distribution function of W , and the
N
same may be used to find WN,a(RN) and YN(RN). However if n is
not very small, the labor involved in this procedure increases
tremendously. To avoid such labor we shall consider the asymptotic
permutation test (next section).
2.4 Asymptotic Permutation Distribution of WN
As in the case of the study of asymptotic theory of rank order tests
for various other problems of statistical inference, we shall impose
certain regularity conditions on
BH~,f3
in (2.2.2) as well as on F(x).
j
Extending the idea of Chernoff-Savage (1958) to the multivariate case,
33
we shall find it convenient to extend the domain of the definition of J~~
in (2.2.2) to (0,1) by letting
J~~
be constant on (
{3=1, ... ,N.;j=l, ... ,r.
J
{3
, {3+1),
J
1'r.+I l'r.+T
J
J
J
Let us now define
')
- 1
Hj [k] (x) - - n
(2.4.1)
F
(2.4.2)
H~,
(")
(k)_
~ x], k-1, ... ,p.,
[Number of X ..
1J
_ 1 ~j
J
J
(X~~)
,X ~lh by F '[k 1] (x,y), and the c.d.f of
1J
1J
J,
We denote the joint c.d.f. of
1J
by F . [k] (x), for k
J
(2.4.3)
(2.4.4)
(2.4.5)
._
~4=lF~,[kJ(x), J-l, ... ,r.
(x) -
J
X~~)
(")
J
* 1= 1, ...
,p.. Further, we define
J
(j)
_ 1
(k)
(k)
F N [k 1] (x,y) - - [ Number of (X .. ,X,,) ~ (x,y)]
j , n
1J
1J
1 k= 1, ... ,p.,
(. * )
~ '5 ~ F ~( .),[k 1] (x,y) ,
J
H~, (x,y) =
J
1~k<r~p,
J'
J
*
i )-1
,
Hj (x)
1 ~.
= ~ 4; 1Fj [k] (x) ,
J
(2.4.6)
*
Hj (x,y) =
(p ,)-1
Li
~k~~p. F j [k,l] (x,y), j=l, ... ,r.
J
Next we define the regularity conditions that will be used throughout
this chapter:
(C.l)
(2.4.7)
lim JH) (H) = J ,(H) exists for all 0 <H <1 and is not
j
J
constant,
n--+oo
34
(C.2)
(2.4.8)
and
f
7JU)(~HU)(X))
- J.( N j H{j)(x))] dFU)
(x)
N. N. + 1 N.
J N. + 1 N.
N . [k]
j
-00
j
j
j
j
_
-0
-~j_
(N.) k-1, ... ,po
P
j
(C.3)
(2.4.9)
Jj (H) is absolutely continuous in H: 0 <H <1, and
I
S
d J. (H)
dH s j
I ~ K[H(1-H)rs -!+<5
for s=O, 1, and <5)0, where K is a constant and j=1, ... ,r.
For the permutational distribution theory, we require two more
mild conditions for the existence and convergence of
CT~. (R~).).
J
we state below:
(C.4)
(2.4.10)
Finally, we define
35
j
These
(2.4.12)
(
k,1) (F.)
J
= fOO
Jl
-00
(2.4.13)
f;·
J
(H. (x)) J. (H. (y))dF . [k 1] (x,y)
J
J J
J,
-00
v~g (F} = FJ/H/x)) J2 dFj [k] (xl, k,l=l, ... ,pj'
-00
(2.4.14)
(C.5)
(2.4.15)
To show the asymptotic distribution of WN we need the following
two lemmas.
Lemma 2.4.1: Define
(2.4.16)
2
A =
o
and
(2.4.18)
f~f(U) du,
-
=
JI.
.]-1
r
J
2
J
( .)
J (F.),
k 1
1~<B:p. '
J
J
"5'"5'
Jl
then, if (C.5) holds,
(2.4.18)
A
2
j
-
Jl
j
> 0,
j=1, ... ,r.
36
e
Lemma 2.4.2: Under regularity conditions (C.l) through (C. 5),
a 2 (R(Nj )), defined in (2.3.5), converges in probability to [A? - ji 0] )0,
N
J
J
J
J
where A ~ and ji. are defined in (2.4.1 7) and (2. 14.18) , respectively.
J
J
o
•
1lleorem 2.4.3: Under the regularity conditions (C.l) through (C.5) and
the
permutational
{n~(T~)
probability
measure
Pn'
the
distribution
of
B~)),
k=l, ... ,p o}, is asymptotically in probability a
j'
j
J
multinormal distribution [of rank (p
1)] with null mean vector and
J
covariance matrix given by (2.3.7) (for j=h) for each j(=l, ... ,r).
k-
0
-
The singularity of the above distribution comes from the fact that
(2.4.19)
Then it follows that
there are at most (p. -1) linearly independent
SW.l·
SW.l,
quantities (TH) ,k Thus, the vector {nf(TH),k k=l, ... ,
J
J
J
J
Pj;j=l, u_. ,r} has 2j=1 (pfl) linearly independent quantities
(TU) N.,k
J
sU)).
No
J
Proofs of Theorem 2.3.4, Lemma 2.4.1 and Lemma 2.4.2 can be
found in Sen (196 7b).
11leorem 2.4.4: Under the regularity conditions (C.l) through (C.5) and
the permutational probability measure Pn' the statistic WN [defined in
(2.3.8)] has asymptotically in probability a chi-squared distribution
with 2j= 1 (p f 1) degrees of freedom.
37
Proof:
First let us write
(2.4.20)
Then, from (2.3.8) we can write W as
N
(2.4.21)
W
N
= L.
~~ wU)
=1 N.
J
J
Using Theorem 2.4.3 and Cochran's Theorem it can be shown that,
under the permutational (conditional) law P ,
n
WH)j
has asymptotically
in probability a chi-squared distribution with (p .-1) degrees of freedom
J
(Sen 196 7b). But we know that under Pn' the r subsets are
permutationally independent. Hence, WN is the sum of r conditionally
independent and asymptotically chi-squared random variables. It
follows that the distribution of WN is asymptotically chi-squared with
degrees of freedom equal to 2:j= 1 (p f 1).
2.5 Asymptotic Multioonnality of the Stardardized Fonn of TN
Using (2.2.2), (2.4.1) and (2.4.2) we can rewrite
TH~,k as
J
(2.5.1)
for each j=1, ... ,r. As in the case of Sen (1967b), (2.5.1) has the
same form as that of Chernoff-Savage type of rank order statistics
related to the multisample case. Next, we define the following:
38
e
.
(")
}.J~ k
(2.5.2)
j'
=
fCD
k=l, ... Pj; j=l, ... ,r,
J (H (x)) dF [k] (x) ,
J J
J
0
0
0
-CD
("h)
(2.5.3)
for j
* h or
CD JCD
f
-CD -CD
<5k1,mq =
[F jh[k,l) (x,y)
(j = hand k :t 1),
(2.5.4)
I
I
- F . [k] (y)) J. (H . (x)) J (H (y)) dF
(x) dF
(y)
J
J J
J J
Jm
Jq
0
I
0
0
[
]
0
[
]
I
- F [k] (y)) J . (H (x)) J (H (y)) dF . [ ] (y) dF
(x) ,
J
J J
J J
Jm
Jq
0
0
0
0
0
[
]
for k,m=l, ... 'Pj; 1,q=1, ... ,Ph; j,h=l, ... ,r. Finally, let
(2.5.5)
0*
jh,kl
= _1_~ j -S-Ph [<5(jh) +<5(jh) _<5(jh) _<5(jh) ]
Pj Ph2.m=l~=l kl,mq
mq,kl ml,kq kq,ml'
j,h=l, ... ,r.
1lleorem 2.5.1 If conditions (C.l), (C.2) and (C.3) of Section 2.4 hold,
then the random vector
[n~(T(j)
k
No,
J
- }.J(j) k)' k=l, ... ,p.; j=l, ... ,r]
N.,
J
J
39
has, asymptotically, a mu1tinorma1 distribution with a null mean vector
and a dispersion matrix
(2.5.6)
P*
P* , where
= ((0 *
'h k1))k-1
J,
- , ... , P .,·J'-1
- , ... , r
J
1=1, ... ,Ph;h=1, ... ,r
Proof:
Following the same steps as Theorem 5.1 of Puri and Sen (1966),
we write
TH~,k as
J
(2.5.7)
TU)
N "k
J
=}.J
U)
N "k
J
+ O(j)
N .,1k
J
+ O(j)
N ,,2k
J
+ ~4 _
c U)
Lm-1 N "mk
J
k=1, ... 'Pj; j=1, ... ,r,
where,
(2.5.8)
(2.5.9)
(2.5.10)
(")
-1
C~ ., 1k = N ,+ 1
J
J
r(.)' (.)
H~ . (x) J j
J
-00
(2.5.11 )
40
(Hj (x)) dF ~ , [k] (x),
J
e.
(2.5.12)
Now, by condition (C.2) of Section 2.4, we have
(2.5.14)
C(j) 4k = 0 (N-fr) ,
N.,
p J
J
j= 1, ... ,r.
Proceeding along the same lines as in Sen (1967a and 1967b), it
can be shown that C(j) . k ' i=1,2,3, are all equal to 0 (N~!). Hence,
N .,1
P J
J
we can write (2.5.7) as
(2.5.15)
n! I (T U)
N .,k
J
- J1 NU).,k ) J
(0 U)
+ 0 U)
)I
N .,1k
N .,2k
J
J
= 0 p (1)
for all k=1, ... 'Pj; j=l, ... ,r. Consequently, it is sufficient to prove
that
{n!(O~~,1k + 0~~,2k)'
k=1, ... ,Pj;j=1, ... ,r} has an asymptotic
J
J
mu1tinorma1 distribution. Now, by partial integration of (2.5.8) and
(2.5.9), we arrive at the following expansion,
(2.5.16)
O(j)
+ O(j)
= _1 L:Pj {--L~~
N.,1k
N.,2k
p. 1=1 n ~=1
J
J
J
[OU)(X.'~))
k:1
IJ
- O(j) (X.(~) )]}
l:k IJ
'
where
41
(2.5.17)
(k) ) -Ok(j) (X ..
:q
1J
r
[F i.[k] (x) - F .[k] (x)] J.' (H . (x)) dF. [ ] (x)
J
J
J
J
j
q
,
-co
(2.5.18)
F~[kJ(X) =C
if X~k)~ x
1J
if X~~»
1J
x
for k,q=1, ... ,p., i=1, ... ,no
j
We will now show that for any non-null real p-vector
a = (a 11' ... ,a
P1
1'· .. ,a 1 ' ... ,a
)
r
Prr
the random variable
(2.5.19)
has, asymptotically, a normal distribution. We can rewrite (2.5.19)
as
(2.5.20)
n~~: ~j ~Pj ~. _1 [OU) (X.(~)) - oU) (X.(~))]
L.J=14=1L.1=1 p.
J
=
n
k:l
l:k
1J
IJ
n~~: ~j J j bU)[_1"".n oU)(X.(~))]
L.J =14=12.1=1
kl
n ~=1 k:1
1J
h
were
b -- (b(1)
11' ... ,b (1) , ... , b(r)
11' ... , b(r)·
) IS also non-nul 1 and
P1 P !
PrPr
real. If we write
(2 . 5 . 21)
~r ~j J j t..U)
O(X i' b) -- L.j=
14=12.1= 1uk1
then (2.5.19) can be written as
42
oU)
(X(k))
k: 1 ij ,
i= 1, ... ,n,
(2.5.22)
which is, apart from the factor n±, the average of n independent and
identical1y distributed random variables {D (X. ,b), i= 1, ... ,n}. Hence,
1
to apply the classical central limit theorem under the Lindeberg
condition, it suffiCient to show that D(X. ,b) has finite first and second
1
order moments. We shall prove a slightly stronger result that for any
1[: 0 <7') <0 (where 0 is defined in Condition (C.3) of Section 2.4).
E{ ID(Xi ,b) I2+0}
<
00,
uniformly in F 1 [l]' ... ,Fr[p]. Now, using (2.5.21) and the inequality
r
~n
12+0 < 1+0 {~n I 12+0}
I L..i=1
si
- n
~=1 si
we have
(2.5.23)
EID(X.,b) 1 2 +1[
1
~ (~~ p?)l+1[~~ ~:j ~Pj
L..J =1 J
(bU))2+1[
L..J =1"'k=1L..l=1 kl
EID{j)(X~~)) 12 +7').
k:l
lJ
Then, we only need to show that
This already has been shown, however, in Lemma 3.2 of Sen (1970).
We can conclude, then, that
o(Xi ,b)
has a finite moment of order 2+7'),
1[)0. This, in turn, implies that the first two moments of
o(Xi ,b)
are
finite. Hence, we arrive at the asymptotic normality of the variable
given in (2.5.19). Thus {n±(TH) k j'
jJ~)
j'
43
k)' k=l, ... ,p.;j=l, ... ,r}
J
has an asymptotic normal distribution.
Next, we are going to derive expressions for the elements of the
variance-covariance matrix
fJ* . Using (2.5.16) we can write
(2.5.24)
- cov(O{j) (X(k~) O(h) (X(q))]
k:m YJ' q:l (3h '
for m,k=l, ... 'Pj; l,q=l, ... ,Ph; j,h=1, ... ,r.
Since we are assuming a random sample,
(2.5.25)
cov(Ok{j) (X (k~ ),O(lh) (X{3(1h))) = 0 (3,y = 1, ... ,n
:m YJ
:q
f3 * y.
For the case when
(2.5.26)
f3 = Y we have
cov(O{j) (X(k~ ) O(h) (X(l)))
k:m YJ ' l:q yh
-0:> -0:>
m,k=l, ... 'Pj; 1,q=1, ... Ph; j,h=l, ... ,r.
44
By Fubini's Theorem, we can bring the expectation inside the integral.
Then we have the following two cases:
Case 1: j=h and k=1
For this case, (2.5.26) can be written as
(2.5.27)
-00 -00
,
,
.J . (H. (x)) J. (H . (y)) dF. [ ] (x) dF. [ ] (y)
J
J
J
- c5 Uj )
Case 2: j
* h or
Jm
jq
m,k and q=1, ".'Pj'
kk,mq'
(j=h and k
J
* 1)
As in Case 1,
(2.5.28)
Thus (2.5.25), (2.5.26), (2.5.27) and (2.5.28) imply that
(2.5.29)
k,m=1, ... 'Pj; 1,q=1, ... ,Ph; and j,h=1, ... ,r.
Now, from (2.5.24) and (2.5.29) we have
45
(2.5.30)
(j)
(j)
(h)
(h),
cov(ON ., i k+ON .,2k ' DNh , 11 +ON ,2l J
h
J
J
I
= _i_~ j ~h (6 Uh ) + 6Uh) _ 0 (jh) _ oUh) )
PjPh2.ITl=iLq=i kl,mq
mq,kl
ml,kq
kq,ml·
Therefore, from (2.5.5) and (2.5.i5) we have
(2.5.32)
·
(j)
(h)
*
11m {n cov(TN. k ' TN l)} = D· h kl
n~
J'
h'
J ,
*
where Djh,kl
is defined in (2.5.5). Hence, proof is complete.
It has already been pointed out that the asymptotic multinormal
aistri oution, aeri ved in Theorem 2.4.3, is singular and is of rank at
most equal to
2:
= i (p f i).
J
If the null hypothesis holds, then it follows
from (2.5.3), (2.5.4) and (2.5.28) that, if j
(2.5.40)
* h,
·
(j)
(h)
11m {n cov(TN. k' TN 1 / HOl} = 0 ,
n~
J'
h'
and if j=h,
(2.5.4i)
( .)
_
2 ·
( .)
11m {N. cov(T~. k,T~. 1 / HOl} - (okl P .-i) (A. - J).) ,
n~ J
J'
J'
J
J
J
k=i, ... ,p.;1=1, ... ,Ph;j,h=1, ... ,r, where A~ and
are defined in
J
J
J
Lemma 2.4.1, and 0kI is the usual Kronecker delta. Consequently, with
v.
the help of Lemma 2.4.i, we arrive at the following.
Corollary 2.5.2: If H holds and Conditions (C.i), (C.2) and (C.3) of
O
Section 2.4 hold, then under Condition (C.5), [Nfr(TNU) k -}J .),k=l, ... ,
J
j'
J
P .; j= 1, ... ,r] has a singular multinormal distribution of rank
1
2:~J--i (p J.-1 ), where }J.J =0 J J. (u) du, j = i, ... ,r.
J
f
46
e
We shall now consider the usual type of Pitman's translation
alternatives. For this we replace the parent c.d.f., F(x), by a sequence
of c.d.f.'s, FN(x), such that the marginal c.d.f.'s of {F[N] (x)} satisfy
the sequence of alternatives {H N}, where
k= 1, ... ,p.;
(2.5.42)
J
j=l, ... ,r,
where H; (x) is assumed to be an absolutely continuous (univariate)
J
c.d.f. having a continuous density function h. (x), and where the
J
assumptions of equality of scales and symmetry in HO are also assumed
for the sequence of c.d.f. 's {F [N] (x)}. Let us then define
co
(2.5.43)
_
~ (H ,) -
I
J
d
-:r.-
ax
J ' (H . (x)) dH, (x) ,
J
J
j=1, ... ,r.
J
-co
Then,
(2.5.44)
(2.5.45)
(2.5.46)
lim
n--+ro
[N~ E{(T~) k J
lim {N.
J
Consequently,
J
j'
ek · ~(H,),
J
J
k=1, ... ,p.,
J
.
(')
(')
_
2 11m [N. cov(T~ k,T~ 1 / HN)] - (0 . . p.-1)(A. - v.),
n--+a:> J
j'
j'
1J J
J
J
k,!=1, ... 'Pj'
n--+ro
{N~ (TH) k -
fJ·) / H }] =
N
j,J
cov(T~) k,T~) 1 / HN)] = 0
j,
h'
k=1, ... ,p,
J
h= 1, ... ,1;j *h= 1, ... ,r.
it follows from Theorem 2.5.1
that, under {H N},
fJ), k= 1, ... ,p .-1 ;j= 1, ... ,r} has an asymptotic
J
(2:j=1 (pf1)) variate normal distribution with mean vector
dispersion matrix
*
~
where
47
e and
~1*
(2.5.47)
~* =
o
o
where
(2.5.48)
*
2
-
.
~J' = [(c5kl P·-1)(A. - v')]k 1-1
,J=1, ... ,r.
J
j
j
, - , ... ,p j
It readily follows that, under {H },
N
(2.5.49)
-H')
~r. -1 ~j-1 (TU)
W* -- nL.
N ., k - B .) / (A.2
- -v,)
N
JJ
J
J
J
has an asymptotic noncentral chi-squared distribution with
degrees of freedom and non centrality parameter
~S=l (p
f
l)
(2.5.50)
where
-
ej
1~'
= p,4~1 ekj .
J
From (2.3.9), Lemma 2.4.2 and (2.5.49), we readily find that
* in
under {H } in (2.5.42), WN is asymptotically equivalent to WN
N
p *
probability. We express this by writing WN """ WN . Hence, we arrive at
the following:
48
e·
lneorem 2.5.3: Under the sequence of alternative hypotheses {H } in
N
(2.5.42), the statistic WN in (2.3.9) has, asymptotically, a noncentral
chi-squared distribution with
centrality parameter
~W
2:J= 1 (pf
1) degrees of freedom and non-
defined in (2.5.50).
At this stage, we may consider some asymptotic distribution-free
test for HO. This may formulated as follows. Let Sj 2be a consistent
estimator of A~ in the sense that
J
J
v.,
(2.5.51)
2 P
J
2
J
S. ---+-A. -
-
J/.
J
for all FOE FO' j=1, ... ,r.
Then it follows from (2.5.49) and under {H } that
N
(2.5.52)
* will be, asymptotically, a distribution-free
Hence, the test based on WN
test of H • It further follows from the last theorem that the test based
O
'"
on W will be asymptotically power equivalent to the one based on WN ,
N
for any sequence of alternatives of the type {HN } defined in (2.5.42).
2.6 Asymptotic Relative Efficieocy
We shall now conSider the asymptotic relative efficiency (A.R.E) of
the proposed rank order tests in comparison to the likelihood ratio test
L
, considered by Votaw (1948). Votaw showed that, under the null
1m
hypothesis,
(2.6.1 )
49
where V 1 and V2 are block symmetric matrices. Using (4.7), (7.1)
and (3.3) of Votaw's paper, it can be shown that
p.-1
(2.6.2)
I V 1 1 = [Ilj=y+1 W
d ] IV3 ,
p.-1
(2.6.3)
I
V 2 1 = [Ilj=y+1
wi j
]
I
V3 1
where y is the number of subsets of cardinality equal to one and
(2.6.4)
(2.6.5)
and the elements of V3 are
1Of
h<
J,-y
°
if
j~y
if
j=h>y
and h> y
(2.6.6)
and
(2.6.7)
Uh)
(7kl
_ ~
U) -U) (h) - (h)
_.
- ~=1 (x ik - x. k )(x il - x. l ), k-1, ... 'Pj'
1= 1, ... ,Ph; j, h= 1 , ... ,r.
50
•
Then from (2.6.1), (2.6.2), (2.6.3), (2.6.4) and (2.6.5) we have
(2.6.8)
L
1m
=
r
TIj =y+1
[
1
1+
A.
p .-1
J
Jf.-J
J
where
(2.6.9)
(2.6.10)
and
(2.6.11)
2
Cl
j
=
1 P.
-4
J
Pj
=1
( , .)
JJ
kk
0;
and
2 -
Cl
~
1
.
(J' J')
' P · = p. (p ,-1)4:¥1 Clkl '
J J
J J
j=l, ... ,r.
Then under the alternative hypothesis that is given in Section 2.5,
-n In L
2:
has asymptotically a chi-squared distribution with
1m
=1 Pfr- y degrees of freedom and non centrality parameter,
J
(2.6.12)
- ~r { 2 1 J(1-)
Cl.
P'
~l - ~. -1
J
i:
j
e
-1 ( k' -
-
J
J
Ji )
t7.
J
2} -
~r
A (j)
~. -1 ~ L .
J-
Then the A. R. E of the WN test defined in (2.3.8) with respect to L 1m
test defined in (2.6.8) is given by
(2.6.13)
if we let
51
(2.6.14)
e(j) (W
L
N' 1m
~W
) = -'7""':"'<"'"-, j=1, ... ,i,
~(j)
L
then from Sen (196 7b) it can be shown that when using normal scores
with normally distributed data, e (j) is close to unity. Then we can
argue that our e(W ,L
) should be close to unity for the same
N 1m
reasons. Hence, rank order tests are as efficient as parametric tests
when using normal scores. For the nonnormal data case, the parametric test is not applicable and is less powerful than the nonparametric
one for obvious reasons.
2.7 RFMARI<S
(1) It is worth mentioning that the score function defined in (2.2.2) is
of no meaning in calculating the exact distribution. One can use the
original data or the ranks. Meanwhile, the scores are of important
value in approximating the distribution, especially, under the
alternative hypothesis.
J~~
J. ],
1
is [M . ' 1J
J
J
where M .=N .+1. When N is large, the domain can be extended to
J
J
(0,1) .
(2) The domain of definition of the score function
52
CHAPTER III
DISfRmtJrION-FREE RANK ORDER TESTS OF
INTERrnANGEABIUTY WIm INCOtv1Pl.ETE DATA
3.1 Introdt.d.ion
In any data analysis, some information is lost when ignoring
incomplete observations, especially when the incomplete observations
form a big portion of the data. In this chapter, we modify the rank
order test to be able to use this information (Le., incomplete data).
Let Y=(Y l' ... ,Yp) be a vector valued random variable that has an
unknown p-variate continuous c.d.r., F(x). Also, let Y i =(Y i1 , ... 'Yip)'
i= 1, ... ,n* , be a random sample from Y, where some of these p
variates are incomplete (we assume that the p.1 missing variates in the
i-th block occur at random and not by design Pi= 1, ... ,p).
As in Chapter II, we are interested in testing the null hypothesis {H a}
of interchangeability within Y for each i=l, ... ,n (including the
i
interchangeability of the missing values).
We also are going to use the distribution-free rank order test of
interchangeability to test H using two different methods (With all the
a
incomplete data). The first method, which we label as Method 1, is
the method of intra-block rankings; the other method, which we label as
Method 2, is the ranking after alignment method. Since we utilize the
same conditional and permutational procedures in both tests, we
formulate in Section 2.2 their exact conditional distribution and
compute their first and second moments using the same method. We
will see later that the exact (as well as the approximate) distributions
have the same nature for the two methods. Scores and ranking
alignments for both methods are gi ven in Section 3.2. These scores
are very much dependent on the alternative hypothesis.
Test statistics and their quadratic forms are defined in Section 3.3.
Also in that section, we give the rejection rule under the exact
permutational distribution. When n is large, however, which is
usually the case, the labor involved in finding the rejection region
increases tremendously. For this reason, Sections 3.4 and 3.5 are
devoted to studying the asymptotic permutational distribution of the
proposed test for both methods (Le., when nO' ... ,n _2 are large,
p
where n is the number of blocks with k missing variates, k=O, ... ,p).
k
Regularity conditions for the ranking after alignment method are given
in Section 3.4. Later, we see that the permutational distribution of
the quadratic forms of both tests converge to chi -squared wi th (p-l)
degrees of freedom. In Section 3.6, we compare these tests under
Pitman's shift alternative. The noncentrality parameters of the above
distributions under the alternatives are given in Sections 3.4 and 3.5
for Methods 2 and 1, respectively. Also in Section 3.6, we compare,
for each method, the power when using the n observations (including the
54
incomplete blocks) with the power when using the complete data (Le.,
using the first nO complete blocks). For both comparisons, we will use
the asymptotic relative efficiency. We also compare these two
efficiencies with each other. As a result, we gain some power from
using some of the incomplete data. Also, we will see that the ranking
after alignment test shows better effiCiency and power than the intrablock ranking test.
3.2 Preliminary Notions and the Exact Distribution
Without any loss of generality in testing for H we can arrange the
O
n* blocks so that the first nO blocks have no missing variates, the
next n 1 blocks have one missing variate each, and so on down to the
last n blocks which have all variates missing, with n* =L.k=Onk and
p
n=~~~nk'
Let us denote these blocks by Xik)where
X (k) = (X (k) X (k)
iii'i2'
(3.2.1)
i=l, ... ,nk;k=O, ... ,po
Hence, X ~k) is the i-th rearranged block in the k-th subset (i.e., in the
1
subset of blocks that have k missing variates), i=1, ... ,n ; k=O, ... ,po
k
Then H is the hypothesis of interchangeability within {x~kh for each
O
IJ
i=1, ... ,nk;k=O, ... ,p-2. (Subtracting from each variate its average
score in that block as given in (3.2.12), the last two subsets of blocks
contain no information in testing HO' and hence will be ignored.)
As mentioned in Section 3.1, two methods for testing H are
O
proposed. Each method will have its own notations (which is slightly
different from the other), but to avoid repetition, and because the exact
55
distribution is the same for each method, the rest of this section will
be devoted to forming the exact distribution for the general case with
simplifications for each method at the end.
Let us now define R~~h) as the rank of the non missing X~~) using
IJ
IJ
method h, h=1,2. Thus, for Method 1, R ~~1) is the rank of the nonmissing X~k) in the i-th block, while R ~~2) is the rank of the non-
IJ
missing
IJ
X~k) in the combined blocks of the k-th subset(combined blocks
IJ
with the same number of missing values), Le.,
f
(3.2.2)
.
,
R~~h)=
IJ
rank of X ~k) using
IJ
otherwise
h=1,2.
method h,
Since there are (p-k) nonmissing variates in any block of the k-th
subset, the total number of nonmissing values in the k-th subset is
_
.
.
(kh)
Nk-n (p-k), hence, If R
is not missing, we can write
k
ij
(3.2.3)
1 ~ R ~~ 1) ~ p-k
(3.2.4)
1 ~ R~~2)~ N
IJ
IJ
k
i=l, ... ,n ;k=O,l, ... ,p-2;j=1, ... ,po
k
Next, we define our two collection rank matrices
follows
56
R~) and R~) as
fR
(Oh)
R(Oh) IR(1h)
R(1h) I
1 1 . .. nO 1 i 11 . . . n1 1 I
I
I
I
I
I
I
I
I
I
I
i
I
(qh)
R(qh)
n 1
q
I (qh)
R (qh)
IIR 12 ... nq 2
IIR 11 ...
(0 h)
R (0 h) IR (1h)
R (1 h) I
R 12
. . . nO 2 I 12 . . . n12 I
=
I
I
I
!
R (Oh) ... R (Oh)!R (ih) ... R (1h) I
1P
nap I 1P
n 1p I
.
I
\R (qh) ... R (qh)
I 1p
n p
j
q
where h= 1,2 and q=p- 2.
Then,
R~) is a stochastic matrix with some missing components.
Now, there are p elements in each block of which k are missing.
Therefore the elements in this block can be permuted among themselves
in (p!fkO ways, k=O, 1, ... ,q. This implies that, for a given collection
matrix
R~), we can have Ilk':a (-tf-) nk possible realizations of such a
matrix, simply by permuting the elements within each row. Denote
S (h) (R ~)) as the set of these possible realizations.
Then, the
unconditional distribution of R~) over the set S (h) (R~))
will
eVidently depend on the unknown parent c.d.f. even under the hypothesis
of interchangeability. Nevertheless, following a similar argument as
in Chapter II", we generate the conditional distribution by defining a
finite group
r n of transformations {gn}
* *'
itself), where gn(Y )=Y * =(Yl'Y
2
N
N
of elements of the vector Y., i= 1,
1
(which maps the sample onto
,Yn* ), and Y i* is a permutation
,no Thus
contains a set of
rn
I1k~~ (it )nk points, Y~, which are permutationally eqUivalent to YN;
this set wi 11 be denoted by S (YN) . It follows from the above discussion
57
that under the null hypothesis of
i~terchangeability, the
ribution of YN' given a set S (Y~),
conditional distk
is uniform over the I1k~20 (~ )n
possible realizations. For each Y
and S(Y }, however, there exists a
N
N
matrix R~} and a set S(h} (R~}), respectively. Hence, the conditional
distribution of R~! given a set S (h) (R~}), is uniform over the
I1k~20(~ )nk
(3.2.6)
pOSSible realizations. This implies that
P {RRJ)
(h)
I S(h) (RRJ})}
= I1k~20(-tt)
-n
k,
h=1,2,
(h)
N ), and nO' ... ,np_2~O. Let us denote this conditional
permutational probability measure by Pan' Next, define the score
for any S
functions
(3.2.7)
(R
B~~~ as
if
(1) _{O
Bk,f3
-
Q_t
~-
.
,
{3
J k (p-k+ 1) otherwise
if
f3 --
t
.
,
(3.2.8)
otherwise
J needs to be defined only at values of
f3 equal to 1,2, ... ,p-k+ 1 for
(h)
method 1, and 1,2, ... ,Nk for method 2. Then, B
(kh) depends on
k,R
..
(kh)
IJ
the rank R.. . Now, if we define
IJ
(3.2.9)
B (h)
N
=
((B (h)
)) ._
k,R~~h) 1-1,
1J
• _
•
... ,nk,k-O, ... ,p-2,
.J= 1, .•. ,p,
58
h=1,2,
the matrix
B~) depends on the collection matrix R~)for each h=1 ,2.
Thus, from (3.2.7) and (3.2.8) we can write
h=1,2.
Now, let
(3.2.11)
Bk,R
(h) (kh) = ± ~P-1B (h) (kh)
.
p-K J- k,R ..
,
1J
1.
o
*(h)
_
(3.2.12) B
(kh)k,R iJ·
{
(h)
- (h)
B
-B
otherwise, i=l, ... ,nk;k=O,
k,R~~h)
k,R~kh)
... ,p-2;j=1, ... ,p;h=1,2.
1J
1.
Then, it can be shown that
(3.2.13)
E (B
(3.2.14)
E (B
*(h)
(kh)
k,R ..
1J
I Pan)
* (h)
= 0, h=1,2,
*(h)
I
(kh)' B
(lh)
k,R. .
1,R R
1J
~y
Pan)
°
= _
1
~p
(B*(h)
p(p-1) L..j=l
k,R~~h)
)2
i*~
if
k*l or
if
k=l,i=~
and j*y
if
k=l,i=~
and j=y
1J
1 ~p
P
L..j=l (B
*(h)
(kh)
k,R ..
1J
)2
i=l, ... ,nk;~=l, ... ,n1;k,1=0, ... ,p-2;j,y=1, ... ,p, for each h, h=1,2.
59
Next, we define C ~~) as
lJ
(3.2.15)
C~~)=
{
lJ
o
1
otherwise
and
(3.2.16)
~ =
(C(O) ,C(1), ... ,C(p-2)) ,
where
(3.2.17)
C(k) = ((C~~))) ,'_
. _
1J 1-1,. .. , nk ' j -1, ... , p
Then, ~ itself is a random matrix which is independent from
Now, for each block in C(k) there are 2 P-p-l =
realizations, where
(~J
~~~ (~)
EN·
possible
is the number of ways to get k missing values
in one block. Hence for k missing values in a block all the
(~J
possible realizations are equally likely. Because of the independence
of the observations, then for each given subset k (k=O, 1, ... ,p-2), there
are
(~J nk equally likely realizations.
Denote this set of realizations by
Sk (~) . Then, given S 1 (~), ... , Sp_2 (~), we have
(3.2.18)
p {~
I Sk(~)'
-2 (P) -nk
k=O, ... ,p-2} = Tlk=o k
for any Sk (~), k=O,l, ... ,p-2. Let us denote this conditional
probability measure by P
bn
. Hence, given P
60
bn
, it can be shown that
e.
p
(3.2.19)
bn
)
= .Jcl.
p
and
(3.2.20)
E (C
~~) C ~k)Y I Pbn ) = {
IJ '
J?:!s.
if
y=j
if
y*j ,
P
1
(p-k) (p-k-l)
p(p-l)
i=l, ... ,nk;k=l, ... ,p-2;j,y=1, ... ,p.
Under H ' and because of the multinomial distribution defined above
O
(Le., P ) , the pattern of the missing values is not of any importance.
bn
Hence under P we assume that
bn
(3.2.21)
\,K=l.
is fixed,
n
k=O,l, ... ,p-2 and
Since the two probability measures Pan and P bn are independent, we
define the overall conditional permutational probability measure P as
n
the product of these two measures (Le. P = P . P ). This overall
n
an
bn
probability measure will be used in the next section to calculate the
moments of the proposed statistics.
3.3 Proposed Test
We start by defining the following statistics
(3.3.1)
T(h) .= ~p-2 ~~k C~~) B*(h)
N,J "1<=0~=1 IJ
k,R~~h)
IJ
61
,j=l, ... ,p
(3.3.2)
(h) _
(h)
(h)
(h)
TN - (TN,1,T N ,2' ... ,TN,p)
, h= 1,2 ,
(h) . is the sum of the nonzero scores in the j-th column over all
Then TN
,J
the p subsets. Now, using (3.2.13), (3.2.14), (3.2.19), (3.2.20), and
and P
are independent, it is easy to show the
the fact that P
bn
an
following:
(3.3.3)
E (T(h).
N,J
IP
(3.3.4)
(h)
var (T N'
,J
I
n
)=0
'
P ) - ~-2 -q-k 2 (R(kh))
-0
'
n
p (IN k N
and
(3.3.5)
cov(T(h) . , T(h)
N,J
N,y
IP
2 (R(Nkh )) ,
) = - ~-2 (p-k)(p-k-1) (JN
4=0
n
[p(p-O]2
k
e'
for j*y=1, ... ,p;h=1,2, where
(3.3.6)
(72 (R (kh))
Nk
N
= ~~k 2 P (8 *(h) ) 2
~=1 J=l k,R~~h)
, k=O, ... ,p-2;h=1,2.
lJ
. .
(h)
(h)
(h)
Now, lf we defme a h= var(T N . I P ) and bh=cov(T .,T
I Pn),
N ,J N , y
,J
n
then the variance-covariance matrix of T~), which we denote by Vh , and
its inverse, V 1 , can be written as follows
h
a
(3.3.7)
V =
h
h
bh
bh
bh
b
h
ah
bh
bh
.
62
and
(3.3.8)
As before, the proposed test is a quadratic form, which can be
written as
(3.3.9)
Then, by using (3.3.1), (3.3.2), (3.3.7) and (3.3.8), we can write W~)
as follows
(3.3.10)
W~)= (ah-bh)[a~+b
h
(p-l)r{[a h+b h (p-l)]2:f=l
-
(T~:}2
bh2:~=l2:r=l T~:j T~:l])
\
v
=0
Substituting for the expressions a h and bh in (3.3.10), we get
63
/
(3.3.11)
where
(h) _
WN -
2
2:)= 1 (T~~
j)
(h)
ON
2
°(h)N =",p2 (p-k) (p -p-k)
4=0
[p(p_l)]2
(J
2 (R (kh)) .
Nk N
(3.3. 11) is the general form of the proposed test using either of the
two methods we discussed earlier.
To simplify this test further, we
discuss it in terms of each method separately.
For Method 1 (Le., method of intra-block rankings), we can write
o~)
as follows
(3.3.12)
2
(p-k) (p -p-k) 2
N - '1<=0 nk
[p (p-1)] 2 (Jk '
0(1) _ ~p-2
where
(3.3.13)
(J~
= 2:f=1 (B * (1lk 1)) 2 = constant wi thin the k-th subset.
k,R ..
IJ
By (3.3.11), (3.3.12) and (3.3.13) we have
(3.3.14)
W(N1 )
2~
=__
(T(1).) 2
=.. . ,;1::.. .-N;.. . :. z. J1~
' __
.....L..1
(p-k) (p -~-k)
0;2
4=0 k
[p(p-1)]2
k
From (3.3.11), we can write
(3.3.15)
W (2) =
N
2
",p- 2 n
W~) as
2:~ (T(2) .)2
1=1 2~---N, J
-----W-~.......
",p-2 (p-k) (p -p-k) 2 (R (k2))
4=0 [p(p-1)]T (JN k N
64
Note that under (3.2.21) ( ~k=Ak is a fixed value under P n) we can
write the following:
0(1)
(3.3.16)
N=
-n
2f- 2 A (p-k) (p2 -!2k)
=0 k [p (p-l )]
(J
2 =constant.
k
To determine the rejection region, it can be shown that if O~) is
finite and nonzero, then under the permutational probability measure
Pn'
w~)
will have
IIk~~ (~ nk possible realizations which are
conditionally equally likely. On the other hand, if HO does not hold and
the p variates have locations not all equal, then W~), being a positive
semi-definite quadratic form in
T~) , will be stochastically larger.
Thus it appears reasonable to base our permutation test on the
folloWing rejection rule:
(3.3.17) !\ (T~)) =
>WN,a (R N(h))
if
W(h)
N
y (R (h))
if
W(h) = W
(R (h))
N
N,a N
0
if
W(h)
N
1
N
N
<W
N,a
(R (h))
N
Thus, if in actual practice n is not large, we can consider the set
TN [5 (R~))] of rrk~~
nk values of T~) (and hence of W~)), which
(-fl )
will provide us with the permutational distribution function of
65
W~) ,
and may be used to find out WN,a (R~)) and YN (R~)).
small,
however,
tremendously.
the labor involved
If n 1s not very
in this procedure
increases
To avoid such labor, we next consider the asymptotic
permutation test.
3.4
Ranki~
After Alignment Mettoi
3.4.1 Asymptotic Permutation Distribution of the Proposed Test
As in Chapter II, we shall impose certain regularity conditions on
B~:bin
(3.2.8), as well as on F(x). Also, we extend the domain of
definition of I
interval
N
in (3.2.8) to (0,1) by letting I
N
be constant over the
(d!-+r,
~:.\), f3=1, ... ,Nk;k=O, ... ,p-2.
k
k
e"
Let us now define the following:
(3.4.1)
F
(k)
_
1
(k)
[.] (x) - -[number of X ..
Nk J
nk
IJ
::5;
x, i=1, ... ,n ] ,
k
(3.4.2)
(3.4.3)
FN(k)[O 1] (x,y) =..1 [number of (Xoj,X. )
k J,
nk
1
1l
Again, let
Ff~~ (x)
and
Ft~:l] (x,y)
::5;
(x,y)], j*l=1, ... ,po
be the c.d.f. in the k-th subset of
Xo . and of (X .. ,X. I) respectively, for j*l=1, ... ,p; k=O, ... ,p-2. We
IJ
IJ
1
also define
(3.4.4)
H(k) (x)
t
= 2:}=1 Ff~j (x)
66
and
(3.4.5)
*k
H
(x,y) =
(P-k) -1
2
(k)
2:
F[. 1] (x,y) .
l=S;j<l=S;p J,
The regularity conditions to be used for method 2 become
(C.l )
(3.4.6)
lim I
n-ko
N
= J (H) exists for all 0 <H < 1 and is not a constant,
J(H) is absolutely continuous in H : O<H<l, and
(C.3)
(3.4.9)
for r=O, 1 and some 0)0, where K is a constant.
Finally, we define
(3.4.12)
IIt)(F)_:[J~(H(k)(X)) J(H(k)(y)) dF[~~ll(x,y)
(3.4.13)
v(k) (F) =
(C.4)
(3.4.14)
j,l=1, ... ,p,
((lI~~) (F))) . 1-1
lJ
J, - , ... ,p
rank of v(k) (F)
~ 2.
The following two lemmas are straightforward extensions of Sen's
67
096 7b).
Lemma 3.4.1 Define
(3.4.15)
(3.4.16)
Then, if (C.4) holds,
(3.4.17)
A
2 - J.lk
>0
, k=O, ... ,p-2.
Lemma 3.4.2 Under the regularity conditions (C.l) through (C.4) ,
~~ (R~2)),
defined in (3.3.6), converges in probability to
P~~kl
> 0,
k
k
where A 2 and iik are defined in (3.4.15) and in
(3.4.16), respectively.
[A 2 - J\]
From (3.3.11), we can write
D~) as
a~ (R~h))
2
(3.4. 18)
k
D(2)
N = 2"(-20 (p-k) (p -~-k) N·
k ---r'Nr---=
[p(p-!)]
k
Using lemma 3.4.2, we have
(3.4.19)
p
_
~p- 2 (p-k) ~p 2 --p-k)
Z A
---+~ D - 4=0
[p (p-!)]
k (A
2 -
-J.l
P-kk1
k) P_
1 2~~~ (p-k-!) (p2 -p-k) (p-k) Ak (A 2 -vk).
[p(p-!)]
-
68
e~
Then we have the following theorems,
Theorem 3.4.3 Under the regularity conditions (C.l) through (C.4) and
the permutational probability measure P , the distribution of
{n-~TN(2).,
,J
j= 1, ... ,p}, is asymptotically in probability a multinormal distribution
n
of rank (p-l) with null mean vector and variance-covariance matrix
given in (3.3.7).
Theorem 3.4.4 Under the regularity conditions (C.l) through (C.4) and
the permutational probability measure Pn' the statistic
W~) given in
(3.3.15), has asymptotically in probability a Chi-squared distribution
with (p-l) degrees of freedom.
3.4.2 Asymptotic Multimrmality of the Stardardized Form of T~)
Similar to Section 5 of Sen (1967) and with the help of (3.2.8),
(3.2.11), (3.2.12), (3.3.1), (3.4.1), and (3.4.3) we can write the
following
(3.4.20)
where
(3.4.21)
(2 * ) _ ~-2 nk . (k) (2)
TN,j - 4=0~= 1Cij Bk,R~~2) .
lJ
As before (Section 2.2.5), (3.4.20) is the sum of statistics that have
the same form of Chernoff-Savage type of rank order statistics related
to multisample case. In this case, however, the p-2 subsets which we
69
sum over are mutually independent.
Let us, then, introduce the
following definitions
(3.4.23)
(k)
f3"JJ, 1m
= f fF [J]
(~) (x) [1-F [J]
(~) (y)] J' (H (k) (x)) /
(H (k) (y)) dF (k) (x) dF (k) (y)
[1]
[m]
-oo<x<y<oo
+f fF [J](~) (x) (1-F [J](~) (y)] /
(H (k) (x) ) J' (H (k) (y)) dF (k) (x) dF (k) (y)
[m]
[1]'
-oo<x<y<oo
and
(3.4.24)
,ln::foo
(k)
f3 gj
r
-00
(k)
(k)
(k)
[F [g,j] (x,y)-F [g] (x)F [j] (y)]
/ (H (k) (x)) / (H (k) (x)) dF (k) (x) dF (k) (y)
[1]
[m]
for j*g=1, ... ,Pi 1,m=1, ... ,po Finally let
(3.4.25)
f3*~ =_1 ~1 1 ~ 1 [f3(~) 1 + f3 (k)
gj
P L,:l=
L.m=
gj, m
. - f31(~)
- f3(k) 1,J
j,mg
mg, j
1m,gj
g,j=1, ... ,po
and
(3.4.26)
fJ*k
*k
= ((13 gJ.)) g,J'-1 , ... , P , k=O, ... ,p-2.
70
e-
1lleorem 3.4.5 If the conditions (C. 1) • (C.2) and (C.3) of Section 3.4.1
~ -1 (2 * ) ,p-2
(k)"_
hold, then the random vector n [n TN"
4=O~kJ.lN ., j-l, ... ,p]
,j
,j
has asymptotically in probability a multinormal distribution with a null
mean vector and a dispersion matrix ~-2
4=O~kP* =
k
P* .
Proof: We proceed precisely on the same line as Theorem 2.5.1 [or
Theorem 5.1 of Sen (1967b)] and write
(3.4.27)
where
(3.4.31)
(3.4.32)
-1 (2 * ) ,p-2 {(k)
(k)
n TN·
="k-O~k
J.lN
.+
01
. Nk+
,j
,j
J,
°2(k)J,, Nk+~1-1- Cl(k)"Nk} '
~4
j,
(3.4.33)
foo[IN(NNk+1 H(k) (x))
(k)
_
C4 · N J, k
Nk
k
-00
-
Nk
(k)
(k)
J(N +1 HN (x))] dFN [,] (x),
k
k
k J
for j=l, ... ,p and k=O, ... ,p-2.
By condition (C.2) of Section 3.4.1, we have
(3.4.34)
C (k) N = 0 (n
4 J, k P
k!) ,
Also, it can be shown that
k=O, ... ,p-2; j=1, ... ,po
dt)J, N k=0P(nk~)' for
all k=O, ... ,p-2, j=i,
... ,p and 1=1,2,3 (see Theorem 2.5.1). Hence, we can write (3.5.7) as
Consequently, it is sufficient to prove that
{n~ 2f~~Ak(0 i~),N;O~~,Nk)'
j= 1, ... ,p} has asymptotically a multinormal distribution. To show
the normality, we need to show that for any non-null p-vector a=(a l'
... ,a ) the random variable
p
(3.4.36)
!
P
a
n 2·-1
J- J.
2f- 2
(k)
(k)
-OA
(01'
N
+
O
, N k)
2
k
J, k
J,
",p- 2 A-!",p
~
(k)
(k)
=4-0
k L. '-1 a .n k (0 1 . N + O2 . N ),
J- J
J, k
J, k
. normal. Sen (1967) has sown
h
h n1",p
(k) N + 0 2'
(k) N)
a .( 0 l'
is
tat
k L.·-1
J- J
J, k
J, k
has asymptotically a normal distribution. We also know that the p
72
subsets (k=0,1, ... ,p-2) and
Ak~O
are finite, and that any linear
combination of normally distributed random variables is also normally
distributed. This implies that the expression in (3.4.36) has asymptotically a normal distribution. Hence, from (3.4.35) we can say that
.. t d'ISt rl.but lOn
. 0 f n-!TN°'
(2 * ) J=
. 1 , ... P IS
. asymp tOll
1
the Jom
otlca y norma.
,J
Since the p subsets are independent, we can write
=n
-2 2 cov(01(k). N +° (k), N '01(k) N +°2(k) N)'
2t--OAk
J, k 2 J, k
m, k
m, k
j,1=1, ... ,po
From Sen (1967), we have
Hence from (3.4.35), (3.4.37) and (3.4.38), we can write
where
11: = (p;r) is defined in (3.4.26).
.
mdependent
then
Since the subsets are
~-2
11* = 4=0
Ak 11k* .
It has already been pointed out that the asymptotic multinormal
distribution, derived in Theorem (3.4.5), is singular and is of rank at
73
most equal to (p-1). If the null hypothesis holds, and if we define
H(x), H* (x,y), A2
and-11 as in (3.4.4), (3.4.5),(3.4.15) and (3.4.16),
k
respectively, then it will readily follow from (3.4.39), (3.4.22),
(3.4.23) and (3.4.24) that
(3.4.40)
*
*
lim {n- 1 cov(T~ ) ,T~ 1)
n~
,J,
I HO}
A (p-k) (p-k-1) 2
_~p-2 k
(A 2 -
4=0
[p(p-1)]2
v)
if
j*l
k)
if
j=l
k
=
~p-2
4=0
A (p-kHp-k-1)
k
p
2
(A
2
-
11
Corollary 3.4.6 If H holds and if the conditions of Theorem 3.4.5
O
hold, then, under condition (CA), [n-~(T~).), j=1, ... ,p] has a singular
,J
multinormal distribution of rank (p-1).
As before, we now consider the usual Pitman-type translation
alternative. For thiS, we replace the parent c.d.f F(x) by a sequence
of c.d.f. 's, F [N] (x), such that the marginal c.d.f. 's of {F [N] (x)} satisfy
the sequence of alternatives {H N}, where
and H(x) is assumed to be an absolutely continuous (univariate) c.d.f.
haVing a continuous density function h(x), and where the assumption of
74
equality of scales and symmetry is also assumed to hold for the
sequence of c.d.f.'s {F[N] (x)}. Now let us define
Then, we can write
(3.4.44)
lim [n- 1 cov(T (2 *) ,T (2 * )
N , JN
n~
,l
2
_~p-2 Ak (p-k) (p-k-l)
4=0 [p(p-l)]2
*
{n-~TN(2),
2
(A
2- Jlk)
if
j*l
if
j=l,
it follows from Theorem 3.4.5,
that under {H },
N
j=l, ... ,p-l} has, asymptotically, a (p-l)-variate normal
,J
distribution with mean vector
2f~20
At
dispersion matrix as given in (3.4.44).
It readily follows that, under {H N},
(3.4.45)
)
P
j,l=l, ... ,p.
Consequently,
(A 2 _ v
k
~p- 2 Ak (p-k) (p-k-l )
4=0
I HN)]
*
W~)
75
(p-k)~k(H).(el' ...
,e ) and a
p
has asymptotically a non-central X2 distribution with (p-1) degrees of
freedom and noncentrality parameter
(3.4.46)
A (2)
W
Now from (3.3.15) and lemmas (3.4.2) and (3.4.45) we find that
*
under {H } in (3.4.41), w~) is asymptotically equivalent to WiJ ) in
N
* p
probability, we write this as W~ ) .... W~). Hence, we arrive at the
following.
1lleorem 3.4.7 Under the sequence of alternative hypotheses {H N} in
(3.4.41), the statistic
W~) in (3.3.15), has an asymptotic noncentral
noncentral Chi-squared distribution with (p-1) degrees of freedom and
noncentrality parameter
A~) defined in (3.4.46).
At this stage, and as Sen (1967) suggested, we now may consider
some asymptotic distribution-free tests of H . This may be formulated
O
2
as follows. Let S~ be some consistent estimator of A -ilk in the sense
that
(3.4.47)
It follows that under {H } in (3.4.41)
N
76
e
(3.4.48)
W(2)
N
Hence, the test based on
w~) will be, asymptotically, a distribution-
free test of H . It further follows from the last theorem that the test
O
based on WiJ) will have asymptotic power equivalent to the test based
.-
on W~) for any sequence of alternatives of the type {H N} defined in
(3.4.41).
Note that the singularity of the distributions in Theorems 3.4.3,
3.4.6 and 3.4.7 (and theorems 3.5.1,3.5.2, of the next section),
comes from the fact that
(3.4.49)
2:P_1TN(h) . = 0 ,
J,j
h=1,2.
It follows that there are, at most, (p-1) linearly independent quantities
(h)
-~
_
(h)
'_
TN " h-1,2. Thus, the vector {n TN" j-1, ... ,p} has (p-1)
,J
,j
linear! y independent quanti ties. Proofs of the above theorems and
lemmas are the same as the proofs in Section 2.4 with some minor
changes.
3.5. Intra-Block
Rari<i~
Method
3.5.1 Asymptotic Permutation Distribution of the PJ'OpOsed Test
0(1)
In equation (3.3.16), we were able to show that the ratio
77
~
is a
n
constant. From (3.3.14), then, we have
(3.5.1)
Now we have the following theorems.
Theorem 3.5.1 Under the permutational probability distribution, the
distribution of
{n-~ T N
(1)., j= 1, ... ,p}, is asymptotically in probability
,J
a multinormal distribution of rank (p-1) with null mean vector and
variance-covariance matrix as given in (3.3.7) for h= 1.
Theorem 3.5.2 Under the permutational probability measure Pn' the
statistic W~) given in (3.3.14) and (3.5.1), has asymptotically in
probability a Chi-squared distribution with (p-1) degrees of freedom.
Proofs of theorems 3.5. 1, and 3.5.2 are special cases of Theorems
3.4.3 and 3.4.4. For more details see Puri and Sen (1971) Chapter 7.
3.5.2 Asymptotic Multioormality of the Standardized Form of
T~P
Following the same steps as in Section 7.2 of Puri and Sen (1971),
we first define the following,
(3.5.2)
(3.5.3)
(1 *.)=_12!-2 n.k C(.k.)S (1)
TN
~ 1
(k 1) , j= 1 , ... ,p ,
,j
n
=O L1= 1J k,R ..
1J
p~~r) = p{R~~1)=r},
1J
1J
p~k~rsl)= p{R~~1)=r,R~k11)=s}, j*l,
1,J
for i=1, ... ,nk ; k=O, ... ,p-2.
78
1J
1
r*s= 1, ... ,p-k,
Notice that the sum over k is from
°
to p-2.
That is, the (p- i) st set
is ignored. the reason is that the scores are all equal (ranks too). We
also let conventional!y
(k,rr) - . !
(kr)
d (k,rs) _.!
(kr)
1 1
Pi,j1 -OJ} Pij
an Pi,jj -ors Pij , j, = , ... ,p;
r,s= 1, ... ,p-k; i=i, ... ,nk ; k=O, ... ,p-2.
(3 .5. 4)
where <5 '1 and <5
J
(3.5.5)
J1N,j
rs
are the usual Kronecker deltas. Further, let
=n -1,p-2 ~p-k (kr)S (1)
4=OLr=1 Pij
k,r
(3.5.6)
and
(3.5.7)
,,(1)=((1'T
,)) , J', 1-1
..::oN
v n,J 1
- , . •• , P.
,
(k)_ (k)
(k),_
_
.
Fmally, let F.1 -(F.1 1 ' ... ,F.1p ),1-1, ... ,n k , k-O, ... ,p-2, let
F*=(F(10), ... ,F(p-l)), and let F*={F* I rank [~(Nl)]~l}. Then we have
n
n 1
n
n
Pthe following.
*
lbeorem 3.5.3 For all {F*} E F*, the distribution of {n-! ( T (1 .) N ,j
n
n
J1N,j)' j=1, ... ,p} converges (as nk '-00, k=O, ... ,p-1) to a singular
normal distribution of rank (p-i) and with null mean vector and
variance-covariance matrix as given in (3.5.7).
The proof of this is omitted, but can be found in Section 7.2 of Puri
and Sen (1971) (Theorem 7.2.2). Note that under HO' Theorem 3.5.3
79
implies Theorem 3.5.1.
Again, we will only consider the following sequence {K} of
n
alternative hypothesis:
(3.5.8)
K :
n
Ff~) (x) = F~k) (x - n-~ej)' j=1, ... ,p; i=1,
k=O,
,nk ;
,p-2,
where Ft)E F 0' i=1, ... ,n k; k=O, ... ,p-2, and F 0 is the class of all
absolutely continuous distributions having continuous density functions
satisfying
(3.5.9)
_""Ffx
F;k) (x)}2 dx
<""
for all F;k)E F O.
Let f~k) (x) = d F~k) and
1
Ox 1 '
(3.5.10)
.
and, conventlOnally, let
(ik)
0
P-Ok)
1 ,p- k- 2 = Pp- k- 1 , p- k- 2 =
c
.
lor 1= 1, ... ,nk ;
k=O, ... ,p-2. Also, let
(3.5.11)
_ n- 1"",p-2 ,nk { (ik)
_ Ok)
}
Yr,n4=0£1=1,8r-1,p-k-2 ,8r-2,p-k-2 '
-2
(k)
_
=~ --0 Ak Yr,n , r-1, ... ,p ,
(k)
-1 nk
(ik)
Ok)
_
where, Yr,n = nk ~=1 {Pr-1,p-k-2- P r - 2 ,p-k-2} , r-l, ... ,p-k .
Then, we have the following theorem.
80
TIleorem 3.5.4 Under the sequence of alternative hypotheses given in
(3.5.8), and under (3.5.9), the statistic W~) in (3.3.14) has an
2
asymptotic noncentral X distribution with (p-1) degrees of freedom
and noncentrali ty parameter equal to
(3.5.12)
where
Cl~
is given in (3.3.13).
Proof is the same as Theorem 7.2.3 of Puri and Sen (1971).
3.6 Asymptotic Relative Efficieocy
In this section, we compare the efficiencies of the distribution-free
rank order tests that are introduced in Sections 3.4 and 3.5, Le., we
compare the efficiency of the ranking after alignment method with the
efficiency of the intra-block ranking method. When we say efficiency
we mean the asymptotic relative efficiency, that is, for each method,
we will compare its power using all the n observations (including the
incomplete data) to its power when using the first nO complete
observations.
Sen (9167b) introduced the ranking after alignment test when the
data is complete. It has the same form as the one we developed in
Section 3.4 for the case when k=O. Hence, substituting 0 for k in
(3.4.46) we get the following noncentrality parameter for the first
complete nO observations
81
(3.6.1)
~W
2
=
2 ~
- 2
{SO(H)} {L.°j=1(e j - e) }
-1 2 P
(A
-
l/ 0)
Under the case of Pitman's translation alternative, the asymptotic
relative efficiency (A.R.E.) is the the ratio of the noncentrality
parameters given in (3.4.46) and (3.6.1). It can be written as follows
Now, we will write A
2
-iik in the form
2
A (1- Pk) where Pk is the
average score-correlation of the p-variates in the kth subset, k=O, 1,
... ,p-2. Hence, we can write (3.6.2) as follows:
Under the assumption that the distribution is normal, (as we will see
later) the left side of the above equation can be written as
(3.6.4)
82
e
Since
Po
is bounded above by 1, we can write the right hand side of
equation (3.6.3) as
(3.6.5)
Using Lemma 7.3.7 of Puri and Sen (1971) Chapter 7, we can write
Pk
~ p-~-l·
(3.6.6)
Z ~
This implies that
1
~=-2 A
2 2
(p-k) (p -p-k)
+ AO
4-1 k p2(p_l)2
Then using (3.6.3), (3.6.4) and (3.6.6), we can write
(3.6.7)
(2)
e(~W '~W
2
) ~ U.Z
P(p-l)
~ ~-2
2
2:k~~ Ak (p-k)
2 2
4=0 Ak (p-k) (p -p-k)
Hence,
e(~~) '~W ) is always greater than or equal to one. This
2
shows us that the test using the incomplete data is more efficient if we
include all the incomplete data in the analysis. The degree to which
the test is more efficient depends on the parameters A . That is, if
O
A is close to one, then the A.R.E. in (3.6.7) is close to one. If AO
O
is small, the A.R.E. in (3.6.7) is Significantly larger than one. Hence,
83
if the number of complete blocks (nO) is not large compared to the
whole sample of observations (n), then it is appropriate to use some of
the incomplete data.
For the intra-block ranking method, we calculate the A.R.E by
following the same steps as above. First, the noncentrality parameter
of the test using the nO complete blocks can be found by substituting 0
for kin (3.5.12) to get the following:
2( -1) [~
(3.6.8)
~w
=
1
p P
(0) 8(1),2
L.r=1 Yr,n
2
0"0
O,r J
Again under Pitman's shift alternative as given in (3.5.8), the
A.R.E is the ratio of the noncentrality parameters given in (3.5.12)
and (3.6.8). It can be written as
~ (1)
W
~W
(3.6.9)
[
1
~=-2 {A-; (p-k) ~=-k Y(k) B (1) ] 2
4-0 k
L.r-1 r,n k,r
~
(0) B (1 )
4=1 Yr,n O,r
p2 (p-1) 0"2
0
. ~- 2 ( k) (2 k) 2
4=0 p- P -p- O"k
By simple algebra it can be shown that
~~~ Ak (p-k)
~-2
2 2
4=0 Ak (p-k) (p -p-k)
P(p-1 )
(3.6.10)
2
~
1 .
Hence, for both methods, the A.R.E. is greater than or equal to
Z.U, which is greater than or equal to one. This tells us that, when
84
AQ is small, the incomplete part of the data should be used.
Puri and Sen (1971) Chapter 7, showed that the A.R.E. when the
data is complete, is larger for the ranking after alignment method in
comparison with the intra-block ranking method. They concluded that
the ranking after alignment method is more powerful when the data is
complete. Next, we demonstrate that thiS is also true for the
incomplete data situation.
First we simplify the A.R.E. parameters in (3.6.3) and (3.6.9) for
gi ven scores. For the intra-block ranking method, we propose to use
the following score function :
(3.6.10)
u
Jk(u) = p_k +1 ' u=1, ... ,p-k.
Hence, the score function is the rank within the block. The test then is
Friedman's X2 test on the incomplete data. Sen (1968) showed that
within each subset k, k=O, ... ,p-2, the intrinsic effiCiency of this
method can be achieved when
(3.6.11)
(k) ,lor
r
-1 , ... ,p- k ,
every h*O , r- t Yr,n
B*(1)k,r
where B*k ( 1)and y (k) are defined in (3.3.12) and (3.5. 11) respecti vel y.
,r
r,n
Also, using (3.6.9) he showed that
(3.6.12)
-k
(k) 2
12
~1 (Yr,n) = (p-k){p-k-1)(p-k+1)
85
[f
oo (k)
-co {f
2
(xl) dx
]2
•
for each subset k, k=O, ... ,p-2. Assuming that f(k) (x) is normal with
mean
fJ; and constant variance 0'~1' we can write (3.6.12) as
(3.6.13)
(k) 2_
3
2 -1
_
L..r=1 (Yr,n) - (p-k)(p-k-1)(p-k+1) . (7T 0'* 1)
,k-O, ... ,p-2.
~-k
Using (3.6.11) and (3.3.13), O'~ in (3.6.9) can be written as
Hence, using (3.6.11), (3.6.13) and (3.6.14) we write e(~~),~W ) in
1
(3.6.9) as
(3.6.15)
~- 2
2
(p -p-k) A.
4=0 (p-k) 2_ 1 k
For the ranking after alignment method, we define the score function
as
(3.6.16)
Using this score function, our test is equivalent to the rank sum test on
the incomplete data. In thiS case, Hodges and Lehmann (1961) showed
the following
86
and
1
Pk ~ - p-k-1 ' k=1, ... ,p-2 ,
(3.6.19)
where
1I~=A2 Pk
as defined earlier.
Since f(k) (x) is assumed to be normal, then (3.6.17) can be written as
S~(H)
(3.6.20)
A
2
=
3
~-=2"'-
, k=O, ... ,p-2.
rro*2
Then, by using (3.6.20), we write
e(~~),~W ) in (3.6.3) as
2
(2)
(3.6.21 )
e (~W ,~W )
2
p (p-l)2(1-PO) {Lk~20A~ (p_k);}2
2
2
~=O Ak (p-k) (p-k-l) (p -p-k) (1- Pk)
= -
From equation (7.3.72) of Puri and Sen (1971) Chapter 7, we can
write Pk as
(3.6.22)
Pk =
rr6
. -1
Sln
-1
(2(p-k-1)) ,k=0,1, ... ,p-2.
By using (3.6.19) and (3.6.22) we finally can write
e(~~),~W ) in
2
(3.6.21) as
87
2
6 sin- 1
1
~- 2 !
! 2
(2)
P(p-1) [1
(2 ill-I) ) ]{4=0 Ak (p-k) }
(3.6.23) e(~W '~W ) ~
-2
:2 2
.
+-rr
2
2"t=0 Ak (p-k) (p -p-k)
Now, we define
(3.6.24)
e(~ (2) ~ (1»_
W ' W -
~ (2)
W
~ (1)
W
which is the A. R. E. of the ranking after alignment method in
comparison with the intra-block ranking method. Using (3.4.46),
(3.5.12), (3.6.13), (3.6.14), (3.6.20), (3.6.22) and the fact that
distribution is normal with variances equal for all subsets, we get
where Pk is given in (3.6.22).
From (3.6.15), (3.6.21) and (3.6.25), it can be seen that for a given
set of values p and Ak , k=O, ... ,p-2, all A.R.E's are greater than or
equal to one. For example, when p=3, AO=0.6, Ai =0.2 and A2=0.2,
e(~ifj)'~W )=1.925, e(~~)'~W )=3.70S, and e(~~) ,t.~»)=1.402.
2
i
Looking at Table 3. 1, we can see that when p=3 and for any values of
AO' Al' and A2' all A.R.E's are greater than one. Also, by observing
the values in Table 3.1, we notice that, as AO increases the A.R.E's
decrease, and for a specific value AO' all A.R.E's increases as A 1
88
Table 3.1
A.R.E's of Method One and Method Two
for the Case When p=3
AD
A1
A2
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
0.4
0.5
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.1
0.2
0.1
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.6
0.5
0.4
0.3
0.2
0.1
0.5
0.4
0.3
0.2
0.1
0.4
0.3
0.2
0.1
0.3
0.2
0.1
0.2
0.1
0.1
(2)
e(~W ,~W
2
)
(1)
e(~W ,~W
4.172
4.181
4.118
4.056
4.003
3.960
3.919
3.886
3.944
4.172
4.200
4.181
4.151
4.118
4.086
3.705
4.068
4.172
4.199
4.196
4.181
3.500
3.944
4.107
4.172
4.196
3.328
3.821
4.028
4.125
3.183
3.705
3.944
3.060
3.598
3.954
2.400
2.654
2.744
2.774
2.779
2.771
2.757
2.740
2.095
2.400
2.561
2.654
2.710
2.744
2.764
1.925
2.223
2.400
2.517
2.597
2.654
1.813
2.095
2.275
2.400
2.492
1.734
2.000
2.176
2.303
1.673
1.925
2.095
1.625
1.864
1.586
89
1
)
e(~~) ,~~))
1.552
1.712
1.798
1.845
1.873
1.889
1.898
1.902
1.433
1.552
1.645
1.712
1.762
1.798
1.825
1.402
1.474
1.552
1.617
1.670
1.712
1.398
1.433
1.494
1.552
1.602
1.405
1.412
1.458
1.506
1.418
1.402
1.433
1.433
1.398
1.449
increases. We also notice that the ranking after alignment method is
asymptoticall y more powerful than the intra-block ranking when the
data is incomplete, but the relative gain in the efficiency over the
complete part of the data is higher for the intra-block method.
3.7 Remarks
Puri and Sen (1 97 1) pOinted out (page 286), that the ranking after
alignment method requires more assumptions than the intra-block
ranking method. Hence, under weaker assumptions it may not be better
as the A.R.E. has indicated.
90
CHAPTER IV
RANK ORDER TESTS FOR INTERrnANGEABILITY
AGAINST NON-LOCATION ALTERNATIVES
4.1 lntrodLction
Let Xi' ... ,X n be n random vectors, each with an unknown distribution function F(x), where
(4.1.1)
X/3 = (X/31' ... ,X/3p) ,/3=1, ... ,n; p ~ 2.
When F (x) is a multinormal c.d.f., it is completely specified by its
mean vector fl = (fl1' ... 'fl p) and dispersion matrix ~ = ((a Ij
. .))
(i,j=1, ... ,pl. It is well-known that a ii 0=1, ... ,p) are measures of
dispersion of the p variates, and p.. =a ./ (a .. a .. ) ~ O;i j= 1, ... ,p) are
Ij
1j
11
jj
measures of their association. The hypothesis of compound symmetry
(H mvc ) as sketched by Wilks (1946) is specified as
where Hand H are given in Section 1.2.1. A brief summary of the
m
vc
likelihood ratio tests Lmvc ,L vc and Lm is given in Section 1.2. 1.
A nonparametric approach for H given by Sen (1967) is explained in
m
Section 1.2.2. In this chapter we propose and study a nonparametric
test to test H which is the hypothesis that the variances are equal and
vc
the covariances are equal. irrespective of the values of the means.
Hence, we formulate our null hypothesis H as the hypothesis of interO
changeability within X{3 for each {3=1, ... ,n against the alternative
hypothesis that Hvc is not true.
Rank scores and preliminary notations are given in Section 4.2.
These scores are very much dependent on the alternative hypothesis.
Section 4.3 deals with the exact permutation distribution and its first
and second moments under H . The proposed test is given in (4.3.2 1) ,
O
with the rejection region rule given in (4.3.22). In actual practice,
however, n is usually large. In this case, the labor involved in finding
a rejection region under the exact distribution increases tremendously.
In Section 4.4 we give a solution when n is large under some regularity
conditions. As a result, we will see that the permutational distribution
of the quadratic test in (4.3.21) converges in probability to a chisquared distribution. The decision rule for the asymptotic case is
given in (4.4.9). However, under the alternative, the means are
unknown. In this case the exact distribution in Section 4.3 will not
apply any more. In Section 4.5 we correct for this problem by
showing that the test will not change asymptotically by centering the
observations around their variate means. Applying that result, the
distribution under the alternative hypothesis is a chi-squared distribution with the noncentrality parameter given in (4.5.67).
Comparing the proposed test with Wilks Lvc ' the A.R.E. in Section
4.5 shows that the two tests are power equivalent under the assumption
that the distribution is normal.
92
e
4.2 Preliminary Notions
Let F (i) (x) and F (i,j) (x,y) be the marginal c.d.f. 's of X{3i and
(X{3i,X{3j) respectively for i(j=l, ... ,po Let J(u) be some absolutely
continuous function of u, defined in the open interval (0,1), and suppose
that J (u) is normalized in the following manner:
Later on we shall impose certain regularity conditions on J(u). Let us
define
(4.2.3)
Y(3i = J(H(X{3i))
and write
(4.2.4)
Y = (Y l' ... ,Y p) ,
Y. will be called the grade functional of X., i= 1, ... ,po
1
1
We will use these grade functionals to formulate the desired class
of tests.
(4.2.5)
Write
e..lJ =cov(Y.,1 YJ.)=[ P(H(X))
J(H(y)) dF( ..) (x,y)
1,J
-00 -00
-L,P(H(X)
dF (i) (xl ]
[_ofJ(H(Y)) dF (j)
(y) ] ,
1,J=1, ... ,po
93
Finally let
(4.2.6)
Thus
of
8= ((eo .)).0 i
1J
e is
e is
1,j= , ...
.
p
the dispersion matrix of the grade functionals. The purpose
to work with a matrix which is invariant under certain trans-
formations of variables, is less sensitive to outlying observations, and
at the same time, is a reasonably informative measure. In particular,
if J(u) is monotone in u (O(u(l), then
e defined
in (4.2.6) will be
invariant under monotone transformations of the coordinate variables.
Now, since the c.d.f. F is unknown, so is the matrix
e.
To estimate
these we follow Von Mises' (1947) approach, and define for any q
(4.2.7)
FN (·11' ... ,1 (xl'
q
0
)
,Xql
~
(4.2.8)
F(o
1
1""
=
*[
i
q
q
fJ 1·j
~ x.; j=l, ... ,q}
J
for all i1r! ... r!i =l, ... ,Pi q=l, ... ,po
q
In actual practice we have a sequence of IN(u) which converge to J(u)
as
N~
for all 0 (u( 1, and i=l, ... ,po Next, if we define
94
p
number of (XPi l' ... ,XPiq)
(Xi' ... ,x ) ; {3=i, ... ,n ]
,1' )(X ' ... ,x) = p{X
q
~
then
eij
defined in (4.2.5) can be estimated by
where
(4.2.11)
TN,!
~(JroJN(HN(X)) dFN(i) (xl
= ~2P=1 I N (H N (x,8i))
; i=1, ... ,p ,
and finally
(4.2.12)
TN
= {TN",1J
, 1~i<j~p} .
Then our proposed test is based on some function of TN. It may be
noted that the class of dispersion matrices considered above contains
many well-known nonparametric measures as special cases. For
example, if we let IN(N!I) = (
1l
)! (,8 - ~), ,8=1, ... ,N, then TN
N -1
reduces to Spearman's rank correlation, while for J (u) = (12)! (u-!),
reduces to the grade correlation matrix.
4.3 Permutation Tests For HO
Since we are studying the interchangeability within X, the exact
95
a
distribution under HO is the same as in Sen (1967) and Section 2.3 of
Chapter II (when r=l). We will refer to the needed results as
necessary. First let us pool the n observations X , f3= 1, ... ,n into
f3
one combined sample of N=np values, and denote the sample point by
I
(4.3.1)
~
I
I
= (Xl' ... ,Xn )
Denote by Rf3i the rank of Xf3i in
~
and by
~
the collection matrix
of these ranks. Hence, we have
(4.3.2)
RN =
R
R
11
12
R 21 R 22
.
R
R n1 R n2
R
1p
~2p
e
np
where by virtue of continuity of the c.d.f. 's, the possibility of ties may
be ignored in probability. Replacing the ranks f3 (=1, ... ,N) in
~
a set of general scores {B N ,f3=JN (N~ 1)' f3= 1, ... ,N}, gi ves us the
corresponding score matrix
B
(4.3.3)
~=
N,R 11 BN,R 12
B N,R
1p
.
BN,R
n1
BN,R
n2
96
BN R
' np
by
Then under the permutational distribution measure Pn that is given in
Section 2.3 (for r=l) we have
p{~
(4.3.4)
I P,) =
(p!)-n
Operationally, it will be much simpler to work with the slightly
adj usted statistics
(4.3.5)
SN,ij =
n~ 1 TN,ij
for i,j= 1, ... ,po
Using the permutational probability law given in (4.3.4) we can
show that the following holds:
(4.3.6)
where 6.. is the usual Kronecker delta,
lj
=
_1-2Pi-12~-1
B R
pn
,.,- N 'pi
97
= constant
(4.3.9)
* ..
SN
,1J
= E (SN ,1J..
I P n)
,p- 1 2
= c5. p-1
1
(R )
(TN N +
1
1
-
~n
n-1L..,8=1
(8
-
8 )
N,R,8.- N
2
i,j=1, ... ,po
Furthermore, SN .. in (4.3.5) can also be expressed as aU-statistic
,1J
in the scores 8 N R ,i=1, ... ,p; ,8=1, ... ,n as follows:
, ,8i
(4.3.10)
1
SN' . = -2
,1J
[n] -1 2:n
2
=
Y <,8 1
(B N R -8 N R )(8 N R -8 N R )
' yl.
',8'1
, YJ.
',8 J,
for i,j=1, ... ,po
Let us now define
(4.3.11)
4
~;1~kl (RN) n(n-1~ (n-2) 2:~=14=12:';.:1
(8
N,R .
KJ
-8
*
N,R,8 .
J
)(8
N,R k
-8
K
(BN,R
Ki
-BN,R
lli
)
)(8
-8
)
N,R k N,R 1 N,R 1
Y
Y
K
*
- SN,ij SN,kl '
(4.3.12)
d
4;i )kl(RN ) =
J,
(8
N,R yj
*
n(n~1)2nY=12:n,8=1 (8N
-8
N,R,8j
)(8
N,R
-8
Yk
R .-B R .)
'Y1 N ',81
N,R,8k
)(8
N,R
-8
Yl
N,R,8l
*
- SN,ij SN,kl '
for i,j ,k,l = 1, ... ,po Then, by an adaptation of the results of Nandi
and Sen (1963) and Sen (1966), it can be shown after some algebraic
manipulations that
98
)
•
(4.3.13)
cov(SN ,1J
.. ,SN , kl l Pn )
for i,j,k,1=1, ... ,po
For small samples we may work with the random variables
(4.3.14)
*
{SN"
- SN";
i~j=1, ... ,p} .
,1J
,1J
By considering the generalized inverse of the (permutation) covariance
matrix of these random variables, we arrive at the desired quadratic
form as a sUitable test statistic.
By virtue of equation (9.3. 19) of Puri and Sen (1971) Chapter 9, we
can write [under HO]
(4.3.15)
1
n
ISN,ij - n2,B=1BN,RpiBN,R,Bj I = 0p(N
-1
)
for all i,j= 1, ... ,po
Now let us define
(4.3.16)
11. .
IJ,kl
(R) = -l.2 n _ B
B
B
B
N
n ,B-1 N,R,Bi N,R pj N,R pk N,R pl
- SN,ijSN,kl'
99
i,j ,k,l= 1, ... ,po
Setting s * =p (p+ 1) /2 and
(4.3.17)
r=[(i-l) (2p-i)/2] + j
for
l~i~j~p,
we rewrite
(4.3.18)
{SN . . , i ~ j =1, ... ,p}
,1J
as
~ = {SN,r ' r= 1, ... ,s * } ,
(4.3.19)
* . . , i ~ j =1, ... ,p}
{SN
,1J
as
* = {SN,
* r ' r= 1, ... ,s * } ,
~
and
Then by using (4.3. 15), the test statistic has the following form
(4.3.21)
(Here
V~ (RN) is the inverse of VN(~) and is assumed to
be positive
defini tel . Thus we have the folloWing rule:
(4.3.22)
if LN
~
LN,a
reject HO
< LN,a
accept HO
LN,a is chosen so that p{L N ~ LN,a I HO} = a, where O<a<1 is the
preassigned level of significance of our test. In small samples, we
100
evaluate LN,a by referring to the exact permutation of LN generated by
the (p!)n equally likely permutations of elements within each row of
(4.3.3). For large samples, we shall show in the next section that
L
N,a
P •
2 *
Xa,[s - 2]
2 is the iOO(i-a)% point of a chi-squared distribution with r
where Xa,r
degrees of freedom.
4.4 Asymptotic Permutation Distribution of
4i
As in the preceding section we define
and extend the domain of IN(u) to (0,1) by letting I N be constant on
(~, ~:l). Furthermore, we shall make the following assumptions:
(A. i)
(4.4.2)
J(u) = lim IN(u) exists for O<u<l ,
n-+ro
~~}3)
-oocn
I N [H N (x)] IN[HN(yl] - J[HN(x)] J[HN(y)]] dF (i,j) (x,yJ
=0 (n-')
p
(A.3)
(4.4.4)
lJ(r)(u)
r
I= Idr
du
J(u)
I~
i,j= i, ... ,p ,
K(u(1-u)]Y r ,
for r=O, 1, where 0 <YO < ~; Y1=-1, and K is some constant. Now let us
write
101
(4.4.5)
J-L . = [fJ[H(X)] J[H(y)] dF(. 0) (x,y) ,
1J
1,J
-00 -00
(4.4.6)
v~,kl :J]]]~[H(X)]J[H(y)] J[H(z)] J[H(wl]
dF(o1,j,
. k , 1) (x,y,z,w) -
jJo . jJl
1J
i<
1 ' for i,j,k,l=l, ... ,po
Adopting the suffiXing system of (4.3. 17), we rewrite v.* . kl as v * for
1J,
rs
r,s=1, ... ,s * and define
(4.4.7)
* ))r,s=1, ... ,s*
v * = ((vrs
(A.4)
(4.4.8)
v * defined in (4.4.7) is positive definite.
The following theorems follow as special cases of section 9.4 of Puri
and Sen (1971) Chapter 9.
(~) defined
N
in (4.3.20), converges in probability to v * , defined in (4.4.7), as
Theorem 4.4.1 Under assumptions (A.1) to (A. 4), V
N~.
l1leorem 4.4.2 Under assumptions (A.1) to (A.4), the permutation
distribution of L
N
(defined in (4.3.21)) asymptotically reduces to a
chi-squared distribution with (s * - 2) degrees of freedom.
By virtue of Theorem 4.4.2, L
N ,a
defined in (4.3.22), converges in
probability to X;,[s*_ 2]' Hence, the permutation test procedure based
102
on LN simplifies in large samples to the following rule:
(4.4.9)
if
L
N
~ X2 , [s * - 2]
a
reject H
O
<Xa2 , [s * - 2]
accept H
O
In order to study the power properties of the test considered above,
we need to study the unconditional distribution of L
under an
N
appropriate sequence of alternative hypotheses. This, in turn, requires
the study of the joint distribution of the rank order statistics
~
defined in (4.3.18).
4.5 Asymptotic Nonnality
of~,
For Arbitrary F
As mentioned in Section 4.1, when testing against the alternative
that Hvc is not true, the means are unknown and may not be equal,
hence the basic permutation argument of Section 4.2 is no longer
tenable. Then it is difficult to derive the permutationally distributionfree tests. However, we shall derive a class of rank order tests by
centering the observations at the respecti ve estimates of the location
parameters. Due to centering, the independence is vitiated, and as a
result, we consider the conditions under which the test based on the
centered observations is asymptotically equivalent to the test based on
the observations centered at the true locations. In this section, we
study the distribution under the alternative hypothesis and show the
noncentrality parameter. For this purpose, we assume that F(x) is
absolutely continuous haVing the continuous density function f(x) and that
the follOWing conditions hold:
103
A. The density f(x) of F(x) is diagonally symmetric about its location
parameter.
-1
B. J (z) ='IJ (z) where IJ1 is symmetric about zero.
C. The functions (d/dx)J [F (i) (x)] are bounded for each i= 1, ... ,po
Recall that
EN,#=J(~) is
assumed to be a non-constant, non-
decreasing, and square integrable function on (0,1).
Let us start by defining the following:
(4.5. 1)
d
= (a l'
...
,ap) =- Jl
,
where Jl = (}11' ... ,}1p) is the location vector of the unknown
distribution F (x). Assume that for a given d there exists a sequence
(4.5.2)
such that
(0
dn = (e5n l'
n~ (cSn
<C <(0),
e
,e5np ),
...
- d)=U is bounded in probability, Le. for some finite C
n
(4.5.3)
Note that IIUnl1 <C produces a compact region Un that contains un with
probability that converges to 1. Now let
(4.5.4)
d _
X #i - X #i + a i ' and
A
d
A
XQ~= X Q. + a.
}-Il
}-Il
m
,#=1, ... ,n; i=1, ... ,p,
A
where a and ani are given in (4.5.1) and (4.5.2) respectively
i
i= 1, ... ,po Also define
104
""
(4.5.5)
"
d_
d
d,d
d
d I
X.1 - (X 1"1 ... ,X n1.) ,X.1 n = (X1~1, ... , X n1~) , i= 1, ... ,p,
~
d
r, . .
"
A
d
d
, x Nn = (X
(4.5.6)
,X;) .
"
cS
Applying the same steps as in Section 4.2 to ~ and X Nn , we have
the folloWing collection rank and score matrices respectively,
"
(4.5.7)
Rd
N
= «R N,th
d .)) _
{i-1,
i=1,
(4.5.8)
BcS
N
= «B N,{i1
cS .)) _
{i-1,
i= 1,
,n
,p
"
dn
dn
, R N = «R N,,...,1
R')) ,...,R-1 , ••• ,n
i=1, ... ,p
,n
,p
where
~
(4.5.9)
B
d
N,yi
cS
= BN R d
B
, N, yi
n
~
.=B
d
N,y1
NR n
, N, yi
"
~
d
cS
d
d
R N . and R Nn . are the ranks of X . and X yn1· in the combined vectors
,y1
,y1
Y1
"
d
_
d
of all {X{ij' {i-1, ... ,n;j=1, ... ,p} and {X/], {i=1, ... ,n;j=1, ... ,p}
respective1y.
"
d
Next, define G d " and G n .. as
N ,1J
N ,1J
(4.5.10)
d
1 n
d
d
B R' BN ,,...,J
G N ,1J
.. = -n 2R-1
R'
,...,- N ,,...,1
,
"
where d=d or dn .
~
dn
d
Our purpose in defining G .. and G .. is to show the follOWing:
N ,1J
N ,1J
105
'
A
(4.5.11)
d ·· - G 6n .. I
n ~ IG N
N ,1J
,1J
p
-~+.
0 , as n - - +
. 1 , ... ,po
co, .1,J=
Without any loss of generality we assume that d = 0, then from
(4.5.3), n1
dn = u n =
(u l' ... ,u
n
np
) is bounded in probability.
Then (4.5. 11) is reduced to show that
_1
A
(4.5.12)
! dn _ 0
_ 1 n un _ 0
n IG N ,1J
.. GN··I
.. GN ,1J
..
,1J - n IG N ,1J
I
p • 0, as n --+ co,
i,j=1, ... ,po Before we show (4.5.12) we note that under (4.2.1)
there exists a value 0 <R <N+ 1 such that
n- 1u
if RN,,B;~ R
(4.5.13)
n-!u
ifRN,,B~< R , i=1, ... ,p; ,B=1, ... ,n,
To show (4.5.12) we need the following lemmas.
n-~u
n
Lemma 4.5.1 Under the assumption that BN,,Bi is a nondecreasing
n-~u
n-~u
function of R ,B.n , G ..n can be written as
N, 1
N ,1J
(4.5.14)
n-~~
where GN,ij,hl is monotone in each unk ' k=1, ... ,p, h=1,2;
1=1, ... ,4 or h=3,4; 1=1, ... ,2 P-2.
106
Proof:
n-~u
For fixed i and j we can split Q n= {(R N ,,9
u
r, ...
n-~u
,RN,,9~)' ,9= 1, ... ,n},
i,j=1, ... ,p, into four quadrants as follows:
n-~u
n-~u
n-~u
u
1) quadrant Qo ~ 1 is the set where (R I)~)(RN I)~r~o and R N f3~~O,
N ,J-I1
1j,
,J-Ij
, 1
U
2) quadrant Q.
n
0
1j,
.
2
1S
n-~u
n
n-~u
n
the set where (R N f3 )(R N I)
, 1
'J-IJ
0
0
)
>0
n-~u
n
and R N f3 <0,
, 1
0
n-~u
n-~u
n-~u
n
o
R
nR
n/
3) quadrant Q.1J, 3 1S the set where ( N , f3'1 )( N , f3'J )~o and R N , f301n <0,
U
0
n-~u
n-~u
n-~u
n
o
n
R
n
4) quadrant QoIj, 4 IS the set where (R N , f301 )( N , f3'J )~o and R N , f301n ~O,
U
0
for all ,9=1, ... ,no
Now, let
1
u h
(4.5.15)
Cf3n.. =
,lj
{
o
otherwise, f3= 1, ... ,n,
for h=l, ... ,4; i,j=l, ... ,po
n-~u
Using (4.5.15) we can write G
on in (4.5.10) as
N ,1J
0
(4.5.16)
where
(4.5.17)
n-'u
u h
n-'u
n-'u
n
1
~n C n (8
n) (8
n
'-1
GN,ij,h=nLof3=l f3,ij N,{3i
N,{3j ), 1,J- ,
h= 1,
107
,p,.
,4.
In the h-th quadrent and for fixed values of i and j i,j= 1, ... ,p,
n-!u
G 0.n is either nondecreasing, nonincreasing or a quadratic function
N , lJ, h
in each u k=l, ... ,p;h=1, ... ,4. These monotonic proporties are
nk
summarized in Table 4.1 for all values of h=1, ... ,4 and unk ' k=1, ...
,po For example, keeping all unk fixed (k 'I- i), from Table 4.1 we see
n-!u
n-!u
that G 0on is a nondecreasing function of u 0' and G N On! (keeping
N ,1J, 3
m
,1J,
u k fixed, k'l-j) is a quadratic function of u 0 with its maximum defined
n
nJ
to be at u 0= u(~). Based on the results in the table, we have the
nJ
nJ
following subregions within Un for the first two quadrants (Le., for
0
h=1 ,2):
Table 4.1
n-!u
Behavior of G N 0onh as a Function of u k' k=1, ... ,p
, lJ,
n
for Any Fixed Values of i and j, h=1, ... ,4.
n-~u
n
G N . i ;.2
n-~u
n
G N •i ;.1
um
0
1'\
maximum at
u( ~)
m
unj
!'\
~
maximum at
u~~)
1'\
n-~u
n-~u
G N . i i~3
GN . i ;~4
1
\
1
\
\1
\1
minimum at
u(3)
nk
minimum at
u(4)
nk
maximum at nondecreasing nonincreasing
u(~)
m
1'\
maximum at nondecreasing nonincreasi ng
(2)
un;
k*i & k*j
\
1
nondecreasing nonincreasing
unk
k=l, ... ,p
a) U. 0 h1 is the subregion within U that u o;S; u (~) and u . ;S;u'~) ,
lJ,
n
m m
nJ nJ
108
b) U.. h2 is the subregion within U that u .>u(~) and u .>u (~) ,
IJ,
n
m m
nj nJ
c) U.. h3 is the subregion within U that u .~u(~) and u .>u(~) ,
Ij,
n
m m
nj nj
d) U .. h4 is the subregion within U that u . >u(~) and u .~u(~), h=i ,2.
Ij,
n
m m
nj nJ
n-~u
G ..n h=i,2 can then be written as
N ,Ij, h
(4.5. 18)
n-~u
GN ,Ij,
··nh =
n-~u
2:1~- i GN ,Ij,
..nh1 , h= 1,2
n-~u
where
u
[GN,Ij,
.. nh if u n EU.Ij,~h1
n- .12 u
G .. n =
N ,Ij, h1
otherwise
n-~u
n-~u
i.e., G ..nh1 is the statistic G N .. nh in the U .. hl-th subregion
N ,Ij,
_~
,Ij,
Ij,
n
u
1=1,2,3,4. Note that GN,ij~hl is a monotone function of each unk '
k=i,. ... ,p within subregion U..
Ij, hI .
u
Similarly, for each Q. ~h' we have 2 P-2 subregions U.. hI h=3,4;
Ij, _~
Ij,
-2
n un
.
1=1, ... ,2 P • Hence, G ,Ij,
.. h h=3,4, can be wrItten as
N
n-~u
(4.5.19)
p-2 n-~u
2
G N ,Ij,
.. nh = 21 = 1 GN ,Ij,
· .nh1 , h=3,4
n-~u
where G .. n is a monotone function in each u k' k= 1, ... ,p, wi thin
n
N,Ij, h1
the I-th subregion of the h-th quadrant. Then by using (4.5.16),
(4.5.18) and (4.5.19) we prove the lemma, that is
109
n-!u
(4.5.20)
G
n-!u
p-2 n-!u
n = ~.2 ~4 G
n + ~.4 2: 2
G
n
N,ij
Lfl= 1£.1=1 N,ij ,hI Lfl=3 1=1
N,ij,h1'
i,j=l, ... ,p, a total of (2 P- 1+8) statisqcs.
Corollary 4.5.2 For a given un and under the assumptions in Lemma
4.5.1, corresponding to (4.5.20) we can write
(4.5.21)
n-~v
i,j=1, ... ,p, where G N
O .. hI is the statistics G N · .nhl for v =0 in the
,1J,
,1J,
n
quadrants and subregions that are defined by un.
Note that there exists a value K such that for any value u ~ [-K,K] f
n-~u
*
n
GN,ijn is less than G N . Let us denote by J =[-K,K]IP the domain of
definftion of u. Since we have a finite number of regions (2 P- 1+8),
n
then from (4.5.20) and (4.5.21) we can write the follOWing:
(4.5.22)
~I n-~u
Sup*n GN ,1J
·· -
uEJ
= Sup* n
uEJ
GO
I
N°'
,1J
~142-1£.1-1
~4 (Gn-~u
GO
N"
hlN"
- ,1J,
,lj, hI)
p-2
-~
~2- (G n u
)I
_ GO
~=3~=1
N,ij,gm N,ij,gm
+ (~4
~
~4 S up n~ IGn-~u
GO
I
4 2-1
- £.1-1
* N',1J,. hl- N",1J, hI
+
uEJ
4
! n-~u
0
1 Sup n IGN . .
-G ..
I,
~=
*
,lj ,gm N ,1J ,gm
2P-2
~"1 3~"~=
uEJ
110
e
i,j=l, ... ,po
Since the sum in (4.5.22) is over a finite number of sets, then showing
(4.5.12) can be reduced to show that for every compact region J* ,
(4.5.23)
0
D.
S up n 'IGn-~u
N'
. hl-GN ,1J,
.. hll~O as n--+oo ,
*
,1J,
uEJ
for each h=1,2;1=1, ... ,4, and h=3,4;1=1, ... ,2 P-2.
For each component of u in that compact region we consider the
following partition:
where
(4.5.25)
I~m - ~m-11
<11
, m=l, ... ,r,
in doing that we partitioned the compact region into r P cubes with each
cube having 2 P vertices. Denote by Dr the set of all such vertices.
Now consider a particular point u = (u l' ... ,up) such that
(4.5.26)
~
q.1
1
s: u.1
~ ~
q.
, q.=O,1, ... ,r; i=1, ... ,po
1
1
Then we have the following lemma.
Lemma 4.5.3 Under the assumptions A, B, C and those in Lemma
4.5.1 and for the point u E U.. hI that is given in (4.5.26) we have
Ij,
111
(4.5.27)
~ I n-~u
0
I
Sup*n GN ,1J,
·· hl-GN ,1J,
.. hI
uEJ
~I n-~v
0
I~ 0 as n
~r max n GN,ij,hfGN,ij,hl
l' ... , r p
.00,
n-!
where v = (~ r , ... ,~ r ) and rk=qk if GN ,1J,
.~ hI is a nondecreasing
1
p
function in Uk' otherwise rk=qk-1 k=l, ... ,p. The maximum in
(4.5.27) is taken over the set of all groups r l' ... ,rp where each r k
runs through 0,1, ... ,r for each k=!, ... ,p.
Proof:
n-!
From Table 4.1 G N ,1J,
.~ hI is either nondecreasing or nonincreasing in
each uk' k= 1, ... ,po Considering the point u = (u i' ... ,up) defined in
(4.5.26) we can write the following
(4.5.28)
! I n-~u
0
I
!
n GN ,1J,
·· hl- GN ,1J,
. . hI ~ n max*
vEO r
IGn-1v
0
N'
. hl- GN ,1J,
.. hI
,1J,
where Dr* is the set of all possible vertices with the i-th component is
or ~
that are defined in (4.5.26) i=1, ... ,p. Hence
qi
qi-l
contains 2 P verticies and O;EO r . To show (4.5.28) we define the
either
~
following:
(4.5.29)
U = (U 1 , ... ,U ) =
P
(~r
L = (L 1 , ... ,L ) =
P
(~s ' ... ,~
' ...
1
!
,~
)
rp
sp
)
,
-!
rk=qk and sk=qk-1 if G~,ij,hl is a nondecreasing function in Uk'
otherwise rk=qk-1 and sk=qk (Le., if nonincreasing) k=l, ... ,po
112
°*
r
e
n-~.~ hl-GNO .. hI is positive, in
First let us consider the case when GN
,lj,
,lj,
this case we have
(4.5.30)
~ n-±u
0
~ n-~u
0
n IG N ,lj,
.. hl-GN ,lj,
.. hi I = n (G N,1J,
.. hi-GN ,lj,
.. hi)
~ n
~
n-~U
0
IG N ,lj,
. . hl- G N ,lj,
. . hi I ,
n -~
similarly for the case when GN ,lj,
.~ hi-GNO ,lj,
.. hI is negative we have
(4.5.31)
~ I n-~u
GO
I - ~ 0 . . hl- Gn-~u
n GN ,1J,
·· hl- N"
. . hI
N ,lj,
,lj, hi - n (G N ,lj,
~
n-~L
0
~ n IG N ,lj,
.. hl-GN ,lj,
.. hl\
,
(4.5.30) and (4.5.31) imply (4.5.28).
From (4.5.28) we conclude that
(4.5.32)
To complete the proof we need to show that for any value of v = (v1'
... ,v ) the following equation is true:
p
(4.5.33)
I
n~IGn-~v
N'
. hl- GO
N"
,lj,
,lj, hI
~0
as n --.
00 •
First, for fixed values of i and j i,j= 1, ... , p, we denote by nh the
number of observations in the Q~. h-th quadrant where v is defined by
IJ,
(4.5.27). For the rest of this proof we will denote by yf1f3=(Xhf31, ... ,
y
y
113
n-!
Xh(3) any original observation Xf3 with R f3v E Q'!. h
N,
yp
IJ,
y=l, ... ,n . Then it is reasonable to assume that
h
(4.5.34)
~= ,1
nh
/\h
f3=1, ... ,n;
is finite .
Next, we define the following:
(4.5.35)
(4.5.36)
vh
_ 1 [
hf3
Xh{3
-~
.
FN(k) (x) - n number of X yk such that yk + n vk ~ x,
h
r- 1, ... ,nh) ,
Oh
_ 1
h{3
h{3_
FN(k)(x) -nh[number of X yk such that Xyk~X, y-1, ... ,nh ],
(4.5.37)
(4.5.38)
Oh
_ 1
h{3 h{3
hf3 hf3
FN(k,l) (x'Y)-n [number of (Xyk,X yl ) such that (Xyk,X yl )
h
~
(x,y), y=1, ... ,n ] ,k,1=1, ... ,p,
h
(4 . 5 . 39)
vh
Nh
h
~ 1FN (k) (x) ,
HVN (x) = N + 14=
h
(4.5.40)
Oh
Nh
P
Oh
HN (x) = N + 1 2 k=1 FN(k) (x) ,
h
114
i,j,k,I=1, ... ,po
Similarly, we define
A?~ (i,j)' A~~(i,j) and A~~(i,j)
by substituting
for v by 0 in (4.5.41), (4.5.42) and (4.5.43) respectively. From the
n-~
proof of Theorem 9.5. 1 in Puri and Sen (i 97 1) we can wri te G N V. hI
,1J,
0
a
and G N ,1J,
.. hI as
(4.5.44)
n-~v _
GN ,1j,
0 0 hi -
(4. 5. 45)
G N,1J,
0 . hi
a
3
vh
2:a-1 Aa N(O1,j0)
Oh
= ",,3
L. -1 A N(. 0)
a- a 1,j
-~
+ 0p(N h )
-~
+ 0p (Nh ) .
Then by defining
(4 5 46)
. .
G (i,j)
1N, yh
(Xh~ Xh~)
Y1' YJ
= J{H(Xh~)
J(H{Xh~)
Y1
YJ
(4.5.47)
(4.5.48)
where
Of o
Xhf3X
5.
l
Y1
otherwise
and
115
'
(4.5.50)
(1)
nh-1 [(i~')
hI' + n-~ v.,X hI'. + n-~ v.)
LN
h .. -_ N-~
G 1 ,y h(X,
h L: y, ,IJ
Yl
1 Yj
j
(4 5 5 1)
. .
(2) .. =
L N,h,lj
N-2~nh
h
L.. y
(Xh~. + n- 2v.)
=1 [G(i,j)
2N, yh Yl
1
1
- G (i,j) (XhB..
2N, yh yi) ,
Fallowing with slight modification the steps in the proof of Theorem
9.6.1 by Puri and Sen (1971) we can show for the given value v that
(4.5.53)
N-h2
3 1 (Avh (· .)-A°N
h
[GnN-~~
O ,lj,
.. hI] = N-h2 [2:a=
,lj, hI - G N
a N 1,j
a (0l,j.))]
i
=
~3
L..a
=1L N(a),h ,iJ'
,
for i,j=1, ... ,p and h=1, ... ,4, and for fixed values of i and j i,j=1,
... ,p that
(4.5.54)
LN(a)h"~
0 for a=1,2,3.
, ,IJ
Hence, from (4.5.53), (4.5.54) and (4.5.34)
(4 .5. 55)
n- 2v
I
n 2IG N'
. hI - GO
N"
,IJ,
,IJ, hI
p
--J;'--fo.
0
•
(4.5.32) and (4.5.55) implies (4.5.27).
Finally, using Lemma 4.5.1, (4.5.22), Lemma 4.5.3 and the fact
116
that the sum in (4.5.14) is over finite number of sets we have the
following Theorem.
1lleorem 4.5.4 Under the assumptions A, B, C, (4.5.3) and that BN ,,8
is nondecreasing and square integrable we write
(4.5.56)
~
n-~u
Sup n IG N ..
*
,1J
uEJ
GD
N ..
,1J
1
~O asn-+oo
The following theorems are special cases of the theorems in Section
9.5 of Puri and Sen (1971). Using Theorem 4.5.4 we have the
following theorem.
Theorem 4.5.5 Under the assumptions (A.3) of Section 4.4, and the
additional assumptions that F(x) is continuous and J(u) is symmetric
d
dn
about zero, the random variables n~[GNn.. - j1.. -}J.}J., T . - }J.;
,1J
1J 1 J N ,1
1
.< . -1 , ... ,p } and n ~ [G N,ij - }Jij - }Ji}Jj' T
< .-1 , ... ,PJ' have
1-JN,i - }Ji;'1-JA
the same limiting distribution. TN' and
,1
and (4.4.5) respectively, and
~
}-l . .
IJ
are given by (4.2.11)
~
d
T n . is the same function as TN . in (4.2.11) for the observations that
,1
N ,1
are centered around d .
n
~
lbeorem 4.5.6 Under assumptions (A.1) to (A.4) of Section 4.4, the
random variables
117
,
n [G
.. N ,1J
}J. . - }J. }J. ,
1J
1 J
}J. ; i ~ j
TN . ,1
1
=1,
... ,p]
have asymptotically a p(p+3)/2 variate normal distribution with null
mean vector and finite covariance matrix.
The theorem is a straightforward application of Theorem 9.5. 1 of Puri
and Sen (197 1) chapter 9 . For thiS reason the proof is om itted and
can be found in the referenced text. From the same reference, we note
that
(4.5.58)
N cov{SN ,.1
"SN
.. kl ,i,j,k,l=l, ... ,p,
J , kl} ----+ 111J,
where
kl is defined in (4.4.6).
11 . .
1J,
We now formulate a class of dispersion alternati ves. For the scalar
alternatives we let
where under HO' all C7 i are equal. As in Sibuya (1959), we consider
bivariate dependence functions Q .. defined as
1J
(4.5.60)
Q: . = F( .. ) (x,y) / F. (x) F . (y)
1J
1,j
1
J
for all i~j=l, ... ,po Clearly, Q ij is a function of F(i) and F{j)' and
will be equal to unity if X . and X . are statistically independent.
a1
aJ
Various properties of such dependence functions are studied in Chapter
8 of Puri and Sen (1 97 1) . These functions appear to have some
118
advantages over other nonparametric measures.
Denote by
n the same function as in
(4.5.60) for the bivariate c.d.f.
F(x,y), and specify
(4.5.61)
n ,
~i.,
IJ
= n+ w"IJ
f or 1'*'-1
J - , ... , p.
~i.
(4.5.59), (4.5.60) and (4.5.61) constitute our desired class of
alternatives. Now we can rewrite (4.2.5) as
(4.5.62)
eij ~co[rF(i ,j) (x,y)
F F
-
(i) (x)
(j) (y)]J' [H(x)]J' [H (y)]
dH (x) dH (y),
for all i,j= 1, ... ,po
For the study of the asymptotic non-null distribution of L N , we shall
consider a sequence of alternatives converging to the null hypothesis in
such a manner that the power of the test based on L N lies in the open
interval (a,1). Thus we replace (I, in (4.5.59) by a sequence {1+n-~K,}
1
1
where Kp i=l, ... ,p are real finite constants. Also, in (4.5.62) we
make the following changes:
(4.5.63)
F(, ') (x,y) - F(.) (x) F( ') (y) -- [H(x,y) - H(x) H(y)] + n-~ ~" (x,y)
1,J
1
IJ
J
i;tj=l, ... ,p, where {. ,(x,y) are real and finite valued functions, and
IJ
(4.5.64)
H(x,y) =
[PJ-l
2: F(.
2 1~i <j~p
,) (x,y).
l,j
We shall denote such a sequence of a1 ternati ve hypotheses by {H N}.
119
Thus. on defining
~IJ.. =
(4.5.65)
[p..
IJ
-co -co
(x,y) J' (H(x)] J' (H(y)] dH(x) dH(y)
for i,j= 1, ... ,p;
~
(4.5.66)
= {~.IJ.,
i ~j
=1,
... ,p} ,
and following Theorem (9.5.2) of Puri and Sen (1971) we arrive at
the following theorem.
l1leorem 4.5.7 Under {HN}, LN asymptotically has a noncentral chisquared distribution with (s * - 2] degrees of freedom and noncentrality
parameter
(4.5.67)
AL =
~
* -1 ~
(JI)
where (JI*)-1 is the inverse of
JI*
defined in (4.4.7).
In the parametric case, Wilks' (1946) likelihood ratio (lor.) test
Lvc for this problem is given in Section 1.2.1. Wilks has only
considered the distribution of the l.r. criterion under the null
hypothesis. However, Using Wald's (1943) methods, one can readily
find out the asymptotic non-null distribution of the l.r. under {H }. If
N
~1 is the dispersion matriX, then under {H }, we will have
N
where
£
is real and finite, and
~O
is the dispersion matrix under HO.
120
Now defining
and adopting the suffixing system of (4.3.17), we write
(4.5.70)
,s*
Note that under the assumption that F (x) is normal and by using normal
scores,
~
and r are asymptotically equal, also"* and
r.
Then we have
the following theorem. The proof is gi ven as Exercise 9.5.3 in Puri
and Sen (1 97 1).
11leorem 4.5.8 Under (4.5.68), the l.r. test L
vc asymptOtically has a
noncentral chi-squared distribution with [s * - 2] degrees of freedom,
and noncentral itY parameter
(4.5.72)
Comparison of (4.5.67) and (4.5.72) will yield the asymptotic
relative efficiency of L
with respect to Wilks' Lvc ' For the normal
N
scores test where J (u) is the expected value of the i-th order statistic
of a sample of size N from a standardized normal c.d.r. and assuming
that F(x) is normal then (4.5.60) and (4.5.65) are asymptotically
equal, and the two tests are asymptotically eqUivalent.
121
CHAPTER V
NUMERICAL EXAMPLES OF 11IE TEST PROCEDURES
5.1 Intr'odtrlion
In this chapter, some real life data are analyzed to numerical! y
illustrate the use of the theory that has been developed in the previous
chapters. Two data sets will be used, both of which are parts of
larger studies that has been conducted by the Lipid Research Clinics
Program.
In Section 5.2, we apply the test of restricted interchangeability
using the nonparametric approach that we introduced in Chapter II. The
resul ts of thiS approach are compared to the results from using
Votaw's parametric test (Section 1.2.1) on the same data.
Section 5.3 is a data application of the theory that was developed
in Chapter IV, that is, we test against a non-location alternative. In
thiS section, the results from using thiS nonparametric approach will
be compared to the results when using Wilks' L
(Section 1.2.1)
m
parametric test on the same data. For this comparison to be possible
and fair, normal scores are used in the nonparametric procedure (the
same is done in Section 5.2).
Finally, Section 5.4 is concerned with applying the methods of
Chapter III, Le., test of interchangeability with incomplete data. In
that section we will use the complete data of Section 5.3 and randomly
create missing values by using a uniform random number generator.
Both the "ranking after alignment" and the "intra-block ranking"
methods are used. The analysis is done using all the observations and
is repeated again using the complete part of that data (Le., what is
left complete after creating the incomplete observations). We end this
chapter considering extensions of this work and examining future
research possibilities.
5.2 Examples of the Restricted Int.ercharJteability Test
In the following example we apply the nonparametric test statistic
(using normal scores) that is given in (2.3.8). The results of this
procedure are compared with the results from the parametric one that
is given in Section 2.6.
Blood samples from 199 patients ( 101 females and 98 males)
were taken at two different visits, with approximately a one month
period between the visits. The patients were not told their cholesterol
and triglycerides values 'Until after both measurements had been made.
Thus unless the patients changed their diet or other behavior, one would
expect the two cholesterol measurements to be interchangeable. The
following are the blood fat levels that are measured at visits one and
two for each patient:
1)
x~ 1)
= CHOL 1 = Cholesterol level at visit
123
1
2)
Xl~) = CHOL2 = Cholesterol level at visit 2
3) Xi1)
4) Xl~)
=TG 1 =Triglycerides level at visit 1
= TG2 = Triglycerides level at visit 2.
Observation
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
Sex
2
1
2
1
1
2
2
1
2
2
2
1
1
1
1
2
2
1
2
2
2
1
2
2
1
2
2
1
2
2
1
2
1
2
2
1
1
1
X(1)
1
182
173
220
212
245
218
207
187
223
220
213
235
157
198
197
178
176
235
196
272
205
254
184
181
215
171
250
260
217
182
217
170
214
150
160
203
170
208
124
X(2)
1
180
173
224
209
261
276
215
202
226
197
237
244
153
198
194
189
181
193
216
290
199
206
173
185
234
168
259
319
208
170
241
161
255
153
163
187
182
171
X(l)
2
53
115
186
159
342
137
111
70
50
113
86
110
60
104
125
220
148
57
39
102
75
144
51
113
120
68
101
1160
57
48
145
52
127
90
83
151
48
136
X(2)
2
57
136
216
103
540
132
78
95
42
57
101
156
92
154
95
155
88
81
40
71
99
92
76
162
104
63
120
166
72
61
176
56
183
96
87
132
73
75
e
Observation
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
Sex
2
2
1
2
2
1
2
1
1
1
1
2
2
1
1
1
2
1
1
2
2
1
1
1
1
2
2
2
2
2
1
1
1
1
2
1
2
2
2
1
2
2
1
2
2
2
2
1
X(1)
1
245
185
185
195
214
217
202
220
196
241
181
202
241
230
164
187
165
161
168
166
160
163
215
192
199
180
155
397
208
166
184
212
158
204
181
194
206
219
. 211
181
178
226
117
203
191
220
180
162
125
X(2)
1
252
162
162
217
239
221
199
190
194
220
173
220
249
205
142
187
159
150
250
200
172
168
206
197
150
187
170
294
234
148
179
207
152
190
208
191
209
196
217
229
188
197
132
184
226
236
191
152
X(1)
2
87
55
127
143
160
58
142
176
91
78
109
70
43
62
48
66
66
219
107
74
37
117
169
124
65
65
65
98
105
134
145
89
135
88
211
106
88
70
120
155
49
62
52
141
82
67
128
65
X(2)
2
91
68
106
110
114
101
128
212
79
105
110
138
69
93
86
86
77
212
143
86
38
82
91
108
60
88
58
118
145
97
245
63
49
80
84
180
120
114
85
181
56
71
76
138
80
88
62
47
Observation
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
Sex
2
1
1
2
1
2
2
2
2
1
2
2
2
2
2
2
1
1
2
2
2
1
1
2
2
1
1
1
1
1
1
2
2
1
1
1
2
1
1
2
1
1
2
2
2
2
2
1
X(1)
1
150
174
232
272
288
199
188
223
177
165
132
168
217
252
147
217
200
240
118
195
238
221
135
213
231
117
216
259
232
162
161
178
240
140
234
142
169
266
169
176
190
202
237
168
161
126
180
150
126
X(2)
1
181
206
234
240
287
209
179
233
178
188
142
209
264
238
153
251
202
253
118
221
231
242
147
249
223
120
197
281
257
166
186
218
250
119
205
128
170
279
181
203
208
222
221
181
148
152
183
160
X(1)
2
75
77
65
83
133
76
81
79
89
172
204
79
111
99
68
73
81
252
93
65
188
110
93
176
80
57
84
253
95
107
55
180
84
72
234
116
56
234
109
110
108
78
97
61
61
72
97
58
X(2)
2
88
79
121
95
175
97
56
75
118
122
230
109
102
116
80
75
69
204
70
66
112
113
134
159
112
51
87
239
196
89
68
502
104
48
147
191
88
172
80
154
147
50
78
50
45
86
174
48
e
Observation
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
Sex
1
1
1
2
2
2
2
1
2
2
2
1
2
2
2
1
1
2
1
1
1
1
1
2
2
1
1
1
1
1
1
1
2
1
1
1
1
2
1
1
2
2
1
1
1
1
2
1
X(1)
1
176
148
134
220
160
198
205
147
190
104
251
185
192
227
217
256
257
225
159
261
163
232
194
203
170
220
189
260
166
219
225
274
216
300
114
201
266
253
159
191
192
199
209
219
230
181
167
240
127
X(2)
1
175
162
138
234
143
216
221
168
193
171
275
174
184
236
236
248
242
224
165
243
204
202
230
197
164
233
237
243
185
199
236
294
206
300
105
200
224
196
193
173
182
220
248
233
194
177
169
214
X(1)
2
120
85
69
71
50
101
39
92
112
118
155
139
71
130
127
230
170
71
61
216
40
200
95
102
38
55
130
196
119
163
197
300
82
190
31
155
129
114
68
132
81
153
72
108
163
49
. 82
130
X(2)
2
176
105
62
56
40
70
40
111
90
99
115
71
96
152
138
226
265
105
78
129
34
171
69
88
61
61
175
163
159
105
158
265
114
144
20
155
70
147
130
144
95
188
96
122
217
60
67
114
Observation
1
2
2
2
2
1
2
1
2
1
1
2
1
2
2
2
2
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
X(2)
1
191
243
305
208
158
228
182
200
171
167
195
172
228
158
192
254
236
X(1)
1
Sex
195
221
268
157
159
220
186
189
178
171
183
173
211
146
235
238
215
X(2)
2
X(1)
2
137
135
160
114
89
199
66
124
69
62
45
50
456
56
75
125
100
123
160
154
94
72
189
63
184
62
66
43
60
336
55
67
103
118
To be consistent with the notation in Chapter II we denote
_ (1 ) (2) (1) (2)
X.1 - (X'1 1 ,X'1 1 ,X'12 ,X'12 ) .
Then our purpose is to study the interchangeability within the
Cholesterol set (X ,X 2 )) and wi thin the Triglycerides set
l1) i
(X11) ,X12 )) simultaneously.
First we give a summary of the statistics
to be used in the nonparametric data analysis as follows:
aU)
(k)
N.,R..
J 1J
(k)
R ij ) ,
= JU)(
N. N.+l
J
gU)
N.
J
J
BN.,R
(j) (.) = .l..2~_ aU) (k),
..
p. J-1 N.,R ..
J
1J
J
J
1J
= _1L:Nj
N.
J
aU)
(3= 1 N.,{3'
J
TU)
= l.~_ aU) (k),
N .,k
n 1-1 N .,R ..
J
128
J
1J
U) _
2
i
n
pj
( ")
-
U)
2
aN"(I~~.) - n.(p.-i)~1=i4=i (B~.,R~k,> - BN.,R~·.» ,
J
J
J J
j
lj
J
lJ
n ~~1 (T(j) - gU) )2
.)
.oC..K-i N .,k
N.
W (tJ ]
J
N. 2 (RU»
J a N . N.
J
J
for k= 1,2; j=i,2 , where R.(~) is the rank of X~~) in the j-th set, and
lJ
lJ
J~~ is the normal score function. Because Pi =P2=2, we note from the
def1nition of TU) that TU)
= - TU) for j=1 ,2.
N.,k
N.,1
N.,2
J
J
J
The ranks of the above observations and their normal scores are
given in Appendix Ai. The following table contains a data summary of
some of the above statistics.
Table 5.1
Data Summary of the Nonparametric Statistics
By Sex
CHOL
TG
1
2
2
T(2)
T(2)
T(1)
T(1)
E(2)
E(1)
aN
aN
N ,2
N2 ,1
N2 ,2
N1
N2
N1'1
1
1
2
-.0305 .0305 .0000 .1487 -.0099 .0099 .0000 .2310
2
-.0922 .0922 .0000
SEX
1&2 -.0612
.0612
.1731 -.0387
.0387 .0000
.2400
.0000 .1666 -.0208 .0208 .0000
.2261
Note that for SEX=2 in the cholesterol set, (T (1) 1-TN(1) 2)2
N l'
l'
>a~
.
1
Statistics to be used in calculating the parametric test are listed
129
below. Table 5.2 give a data summary of some of these statistics.
lU)1
1m -"[1 +(A ./8.) I
J
J
-nln(lU)) -"'X 2
1m
1
where
Cl~~j) and Cl~ij) are given in (2.6.7), j=1,2;k,I=1,2
Table 5.2
Data Summary of the Parametric Statistics
By sex
TG
CHOl
SEX
X (1)
.1
X (2)
.1
X (.)
.1
8
- n1
X (2)
.2
X(2)
.2
X (.)
.2
8
- n2
1
199.49 201.89 200.69 129.16 134.63 127.29 130.96 3014.4
2
198.10 204.11 201.10 133.52
95.30 99.62
97.46 486.86
1&2 198.78 203.01 200.90 264.31 114.64 113.25 113.94 3518.4
Table 5.3 contains test statistics, degrees of freedom and p-values
of both the parametric and the nonparametric procedures. As we
notice, the results match with little variation in the p-values. Both
procedures reject the null hypothesis that the two cholesterol levels
and the two triglycerides levels are simultaneously interchangeable.
130
Looking at the p-values wi thin each set in Table 5.3, we can notice
that this significance is due to the difference in the cholesterol levels
for the females.
Table 5.3
Parametric and Nonparametric Test Statistics and Their p-values
for Each (and Combined) Set (s) of Fat Levels for the
Whole Sample and for Each Sex
CHOL
SEX
N
0
N
P
A
R
P
A
R
A
M
t
1
2
TG
1
&2
1
2
CHOL , TG
1
&2
1
2
1
&2
WN 1.227 9.920 9.012 0.083 1.251 0.758 1.310 11.17 9.770
1
1
d.f.
1
2
2
2
1
1
1
p-val 0.268 0.002 0.003 0.773 0.263 0.384 0.519 0.004 0.007
t
In L 1.084 6.680 6.678 0.393 1.015 0.060 1.481 7.656 6.739
d.f.
1
1
1
1
1
1
2
2
2
p-val 0.298 0.010 0.010 0.531 0.314 0.810 0.477 0.022 0.030
In L is the parametric test -n In(L 1m)'
5.3 Example of the Test of Int.er'charEeability Against a Non-Location
Alternative
The following data are a part of a second study conducted by the
Lipid Research Clinics Program. The data set consists of four
cholesterol levels that are measured on the same person at four
different visits. The first two cholesterol measurements were made
before the patients were instructed in a low cholesterol diet, while the
second two measurements were made after the diet. Thus we would not
131
expect the four cholesterol measurements to be interchangeable. Let
CHOL 1 = Cholesterol level at visit one
= cholesterol
CHOL3 = cholesterol
CHOL4 = cholesterol
CHOL2
level at visit two
level at visit three
level at visit four.
Observation
CHOL1
CHOL2
CHOL3
CHOL4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
306
293
350
291
308
315
293
301
305
267
390
294
307
283
299
274
270
304
303
279
288
284
276
309
329
313
263
278
283
288
310
368
268
257
312
309
291
288
261
291
305
318
336
398
301
375
301
272
333
291
250
281
333
318
281
331
275
253
282
316
308
266
287
282
305
373
353
302
271
304
290
274
265
251
294
290
332
266
344
277
349
301
262
290
291
276
249
275
276
267
276
279
280
273
309
311
261
286
277
277
335
319
256
265
273
283
256
277
261
281
270
302
282
318
282
346
250
264
342
302
263
247
326
296
270
269
250
267
280
314
304
270
274
313
312
298
358
234
266
274
132
e
Observation
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
.71
72
73
74
75
76
77
78
79
80
81
82
83
CHOl1
CHOl2
CHOl3
CHOl4
308
270
239
269
322
329
273
268
279
366
248
286
269
281
283
289
315
304
273
304
321
277
289
277
293
267
284
283
260
308
312
349
285
297
260
281
273
277
275
325
282
285
317
257
300
305
291
259
261
264
268
292
295
289
292
275
310
390
225
285
273
293
240
251
312
302
271
308
261
317
267
287
306
298
288
272
268
286
277
317
296
334
274
265
291
317
277
317
301
263
266
248
288
315
308
259
259
299
257
254
297
262
285
263
295
362
258
272
291
269
247
263
295
292
246
356
294
314
257
294
264
258
264
312
287
280
257
282
250
270
262
265
265
307
258
292
274
280
238
255
278
310
276
280
266
278
269
260
267
293
240
278
289
374
248
252
272
257
254
262
304
285
251
284
333
294
257
263
265
259
266
261
260
287
246
310
255
269
269
282
264
316
250
295
284
269
250
247
316
300
274
316
133
Observation
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
CHOll
CHOl2
CHOl3
CHOl4
385
287
287
331
303
286
268
285
289
280
257
311
309
344
371
392
293
289
290
311
327
274
277
315
268
302
306
256
290
267
330
272
360
299
267
301
350
297
334
269
300
331
375
284
288
300
295
254
358
280
274
288
302
256
249
270
273
289
282
309
306
307
370
406
284
297
321
273
354
269
276
291
272
295
290
251
304
296
265
275
379
297
272
315
292
235
307
286
278
257
342
274
262
314
270
243
357
294
243
297
283
280
252
268
281
252
273
321
297
331
319
376
252
290
272
297
321
260
271
273
276
304
287
234
289
277
293
283
324
270
242
297
307
281
283
263
278
279
341
279
282
258
299
260
346
277
266
296
316
253
261
295
270
253
305
316
278
338
393
374
250
287
277
309
271
261
270
307
256
325
294
248
332
251
300
296
311
294
239
272
295
270
309
303
263
279
306
277
314
270
274
252
134
e
Observation
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
CHOL1
CHOL2
CHOL3
CHOL4
283
287
299
269
283
288
375
355
321
269
455
293
268
274
277
278
267
276
374
321
291
294
280
304
290
335
331
279
318
291
263
268
276
280
303
275
282
307
331
275
286
265
288
292
301
275
297
256
277
283
347
271
302
289
355
309
339
294
449
273
277
244
251
283
277
273
386
277
317
313
275
280
290
343
317
307
321
283
291
256
307
299
326
265
273
280
305
306
333
262
245
289
278
232
311
286
261
265
311
258
287
242
338
308
292
274
430
290
253
256
247
280
278
266
356
269
268
312
278
280
287
321
276
284
304
328
256
257
287
288
275
282
267
286
285
289
322
286
275
278
259
258
311
275
249
274
286
263
304
273
399
274
294
255
409
269
253
278
217
282
274
264
368
253
287
286
284
281
302
306
250
276
292
278
251
270
275
274
307
258
282
296
303
265
310
310
267
288
255
265
294
285
135
Observation
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
CHOll
CHOl2
CHOl3
CHOl4
265
289
291
280
282
283
306
497
272
277
395
276
277
255
340
272
259
273
295
266
285
258
249
238
273
283
283
260
562
270
275
355
260
309
269
307
292
287
275
282
287
291
249
279
289
254
293
244
277
502
243
299
347
253
274
270
255
234
285
252
242
283
275
283
239
272
240
269
230
259
486
238
287
360
295
255
258
264
235
275
249
257
284
277
e
And let
Then our purpose is to test the null hypothesis HO that {X 1 ,X 2 ,X ,X 4 }
3
are interchangeable against the alternative hypothesis that Hvc is not
true. Hvc is the hypothesis that the variances are equal and the
covariances are equal irrespective of the values of the means. Testing
of the above hypothesis will be carried out using the nonparametric
procedure that is given in Chapter IV, and the parametric test lvc that
is given in Section 1.2.1. First we give a summary of the formulas
that are needed to calculate the nonparametric test. From Chapter IV,
RPi is the rank of X
Pi
we have the following:
in the combined sample of all observations, then
136
1 R Pi
BN,R .= IN(R pi ) = t- (N+t)
= normal scores, i=l,
P= 1,
Pl
,p;
,n,
for a given i and j, define r=[(i-l) (2p-i)/2]+j, and s=[(k-l)(2p-k)/2]+l,
then we have the following:
].I
l~n
B R B R B
B
-S
S
n,....= 1 N 'f3i N'f3 j N,Rf3k N ,Rf3l
N,r N,s
(R..) = - ~R
rs - '"N
for i,j,k,l=l, ... ,p , r,s=l, ... ,s * and s * =p(p+1)/2.
Next define the vector
~,
its expected value, and its variance-
covariance matrix as follows:
*
*
*
~ = (SN,1' ... ,SN,s*)' SN = (SN, l' ... ,SN,s*)
and
vN = ((].I rs) ) r,s=l,
... ,s *
,
then the test statistic is
LN
* V-1 (~ -~)
*
= n(~ - ~)
N
I
2
... X *
(s -2)
.
As we notice from the definition of I , normal scores are being used
N
in determining the test statistics. The ranks and their corresponding
137
scores are given in Appendix A2. In this example n=200, p=4,
N=8 0 0 and s * = 1O. Hence, using the above formulas we get the
following results:
~=(. 762
.496 .472 .490 1.018 .571 .575 .930 .719 1.100)
~=(. 988 .539 .539 .539 .988 .539 .539 .988 .539 .988)
3.402 2.813 2.283 2.259
3.217 2.405 2.353
2.536 2.284
2.700
LN = 32.1565 - X2
8
~
2.724
3.092
2.337
2.263
5.406
p-value
~
2.225
2.534
2.236
2.010
3.252
3.135
2.179
2.477
2.018
2.228
3.368
2.743
3.397
2.083
2.044
1.937
1.773
2.513
2.498
2.126
4.206
1.9942.142
1.933 1.965
1.890 1.841
2.007 2.127
2.340 2.609
2.251 2.101
2.315 2.551
3.595 3.402
3.908 4.076
5.758
.0001 .
Next, for the parametric procedure, from Section 1.2.1 we use the
following formulas to determine the test statistics:
1~n
Xi = nL..p=1 XPi
'
_ 1~
s .. - - L.f3t:J. 1 (X Q .
1J
n
=
,....1
-
-
X.)(X Q
1
-
. -
,....J
X.) ,
J
(s .. ) the variance-covariance matrix of X ,
1j
138
e
1 ",p
p(p-l)'-i;z!j=l Sij ,
lSi i I
Lve - -[S-::2-----:2::;-r-]p--'=':-ll.-[--;s2:::;--+-(-p_-l-)s--::2::-"r-]
S
where
-n In L ... X2
ve
s * -2
Applying these formulas on the above data we get the following
results:
1182.05
((s .. ))
lJ
s
=
2 = 1215.53 ,
rs
986.694
1478.46
821.222
973.595
1033.88
852.066
955.816
865.779
1167.73
2 = 909.196 , L = 0.87368
ve
-n In L = 27.008 ... X2 ~ P-value = .0007.
vc
8
Table 5.4
Test Statistics and their p-values for Both the Parametric
and the Nonparametric Procedures for the Non-Location Alternative
Test Statistics
PARAMETRIC
NONPARAMETRIC
df
p-val
-n In Lve = 27.008
8
.0007
LN = 32.156
8
.0001
139
From Table 5.4, we note that both procedures gave very similar
significant results, Le., at a=.O 1 we reject HO the hypothesis of
interchangeability against the alternative hypothesis that the variances
are not equal and/or the covariances are not equal.
In~eabilityWith
5.4 Example of Rank Order Tests of
Ilnlmplete Data
The following data were the same data that are given in Section 5.3,
with missing values generated randomly. The program that was used to
generate these missing values is given in Appendix A4. Note that the
probailities that are used in creating these missing values are
PO=~~~,
188
132
24
d
125
h
. th
b b' lOt f
P1 =625' P2=625' P3=625' an P4=625' were Pi 1S e pro all y 0
generating i missing values within each block i=O, ... ,4.
e
Observation
CHOL1
CHOL2
CHOL3
CHOL4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
306
293
350
309
291
290
274
.
·
251
294
283
256
277
261
.
308
305
318
.
274
.
250
281
333
140
·
·
332
270
302
·
344
·
318
·
262
290
·
264
342
·
276
249
275
·
263
247
326
Observation
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
CHOL1
CHOL2
CHOL3
·
253
280
·
266
·
309
311
261
·
304
270
·
305
·
277
335
·
312
298
·
271
256
265
273
·
·
234
266
·
292
·
257
254
·
269
·
329
·
263
·
310
·
268
257
·
239
·
322
·
268
295
·
292
·
263
279
366
295
362
·
272
·
286
·
·
293
283
289
315
304
240
251
312
·
304
321
·
308
261
·
289
·
293
·
267
·
306
267
284
283
260
·
·
285
312
297
298
288
272
268
286
·
317
·
334
141
CHOL4
·
267
·
240
278
289
374
·
269
247
·
272
257
254
·
295
·
304
·
284
·
257
294
264
258
264
312
287
·
257
282
·
270
·
294
·
265
·
266
261
260
·
255
Observation
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
CHOll
CHOl2
260
281
273
·
265
291
·
325
·
277
CHOl3
269
282
·
258
292
·
·
263
·
257
·
305
291
·
287
·
·
255
·
315
308
259
358
280
CHOl4
·
310
276
·
357
294
243
269
250
247
·
300
274
316
·
277
·
316
303
286
268
·
256
249
·
289
·
273
·
252
268
281
311
309
·
309
·
321
371
·
·
370
316
278
338
393
·
293
·
284
·
252
·
250
·
·
331
·
321
·
354
·
277
315
268
302
·
276
·
271
·
272
·
276
·
256
290
·
330
·
360
299
·
304
296
265
275
·
297
142
·
·
277
·
327
295
·
261
295
270
·
304
287
234
289
277
·
270
271
261
·
307
256
·
332
·
300
·
311
294
e
Observation
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
CHOLl
CHOL2
CHOL3
CHOL4
·
301
272
315
292
242
297
307
239
272
295
334
·
·
300
·
307
331
375
·
342
350
·
288
300
295
254
283
287
·
283
288
·
355
·
455
293
268
274
277
·
267
·
374
·
291
294
280
304
290
335
331
279
318
291
263
·
276
280
274
262
·
·
283
243
·
289
·
309
·
449
·
244
·
309
·
279
341
·
258
·
·
287
·
304
273
·
338
·
430
290
·
277
273
·
278
·
307
·
299
143
·
270
274
·
247
317
313
275
280
290
343
·
277
261
265
251
·
277
·
279
·
·
274
·
409
·
253
278
·
321
·
274
264
368
253
287
286
284
281
302
306
·
·
276
·
356
·
268
312
·
280
284
304
328
256
257
287
288
·
270
·
274
Observation
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
i83
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
CHOLl
CHOL2
CHOL3
CHOL4
303
275
282
307
331
275
286
265
288
326
265
275
·
258
·
305
·
286
285
289
·
296
303
·
262
245
·
286
275
·
·
278
·
259
·
297
·
311
286
258
249
·
275
301
·
289
291
·
306
497
272
277
·
277
255
340
272
259
·
295
266
285
·
283
260
562
270
275
·
260
309
269
307
·
275
282
·
291
·
310
310
267
288
255
265
294
·
279
289
·
272
·
244
277
502
·
230
259
486
·
·
287
·
274
270
·
255
258
·
234
285
·
235
299
347
·
242
283
275
·
249
257
·
277
Let
then we are interested in testing HO' the hypothesis of
interchangeability, against the alternative hypothesis that the variates
differ only in their means. The methods that we will use are given in
144
e
Chapter III: namely, the ranking after alignment method and the intrablock ranking method. First, let us regroup and denote by
all the observations that have k missing values k=O, 1, ... ,4. As in
Chapter III we denote by nk the number of blocks that have k missing
variates Nk=nk(p-k) and n=n O+n 1+n 2 . In these data nO=43, n 1=44,
n =57, n =20 and n =36. Denote by R~~) the rank of X~~) in the
2
3
4
1J
1J
combined sample of all the observations with k missing values (note
that R~~) =t.' if X~~) =t. '). Next we give a summary of the formulas to
1J
IJ
be used in calculating the ranking after alignment test statistics and
its p-value.
R.(~)
8
(k)
k,R ..
(.k.).1S no t mlssmg,
..
th
. 8
0 erWlse
= N 1+J 1 1· f R IJ
k
IJ
8
k,R~k)
(k)= 0 ,
k,R ..
IJ
=_lL:p
8
p-k J= 1 k,R~~)
IJ
1.
8 * (k) =8
(k) -8
if
k,R ij
k,R ij k,Ri~)
Ri~)
is not missing, otherwise 8* (k) =0,
k,R ij
o
TN .
,j
2
erN
-2
(k)
*
= ~ -OC.. 8
(k)
4 - IJ
k,R ..
IJ
(k)
n
C~~)={
IJ
where
1 otherwise
*
k
(RN
)
= 2i=1~=1 (8
(k))
k
k,R ..
2
1J
145
W (2)
N
~1·=1 (TN, 1·)2
2
=- __
- - - - "'- X
.....=....;.;2~-~-
-2 (p-k)
(p -n-k) 2
(k)
-- 2 aN (R )
2f-0
[p (p-1 )]
k N
.1-_
p- 1
t::
For the sample with n=n O+n 1+n 2 we get the following data
summary of the above formulas:
Table 5.5
Data Summary for the Ranking After Al ignment Method
with Incomplete Data
TN ,1
8.402
TN ,2
TN ,3
TN ,4
3.742 -5.196 -6.948
2
aN
0
5.198
2
aN
1
2.97
2
aN
2
2.797
W(2)
N
57.061
e
while for the complete data set with nO observation we have the
following data summary:
Table 5.6
Data Summary of the Ranking After Alignment Method
with the Complete Part of the Data, Le. for the Complete nO
Observations
W(2)
T N ,2
3.393
O
2.168
NO
-1.286
-4.275
1.733
20.856
For the intra-block ranking method we need the following formulas:
146
B
R.(~)
1J 1. f R ij
(k).1S no t m1ssmg,
"
= p-k+l
(.~)
k,R 1J
0
therw1se(k)
. B
= 0,
k,R ..
1J
(k).1S no t mlssmg,
.,
B* (k) = B
(k) - B
(k) 1.f R ij
k,R. .
k,R..
k,R.
IJ
1J
1.
-2
*
(k)
TN . = ~-OC .. B
(k)
,J 4 - 1j k,R ..
1J
2
CT k
~
= L. =l (B
J
W(1)
N
*
(k))
k,R ..
2
where
C~~)IJ ={
0
therWlse
. B* (k) = 0 ,
k,R . .
1J
o
1 otherwise
for any i=l, ... ,n
k
IJ
~1'=1 (TN, ,.)2
- - - ....
=
---=-:::'2~-t...l--
oL-_
~- 2 (p-k) (p -p-k) (n 02)
4= 0
[p (p-1 )] 2
2
X
p-l
k k
For the sample with n=n +n +n we get the following data
O 1 2
summary of the intra-block rankings formulas:
Table 5.7
Data Summary for the Intra-Block Rankings Method
with the Incomplete Data
9.925
4.042 -6.208 -7.758
147
0.2
0.5
0.055 25.584
while for the complete data set with nO observation we have the
following data summary for the method of intra-block rankings:
Table 5.8
Data Summary of the Intra-Block Ranking Method With the
Complete Part of the Data, Le. for the Complete nO Observations
3.8
3.0
-1.0
-5.8
0.2
20. 26
1
All the above cases are distributed as chi-squared with 3 degrees of
freedom. The p-values for each case are given in the following table:
Table 5.9
p-values for the Intra-Block Ranking and the Ranking After
Alignment Methods for Both Complete (n CLObsrvations)
and Incomplete (n Observations)-Uata
Method *
Data
complete
1
1
incomplete
~
complete
incomplete
*
2
2
p-values
<0.0001
<0.0001
<0.0001
<0.0001
Method 1 is the intra-block ranking method. Method 2 is the
ranking after alignment method.
As we see in Table 5.9, all p-values are zero and this indicates that at
a=.O 1 the null hypothesis of interchangeability is rejected against the
alternative that the variates differ in locations.
148
5.5 Recommerdations For Future Research
In Chapter III, for the case of incomplete data, we only considered
testing for interchangeability against a location alternative. However,
one could extend the incomplete data case to test for interchangeability
against non-location al ternatives by using the same techniques as in
Chapter IV. In this case, for any particular two variates, the number
of observations with nonmissing values for these variates might be
different from the number of nonmissing values for another combination
of variates. Also it will be easy to extend the techniques of Chapters
II and III to test for restricted interchangeability with incomplete data
against a location alternative. In addition, using the same statistics
and the large sample techniques of Chapter III, one can form a rank
order test to test interchangeability with censored data.
Restricted interchangeability can be extended further to test against
non-location alternative. In this case, the exact distribution in Chapter
II and the statistic that is used in Chapter III can be combined to come
up with the needed test. This case can be extended further to include
tests with missing data. By studying the exact distribution and its
large sample approximation, one can determine for what values of n
(the sample size), the large sample approximation can be used.
All the above pOSSible extensions and all the techniques that have been
used in Chapters II, III and IV are done for the one sample case.
However one can extend all these to the multisample case as specified
in Puri and Sen (1 97 1) .
149
References
1.
Chernoff, H. and Savage, 1. R. (1958). "Asymptotic normality
and efficiency of certain nonparametric test statistics". The
Annals of Mthematical Statistics 29, 972-996.
2.
Ekbohm, G. (1976). "On comparing the means in the paired
case with incomplete data on both responses". Biometrika 63,
299-304.
3.
Friedman, M. (1937). "The use of ranks to avoid the assumption
of normality implicit in the analysis of variance". Journal of the
American Statistical Association 32,675-699.
4.
Gerig, T. M. (1969). "A multivariate extension of Friedman's
2
X -test". Journal of the American Statistical Association 64,
r
1595-1608.
5.
Gerig, T. M. (1978). "A multivariate extension of Friedman's
2
X -test with random covariates". Journal of the American
Statistical Association 70, 443-447.
6.
Govindarajulu, Z., Lecam, L. and Raghavachari, M. (1966).
"Generalizations of theorems of Chernoff and Savage on the
asymptotic normality of test statistics". Proc. Fifth Berkely
Symp. on Math. Stat. and Prob. 1, 609-638. Univ. of California
Press.
e
7.
Hodges, J. L., Jr. and Lehmann, E. L. (1961). "Comparison of the
normal scores and Wilcoxon tests". Proc Fourth Berkeley Symp.
Math. Statist. Prob., 1, 307-317. Univ. of California Press.
8.
Hodges, J. L., Jr. and Lehmann, E. L. (1962). "Rank methods
for combination of independent experiments in the analysis of
variance". The Annals of Mathematical Statistics 33, 482-497.
9.
Lin, P. E., and Stivers, L. E. (1974). "On differences of means
with incomplete data". Biometrika 61, 325-334.
10. Jureckova, J. (1969). "Asymptotic linearity of a rank statistic in
regression parameter". The Annals of Mathematical Statistics 40,
1889-1900.
i 1. Jureckova, J. (1971). "Nonparametric estimate of regression
coefficients". The Annals of Mathematical Statistics 42, 13281338.
12. Loeve, M.,(1955). "Probability theory". D. Van Nostrand Co.,Inc.
13. Nandi, H. K. and Sen, P. K. (1963). "On the properties of Ustatistics when the observations are not independent. Part II.
Unbiased estimation of the parameters of a finite population".
Calcutta Statistical Association Bulletin 12, 125-143.
14. aIkin, I. and Press, S. J. (1969). "Testing and estimation for a
circular stationary model". The Annals of Mathematical
Statistics 40, 1358.
151
15. Olkin, 1. (1973). "Testing and estimation for structures which
are circularly symmetric in blocks". Mul ti variate Statistica1
Inference. Edited by Kabe and Gupta. North-Holland.
16. Puri, M. L. and Sen, P. K. (1966). "On a class of multivariate
multisample rank order tests". Sankhya, Series A 28,
353-376.
17. Puri, M. L. and Sen, P. K. (1971). "Nonparametric methods in
multivariate analysis". Wiley, New York.
18. Quade, O. (1979). "Using weighted rankings in the analysis of
complete blocks with additive block effects". Journal of the
American Statistical Association 74, 680-683.
19. Salama, 1. and Quade, O. (1981). "Using weighted rankings to
test against ordered alternatives in complete blocks".
Communications in Statistics Al 0 (4), 385-399.
20 Sen, P. K. (1966). "On a class of bivariate two-sample
nonparametric tests". Proc. Fifth Berkeley Symp. Math. Stat.
Prob., Univ. of California Press, 1, 638-656.
21. Sen, P. K. (1967a). "Nonparametric tests for multivariate
interchangeability. Part 1. Problem of location and scale in
bivariate distributions". Sankhya, Series A, 29, 351-372.
22. Sen, P. K. (1967b). "On some nonparametric generalizations of
Wilks' tests for H , HyC and HMyC ' 1.". The Annals of the
M
152
Institute of Statistical Mathematics 19, 451-471.
23. Sen, P. K. (1968a). "On a class of aligned rank order tests in
two way layouts". The Annals of Mathematical Statistics 39,
1115-1124.
24. Sen, P. K. (1968b). "Nonparametric tests for multivariate
interchangeability. Part II. The problem of MANOVA in two way
layouts". Sankhya, Series A, 31, 145-156.
25. Sen, P. K. (1968c). "Asymptotically efficient tests by the
method of n rankings". Journal of the Royal Statistical Society,
Series B 30.,312-317.
26. Sen, P. K. (1970). "On the distribution of one sample rank order
statistics". Nonparametric Tech. Statist. Inf. Cambridge Univ.
Press., 53-72.
27. Silva, C. and Quade, D. (1980). "Evaluation of weighted
rankings using expected significance level". Communications in
Statistics A9(10), 1087-1096.
28. Silva, C. and Quade,. D. (1983). "Estimating the asymptotic
relative efficiency of weighted rankings". Communications in
Statistics 12(5), 511-521.
29. Votaw, D. F., Jr. (1948). "Testing compound symmetry in a
normal multivariate distribution". The Annals of Mathematical
Statistics 19, 447-473.
153
30. Wald, A. (1943a). "An extension of Wilks' method for setting
tolerance limits". The Annals of Mathematical Statistics 14, 4555.
31. Wald, A. (1943b). "Tests of statistical hypotheses concerning
several parameters when the number of observations is large".
Transactions of the American Mathematical Society 54, 426-482.
32. Wei L. J., (1983). "Tests of interchangeability with incomplete
paired observations". Journal of the American Statistical
Association 78,725-729.
33. Wilks S. S. (1946). "Sample criteria for testing equality of
means, equality of variances, and equality of covariances in a
normal multivariate distribution". The Annals of Mathematical
Statistics 17, 257-281.
154
e
Appendix Contents
A1
Ranks and Scores of Section 5.2 Data
156
A2
Ranks and Scores of Section 5.3 Data
160
A3
Ranks for Section 5.4 Data
164
A4
Program to Generate the Missing Values for the Data
Given in Section 5.4
Note: In Appendix A3, RANK) stands for the intra-block ranks.
RANK_A stands for the ranks after alignment.
168
APFcNcrx
_i:: S
F.il
~.K.
eM
"t
..J
71 .5
5 I .5
143.5
4
1~4.5
'"
1 7 I .0
138.5
1 15.0
7 , t.:
152.0
143.5
1 ~ 4.5
158.5
::4.0
98.5
';b.O
57.0
~
6
7
C
:;
1C
I1
1:
13
14
1S
16
17
~
....
5~.5
~4
158.5
95.0
195.5
1 1 1 .5
17E:.0
76.0
67.e
-: l::
...
..,
1~8.5
:'6
46.0
le4.5
1t~. 5
134.5
71 .5
13 1.5
10
19
':C
~ 1
,- .r;
- ..,
'l
...
~7
:'0
:'9
3C
:3 1
...
~
~,
4~.C
,'"
1~6.5
::'4
-..,
.'"
~'t.o
~..,
12.:>
1,<;
:.(,
:. 1
11 C.0
47.0
119.5
101 .0
78.5
f:8.0
52.5
1,6.0
131 .5
106.5
136.5
<;4.0
164.5
61 .5
106.5
179.0
"'~
~~
1~8.0
':3
37.0
72.5
31.0
~E
:7
:8
:<;
loG
.. 1
.. ~
103
1,4
.. =
..,.
1,6
.. 7
.:.e
54
':.5
1
AI:
F<
M.K S
A~C
S::::;R~S
iiAt-.r<TGI
S:ORECHI
SCCriETGI
20.0
103.0
191+.0
147.5
193.0
170.0
141.5
4;: .5
1:3 .5
146.5
77.0
98.0
24.0
b 3.5
1 1::.0
200.0
179.0
19.5
5.0
132.0
73.0
134.0
18.0
14E: .5
10 7.5
53.0
129.0
196.0
29.0
12.0
136.5
19.0
1 16.5
I 10.5
91.5
140.0
9.5
128.5
99.5
21.5
116.5
176.0
188.0
21.5
175.0
161.0
69.5
52.0
95.5
58.5
10.0
29.0
9.5
-0.3793
-0.6390
0.5443
0.3371
1.1171
0.4740
0.1675
-0.3371
0.6706
0.5443
-1.29CC
0.0573
1.7025
0.6705
2.J475
0.9b40
0.5159
-C.7bb7
-1.4300
0.5&7&
-0.0556
-0.0:64
-1.1659
-0.1920
0.2115
2.1760
1.1039
-1.2874
-1.9663
0.3 b6 0
-0.3595
0.4683
-1.3490
0.567&
0.1148
-0.6400
0.3464
2.5706
-1.0676
-1.5623
C.5C41
-1.3189
0.,311
0.111 ..
-0.1238
0.5553
-1.6623
0.3915
-0.0247
-1.2486
0.2311
1.1123
1.4474
-1.2311
1.0897
0.9050
-0.3778
-0.6312
-0.0382
-C.5587
3~.5
47.0
0.~877
0.8581
-1.1655
a.oeco
-0.0318
-0.5805
-0.6476
0.8581
-0.0803
1.7873
0.12:!6
1.2448
-0.3~C3
-0.4398
0.3915
-0.7501
1.3338
1.4495
0.4195
-0.3793
0.4331
-0.8172
0.3642
-1.5417
-1.1839
0.1469
-0.7109
0.2705
1.2352
-0.2879
-0.3984
-0.1114
0.3073
0.4331
0.0618
0.5041
-0.0573
0.9742
-0.4897
0.0618
1.1839
0.6785
-0.8860
-0.3371
-1.02"9
-1.6~21
-1.04S5
-1.6623
-0.9146
-0.7338
156
Far:.
S~CT!O:\
i1A~KCi-12
'
o' -••
~
51 .5
151+.5
121 .5
184.5
198.0
127.::
107.5
158.0
98.0
171 • =
170.0
23.0
98.5
89.0
84.0
67.0
85.3
130.0
199.0
102.5
116.0
50.5
78.5
156.0
36.0
192.0
196.0
1 17.5
42.0
164.5
26.5
177.0
15.5
29.0
72.5
64.C
48.5
188.5
28.0
32.5
134.5
176.0
139.5
102.5
79.0
89.0
136.5
51.5
143.5
182.5
113.5
13.5
72.5
21.5
s •.:.
RANKTG2
29.0
12b.5
199.0
82.0
195.0
lo7.J
79.5
77.5
9.0
29.0
1~9.C
145.0
72.0
14 I .0
77.5
184.5
104.0
58.5
7.0
63.0
125.0
72.0
76.5
19 C.0
83.5
41 .5
160.0
152.0
67.0
35.5
16 I .0
25.0
16~.O
117.5
99.5
123.5
47.0
48.0
112.0
53.0
88.5
139.5
150.0
81 • a
164.5
17f.5
54.5
86.0
98.0
172.0
55.5
74.5
63.5
63.5
78.0
:~TA
SCCfi~CM2
~:OF~T:;:
-0.5019
-0.6390
0.7098
0.2970
1.5264
1.9663
0.3268
0.111+8
0.7665
-0.0432
1.0145
1.0936
-1.1914
0.0000
-0.1212
-0.2178
-0.4398
-0.1662
0.3595
2.0399
0.0123
0.2243
-0.6784
-0.2879
0.8130
-0.9256
1.6055
2.5706
0.1989
-0.8172
0.9742
-1.1238
1.2729
-1.4300
-1.0676
-0.3371
-0.4541
-0.6865
1.4652
-1.0897
-0.9742
0.4195
1.1123
0.5479
0.0123
-0.2507
-0.1212
0.5041
- 0.6390
0.54,.3
1.2760
0.1920
-1.4868
-0.:3371
-1.2486
-1. C676e
0.331;;
2.C599
-O.~ll~
2.3=:07
0.9256
-0.2751
-0.27e5
-1.7025
-1.0676
0.3464
0.6312
-0.3436
0.57C2
-0.~70S
1.3338
0.0305
-0.5332
-1.8186
-0 .... 949
0.294"
-0.:!,.38
-0.3138
1.5217
-0.1920
-O.8~5_
0.800
C.744C
-0.4396
-0.9351
0.9050
-1.1594
0.Sb45
0.1989
-0.0247
0.3237
-0.7109
-0.6946
0.1300
-0.6400
-0.1276
0.4879
0.6400
-0.2245
0.8792
1.2587
-0.5928
-0.1597
-0.0064
1.0249
-0.6026
-0.:310_
-0.461
-0.4612
-0.2944
APPEt'.Dl ... AI:
~~S
-
~.,
:"6
:'J
tC
t 1
MAI\KCnl
~9.5
"".0
32.5
~1t.C
-';1:.
-' _
•
c:.
..,J
l~b.5
05
toe
t7
clt.O
ICO.5
62.5
17.e
2C:'.O
117.5
t:.e
.J~.J
t:~
63
E.4
.... ,.
t:
R/l~KTGl
1 & C.O
C;; 0.5
70.0
1.0
1 C5.::>
153.0
1 13.5
33.0
41+.0
44.0
1::3.0
136.5
1 c 8.0
136.5
67.5
1 ~ 7.0
66.0
19 1).0
b8.5
101+.0
RAt--i<S
SCGR~CMl
-1.0:375
-0.7609
-0.9S41
-1.18:39
-O.9H6
0.3515
-0.lb56
0.0255
-0.5019
-1.3~O3
2.5bl0
0.1589
-0.99"1
-0.4,61
0.3371
At'.:
SC(;R~S
5C;:'H~TGI
1.3639
-0.1020
-0.3953
-2.5bl0
0.0828
0.76C9
O.19~0
-0.9640
-0.7632
-0.7632
0.26e7
o• 44 b 6
C.9447
0.5C41
-0.1+053
0.3710
i:~
66.0
7C
1~4.5
71
~5.e
7~
73
74
75
76
77
78
75
111 • 5
67.e
b9.0
113. :»
140.0
1;,.3.0
61.5
57.0
39.0
t
158.0
3.5
171+.0
0.0~27
0.5,.43
-0.6862
lC;~.C
-0.5019
-0.9742
-1.5417
-0.592&
0.7274
1.7873
1.9535
0.0123
-0.2368
0.6706
-0.6249
-0.8581
-1.9663
C..8792
-0.9640
-0.3595
-0.6626
-0.9640
-0.1238
51
89.0
SO.O
164.5
3:3 .0
73.0
50.0
33.0
91.5
125.0
76.5
86.5
81.:
-0.1863
';(,:
lC9.G
b6.S
143.5
f:2.5
32.5
12.5
54.5
151 .0
1';5.5
O.&OC.l
0.6004
-1.5217
-0.8701
-1.3969
1.0676
-0.1550
eO
b.
E3
c4
05
i6
07
e&
- c.
c"
c;.'
_L
C?
,,-
';4
cc:.
,,-
';E,
r,7
S8
S5
1 CO
1 C1
1(2
lC3
1 ;:. 1+
lC5
ICE.
1C
leo
lee;
110
102.5
82.5
152.0
54.0
38.5
5.C
36.0
134.5
188.5
9.0
134.5
103.0
Ib2.5
2.5
92.5
174.0
1~9.5
lC.O
124.5
58.:
1 b 0.0
11+3.0
13.0
1 6. CI
108.5
156.5
197.0
81.5
141.5
125.0
:3.0
69.0
58.5
1 e 7.0
113.0
41+ .0
195.5
98.0
71t.5
192.0
-1.1412
0.1662
-0.1+398
-0.1212
0.14e7
0.1+91+9
0.2687
-0.4897
-0.5805
0.7665
-2.1022
-0.9256
0.4195
1.4652
-1.7025
0.,.195
0.0573
0.9341
-2.21t71
-0.1114
1.0676
0.5479
-1.6375
0.2875
-0.4261
1.966:3
-0.1:76
0.0309
-0.5587
FOR
RANKCM~
19.0
171+.G
105. C
48.5
44.0
116.0
96.0
19.0
81.0
42.0
200.0
164.5
10.5
59.0
118.0
21.5
79.0
117.5
82.0
121 .0
S5.0
131+.5
11+6.0
82.5
98.0
8.0
76.0
158.0
168.5
86.5
21.5
67.0
-0.3138
-0.1863
116.0
156.0
177.5
191 .0
121 .0
60.0
-0.2495
163.0
0.0865
0.8218
1.8873
-0.21+95
C.5159
0.2941+
-0.6400
57.0
75.0
0.3"38
-0.4127
-0.5332
1.6375
0.11+25
-Cl.7832
1.7873
-0.0064
-0.3103
1.6055
157
St;:TIG~.
6.0
121.0
193.0
174.0
15.5
186.5
107.5
175.0
2.5
148.5
161.5
166.5
15.5
182.5
c:;
...
_e ..
RANKTG':
1 H .• 5
13::.0
97.0
3.0
60.0
69.5
93.0
24.0
104.0
31 .0
156.5
177.0
120.5
186.0
31.0
12.5
56.5
93.5
163.0
160.0
15C .0
95.0
164.0
2~.0
63.0
49.0
172.0
84.0
10,..0
39.0
:ATA
SCQGECH:2
-1.:3021
1.1914
0.01t3~
-0.7098
- 0.7609
0.2245
-0.0318
-1.30:1
-0.2559
-U.oI72
2.1760
0.c.79~
-1.6284
-0.5259
0.~507
-1.2311
-0.2507
0.1989
-0 • .2115
0.2432
-0.0803
0.4155
0.6468
-0.2368
-0.01+32
-1.7437
-0.3203
0.7665
0.95,.4
S:GF=T~4
1.2:1:7
o .44Cl
-0.0::::£
-2.17eO
-0.5113
-0.:3778
-0.C7CC
-1.1655
C.03C5
-1.0~4g
0.741S
1.1355
0.2368
1.5907
-1.0054
-1.5264
-0.5620
-0.0989
0.9440
0.8001
0.640C
-0.0803
0.964C
-1.1554
-0.4945
-0.6785
1.024';
-0.217E
-0.1663
0.030S
-0.87Cl
7.0
-1.2311
-1.80~1
104.0
54.5
109.0
115.5
156.5
120.5
25.0
73.0
156.5
110.5
201.0
138.0
132.0
154.0
84.0
73.0
40.0
175.0
58.5
47.0
144.0
101 .0
12:6.0
186.0
-0.4:398
0.2245
-0.5928
0.8130
1.11+74
I.Es741
0.2432
-0.5372
0.8522
-0.5805
-0.3036
-1.8873
0.2432
1.6521
1.0676
-1.4300
1.3965
0.111+8
1.2177
-2.2471
0.6174
0.8259
1.0160
-1.4140
1.2760
O.03CS
0.1:34C
0.1737
0.8581
0.2368
-1.1Se;,,.
-0.355:
0.7415
0.1533
2.331S
0.4671
0.3860
0.701S
-0.2118
-0.3595
-0.83C8
1.2177
-0.5587
-0.7338
0.551::
0.0318
0.3574
1.3BC~
AP PEt'~LJ i X AI:
_~s
11
1:;
~ 13
114
i 1:
11 C
117
11(j
11';
1:';'0
1~ 1
I
i
l~~
1" 3
1:4
I
~
"-
.. J
l:b
l L7
1 ~e
1:: 9
13C
131
1-'''
-~
RA"'KCH1
1e 1 .5
3.5
13e.o
1i:l.0
151 • C
32.5
~:;,.~
57.0
177.5
1 ~ .0
156.0
13.~
c:
..... 0- ...
~
18t.5
46.0
52.5
79.0
1e7.5
171.5
36.0
~6.5
4.0
•.~--.:
_
J
E.~.5
134
1;5
136
137
1;8
19. a
57.0
17.0
9.0
143.5
:4.0
139
i40
: 41
14~
143
:44
,45
,4Eo
." 7
,48
, '+ 9
.~C
iSl
1~~
_ ...
1'" ~
154
',-1
_:J
l':b
1': 7
1"-JO'
1':9
I~C
IU
Ie:.
lE.3
164
16~
leO.O
111 • 5
15.5
85.0
1 .0
186.5
E.8.C
89.0
160.0
134.5
178.0
179.5
156.0
~6.5
164.5
35.5
151.0
b9.0
1C9.0
42.0
136.5
76.5
182.5
40.5
13:3.5
143.0
F1A~t<T~l
~
Ai\K S AN:' SCORES
SCCRECHI
SCCR~TGI
b 4.0
0.8~59
-0.2176
19.5
61.0
188.0
77.5
90.5
1 7.5
193.0
93.5
1+5.5
18:3 .5
104.0
32.0
183.5
95.5
139.5
93.0
52.0
120.5
35.5
35.5
67.0
120.5
21.5
107.5
62.0
40.0
63.0
15.5
129.0
3.0
72.0
-2.1022
0.4122
1.3969
0.7274
-0.9742
-1.C;75
-0.5805
-1.~874
144.0
ISE..5
1e4.5
131.0
6:3.0
16E.0
163.0
182.0
154.0
63.0
26.5
178.0
'+.0
171+.0
77.5
132.0
3.0
17.5
121.0
170.5
106.0
150.0
172.0
1.147it
-1.5471
C.6130
-1.4068
-0.8792
1.6137
-0.7'274
-0.6476
-0.2507
0.1148
1.0145
-0.9256
-1.1238
-2.0599
-0.5019
-1.3021
-0.5553
-1.3639
-1.6882
0.5443
-1.1839
-C.OI85
0.1238
-1.4140
-0.2052
-2.5810
1.3965
-0.3984
-0.1550
0.8001
0.4195
1.3021
1.3480
0.1338
-1.1053
1.5264
-0.9146
0.7274
-0.1212
0.0927
-0.8172
0.5041
-0.2837
1.~4C;5
-0.8218
0.4612
0.6004
-0.4969
1.6ee~
-O.:70~
FOR
iiANI(TG2
152.0
6. C
96.e
190.0
179.5
14it.C
15.C
65.0
'to.~
-1.34BO
70.0
138.5
104.5
5.0
113.5
-1.00it~
1.486b
-0.C382
0.4879
-0.0700
-0.6312
0.2368
-0.9351
-0.9351
-0.4390
0.2368
-1.2311
0.1148
7.e
42.CJ
109.0
61 .5
lC9.0
119.5
141 .0
148.5
67.0
10.5
14.0
74.0
28.0
56.0
-0.8308
-0.4949
-1.4300
0.3464
-2.1760
-0.3438
32.5
11• 0
164.5
7.0
130.C
148.5
44.0
0.5515
91.0
0.7419
1.3338
0.4261
46.0
197.0
Sit.S
76.0
168.5
168. :3
172.5
166.5
154.5
38.5
168.5
111 .5
107.5
148.0
98.0
30.0
153.5
161 .0
168.5
68.0
100.5
160.0
-O.48~5
-0.it949
0.9068
0.8522
1.4315
o .~7 780
-0.4949
-1.1053
1.3021
-2.0475
1.1914
-0.2705
0.3860
-2.1760
-1.3480
0.2903
1.1053
0.0956
0.7109
1.1412
158
5.2 ;:';TA
RANKCH2
-0.1040
1.6521
-0.0989
-0.7357
1.4e68
0.0700
";~CTICJ~
18:.0
170.5
67.5
37.5
202.0
135.0
9.5
13d.5
16<:;.0
104.0
156.5
5E.5
182.5
130.5
14.0
79.5
1:.5
1 1 .0
97.0
191 .0
9.5
161 .0
8E.0
~Q
~
~
•• v
25.0
7.0
56.5
7.0
100.0
110.5
125.0
153.0
44.0
117.5
180.0
172.0
181 .0
18~.5
136.5
52.0
118.5
3.0
155.0
40.0
104.0
35.5
26.5
158.5
150.0
147.5
86.0
146.0
~
SC:R~CH2
SCOF;E.TC2
0.6706
-1.8741
-0.0318
1 • .1051
1.3480
-1.431-O.44Cl
1.5471
-0.8218
-0.3710
0.4740
1.3338
-1.9535
0.192:C
-1.8051
-O.617Z
1.7437
-0.4697
0.0927
0.2705
0.5702
0.6174
-0.4398
-1.6284
-1.~835
-0.3464
-I.C708
-0.5702
-0.9742
-).5907
0.e792
-1.8186
0.3595
0.6174
-0.7609
-0.1300
-0.7501
1.8873
-0.5928
-0.3203
O.~5_
1.105~
-C.4053
-0.876E:
2.581C
0.4262
-1.562~
0.533~
1.07Ce
0.03CS
o.a2H:
-0.56~a
1.2760
0.533~
-1.4679
-0.2751
-1.430::
-1.6055
-0.05SE
1.56:~
-1.6623
0.9C:C
-0.15.
-1.04
-1.159"
-1.8166
-a.55t7
-1.818£
0.01S1
0.1114
0.294~
0.6862
-0.7ees
0.19813
0.954it
1.20C;~
0.9544
1.1534
1.01CiO
0.7098
1.024«;
-0.8581
1.0596
0.)662
0.1148
0.6785
-0.0432
-1.0it60
0.7695
0.9050
1.0596
-0.3984
0.0255
0.8860
1.3965
1.7735
0.4466
-0.6312
0.2573
-2.J641
0.7954
-0.S3Ge
0.030C;
-0.9351
-1.1053
0.8561
0.7_
0.6
-0.1597
0.6468
.
.APP::~C:X
1.6882
0.3595
2.2356
2.0
-2.3207
1.8741
-0.1550
1.0485
-2.32C7
143.0
1 16.5
1:' o. C
37.5
123.5
0.0828
1.t:131
1 .5217
-1.1053
-0.2'115
-0.1550
0.0123
0.2970
0.4612
0.6785
-0.4897
-0.9641
0.'3341
IE(
188.0
H.9
17e
171
172
173
174
175
1H.
177
17E.
179
loG
lEI
leL
Ib3
164
1t-5
lEo
~6.5
e2.0
09.0
lC2.5
121 .5
13~.5
148.0
t.l.5
34.0
162.5
92.5
148.5
1'3 ... 0
18.0
167
~1.!)
Its
136.5
60.0
76.5
57.0
48.5
f.5.0
18';
150
I~1
Ie;;.
j53
~S4
:0.5
155
123.0
156
1:'7
l~e
le;S
8.a
lE:E:.O
174.0
127.5
S~:M~S
191.0
09.0
168.0
RAt\KTGI
130.0
194.5
2.0
lC5.0
Ib6.5
190.0
Ar--:
SCCMETGI
RAf'.lI..Crol
leo
I=A:".KS
SCOFECHI
~~S
It7
AI:
8~.5
1&I.C
45.5
Cj:3.0
15C.0
1 L.::
b9. C
1::1.0
130.0
169.0
188.0
150.0
10tl.5
173.0
't 7.0
113.5
55.5
29.0
6.0
15.5
194.0
25.0
73.0
16:;:;.0
127.0
-0.CH~4
0.6174
1.7025
-1.3490
-1.2486
0.5041
-0.2687
-0.2837
-0.5805
-0.6865
-0.4401
-0.6784
0.3170
-1.7576
0.9068
1.0676
0.3268
0.6004
0.2513
0.6400
-0.&7£6
0.3237
-0.1863
1.2352
-0.7357
-0.0700
0.7109
-1.:'264
-0.1550
0.29C3
0.4122
0.9641
1.4474
0.6400
0.0865
1.1659
-0.7338
0.19~0
-0.6026
-1.01+85
-1.&741
-1.4300
2.1641
-1.1594
-0.3595
O.l:I346
0.3203
159
FOR
S~:TICr--
s.::
RANKCM2
RANKT:1~
193.0
1817.5
150.0
134.0
1 .0
143.0
42.5
178.0
121 .0
134.0
115.5
195.5
80.0
110.5
179.0
24.0
50.0
102.0
112.0
188.0
182.5
11't.0
67.0
167.0
'tI.5
166.0
35.0
35.5
5.0
33.0
192.0
21 .5
50.0
134.0
15f.5
113.5
194.5
1• 0
103.0
142.0
95.0
65.5
51 .5
71.5
143.5
172.5
153.5
89.0
58.0
38.5
126.5
82.0
180.0
201 • C
117.5
19.5
14~.5
71 .5
103.0
46.0
42.0
92.5
48.5
144.5
19.5
89.0
191 .0
168.5
:'ATA
SCCf;ECM2
2.0473
0.11+87
2.2356
-2.5706
0.0573
0.5853
-0.oa03
-0.1662
-0.6390
-0.3793
0.5443
1.1534
0.7695
-0.1212
-0.5405
-0.8792
0.3642
-0.2115
1.2092
2.3319
0.1989
-1.3043
0.6234
-0.3793
0.0573
-0.7501
-0.7954
-0.0764
-0.7098
0.6234
-1. 3 O~ 3
-0.1550
1.5623
0.9544
I,
SCCF~T:::2
1.77370.c4CC
0.4£83
-2.S70E
0.£004
-0.78b7
1.1594
0.2«;C3
0.460;
0.17~7
1.787:'
-C.237E
0.1533
1.3324
-1.16C:S
-O.68E:2
0.044:
0.172£
1.4474
1.276C
0.15:C
-0.439E
1.02£7
-0.82:5
1.0054
-0.8701
-0.9146
-1.953::
-0.984C
1.55~:
-1.:4B£
-0.6862
0.4127
0.741S
~PPE~.:)
Ci:: S
646.0
501.5
4£6.0
131.0
~.e6Su
489.5
275.::165.0
55.0
532.5
489.5
727.:176.0
746.:325.C
752.5
582.0
130. ::
489.:;
501.5
307.S
40.5
293.C
307.S
106.0
307.S
354.5
3Ec.O
256.5
646.0
66C.(I
1 .3 1 .0
439.0
325.0
3:5.0
7"T,r::.
0<11_--..:
659.5
h7.D
165.0
258.:1 15.0
570.C
57.0
73.5
559.5
138.5
428.5
146.5
543.5
768.C
107.0
244.5
501.5
209.:31.5
146.5
543.5
513.0
28.5
761.5
C.2a22
-0.4C17
-0.6204
-1.485b
0.4256
0.:622
1.3300
-0.7731
1.4905
-O.23E.5
1.5502
0.6025
405.5
87.0
-1.~:!39
3~5.0
-C.2~6S
c::~~.c
C.60~5
c.-
C 1 ~ .5
1&6.0
765.5
c
"'"1:~
.I"" . . . .."
6~S.O
14
1~
16
17
405.5
57C.O
2 7~. 5
18
~~5.5
3E.
604.C
5S6.C
354.5
4t6.C
415.C
307.5
646.0
718.=
e71.C
146.5
343.5
405.5
4tb.C
6:;3.5
770.5
196.S
97.0
666.5
638.0
37
2'C::
'.-_-1c::
3~
39
4(;
41
14.C
2C9.5
7eS.5
71t:.5
4~
25e.~
43
44
45
,.6
47
48
49
19£.5
354.5
7E.9.C
35.5
43S.C
209.5
377.5
405.5
478.0
619.0
c04.C
258.5
6C4.0
~
..,
44
::3
24
... ..,
~,~
2c
':t..7
2i.
~5
3C
..
C.3~15
5E.~.C
1
31
~,
oJ4
33
34
35
~"
oJ"
51
-..,-...
~~
C"7
54
~5
G~T,:l
0.7501
0.3899
1.5713
13
~
5.3
619.5
7
~C
~~:TICN
R~:\jt<4
0
15
CF
SCC;;'~3
0.8252
1.C266
C.3399
10
11
1..:.
SeeR';::)
RANK~
7':4.5
5 C1 .5
63 C • ~
679.C-
c
AI'.:
SCORt:.2
"1:
oJ
Rt.~,KS
NAI\K~
~~2.0
~
:
SCORE1
~
1+
~
RPNK1
...
oJ
IX A.
C.7~14
-C.7316
".0673
0.4256
0.79C1
0.0156
C.5561
-C.4017
-C.S7b3
0.6873
C.6S:;9
-0.1445
0.2C64
C.0579
-C.2953
0.b6:'v
1.2647
0.9851
-C.9044
-C.1793
0.0156
0.2064
0.8997
1.7734
-0.£693
- 1 .1695
0.9£:'4
0.82';2
-0.5703
-2.1089
-C.c386
1.2043
1.2647
-0.4601
-0.6893
-0.1445
1.7513
-1.7026
C.1208
-O.638b
-0.0720
0.0156
0.2450
1.0266
0.6873
-0.4601
0.6873
~Ol.5
612.5
696.5
737.0
7S0.0
5E:£.C
779.0
0.3~1=
O.~Ce4
-C.9tCO
C.3:15
C.7"1'+
1.1242
1.4e57
2.~C4b
C.6C:5
1.9194
58~.0
C.CO~5
~44.5
-C.~094
730.5
5C1.5
47.5
377.5
730.5
69f:.:
377.5
1.3531
0.3215
-1.56C7
-0.0720
1.3531
7~3.5
293.0
6E.~
38~.~
684.5
63e.0
1 76.0
452.5
3E.S:.O
612.5
774.C
756.0
589.5
236.0
604.0
131. C
155.0
156.5
513.0
543.5
478.0
513.0
293.0
653.5
765.5
2.0
428.5
2:8.5
5~2.0
17.0
~5.0
666.5
509.5
236.0
638.0
1.1~42
-0.07:0
1.3003
-0.3430
-1.37,,9
-0.0360
1.0:'62
C.8292
-C.7731
0.16:;5
-0.0300
0.7214
1.8:69
1.5877
0.6309
-0.5379
0.6873
-C.9E.00
-C.E-6S0
-(,.6893
0.3597
O. 4c 36
0.2450
0.3597
-0.3430
0.8997
2.0£73
-2.b074
0.CEo77
-0.4601
0.3ES9
-2.0291
-1.4858
0.9624
0.6309
-0.5399
C.8292
160
-C.~4~7
O.2C~2
C.3215
-0.:9'53
-1.E:394
-C.3430
-0.2953
-0.7.316
-0.2953
-0.1445
-0.1062
-C.4601
C.e650
C.S-306
-0.9600
C.120d
-0.2365
-0.2385
1.3932
1 • 1421
-1.2339
-0.6204
-0.4601
-1.06,+4
0.5561
-1.1695
-1.3300
0.5201
-0.9427
0.C077
-0.9044
0.4636
1.7369
-1.1096
-0.5094
0.3215
-0.t..386
-1.7586
-0.9044
0.4636
0.3597
-1.8044
1.6515
131.0
377.5
:25.5
St9.S
309.0
696.5
3eS.C
74c.5
47.5
155.0
743.5
52S.5
146.5
31 .5
714.5
551.5
2~5.5
2C5.5
47.5
1f6.0
366.0
674.5
604.0
2~5.5
275.5
671.0
6c6.5
565.5
764.5
6.0
17£.0
275.5
17e.0
343.5
2C5.5
122.5
l~£.G
522.0
17.0
343.5
47&.0
776.0
35.5
61.5
244.5
97.0
73.5
1~8.5
604.0
428.5
55.0
419.0
S:C::'='4
0.01S6
e
-o.seOO
-0.072:
-0.57e3
0.6309
-O.0:!6C
1.1:42
-0.036C
1.SC58
-1.5te7
-0.0::50
1 ... 626
C.t30~
-C.9C44
-1.7::E;S
1.:373
0.4516
-0.5783
-0.6366
-1.5607
-O.731f
-0.1062
1.0C30
0.6073
-0.S7b3
-O.4C17
C.~b51
e
•
0.9£24
O.~417
1.6B94
-2.1+328
-0.7731
-0.4017
-0.7731
-0.175:;
-C.6386
-1.0:35
-0.7316
C • .3e99
-2.0291
-0.1793
O.24C::0
1.5633
-1.7026
-1.4271
-0.509'+
-1.1695
-1.3300
-0.9427
0.6(:73
0.0/:,77
-1.4858
0.0579
e
....
JlPF::::t-.:'iX A -: • FANI<. ;;. A.... :.
-
\.
5b
59
6e
61
t:.2
63
64
65
66
f7
i;E
£5
7C
11
7;.
13
14
75
7£
17
18
75
ov
E1
83
84
65
86
f:.1
t&
89
50
r;1
5~
t;3
t;4
95
96
137
C)8
99
131 .0
691.0
- 0 • 9C; C0
C.L45C
1b6.0
-Ci.731t:
-0.2385
452.5
619.5
5£5.5
u.1635
C.15el
0.5417
532.S
674.5
97.0
532.5
155.Co
107.0
155.0
666.5
452.5
366.0
97.C
389.0
47.:'
225.5
138.5
165.0
165.C
6:L9.C
107.0
513.0
275.5
366.0
11• C
19.0
343.5
653.5
307.5
36£:.0
763.0
532.5
23.C
559.5
405.:;
366.0
61 .5
196.5
377.5
61 .5
258.5
104.5
559.5
723.5
699.5
781 .0
61 .5
489.5
244.5
559.5
704.':.
0.4256
1.C03C
-1.169:>
0.4236
730.5
532.5
97.0
146.5
165.0
115.0
176.0
131.0
122.5
452.5
28.:>
653.5
79.0
2C9.5
1 • .3531
0.4,56
-1.1e:;:
-0.9044
-0.0::04
-1.0E:44
-0.1731
-0.5800
-1.0239
G.1635
-1.be44
0.bS97
-1.2eS4
-C.o38E:
-0.£380
1.17~6
~22.0
G.3e~9
196.C
419.0
4C5.5
-0.731b
0.0579
C.01:'0
-1.C2:;:9
Jl.N I< L
1.09~4
4U:.0
C.~Ctl+
~44.5
19t:.5
439.0
-0.5091+
-C.£0<;3
(.1 ~ 08
630.0
666.5
O.8~92
C.96~4
3~:'.O
-O.23e:5
7~2.5
1.550::
C.0&77
0.52Cl
-1.0::39
b91.C
551.5
733.5
~15.5
1.0924
0.4S16
1.::769
-L.4017
-e-.C1:'~
If5.0
-O.~2C4
-C.4bCd
-C.23c.S
-C.3430
501.5
0.3215
1.09:4
-C.2385
1.0924
0.60:5
-0.9044
~
'~
•
.J
371.~
25t.5
325.0
293.0
112.5
3 f:, 9.0
4~b.5
0<;1.0
97.0
576.C
612.5
501.::
115.0
7e3.0
452.5
452.:'
1.2~39
E91.0
325.0
6C:1.0
-C.C3bO
0.0&17
1.0924
-1.1695
5b~.0
'"
-1.70:'6
C.5t.0~
466.0
079.0
636.0
115.0
761+.5
3tf:.J
O.~Of:4
14f:.5
17f:.O
"tl:
... """
....
596.0
439.C
1<;6.:-
0.1214
C.3215
-1.0bIt4
2.0C:'2
0.1635
0.1635
1.3CC3
0.6:>59
0.120&
-O.6e93
4~6.S
O.0E:-71
47e.C
36£..C
<;7.0
660.0
O.~I+:'O
25E.5
-0.10&2
-1.1695
646.0
0.8650
41€. C
389.0
64f:.0
619.5
629.0
7~3.5
746.5
773.C
787.0
leo
52~.0
leI
1C2
103
1(,4
1C5
lC6
1r 7
1
1CY
lltl
410.(
Itf9.5
660.0
716.0
275.5
325.0
t,19.0
196.5
5E.9.5
619.5
CAT':'
SCOF::4
-C.2365
4Le.5
559.5
c;::
1 ,-
c~.~
"t
RAt'c!'o.4
3 ~5.0
478.C
325.0
1~2.5
TIC ~~
seDli::}
704.5
c= -
~t: ~
RAt-tK3
I=<
RAI'.l< 1
CF
SCO;;E2
S.C.C~E1
LES
S::'C;;~S
0.9306
1.4905
1.E:125
2.1C89
0.3899
0.~4:'C
0.2e~2
0.9306
1.2474
-0.4017
-0.23b5
1.0266
-0.6893
0.6309
0.7501
275.5
4E:E:.C
5b9.5
81.0
4C.5
225.5
772.0
792.0
419.0
559.5
704.5
258.5
751.C
209.5
301.5
501.5
244.5
543.5
489.5
-C.7131
1.C~£6
C.E.2c;2
-1.0644
1.Ge54
-0.lCb2
-0.4017
C.2C61+
0.6309
-1.2339
-1.6374
-0.5703
-0.4601
0.2450
-0.0360
0.8650
0.7501
0.7901
1.7965
~.28,,3
0.0519
C.~~OI
1.17~6
-C.4601
1.5ge8
-0.6386
-0.2953
0.:3215
-0.5094
0.4636
0.2622
161
122.~
2:36.C
258.~
307.5
604.0
452.~
-0.b650
-1.1096
-O.Sb50
G.9b24
0.1635
-0.1002
-1.1695
-0.C360
-1.5E:07
-C.:'7f:,3
-C.C;4~1
L09.~
-0.&204
-0.&204
C.1S01
-1.1096
C.3597
389.0
-0.~360
15~.0
-C..8c:-0
1.0562
-1.5e07
-0.~953
be4.5
47.5
543.5
419.0
20<;.5
47.5
31.5
6£4.5
57£.0
275.5
6&4.5
748.5
325.0
176.C
551.5
684.5
6&.0
131.0
543.5
225.5
66.0
612.5
684.5
343.5
130.5
78e.0
176.0
47.5
452.5
325.0
646.0
236.0
1:31.0
225.5
629.0
87.0
0.6873
0.1635
71~.5
-0.98CO
-0.:'783
0.1':101
-1.2339
1.223S
53::.5
O.4~56
-0.~017
-0.1002
-2.2040
-1.2E.94
-0.1193
0.8997
-0.2953
-0.1082
1.E702
0.4256
-1.9COO
0.5'::01
0.ClS6
-0.1082
-1.4Z71
-0.6BS3
-0.0720
-1.1+211
-0.4601
1.1726
0.5201
1.3003
1. 1 4~ 1
1.9605
-1.4211
0.2822
-0.5094
0.5201
1.1126
-1.0239
-0.5399
-0.4601
0.4636
0.0519
-0.6386
-1.5601
-1.7586
1.0562
C.5802
-0.4017
1.0:62
1.5:5'0
-0.23&5
-0.1731
0.4916
1.0562
-1.3729
-0.9000
0.£tE:;E:,
-0.5783
-1.3129
0.7214
1.05f2
-0.1793
1.41e5
2.1387
1.8633
-1.~tC7
0.1E:35
-0.2365
0.8650
-0.5399
APP~r-.:IX
.
A -; •
~
F.A~"S
A~.:
SC Jr,
LE 5
FiAr-.Kl
scenEl
RA~K2
SCORE.:
11 1
11 =
11 3
11 "
115
b7.D
489.5
lee.C
7:0.0
-1.2339
55.0
604.0
551.5
le5.0
L44.~
liE:
766.:-
11 7
lIe
lie;,
~7~.C
=E~.O
I~C
754.5
1~ 1
~L.Q
-1.4b58
0.6873
C.4916
-0.6:;'::4
-0.3430
1.<1c::'3
C'.5::01
-0.5094
1.C:66
C.2597
-2.304-0
C.7901
0.1206
-0.1793
-1.1695
1.4£26
-0.4017
-C.94:::7
l
u
4-4-
1~"
4.-.J
1~4
1 4.-."
1:'6
127
~ ~
l~b
129
130
131
1 _4.-,.
133
134
135
136
137
138
139
140
141
~
14~
143
144
14S
146
147
14b
149
150
151
lC:~
.,,4-
153
154
155
156
157
158
159
leO
1C1
16~
163
164
165
Ibb.C
.- ....
"
.
;;
.,.;
733.5
2C9.::
576.C
723.:
77<1.0
419.0
46E-.O
576.0
543.5
73.5
405.5
452.:570.0
2C9.':
405.5
't66.0
779.0
759.0
704.=
2(,9.5
7C:6.G
522.0
196.5
275.5
325.C
343.5
186.0
307.5
776.0
7(;4.5
501 .5
532.5
366.C
b04.0
489.5
735.5
723.5
351+.5
696.5
501.5
146.5
196.5
307.5
366.C
O.282~
-0.7316
1.2752
-C.5094
1.71cl
C.55d
-0.7316
0.50:'5
1.5713
0.52Cl
1.3769
244.5
679.C
513.0
t.5
629.0
-0.63~6
43~.0
0.58C2
1.3003
1.9194
0.G579
C.2 C64
0.58(;:
0.1t636
-1.3300
0.0156
C.1635
0.5561
-0.6386
0.01::6
0.:064
1.9194
1.6217
1.1726
-0.6306
:.4981
0.3899
-0.6053
-0.4017
-O.23c5
-C.1793
-0.7316
-0.2953
1.8633
1.1726
0.3215
0.4256
-0.1082
C.6873
0.2822
1.3932
1.3003
-0.11+1+5
1.1242
0.3215
-0.9041+
-0.6893
-0.2953
343.5
S7.0
743.5
275.5
1 3e .5
674.5
225.5
23.0
325.:>
405.5
750.5
236.0
589.5
478. C
759.0
64E.0
740.0
-0.10e~
29~.0
782.0
~~9.5
C:~~
c:
-tI..JLe.J
795.0
256.5
325.0
25.5
55.0
405.5
:5 25.0
258.5
784.0
325.0
691.0
671. C
293.0
366.0
4,9.5
745.0
691.0
629.0
7C4.5
405.5
501.5
87.0
629.0
570.0
I.CO~O
-0.5783
-1.9COO
-C.23b5
0.0156
1.5~97
-0.~399
0.6309
C.2450
1.6217
0.&650
1.4314
0.4256
2.4328
-0.4601
-0.2385
-1.e545
-1.4b':.B
0.01~6
-0.2365
-0.4601
~.0291
-0.2385
1.0924
G.9851
-0.3430
-0.1082
0.2822
1.4764
1.0924
0.7901
1.1726
0.0156
0.3215
-1.2339
0.7901
0.5581
162
~
S
OF
~c::TrCI\i
=_3
...
:.;.TA
SCCH~3
RANK't
SC:; ..E 4
6.e
-2.4328
478.0
325.C
O.~450
3S.5
727.5
RMlI<.3
5~2.C
4C5.:71 1 .0
225.5
20.0
559.:
629.0
377.5
405.::
146.5
343.5
354.5
742.C
354.5
369.0
107.0
570.0
122.5
131 .0
165.0
660.e
107.0
452.5
20.0
738.5
636. C
513.C
275.5
794.0
489.5
6ti.O
87.0
31.5
3£6.0
343.5
176.0
761.3
209.5
196.5
666.5
343.~
366.0
1+52.5
704.~
307.5
419.0
604.0
717.0
87.0
97.0
452.5
466.0
-C.~305
S:=,.O
-1.7C:b
1.33CO
-1.4c:£
C.3099
0.0156
1.2141
57~.0
C.SE::2
-0.57~:5
-1.S'6C5
0.5201
0.7901
-C.C720
0.0156
-0.9044
-0.1793
-0.1445
1.4491
-0.141t:-0.0360
-1.105b
0.5551
-1.0239
-0.9bOO
-0.6204
0.9306
-1.1096
0.1635
-1.ge05
1.4185
0.8292
o e3 597
-C.4017
2.3765
0.2822
-1.3729
-1.2339
-1.7Se6
-0.1082
-0.1793
-0.7731
l.e515
-0.6386
-0.6693
0.9621t
-0.1793
-0.1062
0.1635
1.1726
-0.~953
G.0579
0.6873
1.2543
-1.2339
-1.1e95
0.1635
0.2064
5:'1.5
060.0
532.5
14. C
244.5
543.5
..
64tS.O
-,...~
~
""".~
5,,6.0
1 4 e.5
35 ... 5
619.:-
0 ... SI6
C.93Cf:
O.4~5t
-2.10/:9
-C.5CC?4
0.463b
-i).~7E3
O.6c5C
O.C~5S
-0.9C4~
-C.14~::
C.7SCI
3~5.0
-O.~.3oS
674.5
::'5.5
:75.5
~ 1 .5
40.5
275.5
439.0
146.::
604.0
258.5
7'?1.0
1.0C30
-0.5783
-0.4017
-1.4271
-1.6394
-O.4e17
0.1:08
:'75.~
-0.4CI7
0.4256
53:.5
79.0
7C:3.C
2C9.5
68.0
343.5
1 .0
389.0
275.5
155.0
770.5
e.8.0
.. 52.5
439.0
1t19.0
377.5
529.5
619.5
47.5
307.5
513.0
31t3.5
55.0
225.5
293.0
275.5
e
-0.~0"4
0.6873
-(;.4601
~.::419
e
-1.~cS4
2.3,,68
-0.6386
-1.37::9
-0.1793
-3.C237
-C.C360
-C.4C17
-0.6650
1.7734
-1.3729
C.1635
O.1~C8
0.0579
-0.07:0
0.6309
0.75Cl
-1.SeC7
-0.2'::53
0.3597
-0.1793
-1.4E56
-0.5783
-0.3430
-0.4017
e
APpa~~
,.. ...
IX A L •• FA~r< S A r.~ £CJM::S ur~
CE S
RAt'>.K1
SCORE1
RANK2
16'
16.
168
165
17e
171
172
173
174
175
176
177
17e
175
596.0
0.6559
-C.3430
-0.0360
C.7901
1.3003
-C.343C
C.12C6
7H.5
HS.O
293.0
:3 89.0
£.'9.C
7~·3.5
~93.C
439.0
165.0
4E6.C
513.0
5E~.C
~53.C
5 5~. 5
87.e
1£0
165.0
1b 1
.. 7E.C
501 .5
3b6.0
:3 89.C
405.5
619.5
7"8.0
244.5
:5 25.0
71::9.0
307.5
325.0
79.0
741 .0
24'+-.5
115.0
Ih~
18 :3
164
105
186
187
Ib8
169
19C
1~ 1
1~
153
le;4
195
IS6
197
158
195
20e
')
~
~
;:;
.. _0 ....
543.5
176.0
428.5
-o.e~C4
0.2064
0.3597
C.60:5
-0.3430
0.52C1
-1.2335
-C.8:C4
0.2450
0.3215
-0.10&2
-0.0360
0.0156
0.7501
2.6742
-0.5094
-0.~3b5
2.1706
-0.2953
-0.23b5
-1.2894
1.44C2
-0.5094
-1.0b44
-0.4601
C... 636
-0.7731
0.0877
2~8.5
36E.0
612.5
619.5
7;0.5
138.5
27.0
47t.C
343.5
4.0
6EC.0
4319.0
lC7.0
40.5
11.0
25e.5
405.5
405.5
122.5
8CO.O
225.5
293.0
759.C
122.5
646.0
209.5
629.0
513.0
leI'.
:;.,:;
.
:,,:. H
SC~R:2
RANK3
seeR=:3
Rio r~~ '"
S:JF::4
1.~373
293.C
:5 &9. C
-C.3430
-0.C3c.O
-0.7316
0.1200
0.C877
0.2 .. 50
1.20'1-3
O.12C8
-C.3430
-0.1793
629.C
107.0
3t.9.0
551.5
59t.0
lE.5.0
C.75C1
-I.1C';6
-C.03fG
-0.82C ..
-C.4601
-C.1062
C.7214
0.7501
1.3~31
-O.~4~7
-1.8:::69
0.~45J
-0.17S3
-~.5763
C.9306
O.1~O8
-1.10f:i6
-1.6394
-2.2048
-C.4601
O.C15b
0.0156
-1.0239
3.C237
-0.5783
-0.3430
1.6217
-1.0239
0.8650
293.0
3eS.O
452.5
-0.f3e6
C.79C1
C.3597
C.1635
-0.3430
-0.0360
0.1£35
501.5
0.3~15
4~2.5
;)~CT
163
186.C
439.C
.. ~8.~
478.0
709.5
439. ::
293.0
343.5
1 15.0
107. C
6eO.0
293.C
40.5
354.5
47e.C
73.:"
5~2.0
25.:325.0
799.0
23.0
570.C
750.5
66.0
275.:
225.5
79.0
-1.:'t44
-1.1eS6
0.9306
-0.3430
-1.t39 ..
-0.11+.,5
O.~"50
-1.3300
C'.3899
-1.bS45
-C.2305
2.8074
-1.9000
0.5501
1.5297
-1.37:9
-0.4017
-C.57b3
-1.2894
6.e
-~.43::8
428.:61 .5
20.0
405.5
293.0
0.0877
-1.4:.'71
-1.9605
0.0156
-0.3430
S:S~.5
653.5
leb.O
4t-b.D
79.0
165.0
5~:.5
(;.4~lb
C.6:::'3
-O.8~(;4
0.8557
C.eS"7
-O.7~1t:
C.~C64
-1.21:.94
-C.8~C4
O.4:Sf
v.Jt.77
42b.5
403.5
c.: 15e
11t.0
-2.1 CE9
244.5
17.0
-C.SC~'+-
3.0
-":.0'£51
-0.63b6
-2.6742
115.0
-1.064'+-
::~9.5
797.0
11• 0
1+52.S
766.5
543.5
79.0
1C7.0
155.0
8.5
253.0
4C.5
97.0
419.C
325.0
~.~7c~
-2.2048
0.1635
1.7161
0.4E36
-1.2E94
-1.1CSE,
-C.SeSC
-~.3C40
-0.3't30
-1.6394
-1.1655
C.0579
-O.23ES
A3:
APP~NJIX
RANK
Ct S
:3
2
:>
:2
6
•
7
&
9
10
11
1~
13
14
15
10
17
16
19
20
~ 1
•
•
•
•
•
•
•
•
•
•
•
•
•
•
::3
:3
:2
•
•
•
•
•
•
•
•
•
•
RAf\4K
Al
1
137.5
1
I
2
114.0
I 11 .0
•
88.e
I
1
•
•
•
•
•
•
•
•
•
•
•
•
•
2
•
2
1
1
•
•
•
•
58.0
•
•
•
•
•
1
2
•
•
•
•
105.0
•
1
2
•
•
•
1
2
:3
1
It
28.5
•
•
•
•
•
•
•
•
:2
•
•
•
4
•
:3
•
•
:2
:2
1
2
•
2
1
:3
2
•
1
•
•
1
2
•
47
:2
48
49
50
51
52
53
•
•
•
2
•
•
1
3
I
:3
:2
2
4
2
4
1
1
3
1
•
•
•
•
•
•
•
•
•
•
•
3
•
•
1
:3
•
•
•
•
:2
55
RANK 14
S:CTl(~
•
~4
54-
•
•
13
FOR
•
•
•
43
45
46
•
4
1
4~
•
•
1
1
1
3
,. 1
•
•
•
:3
•
39
40
•
•
2
37
30
•
•
•
•
~6
•
•
•
•
•
35
R ~.f\4K
2
2
4
3
it
I~
hA~KS
•
1
•
1
•
•
•
3
•
•
1
3
•
•
1
1
3
•
•
•
•
•
•
•
104.0
•
32.5
16.5
•
•
•
5.5
•
108.0
•
•
32.5
57.5
125.0
•
64.5
•
•
1
:2
•
86.5
69.5
154.0
•
•
•
17.0
•
•
•
I
164
•
96.0
5.4 SATA
RANt< A2
144.0
109.5
•
•
•
85.0
107.0
•
•
•
•
•
•
•
•
6.5
61.5
115.0
•
•
•
•
8.e
•
•
•
37.0
•
•
98.0
•
•
•
50.0
•
106.5
58.0
•
•
39.0
5.5
79.0
•
110.0
•
•
•
26.5
71.~
•
66.5
13.5
47.5
•
•
•
•
57.0
•
89.5
52.5
25.0
•
•
5~.O
1 10. 0
•
17.0
33.5
7.0
•
•
•
76.0
•
19.5
10.5
•
•
•
25.0
•
•
82.0
124.0
•
•
41.5
•
•
•
79.0
1 .0
5.5
34.5
4.0
1:0.0
120.0
•
•
101 .0
24.0
•
114.0
•
•
lit.S
52.5
•
76.0
82.0
86.5
•
•
•
•
•
99.5
•
•
•
30.:
109.0
•
28.5
1 1 .0
109.5
•
•
•
•
•
•
•
83.5
47.5
•
•
10c.0
67.5
•
:.5
37.C
•
•
•
34.5
•
30.0
•
2.0
54.5
73.0
129.5
•
•
6.0
19.5
lo.e
•
13~.C
•
•
66.5
A3:
AFP~NJ1X
OcS
~
57
5b
5';
2.0
•
:3.0
RAt-.K 13
1
•
•
1 .0
2.0
•
2
1
:3.0
4
3
•
~
1
63
:3.0
3.0
6~
1.5
4
•
3
1
•
•
2
1
1
...•
E: 1
62
65
E6
b7
•
;.0
2.0
tb
~.o
69
2.0
1 .0
2.0
7C
71
7~
1• 0
73
74
75
76
•
77
7b
•
2
2.0
•
•
•
•
1
79
•
:3.0
•
bO
•
•
•
C
~
&...
4
1
&3
~4
•
•
t:.5
t:.e
3.0
e7
•
•
eEl
1• 0
b9
9Ci
2.0
4.0
1
c; 1
•
•
1
94.0
•
1 .0
•
•
•
•
2.0
•
•
•
•
•
•
•
•
1 .0
1 .0
2.0
4
1•C
1
•
•
•
3
:2
•
•
:2
1
•
•
•
134.5
109.S
•
•
98.C
42.5
2.0
1 .0
104.5
•
•
•
148.0
•
•
1• 0
89.5
1
2.0
•
3.0
:2
1 .0
lCO
•
~.O
•
1 C1
H,2
•
•
•
2
•
•
•
1• 0
•
•
1.0
1• 0
•
•
•
•
•
,.0
2
2.0
2.C
3
•
•
1
•
1
4
3
1
•
•
1 .0
•
1.0
•
•
•
165
24.0
•
30.0
•
137.5
87.5
102.0
52.0
42.5
10.0
•
•
128.0
•
114.0
•
•
•
1 11 • 0
•
52.0
95.0
42.5
92.0
•
A3
•
•
19.5
13.5
~0.5
23.0
30.5
150.0
98.C
•
16.5
56.0
•
13.5
•
•
•
37.0
25.0
22.0
•
•
•
•
•
1: • C
•
•
36.5
63.C
lIE .5
27.0
73.5
•
52.5
•
•
28.5
•
•
•
•
•
•
•
18.5
76.0
•
•
•
16.0
•
154.0
142.0
20.5
146.5
113.0
112.0
117.0
79.5
•
•
3.0
•
•
RAM:
•
3.0
•
:ATA
•
1
•
=.~
•
•
99
IC9
110
•
82.0
64.5
2.0
1
3.0
•
•
2.0
1(
lCo
103.0
•
9b
1(;3
IC4
IuS
lC6
•
•
•
•
•
•
43.5
19.5
3
~.O
~.O
2
•
•
97
•
2.0
1 .0
1 .0
2
90
•
61.5
85.::
22.0
61 .5
1
1
•
2.0
•
•
~.O
95
•
•
•
93
•
73.0
•
•
c,:z
94
•
•
•
1
•
•
•
101 .5
2.0
1 .0
1 .5
...J
•
•
114.0
30.0
90.0
8t;.5
22.':
~
•
FOR SECT1CN
RANK Al
RANK 12
1
1
1
6e
RA~KS
14.0
5.0
•
54.0
•
•
•
•
127.0
•
90.0
•
101 .5
•
121 .0
•
50.0
•
52.0
82.0
•
66.5
•
1• C
•
•
•
6.5
34.5
b2.5
•
•
158.5
•
106.0
•
•
8.5
•
•
•
•
•
36.5
2.0
11 .0
•
127.0
58.0
96.5
•
71 .5
•
•
•
25.0
80.e
47.5
•
•
156.0
:'6.0
107.0
132.0
•
6.5
•
•
•
•
•
5".5
•
8E:.5
38.5
66.5
96.0
11• 0
•
38.5
4.u
•
14.5
•
•
A3:
AFP~~~IX
RANK
111
11 ~
1 13
2.0
2.0
•
114
3.0
11 :.
lIe
•
117
4.0
116
•
•
12
RANK
1• 0
•
3.0
3.C
4.C
1. a
•
1 .0
2.0
2.0
2.0
3.0
1 .0
•
•
1• 0
3.0
4.0
1~~
3.0
1. C
•
•
1~3
•
•
•
1:'+
1• C
3.C
3.0
•
1~:'
1~6
1:7
1~ 6
1 ~ c;
DO
131
13 ~
133
134
135
136
137
138
139
140
141
14~
143
1'+4
145
146
147
148
145
•
~.c
3.0
1 • (;
2.C
2.0
4.0
•
•
1.0
;;.0
1.0
1.0
1.5
1• 0
•
•
•
151
152
153
154
155
•
3.0
~.o
~.v
4.0
156
1.5
157
159
3.0
1. C
2.0
IcC
2.0
leI
162
163
164
1• 0
2.C
•
1.0
2.0
•
2.0
•
•
•
•
•
•
•
•
1.0
2.0
•
•
•
•
•
•
116.5
•
16.0
•
•
68.0
89.5
15.C
10 • ~
59.5
98.0
•
•
•
2.0
•
3.0
1•C
•
1 .0
•
•
2.0
1.0
1 .0
•
•
1 .0
3.0
1.0
•
•
•
128.5
164.0
•
1.0
•
•
•
4.0
4.0
114.0
125.0
•
•
3.0
2.0
2.0
4.0
4.0
1.0
1.5
1.5
4.0
•
2.0
•
•
•
112.0
1.0
1 .0
3.0
3.0
1'+.0
106.5
11:3 .0
2.0
•
•
1• 0
3.0
•
1.5
•
•
3.0
1.0
2.0
1.0
1.0
2.0
3.0
•
•
•
2.0
1• 0
2.0
1 .0
2.0
1.0
3.0
3.0
3.0
1 .0
•
1•a
•
•
64.5
70.0
•
122.0
•
•
168.0
78.C
34.5
44.5
52.0
•
39.5
122.5
41 .0
154.0
1 12.0
•
99.5
1• 0
104.5
•
•
•
47.5
120.0
•
•
57.5
119.0
•
•
23.0
•
•
•
69.0
•
19.0
•
•
167.0
•
•
8.5
15.0
•
52.5
•
•
•
•
•
•
166.0
71.5
•
•
11• 0
•
75.5
•
123.0
•
109.5
117.0
60.0
132.0
76.5
162.0
18.0
157.0
42.5
152.0
47.5
79.5
150.0
77.0
99.5
14 0.5
73.5
49.0
•
79.5
76.5
163.0
•
•
•
•
•
•
125.0
•
•
57.5
•
52.~
•
•
•
58.0
•
•
•
102.5
•
102.5
•
•
86.5
73.0
-.-=
140.5
•
3.0
•
92.5
11 7 .0
c;
•
•
•
•
•
89.5
7.0
122.5
•
120.0
45.0
26.5
•
161 .0
5~.5
•
•
•
166
•
•
129.5
•
1.0
•
132.0
81 .0
27.0
8.0
71 .5
43.5
28.5
2.0
5.4 :ATA
Al
131 • U
•
•
3.0
SECTIC~
1 .5
•
•
•
1 .0
1 .0
2.0
•
1. C
•
•
•
2.e
•
•
•
•
•
2.0
1• 0
•
4.0
2.0
2.0
2.0
3.0
4.0
•
2.0
3.0
165
•
~.o
•
3.0
150
15b
•
1• 0
•
•
1.0
1• 0
FOR
RA~K
13
3.0
2.0
1.0
1. C
1 1 c;.
1~ 0
1~ 1
•
MA~KS
•
79.5
•
158.5
•
90.0
83.5
104.0
14.0
16.5
67.0
102.0
•
•
96.0
43.0
•
e
44.5
•
•
165.0
•
8.0
54.5
•
•
58.0
30.5
126.0
8.e
96.0
94.5
66.5
02.5
92.0
137.5
•
66.5
•
•
•
39.0
•
58.0
e
....
APP;:t-4CIX
A~'
PAr...t<S F8R
RANK 12
RAt-iK 13
RANK 14
2
3
1
3
2
•
16b
1
169
3
170
171
172
173
114
175
176
177
4
1
1
2
4
•
•
CbS
MAI'lK_l1
1
1b (
17c.
17S
1 t. 0
1t 1
•
1+
•
2
..•
~,
~
oJ
lE.~
:5
Ib3
•
•
•
!t4
165
lee
4
187
188
189
~
190
•
•
~
2
1 .
1._
:3
193
1
194
195
196
IS7
ISb
199
2
:3
~
2"0
1
•
4
1
~
oJ
•
•
-
1
•
3
1
1
2
2
•
2
•
•
.:.
•
1
1
:3
3
4
2
3
2
•
•
•
1
1
1
1
•
•
•
3
2
1
1
•
•
•
3
2
..
•
1
2
2
•
•
•
1
•
•
•
3
1
oJ
~
1
•
•
2
1
1
1
4
3
•
1
•
1
4
3
1
•
•
,"
:3
•
4
•
•
S~CTIC'"
Al
RA~t<
S4.0
47.5
S.O
99.5
160.0
47.0
64.S
33.5
102.0
•
128.=
•
85.:>
•
32.0
73.0
76.C
•
•
•
137.5
170.0
41.5
11 .5
•
•
2
1
4
2
•
1
..'"•
2
•
71 .5
12.0
108.0
41 • C
20.5
2
120.0
•
33.0
•
1
2
1
1
2
167
•
92.5
5.4 CAT A
RANK. A2
-
109.5
27.0
•
•
134.5
•
•
27.0
3.0
•
75.5
•
10:: .0
64.5
18.5
13.5
•
•
•
64.5
22.0
~
RAr...r< A:'
-
47.5
•
•
68.0
92.5
69.5
•
94.5
63.0
•
19.5
•
•
.. 7.0
•
57.5
73.0
•
91.C
146.5
39.5
1L.0
12.0
5.C
80.C
•
•
•
41 .0
•
125.0
•
~O.O
•
58.0
47.5
•
•
•
2.5
61.5
•
2.0
59.5
63.0
109.5
•
84.e
130.C
•
39.0
63.0
47.0
84.0
•
23.0
•
17~.0
•
- A4
•
8.5
71.5
171 .0
3.0
144.0
45.0
86.5
RAt-.K
•
1 .0
19.5
169.0
•
98.0
•
•
12.0
18.0
•
4 .0
•
4.0
16.5
•
71 • '5
AFFE~CIX
A4: PRCG?'AM TO 3ENERATE
I~ SECTION 5.4
TH~
~ISSING
VAL~~S
FC~
TH~
~ATA
GIV:~
*
,.
*
I/CH~FTER3
wC~
L~C.B.~6903,~ERCAC~,PP.TY=2
1/ EXEC SAS
IIIN SC
~S~=L~lISC.SASGATA,:I5P=OLJ
//SY5.I~
DC *
LPTA C~c;A~nAY Y Y: Y3 Y4;
~C 1=1
H, 50(;;
Xl=IC·MA~~NIC~C);
X~=IC*FA~W~I(40);
x3=lC·RA~UNICfC)i
X4=IC*RA~v~ICeO);
Yl=RCL~L(Xl)iY~=RG~NG(X2)iY3=R:u~C(X3);Y4=nCJ~C(X4)j
lLTPIJT;(~C;
LATA l~C;S:i U~E;ARRAY : 22 ~3 24;A~RAY Y YZ Y3 Y4;
[c eVER Y;
IF y=c Cn V=~ CF Y=lO THE~ ~=l;ELSE Z=Oj
L~Di
~1=Z:+Z3+~4j
:RCP Xl X~ X3 X4 22 23 24;
CAlA lHREEiSET T~C;
IF ZI=C;IF Vl=lC THEN Y1=C;
::PCP Zl;
:ATA FCLR;SiT T~R~E;APRAV X Xl X2 X3 X4;ARRAY N
IF _~_ < 2Cl;
~1
~2
N3
Xl=1;X~=1;X3=1;X4=li
IF Yl=l OR Y1=b THEN ~2;
IF V~=l OM Y~=5 THEN XI='.'iIF
IF V~=3 eR V~=7 lHE~ X3='.'iIF
:.~; C i
IF
IF
IF
IF
IF
V1=~
V:=l
V::=3
Y3=1
..J-_
Y"'-'
OR Y1 =7
Y2=5
OF V2=7
CR Y3=5
OR V3=7
THEN
THEN
TrlEN
ThEN
ThEN
co;
Y1 =b
Y2=5
Y2.=1
V3=5
V3=7
V4=5
Y4=7
THEN
co;
OR
E~Ci
IF
IF
IF
IF
IF
IF
IF
VI=3 CR
Y~=l c.R
V:=3 GF
Y3=1 eR
V"l-"t
_-..J
CP
VI+=l OR
V4=3 uF.
X1= , • ' ; I F
X3='.'iIF
X1='.'iIF
X3='.';IF
Y~=2
Y~=4
OR Y2=6
OR V2=8
THE~
X2='.'i
TH~N
X~='.';
Y2=2 OR Y2=6 THEN X2='.';
Y2=4 OR Y2=8 THEN X4='.';
Y3=2 OR Y3=6 THEN X2='.'i
Y3=4 eR Y3=8 TH~N X4='.';
X1='.' IF~Y2=2 (;R
THEN ·X3='.' IF V::=I+ OR
THEN XI='.' IF Y3=2 DR
THEN X3='.' IF Y3=1+ OR
ThEN XI='.' IF Y4=2 OF.
THE~ X3='.' IF V4=4 OR
THE~
E~C;
IF Y1=4 eli YI=9 THEN Dui
XI='.' iX2='.';X3='.';XI+='.';
EM; ;
168
Y2=6
V2=8
V3=6
Y3=8
V4=6
V4=8
THEN X2='.';
THE:N X4='.';
TH~N X2= , • ' ;
THEN X4='.';
THE~ X2='.';
THEN X4='.';
~4i
"
cc eVER XiIF X='.'
lHE~
N=CiELSE
N=l;E~Di
~~ISS=~1+~~+~3+~4;
[ROP
~1
~~
~3
~~i
FRuC PRINT;
FRee FRE.Gi
ll-E>LES ~toIISSi
tAlA I~.toIIS51~G;MERGE FCUR
I~.l~S;
ehOL1=Xl*CHCLl;CHuL~=X2*CHCL~;=hOL3=CHCL3*X3;CHOL~=CHCL4*X4i
PFlce PRINT;
FRoe toIEAN5;
II
169