ON THE MONOTONIC CHARACTER OF THE POWER OF A CERTAIN TEST IN
MULTIVARIATE ANALYSIS OF VARIANCE.
1
By
S. N. ROy
Institute of Statistics
University of North Carolina
Institute of Statistics
Mimeograph Series No. 74
August,
1.
1953
This resea.rch was done in part under a grant from the Ford Foundation
ON THE MONOTONIC CHARACTER OF THE POWER OF A CERTAIN TEST IN
MULTIVARIATE ANALYSIS OF VARIANCE.
1
By
S. N. Roy
Institute of Statistics
University of North Carolina
Institute of Statistics
Mimeograph Series No.
August) 195.3
1.
This research was done in part under a gra.nt from the Ford Foundation
1.
Summarl.
In a previous paper Ll_7 three different tests were oftered for
three types ot hypotheses in m.ultivariate analysis, namely, (1) that ot equality ot the dispersion matrices of two p-var1ate normal popUlations, (1i) that
of equality of the p-dimensional mean vectors tor k p-variate normal populations
(Which is mathematically identical with tl;le,genel'al problem. ot multivariate
\
'
"'.f
;
t,,' i
I
j
<;
•
.'
~.-!
I
!,
analysis of variance ot means) a.nd (i11) tlult o~ a p-set and a q-set (p < q)
-
,""
\
ot variates in a (p + q)-va.riate normal population. Some properties of such
tests (and their power functions) were stated and proved, and some further
properties of their powers were stated Without proof.
I
Among these latter is
the property, proved in this paper, that tor problem (11) the power of the ~
~~
)lmonotonicallY increasing function
of the absolute value of each ot the corres-
ponding "deViation" parameters, that is, a set ot parameters (characterizing
the dev1ation trom the null hypothesis) which alone occurs in the power ot the
test.
This shows, incidentally, that the test 1s an unbiassed one in the
sense of Ne)'lD&n.
In proving these, use is made of an unpublished theorem (and
its corollaries) due to S. Moriguti, a former student of the author,' which,
together with Moriguti IS own proof,
2.
Introduction.
Let us consider the test tor (11), 1.e., for analysis ot
variance, offered in Ll_7.
be the same as in
is· given in Section, of this paper.
J:lJ
The notation and terminology to be used here will
except that X(p x q), tor example, will denote that
the matrix X consists ot p rows and q columns.
2.1 Multivariate analysis of variance test.
Let
t (p x 1) (r
_r
K
1,2, ••• ,
k) be the mean vectors of k p-variate normal populations with the sQme dispersion matrix t
(assumed to be p.d.), from which random samples of sizes
n (r • 1, ••• , k) are supposed to have been drawn.
r
(to be tested) is that
!1
• •••
=
!k.
The null hypothesis H
o
Let S* be the usual sample "between"
dispersion matriX of means (weighted by nr's) and S, the usual sample "within"
-2dispersion matrix (pooled from the dispersion matrices of the k samples).
Then it 1s well known that, almost everywhere, S is p.d. and S* is at least
= min(p,
p.s.d. of rank q
ot S*S·l are
root.
Then
k-l), so tv.at, a,e
. . lye
.
. -lob
pOS 1.1:i
·3.11C!.
... :~
t~e
q of the
Q ,
che.racter~.stic
r.es t , -I_. e ., p-q are zero.
Le t
r.
~
q
ruots
be the 1arges t
c4itical region of the test at a level of significance, say
a, is:
(2.1.1)
9
> c,
q-
where c is g1ven by
(n
=
k
I:
nr ).
r=l
~)
Thus c or Q is the;ta"po!Il'l; of the distribution of
a
Q
q
and depends on ex, p, k..l
and nook.
For a given c and for arbitrary non-null
~(p
x l)'s, it was shown 1n
~1_7 that this critical region (2.1.1) was the same as
(2.1.2)
The region of accept&nce,1.e., Qq < c, was thus the same as
["sup,,-
-
~ '8*'!::./~' s?::-7 <
c.
-3If we denote by H the usual non-null hypothesis and by £* the usual
weighted "between" dispersion matrix of means tor the k populations, it is well
known and has also been shown in
Le., P(Qq?,: Qa
I
L1J that
the power 01' the critical region,
H) is a function of Just the characteristic roots C:\'s
(all non-negative) of the matrix £*£-1, aside 01' course tram the other parameters
Q,
p, k-l and n-k. A lower bound to this power was obtained in
£:1_7.
It is shown in the present paper that for a given c, i.e., 0, this power is a
monotonically increasing function of each of the population characteristic
roots, which incidentally proves that the proposed test is unbiassed.
j.2.2 Canonical form for
J 2.1.
As is well known and also shown in
L1J
and ~2J we can, for the purpose of discussing the problem of the power function under ~; 2.1, start, without any loss of generality, from the canonical
...
-
probability law:
(2.2.1) Constant. exp
P
1=1
j=l
dx
iJ
all the var1ates going from matrix (of means)
(2.2.2)
s*
*
(k - 1)(8)
01'
11'
J
-R
k-l
IT TI
x
~
£:-
~ 2.1
n-k
)( IT dyiJ ,
1=1 3=1
cO to
is
~.
Also the sample "between II diapers ion
now expressed in the canonical form:
,
(1 < l' • 1, 2, ••• , Pi
8
11 ,
= si'i)
-4-
O
where ",-. i' s (i
= 1,
2, .•• , p) stand for the p characteristic roots of I:*-1
I:
and where I: is supposed to be p.d. and I:* is at least p.s.d. of rank r (say)
< min (p, k-l).
It is well-known that in this situation, out of the p
E)1'S,
p-r are zero and r are positive. Without any loss of generality these positive
roots might be taken to be
being necessarily implied.
G l' 02' ...,
Gr' no ordering in magnitude
This means that by varying I:* , i.e., by considering
suitable non-null hypotheses, we could have at the most q(= min(p, k-l»
tive roots.
The sample "within" dispersion matrix S of '
posi-
2.1 will of course
be given by
(i,
(n-k)(S)
11'
i'
= 1,
P
2, ""
p) =
I:
i,i'=l
ThUS, from (2.1l) - (2.1.3) we note that the power of the critical region
(2.1.1) is given by
(2.2.4)
P(Qq
I
? c H)
=
Integral of (2.2. 1) over the domain
The second kind of error is of course given by the opposite inequality throughout (2.,.4), the U being replaced bY~. It should be observed that the
constant factor in (2.,.1) does not involve the ~i'S'
-5-
3. Moriguti's results.
(3.1)
Main theorem.
Let t, u, v and w be independent random variables. As-
sume that t has a symmetrical distribution with the p.d.f. ~ (t) which is a
continuous, decreasing function of
• Assume further that p(u "\ 0)
= 1.
/
Let a be a parameter which does not enter into the distribution of t, u, v and
w.
,
Then for the composite random variable T
peT
'!.. c)
"
is a decreasing function of
+ au)
222
+ v _7 / w ,
for any fixed positive value of c.
2
2
2
2
Let the c.d.f. of u, v and w be denoted by F(u), G(v ), H(w ),
Proof.
respectively.
(3.2)
Ia'
= L(t
p
Then
{(t
OJ
+
au)2~ x }
)
=
o
( IX - au
dF(u)
~(t) dt,
)
(x)
0) •
- 'jX - au
This is an even function of a for fixed x, because of the assumed symmetry of
¢(t). We differentiate (3.1) with regard to a and obtain
)a
p
~ (t + au)2 ~x}
o
(x> 0).
This is negative for every a
¢(t).
>
0 and u> 0, on account of the assumptions on
Hence (3.~) is a decreasing function of
I a(
Next, we have
cw
2
J
(3.4)
(c> 0).
P
i)
!
'-
(t + au)2 < cw 2 _ v2
-
? dG/v2),
)
-6..
The monotonic character with respect to
tion.
1a r
is preserved under the opera-
This completes the proof'.
The assumptions can be weakened. For instance, the continuity ;1f
Remark.
2
2
,(t) is not necessary. Also, v and w can be dependent.
The theorem as
stated will suffice for the following simple applications.
Some simple
applicaticn~.
Under the assumption of normality of the population,
the sample variance ratio with non-central mean-square in the numerator can
.
be written in the form F
= (n2/n1)~(t
+ a)
2
2
+ X.
~ll-
1-7 / Xn2
,
when t is a
2
N(O, 1), X2 's are random variables each distributed as chi-square with the
{(;t'l ,~.; J
number of degrees of freedom indicated by the subscript, and(thfee are independent.
For a p-variate normal population, the sample multiple correlation
coefficient R and the sample partial correlation coefficient r can be brought
into the forms:
2·
2
2
2
r /(1 - r ) ... (t + aXn-p- 1) / Xn-p ,
respectively,
Where the symbols have similar meanings as in F and where a is equal to
.
I
•
p/(l .. p2)1/2, P being the corresponding population correlation coefficient.
The ordinary, i.e., total correlation coefficient is a special case where
p
:or
2.
This implies that the powers of the respective critical regions:
2
2
2
2
R ~ RO and r ~ r O' are each an increasing function of the corres-
ponding "non-central1ty parameter"
Iat.
w·(_
4.
Some further observations on Moriguti's theorem and its uses.
It will be
seen on a close examination of the structure of (3.3) and its tie-up with ('.2)
and (3.4) that the secret of the proof lies in our being able to get 1n ('.'),
after the t-integration, a negative factor in the integrand which leaves the
whole integrand negative for all a > 0 such that the negativeness
set in the subsequent stages of integration.
is not up-
Under the assumptions on the
p.d.f. of t, this negative factor would arise out of the t-integration
when-
ever the modulus of the upper limit of the t-integration is less than that of
the lower limit of the integration. A formal generalization of the theorem
('.1) with weaker (sufficient) conditions could be given but that would not be
necessary for the limited purposes of this paper.
5. Monotonic character of the power of the multivariate anallsis of variance
test.
-
Going back to " 2.3 consider the complement of the power, 1.e., the
-'OJ
second kind of error, of the test, which will be the integral of (2.2.1) over
the domain
or
It will suffice to show the monotonic character of this integral with respect
to variation of anyone parameter,
that At is the. row
~. .'
..
-:.
generality, put
~
vector
=1
.h01..
say
~
''P
and rewrite (5.1) as
To this end, .remembering
), we might, without any loss of
-8.
where 0lj
=0
=
if 1 .,. j and
now the integral of
(2,~,l)
/0 i
if
i
= J,
Consider
over the domain (5.2) and, for given values of the
other v 'riates, perform first the integration over x ll '
the total p.d,f. (2.2.1) made by xII is Just the factor
which satisfies the conditions of theorem
of this xu-integration are
= 1.
and where A.i
t1
The contribution to
const~
,-
1 2 7
eXPLr - '2xll_
(;.1). The upper and lower limits
and t,2 given by
(5.;)
and
n-k
t
J=l
p
(t
1=1
2
A.iYij)
/~l
It is easy to check that the xII-integral is an even function of
and also that, for all positive
\1 1 \
<
It 2 \
/
<3
'
1
Ifl \
~
\li2\
and, a.e.,
• If we now differentiate, with regard to
/0
1
the
integral of (2.;,1) over the domain (5.2) we immediately obtain, through the
XlI-integral, as in (;.;) an integrand which is
-exp
~all positive
/
/8
1
1 12 _7" + exp Lr - 2'/
1 22 _7 ,
L - '21
this is, a.e., negative, since, a.e.,
r.[d < Il21 ·
-9Notice that (5.4) is free from all ~'s and is a pure function of (")i's
(i = 1, 2, ••• , p) and all stochastic variates other than xll • If now we
integrate out over the other stochastic variates it is easy to see that,
~
since for all positive
the integrand is, a.e., negative, therefore
I~
the integral will also be negative for all positive
\
being just the differential coefficient with regard to
lit
I.~
This integral
lof the inte-
gralof (2.2.1) over (5.2), it follows that, for all positive
latter integral (which is also an even function of
function of
J I~ 1
I~
~,
the
1) is a decreasing
I . Thus the second kind of error of the test (2.1. 1 )
is a decreasing and, therefore, the power of the test is an increasing
function of each
}
If)
i \
separately.
We have shown that the power of the test is an increasing function of
each of the parameters(01 ,t)2' •••
,LJ
q
separately, i.e., under variation of
anyone of them, the others being held fixed.
Remembering that the l:)i's
are the characteristic roots of the matrix E*E-1 one can easily show further
that corresponding to two alternatives Hl and H we should have
2
i 0 i(1) > i=li G
(2)
-
i=l
,
1
>
rrG~2)
i=l
J.
i
i 0
D (1) ~ (1) >
i1j=1~ i ~j
i1j=1
, or in other words, if
(2) C' (2) ,
~ j
i
ud··~f
•
-10-
Here E*l ,
~l
1:\ (1) ,
andL:Ji
B correBpond to Hl and
6. Concluding remarks.
*and
' ~i
1\ (2) 's
~2' ~2
to H2 •
By uBing the same method (discussed in sections 3 and
5) it is possible to show that the power of the teBt of independence between
two sets of variates (or, in other words, the teBt for problem (iii»
given in
~1-7 is also a monotonically increaBing function of each of the 'deviation
parameters', i.e., of the population canonical correlation coefficients.
This
will be given in a later paper.
REFERENCES
1.
S. N. Roy, liOn a heuristic method of test construction and its use in
multivariate analysis", Annals of Math. Stat., Vol. 24 (1953), pp. 220-
2.
S. N. Roy, "Lecture notes on multivariate analysis ll ,
(Unpublished) .