Gertrude M. Cox
ON A TEST FOR THE MULTIVARIATE BEHRENS-FISHER PROBLEM
by
Koichi Ito
Work a.t Cha.pe1 Hill sponsored by the
Office of Naval Research under Contract
NR 042 031. Reproduction in whole or
in part is permitted for any purpose
of the United States government.
Institute of Statistics
Mimeograph Series No. 132
June 11, 1955
•
ON
A.
TEST FOR THE MULTIVARIATE BEHRENS-FISHER PROBLEM l
by Koichi Ito
University of North Carolina
1.
Introduction.
As a partial solution for the multivariate Behrens-Fisher prob-
lem, Bennett L-l_7 suggested an extension of Scheff6 ' s procedure
variate case.
L-6_7
to the multi-
The purpose of. this note is to show that Bennett I s test is the most
powerful among all tests obtainable under the procedure, and also to compare its
power with that of the ordinary generalized Student T test, when both tests are
used to test the same hypothesis.
2.
Extension of Scheffels procedure and a class D of tests for the problem.
Suppose that (Yir)' i
= 1,
••• , Pi r
••• , N2 , where it is assumed that N
l
/
=
= 1,
~
=
•• ', Nl and(ziS~ i
1, ••• , Pi S
1,
N , represent two independent random samples
2
from the p-variate normal populations N(.n,I)} and N<,~, £2)' respectively.
.!! and
~
are the p x 1 column vectors whose elements are the means of the p characteristics
of the two populations, respectively, and
~
(1)
= ( 0iJ ) and 1:2
(2)
= (OiJ
),
i,
j =
1,
... , p, are unknown covariance matrices of the two populations, respectively, both
being positive definite.
H:
Suppose moreover that 1:1 '" 1:2 "
The problem of testing a null hypothesis H : 1) = ~ against any alternative
O
not HO' is a Behrens-Fisher problem in the multivariate case. We shall apply
Scheffe's procedure to this case in a general way to obtain a class D of tests for
HO which make use of the generalized Student T statistic, and choose the most powerful test in D.
Consider a p x N matrix (x
), where N is to be determined later on, and whose
it
elements are defined as linear combinations of the elements of (Y. ) and (zi ) as
J.r
s
follows:
1
Work at Chapel Hill sponsored by the Office of Naval Research under Contract
NR 042 031. Reproduction in whole or in part is permitted for any purpose of
the United states government.
2
(2.1)
where i
= 1 , ••• " P · t
= 1 , ••• " N
and (a(i»
rt
and (b(i»
st
are N1 x Nand N2 x N
matrices, respectively, whose elements are non-stochastic constants independent of
popula.tion parameters.
They are determined subject to the following conditions:
and
for i, j = I, ••• , p; t, u
= I,
••• , N; where 5 tu
= 1 is
Then the null hyPOthesis HO: ]
L-2_7.
by the generalized Student T
=1
if t
equiva.lent to H~ :
!
=u
and 5 tu = 0 if t f u.
= 0, and can be tested
If we set
1
=i
1
(2.5)
siJ
where n
=N -
=-n
N
E
t=l
(X it - Xi.)(X Jt - XJ .)
1, then the test consists of the following critical region:
(2.6)
T
2
0:
FO(p,
n-~l)
tribution
= n-~l
np Fr.(P, n.~l),
.....
being the upper 100 a per cent point of the analysis of variance F dis-
Wi~
degrees of freedom p and
n-~l.
The power of this test is given by,
3
p(Alp, n-pt"l, ex)
=1
- e-
A
00
",h
E.,.
h=O h.
I
(1
1
-2P + h, -2(n - p + 1) ),
x
where
Now, taking expectation and variance of (2.1) we see that conditions (2.2) and
(2.3) are equivalent to
N
N
l
(i)
art
E
(2.8)
r=l
Nl
E
r=l
(i)
art
a(j)
r~
2
= 1,
E
s=l
b(i)
st
1,
=
N2
(2)
b(i) b(j)
=
°tuCij
su
6=1 st
(1)
= 0tu c ij ,
E
,
and
o
(2.10)
ij
= c(l) 0(1) + c(2) 0(2)
ij
iJ
ij
ij'
(1)
where i, j = 1, ••• , Pi t, u = 1, ••• , N, and c ij
Following arguments similar to Scheffe's
for
ai~)
and
b;~)
L-6_7,
to satisfy (2.8) and (2.9),
(2)
and c ij
are independent of t.
it is easily shown that in order
N,
ci~)
and
cl~)
must satisfy
(2.11)
and
(1) = c{l) >
c ij
_ !L
N
(2.12)
'
1
where c(l) and c(2) are constants independent of i and j.
Hence, we have obtained
a class D of tests of the type (2.6), each of which is given by a solution of (2.8)
and (2.9) for
a~~) and b~~) when N, c(l) and c(2) satisfying (2.11) and (2.12), re-
spectively, are specified.
specific alternative
1
=
~
We note here that the power of a test in D against any
-
1r
0 is determined by the values of N, c(l) and c(2) ,
and so our next problem is to choose N, c(l) and c(2)
4
~
so as to obtain the most powerful test in D.
3.
Determination of the most powerful test in D.
We shall first observe the follow-
ing two properties of the power function given by (2.7).
P(~
(i)
~,
I a,
a and
b, a) is a non-decreasing function of b for fixed values of
Q.
This is easily proved from Hsu's theorem
the power function of the analysis of variance F test.
fact, tables of p(~
I a,
b, a) (e.g., Tang
L-7-7)
L-4_7
concerning
As a matter of
indicate that p(~la,b,a)
is actually a monotonically increasing function of b for fixed h and a
when a is a usual level of significance, i.e., .05 or .01.
(ii) P(h
I a,
b, a) is a monotonically increasing function of ~ for fixed
values of a, b and a
L-5..1.
We shall next show that the parameter h in P(h
I
a, b, a) is a monotonically
decreasing function of c(l) for fixed c(2), and also a monotonically decreasing
function of c(2) for fixed c(l). We write:
N
h=2
where
l'
p
r.
i,j=l
denotes the transpose of
1.
Now, let I. = M.Diag(e , e 2 , ••• , ep).M' and I.2 = MM', where M is a p x p nonl
l
singular matrix, and Diag(e , e 2 , ••• , e ) is a diagonal matrix whose p elements are
p
l
the characteristic roots (all positive) of I. I.- l • Then, we get
1 2
Again, let
~
,
=
.1' M'
-1
• Then
5
c
(2)
, ••• ,
•
Hence, since e and 9 are independent of c{l) and c(2), our proposition follows.
i
1
It is now clear that the most powerful test in D against any specific alternative is obtained, when we ascribe to N its maximum value satisfying (2.11); i.e.,
N and then to c(l) and c(2) their respective minimum values satisfying (2.12), i.e.,
l
1 and Nl /N 2 • Hence the most powerful test in D is given by solving,
Nl
(3.1)
I.
r=l
and
N
l
...
(3.2)
'I;'
r=l
where i, J
(i)
art
(1)
art
a(j)
ru
=
N2
1,
I.
a=l
N
2
= ~\u'
E
s=l
b(i)
st
= 1,
b{i) b(j)
st
au
= ~\u
Nl
N2 '
= 1,
••• , Pi t, u = 1, ••• , N " Very simple solutions of (3.1) and (3.2)
l
are an immediate application of Scheffels solution which was proposed by Bennett
b~~) =
By substituting (3.3) into (2.1) we obtain
zis -
N
2
1
--N
2
L
a=l
Z4s'
~
6
where i
= 1,
••• , Pi t
= 1,
••• , Nl , thus the critical region of the most powerful
test in D being completely specified.
4.
Comparison of power.
In this section we shall compare the power of the test
given by (2.6) and (3.4) with that of an optimum test for the case where Ll
= L2 •
The latter test is an ordinary generalized Student T test whose critical region
consists of
(4.1)
where
and
The power of this test is given by
pp.. lp,
m-p+l, ex) where
and
The test (2.6) witb N
= N1 and
a
iJ
= ai~) + ; ; ai~) wbicb
S
is specified by (3.4)
is the most powerful in class D of tests for testing H : ] = under the assumption
O
that Ll ~ L2 , but since we deal with a less favorable situation where nothing is
7
known of 1: and r. , this test is presumed to be less powerful than (4.1). For the
2
1
sake of comparison we suppose that both tests are used to test H assuming that
O
~ .. 1: • Then it is easily seen that in the power functions of the two tests, the
2
Parameter ~ takes on the same value for any specified alternative of .!! and
.s.
Hence, given A and 0, we may comPare the two powers for various values of N and N2 •
l
The following tables give the ratios of powers of Bennett's test to that of (4.1)
for several values of
Table 1.
~,
0,
p, N and H • They are calculated by using Tang's table
1
2
Ratio of powers for
H2
°
I:
.05 (above) and for ex ... 01 (below) , p = 2, A =- 1.5
6
10
15
20
00
.730
.566
.669
.473
.633
.422
.615
.394
.547
.307
.847
.710
.818
.664
.801
.628
.724
.507
.904
.805
.889
.784
.817
.650
.936
.864
.863
.729
H
l
6
10
15
20
1.000
1.000
00
Table 2.
N
l
''7
•
10
15
20
00
Ratio of powers for ex = .05 (above) and for ex
N2 .
= .01 (below) , p = 3, A'" 2.
7
10
15
20
.667
.500
.620
.436
.778
.615
.576
.376
.739
.557
.859
.740
.554
.345
.716
.520
.843
.705
.904
.817
CO
.475
.250
.626
.390
.751
.555
.813
.652
1.000
1.000
8
Table 3.
Nl
8
Ratio of powers for ex = .05 (above) and ex = .01 (below)" p= 4 and A == 2.5
N2
10
8
10
15
20
00
.616
.449
.584
.404
.531
.342
.509
.310
.422
.211
.101
.533
.655
.461
.630
.429
.528
.300
.816
.669
.192
.636
.684
.468
.816
.164
.165
.591
15
20
1.000
1.000
00
5.
Summary. As is clear from the present status of the theory of multivariate
analysis" the existence and construction of an "optimum" test for the multivariate
Behrens-Fisher problem is a fornidable affair" but it is of interest to know that
Bennett's test provides an exact test and is the most powerful among all tests obtainable under Scheff€'s procedure. Also its power is fairly high when Nl == N2 , as
compared with that of the optimum test (4,l) for the case of t l = I.2 '
6, Acknowledgment.
The author is grateful to Professor Harold Hotelling under
whose kind supervision this work was done.
REFERENCES
L-l_7 B. M. Bennett" "Note on a solution of the generalized Behrens-Fisher problem,,"
Annals of the Institute of Statistical Mathematics" Vol. 2,
No.2 (1951)" pp. 81-90.
L-2.1
H. Hotelling" "The generalization of StUdent's ratio,," Annals of Mathematical
Statistics" Vol. 2 (1931)" pp. 360-318.
P. L. Hsu, "Notes on Hotel1ing P s generalized T,," Annals of Mathematical Sta...
tistics, Vol. 9 (1938)" pp. 231-243.
L-4_7
P. L. Hau, ".t\.oalysis of va.riance from the power function point of View,"
Biometrika, Vol. 32 (1941), pp. 62-69.
9
L-5_] s. N. Roy,
L-6_7
",h
report on some aspects of multivariate analysis," Institute
of Statistics, University of North Carolina, Mimeograph
Series No. 121, pp. 61-62.
H. Scheffe, "On solutions of the Behrens-Fisher problem based on the
t-distribution," Annals of Mathematical Statistics, Vol. 14
(1943), pp. 35-44.
L-1-1
P. C. Tang, "The power function of the analysis of variance tests with
tables and illustrations of their use," Statistical Research
Memoir, Vol. 2(19}8), pp. 126-151.