..
-
•
A NOTE ON J. ROY IS" STEP-DOWN PROCEDURE
IN MULTIVARIATE ANALYSIS"
by
v.
P. Bhapkar
University of North Carolina
This research was supported partly by the
Office of, Naval Research under Contract No.
Nonr-855(OP) for research in probability and
statistics at Chapel Hill and partly by the
United States Air Force through the Air Force
Office of Scientific Research of the Air Research and Development Command under Contract
No. AF 49(638)-213. Reproduction in whole or
in part for any purpose of the United States
Government is permitted.
Institute of Statistics
Mimeograph Series No. 212
December, 1958
Ii NOTE ON J. ROY' $ ",stEP-DOWN PhOCE.;DlJ.rl.E.
IN NULTIVA.ri.IATE
Al~ALYSIS"
By
V: P. Bhapkar
University of North Carolina
1.
Introduction and Summary.
Test procedures in multivariate analysis are usually based on the
A-criterion or a criterion in terms of the largest and/or the smallest
characteristic roots of certain matrices, each criterion being a special
case of the general union-intersection principle.
An alternative proce-
dure, called the step-down procedure, has been used by Roy and Bargmann
~2_7
in devising a test of multiple independence between variates distrib-
uted according to the multivariate normal law.
This procedure again can
be derived as a special case of the union-intersection principle.
This
procedure has been recently applied to multivariate analysis of variance
by Roy
~1_7
in deriving new tests of significance and simultaneous
confidence-bounds on a number of lIdeviation-parameters.1I
In this note
the same procedure is applied to test multiple independence of normal
variates under a general linear model.
2.
Notation and Preliminaries.
In the notation of £1_7, we have a matrix Ynxp of random variables, such that the rows are distributed independently, each row
*This research was supported partly by the Office of Naval Research
under Contract No. Nonr-855(Oo) for research in probability and statistics
at Chapel Hill and partly by the United States Air Force through the Air
Force Offiaaof Soientific Research of the Air Research and Development
Command under Contract No. AF 49(638)-213. Reproduction in whole or in
part for any purpose of the United States Government is permitted.
having a p-variate normal distribution with the same variance-covariance
~
matrix
= (cr .. )
l.J
p x p
which is symmetric and
positive~definite.
The expected values are given by
e
C9
(1)
Y
nxp
-·A
nxm
® mxp
where A is a matrix of known constants of rank r, r ~ (n-p), and~is a
matrix of unknown parameters. We want to test the hypothesis that the
p-variables are independent, i.e.,
= 0
(1 t j, i,j=l,2, ••• ,p).
'rhe customary likelihood-ra.t~~ test for ~ 0 is based on p x p
matrices of random variables
U)
called respectively the sum of products matrix due to error and the sum
of products matrix due to hypothesis.
Here E and Hare n x n symmetric
idempotent matrices with non-stochastic elements, E of rank n-r and rl of
rank p.
(4)
The test is
accept
1eo' if L =.
otherwise reject
>
c
1eo
where c is a pre-assigned constant depending on the level of significance.
3.
The Step-Down Procedure to Test ~o'
we shall denote the i-th columns of the matrices Y and @ by y. and
-l.
.§:i ,respectively and wri.te Yi
=
LXI' 1.2' ... , 2.i.,-7 and®i= L~l'~?'· .. ,9i _7.
3
We shall also denote the top left-hand i x i submatrix of Z by 1:i •
If Yi is fixed, the n elements of 1i+l are distributed indepen
dently and normally with the same variance a~+l and expectations
given by
(5)
where
~i
is a column i-vector
(6)
and .!li+l is a column m-vector given by
2Ji+l
=
~+l - ®i&
and
a2
(8)
'"
i+l
eJlo is true if and
alIi = 1,2, •• ,(p-l}.
We note that
holds for
f Zi+l t
~
i=1,2, ••• ,p-l
only if the hypothesis
~ 1 that ~i = 0
Now the elements of the vectors
.@i' .!l1+1 in (5) may be regarded as unknown parameters and hence, when
Y
i
is fixed, the hypothesis
1ei
that ~i
=0
is a linear hypothesis in
univariate analysis With the linear model given by (5).
{
NoW Rank (Yi
)
=i
and Rank (A, Yi )
a.e.
=
r+ i
a.e.
•
4
Also
(10)
implies i (0, I)
m i
( ..!li+~
=0
/3
ixl
-i
Furthermore,
(11)
Hence
:::: r + i = Rank (A, Yi ).
Rank
~i is estimable and the hYJ;lothesis ~i testable. Let
estimator of
Of~i+l
~i'
A
~i
be the
the elements of which are linear functions of elements
and also are minimum variance unbiased estimators of the corres-
ponding elements in
~i •
Denote the variance- covariance matrix of
A
~i
by
Ci &i+l where C
i is an i x i positive-definite matrix. Let si/n-r-i
denote the usual error mean square giving an unbiased estimator of
CT~+l. Then it is well known that
A
I
(~i - ~i) C~
(12)
2
8
i
1
/
A
(~i - ~i
..
Vi
' i::::l,2, ••• ,{p-l),
n-r-J.
is distributed as a variance ratio with i and n-r-i degrees of freedom.
Thus the conditional distribution of F , given Y , does not ini
i
volve Yi and hence F1 , F2 , "', F _ • Therefore, the statistics
i l
Fl' F2' ""
Fp-l are independently di stributed as vari ance ratios
with degrees of freedom i and n-r-i respectively (i=1,2, ••• ,p-l).
For a preassigned constant a., 0 < a < 1, let f. denote the
J.
J.
i
upper 100 a. percent point of the variance ratio distribution with
J.
i and n- r- i degrees of freedom.
Then the probability that simUltaneously
5
F.J.-< f i ,
1=1,2, ••• ,p-l ,
p-l
is equal to
.1\
(1-a ).
1
~-a.t
ffO
Since 1{ 0 4
~i= 0
>X 1 :
1=1,'2, ••• ,p-l, we utilise (12)
and set up the following test procedure for 1Q 0 :
accept H ' if
o
(14)
for all 1=l,'2, ••• ,p-l
otherwise reject
1-e O·
To carry out the test one should first compute u •
l
ul
> fl
1e 0 is rejected.
,
It U1'S. f l' u
2
If
1s computed.
is reJected.
If u2 ~ f 2 , u 1s computed and so on. The level of significance
3 p-l
for this test 1s obviously l ..
(l-ap. One PPfJ8:ibility is 0: ..,a2;;::"'=O: 1" We
1
1-1 .
p-
rr
would preter choosing a's
4.
i
l
= t'2
= ••• = f p _l
for reasons discussed
Confidence bounds associated with the test.
Now F i ::; f i
r/;
that t
80
.
~
J.
J.
A
->
,
A
(~i -~i )(~i-~i) ~ A {C i)
,2
r1
2
s 1 where
mx
i
= n-r-1·
(15)
•
••
, A
. 1/2
~i ~i-fi si A
(C 1 )
mx
:s ~i, ~i ~ ~1, (31 + f i
A
1/2
s1 A
(C i )
mx
tor all non-null ~1 (ixl) such that ~i~i=l. This, theretore, implies
(16)
~, ~ )1/2
( -i
-i
_
1/
Xi
s. Al
J.
/2
mx
(C ) < «(3' (3 )1/2 < (~, ~ )1/'2 + II s Al / 2 (C.).
i - -i -i
- -1 -i
Xi i
1
We may obtain partial statements by choosing some elements of
(15) to be zero.
~
~i
in
Thus we have the simultaneous confidence bounds given by
(16) for all possible subsets of ~i for all i=l,2, ••• ,p-l with the contil> -1
dence coetfecient > 1 (l-et ).
i
1=1
rr
6
5.
Remarks.
(1) It can be easily seen that when Y represents a random
sample of size n from N
(17)
(~,
~ (Yi+l,k t Y1 ) = ~1+1
where ~{ =
(5) takes the form
E),
1
+j:ll3 ij (Yjk -
[Y i l ' Y i2' ••• , y in7!?i
=
~j)'
Lf311 ,
l3 i2 , ••• , I3 U J,
then it is well to know that
= -i
b
so that
2
_ r i +l •l ,2, ••• ,i
- l_r2
i+l.l,2, ••• ,1
n-1-1
i
where r.J.+ 1 • 1 , 2 , ••• , i denotes the multiple correlation coefficient of
(i+l) with (1,2, ••• ,1); thus g1ving as a special case the test procedure
already obtained 1n ~2_7.
(i1)
This is, of course, as 1t should be.
In this set up, it is of interest to investigate whether
(a) the test of the usual multivariate linear hypothes1sof the type
(18)
'S{b :
~
..
B
®
txp txm mxp
=0
(Rank B ""
t),
7
where
~
is estimable, and (b) the above test of independenceJare
quasi-independent. As shown in Cl J, the step-down test procedure
for (18) gives, when Yi is fixed,
1\
(!i+l
2
si/n-r-i
where !i+l
D i +l
= B ]i+l
(i a O,1,2, ••• ,p-l)
1\
and the variance-covariance matrix of !i+l is
2
O'i+l·
Fi given by (12)and Fi given by (19), for fixed Yi , are quasiindependent if the numerators, which are marginally distributed as
X~ iO'~+l
X~
{+l
t
respectively, are independent.
2
2
It can be easily verified that Xi and Xt are not independent and
hence the tests for }eO and
are not quasi-independent. It may be
and
nO
=0
noted that when Yi is fixed, the test of
~i
significance of covariance, as seen from
(5), while the test of !i+l
is in the nature of covariance-analysis.
is the nature of testing
=0
These two are not quasi-
independent.
6. Acknowledgment.
I am indebted to Professor S. N. Roy for suggesting this problem
I
and for
suggesting improvements.
References
[ i ] Roy,
J., "Step-down procedure in multivariate analysis," North
Carolina Institute of Statistics Mimeograph Series No. 187
(1957).
(
~,
!.
~
,
.
C2J
Roy, S.N., and Bargmann, "Tests of multiple independence and the
associated confidence-bounds,"
(1958) pp. 491-503.
~. ~. ~.,
Vol. 29