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A MULTIVARIATE EXTENSION OF
FRIEDMAN'S X2-TEST
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by
T. M. Gerig
University of North Carolina
Institute of Statistics Mimeo Series No. 550
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September 1967
Work supported by the National Institutes of
Health, Public Health Service, Research Grant
GM-12868
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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A MULTIVARIATE EXTENSION OF
FRIEDMAN'S x2-TEST*
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by
T. M. Gerig
Abstract
This paper deals with a multivariate extension of Friedman's
X;-test.
A rank permutation distribution and the large sample
properties of the criterion are studied.
efficiency (A.R.E.) for a sequence of translation alternatives is
studied and bounds are given for certain special cases.
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It is
shown that, under specified conditions, the A.R.E. of this test
with respect to the likelihood ratio test is largest when the
block dispersion matrices differ and can be greater than unity
when the differences are large.
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The asymptotic relative
1.
INTRODUCTION
Suppose we are given p-variate data which form a complete two-way layout.
Let
~ij
=
(X~j,X~j, ••• ,Xij)
that received the j
th
be the response from the plot in the i
treatment; i=1,2, ••• ,n; j=l,2, ••• ,k; (k
~
th
2).
block
Assume
that the cumulative distribution function of ~ij is Fij(~)' ~ERP.
We wish to test the null hypothesis
(1.1)
for i=l,2, ••• ,n,
against translation type alternatives of the form
HA:
for
Fij(~) = Fi(~-~j)'
j~1,2, •••
,k and
i=l,~,
(1.2)
••• ,n,
* Work supported by the National Institutes of Health, Public Health Service
Grant GM-12868.
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where
~j
1 2
P
"" (a j ,aj , ••• ,aj ).
Under this type of alternative, we can write the
null hypothesis as
(i.3)
In standard parametric tests of this hypothesis, it is usually assumed
that
~ij "" ~ + ~j + ~i + .tij ,
where
~, ~j
~i
and
and block effects.
are the constant vectors of mean effects, treatment effects
The vectors .tij are nk independent random vectors follow-
ing a common mu1tinorma1 distribution with zero mean vector and a common dispersion matrix
t
~
s
s'
= «cov(eij,e
ij »).
Several such tests are discussed in
Anderson [1], in Rao [7], and elsewhere.
Frequently, some or all of these assumptions cannot be justified.
The
subject of this investigation is a non-parametric test which allows the assumptions of block additivity and multinormality to be relaxed.
Friedman [5] pro-
posed the X:-test for the univariate case and its asymptotic efficiency was
later studied by E1teren and Noether [4].
A class of non-parametric tests,
which do not require the mu1tinorma1ity or independence assumptions, has been
t
proposed and studied by Sen [8].
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depth by Chatterjee [3] for k=p=2 and by Sen [8], will be presented in the
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••I,
A brief outline of the rank permutation principle, which was studied in
next section.
2.
PERMUTATION ARGUMENT
The t~st to be proposed is based upon intrab10ck rankings.
Ranking will'
be carried out over the k treatments within each block and done individually
for each variable.
{X~t' t=1,2, ••• ,k}.
Thus, Rs will be the rank of Xs among the observations
ij
ij
I...
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To illustrate the permutation
argument, form the n matrices
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R1
ik
R1
il
R •
",i
Each row of
~~
~i
•
p
R
il
, i=1,2, ••• ,n.
Rik
will be some permutation of the numbers 1,2, ••• ,k.
to be the matrix derived. from
~i
Define
by permuting the columns in such a way
that the numbers 1,2, ••• ,k appear in sequence in the first row.
We say that
two matrices, A and B, are permutationally equivalent if A can be obtained
from
~
'"
'"
by a finite number of permutations of the columns
'"of~.
the set of matrices which are permutationally equivalent to
~~.
Let
S(~~)
Hence,
be
S(~t)
contains k! elements.
The distribution of ~~ over all its (k!)p-l realizations will depend upon
the parent distributions, even under H.
o
zation of
~~,
the distribution of
~i
However, given a particular reali-
over
S(~t)
will be uniform under H •
o
In
fact, if R E S(R*i)' then P{Ri - R IS(R*i)' H } = 11k!, no matter what be the
",0
'"
'"
'Va
'"
0
parent distributions.
Finally, if
~i E S(~t),
i-l,2, ••• ,n, then
- R , i=1,2, ••• ,nIS(R~), i=1,2, ••• ,n; Ho }
'Va i
'"
(2.1)
n
=
n P{R = R IS(R*i)' H }
i=l '"i
'V° i '"
0
=
(1
k!
)n
because intrablock rankings are used and complete independence is assumed
from block to block.
We are.now in a position to select a test function which depends upon
R , i-l,2, ••• ,n.
i
"'.
.
.
~
Such a test will be completely. specified by the conditional
probability law (2.1) and, hence, will be a similar test of H •
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Henceforth, we will let
3.
s
Define T
j
=
@n
denote the probability law given by (2.1).
THE PERMlITATION TEST
n s
(l/n)Ei=lRij for s=1,2, ••• ,p; j=1,2, ••• ,k.
s
Note that Tj
depends upon n.
It is easily seen that
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and, after some essentially simple steps, that
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k' s s'
{(k-1)/k}[Lt=lRitRit - (k+1)2/4], for i=i', j=j'.
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Finally, we find that
(3.1)
where
a[s,s'](~*) = (1/n(k-1»Li:1L~lR~tR~~-(k(k+1)2/4(k-1»,
the Kronecker delta.
Define L(R*)
~ tV
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Again, we note that a[
= «a[ s,s ,](R*»)
tV
T'
tV(lxp(k-1»
=
s,s
and 0jj' is
,](R*) depends upon n.
tV
and
1
1
1
P
P
(T1'···' TP
l' T2'···' T2'···' Tk-1'···' Tk-1 ) •
Thus,
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for i7'i'
Var(T) = !:J. ® L(R*l = V(R*) say
tV
tV tV tV
tV tV
'
where ~
=
«(Ojj ,k-l)/nk» and
as defined in [1, p. 347].
® denotes
the 'direct product of the two matrices
If we assume that
L(R*) is positive definite," then
tV tV -
5
Finally, using the well-known formula
where u' is (lxr) and A is (rxr), it follows that
'"
'"
21
12
• n
11 ••• 2
Now, for our test function we take the quadratic form
(3.2)
where T~lxp(k-l»
'"
"" «k+l)/2, ••• ,(k+l)/2).
After some simple steps, we
obtain
t
= n
n
~
~
O'[s,S'](R*)! (T s _ k+l)(T s ' _ k;l),
s""l st""l
'" j=l j
2
j
where «O'[s,s'](R*») = «O'[
.
'"
(3.3)
,](R*»)-l = E-l(R*).
s,s
'"
'"
'"
It will be demonstrated later that, for large n and with some mild
restrictions on F , i=1,2, ••• ,n, E(R*) is positive definite with high probi
ability.
"''''
If, in practice, it is found to be singular, then elimination of
the proper variables will remedy the problem.
Following the arguments of Section 2, we can now compute the exact permutation distribution of
1.n
and thereby, determine a randomized test of H which
0
is strictly distribution-free and has exact size a.
'\
However, when p, k and n
are large, these computations may be too lengthJ.y to be practical.
In the
next section, we will study the asymptotic permutation distribution of ~
".
n
This
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will provide us with a large sample approximation to the permutation distribution of
Ln
and, thereby, increase the usefulness of the test.
4.
ASYMPTOTIC PERMUTATION DISTRIBUTION OF ~N
s
Define Fij[s](x) and Fij[s,s'](x,y) to be the marginal cdf's of Xij and
s
s'
(Xij,X ) for s,s'=l,2, ••• ,p; i=l,2, ••• ,n; j=l,2, ••• ,k.
ij
Define
1 { kkk
O[s s,](F io ) = k-1
,~
j~t~tr=l
co
I I Iff
co
Fit[s](X)Fit'[s,](y)dFij[S s,](x,y)
-co -co
'
j~t'
fco fcoFit[s
kk
+ L~ L~
j~t=l -co - c o '
for
s~s'=1,2, •••
s,](x,y)dFij[s s,](x,y)
- k(k~1)2 }
' .
,p; i=l,2, ••• ,n
= k(k+1)/12
for s=s'=l,2, ••• ,p; i=l,2, ••• ,n.
(4.1)
Let
L(F
~ ~
Theorem 4.1:
i
0
) =
«o[ s,s ,](F~i 0 »), s,s'=l,2, ••• ,p.
(4.2)
If Fij is continuous for i=1.2 ••••• n; j=1.2 ••••• k. then
L(R*)-(1/n)L
~ ~
n
U;(F-. )}
i = 1 ~ ~l.O
converges in probability to the null matrix"as ntco.
Proof:
Define the function
1
for
x > y
w(x,y) =
(4.3)
o
for
x
~
y.
k s s'
,](R*) =
- k(k+1)2/4(k-1), then o[
Let o[ S,S ,](R~)
=
[1/(k-1)]L
j
s,s
~
~l.O
=1Ri.Ri.
J J
s
1
~
k
~(s
s)
(1/n)Lin10[
NoW using (4.3) we can write Rij = +~t=l Xij,Xit
= S,S ,](R~).
~l.O
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and, hence,
From (4.3), it follows that
0Cl
~ Fit[S] (x)dF ij [s](x),
for
0,
j#t
for j=t,
that
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and that, if s#s', then
0Cl
0Cl
~ ~ Fit[S](X)Fit'[S,](Y)dFij[S~S'](x,y),
0Cl
for
t#t', t#j, t'#j
for
t=t',
0Cl
= ~~
Fit[s,s,](x,y)dFij[,s'](x,y),
0,
Using the above results, we can show that
t~j
. otherwise.
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and hence, that E{O[
= (lIn) Ei : 10[ s,s ,](f~).
s,s ,](R*)}
~
'ULo
s
Since R is bounded above and below for s=1,2, •••,p; j=1,2, ••• ,k;
ij
i=1,2, ••• ,n, it follows that var{o[
independent of i and n.
••
I
,](R~)}
~J.O
has an upper bound that is
Thus, (1/n 2 )E i : 1 Var{o[s,s,](R!o)}~O as nt oo and so,
by the weak law of large numbers o[ s,s ,](R*)-(l/n)Ei~lo[
~
s,s '](¥~)
,~o
~ 0 as
ntoo , which completes the proof.
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Under Ho , (l/ n )E i =l o [s,s ,](F
) = o[ s,s 'l' where
~i 0
Theorem 4.2:
a
...
[s,s']
(4.4)
k{k+1) 112,
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s,s
't
i[s,s']
for s=s',
=4 J J
co
Pgi[s,s'] = 12
co
J J
-co
-00
[Fi[s](X)Fi[s,]{Y)-~]dFi[S,s'](x,y),
th
th
and where Fi[s]{X)' Fi[s,s']{x,y) are the sand (s,s')
marginals of Fi{~)
given by (1. 2) •
Proof:
The theorem follows from Theorem 4.1 after setting
Fij(~)
=
Fi(~)'
If Fl ,F 2 , .•• ,Fn are continuous and ~
E = ({o[ s,s ']» is positive
l
definite, then the joint distribution of {n:(T; - k1 ); s=1,2, ••. ,p;j=l,2, .•. ,k-l}
Theorem 4.3:
-
is asymptotically p(k-l)-variate normal with mean vector-zero and dispersion
matrix t. 0
tV
r
~
where t. =. ({OJ. ,-11k».
~
J
\
Proof:
Let a j; s=1,2, .•• ,p; j=1,2, ••• ,k be
k
s
l a j=O for s=l,2, •.. ,p. Then write
j=l s
~ny
sp constants such that
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where
Ui -
! I aSjR~j'
8-1 j=1
Now, we find that E{Uil@n} = 0 and that
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k(k+l)2}
4(k-1)
- -L
k-l
f ff r 1{~
8=1 s'-l~=l sj S :f t=l it it
a
err f I
s-l
RS RS '
a,
:f'
_
Jf
k(k+1)1
4
8 'a S 'j"l[S s'j(Rto>'
s'=l~""l Sj
Thus,
f f
~
{I
.! I.E{U i2 /CP} ""
a ja , } [
,](R*).
n i""l
n
- 1S ' "" 1 j-1
S s
s,S . ~
sBut,
k(k+1)
so that
12
*
~ a[s,s'](~ ) <.
k(k+1)
12
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Thus, the conditions for the Liapounoff central limit theorem are satisfied, and we may conclude that Y is asymptotically normal.
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An application
of some well-known results from multivariate analysis completes the proof.
Theorem 4.4:
Under the conditions of Theorem 4.3, the permutation distribu-
tion.of ~n is asymptotically chi-square with p(k-l) degrees of freedom.
Proof:
The theorem follows as an immediate consequence of Theorems 4.2 and
4.3.
Using Theorem 4.4,·we can now write the asymptotic permutation test of
H , given by (1.1), as
o
Reject H
o
if
tn
Accept H
o
if
S,n < Xp2 (k-l),cx,
2
-> X
'-p(k-l),cx
(4.5)
where
p{x 2 <
t -
X2t,cx } = l-a, O<cx<l.
We need to know condition3under which
we examine o[
s,s
r
'"
is positive definite (pd).
If
,'] in (4.4), we see that we can write
(4.6)
. where CI and Ci are constants, ti
= «Ti[s,s']»
and ~i
=
«Pgi[s,s']»·
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Since T and G are positive semi-definite, E will be pd when either terms
tVi
tV i
tV
I 2
P
on the right hand side of (4.6) is pd. If we let (Xi'Xi' ••• 'Xi ) be a random
vector whose cdf is
Fi(~)'
then it can be shown that
~i
is the dispersion
Thus, ~i is pd unless there exists a
s
linear relationship between the variables Fi[sj(X ), s=1,2, ... ,p that holds
i
matrix of (Fi[lj(Xi), ••• ,Fi[pj(Xi».
a.s.
Now, if there are m=m(n) values of i for which this relationship fails
to hold and if m(n)/n+A, where O<A<l as ntoo , then (l/n)Ei~~i is pd a.s. and
so is tVE.
Finally, we can show that the sequence of tests given by (4.5) is consistent, since,
1n -> n
ma x{T s _(k+1)/2}2/ a [
j(R*) = (12n/k(k+I»max{T s - (k+1)/2}2 = 0 (n),
s, j j
s,s tV
s,j j
P
and so
pitn -> Xp2 (k- 1) ,a IH 0
5.
Theorem 5.1:
not true}+l as nt oo •
THE NON-NULL ASYMPTOTIC DISTRIBUTION OF {T;}
If F
is continuous for i=I,2, ••• ,n; 1=1,2 ••••• k. then
ij
{n~(T; - ~;), j=1,2, ••• ,k; s=I,2, ••• ,p} is asymptotically normal with mean
vector zero and dispersion matrix
~
ri I I [
i=l
t~t'=l
~*
a*
II
=
«aj j '-[s,s' ]»,
where
•
j:;::::Y)dF [s](x)dF [sl(Y)
-oo<y<x<oo
it
it
,
t~j,t'~j
+
If.
-00< x<y<00
A~j(X'Y)dFit[Sj(X)dFit'[Sj(Y_)]+.
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12
}
II ~I~~I~A~j(x,Y)dFOt[
](x)dF
(y)
1 S
it[s]
k
+ t=
,
t"j
a*jj'[S,S]
+
II
~y<k<~
A~t(y,X)dFij[ s ](x)dFij'[s] (y)]
k
- t=l
l [~<x<y<~
II ASij (x,y)dFit [s] ( x)dFij' [s] (y)
t"j
t"j'
+
II
~<y<x<~
A~j(y,X)dFit[ s ](x)dFij' [s] (y)]
k
- tll [
II
=
~<X<y<~
t"j
t"j'
+
II
~<y<X<~
A~j' (x,y)dF1OJ [s ] (x)dF1t[S] (y)
°
A~j,(y,X)dFij[ s ](x)dFit[s] (y)]
1
~ ~
A~j(X.Y)dFij'[SJ(X)dFij'[Sl(YJ
-1. 1.
n{
a*jj[s,s'] -- -n1 ill lk
=
t=l
t"j
for
~ ~Bit'
s s'
(x,y)dFio[
I I
~ ~
Hj',
J s,s
'](x y)
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00
00
,
/ B~jS (x,y)dFit[S](x)dF ij , [S,](y)
/
_00 _00
_
s,s'
]
Bij , (x,y)dFij [s] (x)dF i t [s' ] (y)
/00 /00
_00 _00
00
-
/
00
/
_00 -00
,
B~j~ (x,y)dFij[s,s'](x,y)
where
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Proof:
Let a
sj
' s=1,2, ••• ,p; j=1,2, ••• ,k be pk arbitrary constants.
Write
where
~~j
k
=
00
~ + t~l
-L"Fit[s](X)dFij[s](x).
Now, it is easily seen that EUi=O and that
R
R
k
k
s
s'-
EU~ = 2
·2 I
I a ja ~,Cov{Rij ,Rij ,} •
. s=ls'=lj=lj'=ls S..JO
s
Since Rij , i=l,2, ••• ,n; s=1,2, ••• ,p, j=1,2, ••• ,k are bounded above and below
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s
by 1 aRd k, there exists upper and lower bounds for COV{Rij,R
n
not depend upon i, j, j', s, s' nor n.
Thus,
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s'
ij
,} which do
E{U~} = O(n) and similarly
i=l
lim E n E{lu .I~}/[I n E{U 2 }]3/2 = O.
i=l
i
i=l
i
n~
Hence, by the Liapounoff central limit theorem and some well-known results
s s
is
from multivariate analysis, the asymptotic normality of {n~ (Tj-~j)}
established.
Finally, we obtain 0jj'[s,s']
Using (4.3), it can be shown that
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(5.1)
After some very tedious calculations, the first term on the right hand
side of 5.1 can be evaluated, and the expressions for 0j*"[
J
the theorem, can be obtained.
s,s
'l' given in
This completes the proof.
It is useful to note that, under Ho (1.1), we find that 0*jj '[ s,s 'l =
(Ojj,-l/k)O[s,s'] where O[s,s'] is given by (4.4).
The rank of the asymptotic multinormal distribution can be at most p(k-l),
k
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T~ = k(k+l)/2 for s=1,2, ... ,p. In fact, when H · (1.1) holds and
o
j=l.
\
F ,F , ••• ,F are continuous and are such that ~.is pd, the rank is p(k-l) and
1 2
n
since
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1*n
where «a
[s s']
'
»
=n
(5.2)
= «a[s,s']»
-1
, is asymptotically chi-square with p(k-1)
degrees of freedom.
6. ASYMPTOTIC DISTRIBUTIONS OF i AND 1*
n
n
UNDER A SEQUENCE OF TRANSLATION ALTERNATIVES
For the purpose of studying asymptotic power efficiency, we shall concern ourselves with the sequence of translation alternatives, {H }, given by
n
F (x1 ,x 2 , ••• ,xp ) = F (x1 ,x 2 , ••• ,xP )
nj
ij
i nj nj
s
where x
= x
nj
-~ s
s
(6.1)
+ n a j , for s=1,2, ••• ,p; j=1,2, ••• ,k and where
~1=~2= ••• =~k
fails to hold.
Throughout this section, we will write
are absolutely continuous and are such that
{E~
to mean that F ,F 2 , ••• ,Fn
l
r is pd.
tV
We note that if ~l=~2=."=~k' then Fij(~) = Fi(~) for i=l,2, ••• ,n;
j=l,2, ••• ,k.
Theorem 6.1:
.
.
~
If FEt, then, under {Hn }, {n
-tV
s
s
(Tj-~j);
j=1,2, ••• ,k; s=1,2, ••• ,p}
is asymptotically normal with mean vector zero and dispersion.matrix 6€D
.
where
~
- - tV
= ((OJ.,-l/k)) and
Proof:
J
--
tV
r
tV
r
tV
is given in Theorem 4.3.
Using 6.1, Taylor's theorem and some routine analysis, the
theorem follows from Theorem 5.1.
Theorem 6.2:
.!i{E~,
~n! defined by (3.3), is
then, under {H }, the asymptotic distribution of
n
non-central chi-square with p(k-1) degrees of
freedom and non-centrality parameter
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r r
O[S,S']ASAs '
(6.2)
s=l s' ..l
where
co
AS = (k/n)E i : l ~ [f i [s](x)]2 dx , s=l,2, ••• ,p,
(6.3)
«0[· s,s ']»
Proof:
-1
Under {H land FEr, it can be shown that (l/n)Ei"nl{E(F
n
=
I\"
tends to the null matrix as nt co •
7.
!
O.
i
I\" I\" 0
)} - E
I\"
Thus, we have, by Theorem 4.1, that E(R*)-E
I\"
1\"1\,,
converges in probability to the null matrix as ntoo.
IE{n~(T;-(k+l)/2)}-a;Asl
.
Also, under {H },
n
The theorem now follows from Theorem 6.1.
ASYMPTOTIC POWER EFFICIENCY OF
tn
For the purpose of studying asymptotic relative efficiency (A.R.E.), we
will confine our attention to the well-known likelihood ratio test (l.r.t.)
discussed in Anderson [1, pp. 187-210].
Unlike many of the parametric tests
of H , the non-null asymptotic properties of the l.r.t. are known.
o
In fact,
under the sequence of alternatives {H }given by (6.1), the l.r.t. statistic,
n
U , can be shown to be asymptotically non-central chi-square with p(k-l)
n
k
--1
degrees of freedom and non-centrality parameter AU2 = Ej=l~j
~ ~j' where
--1
-1 n n -1
.
~j"
j=l,2, ••• ,k is given by (1.2), ~
= (n Ei .. lAi)
, - and 1\11.
A. = Var(x
.. ) •
.u
.u
1\1
1\11.J
Thus,·the A.R.E. of
tn
with respect to U can be taken as
n
(7.1)
In the univariate case, Sen_[9] has shown that e£ U ~ O.864k/(k+l)
n' n
uniformly in ~l,F2, ••• ,Fn' for absolutely continuous Fi , i=l,2, ••• ,n, with
finite, non-zero variances.
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In the multivariate case, e~ U depends upon a~, ~,
"'oJ
n' n
·v
Fand
AS, and,
·v
hence, a bound would be very difficult to establish.
For the special case when ~i = A~ ~ and ~j has identically distributed
components, it is easily shown that eL u reduces to that of the univariate
n' n
case and, hence, the results given by Sen [9] apply.
Sen [9] studied the A.R.E. of Friedman's X:-test for the case Fi(x)=F(x/a ),
i
i=1,2, ••• ,n. These results may be extended to the multivariate case. We have
k
2
AL ~ Lj=l~jt
s s'
-1
~j' where { .. ((a[s,s,]/A A
n
»
and, under Fi[s](X)=F[s](X/Ai[s])'
we find that
where
co
S
B
..
k_L
[f[s](x)]2 dx and A~[s] = (~i)s,s.
2
2
~-l
_-1 -1
2
= c ~ , where As =
Now, if ~i = ci~' c i > 0, then Ai[s] = ciA s and ~
-
and c = n
-1
n
Li=lc i •
Thus,
- -1 n
r- 2 k
-1
k
-1
= c[n Li l(l/vc i.)] Ej la~r* a~/Ej laj'A a.,
..
= "'oJ'"
"'oJ
= '" '" '"J
where
But
=
h- l
and hence,
with equality holding only when cl=c2= ••• =cn~ Thus, under the stated con.
.
ditions, e
u
is a minimum when c =c =••• ·c and can be made larger than
n
l 2
t n' n
one by a suitable choice of c ,c , ••• ,c •
l 2
n
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Turning attention to another aspect of interest, suppose we make the
assumption of normality and equal block dispersion matrices, then for p=2
we have
where
is the common dispersion matrix.
It can be shown that the extreme values of
etn,u are obtained when 0jl = 0j2 for j=l,2, ••• ,k or when 0jl = -Oj2 for
n
j~l,2, ••• ,k.
Thus
max
°jl,Oj2
j=l,2, ••• ,k
3k(l+p)/TI(k+l) (l+p*),
for p
~
0
3k(l-p)/TI(k+l) (l-p*),
for p
~
0,
3k(l-p)/TI(k+l)(l-p*),
for p > 0
3k(l+p)/TI(k+l) (l+p*),
for p
and
min
°jl,Oj2
j=l,2, ••• ,k
~
o.
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It is easily seen that, for fixed k,
and is concave for -1
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P
~
Using the inequality -1
~
~
0 and 0
3k/n(k+l) ~ eM(p,k) ~ .966.
p
~
is symmetric about p=O
It can be shown that
1.
The lower bound is obtained when Ipl~l or p=O.
~
3T-2 P
g
1, .given in Kendall [6, p. 12], it can
In view of this, and
be shown that eM(p,k) is monotonic increasing in k.
max lim
p k~ eM(p,k)
after some numerical calculation, we find that
occurs when p=.52.
The following table gives the quantity
=
.966 and
max
p eM(p,k) for
some values of k and gives the corresponding A.R.E. for the univariate
k
2
3
4
5
6
.725
.790
.828
.852
.869
P
.69
.68
.67
.67
e X2 F
r'
.637
.716
.764
.796
max
0
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I-
~
~(p,k)
eM(p,k)
10
20
100
.881
.905
.934
.960
.966
.66
.66
.64
.61
.54
.52
.819
.836
.868
.909
.945
.955
7
It is worth noting that the A.R.E. for X2 is smaller than that for
r
00
tn ,
under these assumptions.
Turning our attention to e (p,k), we see that, as a function of p, it
m
is symmetric about p=O.
Also, we see that e (p,k) is monotinically decreasing
m
in Ipl and increasing in k.
o~
em(p,k)
~
.
Finally, we find, by direct analysis that
J::'"
~3/4 ~
3k/n(k+l) and that
lim
k+oo em(p,k)
are obtained when Ipl~l and the upper when p=O.
~
3/n.
The lower bounds
In practice, having Ipl near
unity will lead to degenerate test statistics.
Hence, more meaningful bounds
can be obtained by restricting the range of p.
Chatterjee [3] has studied
the case p=k=2 so we will -confine our attention to k
table gives values of p
o~
Ip I ~ Po and k ~ 3.
o
~
3.
The following
and m(p ) where m(p ) < e (p,k) < 3k/n(k+l), when
0
0
-
m
-
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Po
m(p )
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
.69
.67
.64
.61
.57
.52
.47
.40
.30
.00
These results follow from the stated monotonicity properties and some direct
calculations.
Results similar to these have been obtained by Bickel (2].
Finally, if we assume that
~ij
follows a p-variate normal distribution
with dispersion matrix
1 P
A
'"
2
=A
p l ••• P
p
then e
t n' un = 3(1-p)/n(k+l)(1-p*)
P
P
1
and hence the bounds developed above may
be applied.
8.
NUMERICAL EXAMPLE
To illustrate the use of the test, bivariate normal data was generated
and recorded in a two-way table with k=3 and n=lO.
Then 1 was added to
each observation (both variables) in the second group and 2 was added to
each observation in the third group.
table.
The results appear in the following
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Groups
1
Obs.
0.547
1
-0.575
2 1. 706
1.252
-0.288
3 -0.310
1.417
4
0.932
0.878
5
0.819
-0.680
6
0.497
0.056
7 -0.285
0.711
8
0.089
-1.335
9
-0.349
1.635
10
0.845
2
Rks.
1
1
2
1
1
1
3
1
2
2
1
1
1
1
2
1
1
1
2
1
Obs.
1.811
1.840
2.509
1.574
2.524
1.553
0.703
1.390
0.094
0.045
2.077
1. 747
0.542
0.760
0.269
1.076
1.545
1.471
0.200
1.480
3
Rks.
2
2
3
2
2
3
1
2
1
1
2
3
2
2
1
3
2
2
1
2
Obs.
2.561
2.399
1.414
3.059
3.310
0.560
0.961
3.083
1.682
3.348
3.181
1.355
2.983
2.332
1.662
0.960
2.920
4.121
2.065
3.391
Rks.
3
3
1
3
3
2
2
3
3
3
3
2
3
3
3
2
3
3
3
3
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that
(k-1) , is
distributed as F
when p=2.
2c 2 ,2(c1-1)
Thus, we find that F "= 6.807 and is
0
significant at the .0005 level.
To compute ~ , we have
n
E(R*) = [1
0.4
~ ~
and so
0.4] E-1(R*)=[ 1.1905
1 '~~
-0.4762
1n = 10[(1.1905)(2.08)-(2)(0.4762)(0.79)]
so that
tn
~ext,
-0.4762]
1.1905
= 17.238.
But ~ ~ X~(k-1)
is significant at the .005 level.
using the same bivariate data, the observations in each block
were multiplied by a constant.
In fact, the multiplier for block 1 was 1,
for block 2 was 1.2, for block 3 was 1.4, and so on.
added to the observations of groups 2 and 3 as before.
Then constants were
In this case, the
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parametric analysis gave Un • 0.4854 and F0
= 3.700
which is significant at
the 0.25 level, whereas the non-parametric analysis gave
1n = 16.044
which is
significant at the .005 level.
These figures verify the results obtained in Section 7 pertaining to
ACKNOWLEDGMENT
The author is indebted to Professor P. K. Sen for suggesting this prob1em and for his help and guidance.
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REFERENCES
[1]
Anderson, T. W., An Introduction to Multivariate Analysis.
John Wiley and Sons, Inc., 1965.
New York:
[2]
Bickel, P. J., "On Some Asymptotically Nonparametric Competitors of
Hote1ling's T2 ," The Annals of Mathematical Statistics, 36(1965),
160- 73.
.
[3]
Chatterjee, S. K., "A Bivariate Sign Test for Location,"
of Mathematical Statistics, 37(1966), 1771-82.
[4]
Van E1teren, P. H. and Noether, G. E., "The Asymptotic Efficiency of
the ~-Test for a Balanced Incomplete Block Design," Biometrika,
46(19!59), 475-7.
[5]
Friedman, M., "The Use of Ranks to Avoid the Assumption of Normality
Implicit in the Analysis of Variance," Journal of the American
Statistical Association, 32(1937), 675-99.
[6]
Kendall, M. G., Rank Correlation Methods.
Company, Limited, 1955.
London: Charles Griffin and
[7]
Rao, C. R., Linear Statistical Inference.
Sons, Inc., 1965.
New York:
[8]
Sen, P. K., "On a Class of Nonparametric Tests for MANOVA in Two Way
Layouts," To appear in Sankhya.
[9]
Sen, P. K., "A Note on the Asymptotic Efficiency of F.riedman' s X:-Test,"
To appear in Biometrika, 54(1967).
The Annals
John Wiley and