PARAMETER CONSISTENCY OF INVARIANT TESTS FOR MANOVA

AND RELATED MULTIVARIATE HYPOTHESES

by

T.W. AndersonMichael D. Perlman

TECHNICAL REPORT No. 239

October 1992

Department of Statistics, GN-22

University of Washington

Seattle, Washington 98195 USA

Parameter Consistency of Invariant Tests for MANOVA

and Related Multivariate Hypotheses

T. "V. Anderson* and Michael D. Perlmant

Abstract

A statistical test is said to be parameter consistent (PC) if, for fixed sample

size, its power approaches one for any sequence of alternatives whose distance

(in the Kullback-Leibler metric) from the null hypothesis approaches infinity.

Necessary and sufficient conditions are given for the parameter consistency of

invariant tests for the multivariate analysis of variance (MANOVA). The Roy

maximum-root test, the Lawley-Hotelling trace test, and the likelihood ratio test

(Wilk's criterion) are PC, whereas the Bartlett- Nanda-Pillai trace test, although

consistent in the usual sense as the sample size approaches infinity, fails to be

PC unless the significance level or error degrees of freedom n is sufficiently

large. These results extend to the problems of testing independence of variates,

testing equality of two covariance matrices against one-sided alternatives, and to

other multivariate l1YI)otl1e~'es.

Key words and phr~ies: mtlltivalria1te analysis of variance, MANOVA, invariantconsistency, characteristic stochastic bounds, tests

covariance ma,tm:es.

of Stanford Unive,rsitv. S:tanfor,d, CaliforniaNational Science Foundation Grant No. DMS-89-04851.

Research SUj>pc,rte,d U.S.

Research In

1 Introduction.

The general multivariate analysis of variance (MANOVA) problem can be reduced

to the following canonical form by means of sufficiency and invariance (d. Anderson (1984),

§8.3.3; Lehmann (1986), §8.1-8.2): observe the independent random matricesl

(1.1 )

and test

(1.2) Ho : e= 0 vs. HI: e=f. 0

with E unknown. Thus the r + n columns of X and Yare mutually independent, normally

distributed column vectors (p xl) with common covariance matrix E. It is assumed that E

is nonsingular and n ~ p, so that the unbiased estimate n-lyy' of E is nonsingular with

probability one. The matrix e(p x r) is the matrix of unknown mean vectors of X.

For any real symmetric matrix S let chl(S) ~ ch2 (S) ~ ... denote its ordered character­

istic roots. Define

(1.3)

t pA r,

Ci =Ci(X, Y)

c Y)

s p Vr

chi[XX'(yy')-I], 1:S; i :s; t,

... ,

so that CI ~ ••. ~ Ct ~ O. The ran.(10111l vector c is a rep,res~enltatlon of the maximal invariant

statistic

IS nOltlSl]tlgll1ar are ort,hogOilaL

.5) ),

A _ A(e,E) = (At, ... ,At),

1 '5: i '5: t,

so that Al ~ ... ~ At ~ 0 and A is a representation of the maximal invariant parameter.

Then MANOVA problem (1.2) is equivalent to that of testing

(1.6) Ho : A=0 vs. HI: A =1= o.

An invariant test for (1.2) = (1.6) depends on (X, Y) .only through CI, •• . ,Ct, while its

power function depends on (e, E) only through the noncentrality parameters AI, ... ,At. Since

Ci estimates Ai, the appropriate invariant tests are those that reject Ho for large values of

CI, .. . , Ct. More precisely, Schwartz (1967b) showed that every admissible invariant test must

have a monotone acceptance region in terms of Ct, ... ,Ct. In the present paper, therefore, we

restrict attention to this class of monotone invariant tests and study their power functions

as one or more Ai -+ 00 .

Perlman and Olkin (1980) showed that monotone invariant test is unbiased for

the testing problem (1.2)=(1.6), so criterion does not differentiate among admissible

invariant tests. Likewise, neither the usual notion of consistency, which we shall call

n.-..,ATP,. to one every

(l,1\,e111(1,\,1 ve as the we introduce of parameter

IS aellne:a to

nrnxrp,. apprc)actles one

Definition 1.1. r,n, a, an inv·ari.ant a test with ac(~eptaIlce ref!IOn

for .6) is 1JanrLmt;ier cons1,si:ent if approa(~nes 1 as HAll ~ 00,

IlA ).2 For i = 1, ... , t, such a test is parameter consistent of

degree i (PG(i)) if power at A approaches 1 as 1,; ~ 00. Since Al ~ ... ~ 0, PC ¢}

PC(l) => ... => PC(t). 0

Consider a sequence of alternatives He(m) , E(m»)Im = 1,2, ...} such that llA(m)1l ~ 00

as m ~ 00, where A(m) = A(e(m) , E(m»). For i = 0,1, ... , t, such a is said to

asymp.to.tic ranki if (A(m)}j ~ 00 as m ~ 00 for j = 1, ... , i but (A(m»)j remains bounded

as m ~ 00 for j = i + 1, ... , t. Thus, (e(m), E(m») has asymptotic rank ~ i iff (A(m»),; ~ 00

as m ~ 00. Clearly, an invariant test is PC(i) iff its power approaches 1 for all sequences of

asymptotic rank ~ i. (Also see Remark 3.7.)

Example 1.2. (One-way MANOVA, equal sample sizes - cf Muirhead (1982), §10.7). Sup­

pose that {Xjk l1 ::::; j ::::; J, 1 ::::; k ::::; I<} is an array of independent random p-vectors with

Xjk ......., N(eiJ E). Consider problem of testing

with E unJ:mo,wn

J 1, n J(I< 1),

can be reduc(~d to the canonical MANOVA model (1.1) with r =

x

1

~=

- , ... ,t =p matI~lX e

A secruence m ex-

ists an H(m) in RP that sum

from (1£1 )(m), ..• (1£] )(m) to H(m) remains bounded as m --t 00 while the dispersion among

(p,l)(m), .. . ,(p,J )(m) along any direction within H(m) approaches 00, where

If a monotone invariant test fails to be. PC(i), then for fixed p, J, J(, and £x, its power remains

strictly below 1 for every of even though it may be admissible

in the decision-theoretic sense. Injudicious or routine use of such a test (for example, in a

statistical computer package) could result in failure to detect a sizeable departure from the

nun hypothesis Ho. 0

In Section 3 we present necessary and sufficient conditions for the parameter consistency

of monotone invariant tests for MANOVA. These results are based on a series of stochastic

bounds for the characteristic roots of a noncentral Wishart matrix, first in Section

2. Specific tests are discussed in Section 4. Whereas the Roy maximum-root test, the

Lawley- HoteUing trace test, and the likelihood ratio test ("Vilks' are PC, the

Bartlett-Nanda-Pillai trace test, although admissible, proper Bayes, locally most powerful

invariant, sense as

treedClm n IS sulllCJLently

Catltlo,n 18 <>1"{,.1'O"''; ht"tnl'p

be PC un 1~!'lS the Slg'llltl.callce level £x or the error de~~r~~s

test crrterlon IS ch()seJIl.

mc,notorle IIlvalrlaIlt tests

two co'rarialice maltnces VS. a one-

l'JxteIIS1(}nS to

2 Stochastic bounds for the characteristic roots of anoncentral Wi$#hart matrix.

In order to study power of an invariant test for the MANOVA problem (1.2)=(1.6) as

a function of the noncentrality parameters A = (AI,"" At), we may assume that (e, E) =

(Il, Ip ), where Il : P X l' satisfies

Under this assumption XX' and yy' have standard Wishart distributions, noncentral and

central, respectively, and are independent. Let

so that i l :2: ... :2: it :2: O. Our results on parameter consistency are based on a series of

stochastic bounds for il, ... ,it . In Lemmas 2.1-2.3, p, 1', and n remain fixed, while X~(u)

denotes a noncentral chi-squarerandom variable with m degrees of freedom and noncentrality

parameter u.

Lemma 2.1. For 1 :::; k :::; t ana a > 0,

(i) P{il a},

P{lt-k+l +... + it :2: a} :::; P{X~s(At-k+I + + At) :2: a}.

that for k t, it + + It + ... + .)

Proof. Jje(:aUl~e

S 'f".

1 :::; i :::; t, we p:::;r,sot

r : k x p ranges over IS

u, we

P[ll lk :::; aj < P{tr(rXX'r'):::; a}

- i~f P{X;r(tr(rpp'r')) :::; a}

- P{X;"(sup tr(rpp'r')) :::; a}r

- P{X;"(AI + ... + Ak) :::; a}.

The desired stochastic bound for It-k+l +... + It follows similarly, starting from the relation

(2.2) It-HI +... + It inf tr(rX x'r'). 0r

Now set

v = X - p '" N(O,Ip Ir ),

Vk chI(VV') +... +chk(VV').

Note that the distribution of Vk depends on p and r but not on A, since VV' has the central

Wishart distribution Wp ( Ip , r). For u, v ~ 0 define

G(u, v) = (UI

/2

VI

/2

)2,

H(u, - VI

/2

) V Oe.

Lemma 2.2. For 1 :::; k :::; t and a> 0,

>

<

Proof.

< +

As G(u, IS lDC1'easan,e; u and v, )

11 +'" +h < G(suptr(rJt1t'r'), suptr(rVV'r'))r r

- G(AI +... + Ak' Vk),

frolD which (i) H.JH~'VV",

(ii) Again by the Cauchy-Schwarz inequalitYl

tr(rX X'r') > tr(rVV'r') + tr(rJtI/r') - 2[tr(rVV'r')tr(rJtJt'r')p/2

- [(tr(rJtJt'r'))1/2 - (tr(rVV'r'))1/2J2

> H(tr(rJtJt'r'), tr(rVV'r')).

Since H (U lv) is increasing in u decre~nn,!!; in v, it follows from (2.2) that

It-k+l +, ,,+ It > H(inftr(rftJt'r'), sup tr(rVV'r'))r r

+ ,.. + Atl

o

sto,::ha:stlc Dounas ll1 LeEl1m,'LS 2.1 2.2 are sharp, lead to

Ui rfn" rln '" I cmtra<:tel'lst:IC roots are not as

to det,errrnne par.amt::ter COlllsIsl;en<:y

Lelnmla 2.3. < <

>

Proof.

(2.3)

Lemma 2.1(ii) then that

P{lt-k+l ~ a} < P{lt-k+l +... + It ~ a}

< P{X~s(At-k+l +... + At) ~ a}

< P{X~S<kAt-k+l) ~ a},

while Lemma 2.2(ii) yields

P {It-k+l ~ a} > P{It-k+l + ... + It ~ ka}

> P{H(At-k+l +... +At, Vk) ~ ka}

> P{H(At-k+b vd ~ ka}.

Remark 2.4. The inequality Ik :::; (11 + .. ·Ik)/k and Lemma 2.2(i) together imply that

o

but this does not lead to an upper bound in terms of Ak alone. Similarly, Lemma 2.1(i) does

not yield a lower bOlInG of o

-+ooas'U-+oo IS stoCh<tStlcaIly llJlCn~as:mg in 'U, and

'Tni'",h.,,,, A, Lel:nIrta 00 if

if -+ 00, umtorlrnly to Get;erlWfte

where each numerator is independent of the corresponding denominator. Furthermore,

yy' t'V Wp(Ip, n), so its distribution does not involve A. Finally, yy' is nonsingular with

probability 1 since n ~ p, and ch1(YY') ::;; tr(YY') t'V X;,p so

(2.5)

It follows that Ci 00 if and only if Ai -+ 00, uniformly in {Ai Ij =J. i}.

3 Parameter consistency of monotone invariant testsfor MANOVA.

In order to avoid consideration of points at infinity in the sample space, it is convenient to

characterize parameter consistent invariant tests for the MANOVA problem (1.2) =(1.6) in

terms of

(3.1 )

where the

d

of dis

d(X, Y) =

bounded set

(3.2) , ...

+

IS a 8L£tCUY InCre<liSlDlg tUlnct,lon d is ~(l1nV;,tlelnt to as

(), non.central (). 0) distriblltiOJElS of dare absolultely con'l;inu.ous respe<:t to

Le,beEIJ,(U.e measure on the int,eri()r

For any subset

(3.4)

V t , let A denote its closure in V h where

(3.5) x E A, y E V t (y E V t ), x ~ y :::} yEA.

Here, x ~ y means that Xi ~ Yi for i 1, ... , t. If A is monotone in V t , then A is

monotone in Vt . Schwartz (1967b, Theorem 2) noted that the acceptance region (in terms

of d) of any admissible invariant test is Lebesgue-equivalent to a monotone subset A of

V t ; hence, since Leb{A\A} = 0 by Lemma 3.2(ii) (to follow), such an acceptance region is

Lebesgue-equivalent to a dosed monotone subset A ~ Vt • Therefore Theorem 3.1, our main

result, although stated for closed monotone invariant acceptance regions, in fact determines

necessary and sufficient conditions for the parameter consistency of any monotone acceptance

region in V t , hence for the parameter consistency of any admissible invariant test.3

Denote the ver·tl('J~S of V t by 0), eI, ... , et, where

= (1, ... ,1,71" .. ,71)·--..-- --..-­t-i

< IS m(mcltOEle,

COI1V(mH~n(:e we a test is if

if a.e.

of an admissible invariant is not

Theorem 3.1.

testing problem

(i) If ei(fJ) ¢ A for all fJ > 0

be a

O!, p, n, and i, 0 5 i 5 t.

particular, if ei ¢ A), then

as Ai -+ 00, i.e., the test with acceptance region A is PC(i).

(ii) If ei(fJ) E A for some fJ > 0, then the test is not PC(i).

(iii) Hence, for 1 5 i 5 t a and sufficient condition that the invariant test with

acceptance region A be PC(i) is that ei(fJ) ¢ A for all fJ > O. A sufficient condition is that

ei ¢ A.

(iv) For 1 5 i 5 t, if ei(fJ) ¢ A for all fJ > 0 (in particular, if ei ¢ A) but ei-l(fJ) E A for

some fJ > 0, then P>..{d ¢ A} -+ 1 if and only if Ai -+ 00, i.e., the test is PC(i) but not

PC(i - 1).

Proof. (i) The result trivial for i = 0, so assume that 1 5 i 5 t. We must show that for

e > 0, there exists N e N(e, A) such that

(3.6) P>..{d¢ A} ~ l-e

and

'Dt .

E'Dt\A. the

mCino'Lonlc11;y of

> -1

> >

~l­

~l-

a rorLsecluelllce

> = 1, e>O >0

(3.8)

(2.4) and (3.3),

P:ddi ~ 1 - P(lle)} ~ P {........,:...------..;.. ~ (t - i

Because H(u, v) ----t 00 as u ----t 00, exists Ne such that

wh,F>npvpr 'xi ~ Thus, 'xi ~ Ne implies that (3.6) holds.

(ii) The result is trivial for i = t, so assume that 0 ~ i ~ t - 1. We must show that for

any 0 < M < 00 there exists fiM =fi(M;A) such that P>.{d ¢ A} ~ 1 fiM < 1 whenever

'xi+l ~ M. Since A is monotone and ei(ll) E A, A;2 {d E Vtldi+I ~ 1l}. Then as above, it

again follows from (2.4) and (3.3) that

(3.9)

(ii). o

If test

if

cases are IlltlstI'atc~d if >

Fig. 3.1a: test is PC(l). Fig. 3.1b: test is PC(l). Fig. 3.1c: test is not PC(l).

An invariant level a acceptance region A 'Dt usually is defined in terms of an invariant

test statistic f = f(d) as follows:

(3.10)

where Ca ca(p, r, n; f) satisfies

(3.11)

If f is monotone on 'Dt (i.e., nondecreasing in each di ) then Af(ca ) is a monotone invariant

acceptance region with power function given by P>.{f(d) > ca }. In Corollary 3.3 we shall

restate Theorem 3.1 in order to characterize PC in terms of the test statistic f. First we

show that without loss of generality, we may restrict attention to statistics f that are lower

semicontinuous as well as monotone.

Let Ft denote the coJlec:tlc,n of

on 'Dt . If f E Ft, C E

exten(led real-valued, monotone test statistics f defined

IS a mcmotorte a,ccept''l,nc:e

some f E :Ft

lUIlCtlOlJtS d,enIled on as follow,,:

tolllowm2: propertIes are real:my veriJ:ied:

Af(c) {x E Vtl/(x) ::; c} is closed and monotone.

Af(c) =

l(x) ::; f(x} for x EDt.

Aj(c) ~ Af(c).

If f 1 lA, then 1= 1 - {-a.

If f E :Ft., 9 E Ft, and 9 ::; f on 1Jt ,

1 is lower senrncl:mtinuous on V t

1 is on

=/(x) for x E Vt •(3.

(3.15):

(3.16):

(3.17):

(3.18):

(3.19):

(3.20):

(3.21 ):

Lemma 3.2. Suppose that f E :FtJ A is a monotone subset of1JtJ and -00 ::; c ::; 00.

(i) Leb{x E 1Jt l/(x) ::f. f(x}} O.

(ii) Leb{A\A} = O.

(iii) Leb{Af(c)\Aj(c}} = O.

lranstl)rm to polar coordinates and apply Fubini's Theorem as follows. ForProof. (i)

x E 1Jt (x ::f. 0) define

E

so

}= }

(ii) (3.17). o

Lemma 3.2 shows that for every f E :Ft there exists ! E :Ft such that 0' tests

determined hy f and I are equivalent, i.e., identical power functions. the

following corollary to Theorem 3.1, although stated for a monotone lower semicontinuous

statistic f E :Ft , also determines necessary and sufficient conditions for parameter consistency

of a test determined hy any monotone test statistic 9 E :Ft (Just apply Corollary 3.3 with f

replaced hy g).

Corollary 3.3. Suppose that f E :F.t and that Ca satisfies (3.11). Fix 0', p, r, n, and i,

o::; i ::; t.

(i) If f(ei("')) > Ca for all", > 0 (in particular, if f(ei) > ca ), then

as Ai -+ 00, i.e., the level O! test based on f is PC(i). If", 0 is a point of increase of

f(ei("')) (i.e., ", > 0 =} f(ei("')) > f(ei)), then f(ed ~ Ca suffices.

(ii) If f(ei("'))::; Ca for some", > 0, then the test is not PC(i). If f(ei("')) is continuous at

",

(iii) Hence, for 1 ::; i ::; t, a and sufficient condition that the level 0' based on

f be PC(i) is that > Ca ", > O. If", = 0 is both a of and a

<>",>0>1

some", >

zs c01l~tznU01'LS at f/ = 0,

a point of increase of

Proof. Since f(ei(f/)) > Coo if and only if ei(f/) E Af(coo ), which by (3.16) is a dosed monotone

subset of Vt , Theorem 3.1 applies directly with A = Af(coo ).

For the next remark, if f E :Ft we define

o

(3.22)

Remark 3.4. Let f and Coo be as in Corollary 3.3.

(i) If a > 0 then Coo < f(et) = fo(1-), so by Corollary 3.3(i), a sufficient condition that the

level a test determined by f be PC(i) is that f(ei(f/)) = f(et) for all f/ > O. In particular,

f( ei) f( et) suffices. Consequently, if f( d) r.p(db" ., di) for some r.p E Fi , then the level a

test determined by f is PC(i). (This does not rule out the possibility that it may be PC(i')

for some if < i.)

(ii) If a < 1 then Coo 2: fo(O+), so by Corollary 3.3(ii), a necessary condition that the level

a test determined by f be PC(i) is that f( ei(f/)) > fo(O+) for all f/ > O. In particular, if

f(ei( f/)) is continuous at f/ 0, then f(ei) 2: fo(O+) is necessary for PC(i). If in addition

PC(i). Also by Corollary then level

a test deternlln(~d not out it fail

to.1

some z >

If a> 0 00 >0 pal~tlcular, if

test deternllll(~d

a<1 some > 0 if f 2: 0 on =0

77=Oisa of mcreal5e of = -00 ImplH~S that IS

not if IS reJ:Ha,(~ea ~ 0 on

o

It follows from Remark 3.4 (i) and (li) that if fed) = t.p(dk+b ... , dz) for some t.p E Ji-k'

where 0 S; k < 1 S; t, then for 0 < a < 1, the level a test based on f is PC(l) but not PC(k). It

mayor may not be PO(i) for k < i < 1: for example, the level a test based on f(d) =niH dj

is PC(I} but not PCfl-I), while based on niH dj (l- dj )-1 and EiHd;(1 dj )-1 are

PC(k +1) but not PC(k). Furthermore, parameter consistency may depend on the value of

a: when k < i S; 1, the level a test based on f(d) =EiH dj is PC(i) but not PC(i-I) for

a satisfying i - k - 1 < Co< S; i - k. These and other examples are presented in Table 3.1,

where Corollary 3.3 is applied to determine the parameter consistency of the level a tests

(0 < a < 1) determined by several monotone test statistics f E :Ft. (Recall that if f E :Ft\:Ft ,

it is necessary to apply Corollary 3.3 to 1E :Ft.) Furthermore, it is straightforward to apply

Corollary 3.3 and Remark 3.4 to determine conditions for test statistics of the more general

forms E!.pi ( di ) and n!.pi ( di ).

Remark 3.5. When .A = 0, each di =chi[XX'(XX' +YY')-I] is stochastically increasing

in r and in n. Arguing as in Perlman be ",hr...r"

that Co< co«p, r, n; f) is mCrealSlIJlg in p and rand de(:re4't8Irlg in a and n. It follows that

r,

IS,

,r ,

iflSLi:tLU,LIC f Efor a

l<P - r' S; r. o

Rem.ark 3.6. <p< + a < 1, can

3.1: PaI'am,eter cOl1lliste,ncy monotone test statistics f E :Ft. Here k,1 are integers satisfying 0::; k < I::; t.

0:'<1 f(ei) (or f(ei(7J)) if 7J appears) 3.3

o Ca < 1 {0, i < 1 }1, i ~ 1

PC(l), not PC(l 1)

Ca < 000J-)I-k\l~II '

00,not PC(k)

+ 1 ::; i ::; I, i - k - 1 < Ca ::; i ­not PO(i

+ 1) iff Ca ::; L

+

{

~::; k }7> k

i < k }k+l::;i::;1

i > 1{

0,. i - k,

1 k,1 k

o

o

+ r) iff Ca ::; L

not PC(/- 1)

, i k 1 i k+ r ::; 7 ::; I, ( r ) < Ca ::; ( )} .

PC(i), not PO(i-

+ 1), not PC(k)

+ 1), not PC(k)

1

{0, i < 1 }

• (l-k)~l, i~1

{0, ~::; k }00, 7> k

{O, i::; k }1, i > k

I0, i <k+ r )

(i ~ k), k + r $ ; $1

(I k), i>1r

~ k)o

0

1 , 1 ) 0 CO' 1

0 Ca < (I

o Ca 00 {( I k)(...!Lf)r i < k}l' 1-11' -

'Xl, i> k+ 1), not PC(k)

I o 1 k( 7' ) I

0,

(I~k)_(l r'),

(I ~ k),

i<k )k+l$/$I-r+l

i>I-1'+1 .

{(i,O:')lk+l::;i::;1 r+l, (I 7,k)_(1

1 kl-i ,< Ca ::; ( r ) ( l' )}: PC( 7t not

PC(k + 1) iff Ca ::; (I - k ) ( 1 k - 1)r r

1)

( k 1l' 1

The surmnat.j()ll extends over 7'~tuples it, ' .. ,ir that k + 1 ::; it < ' . , < ir ::; I, where 1 ::; l' ::; 1- k. We define (;) °if s < r.

probleIllS 1 o

Remark 3.'1. It toll1[)WS the of 'heorem 3.1 proofs of Lemmas 2.1-

iff its power2.3) that a monotone mv'anant test is PC(i) iff it is uniformly PC(i),

at A = (Al, ... ,At) apl>rO,:l,chies 1 unltorllllyin {Aili =1= i} as Ai -+ 00. This equivalence.

would also follow from the conjecture of Perlman and Olkin (1980) that the power of any

monotone test is mc:recLSIrl,ll;

be no less that· power at

for then the power at (A1,' .. ,Ai, Ai+l, ... , At) would

... , Ai, 0, ... ,OJ. Furthermore, this conjecture would imply that

a monotone invariant test is PC(i) iff its power approaches 1 for all sequences of alternatives

o

4 Parameter consistency of specific invariant tests forMANOVA.

"\tVe now eX<l,mllle the parameter consistency of the tests based on six familiar statistics:

(S.N. Roy)

(Lawley-Hotelling)

(Likelihood ratio (\;Yilks))

statement is not true. As will be seen below, the res:uU;s in Section

that nYJ>Ut.neSIS Ho with positive

on IS

test on ael:>eIlas IlOlltfliVlG.Uy on 'r, n, a:

test

(4.1)

In particular, for 0 < a < 1

i-l<

BNP

14) ::; i.

is PC PC(I)

(4.2)

It is of some int~erelst to determine those values of (p, 'r, n, a) for which (4.2) fails and,

III to III of the BNP test

alternatives of 1. the remainder of this section some approximate answers

to these questions are obtained.

Because ca(p, 'r, n; 14) is decreasing in a and n,4 for fixed (p, 'r, n) [respectively, fixed

(p, 'r, a)] there exists a*(p, 'r, n) > 0 [resp., n*(a, p, 'r) ~ pJ such that (4.2) holds iff

(4.3)

If (4.3) does not hold,

Values of a*(p,

(4.4)

or equivalently (see

a ~ a*(p, 'r, n) [resp., n ~ n * (a, p, 'r )J.

BNP test to be PC.

and n"'(a, p, 'r) are Clel~er]mllled from the defining equation

Ca " 'r, n; h) 1,

r, > 1

n

9) or Perlman §2)), so

(4.8)

(4.9)

at(p, r, n) = a*(r,p, n + r - p),

n*(a,p, r) = n*(a, r,p) - r +p.

It can be deduced from (4.5)-(4.9) that a*(p, r, n) is increasing p and r but decreasing in n,

while n*(a, p, r) is mc:rec~SllJl~ p and r but decreasing in a. For a = .05 and .01, respe<:ti,,'ely

Tables I and II in Perlman (1974) list values of n*(a,p,r) for 2 ~ p,r ~ 16; typical values

for a .05 are n*(.O,l),2,2) = 7, n*(.O,l),4,4) = 20, n*(.O,l),8,4) = 40, n*(.O,l),8,8) = 73, and

n*( .05,16,8) = 14,1), while values for a* = .01 are slightly larger. In fact, n*(a, p, r) -+ 00 as

a -+ o.When A = 0 and p, r are fixed,

(4.10)

as n -+ 00, so d(p, r, n)

p and r fixed. If, hovveVier

",-1 XX' ~ 0 d 1 yy',~ -r an n- .

O. By (4.,1)), this implies that a*(p, r, n) -+ 0 as n -+ 00 with

p, r, and n are all of the same order of magnitude then the

approximations (Pillai (19,1),1)))

~ B (pr t(n-p+t)),- eta -, ') ,2 - ~ ,

r,

-+ 1 as n

mCreal3e at same rate. r,

approximate ,,~I"llPQ tables ScEluu:rm.tnn et a1.

(1973) for the central of f4(d), are in Table 1. This table can be

extended by means of relation (4.8), e.g., a*(3, 2, 8) = a*(2, 3, 7) ;:::: .05. Furthermore,

when '1' = n;:::: p, tr[XX'(XX' +YY')-lJ and tr[YY'(XX' +YY')-lJ have identical central

distributions, while

(4.11)

so their common central distrHmtlon must syrrrmetrJlC about p/2. Thus, by (4.5),

(4.12)

(4.13)

a*(2, n, n)

a*(p, n, n)

a*(n, 2, 2n 2) .5 (n;:::: 2)

a*(n,p,2n - p) >.5 (n;:::: p ;:::: 3).

Together with Table 4.1, this paragraph indicates that in many cases the BNP test fails to

be PC for the standard values of a used in practice.

When the BNP test fails to be PC, i.e., when (4.2) fails, it is of interest to examine the

maximum power shortcoming for distant alternatives of asymptotic rank 1. If the power of

the BNP test is monotonically increasing in each Ai, which would follow from the conjecture

of Perlman and Olkin (1980) that every monotone invariant test has this power property5,

then this maximum shortcoming would be given by

(4.14) 1-

n '1'-

nr(}vI,c1es a .e;eIler,al '1', a tl.e;J:Lter

'1'-

a*(p, r, n) > .05 a*(p, r, > .10 a*(p, r, n) > .25"'" "'" "'"

p,r,n: 2,3,7 2, 3, 6 2, 3, 42, 5, 10 2,5, 9 2, 5, 72, 7, 13 2, 7, 12 2,7,10

3, 4, 14 3,4, 12 3,4, 103, 6, 20 3, 6, 18 3, 6, 153, 8, 26 3,8, 23 3,8, 19

4, 5, 24 4, 5, 21 4, 5, 184, 7, 32 4, 7, 29 4, 7, 254,9,39 4, 9, 36 4,9,32

Table 4.1: Approximate values for a*(p, r, n). The level a Bartlett-Nanda-Pillai trace testfails to be parameter consistent when a < a*(p, r, n).

as follows. Partition X as (Xll X2 ) where Xl is p x 1 and X2 is p x (r - 1). Then

(4.16) tr[XX'(XX' + yy't l ]

tr[XIX~(XX' +yY')-l] tr[X2X;(XX' +yy't l ]

:::; 1 + tr[X2X;(X2X; + yy't l ].

When A (AI, 0, ... ,0) we may set (e, and {Lij = 0 otherwise.

= 0, it toU,ows

r, > J"\~\F r - 1 < r,n; - I}

.10 ~ p, r, n) ~ .25 r, n) ~ .01 < r, n) ~

2,3,3 2, 3, 6 2, 3, '"'p,r,n: (

2,3,5 2,5,5 2, 5, 72, 7, 7

5, 5 4,5, 11 4, 5, 134,5,9 4, 7, 11 4, ,5, 154, 7, 7 4,9,9 7, 13

7,9 4,9, 11 7, 174, 9, 134,9, 17

6, 7, 7 6,7, 17 6, 7, 216, 7, 15 6, 7, 19 6, 7, 236, 9, 9 6,9, 17 6, 9,216, 9, 15 6, 9, 19 6, 9, 25

Table 4.2: Approximate values of 8.os{p, r,n), a lower bound for the maximum power short­coming s.os{p, r, n) of the level.05 Bartlett-Nanda-Pillai test.

Some approximate values for the lower bound 8.os{p, r, n) are given in Table 4.2, again

obtained from ::idluulrm.aIlJG, et al. (1973). values indicate that maXimum power

cannot stated COl1dusively S .05, Bondar (1991) reoentlv

OOlJaUleu more accurate ,,~IHP"l

can

tests ~ IW',nl"l

are

5 Invariant tests for independence of variates.

problelrn of test;Injl; Indep<mden<:e two

population be reducc:~d to

of VaJrlaltes In a mtlltivalrialte normal

carlOnllcai MANOVA probleJ[]1 by COIllditionin~ on one set

of all of the preceding results regarding parameter consistency (and sample

consistency) of invariant tests for MANOVA remain valid for the corresponding tests for

independence. This correspondence and the conditioning argument are now outlined. The

reader is retc~rr<~d to Anderson and §9.1O), Anderson and Das Gupta (1964a), and

Perlman (1974,§2) for further details.

Based on a sample of size m from the (PI +pz)-variate normal population

N [( ~ ) , (~:: ~::) =E], Eij : Pi x Ph

we may construct a (central) 'Wishart random matrix

8 (811 ~::) ~ WP1 +P2 [( Ell ~::), m]with m degrees of freedom. Assume that E is nonsingular and that m ~ PI +Pz, so that 8

is nonsingular with probability 1. \Vithout loss of generality, also assume that PI ~ pz. The

problem of independence is equivalent to testing

(5.1) # 0

with and

l~i~t ___ PI

, ...

. ,

1 > 1'1 > ... > 1't > 0 •.. ;:: Pt ;:: 0) are sarrrple (population)

call1oIlicial correlation COl~mClents and represent the maximal mv'anant st8LtIS:t1C

under the group of transformations

(5.5) s -+ (B01 0) S(B1 0) I

B2 0 B2 '

where Bi(Pi x Pi) is nonsingular, which group leaves the testing problem (5.1) invariant.

Any invariant test depends on S only through r2, while the distribution of 1'2 depends on E

only through p2, or equivalently, only through I' Then (5.1) is equivalent to the problem of

testing

(5.6) Ho : I 0 vs. HI: I =F O.

As in the MANOVA problem, every admissible invariant test for (.5.1)=(5.6) must have

a monotone acceptance region A in terms of r 2 = (1';, ... , r;) and every such monotone

invariant test is unbiased. (The range of r2, like that of d in Section 3, is the set 'Dt .) For

fixed Pll P2, m, and Ct, an invariant level Ct test with acceptance region A is PC if its power

1, ... ,t PI)' such a

test is said to be PC(i) if its power at

PC ¢} PC(l) =} ••• =} PC(t).

alternative I approaches 1 as Ii -+ 00 . Again,

All results in Sections with monotone

monotone ac(;eptarlce reJ!:10Irs of

aC(3eDtaIlce regions of the

carryover to mvan,ant tests

{d E A} {f(d) < ca } for MANOVA problem

{r2 E A}

<

ret>1a<;e P, n m,

mlllimum .Knllba:ck-Lelltller distauce from the alt,ern:atnle

of

E is same as

(5.7)

where ~(p) : PI X P2 is <1eltlll€lG

(Ip ~(p) ).

E = ~(p)' Ip2

'

~ij(p) = PiOij with Oij the Kronecker delta function. Under

(5.7), it can be shown that the coniliti.onal distribution of r 2 given 822 is the same as

distribution of d in (3.1) following correspondences:

(5.8)

(5.9)

(5.10)

where D(1) = Diag(11"'" 1t) is PI X PI and where 822 is the upper left-hand PI x PI block

of 822 , The unconditional distribution of r 2 is determined by this conditional distribution

and by the marginal Wishart distribution of 822 :

(5.11 )

In particular, the central h 0) distribution of r 2 r 2(Pb P2, m) does not depend on 822

and is identical to the central (A 0) distribution of d =d(Pb P2, m - P2)'

12)

1

to . = 1, ... ,t

1)

P{O < < <oo}=I,

so for every fixed

(5.16)

Thus the same necessary and/or sutl.icient conditions on A are valid for the implication

(5.17)

to hold for every fixed facts to establish modified version of Theorem

3.1, and similarly Corollary ;3.:3, for the independence problem.

Thus the PC properties of an invariant test based on a monotone statistic f(r 2) for the

independence problem (5.1)=(5.6) are the same as those of an invariant test based on f(d)

for the MANOVA problem (1 1.6). In particular, by (4.2) and (5.8), the BNP test based

on

(5.18)

is PC(1) PC only if

(5.19)

7 By contrast,

andon

IIIsh()rtcOIn1I1g 1Sand its ma,XlInUlm

are ,,111'1:;'1:''''

We return to the U1I1P.'V

6 Multivariate components of variance.

testing problem (1 .2) but now assume

random. That is, relZlla(;e (1.1) with distributional asSUIIlpt:lons:

{(p x r), and Y(p x n) are mutually independent, and

{ is

X(p x r),

Here X, Yare observed but { is unobservable, E is positive definite and (j) is positive

semidefinite. Thus, unconditionally, X and Yare independent with

(6.2)

so we have a multivariate components of variance (= random effects) model (d. Anderson

(1984), §10.6). We assume that p S min(r,n) to insure that E and (j) are estimable, so now

t pAr = p.

The testing problem (1.2) is replaced by that of testing

(6.3) Ho : (j) = 0 vs. HI: (j) i= 0

with E unknown. As in Section 1, problem is invariant under the group of transfor-

mations given by (1.4). The statistics c =(Cl""'Ct) in (1.3) and d =(d1, ... ,dt ) in (3.1)

agam are eOlllV,aleJot r(~presE':~ntiCtti()ns now maximal

invariant parameter is aellnE'::a as foHows:

w·,

w

), 1 SiS t,

UT h,n".. distril:>ution de1=len<ls on

IS eOlllVialeJl1t

adIlCllSSlble Illv.'),rtant test for (6.3)=(6.5) must a mO'not,one acceD-

tance ref!lOn A in terms of a (aI, ... ,at) E V t, and every such monotone IIIvanant test is

unbiased (Anderson Das Gupta (1964b)). For fixed p, .,., n, and an invariant level a

test is PC Hits power Pw{a ¢ A} -+ 1 asEwi = tr(<I)!j-l) -+ 00.8 For i = 1, ... ,t p),

such a test is said to be PC(i) if its power at the alternative W approaches 1 as Wi -+ 00.

Once again, all results in Sections 3 and 4 regarding parameter consistency of tests with

monotone invariant acceptance regions of the forms {a E A} and {f(a) ~ cal for the

MANOVA problem (1.2)=(1.6) over to present problem (6.3)=(6.5); the only

notational change required is to replace Aby w. We briefly sketch the proof of Theorem 3.1

in the present context.

By invariance, the distribution of d under (<I>, E) is the same as under (D(w), Ip ), where

D(w) Diag(WI, •.. ,Wt) : p X p. In this case, by (6.1) the conditional distribution of agiven

e is the same as the distribution of d in the MANOVA problem with noncentrality parameters

A (Ab"" At), where Ai chi(e(E-1) (d. (1.5)) . In particular, since pw{e = O} = 1

when W 0, the central distribution of ais same as its central distribution in MANOVA.

Theorem 3.1 gives necessary and/or sufficient conditions on A for the implication

(6.6)

assumption (<I>,to hold for every ( Furthermore, by (6.1) and

same distribution as chi[D(w)UU'], n.h"n''''

(6.

i = 1 ... t

<<

can he shown that the minimum Kullback-Leibler distance from the alternativeto the

P 5 r,

P{O < < <oo}=l,

so for

(6.10)

fixed U (i.e.,

Thus the same necessary and/or sufficient conditions on A are valid for the implication

(6.11)

to hold for every fixed e. These facts are SUlnCl,em; to establish Theorem 3.1, and similarly

Corollary 3.3, for the testing problem (6.3)=(6.5), provided that A is replaced by w.

Therefore, the PC properties of an invariant test based on a monotone statistic f(d) for

the components-of-variance problem (6.3)=(6.5) are the same as for the MANOVA problem

(1.2)=(1.6). In particular, the BNP test based on

is PC(l) only if (4.2) or equivalently (4.3). The maximum power shortcoming of the

BNP test is again given by sa(P, r, n) (4.14).9 The Roy, Lawley-Hotelling, and vVilks tests,

based

It "liVU~U that on Wilks sta:tlstlC f3(d) is not the

IS

IS "hr,u'n to

conJed;nre has been established for the cOfnpcmeJ:ltg.·ot:'varianceprei,ent context

7 Referen·ces.

Ande];SO]ll, T. An Introduction to Multivariate Statistical Analysis

(second John Wiley :& Sons, New York.

Anderson, W. and Das Gupta, S. (1964a). Monotonicity of the power functions

of some tests of independence hetween two sets of variates. Ann. Afath.

Statist. 35, 2'06-208.

Anderson, T. \V. and Das Gupta, (1964h). A monotonicity property of the

power functions of some tests of the equality of two covariance matrices. Ann.

Math. Statist. 35, 1059-1063.

Anderson, T. W. and Perlman, M. D. (1987). Consistency of invariant tests for

the multivariate analysis of variance. Proc. Second Int. Conf. in Statist. (T.

Pukkila, S. Puntanen, eds.), 225-243.

test for means in MANOVA. Inpower ofBondar, J.V. (1991). On

preparation.

L"'X,~xv.L, J. and R. (1965). Admissible Bayes character of T2, R2, and

other fully invariant tests for classical multivariate normal problems. Ann.

Math. Statist. 36, 747-770.

L. ( John

tests

\JJ."VU, C.L. ch()oSJtn~ a sta:tlstlc tn mtlltlval71at;e all1a1'VSIS of

ance. r """:IL Bulletin 83,579-586.

AdrrnsSlible tests in multivariate analysis of variance.

Perlman, M. On the monotonicity of the power functions of tests

based on traces of multivariate beta matrices. J. Multivariate Analysis 4,

22-30.

Perlman, M. D. and Olkin, 1. (1980). Unbiasednessofinvariant tests for MANOVA

and other multivariate problems. Ann. Statist. 8, 1326-1341.

Pillai, K.C.S. (1955). Some new test criteria in multivariate analysis. Ann. Math.

Statist. 26, 117-121.

Schuurmann, F.J., Krishnaiah, P.R., and Chattopadhyay, A.K. (1973). Exact

percentage points of the distribution of the trace of SI(SI + S2)-I. Report

No. ARL 73-008, Aerospace Research Laboratories, Wright-Patterson AFB,

Ohio.

Schwartz, R. E. (1967a). Locally minimax tests. Ann. ;Math. Statist. 38,

340-360.

Schwartz, R. E. (1967b).

~tatzst. 38,