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A NOTE ON STEP-DOWN PROCEDURE IN MANOVA
PROBLEM WITH UNEQUAL DISPERSION MATRICES
By
S. R. Chakravarti
Department of Biostatistics
University of North Carolina at Chapel Hill
and
University of Calcutta, India
Institute of Statistics Mimeo Series No. 910
,
"
FEBRUARY 1974
•
A NOTE ON STEP-DOWN PROCEDURE IN MANOVA
PROBLEM WITH UNEQUAL
D~SPERSION ~TRICESI
By S. R. Chakravorti 2
University of North Carolina, Chapel Hill
In this paper we have considered step-down procedure
in multivariate analysis of variance problem when dispersion
matrices are different and unknown.
The distribution
problem of the test criterion has also been studied under
the null hypothesis.
1.
Introduction.
'\IVV\fV\/\.IV\
Step-down procedure in standard MANOVA problem has
been considered by J. Roy [7].
The essential feature of this procedure
is that if on some a priori grounds the variates are arranged in
descending order of importance, then the test procedure can be carried
out sequentially by considering marginal and conditional distributions
of the variates concerned.
At each stage F-statistic can be used which
are independently distributed under the null hypothesis so that the
overall hypothesis can be tested by combining the component tests.
Optimum properties of this procedure have been discussed by Roy [7]
and Roy et a1. [9].
lWork sponsored by the Aerospace Research Laboratory, U. S. Air Force
Systems Command, Contract F336l5-71-C-1927. Reproduction in whole or
in part permitted for any purpose of the U. S. Government.
20n leave of absence from the University of Calcutta, India.
•
2
In this article the problem has been considered when the dispersion matrices are not equal.
It has been shown that by transforming
•
the original vector variables to Scheffe-vector-variab1es (Eqn. (2.2)
below) Hote11ing's T2 test can be constructed at each stage of the stepdown procedure and these T2 ,s are shown to be independently distributed.
It may be noted that the distributions of the Scheffe-vector-variab1es
chosen from the original vector variables at different stages (vide,
Remark at the end of Sec. 2) may not, in general, be the same as those
considered earlier.
This is possibly true under certain restrictions
(to be shown in Sec. 3), when the test constructions will be difficult
in this situation.
However, a reasonably satisfactory solution can be
attained only in Bhargava's [3] procedure, which is a
n~dification
of
Anderson's [2] procedure in generalized multivariate Behrens-Fisher
problem.
2.1.
Hypothesis and Test Construction.
Let x(t) (lxp) be the a-th
-a
observation vector in t-th population (a=l, ••. ,n , t=l, .•. ,m) and
t
distributed as N (n(t), E ), where n(t) (lxp) is the mean vector and
p -t
~t(pxp) the dispersion matrix in t-th population.
The problem is to
test
(2.1)
against the alternative of at least one inequality among n(t),s.
Anderson [2] proposed a Rotel1ing's T2 test for (2.1) by assuming
n l < •••
~ nm and transforming the variables ~~t) to SCheffe-vector-variabletlll
3
•
(2.2)
for r=2, ••. ,m, a=l, ••• ,n , where U = (u(2) , .•. ,u(m»
-a
-a
-a
l
jointly follow a
p(m-l)-variate normal distribution.
Now suppose the p variables in the vector x(t) are arranged in
-a
descending order of importance and the ordering remains the same for
each t=l, ... ,m and is given by
(t)
(2.3)
= (xl
a
, ••• ,x
(t)
pa
)
After transforming the vectors x(t) to (2.2) if we order the variables
-a
in u(r) (lXp) as (Ul(r) , ••. ,u(r», these new variables correspond to the
-a
a
pa
ordered variables in (2.3) through this transformation.
!!a(lXp(m-l»
= (U , .•• ,U ), where U.
-la
-pa
-la
(2)
Now writing
(m)
(U . , ..• , U. ), we have the
la
la
distribution of U as N ( 1)(8,f), where for r=2, ..• ,m, i=l, ••• ,p
-a
p m-(2.4)
[(p(m-l)Xp(m-l»
(2.5)
= (Ii.)' I· . (m-lxm-l) =
J.~
n
(2)
n
a~:)J
~ (m)
l
+ diag ( -l a.. , ... , a . )
n 2 1J
nm i J
where i,j=l, •.. ,p,
~(m-lxm-l)
a~:) is the (i,j)th element of Et , t=l, ••. ,m, and
1J
is the matrix with all the elements unity.
Now if we consider j-th step-down 9rocedure, then we are to
consider the conditional distribution of U. (lxm-l) for fixed
-Ja
U(. 1)
- J-
a
= (U l , .•• ,U. 1 ), which is an (m-l)-variate normal distribu- a
-J- a
tion with mean vector W. and residual dispersion matrix f. 1
-J
where
. l'
-J. "",J-
•
4
•
(2.6)
~J~-l
(2.7)
=
(13~
. 1)
-J, 1,···,13~
-J,J-
is a matrix of order (m-1)x(m-1) and B. 1 is the (j-1)th
where I3
- j ,s
-J-
order step-down regression matrix.
Under this set up, the hypothesis (2.1) can be written as
follows
.
(2.8) H [n
(1)
= .•• =n
o -
-
(m)
]
~
n
H :[8=0]
0
I
H(j ) [8 . =0 8.
-.
--
J=l
=0 ]
-J - -(J-1) -
0
-
Thus the component hypothesis R(j)[s.=O] can be tested from model (2.6)
o
-J2
by Rote11ing's T (Anderson [1], Page 187), where
A -lA
(n -(m-1) (j-1)-1)n s .S s~,
1 -J - U.-J
1
(2.9)
J
where
A,
A,
(13.
. . 1)
-J, 1,···,13
-J,Jand
n
~U
This
j
=
~U.
J
1
a~l (~ja-§j-~(j-1)a~j-1)'(~ja-§j-~(j-1)a~j-1)
is distributed as Wm_1 (n 1-(m-1)j, r.
1
. 1) and
. . . J. , ... ,J-
(n -(m-1)j) (m-1)
1
-1
. -1
(n -(m-1) (j-1)-1)
1
F(m-1,n -(m-1)j), j=1,2, ... ,p.
1
2
T is distributed as
j
•
5
~u.
T2 ,
1
It is clear that for fixed j,
are independently distributed (Anderson [1]).
J
•••
•
A
2
Independence of Tl,
..• ,T p2 .
2.2.
~.
~J
and
To prove that
,T 2 are independently distributed under H , let us consider a vector
p
.
0
~(lxm-l) of real elements so that ~~u.~'/~[j.l,•.• ,j-l~' is distributed
J
as xj(nl-(m-l)j).
this
x~
not.
J
Now following Roy et ale [9], Page 47, it follows that
is distributed independently of U(. l)t', whether H(j) is true or
- J-
-
0
So that X~, ... ,x~ are independently distributed.
therefore, (Rao [61, Page 453) that
distributed.
Also under H(j)
o
'
t.t'
~J~
~U
It follows,
for j=l, •.• ,p are independently
j
is distributed independently of
A
u(.
1) -t' and is true for every
~ Jof "Q(j-l)'
(2.10)
~J
so that
~.
~J
is distributed independently
Since T~ in (2.9) can be written
J
[nl-(m-l) (j-l)-l]
where the distribution of
e.
~,
(Rao [6], Page 458).
-1 2
A
A
-1
A
-IA
A
-1
A
J
-IA
A
-1
~.SU ~~/~.r.
~J-
A
nl~·r.
1 , ••• ,J. l~~(~'Su
~~/~.r. 1 , ..• ,J. l~~)
-J~J'
~J ~J
.~J ~J~J'
~J
Tj
.~J
-J~J.
1
A
. l~~ does not depend on
, ••• ,J- ~J
HenceJunder H(j), the conditional distribution
0
of T: for fixed U(. 1) does not depend on U(. 1) (j=2, ••• , p) and T12
J
~ Ja
- J- a
is distributed as (nl-m+l)-l(m-l)F-distribution with d.f. (m-l, nl-m+l).
Hence unconditionally also T~, ... ,T~ are independently distributed.
The test criterion for the overall hypothesis (2.8) can be constructed from the component tests either by using union-intersection
principle (Roy [8]) or by considering the test criterion A =
~ A(j) ,
j=l
is the product of
(m-l) independent beta variables.
The exact as well as asymptotic null-
distribution of the statistic A are available (Chakravorti [4], eqn.
4.39 and 4.55 for zero non-centrality parameter).
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6
REMARK.
It may be noted that if we start with the marginal and con-
'
1 d'1str1'b ut i ons 0 f xl(t) ' x ( t )g1ven
.
(t),
di t10na
xl(t)
a
a , .•• ,x pa g1ven
2a
xl(t) , ••• ,x(t ) from the distribution of x(t), and then choose the
a
p- l a
.
-a
Scheffe-variables in each stage of the step-down procedure, the distributions of the resulting vector-variables will not be the same as
those obtained from (2.2).
Since these variables are correlated both
variate-wise and group-wise, we are to impose a number of restrictions
on the regression coefficient matrix B. 1 to satisfy this requirement,
-Jin which case a satisfactory solution of the test construction is not
easily available by Anderson's procedure.
However, if we assume that the regression coefficients of
x~t)
J
on x(t) (s=1,2, ••• ,j-l) are same for t=1,2, ••. ,m, we can apply the
s
procedure of Bhargava [3], which is the modification of Anderson's
procedure.
3.
Following Bhargava [3] let us assume that
nl=(m-l)n', n = .•• =n =n' and consider the following transformation on
2
m
x(t)
-a
'
= x(r)_x(l)
-a
-a+(r-z)n' ,
(3.1)
r=2, ... ,m; a=l, •.• ,n'.
Then under the set up considered in Sec. 2.1, we can choose the vector
l!a(l Xp(m-1»
= (U l , .•. ,U ), which is distrubuted as N ( 1)(8,r*),
- a
-pa
p m--
where
(3.2)
(2)
(m)
8(r)_ (r)_ (1)
8= (8 , ... ,8 ); 8, = (8 , , ... ,8 . ) ; . - n.
n,
- -1
-p
-1
1
1
1
1
.1
(3.3)
r*=(t~,);
t~,=a~:)I
1 + diag(a~:) , ••. ,a~~»; i,j=l, •.. ,p.
1J -m1J
1J
t
t
1J
.
1J
Then j-th step-down procedure will lead to the model (2.6), where 8. 's
-J
•
7
are defined according to (3.2) and submatrices of B. 1 given by (2.7)
- J-
are for s=1, ••• ,j-1
(3.4)
e.
-J,s
•
(m-1 m-1) = diag(S~2,2) , ••• ,S~m,m».
.
J,s
J,s
When S~r,r) remains constant for r=2, ••• ,m,(Sj (say» the conJ,s
,s
ditions stated in the remark of Sec. 2 are satisfied. Now since the
components of U. = (U (2) , ••• ,u~m»
ja
-Ja
Ja
are independently distributed let us
write the null hypothesis (2.8) as follows from (3.2),
(3.5)
m (r) (r)
(r)
(r)
\--}
~
(l HoJ' [8 J. =ol~(J'-l)=Q]; ..§(J·-1)=(8 1 ,···,8 J·_1 )
j=l r=2
p
- n
Thus we can construct test for H(:) by using individual estimates to
oJ
m
S.
for r=2, ••• ,m and hence the combined test for H • =
H (:) by
J,s
OJ
r=2 OJ
n
(3.6)
L
j
=
m
II
r=2
[l+j(ll-j)-l E ]-1
jr
where Fjr fo110wsCeMMJ . f-distribution with (j;"~Jd.f. Obviously
L. is the product of (m-1) independent beta variables, the distribution
J
of which is well-known (Anderson [1], Rao [5]).
Test criterion of the
p
overall hypothesis can therefore be constructed by considering L = II L .•
j=l J
The independence of L. for j=l, ••• ,p can be verified by the similar
J
arguments as in Sec. 2.2.
~.
The author is grateful to Professor P. K. Sen for
helpful discussion.
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8
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REFERENCES
[1]
ANDERSON, T.W. (1958).
John Wiley & Sons, New York.
Analysis.
[2]
An Introduction to Multivariate Statistical
ANDERSON, T.W. (1963).
A test for equality of means when
covariance matrices are unequal.
Ann. Math. Statist. 'VV
34
671-672.
[3]
BHARGAVA, R.P. (1971).
A test for equality of means of multi-
variate normal distributions when covariance matrices are
Cal. Stat. Assoc. Bulletin 20 153-156.
unequal.
[4]
'VV
CHAKRAVORTI, S.R. (1973).
On some tests of growth curve model
under Behrens-Fisher situation.
Institute of Statistics
Mimeo Series No. 870, University of North Carolina,
Chapel Hill.
[5]
RAO, C.R. (1951).
An asymptotic expansion of the distribution
of Wilks' A-criterion.
Bull. Inst. Internat. Stat.
~
Part II 177-180.
[6]
RAO, C.R. (1965).
tions.
[7]
Linear Statistical Inference and Its Applica-
John Wiley & Sons, New York.
ROY, J. (1958).
Step-down procedure in multivariate analysis.
Ann. Math. Statist.
[8]
ROY, S.N. (1953).
~
1177-1187.
On a heuristic method of test construction
and its use in multivariate analysis.
kit
[9]
Ann. Math. Statist.
220-238.
ROY, S.N., GNANADESlKAN, R., SRIVASTAVA, J.N. (1971).
Analysis
and Design of Certain Quantitative Multiresponse Experiments.
Pergamon Press, New York.
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