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This research was supported by the National Institutes of Health,
Institute of Medical Sciences, Grant No. GM-12868-04
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A GENERAL METHOD FOR OBTAINING TEST CRITERIA FOR
MULTIVARIATE LINEAR MODELS WITH MORE THAN ONE DESIGN MATRIX
AND/OR INCOMPLETE IN RESPONSE VARIATES
David G. Kleinbaum
University of North Carolina
Institute of Statistics Mimeo Series No. 614
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ABSTRACT
KLEINBAUM, DAVID GEORGE.
A General Method for Obtaining Test Criteria
for Multivariate Linear Models with More Than One Design Matrix and/or
Incomplete in Response Variates.
In this paper a test of a general linear hypothesis for the
More General Linear Multivariate Model (MGLMM) is obtained under normality assumptions on the original data by using a test criterion given in
a general form by Wald [12].
The asymptotic distribution of the test
statistic under the null hypothesis is a central chi-square variable.
As a special case of the above test, we obtain a test of the standard
MANOVA linear model which is Rotelling's trace criterion.
Other special
cases of the general model include the RM, GIM and MDM models given by
Srivastava [4J, [5], [8J, [9].
The test statistic uses estimators of the parameters which need only
be asymptotically equivalent in probability to the maximum likelihood
estimators.
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A GENERAL METHOD FOR OBTAINING TEST CRITERIA FOR
MULTIVARIATE LINEAR MODELS WITH MORE THAN ONE DESIGN MATRIX AND/OR
INCOMPLETE IN RESPONSE VARIATES
David G. Kleinbaum
1. INTRODUCTION AND SUMMARY
The standard multivariate linear model (SM) given by:
E(Y)
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Var(Y)
A l;
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n
0 I
where yen x p), A(n x m), l;(m x p), I(p x p)
contains two assumptions inherent to the experimental situation which are
not always met in practice.
(i)
These are
each response variate is measured on each experimental unit
(i.e., on each experimental unit is observed a p-variate vector).
(ii)
the design matrix, A, is the same for each response (e.g., the
same blocking system is applicable to each variate).
Situation (i) will, in general, not hold whenever it is physically
impossible, uneconomical or inadvisable to observe each response variate
on each experimental unit.
Situation (ii) will not hold when different
blocking systems are applicable to different response variates and when
some of the response variates are insensitive to certain treatments.
This paper is concerned with testing linear hypotheses under normality assumptions on the original data for multivariate linear models in
which assumptions (i) and/or (ii) are relaxed.
In this connection, we
define a model, called by the author the More General Linear Multivariate
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Model (MGLMM) .
Special cases of the MGLMM are the HM, GIM and MDM
models of Srivastava ([4], [5], [7], [8] and [9]) and the incomplete
variable designs of Monahan [2].
A test of a general linear hypothesis
for the MGLMM is proposed, which uses a test criterion given in a general
form by Wald [12].
The Wald test statistic uses estimators of the un-
known parameters which need only be asymptotically equivalent in probability to maximum likelihood estimators.
The asymptotic distribution of the
test statistic under the null hypothesis is a central chi-square variable.
When restricted to the SM model, we obtain Hotelling's trace criterion as
a special case.
2.
THE MORE GENERAL LINEAR MULTIVARIATE MODEL
Assume there are n experimental units and p-response variates
VI"'"
V in total.
p
The n experimental units are divided into u dis-
joint sets of experimental units Sl' S2"'"
On each unit in the set S., we measure q.
J
Vo
"'jl
,V o
, ••• ,
J
Su with n j units in Sj'
« p) responses
-
(The remaining p - q. response variates are not
J
"'j2
measured in S,).
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lThere are meaningful multivariate linear models which are not special
cases of the MGLMM (e. g., growth Curve models). Hence we have used "More"
instead of "Most" in the model name. Nevertheless, the technique used in
this paper for obtaining test statistics can be used for linear models which
are not special cases of MGLMM.
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Then the MGLMM is given by
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Var(Y. )
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~B'l;B.
n.
J
j
J
:::"'i jq .
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and for each i, i
that
~'i'k = ~,
= l, ••. ,p
there exists at least one pair (j,k) such
i.e., 'i jk = i (so that the total set of unknown parameters
J
is (~l'"'' ~, l;)).
Y.(n. x q.) is the matrix of observations for the jth set S.
J
J
J.
J
(n. x m'i
A.'i
J jk
) is the design matrix for response V'i
jk
J
in the
jk
set S., and rank A. n
J
~
'i
(m
jk
'i
JX"jk
x 1)
jk
is the parameter vector for response V
'i
jk
(and mn
= mn
whenever 'i '
Jk
Njk
Nj 'k'
Bj(p x qj) is the incidence matrix for response variates consisting
= q .•
of O's and lIs and rank B.
J
J
If the variates V , V are not measured together in any set Sk'
r
we set 0
rs
:: 0 and 0
sr
s
:: 0 in l; = ((0 .. )).
1.J
For testing the hypothesis H :
o
is of full rank
(1)
j=l
c. {J'
J
= c. we will assume that
Y. and
J
A
J
Y~
J
are independent if j I j'
=
0 where C. (c. x m.)
J
J
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(2)
the rows of Y. are also independent and distributed as a
J
q.-variate multinormal vector with variance covariance
J
rna t rix B ~ I B.•
J
J
From (1) and (2) we can write the log of the likelihood function for
MGLMM as
n.q.
u
log
cjJ
= -
I
. 1
-~
2
J=
1
-"2
u
tr[B
I
j=l
We must use log
u
10g2TI -
~
I B.]
J
J
n.
I
j=l
-1
-t log
IB ~
J
I
B.I
J
[Y. - E[Y . ] ]' [Y. - E[Y . ] ]
J
J
J
J
to obtain our test statistic.
cjJ
The technique used is
described in section 4.
2.1.
2.1.1.
Special Cases of MGLMM
Multiple Design Multivariate Model - MDM (Srivastava [4], [8],
[9]) u = 1; n l = n; ql = p; Yl
= Y;
Ap
Var(Y) = I
2.1.2.
n
Bl - I .
P
~] (n x
p)
Hierarchical Model - HM (Srivastava [4], [8], [9])
B.
J
SI, j k
= k, k= 1, •.• ,j ;
1.
J
ap-j ,j
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Thus we get
3 I
u = p; qj = j, j =1, ••• ,p;
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L_u
Thus we get
E[Y .]
J
Var(Y.)
J
[Aj 1 S.l
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n.
J
A S.2
j2
® B' 1: B.
J
J
A .. s.J'] (n. x J')
JJ
J
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.,
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In [8] and [9], Srivastava assumes A
jl
U = the set of all experimental units on which the response
r
variate V is measured
r
then for the HM model we have
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A .. - A .•
JJ
J
Note that if we let
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I
= Aj2
The HM model thus deals with a situation in which the response variates
can be graded in descending order of importance, and, further, if
V is more important than V , then V is observed on each experimental
r
s
r
unit on which V is observed.
s
General Incomplete Multivariate Model - GIM (Srivastava [5], [7],
2.1.3.
[8]; Monahan [2])
A. n
JN
= A· t
jl
J jq.
J
- A"
J
j
1 , ... , u
is (m x 1), where m is independent of j and t '
~~
Jk
jk
Thus
Var(Y.)
J
=
I
n
~ B~ ~
j
J
B.
J
Monahan ([2]) considers the special case q. - q; n. - n/u;
J
j = 1, ...
,u.
J
A. - A •
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3.
REVIEW OF THE LITERATURE
A number of different approaches have been used to obtain tests of
hypotheses for the MDM, HM and GIM models.
The tests obtained are generally
impractical with regard to computation and require more assumptions with
regard to the hypotheses and design matrices than the usual estimability
requirement.
For example, the necessary and sufficient conditions given by
Srivastava in [9] for reparameterizing the MDM model to the SM model (for
which there are several good test criteria) is that the vector spaces spanned
by the columns of the different design matrices are all the same.
In [5],
Srivastava restricts the GIM model (actually a slightly more general form
of the GIM model in which
~.
-J
is (m. x 1) and A. is (n. x m.)) to be what
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he calls a "strongly regular" design.
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J
In [4], he proves to be not strongly
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regular a particular example of the kind of design likely to be considered
in practice.
In [7], however, he works through a GIM model which is strongly
regular and specifies the general computational method for reparameterization.
Nevertheless, the class of "strongly regular designs" needs to be better
described in terms of GIM models likely to be used in practice.
This problem
seems formidable since the computations needed for validating the "strongly
regular" property are quite complex.
Furthermore, the reparameterized form
of the model has a E matrix which is restricted by
Ip ij I ~
A ~ 1 where
ij
A.. may be < 1 for some i # j and p .. is the correlation between variates
1J
1J
V. and V.•
1
J
The problem of testing a hypothesis in a SM model with the above
type of restriction on E has not yet been solved and is likely to require a
laborious computer procedure.
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In [11] Trawinski (i.e., Monahan) and Bargmann summarize the main
results of Monahan's thesis [2] in which is obtained a likelihood ratio
test for the very special case of the GIM model in which u = p, n.
J
q.
J
~
q and A.
J
A, j=l, .•• , p.
~
~
n,
The maximum likelihood equations require
for solution an iterative technique based on the Newton-Raphson method.
In demonstration of the method for a particular set of data, the initial
estimate
~
o
of
~
used in the iteration turned out to be very comparable
to the final estimate obtained after several iterations.
(or a more general form of
~
o
ars
O
~
o
estimate
) is a likely candidate to be studied for use
in the Wald statistic proposed by this author.
in which the(r - ~element
The
It is a pooled estimate,
is obtained by averaging estimates of a
rs
from all the sets S. in which both V and V are measured.
J
r
s
In [10], Srivastava has suggested several different union-intersection
type tests for the HM model, one of which was obtained in his earlier paper
[4] and which involves a generalization of J. Roy's step-down procedure [3].
The model that he uses in [4] includes the MDM model as a special case.
The testing procedure involves making a sequence of independent F tests and
rejecting the hypotheses if any of the F tests are rejected.
As is the case
in general with union-intersection type tests, little can be said about the
properties, asymptotic or otherwise, of the step-down method.
The Wald statistic proposed in this paper was used by Allen [1] in
connection with nonlinear multivariate models, a special case of which is
the SM model.
Allen showed that the likelihood ratio criterion and the Wald
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criterion for his nonlinear model are asymptotically equivalent.
In small
samples, however, Wald's criterion proved not to be a good approximation
of a chi-square variable.
Nevertheless, this may have been due to the use
of a linear approximation of the test criterion (which is not necessary for
the linear MGLMM).
In [8], Srivastava considers distribution-free estimation of linear
functions of the design parameters for the HM, GIM and MDM models. Srivastava
gives necessary and sufficient conditions for 8 =
P
L
i=l
c'.L
i --..L. to have a BLUE.
These are conditions on the vector spaces spanned by the columns of the
design matrices corresponding to the different response variates.
The
II
proofs have been simplified considerably by this author.
4.
4.1
~-'
-n -
!U'
variables !l"'"
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DERIVATION OF RESULTS
X ,8) be the joint density of independent random
Let ¢(!l"'"
where
!.:i - Pi (!,
i2,), i=l, •.. ,n.
which mayor may not be of the same length.
The X. are vectors
~
Assume that the total number
of parameters, say u, in 8 is independent of n.
Let
"8 = M.L.E. of 8
and
B:(~ ~ lin
=
(u x u)
Then' under suitable regularity conditions on the p.1 (X,
-
"
(i)
1,(
vU' B*1/2 (8)
(ii)
X(
(8)
n(~-iD' B*
n-
(iii)
X<
n(~
n
- iD'
-
(~
- ~»-+
(i2, -
B*(8) (~
n-
iD)
- ~»
MN u(0, I)
--r
2
Xu
_
2
Xu
..+
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iD
we have that
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provided the following conditions are satisfied:
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(1)
n
~
----+-
o as
n~
for all j,j=l, ... ,u, and 0<0<1
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and
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r
(2)
{
nEl-! ~
!
Comments:
2
_~(l+ ~)
a log¢!
n i=l
ae 2j_'
1-
I
n
I ()log
L EI
. 1 '
1=·
P. (X. ,e) j 2+0
a
1
e.J
-~
-I
.-+ 0
as n~ for o>O,j=l, .•• ,u
This is a direct generalization of a well known theorem for
maximum likelihood estimators in which the X. are i.i.d. random variables.
-:L
The statistic given in (iii) was first suggested for use in hypothesis
testing by Wald [12].
Conditions (1) and (2) are not severe restrictions
for the models discussed in this paper;
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IIHo
2
. a log p. (x.
,e)
-1 , ~7---1+0 L E
ae 38.,
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n
i=l
J
J
1
the conditions merely require
that each design point be repeated often enough and each distinct variate
(e.g., blood pressure) observed often enough
4.2
Corollary.
Given the same situation as in Lemma 4.1, and suppose that
A
e'=
(~',
2:.')
in large samples.
so that
A
r = M.L.E. of
rand
cr
= M.L.E. of cr.
Let
(u x u)
Then if (1) and (2) of Lemma 4.1 hold, and, in addition,
=0
we have that
for all
i,j
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10
MN (Q , 1)
(i)
. (ii)
(iii)
asymptotically
equivalent
i.e.,
to~,
Irl (.!.t-
i)- 0,
and
(J''<+ (J
P
4.3
Theorem
•
P
Given the standard MANOVA model:
E (Y)
A F;,
=
Var(Y)
In
=
~
l.
where yen x p), A(n x m), F;,(m x p), L:(p x p), rank A = m.
Suppose the rows of Yare distributed p-variate multinormal.
=
Let Ho : C F;,
Let F;,
[~l
=
°where C(q
Sp]
.•.
x m) is of rank q.
and L: be the usual M.L.E. of F;, and L: respectively.
Then under H we have
o
_. ~ ~ -, ,{ C·_-1
'.
.,
I 0
C
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I
Spl·.J
:
A
C ~l
.
.. 0
"
'A
C
-'
c_
2
is distributed asymptotically as X '
qp
Proof:
logcjJ(Y,
F;"
L:)
=-
nn
~ log2~
n
1
- 2
log 1_ L: I - 2
tr L: -1 (Y - AF;,)' (Y - AF;,)
1 dtr L: -l(Y-AF;,) , (Y-AF;, )
-2
dF;,
= A'(Y-A~)L:-l
(k_£)th element of A'(Y-AF;,)L:- l
'"
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11
where A(n
x
m) = «a
uw
»'
t; (m
x,
p)
Y(n x p)
Now
d dlog¢·
_ dt;
.
da
A I (Y - At;)
qr
so that
-1
A'E(Y - At;) ~
da
qr
E
o
(m x p)
and
E
[a(~~~:J
=
k=l, ... ,m
0
q,r,R-=l, ... ,p
Also
n
rRL: a ka a
u=l u uq
- a
rR-
{(q
th
col. of A) . (k
th
col. of A)}
- arR- {(q_k)th element of A'A}, which is a constant
independent of Y.
Thus
B (t;,
L:)
=
=
[L:- l 3 A' A]
(mp x mp)
Now from Lemma 4.2(i) we have
-
r.t;ll.
: I
l
.!
•
I
~J
is asymptotically MN
mp
(Q
,n[L:-
l
~ A'A]-l)
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Similarly
- "l)
I C ~lll
,~is asymp to ti co 11y MN
q
:
.
.". \
L~ ~ i
j
t'1-. n ~
o
o
C
Thus if H is true we amy use Lemma 4.2(iv) to obtain
o
W
-c.i~!'[G
I
jC ~
0
0
[2:- 1
@
A'AJ- l
:1 '1- . ~ iJ
1
IC,
I
0
LC:~
C, )\
C
2
.
is asymptotically distributed as X
pq
q.e.d.
We will show later that W is equivalent to Hotelling's trace criterion.
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4.4
Main Theorem
r'\ c.
(A test of H
o
j=l
J
o
~.
for MGLMM).
J
Suppose we have observation matrices Y.(n
J
a MGLMM.
x q.), j=l, ... ,u from
J
j
Suppose also that Y. and Y., are independent if j f j' and
J
J
that for a given j, the rows of Y. are independent and distributed
J
q. - variate multinormal.
J
Let 2: be an estimate of 2: such that 2:
mates
of~,
i=l, ... ,p such that
~;~'
2: and
+
p
~.,
-1
i=l, ... ,p be esti-
i=l, .•. ,p, as in Corollary 4.2(iii).
Let
nB (I;, 2:)
A'.
J rs
r
Aj
s)) (
rs
tm
r=l r
x
~
m )
r=l r
where
1)
J. rs runs over all integers k for which both responses Vr and Vs
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are measured in Sk'
a
2)
rs
is the element of [B~ ~ B. ]-1 which corresponds to
jrs
J rs
J rs
the variate pair (V , V ).
r
s
sr
(Note (1)
- a.
J rs
(2)
If the variates V and V are not measured together in any
r
s
Sk' then we set
Or,s
J rs
A'.
A.
Jrsr
-
= Oem
Jrss
x m) •
s
r
x m )
.)
r
p
Then under H :
o
nj=l C,J
~, = 0
J
where
c.
J
(c. x m.) is of full rank (
J
J
=
c.)
J
we have
A
Cl ~l
C2 ~2
-.
A
W= n
~l
\
0
0
lI
;
,
'A
Cp
4
Cpi
:c..J
l ')
~l
B
-1
A
(~
A
,
~)
0
\
O! ~
'
c~ )
A
-1
Cl ~l
C2 ~2
A
'A
Cp ~
,
is asymptotically distributed as a central chi-square variable with
p
~
j=l
Proof:
c,
J
d. f.
This follows as a direct extension of Theorem 4.3.
Detailed proofs
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are given for the MDM. GIM and HM special cases.
4.5
4.5.1
Special Cases of the Main Theorem
MDM:
W
=
fe l .illI' r~ll~l1AiAl'"
,0.
:
C
!_ p
--'
-1
where E
4.5.2
0 '
~I
=
:
i.
i: 'f
•
'1- i;'
1
(
\
I
C
p
I
)
IC l
IC : A •
,I.u p
«0 rs )).
:.
,
0
I
0
ApAl ••. &PPA p'Ap
~.a
..
n
J.>V
IC l
1 p
-
~1
'c 'Apl ,
HM and GIM (assuming A'
w=
~lp A'A!-l
A
C t: i
p 1>,
_..
-
I
.
jl
= A
j.Q,j2
=A . .Q,
=A.)
J jq.
J
J
-ql"i
u
'. O. [ E {B. [B ~
0'. .
. -1 J J
JC
p
Proof of 4.5.1
log
~
n log IE I - 2
1 tr E-1 [ Y-E (
log 2TI - 2
Y)] , [ Y-E (Y) ]
= -np
2
,
where E(Y)
=
I
[Ai ll; A2 l2 ' ••• ' A ~] and
I
,I
p-p
« (1..Q,
since [Y-E (Y) ] , [Y-E (Y)]
- A.Q,l.Q,) '(Yk - ~)))
Now
dlA~~
~ - -
diP
P
[- E
2:
2 k=l
9,,=1
dSw - dSw
0
k.Q,
(v
n
~
-
An ~ n)' (v.
.>V~
~
- A £) ]
k,~
~I
_
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1_
15
"'I
rh
a og~ _
d~W
where A (n
w
-
x
P
L
wnN
La
,Q,=1 t=l
P
a
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m,Q,
L
a
,Q,=1 t=l z=l
~
a
(w)tk (,Q,)tz-z,Q,
aw,Q,
m)
w
= {(k_q)th element of A'A } . a wr
w r
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( L q. n. ) 10 g 27T
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1
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i=l
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L
B.1 I
(Y. - A.~B.)'(Y. - A.~B.)
1
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where
~
I
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-1
1
- -
U
L:
2 i=l
a trU.1P.P
~
11
where p.
(Y.A. B.)' , which is (ql' x n)
111
1
Ui i
B' i
L: B' which is ( qi x qi )
i
i=l
~
ai:"
sk~
A.~B,)U~lB~
A! (Y,1
1
,
1
:'Ith
=k-~)
1
1
element of
p
q.
ni
qi
L.
L.
L.
L.
~
~l
~
~
i=l z=l u=l t=l
1
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L:
i=l
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aiukYiukU
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qi
P
ni
qi
m
L:
L:
L:
L:
L:
L:
-
i=l z=l c=l u=l t=l u=l
2
a log
cjJ
a'~~a~qr
=
-
p
qi
n.
qi
L:
L:
L:
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1
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a.lUka.luq )
1
b.~
1
b.lNt
o
_I
t
qi
( L: u izt b
)}
b.
Ht
z=l lrz t=l
{ L:
qi
{L:
z=l
b.
~z-~fh
lrz_
element of U:-1B.J}
1
u
1 1
1
izt
izt
. ,th element of B,U.-1 B.-1 J
-q th element of A!A.J [~-~
P
-1
L: {B.D. B! ~ A!A.}
i=l
B'
i
a.lUka.lUW ~WCb.lCZ u
1 1
p ~
1
b~t
t [~_qfh element of A!A.J
i=l
-
-1
U.
1
qi
1
( L:
L:
1
a.lUka.luq b.lrz u
n.
p
A.~B.)
1
1
1 1
1 1
1
1
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••
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17
I.
I
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I_
I
I
I
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I
I
I
I·
I
P
l:
Le., nB(1;; ,l:)
B.[B~
l: B.]-l B~ ~ A~A.
111
111
i=l
q.e.d.
5.
THE EQUIVALENCE OF THE WALD STATISTIC AND
HOTELLING'S TRACE STATISTIC
5.1
Theorem
The Wald Statistic (W) obtained for SM is equivalent to Hote11ing's
2
trace criterion (T ).
o
Proof:
T~ = trs s;l where SH = (cg)'[C(A'A)-l C,]-l (Cb
H
S = Y'[I - A(A'A)-lA'] Y
E
Now
A
l:
1
= -n SE is the M.L.E. of
l:
Thus
Now
iC
.. , OJ -1
! \
!
I
.. 0
10 "
L
'c
gl
C
-1
.
• A
C
I;;
-p
)!
/
I
.1
18
C
E;,
.-1
,
C~
p
L:
P
L:
i=l j=l
1
1
A_I
tr (cE;,) , [C(A'A)- C']- (CO L:
q.e.d.
6. CONCLUSION AND FURTHER RESEARCH
In Theorem 4.4, we have obtained a general test criterion for testing
linear hypotheses in multivariate linear models which are incomplete in
response variates and/or in which different response variates have different design matrices.
When restricted to the standard MANOVA model, our
test criterion becomes Hotelling's trace criterion.
The general test
criterion uses estimators of the unknown parameters which are asymptotically
equivalent in probability.
Maximum likelihood estimators could therefore
be used, but even for simple cases of the general model (e.g., Monahan
[2]) outslde of the standard model, an iterative method is required.
Computationally simpler estimators can probably be obtained, nevertheless,
since we only require them to be asymptotically equivalent in probability
to maximum likelihood
done here.
estimators.
In any case, further work needs to be
Also it is necessary to study the small sample properties
of the test criterion.
I
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REFERENCES
1.
Allen, D.M. 1967. Multivariate analysis of nonlinear models. Institute
of Statistics Mimeo Series No. 558, Univ. of North Carolina, Chapel Hill,
North Carolina.
2.
Monahan, I.P. 1961. Incomplete-variable designs. Unpublished thesis.
Virginia Polytechnic Institute, Blacksburg, Virginia.
3.
Roy, J. 1958, Step-down procedure in multivariate analysis.
of Mathematical Statistics 29, 1177-1187.
4.
Roy, S.N. and J.N. Srivastava. 1964. Hierarchical and p-blick multiresponse
designs and their analysis. Mahalanobis Dedicatory Volume. Pergamon Press,
New York.
5.
Srivastava, J.N. 1964. On a general class of designs for multiresponse
experiments. Inst. Statistics Mimeo Series No. 402, Univ. of North
Carolina, Chapel Hill, North Carolina.
6.
Srivastava, J.N. 1965. A multivariate extension of the Gauss-Markov
theorem. Ann. Inst. Statist. Math. 17, 63-66.
7.
Srivastava, J.N.
8.
Srivastava, J.N. 1966. On the extension of the Gauss-Markov theorem to
complex multivariate linear models. Aerospace Research Laboratories
Publication.
9.
Srivastava, J.N. 1966. Some generalizations of multivariate analysis of
variance. Symposium on Multivariate Analysis at Dayton, Ohio, 1965,
pp. 129-145.
1966.
Incomplete multiresponse designs.
Annals
Vol. 26, Sankhya.
10.
Srivastava, J.N. 1968. Some studies of intersection tests in multivariate
analysis of variance. Paper presented at Symposium on Multivariate Analysis
at Dayton, Ohio, 1968.
11.
Trawinski, I.M. and R.E. Bargmann. 1964. Maximum likelihood estimation with
incomplete multivariate data. Annals of Mathematical Statistics. 35,
647-658.
12.
Wald, A. 1943. Tests of statistical hypotheses concerning several parameters when the number of observations is large. Transactions of the
American Mathematical Society 54, 426-482.