Download Roy, S.N.; (1954)A new test of compound symmetry." (Navy Research)

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NEW TEST OF COMPOUND SYMMETRY
by
S. N. ROY
INSTITUTE OF STATISTICS, UNIVERSITY OF NORTH CAROLINA
Institute of Statistics
Mimeograph Series No. 97
March 11, 1954
1 Work sponsored by the Office ot Naval Research under Contract No.
N7-onr-28402.
A NEW TEST OF COMPOUND SYMMETRY
by
S. N. ROY
Institute of Statistics, University of North Carolina
1.
Summary.
If xl and x 2 have a bivariate normal distribution with
a correlation coefficient
p
and the same standard deviations a tor both,
then it is well known and easy to check that xl + x 2 and xl - x 2 are uncorrelated, which forms the basis of Pitman's well-known test of H :
O
0
1 = O2 ,
tor a bivariate normal population, in terms of the correlation coefficient
r between xl + x 2 and xl - x 2 in a random sample of size, say, n, from this
population.
Starting from this test which has a number of reasonably good
properties, and then using the union-intersection principle ~l,
test is obtained for compound symmetry, i.e., for HO: all
and all 01j'S are equal (1
1j
2-7,
= 0 22 = •••
a
= app
= 1, 2, ••• , p), where 01j is any element
of the covariance matrix L of a p-variate normal population.
2.
Test Construction.
Let p(x , x ) denote the population correlation
2
l
coefficient between xl and x2 and r(x l , x 2 ) the same in a random sample
·of size, say n, from that population.
variables !'(l x p)(= xl' ••• ,
Then, for a set of stochastic
p ) having a p-variate normal distribution,
X
notice that tor any arbitrary non-null vector !'(l x p), !'(l x p) !(p x 1)
p
p
and L xi have a bivariate normal distribution and, now letting L a i = 0,
i=l
i=l
-2p
consider the hypothesis: ~( ~ xi' !'!)
i=l
(2.1)
= 0 = Rea
(say).
Next notice that
HO: all 0ii's are equal and all 0ij'S are equal (i
#
j
= 1, ••• , p)
p
where !'(1 x p) is any arbitrary row vector subject to
L a = O.
i
i=l
Now going back to HCa ' we have for this hypothesis the Pitman
critical region, say Wa (a), of size a, given by
(2.2)
2
wa (c.):
P
r ( L Xi' a'
i=1
2
? r~
!)
(n-2),
where r c (n-2) is the upper a/2-point of the central r-distribution in
random samples of size n.
Hence, by the union-intersection heuristic principle ~1, 2-7
we have, for HO( =
~aHca)'
the critical region
W(~)
of size
~
given
by
w(~):
n
a
-
2
P
> r C:2,(n-2) .J7
a'x)
Lr r ( . 1. Xi' - 1.=1
2
P
i.e. , Sup r ( L Xi' !'~)
a
i=1
i.e. ,
remembering that
p
I:
,
2
? rr(n-2)
v.
2 P
p-l
2
Sup r ( L Xi' I: ai(x. - x ) ) > r(n-2) ,
a
i=l
i=l
1.
P
- c
a.
i=l 1.
=
0, i.e., a
p
=-
p-l
L ai
i=l
-3It is easy to check that
2 p
p-l
Sup r ( ~ xi' t
a (x. - x ) )
a
i=l
1=1 i
~
P
(2.4)
= square
p
L xi and the (p - 1)-
of the sample multiple correlation between
i=l
2
P
R ~ L x. and (Xl - x ), (x2 - X ), ••• , (x 1 - x )
i=l ~
P
P
pP
-7
2
= R (say).
Notice that, since the new p-set also has the p-variate normal distribution,
this R has the well-known multiple correlation distribution w1th degrees of
freedom p - land n - p and a non-centrality parameter which is the population mUltiple correlation between
p
L
xi and the (p - l)-set above and
i=l
which let us call p2.
It is easy to check that p = 0, i.e., that R has
the central multiple correlation distribution, if and only if
p
p(
L Xi' Xi - Xp )
i=l
p
p (
lXi' !'!)
i=l
=0
=0
(1
= 1,
2, ••• , p - 1), i.e., if and only if
p
(for all non-null! subject to
~
.
L a = ~.
i=l i
We have thus, for testing compound symmetry, the critical region
W(~)
of size
~
given by
P
R ~ ~ Xi and (Xl - x ), ••• , (x 1 - x )
i=l
P
pP
when
R~
is the upper
distribution.
~-point
-7 >- R
Q
~
(p - 1, n - p),
of the well-known central multiple correlation
-4It can be checked after some little algebra that in terms, respectively,
of the elements of the sample and population covariance matrices Sand E,
Rand
p
will be given by
2
R
(2.6)
p
2
P
1
= 1 - P E z.z I
1=1
=1
J.
P
(£ zi)
1=1
P
( E
i
z ),
1=1
PiP
- P t ti~ I( t ~.)
i=l
1=1 1.
when
P
zi = r. Sij'
j=l
p
r.
=
~i
°iJ
J=l
Note that
8
ij = sJi
,
P
1j
1
z = I; s
J=l
p
and r i = l. oij
"
J=l
slJ = sji
,
°ij = 0.11 and
ij
(J
= oji •
The power properties of this test will be discussed in a later note.
References
1.
S. N. Roy, "On a heuristic method of test construction and its use 1n
multivariate analysis," .-100als of Mathematical Statistics, Vol. 24
(1953), pp. 220-38.
2.
S. N. Roy and R. C. Bose, "Simultaneous confidence interval estimation,"
Annals of Mathematical Statistics, Vol. 24 (1953), pp. 513-36.