ON A CLASS OF NONPARAMETRIC TESTS FOR
INTERACTIONS IN FACTORIAL EXPERIMENTS
by
Pranab Kumar Sen
University of North Carolina
Institute of Statistics Mimeo Series No. 525
May 1967
Work supported by the National Institute of Health,
Public Health Service, Grant GM-12868.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
ON A CLASS OF NONPARAMETRIC TESTS FOR INTERACTIONS IN FACTORIAL EXPERIMENTS*
By
Pranab Kumar Sen
University of North Carolina at Chapel Hill
1.
Summary and introduction.
This paper deals with a class of permutationally
distribution-free aligned rank order tests for interactions in factorial experiments replicated in complete blocks.
The asymptotic power-efficiencies of the
proposed tests with respect to the classical analysis of variance test are also
studied.
Nonparametric analysis of variance tests, available in the literature,
mostly relate to one way or two way (without interaction) layouts.
Though the
approach of Lehmann (1964) (see also Puri and Sen (1966)) can be adapted to construct tests for interactions in factorial experiments, the necessity of avoiding
incompatibility of the unadjusted estimates as well as of estimating some functional
of the parent distribution ( appearing in the expression for the dispersion matrix
of the estimators) makes such tests only asymptotically distribution-free and
somewhat tedious to apply.
In the present paper, the theory of aligned rank order
tests based on Chernoff-Savage (1958) type of statistics, developed in Sen (1966b),
is further extended to provide suitable tests for interactions in factorial layouts
with equal number of observations per cell.
Under certain permutational invariance
arguments the nonparametric structure of the proposed tests is established.
tests are also free from the other two difficulties mentioned earlier.
These
Further,
using a generalization of Chernoff-Savage (1958) theorem on the asymptotic normality
of rank order statistics to aligned observations, the asymptotic power-efficiencies
of the proposed tests (along with certain bounds are studied.
2.
Preliminary notions.
We shall consider in detail only the case of replicated
~
two factor experiments with one observation per call and indicate briefly the theory
* Work supported by the National Institute of Health, Public Health Service, Grant
GM-12868.
-2 -
for the case of several factors and/or observations per cell.
The chance variable
Y. 'k associated with the yield of the plot placed in the ith replicate and receivq
ing the combination of the jth variety (or level of the first factor) and the k-th
treatment (or level of the second factor), is expressed, in accordance with the
usual fixed-effects model, as
(2.1)
Y
= Il +V + 'k + Tjjk + U
'
j
ijk
ijk
i
where Ill" .. ,Il
'I"
n
i=l, ... ,n; j=l, ... ,p; k=l, ... ,q,
stand for the replication-effects, VI" .. ,V
""q for the treatment-effects, Tlll'" "Tl
pq
In (2.1), we may put
q
P
p
q
q
L: V.=O,
L: 'k=O,
L: Tj'k=O,
j= 1 J
k= 1
j=l, ... ,p, and
k= 1 J
It is assumed that (U.
~
ll
for the variety-effects,
for the variety x treatment inter-
actions, and U. 'k's are the residual error components.
(2.2)
p
L: Tj'k=O, k=l, ... ,q.
j= 1 J
' ... ,U.
~pq
), i=l, ... ,n are independent and identically dis-
tributed stochastic vectors having a common continuous (joint) cumulative distribution function (cdf) F(x
ll
, ... ,x
pq
) which is symmetric in its pq arguments; this
includes the conventional assumption of independence and identity of distributions
of all the npq error components as a particular case.
(2.3)
We want to test
H:
o
against the set of alternatives that ....
T is non-null.
By means of the following
intra-block transformations, we eliminate the nuisance parameters Vj's and 'k's.
Let us consider the p x q matrices
(2.4)
where we define
(Z"k) and E. = (E. 'k)' i=l, ... , n,
~J
.... ~
~J
-3 -
(2.5)
(2.6)
I
Nt
1
i= 1, ... , n;
q "" q"'q ,
Z. = (I
"'p
llP£ )Y.(I
- £ 1£ )
E. = (I
l£I£ )U.(I
l £ 1£ )
N~
"'p
"'~
P ... p;Jp .... ~ ....q
being the identity matrix of order t and £
... t
elements equal to unity, t 2 1.
i=l, ... ,n,
q "'q"'q ,
p "'p"'p .... ~ "'q
the (row) t-vector having all the t
Then from (2.1) through (2.6), we obtain that
Z. = "oJT + ....E., i=l, ... ,n.
(2.7)
,..,~
~
In the sequel, we shall work with the nuisance parameter-free model (2.7).
we will only consider the case when p, q 23.
Also,
If either of them is less than 3,
Suppose q=2, p > 2, then from (2.1) and (2.2)
the situation simplifies as follows.
we have Tljl = -Tlj2 = Tlj (say), j=l, ... , p; thus (2.3) reduces to H~:
Tll=" '=Tl
p
= O.
Again from (2.5) and (2.6), we have
(2.8)
= Z.. (s ay) ,
~J
= e .. (say) for all i=l, ... ,n.
e .. 1 =
q
~J
It follows from lemma 3.1 [to be proved in section 3J that (e. , ... ,e. ) are
~l
~p
symmetric dependent or interchangeable random variables for all i=l, ... ,n.
Con-
sequently, based on the set of observations {Z .. , j=l, ... ,p; i=l, ... ,n}, the problem
~J
of testing H
o
in (2.3) reduces to that of testing the interchangeability of
(Z'l" .. ,Z. ) (for all i=l, ... ,n), against shift alternatives.
~
~p
of Sen (1966b) will directly apply, and the details are omitted.
As such, the results
If p=q=2, we have
Tlll = -Tl12 = -Tl2l = Tl22 = Tl (say), and
= Z.
(2.9)
~
(say), i=l, ... ,n,
and it is also easily seen that the distribution of Z. is symmetric about Tl'
~
the problem of testing H
o
So
in (2.3) reduces to that of testing the symmetry (about zero)
-4-
of the distribution of Z.; this is the well known one sample location problem, and
~
hence, is not discussed.
3.
We define U. and ....E. (i=l, ... ,n) as in (2.4)
The basic permutation principle.
...~
~
and (2.6), and let F*(E) be the joint cdf of E. for i=l, ... ,n.
....
and~,
be any permutation of (1, ... ,p)
so that j
'"
E
J.
I
... p
rs
....
p
the set of all possible (p!) permutations,
Also let
...
(3.1)
where 5
Let j = (jl'" .,j )
~
2 =
(.)
(5 ~Jk
.. ).~, k-l
- , ... , p '
is the usual Kronecker delta.
Now for any j
E
N
J, I (j) is non-singular
.......
p ...
and has a unique reciprocal I (j*) (say), which also belongs to the set
"'p ....
(I (j): j
""p ""
...,
E
J}.
~
Further, it is easily seen that if I (jl) and I (j2) be defined
"'p ...
E ~,
as in (3.1) for 21
Thus the set (I (j): j
"'p...
....
E ~,
22
E J}
...
IVp ...
then I (jl)I (j2) also belongs to (I (j): j
"'p... -p...
""p ru
E
J}.
ru
IV
forms a finite group of elementary transformations.
It can also be verified that
(3.2)
I (j) (I - 1:. t 't ) I (j*) = I
"'p.... "'p
p ... p"'p "'p ...
"'p
Similarly, let ...
k = (kl, ... ,k q ) be any permutation of (1, ... ,q); the set of all
possible (q!) permutations is denoted by K.
....
A second group of elementary trans-
formation matrices is then defined by
(3.3)
(I (k): k
",q'"
E
K}
where I (k) = (5' k ). n-l
.
..q'"
~t~,,lj-,···,q
......
Let us then define a finite group
1?i
!i
of transformations (g.(j,k): j
1""",
N
by
E.(j,k) = g.(j,k)E.
N
~
....
,...
~
"'
....
"'~
(3.4)
= I (j)E.I'(k), j
I'lP '"
"'~'\)q'"
'"
E
J, k
",'"
E
K, for i=l, ... ,n.
N
E
J, k
'"
E
K}
.....
-5 -
Finally, the group of all the (p1q,)n transformations in (3.4) is denoted by
i.
~~
e.,
(3.5)
As before, we denote the cdf's of U. and E. by F and
"'~
"'~
~~,
respectively.
Let now
~ be the class of all pq-variate continuous cdf's for which the pq variates are
By definition (in section 2), F E ~ .
interchangeable.
LEMMA 3. 1
PROOF.
If FE:!, F* is i-invariant.
On defining U., I (j) and I (k) as in (2.4), (3.1) and (3.3), we let
"'~
",p rv
ruq ru
(3.6)
U.(j,k) = I (j)U.1'(k), j E J, k E K.
IV ~
Since F E
~,
I\J
f"\J
....
p
'\J
N~"'q
""
II.>
'U
rv
tV
it remains invariant under row (or column) permutations.
U.(j,k) has the same cdf F for all j E J, k E K.
1\11.
1\1
IV
'V
IV
fI"U
Hence,
Now, from (2.6), (3.2) and (3.4),
f"\.)
we obtain that
E.(j,k) = I (j)R.1'(k)
""~
'"
"'p
tU
tV
~~.vq
#oJ
=1 (j)(1 - 1. £ '£ )U. (I
....p N rvp
p ,..., p"'p ""1. ""q
(3.7)
= I (j )( I
""p rv
p
=
(I
"'p
- 1.£'£
)1'(k)
q ..... q"'q '" q
tV
_.!.p ....£ p...p
'£ ) I (j *) I
rvp
~p
i<J
(j) U. I ' (k) I , (k*)( I
.... "'~"'q '" ... q....
~q
_.!.p ...£ p"'p
'£ )U. (j k)(1
"'~ ,../...;
... q
Thus, the invariance of the cdf of U.(j,k) (under 1?) implies the invariance of
"'~
the cdf E.(j,k) under~.
"Wl.
IV
Oi
'"
IV
(ti
,.,
Hence the lemma.
Let now z* be the npq-dimensional (Euclidean) space of the sample point
n
Z* = (Zl""'Z),
'Un
"V
-n
Then the finite group (~*n) of transformations in (3.4) and
T
(3.5) maps the sample space onto itself, and under H
o
in (2.3), the distribution
-6 -
of 'Vn
Z* is ...,.
~*-invariant.
n
Thus, proceeding as in Hoeffding (1952, pp. 169-170), we
for all F
E
:l
(0 <
~
may prove the existence of similar size
~
< 1) tests for Ho in (2.3), valid
These tests are essentially conditional tests based on the considera-
tion of equiprobable all possible row and column permutations of the matrices
Zl" .. ,Z.
'"
Such a conditional test is termed a £ermutation test.
-un
In this paper,
we shall study permutation tests based on a celebrated class of rank order statistics
due to Chernoff and Savage (1958).
4.
Formulation of the rank order tests.
Let c(u) be 1,
t
or 0 according as u is
>, = or < 0, and let
(4.1)
R. .k =
q
t
+
n
L:
p
L:
q
L: c (Z. .k - Z
r=l s=l t=l
~J
rs
t)' i = 1, .. . , n ; j= 1, .. ., p; k= 1, .. . , q ;
by virtue of the assumed continuity of F, ties among Z. 'k's may be ignored, in
~J
probability.
We define a sequence of real numbers ~N = (SN1"",SNN)' where
S~ = IN(~I(N+l)),
(4.2)
the function I
N
1 < ~ $ N;
is defined as in Chernoff and Savage (1958) and is assumed to
satisfy the regularity conditions of theorem 1 of Chernoff and Savage (1958).
Let
n
(4.3)
L: SNR
i= 1
Also, let
(4.4)
for j=l, ... ,p, k.=l, ... ,q.
ijk
-7-
s"1(
(4.5)
NR.~J'k
= S
NR.~J"k
P
1 L: S
P j=l NR ijk
for i=l, ... ,n; j=l, ... ,p, k=l, ... ,q.
is
1:.
q k=l NRijk
+-.1:.. ~
is
pq j=l k=l NR ijk
Then, from (4.3), (4.4) and (4.5), we obtain
n
~.jk
(4.6)
= 1
n
L: S"1(
. 1
~=
NR.~Jok'
j=l, ... ,p, k=l, ... ,q.
We denote by ~N' the permutational (conditional) probability measure induced by
the (prqr)n equally likely transformations in ~~, defined by (3.4) and (3.5).
Then, by simple arguments it follows that
(4.7)
Also, let
1
(4.8)
and let
n(p-l)(q-l)
®
n
L:
p
q
L:
L: ( S"1(
i=l j=l k=l
)2
,
NR ijk
stand for the symbol for Kronecker product of two square matrices.
Then
by routine computations, we obtain that
(4.9)
Thus considering the generalized inverse of the pq x pq matrix in (4.9) and employing it to construct a quadratic form in the elements of
test statistic
(4.10)
~,
we derive the following
-8 -
which is analogous to the classical parametric test based on the variance ratio
criterion [cf. Scheffe (1959)J.
For small values of n, p and q, the exact permutation distribution of ~N
can be obtained by considering the (plql)n (conditionally) equally likely row and
column permutations of the matrices
~~i)
=(SNR
), i=l, ... ,n.
ijk
becomes prohibitively laborious for large values of n, p or q.
This procedure
For this reason,
we consider the following large sample approach.
Let us denote the marginal cdf of Zijk by F[jkJ(x) and the joint cdf of
(Zijk,Zijlk l ) by Fejk;jlkl](X'y) for all j,jl=l, ... ,p; k,kl=l, ... ,q.
H(x)
(4.11)
(4.12)
(4.13)
=
1
P
q
I: Fe 'kJ (x) ;
pq j=l k= 1 J
1
HlO(x,y) = qp(p-l)
1
HOI (x, y) = pq(q-l)
1
= p(p-l)q(q-l)
(4.14)
Let then
I:
q
p
I:
I:
k=l j 'f j= 1
p
q
I:
I:
j=l kfkl=l
P
I:
F['k"'k](x,y);
J ,J
F[-k' 'kl](x,y);
q
I:
J ,J
jfjl=l kfkl=l
F*[ 'k' 'Ikl](x,y).
J ,J
lim
We denote by J(u) = N~ IN(u): 0 < u < 1, and define
(4. 15)
00
00
J J
-00
J[H(x)]J[H(y)]d[HlO(X'y) + HOl(x,y) - Hll(x,y)].
-00
Then, proceeding as in the proof of theorem 4.2 of Puri and Sen (1966) and
omitting the details, we obtain the following theorem.
-9 -
THEOREM 4.1
[a 2
(
P N)
-
Under the conditions of theorem 1 of Chernoff and Savage (1958),
converges in probability to zero.
(52J
Now, if we assume that
(4.16)
for at least one pair of jr jt and krk', then as in theorem 4.1 of Sen (1966b), it
can be shown that 0 2 ,
defined by (4.15), is strictly positive.
(4.16) will be
termed, in the sequel, as the non-degeneracy condition of the cdf F*,
The main
theorem of this section is the following.
THEOREM 4.2
Under the conditions of theorem 1 of Chernoff and Savage (1958), the
LN converges
permutation distribution of
~1)(q-l)
PROOF.
to a chi-square distribution with
degrees of freedom (d. f.).
By virtue of (4.7), (4.9) and (4.10), it suffices to show that for any nonp
1
q
null A = (a 'k)' Y = n2 I:
I: a 'k~ 'k converges in law (under ~) to a normal
IV
J
n
j= 1 k= 1 J
, J
distribution as n~. Now, using (4.4), (4.5), (4.6) and the first two conditions
of theorem 1 of Chernoff and Savage (1958), we write
Y
(4.17)
n
=n
_.1.
2
n
~
Y , +
, 1
~=
n~
0
p
(1); Y .
n~
=
p
~
q
~
.
a~:
j= 1 k= 1 J k
J(Rijk )
N+1
i= 1, ... , n,
'
where a* 's are linear functions of ajk's and they satisfy the constraints that
J'k
q
p
tP.N, Y .
a* = 0, j=l, .. "p and ~ a'1: = 0, k=l, ... , q. We note that under
j= 1 J k
n~
k=l jk
can have only p1q1 (conditionally) equally likely values obtained by permuting the
~
rows and columns of the matrix ""~
R. = (R.~J·k).-l
k=l
'
J-, ... , p, - , . , . , q
are all stochastically independent (under
E(Y n .'
~
CPN)
=
0,
and
dPN).
and Yn 1'"'' Ynn
Thus, it readily follows that
-10-
(4.18)
ijk
i
(i
J/
P k=l j=l N+l
1:.
))2
Thus, by routine analysis, it follows as in theorem 4.1 that
(4.19)
where 6 2 is defined by (4.15) and is positive by (4.16).
Further, using the
growth condition of theorem 1 of Chernoff and Savage (1958), it follows that
1 ~ E(/Y ./2+6 1 (9N) <
n i=l
n1
(4.20)
00,
for some 6 > O.
Consequently, using the Berry-Esseen theorem (cf. [ 4 , p. 288]), the asymptotic
normality of Y
n
follows from (4.19) and (4.20).
Hence the theorem.
By virtue of theorem 4.2, an asymptotically size
~(O
hypothesis of no interaction may be proposed as follows.
If
~N
(4.21)
where X2
t,~
5.
is the upper
>
-
X2(p-1)(q-1),~'
<
X (p-1)(q-1),~ ,
10~i.
reject H
2
a
<
~
< 1) test for the
in (2.3),
accept H ,
o
point of a chi-square distribution having t
Asymptotic efficiency of the test based on~.
the test in (4.19) is consistent for any non-null! =
d.f . .
It can be easily shown that
(~jk)'
For the study of
-11-
the asymptotic efficiency of the test based on ~N' we shall therefore consider
the following sequence of
specified by
P~tman-alternatives,
(5. 1)
T = T
IL:
-""N
'" N
.....
= N
_l..
2
A, A = (A
",.v
)
jk '
where Ajk'S are real and finite and they satisfy
p
(5.2)
q
L: A. k = 0, k= 1, . . . , q ; L: A. k = 0 for j= 1, . . ., p.
j=l J
k=l J
-l..
Thus, under (H }, the cdf of Z. (defined by (2.5),) is specified by F*(x - N 2A)
-""N
I\>
~
'"
"V
'
(where x is a p x q matrix), and F*(x) is invariant under the row or column per~
.....
mutations of~.
_1
Thus, the univariate marginal cdf Ftjk](x) (of Zijk- N 2 Ajk ) is indepen-
dent of (j,k) and is denoted by H(x) [cf. (4.11)].
Similarly, the bivariate
marginal cdf F[jk,jk'](x,y) will be independent of (j,kfk') and is denoted by
H01(x,y) [cf. (4.l3)]J
FLk,j'k](x,y) will be independent of (jfj',k) and is denoted
by H10(X'y) [cf. (4.12)], and F[jk;jtkt](x,y) will be independent of (jfj',kfk t )
and is denoted by Hll(x,y) [cf. (4.14)].
1
Now, for arbitary F*(~), the asymptotic normality of N2(T
-~ ) (where
'lIN .... N
'v
00
~N = (~N, jk)' ~N, jk =
_£ J[ H(x)] dFt jk] (x),
j= 1, ... , p, k= 1, ... , q) can be proved
along the same line as in the proof of theorem 5.1 of Sen (1966a) (with directextension to the matrix case).
holds.
We shall specifically consider the case when (H }
N
For this, we define
00
(5.3)
f
B(H) =
d
dx J[H(x)]dH(x),
-00
1
(5.4)
A2 =
J
0
1
J 2 (u)du -
[J
0
J(u)duJ2,
-12 -
00
= A;
(5.5)
-00
for (i,j)
=
00
{J
J J J[H(x)]J[H(y)]dH .. (x,y)
[
~J
-00
1
J(u)du }2J,
°
(0,1), (1,0) and (1,1), where H.. 's are defined earlier.
~J
same technique as in theorem 5.1 of Sen (1%6a), we obtain that under
Then, by the
(~},
_1
[(pq) 2B(H)] A + 0(1)
(5.6)
tV
(5.7)
n E(.!]
® ~I~}
= (I
.up
-
1:.p ....Pt
) ®
P"'P
(I
",q
where B(H), A2 , p .. 's are defined in (5.3), (5.4) and (5.5).
~J
Again, using (4.15),
theorem 4.1 and some routine computations, it follows that under
(~}
in (5.1)
(5.8)
1
(5.6), (5.7), (5.8) and the asymptotic normality of n2 TN .k lead to the following
,.. ,J
theorem.
TH~O~~
If (i) (~} in (5.1) _holds, (ii) the conditions of theorem 1 of
Chernoff and Savage (1958) hold and (iii) the conditions of lemma 7.2 of Puri (1964)
hol~
Sv N,
defined by (4.10), has asymptotically a non-central chi-square distri-
bution with (p-1)(q-l) d.f. and the non-centrality parameter
(5.9)
Referring back to the model in (2.1), let
p
u
F
02
u
be the variance of U..
~J
be the correlation between any two U.. k's belonging to the same block.
~J
(p-l)(q-1), (n-1)(pq-1)
and
Let
be the classical analysis of variance ratio test statistic
for testing H in (2.3) when the parent cdf is assumed to be normal.
o
k
Then, it can
-13 -
be shown that under {HN} in (5.1), QN = (p-l)(q-l) F(p-l)(q-l),(n-l)(pq-l) has
asymptotically a non-central chi-square distribution with (p-l)(q-l) d.f. and the
non-centrality parameter
(5.10)
Let now o~ be the variance of e
ijk
, defined by (2.6).
Then, after some simplifi-
cations, we obtain from (2.6) that
02
e
(5.11)
=
[(p-l)(q-l)!pq]02(1-p ).
u
u
Consequently, from theorem 5.1, (5.10) and (5.11), we arrive at the following.
THEOREM 5.2
When the conditions of theorem 5.1 hold, the aSymptotic relative
~fficiency (A.R.E.) of the
~-test with respect to the classical analysis of
variance test is given by
(5. 12)
We note that 0 2 is the variance of the cdf H, and hence the second factor on the
e
right hand side of (5.12) resembles the usual efficiency factor for the well-known
Chernoff-Savage (1958) type of test statistics.
Also, it follows from lemmas 4.4
and 4.5 of Sen (1966b) that
(5.13)
P10 2: -l!(p-l)" POl> -l!(q-l),
where the equality sign holds iff J
= H- l
from (5.12) and (5.13), we obtain that
(apart from an additive constant).
Thus,
-14-
(5.14)
This leads to the following corollary.
COROLLARY 5.2.1
as
[a~B2(H)!A2J
e({~N}'
A sufficient condition for
{QN} to be at least as large
is that Pll< l!(p-l)(q-l).
We shall now consider two special ~N-statistics, namely, the Normal Scores
and Wilcoxon scores statistics.
In the first case, IN(N~l)' defined by (4.2), is
the expected value of the a-th smallest observation of a sample of size N from a
standard normal distribution, for
1, ... ,N.
OF
In this case, it is well-known
[cf. Chernoff and Savage (1958)J that a~B2(H)!A2 is greater than or equal to 1,
where the equality sign holds only when H is also normal.
Thus, the minimum
A.R.E. of the normal scores test with respect to the classical analysis of variance
1
1
test is equal to l!{l - -- [l-(p-l)(q-l)PllJ}>
1
1 >~.
pq
2-(- + -)
P
for the class of parent cdfrs for which P
ll
~
ll
=
l!(p-l)(q-l) and a:B 2 (H)!A2
=
In particular, if H(x) is normal,
1, so that the normal scores test and the
QN-test become asymptotically power equivalent.
N~l:
1 < a < N.
to 0.864 for all H.
q
l!(p-l)(q-l), the normal scores test
will be at least as efficient as the QN-test.
P
On the other hand,
For Wilcoxon scores, IN(N~l)
=
In this case, [a~B2(H)!A2J is known to be greater than or equal
Consequently, the A.R.E. of the Wilcoxon scores test with
respect to the QN-test is bounded below by
1
(5.15)
0.864!{1 - pq [l-(p-l)(q-l)PllJJ > 0.432 .
For normal F, it is known that
(5.16)
PlO
=;6 S'1n -1
(
-1
)
2(p-l)' POI
=7r6
S' -1 (
1n
-1
)
2(q-l)' Pu
1
)
=:rr6 S in -l( 2(p-l)(q-l)'
-15 -
and hence from (5.12), we obtain that for normal distributions the A.R.E. of the
Wilcoxon-scores test with respect to the QN-test is given by
3pq
l
§. {S1'n1
+ S1'n- l
1
S' -1
1
}]
( 1)( q- 1)[1 + IT
IT p2(p-l)
2(q-l) + 1n
2(p-l)(q-l)
(5.17)
The following table illustrates the numerical values of (5.17).
~
2
2
.955
3
4
5
6
7
8
9
3
.966
4
.965
5
.963
6
.962
7
.961
8
.960
9
.960
10
.959
15
.958
20
.957
.955
.975
.974
.972
.971
.961
.970
.970
.969
.968
.968
.966
.972
.971
.970
.970
.969
.969
.968
.967
.966
.965
.970
.969
.968
.968
.967
.967
.966
.965
.963
.969
.967
.966
.966
.966
.964
.964
.962
.966
.966
.965
.965
.964
.963
.961
.965
.964
.964
.963
.962
.960
.964
.964
.962
.962
.960
.963
.962
.961
.959
.961
.960
.958
.959
.957
10
15
20
00
.955
00
Thus, the efficiency is bounded below by 3/IT and may be as high as 0.974.
If we have more than one observation per cell, we may still work with the
aligned observations obtained by making adjustments for row, column and grand means.
The permutation argument is essentially the same (with p and q replaced by pr and
qr, respectively, r being the number of observations per cell).
For more than two
factors" summing over a subset of factors, we may arrive at the desired nuisance
parameter-free model and proceed as in this paper.
omi tted.
For brevity, the details are
-16 -
REFERENCES
[ 1J
CHERNOFF, H., and SAVAGE, I. R. (1958).
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[2J
HOEFFDING, W. (1952). The large sample power of tests based on permutations
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[3J
LEHMANN, E. L. (1964). Asymptotically nonparametric inference in some
linear models with one observation per cell. Ann. Math. Statist.
~ 726-734.
[4J
LOEVE, M. (1962).
[5J
PURI, M. L. (1964). Asymptotic efficiency of a class of c-samp1e test.
Ann. Math. Statist. ~ 102 -121.
[6J
PURI, M. L., and SEN, P. K. (1966a). On a class of multivariate multisample rank order tests. Sankhya, Ser A. ~ 353-376.
[7J
PURI, M. L., and SEN, P. K. (1966b). Some optimum nonparametric procedures
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[8J
SEN, P. K. (1966a). On some nonparametric generalizations of Wilks' tests
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and ~C' I. Inst. Statist. Univ. North Carolina Mimeo
VC
Probability Theory.
Asymptotic normality and efficiency
Ann. Math. Statist. ~ 972-994.
D. Van Nostrand Co.
Report No. 468, March 1966.
[9J
SEN, P. K. (1966b). On a class of aligned rank order tests in two way
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