-SOME ASPECTS OF ESTIMATION AND HYPOTHESIS TESTING
FOR GENERALIZED MULTIVARIATE LINEAR MODELS
by
Yang Chyuan Yuan
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1476
December 1984
SOME ASPECTS OF ESTIMATION AND HYPOTHESIS TESTING
FOR GENERALIZED MULTIVARIATE LINEAR MODELS
by
Yang Chyuan Yuan
A dissertation submitted to the faculty of the
University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for the
degree of Doctor of Philosophy in the Department
of Biostatistics.
Chapel Hill
1984
Approved by:
/Advisor
ilqn JlA!/iL
Reader
Reader
YANG CHYUAN YUAN.
Some Aspects of Estimation and Hypothesis Testing for
Generalized Multivariate Linear Models.
(Under the direction of P.K. Sen)
Covariate adjustment procedures have been used to analyze the standard growth curve model.
The procedure requires the selection of a non-
singular matrix H satisfying a set of conditions to reduce the model into
a standard MANOCOVA.
We show the resulting confidence procedure is in-
variant under different selections of the matrix H satisfying the same set
of conditions.
We then extend the covariate adjustment procedure to the
generalized growth curve model where in each disjoint set a subset of
the total responses were measured.
Again.
the confidence procedure is
invariant under different selections of the matrix H when the set of conditions are satisfied in each disjoint set.
In a hierarchical mUltiresponse model. the p responses are ordered.
The model can be analyzed through a step-down procedure.
With good esti-
mates of the variances of the p responses. the resulting confidence set
can be constructed from p quadratic forms of chi-square statistics. We
optimize the confidence procedure by minimizing suitable norms of the confidence set.
The incomplete multiresponse (1M) model defined by Srivastava is
based on a lcss-than-full-rank model in each of the disjoint datasets.
For a design satisfying a set of specified conditions. this model can be
analyzed by transforming the data into the framework of a standard multivariate general linear model.
We propose a corresponding full rank 1M
model. show some simplification of the procedure. and compare these two
models.
We then study the relative performance of the proposed 1M model
and the corresponding complete mUltiresponse model.
iii
Finally, we consider the intraclass correlation (Ie) model.
With
the same p responses measured at each of t distinct time points, a natural extension of the IC
mod~l
is the block version of the Ie model. We
examine the hypotheses of the block IC model as well as some special
cases through likelihood ratio and Wald statistics.
We then derive the
null and non-null distributions for the corresponding test statistics.
e-
ACKNOWLEDGEMENTS
.
I wish to express my sincere thanks to my advisor, Professor
P. K. Sen, for suggesting this research topic.
His constant guid-
ance and support throughout the duration of this study are greatly
appreciated.
I also wish to thank Drs. R. W. Helms, J. L. Lamb,
K. L. Lee, and M. J. Symons for serving on my doctoral committee.
Special thanks to Dr. Symons with whom I have had the pleasure to
serve as a research assistant during my graduate studies in the Department of Biostatistics.
The financial assistance from the Department of Biostatistics,
Lipid Research Clinics Program, Cancer Research Center, Northrop
Services, Inc., and Occupational Health Studies Group during my stay
in Chapel Hill are gratefully acknowledged.
I would like to thank Dr. K. Boyle and Ms. V. Kasica for their
friendship and many helpful comments for this study.
Finally, for
their excellent typing of the manuscript, Ms. R. Bahr and Ms. J.
Capparella are warmly thanked .
..
TABLE OF CONTENTS
CHAPTER
Page
I
LITERATURE REVIEW AND OUTLINE OF THE RESEARCH .
1.1
1.2
1.3
Introduction· . . . . . . . . .
Invariant Confidence Procedure.
The Standard MGLM Model. . .
1.4
Confidence Procedures in the Standard
MGLM Model. .
The Growth Curve Models
1
1
2
4
1.3.1
1.4.1
1.4.2
-e
1.5
1.6
1.7
II
The SGCM Model
The GGCM Model
The HM Model
.
The 1M Model
.
Outline of the Research
ON CONSTRUCTING THE CONFIDENCE SETS IN THE
GENERALIZED MULTIVAIRATE LINEAR MODELS
2.1
2.2
2.3
13
16
19
22
.22
.23
2.2.1
2.2.2
Covariate Adjustment in the SGCM Model
Invariant Confidence Procedure. . . .
.23
.25
The Generalized Growth Curve Model. . . . . .
28
Covariate Adjustment in the GGCM Model.
Invariant Confidence Procedure.
The Hierarchical Multiresponse Model
2.4.1
2.4.2
2.5
9
12
Introduction . . . . . . . . .
The Standard Growth Curve Model
2.3.1
2.3.2
2.4
. 7
9
TheHM Model with p=2. . . .
Optimum Confidence Procedure
Examples
2.5.1
2.5.2
2.5.3
An SGCM Model
A GGCM Model
An HM Model
28
.35
.37
37
45
.54
.56
.58
.62
CHAPTER
III
INCOMPLETE MULTIRESPONSE MODEL
.
70
..
.70
3.1
Introduction.
3.2
A full Rank Incomplete Multiresponse Model
.71
3.2.1
.75
•
3.3
Optimum Confidence Procedure.
3.3.1
3.4
IV
Invariance in the Optimum Confidence
Procedure..
...
Some Comparisons between Incomplete Multiresponse Models and Complete Multiresponse Models.
3.4.1
3.4.2
3.4.3
3.4.4
3.5
Removing the Singularity of LL'
Generali zed Variance.
Asymptotic Relative Efficiency.
Cost .
.
Minimum Risk
Numerical Examples
ON COVARIANCE STRUCTURES
4.1
Introduction. . . . .
4.1.1
4.1.2
.77
82
84
85
87
94
97
.103
.109
. 109
Asymptotic Null Distributions of Certain
Likelihood Ratio Statistics
112
Asymptotic Distributions of the Wald
Statistics. . . . . . . . . . . . . . .113
4.2
Testing Block Intraclass Covariance Structures
for a Covariance Matrix: Hl:Eo=I~ E + (.J-I)~E2' .115
l
4.2.1 The Ayrnptotic Null Distribution of the
Likelihood Ratio Statistic. . . . . .
118
4.2.2 Asymptotic Non-null Distributions of the
Likelihood Ratio Statistic when t=2 .
119
4.3
Testing Proportionality for the Two Matrices,
L and L , in HI: H : L =wL
1
2
2
2
l
4.3.1 The Wa1d Statistic When p=2. . . . .
4.3.2 Asymptotic Distributions of the Wa1d
Statistic when p=2
.
120
124
128
e-
IV
(continued)
4.4
•
Testing Intraclass Covariance Structures for
the Two Matrices, II and I 2 , in HI:
H3 : Il=al[I+Pl(J-I)], I 2 =a 2 [I+P 2 (J-I)]
4.4.1
The Asymptotic Null
Likelihood Ratio
Asymptotic Non-null
Likelihood Ratio
4.4.2
4.5
132
133
in H :
3
H4 : Il=al[I+Pl(J-I)]. 1: 2 =a l w J . .
~2
138
The Asymptotic Null Distribution of the
Modified Likelihood Ratio Statistic.
Asymptotic Non-null Distributions of the
Modified Likelihood Ratio Statistic.
4.5.2
•
Distribution of the
Statistic
...
Distributions of the
Statistic when p=2
Testing Equality of Elements of
4.5.1
.-
129
.140
.141
Numerical Examples.
.143
V SUGGESTIONS FOR FUTURE RESEARCH
.149
4.6
BIBLIOGRAPHY
.
.
....
.
152
CHAPTER I
LITERATURE REVIEW AND OUTLINE OF THE RESEARCH
1.1
Introduction
In multiresponse experiments, the responses are commonly assumed
to have a multivariate normal distribution.
With the same responses
being measured on each experimental unit, the most commonly used model
is the standard multivariate general linear model (MGLM).
Once the data of a repeated-measurements experiment have been collected at known times, a natural sequel to hypothesis testing is the
fitting of a single polynomial function to the sample means by some form
of least squares.
In this way. the model for the responses might be tested
more concisely by fewer coefficients as compared to the entire vector.
A standard growth curve model (SGCM) can then be used to analyze this
type of data.
Often the data in the SGCM model are missing either by chance or by
des i gn.
In such a case, the data can be analyzed through a generali zed
growth curve model (GGCM).
Many situations occur
where considerations based on prior information
about the nature of responses and their relative importance in the investigation at hand enable the experimentor to decide which one, among a
•
given pair of responses, needs to be measured on a larger number of experimental units.
In such a case, he could arrange the responses in a de-
scending order of importance into a hierarchical multiresponse (11M) model.
2
In some classes of multiresponse design, not all the responses are
measured on each experimental unit.
Such situations may arise when it is
either physically impossible or uneconomical to study all responses on each
unit.
In this case, we turn to the incomplete multiresponse elM) model.
The covariance matrix is of interest mostly as part of the model
requirements that
it be a known positive definite matrix or that it have
a particular pattern with unknown elements.
In some instances, it is nat-
ural to assume that the covariance matrix of the responses possesses certain symmetry properties that are suggested by the sampling situation.
The main purposes of this study are
1)
To investigate the confidence procedures used in the generali:ed
multivariate linear models.
These models include SGCM, GGCM, HM,
and 1M models.
2)
To test and estimate various covariance structures in blocks.
In this chapter, Sections 1.2 to 1.6 review the related literature
and Section 1.7 out 1ines the research.
1.2
Invariant confidence procedure
Many statistical problems exhibit symmetries, which provide natural
restrictions on the statistical procedures that are to be used.
The in-
variance principle basically states that if two problems have identical
formal structures (i.e. have the same sample space, parameter space, densities), then the same decision rule should be used in each problem.
For
a given problem, this principle is employed by considering transformations
of the problem which result in transformed problems of identical structure.
The restriction that the decision rules in the original and trans-
•
3
formed problems be the same leads to the invariant
decision rules. An
optimal invariant decision rule may be approached from a viewpoint with
proper specifications.
Let X be a random variable, taking values in the sample space X, and
having distribution Pe , 8€
sample space X onto X.
e.
Also let g be a 1-1 transformation of the
Denote by gX the random variable that takes on the
value gx when X = x, and the distribution of gX is Pe"
where
g
with e'
is the corresponding induced transfQrmation of g on
E
8, 8'
=i8,
e.
Suppose one desires inference about a function y of 8, y(8), taking
values in a space
r.
Then, with a constant probability I-a of covering
the true value of y(8), a confidence procedure for y, denoted See), is a
--
function from X into the collection of subsets of
r.
When x is observed,
S (x) wi 11 be the suggested confidence set for y such that P8 (y
for all 8 €
€
S(x))
I-a
~
e.
A confidence procedure given by a class of confidence set
Sex) is
invariant under the group of transformations G if gS(x) = S(gx) for all
XE X, gE G, where gS(x) = {gy: yE S},
With a natural measure of the size of Sex), an optimal invariant confidence procedure can be approached by choosing the invariant confidence
procedure with the smallest measure of Sex) .
Let {ljJ. : i
1
E
I} be a family of parametric functions ljJ.: f-+\j1, where I
1
is an index set and \j1 is an arbitrary space.
regarded as a single function ljJ:
rx
1-+ \j1.
Then {ljJ.: i
1
E
I} can also be
Simi larly, a simultaneous con-
fidence set (SCS) for {ljJ.} can be regarded as a function T on X*1 whose
1
values are subsets of \j1.
Consider a confidence set S(x) of y (8), when the two events {y (8)
and {ljJ. (y (8))
1
E
T(x, i): i
E
E
S(x) }
I} are identical; then a family of exact SCS with
4
respect to 5 is {T(x, i); i E I}.
If there exists a fami ly of exact 5CS with
respect to 51 {T* (x, i): i E I}, such that
T*(x,i)
c
T(x,i) for all i E I;
then {T*(x,i): iE I} is a family of smallest exact 5CS with respect to 5.
Assume that an action of the induced group
g E G,
that y(ge) = iY(S) for all
can be defined such
Also assume that an action of G on I,
e E 0.
and for each i E I, an action of G on
G on r
I{I
(denoted ljJ ~ i.tV) can be defined such
1
that
g. ljJ(y,i) for all yE r, i E I, gE G.
(1.2.1)
1
Then a simultaneous set estimator T for ljJ is equivalent if
(1.2.2)
1.3
T(g ,g.) = g.T(x,i)
x 1
1
for all XE X, iE I, gE G.
The standard MGLM model
The standard MGLM model is given by
(1.3.1)
E(Y) = XB,
and
V(Y) = I
where
Y
is a data matrix,
(nxp)
X
(nxr)
n
~
L,
is a design matrix,
B(rxp) is a matrix of unknown parameters, and
the rows of Yare independently normally distributed with a
positive definite covariance matrix L(
).
.pxP
The likelihood function of the above model is
e-
5
(1.3.2)
The maximum likelihood estimators (MLE's) of Band L are
(1.3.3)
B
= (X'X)-l
L
=~
X'Y
and
Y' [I-X(X'X)-lX']Y.
It is of interest to estimate a linear set of B,
e (gxv)
(1.3.4)
= CBU, where C( gxr ) is of rank g
and
U(pxv ) is of rank v.
The MLE of
--
e
is
e
(1.3.5)
= CBU = C(X'X)-l X'YU.
Three testing procedures are frequently used to test the null hypothesis HO:
e = o.
These procedures include Hotelling' s generalized trace,
Roy's largest root, and Wilks' likelihood ratio.
Letting
(1.3.6)
SH
=
e' [C(X' X) -lC' (1 e and
SE
=
U'Y[I-X(X'X)-lX']YU
=n
U'LU,
then under H ' SH has a Wishart distribution W'y(q,U'LU), SE has a Wishart
O
distribution Wv(n-r,U'LU), and SH and SE are independent.
Together with \j = Chj(SHSE-I), the jth largest eigenvalue of SHS~l,
j=l, ... ,v, these three test procedures can be described as follows:
(1)
(1. 3.7)
Hotelling's generalized trace statistics
?
1,-O
= trace
(5 S- I
HE) =
y
L
j=1
A..
J
6
The percentage points of the distribution of T~ has been tabulated
by Davis (1970).
Small sample F approximations to T; and tables for their
use have been found by Hughes and Saw (1972).
is asymptotically distributed as a
?
For large value of n, n TO
2
X variate with gv d.f.
We reject H
O
when T~ is too large.
(2)
Roy's largest root statistic
(1.3.8)
The percentage points of the distribution of Al can be found either
from tables by Pillai and Bantegui (1959) or charts by Heck (1960) through
the transformation
Al
es = 1+1..
and parameters
1
So = min(g,v),
m
O
= ( \g-vl-l)/2,
nO
= (n-r-v-l)/2.
We reject H when Al is too large.
O
(3)
Wilks likelihood ratio statistic
(1.3.9)
For a wide range of values of v and
So = min (g, v), trans formations to
exact chi-squared distributions and tables of the necessary multiplying
factors have been given by Schatzoff (1966), Pillai and Gupta (1969), and
Lee (1972).
For large value of n,
-[n-r-~(v-g+l)]
tnA is distributed as
a chi-squared variate with gv degrees of freedom.
We reject H when A is too small.
O
When
So = min(g,v)
=
1, all three statistics are equivalent.
e-
7
1. 3.1
Confidence procedures in the standard MGLM model
In the standard
MG~~
model, a (I-a) confidence set for
e
can be con-
structed by replacing SH with
(1.3.10)
in the corresponding test statistic.
Of all three confidence procedures considered, only Roy's largest root
gives the exact SCS for
e.
Letting Al,l-a be the (I-a) point of the largest root distribution under
Ho' then a (I-a) confidence set for B is given by
0_
(1.3.11)
Wi th
ChI CSI1S;1) = max
a
an equivalent confidence set is
(1.3.12)
for all (vxl) vector
~ {Al,l_aca'SEa)(b'[C(X'X)-1C']b)}~ for
That is, Ib'Ce-B)a\
a.
all
vectors a(v x 1) and b(gxl).
e
Thus an exact (I-a) confidence set for
(1.3.13)
b'Ba - d
~
b'Ba
~
b'8a
+
d
is
for all vectors a(v x 1) and b(g x 1),
where d={A1 , l_a(a'SEa)(b'[C(X'X)
-1
Schatzoff (1966) used simulation techniques to compare power among
some tests of the general linear hypothesis.
k:
C']b)}2.
The results indicate that
8
Wilks' A and Hotelling's T~ provide about the same levels of protection over
a wide spectrum of alternatives, whereas Roy's Al is shown to be rather insensitive to alternatives involving more than one non-zero root.
But for
certain restricted alternatives (e.g. with single non-zero root), Roy's Al
is somewhat better.
That is, the relative powers of the tests varied according to the type
of alternative (the nature of the population).
No single test procedure is
best against all types of alternatives (uniformly most powerful).
From the null hypothesis H : 6=0, we can construct an equivalent family
O
of hypotheses
n
=
{HOi' i
C. (g.xg) is of rank
1
1
€
I}, where I is an index set and HOi: C 6U = 0,
g.(~g),
1
i
and U. (vxv.) is of rank v.
1
1
1
i
(~v).
-1
Based on increasing root functions of SHSE ' which inc! ude Hote 11 ing' s
2
TO' Roy's AI' and Wilks' A.
Gabriel (1968) showed some useful properties of
these simultaneous test procedures (STP).
(1)
Coherence:
If HOi' HOj
€
n, and HOi implies HOj ' then the test pro-
cedure rejects HOi whenever it rejects H '
Oj
so is H '
Oj
(2)
Conversely, if HOi is accepted,
The probability of making one or more type I errors in all the tests
of an a-level STP is exactly a if H holds.
O
(3)
These properties include
Otherwise it is no greater than a.
Among all STP given identical decisions on all one-root hypotheses
(i.e. having equal critical values since on such hypotheses all the statistics are equal), Roy's Al procedure is the most parsimonious in that it has
the lowest level.
In other words, it provides the same single root tests
at the smallest probability of any type I error.
(4)
Among all STP of level a, Roy's Al procedure is the most resolvent
9
in the sense that in testing everyone-root hypothesis it will reject every
such hypothesis rejected by any other 5TP, and, possibly, some more.
(5)
Among all simultaneous confidence bounds of joint confidence I-a,
Roy's Al procedure gives the narrowest confidence intervals for each parametric function.
Roy's intervals are contained in the corresponding inter-
vals of any other simultaneous confidence bounds.
For a family
est y
=
{~. (y)}
1
of parametric functions of the parameter of inter-
y (8), Wij sman (1979) presented a method for the construction of all
families of the smallest 5CS in a given class.
The method is applied to the
MANOVA problem (in its canonical form) of inference about M= E(Y 1)' where
Yl(gxP) has rows that are independent multivariate normal with common covar-
--
iance
matrix~.
Let 5 be the usual estimate
of~.
Wijsman then showed that
the smallest equivalent 5CS for all a'M. where a is a (gxl) vector, are necessarily those that are exact with respect to the confidence set for M determined by Ch 1 [M-Y 1) 5
-1
(M- Y1) ']
~
constant, i. e. derived from the acceptance
region of Roy's largest root statistic.
1.4
The growth curve models
1. 4.1
The 5GCM model
The 5GCM model is given by
(1.4.1)
E(Y 0)
= XBP,
and
V(Y )
O
= In o
where
Y
is a data matrix,
O(nxg)
X
(nxr)
~O'
ia a design matrix,
B
is a matrix of unknown parameters,
(rxp)
10
p( pxq ) is a known matrix of rank p, and
the rows of YO are independently normally Jistributed with
common variance matrix LO(qxq) .
The hypothesis of interest is HO : 8=0, where 8( gxv ) = CBU, C( gxr ) is of
rank g. and U( pxv ) is of rank v.
To analyze the above model, Potthoff and Roy (1964) suggested the transformation
(1.4.2)
where G( qxq ) is any positive definite symmetric matrix either nonstochastic
or independent of YO such that PG-lp, is nonsingular.
Using the above transformation, we can reduce the SGCM model to a standard MGLM model on the data matrix Y.
(1.4.3)
E(Y)
e-
= X8,
and
where
To avoid the arbitrary choice of the matrix G in the above procedure,
Rao (1966, 1967) and Khatri (1966) used covariate adjustment to reduce the
SGCM model to a multivariate analysis of covariance (MANOCOVA) model.
the construction of the nonsingular matrix H(
such that
(1.4.4)
PH
1
=I
and
PH
2
= 0,
we have the following MANOCOVA model:
(1.4.5)
I
E (Y Z)
= X8 + zr,
qxq
) = (H
l(qxp)'
With
H
)
2(qx[q-p])
11
and
where
Y(nxp)
= YOH I
Z(n x [q_p])
=
is a data matrix,
YOH 2 is a matrix of covariates,
X( nxr ) is a design matrix,
B(rxp)
is a matrix of unknown parameters, and
rC[q-p]xp) is a matrix of unknown regression coefficients.
The procedure used in analyzing the SGCM model depends on
1) the method used to transform the data into the framework of a standard
MGLM model, and
2) within each method, the statistic used to analyze the reduced standard
MGLM model.
As described in Section 1.3.1, for each method used, no test statistic
is uniformly most powerful.
Estimation of 6 in Rao-Khatri's method is more efficient than Potthoff
and Roy's approach only if Yol\ and Y H are correlated.
O2
Selected subsets
of the covariates may, in some cases, yield a better estimate for
B than
the
complete set, as indicated by Grizzle and Allen (1969).
GIeser and Olkin (1970) introduced a canonical form for the SGCM model
from which the maximum likelihood estimator
B can
easily be obtained.
In
the process, they also considered transformations which leave the testing procedure invariant.
Kariya (1978) then applied further invariance reduction to
the canonical form for the SGCM model and derived a unique locally best invariant test.
Under certain conditions, Hooper (1!::l82) derived a method to construct
a confidence set haVing smallest expected measure within the class of invar-
12
iant level (I-a) confidence sets.
He then applied the method to the SGCM
model in the form of a MANOCOVA model.
1.4.2
The GGCM model
To incorporate possible missing responses in the response curve de-
sign, Kleinbaum (1973) defined the following GGCM model:
(1.4.6)
E(Y.) = X.SPB.,
~d
V(Y.) = I n.
J
J
J
J
J
where
~
E.,
J
j=l, ... ,u,
Y'(
) is a data matrix,
J n j xq j
X.(
) is a design matrix,
J n.xr
J
S
is a matrix of unknown parameters,
(rxp)
P
is a known matrix of rank p,
(pxq)
Bj(qxq.) is an incidence matrix of a's and l's, and
J
L
j(qjxqj)
= B' E B is a positive definite matrix.
j
j
The hypothesis of interest HO: 6=0, where 6(gxV) = CSU, C( gxr ) is of rank g,
and U( pxv ) is of rank v.
Let S*(rpxl) be the vector constructed from Sby putting the columns
of S underneath each other.
Then by also rolling out the columns of the ob-
served data, the GGCM model can be analyzed through an equivalent univariate
linear model.
Letting 6*(gv x l) = HS*, where H(gvxrp) = U' ~C, 6* and 6 have the same
gv elements.
A best asymptotically normal estimate of 6* can be constructed
from 6* = HS*, where S* is the weighted LSE of S* in the univariate model.
e-
13
Then the null hypothesis H can be tested through a Wald statistic inO
volving e* and its estimated variance.
1.5
The HM model
In an HM model, the responses vl •...• v are arranged in a descending
p
order of importance. If V is more important than V.• then V should also
r
J
r
be observed on each experimental unit on which V. is ohserved.
J
Let n. be the number of experimental units on which the response V.
J
J
Then the response data has the structure
is observed.
.' 1
YI
y2
1
Y =
y2
2
-e
where yj( , x 1) is a data matrix.
r n.
yp
yp
2
p
j=l •...• p.
r=l •...• j.
J
j=l •...• p-1.
and
n' = n .
p
p
Let Yj(n!xj)
J
=
j
j)
(yl····'y·
,
J ,
j =1•...• p, and
- [~~ 1•
-.
yp
r
r= 1, ...• p.
~
Then the HM model can be described as follows:
(1.5.1)
E(Y r ) = Xr (3 r ,
and
j
V(y )
= I n!
J
where
Xr(nrxm r )
18
r= I •... , p •
1:.,
J
j=l, .... P,
is a design matrix,
14
8 (m xl) is a matrix of unknown parameters,
r
r
L
is the (jxj) submatrix in the top left corner of Z,
'j(jxj)
Z
- (0
(pxp) -
r,s
) is the covariance natrix of the p res!)onses,
and the rows of y j are independently normally distributed with common covariance matrix Z..
J
The hypothesis of interest HO is
P
(1.5.2)
HO: n H "
j=l oJ
and
c.(
) is of rank g .•
J gjx mj
J
where H "
OJ
e. = O.
J
with
e. (
J gjx l )
=
c. 8 . ;
J J
Roy, Gnanadesikan and Srivastava (1971) descirbed the following procedure
in analyzing the above HM model:
For j=l, we have a univariate linear model
(1.5.3)
For j < p, Y. 1 can be analyzed through the conditional model
J+
(1.5.4)
E (Y.
J+
11 Y~)
=
J
(X~J+1 Y~) l' E;:
j
J
l
Y + ]. ,
J
2
V(Y. lIY~) = I
~ O(j+l)'
nj
J+
J
and
where Y
j +1 (n. lXl)
j=1, ... ,p-1,
is a data matrix,
J+
Y1j+l . . . yj+l]
j
Yj (n.J+ lXj)
=
Yl
Y~
is a matrix of covariates,
e-
15
and
is a design matrix with
=
is a matrix of unknown parameters,
r= 1,
-1
t;;j(jXl)
= Lj
0
0:
1 ,J+
. 1]
0
0
0
,
j ,
oj
=[t;;
: '1]
[ o. . 1
J,J+
t;; ..
JJ
is a matrix of unknown regression coefficients, and
With models (1.5.3) and (1.5.4), the null hypothesis H can be transO
formed into an equivalent hypothesis HOo
(1.5.5)
is of rank g ..
J
Let
y. be the least squares estimate (LSE) of y.,
J
J
2
O(j)W j be the covariance matrix of CjYj' and
2
s(j) be the resulting sum of squares of residuals in the
least squares estimation, j=l, ... ,po
Then under H ' the statistic
Oj
(1.5.6)
F.
J
(C~~')'W~I(C~~.)/g.
= ~J~J_~J:--_J~J:--..,...J,,--_
S~j)/(nj-[(j-l)+It=lmk])
16
p
Under HO: j~l HOj ' Fj'S are independently distributed.
thus, HO can be
tested by the step-down procedure:
(1.5.7)
Accept H* if and only if F.
0
P
IT
where
j=l
1.6
J
~
f.,
J
j=l, ... ,p,
P (F . ~ f·1 H ' ) = 1-0..
J
J OJ
The 1M model
Assume that the experimental units can be divided into u disjoint sets
such that the same set of responses are measured on each unit in the same
set.
Srivastava (1968) defined the following 1M model
(1.6.1)
and
where
E (Y .)
J
= X.(
V(Y. )
J
= I
r j ),
e
J 8*B.
J
n.
J
Y.
J (n(Pj)
~
Eo,
J
j=l, •.. ,u,
is a data matrix,
Xj(njx[rj+t]) is a design matrix of rank rj+t-l,
r.(
) is a matrix of unknown parameters,
J r .xp.
J J
B* (txp)
is a matrix of common unknown parameters such that
J(l xt)8*=O, J(lxt) is a (lxt) vector of l's.
B.(
J PXPj
) is an incidence matrix of O's and l's,
Ej (p. xp. ) = B!J E B.,
J
J
and
J
E
is a positive definite covariance matrix.
(pxp)
The hypothesis of interest is H : C*B*U = 0, where
O
17
C*( gxt ) is of rank g and U( pxv ) is of rank v.
Prior to analysis it is necessary for the data to be transformed into the
framework of a standard MGLM which meets the specified conditions.
Srivastava
suggested the following procedure:
(i)
First construct a linear set of Yo, Q
J
j(txp.), such that
J
=A.S*B.,
J
J
(1.6.2)
V(Q.) = C.
J
(ii)
J
~
E.,
J
j=l, ... ,u.
The design is homogeneous, i.e., there exists a set a known (txt)
matrices, {Fi, ... ,F;}, such that
-e
(1.6.3)
m
Ao = I aJokFk
J k=l
for some known scalars a.l, ... ,a. , j-l, ... ,u.
J
Jm
We then have
m
=
(iii)
(FkB*) (a'\B.),
J
J
where aJ~k = a ok + 8 . for a fixed 8 ..
J
J
J
Combine Ql"" ,Qu'
Q( t x I U
0
]=1
(1.6.4)
I
k=l
p j) = (Q l' ... , Qu ) ,
E(Q)
= (E(Ql),'"
we have
,E(Ou))
m
,I
k=l
(Fk*B*) (a~kB ))
J
u
18
=
where
and
L ( tU
k pXL.j=lPj
= (a*lkBl'·· .,au*kBuJ.
)
(iv) Choose {8 , ... ,8 }' such that LL' is nonsingular.
u
l
Let Z*(
) = QL(LL')
txmp
(1.6.5)
•
-1
= (Zi' •.. ,Z*) and define
~.l],
m
Z(m[t-l]xP)
Zm
where Zk([t-l]Xp) = (O([t-l]xl) l(t-l)) Zk'
k=l, ... ,m.
Then we have
(1.6.6)
-J
where
(
(v)
The factorization of V(Z) is possible, i.e.
(1 [t -1])] and
x
let_I)
there exists a known
positive definite matrix W(m[t-l]xrn(t-l])' such that
(1. 6.7)
(vi)
V(Z) =
w~
E*, where E*( pxp ) is a positive definite matrix.
If the above conditions are met, then there exists a nonsingular
matrix R(m[t-l]xm[t-l])' such that RW R' = lm(t-l)·
Thus we have a standard MGLM model
(1.6.8)
E(RZ)
= [R
~l]]~,
F
m
, V(RZ) = lm(t-l) ~ E*.
19
With C*S* = CS, where CCgx[t-l]) = cfOC1X[t-l])]
Ct - l )
LI
- [C
[~( 1 l ) J]
x
0
.J (l x [t - 1]) ,
([t-l]xI)
the null hypothesis H is equivalent to H : CSU = 0.
O
O
Then the usual analysis procedures used in the standard MGLM model become
available.
1.7
Outline of the research
Chapter II deals with the construction of confidence procedures for the
generalized multivariate linear models.
These models include SGCM, GGCM,
and HM models.
In the seCM model covariate adjustment
the model into a standard MANOCOVA model.
procedures have been used to reduce
The procedure requires the selection
of a nonsingular matrix H satisfying a set of conditions Cl.4.4).
Although
the selection of the matrix H is not unique, the resulting confidence procedure is shown to be invariant under different selections of H satisfying the
same set of conditions.
In the GGCM model, with enough observations and responses being measured
in each disjoint set, we extend the above covariate adjustment procedure to
each disjoint set.
A univariate weighted least squares model is then construc-
ted to simultaneously utilize the information from each disjoint set.
Again,
the resul ting confidence procedure is shown to be invariant under di fferent selections of the matrix H when the set of conditions is satisfied in each disjoint set.
In an HM model, the p responses are ordered.
Using the correlation in-
20
formation among the p responses. the jth response can be analyzed through a
covariate
model with the covariates being the (j-l) previous responses,
j=2 •...• p.
With good estimates of the variances of the p responses, the re2
sulting confidence set can be constructed from p quadratic forms of X statistics.
W~
demonstrate the confidence procedure when p=2.
To optimize the
confidence procedure. we propose two criteria involving suitable norms of
the confidence set.
These criteria are to minimize either the weighted sum
of the p quadratic form lengths or the maximum of the p weighted quadratic
form lengths.
freedom.
The weight is the inverse of the corresponding degrees of
For the maximum weighted length criterion. the resulting optimal
confidence procedure gives a larger confidence coefficient to the variate
with a larger d.f.
The result is also true for the weighted sum criterion
when the degrees of freedom are less than five.
Chapter III considers the 1M model.
The model defined by Srivastava is
based on a less-than-full-rank linear model in each of the disjoint datasets.
For a design satisfying a set of specified conditions. this model can be
analyzed by transforming the data into the framework of a standard MGCM model.
We propose a corresponding full rank 1M model and show some simplification of
the procedure.
Consider a special 1M design upon which the full rank model procedure·
can be performed.
We construct an optimum confidence procedure to minimize
the generalized variance of the parameter estimate 8 and show some invariance
property of the procedure.
We then study the relative performance of the pro-
posed special 1M model and the corresponding complete mUltiresponse (CM) model.
These comparisons include generalized variance. asymptotic relative efficiency.
cost. and minimum risk.
In each comparison, we assume the covariance matrix L
has either an intraclass covariance structure or an autocorrelation with p=3.
Zl
Chapter IV examines various covariance structures in blocks.
When ob-
servations in a design arise in a structured form, the resulting covariance
matrix may have a special pattern.
A common example of patterned covariance
structure is the intraclass correlation eIC) model.
With the same p responses
measured at each of t distinct time points, a natural extension of the IC
model is the block version of the IC model.
We examine the hypotheses of
the block IC model as well as some special cases in the standard MGLM model.
These hypotheses are
f'I
H4: 1>-'
2 = 1
In each hypothesis, the LR test is derived.
For the hypotheses HI' H3 , and
H , we have a closed form expression for each test statistic.
4
Moments of
the test statistics are obtained and used to find asymptotic null and nonnull distributions.
For the hypothesis HZ' there is no closed form expres-
sion for the LR statistic.
We construct the Wald statistic to test the hypo-
thesis and then derive the asymptotic null and non-null distributions.
We outline proposals for further research in the concluding chapter
of the dissertation.
CHAPTER II
ON CONSTRUCTING THE CONFIDENCE SETS IN THE
GENERALIZED MULTIVARIATE LINEAR MODELS
2.1
Introduction
In this chapter we construct confidence procedures for the general-
ized multivariate linear models.
HM models.
These models include SGCM, GGCM, and
We then derive some optimum properties related to these con-
structed confidence procedures.
Section 2.2. examines the SGCM model.
Using the covariate adjustment
procedure, it is necessary to construct a nonsingular matrix H(qxq)
=
(Hl(qxp)' H2 (qX[q_p]))' such that
(2.1.1)
PHI is nonsingular,
PH
2
= o.
Section 2.2.2 shows that the covariate adjustment procedure is invariant under different selections of the nonsingu1ar matrix H satisfying
the conditions (2.1.1).
Section 2.3 examines the GGCM model.
servations in each disjoint set (n
on each observation (q.
J
~
Assume that we have enough ob-
r+q , j=l, ... ,u) and enough responses
j
p, j=l, ... ,u).
j
<!:
Section 2.3.1 extends the covariate adjustment procedure in the SGCM
model to the GGCM model by using the covariate adjustment in each of the u
disjoint sets.
A univariate weighted least squares model is constructed
to simultaneously utilize the information from each of the u disjoint sets.
23
The confidence set in the GGCM model can then be constructed from the Wa1d
type statistic in the univariate model.
The covariate adjustment procedure in the GGCM model requires the construction of nonsingular matrices H
j(q.xq.) = (Hjl(q.xp)· Hj2 (q.X[q._p]))·
J
(2.1.2)
PB j Hjl
J
J
J
J
= I.
PB j Hj2 = O.
j=l •.. .• u.
Section 2.3.2 shows that the covariate adjustment procedure in the
GGCM model is invariant under different selections of these nonsingu1ar
matrices H.• j=l •...• u. satisfying the conditions (2.1.2).
J
Section 2.4 examines the HM model.
Using the correlation information
.
among t he p responses. t he J.th response can b e ana 1yze d t h roug h a covarlates
model with the covariates being the (j-l) previous responses. j=2 •...• p.
The resulting confidence set can then be constructed from the p independent confidence procedures involving the F statistics.
Assuming we have
a good estimate for L. the resulting confidence set can be constructed
2
from the p independent confidence procedures involving the X statistics.
Section 2.4.1 constructs confidence procedures in the
p=2.
~1
model when
Section 2.4.2. proposes two criteria by which the resulting p confi-
dence sets can be optimized.
We then derive the optimum solutions for
these two criteria.
Section 2.5 gives numerical examples to illustrate the procedures derived in this chapter.
2.2
2.2.1
The standard growth curve model
Covariate adjustment in the SGCM model
With the construction of the nonsingular matrix H(qxq) = (H l(qxp)'
24
H
) such that PHI is nonsingular and PH 2 = O.
2 (qx[q-p])
SGCM model through the following conditional model:
I
(2.2.1)
where Y(nxp)
E(Y Z)
= xn + zr,
V(YIZ)
=I
= YaH l
z(nx(q_p))
~
We can analyze the
E,
is the new data matrix,
= YOH2
is the matrix of covariates,
X(nxr) is the design matrix,
n(rxp)
= SPH I
is the matrix of unknown parameters,
r ((q-p)Xp) is the matrix of unknown regression coefficients, and
E
(pxp)
= H'P'(PE-lp,)-l
1
0
PHI is the unknown conditional covariance.
The LSE's of n, r are given by
(2.2.2)
[~J
=
e-
(X' X X' Z) -1
Z'X Z'Z
(X'L X)-l
• [- (Z'
z~lz'
-(X'L X)-lX'Z(Z'Z)-l
X(X 'L1xf
1
(Z
'Z~
1
+(Z' zflz' X(X' LiO- 1X ' z (Z'Zf 1
]
[X,y]
Z 'y
(X'L X)-lX'L Y
=[
(Z'Z~-lZ'[I-~(X'L
1
X)-lX'L
1
lY] ,
where L = I - Z(Z'Z)-lz'.
l
For the standard MGLM model, three frequently used confidence procedures
are Hotelling's trace, Roy's largest root, and Wilk's likelihood ratio.
resulting confidence sets for 8
= cau = Cn(PH l ) -1
where
(2.2.3)
S* = (8-8)' (CRC,)-l (8-8),
H
The
-1
are functions of SA SE '
25
SE
= [(PHl)-lU]
"
e = en [PHI)
-1
I
S[(PHl)-lU],
U,
R
=
(X'LlX)-l,
S
=
(Y-Xn-Zr) I (Y-Xn-Zr)
=
YI {r - (XZ)
(X XOZ]
Z'X z'z
XI
[zo]lY,
is given by (2.2.2).
n
2.2.2
-1
Invariant confidence procedure
The selection of the matrix H in the above procedure is not unique.
The nonsingularity property of PHI includes the following special cases:
-e
(1)
The columns of HI form a basis of the vector space generated by
the rows of P, which was used by Rao,
(2)
PHI
~
I, by Khatvi.
Assume both matrices H
= (H l ,H2 )
and H*
= (Hi,
conditions, namely PHI and PHi are nonsingular,
Hi) satisfy the specified
and PH 2
= PHi = O.
Then
there exist matrices GI(pXP)' G2 ((q_p)Xp)' and G3 ((q_p)X(q_p))' such that
G and G are nonsingular, and
l
3
(2.2.4)
Thus we can examine the effect of different selections of the matrix H
on the resulting confidence sets for
(2.2.5)
e through
the transformation
26
Letting
(2.2.6)
y*
=
YG
Z*
= ZG 3 ,
1
+ ZG ,
2
we then have
(2.2.7)
E(Z*)
= E(Z)G 3 = 0,
(2.2.8)
(2.2.9)
(2.2.10)
Lemma 2.2.1
With the transformation given by (2.2.5), in the parameter
space, we have the following induced transformation:
(2.2.11)
From the transformation (2.2.5), we have
(2.2.12)
Li
= 1-
Z*(Z*'Z*)-l Z*'
= 1=
L*Y*
1
ZG [(ZG ) 1 (ZG )
3
3
3
r 1 (ZG 3)'
1- Z(ZIZ)-1 ZI .
= L1 (YG 1+ZG 2) = L1YG 1 + L1ZG Z
= L1YG 1 + [I
- Z(Z I Z)-1 ZI ] ZG
Z
= L YG ,
1
~* = (X L*X)-1 X'L*Y* = (XILX)-1 X'L YG
1
1
1
1
111
=
'"
e-
27
PHi = P(HI G1 + H2G2 ) = PHI G1 '
e* =
Thus
SH
CB*
=
C~*(PHi)-lU
C~Gl
=
CPH G )-lU = 8.
1 1
= (8-8) 'CCRC,)-l C8 - 8) is invariant under the transformation.
Furthermore , after some simplification,
we have
(2.2.13)
Thus, after the transformation,
(2.2.14)
S* = Y*'[L - L X(X'L X)-l X'L ] y*
1
1
1
1
= Y*'L y* - Y*'L X(X'L X)-l X'1 y*
1
1
1
'1
= Y*'L YG - Y*'L X(X'L X)-l X'L YG
1
1
1
1
1
1
= G'Y'L YG
- G'Y'L X(X'L X)-l X'L YG
11111
1
1 1
= Gi SG ,
1
S* = [(PH*)-l U]' S* [PH*) -1 U]
E
1
1
= [ (PH 1G1) - 1 U] , (GiSG ) [(PHI G )
1
1
-1
U]
= S .
E
Since both matrices SH and SE are invariant under the transformations
C2.2.5), (2.2.11), the confidence procedures based on the information
from
S* S-l are invariant under these transformations.
H E
Theorem 2.2.2.
The confidence procedures based on the information from
-1
SH SE ' where SH and SE are given by (2.2.3), are invariant under different
selections of the matrix H = (H ,H ) such that PHI is nonsingu1ar and PH 2 = o.
1 2
28
2.3
The generalized growth curve model
2.3.1
Covariate adjustment in the GGCM model
Recalling Section 1.4.2, the GGCM model is given by
(2.3.1)
E(Y
V(Y
Oj
Oj
) = X.BPB. ,
J
J
)
= I.0B!E B.,
nJ
J o J
j=l, ... ,u,
YO'J is (n.xq.),
J J
where
X.
is (n.xr),
B
is (rxp) ,
P
is (pxq) ,
B.
J
is (qxq.),
Eo
is (qxq) .
J
J
J
e
The null hypothesis of interest is
(2.3.2)
HO:
e = 0, e = CBU
where
C( cxr ) is of rank c. lJ(pxv) is of rank v.
Assuming
q.~p,
J
n.~r+q.,
J
J
j=l, ... ,u, then the covariate adjustment proce-
dure used in the SGCM model can be extended to the GGCM model.
For each
j, j=l, ... ,u, we can construct a nonsingular matrix
such that
(2.3.3)
(2.3.4)
Lemma 2.3.1.
tional models:
PB.H'
J J2
= O.
The GGCM model can be analyzed through the following condi-
-
29
X.8
(2.3.5)
z.r.,
+
J
J J
v(Y·lz.) = In. ~ E.,
J J
J
J
j=l, ... ,u.
where
Y
= Y H
j(n.xp)
OJ jl'
(2.3.6)
J
Zj(njx[qj_P])
=
YOj Hj2 ,
E. = [PB.(B!LOB.)
J
J J
J
-1
(PB.)']
-1
J
,
H ,H
are given by (2.3.3).
jl j2
Lemma 2.3.2.
The conditional models given by (2.3.5) are equivalent to the
following univariate models:
(2.3.7)
E
(y~)
A
J = j
v
l]
,8
r~
J
v
V(Y .)
J
v
v
J
j =1, ... ,u, where
A. (n.pxp[r+q.-p]) = (X.
(2.3.8)
Y., 8
= I nJ. ~ LJ. ,
JJ
J
J
~
I , Z.
P
J
~
I ) ,
P
v
and r. arc vectors constructed from Y., 8 and r. by putting the colJ
J
J
umns of the matrix underneath each other.
Since L. 's are different from each other, the models given by (2.3.7)
J
can be analyzed through a weighted least squares procedure.
Theorem 2.3.3.
The GGCM model given by (2.3.1) can be analyzed by the follow-
ing weighted least squares model:
(2.3.9)
v
E (Y ) = At.,
v
V(Y ) =
where
n,
30
v
(2.3.10)
Y (np x 1) =
u
L (q.-p)]) =
A(npxp[r+
j =1
J
x
u
~(p[r+
~
I
zu ~ I P
P
u
L (q.-p)]x1)
j=l
=
J
en(npxnp) =
I
n
~
u
E
u
•
From the GGLM model (2.3.1), an unbiased and consistent estimator of
EO
= (ak,~)'
EO
= (ak,~)' can be obtained by the pooled estimate of
ak.~
from
those experimental units on which both responses V and V£ are observed.
k
That is,
(2.3.11)
where
Yk(~)(Nk~x1)
is the observation vector on Vk corresponding to those
experimental units on which both Vk and V£ are observed,
design matrix correpsonding to Yk (£).
Xk~(Nk~xr)
is the
31
Denoting
(2.3.12)
n = nO:.=L) =
J
J
then a weighted least squares estimate of
~
is given by
(2.3.13)
Using the matrix formulae involving Kronecker products, we have
(2.3.14)
.
e
XI 3 I
A'
n-
1
1
=
Zl
1
~
I
P
XI 3 I
2
XI 3 I
u
P
)
P
I
n
3 E- 1
1
1
p
ZI
2
~
I
P
I ~
ZI 3 I
p
u
X ' 3E- 1
1
=
1
Zl ~ ;:-1
1
1
X ' 3E- 1
1
2
u
E-u 1
ZI ~ ;:-1
2
2
Z'~E-l
l
u
u
"-1
L (X!X.)~E.
j=l
f
X 10
nu
J J
J
u
X'Z ~E-1
u u u
(sym. )
l
Z'Z ~~-1
u u
u
E- 1
u )
32
"'-1 v
L (X!0L
)Y.
u
J
J
j=l
J
(ZI 0 ~-l)yV
III
(ZI 0
~-l)yV
222
Denoting
(2.3.15)
u
=
L (X!X.)
j=l
J J
s
= (X!Z.)
J J
R .. = Z!Z.
JJ
"'-1
L
J
"'-1
E. ,
0
J
.... -1
l:. ,
J
IS
J J
= {R 00 - . ~L.
R R- 1R ,}-1
J= 1
O· j J' O·
J
J
•
-1
=
then with appropriate D..
IS
JJ
(2.3.16)
A'
n-
e-
-DOOROjRjj'
(ROO
1A
we have
R
01
R
.
R
.
02
R
U
=
12
l
(sym. )
DOO
{A'Q-1 A}-1
=
001
0
0
02
(2.3.17)
A,
o )
Ou
11
22
(sym. )
= {A ' n- 1A}-1
Ou
R
uu
0
With E;
R )
n- 1y v. we have
ouu
33
where
L.
J
Theorem 2.3.4
estimate of
--
is given by (2.3.17) and an estimate asymptotic variance for
SV
is
V(~v)
IJ
(2.3.18)
- Z.(Z!Z.)-lZ!.
J J J
J
For the model given by (2.3.9), a weighted least squares
BV
the estimator
=I
~
A_I
=Doo={L,(X!L.X.)@E.}
j=l
J J J
-1
J
where 000 is given by (2.3.15),
L.
J
L.
J
= r - z. (Z!Z~l)Z!,
J
J J
J
is given by (2.3.12).
Letting H (cvxrp) = C@ll' , the null hypothesis H : CSll = 0 is then equivalent
O
v
to the hypothesis H : HB = o. With large values of n , ... ,n u ' the sample sizes
l
O
.
A
QV
in the u disjoint sets, the resulting weighte d least squares estlmate HIJ
asymptotically distributed as a normal variate with mean HB
v
is
and variance
HV(SV)H' •
Corollary 2.3.5.
A (I-a.) confidence set for HB
v
is given by
(2.3.19)
With the estimate variance for
set for HB
(2.3.20)
v
is given by
2
W~ ::; X cv,l-a.'
HS v ,
~(H(6v), an approximate (I-a) confidence
34
where
Putting
HS
v
= 0 in W*, we have the following corollary:
n
Corollary 2.3.6.
A test statistic for the null hypothesis H given by (2.3.2)
O
is
(2.3.21)
2
Under HO' Wn has an approximate X distribution with cv d.f.
We reject
H for large values of W .
O
n
When u = 1, the GGCM mode 1 reduces to the 5GCM mode 1 .
The above procedure
then gives
1
(2.3.22)
5,
n-r
E
=
DOO
(pi~lpI)-1,
= { ROO
-I
- RO1Rll R()l}
=
{[(XlX)~E-l]
=
(X 1 LX) -1
~
_
-1
[(XIZ)~E-1][(ZlZ)~E-lf1[(ZlX)~E-1]}-l
E,
where
5
= Ya[I
L
= 1-
- X(XIX)-lXl]Y '
O
Z(ZIZ)-1 zI •
Thus we have
(2.3.23)
BV = D [XIL~i-1]yV
00
= [( X 1 LX) - 1XI L ~ I
P
] Yv ,
which results in the same estimate for
procedure in the SGCM model.
B obtained
by covariate adjustment
35
With
(2.3.24)
H~(Bv)
HI =
[C~ U'][(X'LX)-l~E][C' ~
= [C(X'LX)-lC']
~
U]
[U'E U],
we have
= {[ C ~ UI ]( BV -13 v)} I {(C(X I LX) - 1CI ] -1 ~
• { [C
~ lJ I
](
8
V
_
[U I EIJ
r 1}
13 v) }
-1
= (n-r) trace SH SE '
where
SH
-e
,
= [C8 U - CI3U]'[C(X ' LX)-lC ]-1[C8U- Cl3lJ]
SE = (n-r) U'EU= U' [PS-1 p, (lU.
Corollary 2.3.7.
When u= 1, the covariate adjustment procedure in the GGCM
model reduces to the trace criterion used in the covariate
adjustment pro-
cedure in the SGCM model.
2.3.2
Invariant confidence procedure
In the above covariate adjustment procedure for the GGCM model, it
is necessary to construct a set of nonsingular matrices, {H , ... ,H u }' such that
1
(2.3.25)
H. = (H'l,H'2),
J
J
J
PBjH jl
= I,
PB.H'2 = 0,
J J
j=l, ... ,u.
The selection of the matrices {Ill"" ,H u } is not unique.
Assuming both
sets of matrices {Ii , •.. ,H } and {II*, ...• H*} satisfy the specified conditions
1
u
1
u
(2.3.25), then there exist matrices G , G , where G is nonsingular,
j3
j3
j2
36
j=l, . .. ,u, such that
(2.3.26)
j=l, ... ,u.
Thus we can examine the effect of different selections of the set of matrices
{Hl, ... ,H
U
}
(2.3.27)
on the resulting confidence sets for
e
through the transformations
(Y . , Z.) -+ (Y. + Z. G. 2' Z. G. 3) ,
J J
J
JJ
JJ
j=l, ... ,u.
Letting
(2.3.28)
j=l, ... ,u,
then
(2.3.29)
E(Y~IZ~) = E(Y. + z.G.2Iz.)
J J
J
J J
J
= X.B+ z.r. + z.G'
J
J J
J J2
=X.B
+
J
v(Y~IZ~)
J J
= V(Y.
J
-1
J J
Z~G·3(r. + G· ),
+
J
J2
z.G.2Iz.)
J J
J
= V(Y.I Z.)
J
J
j =1, ...• u.
Lemma 2.3.8.
With the transformations given by (2.3.27),in the parameter
space, we have the following induced transformations
(2.3.30)
From the transformations (2.3.27),
(2.3.31)
j=l, ... ,u.
37
= 1- CZ.G. )[CZ.G. )'CZ.G. )]-lCZ.G. )'
JJ 3
JJ 3
JJ 3
JJ 3
= 1- Z.(Z!Z.)-lZ!
J
=
J J
J
j=l, ... ,u.
L.,
J
The estimator EO does not depend on the information from the selected
set of matrices {Hl, ... ,H u }'
Thus the estimators
'"
-1
-1
E. = [PB.(B!EoB.)
(PB.) ']
J
J
J
J
j=l, ... ,u,
J
are invariant under the transformations.
With L.Z. = 0, j=l, ... ,u,
J J
(2.3.32)
"-1
J
v
v
+ (Z.G. ) ]
2
J
J J
[(X!L.) 0E. ][Y.
J J
v
J
"'-I
J
= [(X!L.)0E. ]Y ..
-e
J J
'"
"1
u
Both the estimator SV = { ~ CX!L.X.)
j=l J J J
estimate variance V(Sv) = {
u
I
j=l
0
(X!L.X.)
J J J
1
'" 1
u
E~ }- { ~ [(X!L.)
J
j=l
J J
0
E~ ]y~} and its
J
J
"
0
E~l}-l are invariant under the transJ
formations (2.3.27).
Theorem 2.3.9.
The covariate adjustment procedure for the GGCM model in
Section 2.3.1 is invariant under different selections of the set of matrices
{Hl, ... ,Hu } satisfying the conditions (2.3.25).
2~4
The hierarchical mUltiresponse model
2.4.1
The HM model with p=2
Section
data
(2.4.1)
1.5 describes the HM model.
When p=2 we have the response
38
The resulting model is then given by
(2.4.2)
E (Y)
__ [E(Y 11)
E(Y 12)
8 l (m x l), 8 (m x l) are matrices of unknown parameters, and
l
2 2
E2 =
[all
OlZ] is the covariance matrix.
°22
° 12
The hypothesis of interest HO is
2
(2.4.3)
HO :
n
j=l
HO " where Ho],:e].=o, with e,=C.8.,C.(g.xm.) is of rank g ..
]
] J J J J J
J
Assuming the design matrices for Y12 and Y22 are the same, X :: X22 :: X2 ,
12
and utilizing the correlation information between Y and Y , then the
22
12
data Y can be analyzed by the conditional model
22
(2.4.4)
E(Y
22 IY12 ) = E(Y 22 )
+ [Y
-1
12 - E(Y 12 )] °11 °12
-1
-1
= X28 2 + Y12°11°12 - X2131011012
-1
X2 [8 2 - 8 1°11°12]
8*
=
[v:l
(XZV lZ )
= I
2
n2
@0(2)
where
-1
8*2 = 82 - 81 °11
°12 =
-1
Y2 = °11 °12'
2
-1
2
°(2)= °22 - °11 °12
•
-1
+
Y12 0 1 °12
e-
39
Together with the unconditional model for the first variable
(2.4.5)
[Y11'
12)
=
EY
r
[::1]
1l1
V Y
12
B]
2
2
InleO(l)' O(l)=OU'
J-
the unknown parameters 8 , 8 , E2 can then be estimated.
2
1
Le nuna 2. 4 . 1.
(2.4.6)
The LSE's for the parameters in models (2.4.4) and (2.4.5) are
81 = (XilXU +X ZX2 )
-1
(XilYU +X ZY12 )·
B2 = (X ZLI X2)-1 X Ll Y22 ,
Z
Y2 = (Yi2 L2Y12)
-e
~~l) =n ~m
1
1
-1
Yi2 L2Y22'
[(Y U-XUB1)' (Y U-X U 8 1 ) +(Y 12- X2!\)' (Y 12- X281)]'
where
Ll
= 1- Y12(Yi2 Y12)
-1
L2 = I - X (X ZX2 ) -lX .
Z
2
If we further assume that there exists a matrix G(
)' such that
g2 xg l
C = GC , then the testing of H is equivalent to the testing of the hypoO
1
2
thesis HO'
2
(2.4.7)
H*:
o
n
j=l
H*OJ" ,
where
with
HOj ~ 4> j
= 0,
4>1 = C1 B1 ,
4>2 = C2 B2
Theorem 2.4.2.
2
.
With a good estimate of o(j)' Wi is approximately distributed
40
as a X2 variate with g. d.L, j=1,2,
J
,.,
"2-1
(<P.-<p.)'
= (<P. - <p.)' [0(') V.]
W~
(2.4.8)
where
J
J
J
J
VI = Cl [XiI XII
+
XP2]-1
V = C2 [Xi Ll X2]-1
2
J
J
j=1,2,
J
ct,
Ci·
Setting <p. = 0, j=1,2, in (2.4.8) we have the following corollary.
J
Oj '
Under H
Corollary 2.4.3.
"''''2
(2.4.9)
W. = <P![o(.)V.]
J
J
J
J
j=1,2, the statistic
-1 ,.,
<p.
J
.
. h g. d .f. We reject H
is approximately distributed as a X2 varlate
at
Wlt
J
level a. if W. > X2
, the upper a. point of the X2 distribution with
J
J
g. , I-a.
J
J
J
g. d.f.
Oj
J
O' Wi
Under H
and Wi are independently distributed.
With the constraint
I-a = (I-a l ) (1-a 2), the (I-a) SCS for <PI and <P can be constructed from
2
(2.4.10)
W~
J
2
::;; X l '
gj' -a j
=
j 1,2.
That is, a (l-a l ) confidence set for <PI and a (1-a ) confidence set for 4>2'
2
The hypothesis H will then be rejected if we either reject Hal at level a
O
l
or reject H at level a .
2
02
The above confidence procedure for <PI and <P
SCS for 8
1
and 8 ,
2
2
does not directly give the
From (2.4.4) and C = GC , we have
2
l
(2.4.11)
Thus the confidence set for 82 depends on the values of 8 , <P 2 and Y2'
1
41
Two different procedures are used to construct the confidence set for 8 ,
2
Procedure I:
o
Giving the value of 8 , 8 , then from (2.4.4) and (2.4.11) we have
1
1
(2.4.12)
8 2 = <1>2
+
G8 oy 2 = (C G8 0 )
1
2 1
*],
[yB 22
(2.4.13)
'e
=
~~2){C2(X2L1X2)-lC2 - G8~(Yi2Y12)-lYi2X2(X2L1X2)-lC2
-
C2(X2L1X2)-lX2Y12(Yi2Y12)-I(G8~)'
o
Given the value of 8 , 8 , the quadratic form
1
1
Theorem 2.4.4.
(2.4.14)
2
is approximately distributed as a X variate with &2 d.f.
o
Given the value of 8 , 8 , a (l-a ) confidence set for
2
1
1
Corollary 2.4.5.
(2.4.15)
Theorem 2.4.6.
W
3~
X2 1
,Where
g2' -a 2
w*
3
is given by (2.4.14).
With (I-a) = (1-a ) (1-a ), a (I-a) SCS for 81 and 82 can be
2
1
42
constructed by the following steps:
1)
A (l-a ) confidence set for 6 given by (2.4.10),
1
1
2)
For a given value of 8 , a (1-a ) confidence set for 8 given
2
2
1
by (2.4.15).
When g2
o
1, with the value of 8 , 8 , a (l-a 2) confidence set for 8 is
2
1
1
=
(2.4.16)
where V(8 ) is given by (2.4.13).
2
Procedure II:
"
Wl"th a goo d estlmate
Theorem 2.4.7.
0
f 0(2)'
2
e-
(2.4.17)
2
is approximately distributed as a X variate with one d.f.
A (1-a 3) confidence set for Y2 is
Corollary 2.4.8.
"
2
2
"2
(2.4.18)
1Y2-Y21 ~{X1,1-a3 °(2) (Yi2 L2Y12)
Theorem 2.4.9.
With (I-a)
SCS for 8
1
and 8
2
= (1-a 1)(1-a2 ),
2)
A (1-a ) confidence set for ct>2 ; and
3
3)
A ( 1-a ) confidence set for Y ·
3
2
=
(2.4.19)
2
= ~'
a (l-a) conservative
3
can be constructed from the following confidence sets:
A (l-a ) confidence set for 8 ;
When g2
}
a
a
1)
l
-1 ~
1
1, a (l-a 3) confidence set for ct>2 is
43
Together with a (1-a ) confidence set for Y given by (2.4.18) and a given
3
2
value of 8 1 , 8~, the resulting confidence set for 82 = 1>2
+
G8~Y 2 is
(2.4.20)
_ {"'2
2
"'2
2
, - 1 ,}~
- C\2)X1,l_a3C2[X2L1X2]
+
=
{;~2)
C2
0
1
0
1
{(1(2)X1,1_a3G8l(Yi2L2YlZ)- (G8 l ) '}'2
xi, 1-a3}~{ [C 2 (XiLl X2) -lC2]~ + [G8~ (Y i2 L/
lZ)
-lG8~]~}.
With (I-a) = (l-a 1) (l-a ), a (l-a) conservative SCS for 8
2
1
and 8 2 can be constructed by the following steps:
Corollary Z.4.10.
1)
A (l-a l ) confidence set for 8 1 given by (Z.4.l0);
2)
Given the value of 8 , 8~, a confidence set for 82 given by (2.4.20).
1
Both procedures (I) and (II) give a (l-a ) confidence set for 8 , Howl
1
o
ever, for every given value of 8 , 8 , the confidence procedure (II) ignores
1
1
the correlation between 1>2 and YZ' This results in a larger confidence coefficient
c~
l-a
z)
for the reSUlting confidence set of 8 ,
2
o
When gZ= 1, for every given value of 8 ,8 , the length of the resulting
1
1
confidence set for 8 from the confidence procedure (I) is
z
(Z.4.21)
L1 = 2hi,l-a2
:II
vcez)}~
Z{;~2)X~,l_az}~{C2(X2LlXZ)-IC 2 - 2G8~(Yi2Y lZ) -lYi2XZ(X2LlXzflCZ
+
G8~(Yi2LzY12)-IG8~}~.
Similarly, the length of the resulting confidence set for 8Z from the
confidence procedure (II) is
44
(2.4.22)
L2 = 2 {;~ 12) X~ ,l-a/ ~{ [C 2 (XZL 1X2 ) -l CZ ] ~ +
[Ge~ (Y i 2L2Y12f1c.;6~]~}'
°
O~
For a given value of 6 , 6 , the correlation between ~2 and G6 Y2 is
1
1
1
.
(2.4.23)
The comparison between L and L depends on the value of R.
1
2
1)
For R < O.
2{;~2)X~,1_a2}~{C2(X2L1X2)-lC2 - 2G8~(YizY12f1Yi2X2(XZL1X2flC2
L1 =
+
2
< 2{;2 X
(2) 1,1-a
2
0,
-1 o}~
G81(Y12L2Y12) G6 1
}~{C 2 (X'L
X )-l C ' +
2 1 Z
Z
G60eY1'zL2Y12)-lGS01}!2
1
Z{;~Z) X~ ,l-a }~{ [C Z eXZLl Xz ) -lC2]~ + [G8~ (Y izLzY12) -lG6~]~}
<
z
=
2)
For R =
{
°
X1, 1-a
2
2
2
X1 ,1-az/2
f2
J
L2 ·
0
(6 =0).
1
_
"2
2
~
1
~
2{o
} ~{C (X'L X )- CI}~
L
1e2) X1, 1-a
2 Z 1 2
2
2
•
3)
For R > 0.
L1 =
2{;~2)X~,1_a2}~{C2eX2L1Xzf1c~ - 2G6~eYi2Y12f1Yi2X2eX2L1x2f1C2
+
°
°
G81eYizLzY12) -1 G8 1 }~
45
"'?
x2
< 2{cr(2) l,l-a
1
}~ {[C 2 (X'L
X )- C' + 2[C (X'L X )-lC']~
2 1 2
2
2 2 1 2
2
1
2
[Ge~(Yi2L2Y12)-lGe~]~+ [Ge~(Yi2L2Y12)-lGe~)}
•
= 2{"'2
2
cr (2) Xl, 1-a
}~{[C
2
2
(X'L X )-1
2 1 2
C
2]~ +
[eO(Y'
)-lGeO]~}
G 1 12 L2Y12
1
Thus we have
(2.4.24)
< 1.
That is, given the value for 8 , the confidence procedure (I) gives a smaller
1
e2
confidence set for
2.4.2
than the confidence set from procedure (II).
Optimum confidence procedure
With a good estimate
a~j)' j=l, ... ,p, a (l-a j ) confidence set for
?
1
' where W~ is given by (2.4.11).
gj' -a j
J
Wj'g are independently distributed. A (I-a) SCS for <Pj's
¢. can be constructed from
J
Under HO'
of
W~SX-
J
can be constructed from
j=l,2, ... ,p,
(2.4.25)
with
P
I-a = IT (I-a.).
j =1
J
The above confidence procedure is not invariant under different selections
p
of a.'s satisfying the condition IT (I-a.) = I-a. Letting y. be the upper a.
J
j=l
J
J
J
2
point of the X distribution with g. d.f., then the SCS for <p .• j=lJ .. 'JP'
J
can be optimized by either criterion:
J
46
1)
Mi.1imizing
y.
__
J , the largest weighted length of the p quadratic
Max
I:5j:5p
gj
forms, or
y.
~, the sum of weighted lengths of the p quadratic
j =1 gj
p
forms; with both being subject to the constraint IT (I-a.) = I-a,
j=l
J
with p[l :5 y.] = I-a ..
gj
J
J
~
2)
L
Minimizing
(2.4.26) Criterion 1. Minimizing
P
...
y.
2
Max ~, where Y. = X l ' subject to
g.
J
g., -a.
I:Sj:Sp J
J
J
II (I-a.) = I-a.
J
j=I
Let
Y.
-L
Z = Max
(2.4.27)
g. '
I:Sj:5p
J
where
2
Y. is a X random variable with g. d.f., j=l, ... ,p,
J
J
Y.'s are independently distributed.
J
Then the distribution function of Z is
Y.
= P [Z :5 z] = p [Max -L S z] = plY.
l:SjSp gj
J
(2.4.28)
=
p
II
:5
zg.. j = 1, ... ,u j
J
p [Y . S zg.] .
J
J
j=l
Since Z has a continuous distribution, a unique value of Z, zo' can be found
Theorem 2.4.11.
Under Criterion 1, the optimal procedure gives
(2.4.29)
a. = 1 - P [Y . :S y.] = 1 - P [Y . :5 zag. ],
J
where
Zo
J
J
J
J
j =1, ... , u,
is the upper a point of the random variable Z given by (2.4.27).
•
47
When g.<g., we have I-a. = Fy (zOg.)<F y (zog.)=l-a., the variate
1
J
1
i
1
j
J
J
with a larger d.f. will have a larger confidence coefficient.
For the special case g. = g., Fy (zag) = Fy (zog), a. = a .. That is,
1
J
i
j
1
J
two variates with the same d.f. will have the same confidence coefficient.
Extending this special case to all p variates, we have
Corollary 2.4.12.
When gl = g2 = .•. = gp = g, the optimum procedure gives
a. = 0* ,
(2.4.30)
J
2
y. =
Xg,l-a*'
J
j=l, ... ,p.
where
a* = 1 - (l_a)l/P.
(2.4.31)
--
(2.4.32) Criterion 2.
r
Minimizing
Y.
-1gj
j=l
p
IT (l-a.)
. 1
J
J=
where Y. =
J
J
= I-a.
2
Letting Y. be a X random variable with g. d.f., then the distribution
J
J
function of Y. is
J
(g/2) -3/2_
(2.4.33)
F. (y) = 2 ~
J
Uy) -
1
L
1 -
e ~y
k=O
J
J
L
k=O
_L
e"2Y
k+~
for odd g.,
J
(g./2)-1
F. (y) = 1-
1
)
( ~y
(~y)
k
for even g.,
r (k+ 1)
J
and the density function of Y. is
J
1
(2.4.34)
1
~g.
Y
J-
1
e
-~y
P
By introducing a Lagrangian multiplier A for the constraint
IT 0-0..) = I-a,
j =1
J
48
criterion 2 is equivalent to minimizing
(2.4.35)
=
p
y.
1
~ - ,\ [ IT F. (y .) - (1-0.)].
gj
j=l
j=l J
J
The optimum procedure can be obtained by solving the following system
of first partial equations:
p
IT F . (y .) - (1-0.) = 0, and
j=l J J
av
ax
(2.4.36)
P
av .
aYk'
(2.4.37)
Replacing
,\
gk
f k (Yk) = 0,
k=l, ... ,p.
P
IT F.(y.) by (I-a) in (2.4.37), we have the following p equations:
j =1 J
(2.4.38)
-1 -
IT F.(y.)
j =1 J J
Fk (Yk)
J
...!....
-,\
gk
I-a
Fk(Yk)
f
(
k Yk
)
= 0'
k 1
= , ... ,p.
e-
From the derivative
(2.4.39)
Fk(Y)
=
FklY)
we know that fk(y)
1 - (gk- 2 - y ) 2Y£k ( y) ,
is a monotonic increasing function of y under practical
consideration (i .e. y> gk-2)'
From (2.4.5) we have
(2.4.40)
1
Fk(Y k )
gk
f k (Yk)
= A(l-a), a constant for k=l, ... ,p.
Thus, for the special case g. = g. = g, we have the same confidence coefficient.
1
F. (y.)
(2.4.41)
1
1
f. (y.)
1
1
=
F. (y.)
J J
f. (y.)
J J
J
,
...
49
y.
y.,
J
1.
a. = a ..
J
1.
Extending the above special case to all p variates, we have the following
lemma:
Lemma 2.4.13.
When g1 = g2 = ...
= gp = g,
the optimum procedure corresponding
to criterion (2) gives
a.
(2.4.42)
J
= a* ,
2
y. = X&,l-a*'
1.
a*
= 1-
j=l,. .. ,p,
where
(l_a)l/p.
This is the same result as from optimum procedure (I).
Now we consider the case &. < &..
1.
J
When P = 2, the system of equations
(2.4.36), (2.4.37) reduces to
(2.4.43)
F
-1
= I-a,
l (Yl) F2 (y2)
&1
F (Yl)
l
= ,\ (1-a) , and
f 1(y 1)
With the constraint Fl (Y1) F (Y ) = I-a, Y2 is a decreasing function of Y1·
2 2
Defining the function
I-a,
(2.4.44)
and applying (2.4.26) we have the following theorem:
The function h(y ) given by (2.4.44) is a monotonic inl
creasing function of Yl. The optimal solution corresponding to criterion (2)
Theorem
2.4.14.
50
(2.4.32) can be obtained by solving the equation
If gl < g2 and the function h
(l
1 )
gl,(l-a)
yi
<
2
X
gl' (l-a)
'~
> 0, we have the optimal solution
1
That is, (l-ap < (l-a)~< (l-a
~.
2),
the variate with a larger
d.f. will have a larger confidence coefficient.
Theorem 2.4.15.
When p=2, 1 ~ gl < g2
~
4, the optimum procedure corresponding
to criterion (2) results in a larger confidence coefficient for the variate
with larger d.f.
Proof:
Letting aO = 1 - (l-a)\
and
j=1,2,
then
(2.4.45)
(2.4.46)
2
2f 2 (X 2 ,1_aO)
2
f 1 (Xl,I-aO)
2
3f 2 (X 3 ,1_aO)
> 1,
2
2f 1 (X 2 ,1-aO)
2
3f 2 (X 3 ,1_aO)
2
f l (X 1 ,1-aO)
2
4f 2 (X 4 , l-aO)
> 1,
2
2f l (X 2 ,l_aO)
> I,
4f 2 (X;,1_a O)
2
3 f 1 eX 3 ,l_aO)
> 1,
we have
(2.4.47)
g2
That is,
-
Case 1)
&1
gl
0
f 2 (Y2)
0 - 1 >
f 1 (y 1)
= 1,
g2
°
for 1 ~ gl < g2 ~ 4.
= 2.
0
0
From F1 (Y1) = F2 (YZ)
we have
> I,
2
4f 2 (X 4 , l-aO)
flex~,l_aO)
>l.
51
(2.4.48)
2 <t> (( Y~)~) - 1 = 1- e
_~yO
2
e
=
2[1-
-~J!
2.
<t>((y~)~)] =
00
2
f
(yO)~
1
e-~Y dy.
Then
(2.4.49)
o
.fiTT e-~Y 2
=
- 1.
o
(Y~) -~e-~Y 1
Using integration by parts,
00
fo y
C2.4.50)
-3/2
Y1
Thus
(2.4.51)
00
f0
0
h (y 1)
=
Y-~e-~y dy
Y1
- 1
0
o -~ -~Y1
(y 1)
e
-~y
2(yO)-~e
>
1
0
1[1_ Cy O)-1]
1
- 1
1
0
o -~ -~Y1
(y 1)
c
=
2[1_(y~)-1]
.:. 1
c
-~y
dy
52
under practical consideration (Y~ > 2).
> 0
Case 2)
&1 = 2, &2= 3.
1
(2.4.52)
1-e
0
-~Y1
.
=
2¢
0 ~
«y Z)
L
1
) - 1- f(3/2) e
0
-~Y2
Then
(2.4.53)
o =
h(Y1)
=
-
3
e-
ffIT
3
=
"2
Using a change of variable and integration by parts, we have
(2.4.54)
Thus,
(2.4.55)
>
1.
z
o
[1+(Y2)
-1
]
53
> 0
.-.
Case
3)
Fl(Y~)
=
F2(Y~)' we have
o
o~
1
-~Y1
0 ~
2~((Yl) ) - 1 - f(3/2) e
(~Yl)
Then
4
(2.4.57)
o
(Y2 > 2) .
&1 = 3, &4 = 4.
From
(2.4.56)
under practical consideration
a
hey 1)
1
0
Y2 c
2
2 r(2)
=
23/2 r (3/2)
1
=
= 1-e
0
2
0
(l~Y2)'
_~yO
2
_~o
1
3
-~y
0
-~Y2
(y~)~ e
1
-1=
hTv Y~ e
0
o ~ -~Y1
3 (y 1)
e
2
- 1
3" (1/2+1/Y~)
o
=
4Y2
o
{l2n
+ I} - 1
3Y2 + 6
4Y
>
o
a2 -
1
3y +6
2
.
> 0 un der practical
conSl°d eration ( Y2() > 6).
Thus we have
- 1
54
heX
2
1»0,
gl,(l-a)~
2
The optimal solution Yi < X I '
gl, (l-a)~
(l-ai) <
(l-a)~
< (l-ai).
That is, the variate with a larger d.f. will have a larger confidence
coefficient.
Extending the above result to the general case p> 2, we have the following corollary:
Corollary 2.4.16.
For 1 S g.
J
S
4,
j=l, .... p.
The optimum procedure corre-
sponding to criterion (2) gives a larger confidence coefficient to the variate
with a larger d.f.
The usual induction method is not applicable to larger values of g ..
J
The exact optimal values of {al, .•. ,ap } cannot be easily obtained from the
above derivations.
A numerical method of trial and error can be used to
solve the system of equations (2.4.36) and (2.4.37).
2.5
Examples
In this section, three examples of SGCM, GGCM, and HM models are presented
to illustrate the procedures derived in Sections 2.2 - 2.4.
The data set for the three models is the RATS data set collected by
Gaynor and Grizzle (1980).
The data was generated
to assess the
eff~ct
of neo-natal lead exposure in Sprague-Dawley rats on the growth of the myelin
accumulation in the brain.
Four different treatments were used in the data:
55
1)
Control
2)
Lead treated
3)
Starved control
4)
Super starved
The data set is comprised of 5 replications of the same experiment.
For each replication 16 rat
pups from 4 litters, all born on the same day,
were randomized into 4 groups of 4 animals.
Each group was then randomly
allocated back to one of the four mothers for nursing.
Each of the four
treatments were then randomly assigned to one of the 4 maternal litters,
so that the rat pups feeding from a particular mother would receive the same
treatment.
Four groups of one rat within a maternal litter were randomly
chosen to be sacrificed at each of 30, 60, 90, and 120 days of age, respectively.
Therefore, within a given replication of the experiment, one mater-
nal litter received one of the four treatments, and then measurements were
taken from one of the animals in that Ii tter at each of 30, 60, 90, and 120
days of age.
The measurement made on each animal was the natural logarithm
of myelin content in the brain (mg/brain).
Assuming
1)
the measurements from different litters are uncorrelated,
2)
the measurements from the same litter may be correlated, and
3)
there is no litter effect,
then we have the following standard MGLM model:
(2.5.1)
where Y
, each row of YO has 4 measurements from a litter,
O(20 x 4)
56
The hypotheses of interest are the difference between the lead treated
group and control group (6 -8 ) and the difference between the lead treated
2 1
group and starved control group (8 -8 ),
2 3
2.5.1
An SGCM model
Using the RATS data, the SGCM model is
(2.5.2)
where
P:::[-~-~ ~~]
1 -1 -1
e-
1
The hypothesis of interest is H : 8 ::: C8 ::: 0, where C'::: [ -I
O
0
Letting
1 -3 1
(2.5.3)
HI
:::
pI
1 -1 -1
:::
1
1 -1
1
3
1
r1
1-3
H
2
:::
I
, 3
.-1
,
the resulting conditional model (2.2.1) is
(2.5.4)
E(yIZ)
I
V(Y Z)
:::
Xn
:::
I
+
!l'
zr,
E,
1
()
1 -1
01
0
57
where
The LSE's of fl, rare
(2.5.5)
T1
=
9.. 73541
8.21822
1.29433 -0.4381721
3.72421 -0.326930
8.99111
3.56806 -0.476797
r
l7.66623
r
=
I
5.29237 -0.860431)
(0.085616 -0.0226278
0.0658045).
We then have
(2.5.6)
e= [-0.379298
-0.193222
-e
0.0214941
0.00780753
(CRC,)-l = (3.33328
r1.66646
SE
= [0.0949042
-1.66646J
3.33258 ,
-0.0274339
0,0430596]
0.0318341 -0.0285718
(sym.)
S* H
000278105)
0.0374669
0.0749477
'
(~-e)'(CRC,)-l(~-e).
-1
The confidence set can then be constructed from SHSE .
Using Roy's largest root procedure, the (I-a) confidence set for
e
is
given by
(2.5.7)
Ch (S*S-l) S A ,
1 H E
a;s,m,n
where A
is the upper a point of the distribution of
a;s,m,n
-1 Wit
. h the parame t ers
of SHSE
~he
largest root
58
s = 2,
(2.5.8)
m=
~=
n =
20-5-3-1
= 5.5.
2
0,
2
The exact SCS for e can then be constructed from
a'S*a
(2.5.9)
H
a'S a
:<::;
E
Aa,s,m,n,
.
for all vectors a,
or from
(2.5.10)
Ib'(8-e)al
{A.
a,s,m,n [a'SEa][b'(CRC')b]}~
:<::;
for all vectors a, b.
From Heck's chart, the upper 5% point of the distribution of the largest
root of SH (SH
+
SE)
(2.5.11)
-1
under HO is
A
a;s,m,n
= 0.537.
A
Thus
A
a;s,m,n
=
a;s,m,n
l-A a;s,m,n
=
1.160.
When 8 = 0,
(2.5.12)
=
[0. 359703
-0.0203465
0.00118379
SH
(sym. )
-0.02664980 ]
0.00126350
0.00378339
and
(2.5.13)
Thus we reject the hypothesis H by Roy's largest root procedure.
O
2.5.2
A GGCM model
The data for this model were obtained from the RATS data with the
e-
59
4th response for the first 12 observations and the 3rd response for the
other 8 observations deleted.
The resulting GGCM model is given by
(2.5.14)
j=l,2 ,
where
n1 = 12, n 2 = 8,
Xl
14 , 1 )'.
4
= (I4'
X2 = (I4' 14 ) , •
= f-3I
P
1
1
-1
1
1 -1 -1
--
BI
=
1
0
0
0
1
0
0
0
0
0
0
1
B = 1
2
0
0
0
I
0
0
0
0
0
0
1
~]
The hypothesis of interest is HO: C13 = 0, C
Since both matrices PB
used in the model.
1
With
and PB
2
(2.5.15)
H
l1
= (PB)
1
=-1 [ -1I -1
2
2
1
1 -1
(PB
. 2 ) -1
=.!.6
r3
-~ 1
2
0 -3
t
l
r-~
0
1 -1
l
are square mqtrices, no covariates will be
0
-1
=
1
2
1
1 ...
60
the weighted least squares model in (2.3.9) gives
(2.5.16)
where
A
=
n =
['1&1 3]
X & 1
2
3
ll12 &"1
IS &
EJ .
61
2.43011
2.08333
2.28926
1.96445
0.156328
0.195738
=
0.191757
. 0.265447
.
: -0.0861451
!
I
-0.0669383 .
: -0.0668931
\. -0.120601 )
-0.346787
-0.205938
HaV
-e
=
0.0394105
0.0192067
l -0.000045205
1. 00849
.504244
1.00849
= .001
- .105946
-.52973
-.186797
- .093399
-.52973
-1.05946
-.093399
-.186797
.396029
-.371661
-.18583
.792057
-.18583
-.371661
.792057
.0855371
.171074
.171074
(sym.)
)
At a=0.05, a (I-a) confidence set can then be constructed from (2.3.19).
That is,
2
X
v
When H8 = 0, we have
6,.95
=
12.592.
62
= 33.5508
> 12.592.
Thus we reject H at level 0.05.
O
2.5.3
An HM model with p=2
The data for this model were ohtained from the first two responses
(30 days, 60 days) of the RATS data with the first 4 observations of the
second response (60 days) deleted.
As per the notation used in Section 2.4.2, the model is given by
(2.5.19)
E(Y11(4X1)) = XllB l ,
e-
E(Y 12 (16 Xl)) = X2 Bl ,
E(Y 22 (16 x 1)) = XZB2 ,
where
The hypothesis of interest is
(2.5.20)
2
HO: n HOj ' with HOj : 8.=C.B.=0,
J
J J
j=1
where
C1 =
C:
(2 =
(-1
1
0
1 -1
2 -1
:]
0)
.
With the models (2.4.4) and (2.4.5), we have the following LSE's:
63
(2.5.21)
1.8084
....
81 =
1.39318
1. 56946
0.825473
....2
° (1) =
0.0607718,
2.2028
....
8*2 =
1.02011
2.08189
1.89668
Y2 =
0.0832716
....2
°(2) =
0.019233.
At a= 0.05, an optimal (I-a) confidence set for ¢1 and ¢2 corresponding
-e
to criterion (1) given by (2.4.26) is a (I-a.) confidence set for ¢., j=1,2,
J
J
where
(2.5.22)
Ct.
U
.
1
= 0.0131213,
2 = 0.0373690 .
....2
2
Assumlng O(j) is a good estimator for o(j)' j=I,2, then an optimal (I-a)
confidence set for (¢1'¢2) can be constructed from a (l-a ) confidence set
1
for ¢1 and a (1-a ) confidence set for ¢2'
2
(2.5.23)
That is, from (2.4.10), we have
W* = [[-.415218]_ ¢1],[(.0607718)[0.4
1
-.176277
0.2
:'>
0.2J -1 [-.415218]
] [
- ¢ ]
0.4
-.176277
1
2
X2 I-a •
•
1
[[ .415218J+ ¢ ]' [2
. 176277
1
-1
-1][[.415218~ +
2
.176277)
q, ] ::; .316027 .
1
64
Wi
(2.5.24)
= [-.644472 - <1>2] [(.019233) (2.09948)(1[-.644472 - <1>2]
2
s X1 ,1-a '
2
1.644472+<1>21::; .418311,
- 1.06303 s <1>2 s -.22641.
Setting <1>1
= <1>2 = 0,
W*
1
(2.5.25)
=
=
we have
A A2
-1 A
<l>i[°(l)V 1 ]
<1>1
[-0.415218) ,[(0.0607718) (0.2)
-0.176277
=
=
2
-0.176277
7.14621
< 8.66704
W*
2
l:
1) ]-1 r-0.415218j
2
X2 ,I-a 1
=
and
e-
Al A2
-1 A
<1>2 [O(2)V 2 ]
<1>2
= (-0.644472)[(0.019233)(2.09948)]-1(-0.644472)
= 10.2861
> 4.33352
= X21,1-a
2
Thus <1>1
= <1>2 = 0
is not in the above (I-a) SCS for <1>1' <1>2'
At level a, we
O: <1>1 =<1>2 = O.
will reject the hypothesis H
A (1-a ) confidence set for 8
1
(2.5.26)
[
Both 8
1
=0
.415218 ]
[ . 176277
and 8
o
1
=81
+ 61 ]'
t
1
is given by (2.5.23).
2 -11
-1
2
[
( . 415218 ]
+ 811 :s .316027 .
.176277
are in the above confidence set for
value of 8 , 8 , we have
1
1
That is,
e1.
For a given
65
82
(2.5.27)
=
<P
2
G8~
+
Y2 = -.644472+(11)
8~(.0832716).
Two different procedures are used to construct the confidence set for 8 .
2
Procedure 1:
A (l-a ) confidence set for 8 is
Z
2
(2.5.28)
At
o
81 =
(0)
~
,
(2.5.29)
8
2
"
1 82
= -.644472,
2
"2
~
- 8 2 1 ~ {X l ,.962631 0"(2)V 2 }
=' {( 4.33352)(.019233)
(2. 09!)48) }~
• .418311.
Thus
-1.06278 s 8
0"
At 8 ::: 8 =
1
1
::; -.226162.
[- .415218] ,
-.176277
8 ::: - .693727,
2
(2.5.30)
Thus
2
-1.04872 s 8
2
~
"
18
2
- 821 s .354989.
.338738.
Procedure II:
At a
(2.5.31)
2
= .037369,
a (1-a /2) confidence set for Y2 is
2
"
IY 2 - Y2 1
S
.366596,
-.283324
S
Y ::; .449868.
2
Similarly, a (1-a 2/2) confidence set for ¢2 is
66
A
(Z.5.3Z)
I<pz-<pzl
~ .472577,
- 1.11705
~
<P
Z
~
.171895.
o
At 8 = 0, the confidence set for 8 Z is the same as the confidence set for
1
<P
2
.
That is,
(Z.5.33)
-1.11705
~
8
z
$
-.171895.
8 Z = -.693727,
(Z.5.34)
A
1
8z - 8z1
-1.38314
~ .689417,
~
8Z
~
-.0043099.
Procedure (II) gives a longer confidence interval for 8 Z'
criterion (2) given by (2.4.3Z),
(2.5.35)
0.
a
1 = .0187951,
Z
=
.0318027,
= 7.94832,
= 4.6091,
=
5.81375.
A (1-0. 1) confidence set for <PI is
(2.5.36)
A (1-0. 2 ) confidence set for
<P
2 is
Similarly, with
~
-
67
(2.5.37)
~21 ~ .431407,
1·644472
+
-1.07588
~ ~2 ~
-.213066.
Thus criterion (2) gives a smaller confidence set for
fidence set for
~1
and
~2'
~2'
Since
~1::: ~2
and a larger con-
~1
::: 0 is not in the above (I-a) SCS for
we will reject the hypothesis Ho at level a.
A (1-a ) confidence set for 8 is given by (2.6.36).
1
1
[
(2.5.38)
j
.415218)
(
.176277
+
81l
,[2 -11
-1
2
[
[.415218]
+
.176277
That is,
81l ~ .28982.
Procedure I:
80 ::: (0)
At
1
0'
8
(2.5.39)
'"
18
2
2
::: -.644472,
- 821 ~ .431407,
-1.07588
::: 8
1
~
8
$
2
-.213066.
::: [-.415218) ,
-.176277
8
(2.5.40)
2
::: -.693727,
A
18
2
- 8
2
1 ~
.366103,
-1.05983 s 8
2
s -.327624.
Procedure I I :
At a
2
::: .0318027, a (1-a 2/2) confidence set for
y~
is
68
A
1Y2-Y2' ~ .375856,
(2.5.41)
-.292585
~
~
Y
2
.459128.
A (1-(a /2)) confidence set for
2
~2
is given by
A
1~2 - ~21 ~ .484515,
(2.5.42)
-1.12899
At
8~ = (~),
8
2
(2.5.43)
$
~2 ~
-.159958.
= ~2' we then have
8
2
= -.644472,
A
182-821 ~ .484515,
-1.12R99
At 8
~ ~2 $
-.159958.
0
=
1
8
(2.5.44)
2
=
-.693727,
-1.40056 s 8
2
$
0.0131048.
For both criteria, the exact confidence set for 8
2
depends on the value
of 8 . By both criteria, procedure (II) gives a larger confidence set for
1
8~ than procedure (I) .
.
Criterion (1) minimizes the maximum of the weighted lengths of the two
quadratic forms.
Criteriqn (2) minimizes the sum of the weighted lengths of
the two quadratic forms, where the weight is the inverse of the corresponding
d.f.
At a
=
.05, from criterion (1), we have
69
(2.5.45)
2
XZ,l-a
= 8.66704,
l
2
X1,I-a
= 4.3335Z,
2
2
X2,1-a
Max{
i 2,I-a
l
2
l
2
-~- +
2
X1, 1-a
2
= 8.66704.
From criterion (2). we have
(2.5.46)
2
X2,I-a
= 7.74832.
1
2
X1.1-a
2
= 4.6091
-e
,
2
X1. 1 -a } = 4.6091,
2
2
Xz I-a
•
2
I + X2
1,1-a
=
8.58326.
Z
Thus for the set with larger d.f. (two d.f.) criterion (1) gives a larger
confidence coefficient (.9868787) than criterion (2) (.9812049).
And for
the set with smaller d.f. (one d.f.), criterion (1) gives a smaller confidence
coefficient (.962631) than criterion (2) (.9681973).
CHAPTER III
INCOMPLETE MULTIRESPONSE MODEL
3.1
Introduction
In multiresponse design, if the same p responses are measured on each
unit of the experiment, we may have a standard MGLM.
it
is either
However, in some cases
physically impossible or uneconomical to examine all the re-
sponses on each experimental unit.
In these cases, it is important to have
a procedure that can be used to analyze the experiment where ohservations
on some of the responses are missing not by accident, but by design.
One class of designs with missing responses is the 1M model proposed by
Srivastava (1968).
the model definition and the general procedure to analyze
the data are outlined in Section 1.6.
The 1M model defined by Srivastava is based on a less-than-full-rank
model in each of the disjoint data sets.
sponding full rank 1M model.
In Section 3.2, we propose a corre-
We then derive the procedure to analyze the
data and show some simplification of the procedure.
In Section 3.3, we consider a special type of 1M full rank model upon
which the analyzing procedure can be simplified.
We then construct an opti-
mal procedure to minimize the generalized variance of the estimated parameters,
and demonstrate the invariance property of the proposed optimal procedure.
To evaluate the 1M design, Section 3.4 compares the 1M model derived in
Section 3.3 to the corresponding complete multiresponse (CM) model with respect to generalized variance, asymptotic relative efficiency, cost, and minimum risk.
71
Section 3.5 gives a numerical example to illustrate the 1M full rank procedure derived in this chapter.
3.2
A full rank incomplete multiresponse model
Recalling Section
the given 1M model has the restriction that the
1.6,
sum of all elements in each column of the unknown matrix of parameters S* is
zero.
To simplify the procedure in analyzing this type of data, we construct
an 1M model without the restriction on the unknown parameters.
The resulting
full rank model is given by
(3.2.1)
v(Y j)
where
= I
j:::l, ...• u.
02~.,
J
n.
J
Yj(n.xp.) is a data matrix,
J
(X
J
X
jl(n.xr.) j2(n. xr)
J
r
J
j(r/ P j )
)
is a design matrix of rank (r. xr) ,
J
J
is a matrix of unknown parameters,
S(rxp) is a matrix of common unknown parameters,
B
j (pxp. )
J
L
Hp
is an incidence matrix of O's and l's,
/P j) ::: B'j
L B., is the covariance matrix for the p. responses
J
in Y.,
J
J
L
•
(pxp)
is a positive definite covariance matrix.
The hypothesis of interest is
where C(gxr) is of rank g.
lJ(pXV)
is of rank v.
Prior to analysis it is necessary for the data to be transformed into the
72
framework of a standard MGLM which meets the specified conditions.
A pro-
cedure hy which this may be performed is as follows:
( i)
(3.2.2)
First construct a linear set of Y.•
J
Q
j(rxp.)
= M!Y.,
J J
J
j =1, ... , u ,
where Mj (njxr) is a known matrix of orthogonal columns such that
M!M. = I
J J
r
, M!X'
J J1
= O.
Then we have
(3.2.3)
= A. BB. ,
J
J
V(Q.) = I
J
r
where A. (
J rx r
(ii)
)
@
L:.,
J
= M!X.?,
J J~
The design is homogeneous, i.e., there exists a set of known (rxr)
scalars a'l"" ,a. ,
J
Jrn
m
E(Q.) = [
J
(iii)
(3.2.4)
(3.2.5)
a known matrix.
L
k=l
Combine Ql' ···,Qu'
j=l, .... u.
We then have
73
where
l.:
U
=
U
( O)(E. IP.xL:. IP,)
J= J J= J
l.
(iv)
LL' is nonsingular.
Denote II
J=
where
=
U
0::·
(v)
IP.xmp)
(H
jk
) = L' (LL,)-l,
J
lI'
is (p.xp),
Jk
J
j=l, ... ,u,
Z*
(rxmp)
:;: QH
U
=
(L
V(Z*)
U
I
Q.11. ,· .. ,
Q.II.),
j=l J J 1
j=l J Jm
(3.2.7)
=
Ir
0
[H' E(O)H]
=
Ir
0
(E(k ,k ))'
1 2
where
U
L
,k ) = I HJ!k EJ.H J· k
Ck1 2
j =1
1
2
(3.2.8)
k=l, ... ,m.
Let
(3.2.6)
(vi)
u
Rearrange the clements in Z*.
then
74
where
u
lk =
j
L Q.H· k ,
=1
k=l, ... ,m.
J J
Then we obtain the expectation of Z
(3.2.9)
(vii)
E(Z) "
[U
s.
The factorization of Vel) is possible, i.e., there exists a known
positive definite matrix W( mrxmr )' such that
Vel)
(3.2.10)
=
W 3 L:*,
where L: *
is a positive definite matrix.
(pxp)
(viii)
If the above conditions are met, then there exists a nonsingular
matrix R
(3.2.11)
such that RWR' = I.
(mrxmr)'
E(RZj "
[R
V(RZ)
r
Then we have
[U 1
mr
s,
3
L:*.
The rows of RZ are independently normally distributed with common covariance matrix L:*.
Thus we have a standard MGLM model, by which the usuai art-
alysis procedures are applicable.
A critical condition in the above procedure
1S
the factorization of V(Z) .
With
u
Zk =
the covariance between Zk
and Zk
1
(3.2.12)
Cov(Zk ,Zk )
1
2
I
Q. H . k '
j =1 J J
k = I, ... , m,
is given by
2
= I
r
3
L:
k) ,
(k l' 2
75
2
If all the m covariance matrices L(k ,k )' k k = 1, ... m, are proportional
l 2
l 2
then we can factorize V(Z) into matrices with proper
to a fixed matrix, 2:*,
orders.
Theorem 3.2.1.
If there exist a positive definite (pxp) matrix E* and known
scalars A ,k ' k l ,k = 1, •.. ,m, such that
2
kl 2
(3.2.13)
factorization of V(Z).
then we have a suitable
(3.2.14)
V(Z)
where
W= f r
=
W 0 L*,
A,
0
l~l1·
=
A
(mxm)
-e
3.2.1
.. Aim]
Aml'
Amm
Removing the singularity of LL'
For the fM model with the restriction on the matrix of unknown para-
meters S* , application of the procedure transforming the data back into the
framework of a standard MGLM model gives
m*
L F*
(3.2.15)
k=l k
where
8. is a fixed constant,
J
m*
•
\"
k:l
(3.2.16)
where
F
k
J
= .
([r+l]x[r+l]).
S*(a· k +8.)B.,
J
J
J
j=l, ...
,u,
76
By using suitable choices of e.'s, we aim to remove the singularity of
J
L*L*'.
Once done, the resulting matrix L* is of full row rank mp.
A matrix of unrestricted unknown parameters
the last r rows of 13*.
(3.2.17)
13
(rxp)
B can be constructed from
The resulting full rank model procedure then gives
= (0
I )13*
(r* 1), r
'
Qj (rxr) = (0 (rxl)' I r )Q*j '
J·=l, ... ,u.
Thus
j=l, ... ,u.
(3.2.18)
where
Fk (rxr) =
m*
L Fk
= 0,
k=l
(3.2.19)
E(Q
u
) = (F1B, ... ,Fm*B)L*.
(r x \'. lP·)
LJ= J
m*
With m= m* - 1, the restriction of
LF
k=l k
= 0 can be removed by replacing F *
m
Then
m
m
(3.2.20)
E(QJ') = L \B(a·k+e.)B. - I FkS(a. *+8.)B.
k=1
J
J J k=1
Jm
J J
(3.2.21)
E(Q
u
(r x \'.
1 ·)
LJ= P J
) = (F B, ... , FmS)L,
1
e-
77
L
where
(mpxL~=lPj)
=
k=l •... ,m.
Thus LL' is nonsingular if L*L*' is nonsingular.
Theorem 3.2.2.
Without the restriction on the set of matrices {Fl'" .,F m}
in the full rank model procedure. L~=IFk = O. the nonsingularity of LL' cannot be removed by a suitable selection of {8 , ... ,8 } in the corresponding
1
u
restricted model procedure.
3.3
Optimum confidence procedure
For simpl iei ty. we cons ider an 1M design upon which the full rank
model procedure can be performed.
'rhe lM design satisfies the following
conditions:
(3.3.1)
(1)
u is even, even number of disjoint sets;
(2)
Bj +u/2 = B.J ,
(3)
A. = Fl'
J
Aj +u/2 = F2 •
(4)
j = 1 , ...• ul 2 •
j = 1, ... , ul 2,
All responses are symmetric w.r.t. the
inVOlved.
U
disjoint sets
That is, the number of disjoint sets for which
a specific set of exactly j responses are measured is
~"
J
the same for all (~) different combinations of j responses.
J
j=l ..... p.
From the condition (4) we have
(3.3.2)
(1)
r
j=l
L(~) = u.
J J
78
(2)
Each response is measured on k
k
(3)
=
O
I
(~-ll)'
9-.
J J-
. 1
J=
O
disjoint sets where
and
Each pair of responses is measured on k
l
disjoint sets
Q,
,
p-2
where k l = L Q,.(. 2)'
j =2 J JThe resulting full rank model procedure gives
(3.3.3)
E(Q.) = F I3B.,
l
J
J
E (Q .
J+u
/2) = F213 B.,
J
j = 1 , ••• , u/ 2 •
We then have
(3.3.4)
V(Q) = I
where L
With
,u
(2 pX L.j=lPj)
e-
~
r
=I Z ~ (B1,···,B u / Z)·
LL'
=
[l z ~ (B1,· .. ,Bu/z)][Iz ~ (B , ... ,B / )J'
1
u 2
= 1
=
r
2
2
~
u/2
I
B.B!
J J
j=l
~
k
O
=T
k
[...2.
2
1 2p '
I ]
P
and
z* (rxZp) = QL' (LL,)-l
u/2
=
(I
u
I
Q. B! ,
Q. B!)
j=l J J j=u/2+l J J
k
O
(T
I 2p) -
1
79
u/2
2
=k
u
\'
I
Q.B!,
Q.B'.l,
J J j=u/2+1 J J
(L
o
j=l
we have
(3.3.5)
E(Z*) = (F
I
B,
V(2*) = 1 0
r
=
F (3) ,
2
u/2
4/k~
2
I
r
k
O
0
1
2
[I2
o [
L (B.B!)
j =1
E(B.B!)]
J J
J J
E*],
0
where
2 u/2
E*
L
=-
kO j=l
(B.B)'.)
J
r.
(B.B!)
J J
= Diag(E) + (k1/kUJ Off Diag(E) .
-e
I~/2Q.B!
]=1 J J
2
With
j
.
Z =-
k
(3.3.6)
E(Z)
O
=
we have a standard MGUM model.
Q.B!
j = (u/ 2) + I J J
u
L
[::1
B,
V(Z) -- .1k
o
The confidence set for CBU can be constructed through the usual confidence procedures used in the standard
is given by
(3.3.7)
with
(3.3.8)
V(B)
MGL~1
model.
A usual estimate of B
80
The full rank 1M model procedure begins with the selection of the set
of matrices {M , ... ,M } such that
u
l
(3.3.9)
M!M.
= Ir ,
MjX
= 0,
J J
jl
j=l, ... ,u.
The above selection of {Ml, ... ,Mu } is not unique,
An optimal confidence procedure can be used to derive the set of
matrices {M , ... ,M } such that the generalized variance of B is minimized.
u
l
With V(S)
=
~
k
O
(F' F + F' F ) -1
1 1
2 2
~ E*,
the generalized variance is
A
IV (8) I
(3.3.10)
Since (f-)rpIE*l
r
o
is invariant under different selections of the set
{Ml",.,~lu}
satisfying the conditions (3.3.9), the optimal criterion is equivalent to
maximizing IF?l
Letting U
+ FZF2
1 .
= {u 1 , ... ,u r 1 ,}
be an orthogonal basis of the vector space
generated by the columns of XII;
then an additional set of orthogonal vec-
{VI'" .,v r } can be constructed such that {U,V} is an orthogonal
basis of the vector space generated by the columns of Xl = (X ,X ).
1I 12
tors V
=
=
Denoting the matrices U
0(n1xr )
1
we then have
(3.3.11)
(u , ... , u ) and V (
I
() n1xr)
r
= (v l' .. , ,vI')
l
X 1.' 2 = UOT u + VOT v fo I' some known ma tri ccs Tu(rtxr)' Tv(rxr)'
Comparing the column space of M with that of YO' we have two different
I
situations:
,
81
1)
Type I:
the column spaces of M and Vo are the same.
l
2)
Type II:
the column spaces of M and V are not the same.
l
o
Under Type I situation, there exists a known orthogonal matrix T
l(rxr)
(3.3.12)
Letting K = {k , ... ,k
} be an orthogonal basis of the vector
l
nl-rl-r
space which is orthogonal to the column space of Xl' we then denote
K)(
( n x [ n -r -r ].) =
1
1
1
(kl, ... ,k n -r -r ).
1
1
Under Type II situation, there exist known matrices T (r x r) and
2
T
3([n -r -r]x r )
l l
~
0, such that
We then have
TIT
(3.3.13)
2 v'
=
T'T
2 2
+ TIT
3 3
and the condition MiMI
=
I,
we have
(3.3.14)
Together with (3.3.13), we have
82
C3.3.15)
IF1(II)Fi(II)
I
::
IT'T
v 21"1'
2 vI
::
11"1"1'
v 2 2l' v I
:0;;
11"1"1' T
::
IT'v CT'T
2 2 + 1"1'
3 3)1'v I
::
11"1'
v vI
::
1Fio/lCI)I.
v 2 2 v + T'T~T
v ~ 3l' v I
Extending the above result to all u disjoint sets,
IFiCII)FICII) I s IFiCI)FICI)
I,
IF~CII)F2CII)1 s IF;CI)F 2CT ) I,
C3.3.16)
1FicII/l(II)
Theorem 3.3.1.
+
FZO I)F 2 (II)I
1FiCI)FICI)
+
FZCII)F2CII)1
s 1FiCI)FICI)
+
FZC I)F 2C I)
S
I·
e-
To meet the criterion of minimizing the generalized variance
of the parameter estimator B given by (3.3.7), an optimum confidence procedure for the 1M model satisfying the conditions C3.3.1) can be achieved by
using the Type I procedure to construct the set {M , ... ,M u } in the full rank
l
model procedure.
3.3.1
Invariance in the optimum confidence procedure
In the above optimum confidence procedure, the selection of the set
{M , .•• ,M } is not unique.
1
u
The effect of different Type I selections of
the set {M , ... ,M } on the resulting confidence set can be examined through
u
l
the transformations
•
83
(3.3.17)
M,J+u /2 ~ MJ'+u/2 PZ'
j
=1, ... , u/ 2,
where PI and P are (rxr) orthogonal matrices.
2
Under these transformations,
(3.3.18)
Q'J+u / Z ~ P2' Q'J+u /2' j = I , •.. , u/
' Z
-e
Zz
-to
P'Z
Z2
F
~
P'F
1
1 I'
The induced transformations for Band L* on the parameter space
are identity transformations.
(3.3.19)
B -+ B ,
E*
-+
E*.
From (3.3.18)
(P' F )'
(P' F )
= F'F
1 l'
(P'F )'
(P'F )
= F'F
2 2'
(I" F )'
(P'Z )
=
(I" F )'
(PZZZ) =
1 I
2 2
1 1
2 Z
1 1
2 2
I 1
i
F Z1 '
1"7
'rz·
Thus both 8 and V(S) are invariant under the transformations (3.3.17), and
84
we have the following result.
Theorem 3.3.2.
The optimum confidence procedure given by Theorem 3.3.1
is invariant under different selections of the set {M ,··· ,M u } satisfying
1
the Type I conditions.
3.4
Some comparisons between incomplete mu1tiresponse model and complete
mu1tiresponse model
The full rank model satisfying the set of conditions (3.3.1) in Sec-
tion 3.3. is
(3.4.1)
E(Q.)
J
= F1BB.,
J
and
j =1, ... ,u/2 .
e-
The resulting estimate of 6 and its variance are
(3.4.2)
2
6 I M= -k
a
u/2
u
I
( F I F + F.' F ) - 1 [F'
Q. B! + F'
1 1 2 2
1 j =1 J J
2
z
V(SIM) = k: (Fi F 1+ F F2)-1 0
E*,
k
E* = Diag(E) + (-1) Off Diag(E).
where
k
The corresponding
eM
o
model is given by
(3.4.3)
E(Qj+u/2)= F26,
V(Q.) = I
J
r
A resulting standard MGLM is
0
E,
j=l, ...
,U.
L
Q.B!],
j=u/2+1 J J
and
85
XB,
(3.4.4)
E(Y)
and
V(Y) :: I
where
Y (urxp) =
X
r::
'u1
l
(F
.
=
(urxr)
and
L:,
~
ur
1
0 J
.
]
( (ul 2) x 1
r· 2 0 J ( (ul 2) xl)
,
J((u/2)xl) is a ((u/2)xl) column of l's.
The estimate of (3 and its variance are
2
(3.4.5)
U
u/2
-1
(Fl'FI + F 2'F )
2
[F '
I
I
u
I
Q. + F~
Q.],
j=l J
... j=u/2+1 J
and
3.4.1
Generalized variance
The ratio of the generalized variance for SIM to that for SCM is
A
I
IV(SCM) I
IV(SIM)
(3.4.6)
x
Thus, we can compare these two mode Is by
"'-
(3.4.7)
R = {
)l/r
IV(~IM)I }
IV(B cM ) I
= (~)p
k
o
#f
L:
•
Depending on the structure of L:, we consider the following two special
cases:
1)
Intraclass covariance structure
86
In this case,
2
croP
and
L:* =
where
r l = k~ < 1,
+
rlP(J - IJ],
k
-l/(p-l)<O<l.
Then we have
R
(3.4.8)
(I-rIP)
p-l
[l+(p-l)TlP]
p-l [l+(p-l)p]
(1-0)
= R(p)
After some simplification,
3R(0)
3Q,
which has the same sign as p.
As 0 increases from -l/(p-l) to 0, R(p) decreases from R(-I/(p-l))
=
+00
to R(O)
p
= (~)
k
O
p
As p increases from 0 to 1, R(O) increases from
> 1.
(~) to R(I) = +00. That is, the generalized variance from the lM model is
kO
greater than the generalized variance from the eM model. As the absolute value
of 0 increases, R(p) increases, making the 1M model less desirable.
2)
Autocorrelation when p=3
In this case,
L: =
2
1
cr o
2
o
p
1
o
1
1
1
-1<0<1.
L
87
Then
2 2
(3.4.9)
R(p)
1 .
Theorem 3.4. 1.
2 4
1- 2r l P - rIP
2p
2
+ P
+
3 4
2r 1P
4
A comparison of the generalized variance between the 1M
model and its corresponding CM model is given by R in (3.4.6).
the intraclass covariance structure and autocorrelation when p=3
Considering
R attains
p
its minimum (~) > 1 when there is no correlation among all p responses.
kO
R(p) is an increasing fWlction of Ip I· As I p I i.ncreases, the 1M model is
less desirable.
3.4.2
Asymptotic relative efficiency
A practical comparison of two different procedures is through a
.-
measure of thei r re lati ve efficiency.
When both procedures are at the
same Type I error level (a), the relative efficiency is the inverse ratio
of the two sample sizes required to achieve the same power against a specified alternative.
In this section we compare the 1M model with the corresponding eM model
through a measure of asymptotic relative efficiency.
For both models, the
null hypothesis is liO: C(gxr)B J(pXl)=O, where C is a matrix of rank g. The
Wilks likelihood ratio statistic will be used to calculate the power under the
fixed alternative Ii a :CB<J = ao.J( gx 1)' where
u(
.)
is a fixed nonzero constant.
To improve the precision of the 1M design (3.3.6) we repeat the same
design.
By repeating the
following standard MGLM.
(3.4.10)
I~1
design n
l
times independently, we have the
88
where
=
Z0(2n r x p)
1
and
The estimate of S and its corresponding variance are
(3.4.11)
n
1
X
j=1
Z(JO)'
and
= (F'F )-1 0
o0
jL E*
k
O
e-
2
= n k
(Fi F + F F ) -1 0 E*.
1
2
1 o
2
For the null hypothesis
H : CSU = 0,
O
(3.4.12)
ISH + SE
1
where
S
HI
= (C8
lM
1
the likelihood ratio
statistic is
I
U)'[C(F'F )-l C , ( l (C8lMU)
0 0
= U'ZO[I - FO(FOFO)-lFO] ZOU.
SE
1
1
Letting T = 1 - - 1
2n 1 r
[r-r+~2(v+r+1)J
then for large value of N
1
=
1 _ v+r+1
4n r '
1
= (2n 1r)T 1 = 2n 1r
-
~(v+r+1),
the asymptotic
power function of the statistic WI at Type I level a. against the a1ternative Ha is
89
(3.4.13)
1- B1
r
= p(-21 1 Q.n Wnl
1
Z
= p(X f (0 1 )
where
~
~
Yf ,l-aIHa)
1
Nl
+ 0(-)
1
Yf,l-c/.)
+ 0(N 1 ,
1
Z
Yf,l-a is the (I-a) point of the X distribution with f d.f.
=
f
gv,
x~(al) is the noncentral XZ distribution function with
noncentra1ity
0
1
= trace
0
1
,
nl ,
and
-¥ [U'L:*LJI-1(CBlJ)'[C(Fil\
n k
=
.
-
Z 2 )-I C,(I(CBU) .
+ F F
A similar procedure can be performed for the corresponding CM design (3.4.4).
By repeating the design n Z times, we have the following standard MGLM .
l3.4.14)
where
YO
(nZurxp)
=
[t ]
(n Z)
and
X
O(nZurxr) =
[ F1 .. . \ (u/ 2) d ) ]
o J( n x 1 )
Z
FZ
0 .J (
(u/ Z) xl)
We then have
(3.4.15)
and
= (X'X
o 0 )-1
0
L: =
Z
nZu
(F ' F + F ' F ) -1 0 L
l 1
Z Z
90
= 0,
For the hypothesis HO:CBU
the likelihood ratio
statistic is
(3.4.16)
where
SH
and
SE
=1
Letting
T
then for
1ar~e
Z
-
Z
z
Xo(Xc)Xo)-IXo1
= U'Yo[I -
1_
nZur
[r-r+~(r+v+l)]
values of N
Z
= 1 -
(n ur) T
Z
2
=
=
YOU.
r+v+1
ZnZur '
r+v+1)
n ur - (
, the asymptotic
Z
Z
power function of the statistic W at Type r level a against the alternative
2
Ha is
= p(-ZTZR.n Wz
1 - B
(3.4.17)
2
2
0z = trace
S"t
~ X~ , I -a. IHa )
1
= P(Xf(OZ)
where
(nZur)/Z
~ Yf , I-a.) + O(rr)'
2
2,
For the two designs to have asymptotically the same power, it is necessary that
0
1
= OZ'
The asymptotic relative efficiency for the 1M model
against the corresponding complete mUltiresponse model is then
(3.4.18)
ARE
nZ
=n
1
k
=
-o
u
1
trace{[U I E*U] -1 (CBU) , [C (F iF 1+ F:/2) - 1C' 1- (C8U) }
trace{ [U'W(l (CBlJ) I [C(PiFl +F?.2) -leI (I(C8U)}
Under the fi xed al ternati ve
11 : C8U = a .] (gx I) ,
O
a
(C8U)'[C(F'F +F'F )-Ic:,,-l(CBU)
1 1
2 2
'
.
2
o b O for a fixed positive
constant bOo
= U
e-
91
I t follows that
trace { [U'E*U] -I a 2 b }
ARE = k O
u
trace{[U'EU]
-1
a
O O
2
O
b }
a
trace{[U'E*U]-I}
=
trace{[U'EU]-I}
We consider the following two special cases of the structure of E.
(I)
Intrac1ass covariance structure. U = J (p x 1) .
In this case we have
1
---<p<
p-l
trace{[U'EUj -I}
1
=
-2
poO
and
trace{[U'E*Uj-l}
=
1
-2
poO
1
[1 + (p-1)p]
•
1
[1+(p-l)r DI
1
Thus the asymptotic relative efficiency is given by
k
o
(3.4.19) ARE(O) = u
. .
-1 }
trace{[U'E*U]
trace{[U'EU] -I}
=
k ()
u
[l+(p-l)O]
[1 + (o-1)r P]
1
•
1
which is a continuous monotonic increasing function of O. - --1 $ 0
p-
At D=
1
p-l
$
1.
we have the lower bound of the function ARE(- --1.-) =
. p- 1
That is. as D approaches - P:l.V(CSCMJ(PXl)) approaches D.
1M model less desirable.
o.
This makes the
When P = 0, there is no correlation among all p
k
responses and ARE CO) = - O < 1.
k
u
_ 0
P
is ARE (l) - 1 ( 1)
u
+ pr1
rrom (3.3.2) we have
At P = 1, the upper bound of the function
and
92
_ ~
k1 -
L
p-2
Tage th er
L ( . -2) .
j=2 J J
the identity (~-1)
)-1
Wl·th
=
!P (~)
J '
we then have
ARE (1)
P
=
e-
=
But
[.I
(~)
L.
R,·(1)j]2 =
[L
]=1 J
j1,J
J 1 J1
2
=
HL(~)j]2
+
(~)j]2
+
j
~
=
)'
j1~j2
J J
HR,.
.
J
J J
~
j1,j2
= [; R,j
~
. #.
J1 J2
[~. (~
J 1 J1
R,.
[~. (~ )
J1 J1
(3)] [r
R,j
(~
J1 J1
(~) j2 L
)
R,.
(I~)
jl j 2]
J2 J2
L
(~
)2 J2
) £.
) j1 j 2 J
(~ ) [~(j~
J2 J2
+
j~)]
93
so,
and
k
(3.4.20)
ARE (1)
Theorem 3.4.2.
O
~
O.
O
u
p
l+(p-l)r
s 1.
l
Assume intraclass covariance structure.
thesis HO:CS.J(pXl)
a
=
= 0
against the alternative Ha :
C(3J =
Consider the hypoaOJ(gxIl' where
Then the asymptotic relative efficiency between the 1M model and
its corresponding
eM
model is given by ARE in (3.4.19), where ARE is an in-
creasing function of p and has value less than one for - I/Cp-I) < p < 1.
.-
1
ARE (- p-l) = 0 •
k
k
ARE (0)
= .-Q.
u '
ARE (I) -=
O
PSI
I+(p-I)r
u
l
(Ir) Autocorrelation when p=3. U=.J(3 XI)'
In this case, we have
P
I
and
-I<p<l, O<r <1.
1
Purthermore,
I
=
and
tracc{[UlL:*U]-l}
2
00
1
3+4p+2p
= [J(lX3)Z*
J
2 •
C3xl
) ]-1
94
Thus we have
k
(3.4.21)
ARE(p) =
O trace{ lUI purl}
u
trace{[U'W] -I}
which is an increasing function of p,
Theorem 3.4.3.
=
3+4p+2p
2
u
-1 < P < 1.
Assume autocorrelation when p=3.
Consider the hypothesis
HO:CBJ (pxl) = 0 against the alternative Ha :CB.J = a O'] (gxl)' where an f O.
Then
the asymptotic relative efficiency between the rM model and its corresponding CM model is given by ARE in (3.4.21).
ARE is an increasing function
of p and has value less than one for -1 < P < 1.
k
ARE(-l) =
3.4.3
O
u
1
3-2r
k
l
' ARE(O) =
uO
k
ARE(l) :: ~
U
Cost
Following the comparison of the asymptotic relative efficiency be-
tween the 1M model and its corresponding eM model, we now consider the cost
factors involved in the two models.
Assuming that in the 1M model, q individual units were used on each of
the u disjoint sets; then we have a total of uq experimental units. Letting
p be the number of complete responses, then the total number of response
measurements in the experiment is
(3.4.22)
For the corresponding CM model, we have the same total numher of expcrimental units, uq.
With p responses for each unit, the total number of re-
sponse measurements is
(3.4.23)
N2 = uqp.
e-
95
Let
Co
be the participant cost for each unit in the experiment and c.
J
be the additional cost per measurement for each unit on which exactly j responses were measured. j=l •...• p.
Then for n
l
repetitions of the IM de-
sign. the total cost is
(3.4.24)
And for n
Z
repetitions of the corresponding CM model. the total cost is
(3.4.25)
Assuming the cost per measurement increases as the number of responses
measured on the same unit increases, then we have
and
where
c
Ij L(~)c.
I (~)
= j=l . J J
A
J =
jL
j =1
J J
is the average cost per measurement for the IM model.
Assume also that in comparing with the cost per measurement the unit
participant cost is negligible.
Then. practically,
(3.4.26)
and
We now examine the asymptotic relative cost (ARC) for the 1M model and its
corresponding
eM model, where ARC is the relative cost of the two models
when both have asymptotically the same power.
The relative cost is
96
=
(3.4.27)
n
When both models have asymptotically the same power,
totic relative efficiency.
n
2
is the asymp-
1
Thus
T1 _ 1
- ARE
2
ARC = T
(3.4.28)
As in Section 3.4.2, we consider the null hypothesis Ha:CS.J = 0 against
the alternative Ha:CBJ= aa.J, where aa is a nonzero constant.
Following
Section 3.4.2, we have the following special cases:
(I)
Intraclass covariance structure
k
O
In this case, ARE(p) = -u
l+(p-l)p
The asymptotic relative cost is then given by
ARC(p) =
(3.4.29)
e·
1
~~~
ARE(p)
l+(p-l)r P
1+(p-l)p
l
= ---,,-..,....,....a continuous decreasing function of p, - 1/ (p-l) < P < 1.
c
l+(p-l)r
1
1
Thus, ARC(- - ) = +00, ARC(a) = A ~ 1, and ARC (l) =
p-l
P
cP
Thus when p >
a,
the cost for the 1M model is less than the cost for the
corresponding CM model.
(II)
Autocorrelation when p=3
k
In this case, ARE(p) =
cost is
a
u
3+4p+2p 2
2
3+4r P+2r l P
l
The asymptotic relative
97
(3.4.30)
1
ARCUJ) =
ARE CP)
2
k
3+4r P+2r P
A O
I
I
=
c u
2
3+4p+2p
P
I.:
a continuous decreasing function of p, -l<p<l.
c
c
A
P
It fallows that
c
c
c
ARC (-1) = (3-2r ) -A> - A, ARC (0) = .3-::; 1,
l c
c
c
p
p
p
1+2r
c
c
1 -A< A
ARC(l) =
-~ l.
3
c
c
P
P
and
As in case (I), when p > 0, the cost for the 1M model is less than the
cost for the corresponding eM model.
3.4.4
Minimum risk
In parametric estimation, the loss function often can be used as
a measure of the discrepancy between the parameter value and its estimate
value.
Consider a p-variate multinormal population with unknown mean
I1(pX 1) and covariance matrix L (pxp)·
Having recorded a sample {y 1'· . . ,yn}'
the loss function in estimating 11 by
Ycan
(3.4.31)
L =
n
(y - 11)'
1
y = n
where
and
i\
i\ (y -
n
L
j=l
be defined as
11),
y. ,
J
is a known pxp positive definite matrix.
If we also consider the cost factor involved, then the loss incurred
in estimating 11 by Y is given hy
(3.4.32)
where
L = (Y-I1)'A(Y-I1) + c .
n
n
c(> 0) is the known cost per unit sample.
Thus the risk function is
98
(3.4.33)
R ; E(L ) ; n- 1 trace(A~)
n
n
+
cn.
~,
R attains a minimum at n* , where
n
(3.4.34)
n*
=
and
R*
n
For a given
;
{.!.. trace(AE)}~,
c
Zcn* .
Extending the above derivation to the general case of estimating a
matrix of unknown parameters , 8 (rxp) , the loss function in estimating
S by S is
A
(3.4.35)
A
L ; trace{(8-8)'A(S-8)}
n
c*(n),
+
where c*(n) is the corresponding total cost in the experiment.
In 1M design, the estimate 8
1M
given by (3.4.11) is an unbiased esti-
mate of S, and its variance is
v(S 1M)
= _Z-( F'F + F' F ) - 1
1 1
2 2
n k
1 O
~ E*
Consider the total cost T given by (3.4.28), T = n1q CAPk ' Then we have
1
1
o
the risk
A
(3.4.36)
RIM (n 1) = E{trace[(BIM-S)'A (8 1M -B)]}
+
n1q c A pk O
"
"
= E{trace[A(BIM-B)' (BIM-B)]} + n1q c
A pk o
= trace{A __
2_ trace[(F'F + F'j: )-lJ[*}
n k
1 1
2 2
1 O
h
= nlkO
o
where hO = 2 trace [(FF 1 +
F:ZFz)
trace(AP)
-1
+
n 1q c A pk o'
].
lbe above risk R (n ) attains a minimum at
IM 1
99
(3.4.37)
Note that
trace (AP}
=
R1M(n )
l
=
Z q c A pk O n
l
~
.
1
Z{h o q c
A
p trace(AL*)}~.
Similarly, for the corresponding CM design, the estimate 6
given by
CM
(3.4.15) is also an unbiased estimate of 6, and its variance is
V(6
CM
) =
Z
nZu
(F'F
L
+ F'F )-1 0
1 1
2 2
.
The total cost T is given by (3.4.26), T = nzqupc ' Thus, the risk is
Z
Z
p
(3.4.38 )
RCM(n Z)
"
"
=
E{trace[(SCM- B)' A (BCM- S)]}+nzqupc p
=
trace{AE [(8 1M - B) , (SCM - 6) ]} + nzqu pCp
"
"
h
O trace(AE) + n qupc .
2
p
nZu
=
The risk R (n ) attains a minimum at
CM 2
h
(3.4.39)
n
Z
={
o
1
trace (AL)}~
2
u qpc
RCM(n Z)
=
P
Z qupc n2
p
= 2{h o
h
1
1 {O
- trace(AL)}~
=4
qpc
qpc
trace(AE)}~
p
1
p
Since both designs have q experimental units in each replication, the
numbers of repetitions, n
l
and n , can then be used as a comparison of the
2
relative sample size.
The relative risk can be evaluated by comparing the optimal sample
size together with the corresponding relative minimum risk.
c 1 = c z = ••• :: c p = c A' the" relative optimal sample sizf' is
(3.4.40)
RN(n 1 , n z )
=
~
"
n
1
_ u
k
Z
with the relative minimum risk
O
{traCe(AL*)}~
trace(AZ)
Assuming
100
{traCe(Ar*)}~
::
(3.4.41)
trace(Ar)
Now we assume that the matrix A has an intraclass covariance structure,
A:: [I+s(J-I)], - P:l
s
<
1, and we consider the following special
<
cases based on the structure of [.
(I)
Intraclass covariance structure
In this case
2
L: :: GO [l
+ P (J - I) ] ,
2
= GO[I
[*
+ r
trace(AL:) =
1P
G
trace (AL:*) =
and
(J - 1)],
__1_ <
P < I,
p-l
2
[I+(p-l)sp],
O
G
2
[1+(p-l)r s p ].
O
1
We then have
(3.4.42)
and
RN (n 1 ' n 2 )
RR(n l ' n 2 )
~ {
=k
=
U
1+(p - 1) r 1 s p
1+(p-1)sp
}
e
l~
1+(p-l)r s p ~
1
{ 1+(p-1)sp }
Both RN and RR are decreasing functions of sp, - -1p-l
1
RR(sp
=-
RR(sp
= 0) = 1,
= +00,
p-l)
1+(p-l)r
RR(sp :: 1) = {
and
-
RN(sp =
0)
~
I} < I,
P
1+(p-l)r ~
RN(sp=l) :: ~ {
I}
kO
k
From (3.4.20),
RN(sp =
O
u
P
1+ ( p-1 ) r
P
1
:s.
1
< sp < I,
1
p-l)
::::
u
= -k
O
+00,
> 1,
where
~
101
o
~
l+(p-I)r
U
RN(sp:: 1) :: k
Thus
{
~
1+(p-1)r
U
> {k
1}
I}
POP
That is, when sp < 0, we have RR > 1, RN > 1.
~ 1.
The 1M model has a large
minimum risk with a larger sample size.
When sp
= 0,
which includes the case p:: 0, i.e. no correlation among
all p responses, we have RR :: 1, RN > 1.
Both models have the same minimum
risk, but a larger sample size is required for the 1M model.
When sp > 0, we have RR < 1, RN > 1.
The 1M model has a smaller min-
imum risk, but needs a larger sample size.
(II)
Autocorrelation when p=3
In this case
.
e
l. = 0
E* =
0
2
0
2
0
[I
[I
2
P
'\
I
:2 j
1
rIP
1
2
02
1
r
rIP ]
1
-l<p<I,
2
trace(AE) = pOo[l+SP+sP ],
and
trace(AE*)
2
2
= poO[l+TlsP+rlsP
].
We then have
')
l+r sp+r sp" ~
{
1
~
},
l+sp+sp
(3.4.43)
From
aRR
ap
and
= -
we know that RR is an increasing function of p i f s (1+2p) < 0 and is a
102
decreasing fW1ction of p if s(1+2p) > O.
RN (p = -1) =
RR (p = -1) = 1.
RR(p
RR(p
1
= - -)
2
4-r s ~
l
= { 4-5 } • RN(p = -
5
= 0) = 1.
RN(p
=
4-r s
.!..)
=~
2
k
{
1}~
4-5
= 0) =ku .
o
and
RN(p
= 1)
> O. RR is an increasing function of p for p < -
decreasing function of p for p > p
'
o
O
1+2r s ~
1
RR(p = 1) = { 1+25 } •
When
u
k
21
1
2'
and is a
Thus RR attains its minimum at
1:
RR (p = 1)
RN(p
1+2r s ~
I} < 1;
1+25
={
and
1+2r s ~
1
-1""'+-:2";;;s-}
= 1)
We see that when
the 1M model has a smaller minimum risk only for large
5 > 0.
values of p (near one) • but a large sample size is needed.
When
I
s < O.
- 2<
RR(p = - .!..)
2
RN (p = _
RR attains its minimum at p
= {
4-r s ~
I} < 1 .
•
4-s
and
4-r s ~
=~ {
I} > {~
2
k
4-s
k
.!..)
O
= - .!...
2"
O
~
4-r,s
u
> {-k
...}
4-5
u
> {-
kO
o
1+2r
3
l
4+.!..r ~
2 I}
1
4+2
~
}
> 1.
We see that when 5 < O. the 1M model has a smaller minimum risk only for p
103
near -
t, but a larger sample size is needed.
When p = 0, both model s have the same minimum rif>k, but a larger samp Ie
size is required for the 1M model.
3.5
Numerical example
Fisher's IRIS dataset consists of random samples of 50 flowers from
the iris species: virginica, versicolor, and setosa.
For numerical illus-
tration of the 1M procedure, we use the responses of three measurements:
petal length (VI), sepal length (V2), and sepal width (V3).
Assume the re-
sponses for each flower are multivariate normally distrihuted with a conunon
covariance matrix.
We will test the hypothesis of a common mean vector for
the three responses in the three species populations.
With suitable arrangement of the data, we divide the data into u
disj oint sets where each set S. consist::. of
J
species.
:;<lr.lJl
= 10
ies of 5 flowers from each
We then assume the following response-wise design:
Set:
Responses:
2 ,3
1 ,3
1,2,3
1.2
That is, each response is measured on k
1,2,3
8 disjoint sets, each pair
k
1 = 0.75.
of responses is measured on k = 6 disjoint sets, and r 1 =
k
1
O
In each disjoint set, we have a standard MGLM.
(3.5.1)
E(Y.)
J
=
X.BB.,
J
J
and
where
l: j (p . xp . )
J J
=B!l:B.,
J
J
o=
104
01
1 JI
o
,
and
Following the procedure derived in Section 3.2, we first construct
j=1, ... ,10,
Q'(3
)=WY.,
J xp j
J J
where
Mj (l5 x 3)
=
is1
=
Xj
Then we have
(3.5.2)
= IS
E (Q .)
J
V(Qj) = 1
3
S B .,
J
0 L
j
and
j=1, ... ,10.
,
Thus, the above 1M design satisfies the set of conditions (3.3.1).
Letting Q
=
(Q1"" ,Q10)' we have
=
IS
(S,S)L,
Now
Z*(3 x 6)
=
~
3.29820
11.33690
7.81506
3.23112
11.04620
7. 56909
9.51447
13.59530
6.30571
9.27968
13.22630
[ 12.58910
14.61270
6.59640 12.24250
14.63510
6.12683j
6.69702
Letting Z(6 x 3) = [ : :
(3.5.3)
= (Zl,Z2) = QLl(LLl)-l
E(Z)
J ' we have a
= IS
(::1
6•
standard MGLM with
105
=
and
VeZ)
where
L:* = Diag(L:) + (0.75) Off diag(L:).
1.0L*=.!..1
6
4 6 0 L: *,
Thus it follows that
(3.5.4)
B1M =
V(SIM)
[ 1.4600
4.2025
5.0050
3.4400
5.9975
5.5525
6.5400
2.7800
2.9725
2 (21 )-1
o E*
3
O
=k
1
= 40
I
")
j.
and
1 3 o L:*.
In testing the hypothesis of species effect,
H : 8 = CB
O
=
0, where C =
(3.5.5)
l'l
1
[
-1 ' 0]
0-1
we have
-2.7425
-0.9925
0.6600J'
-4.0925
-1. 5350
0.4675
The matrix of sums of squares and products among species is
[86.9745
SI!
=
32.4543
-11.54470 I
12.1186
- 4.22744
2.30404.
J
The error matrix is
[0.089875
SE
=
0.0491875
0.1105630
0.0118125
0.0698750
J
0.0513125
The Wilks LR statistic is
(3.5.6)
0.000002579.
To evaluate the signi ficance of the statistic
F statistic
1\,
we use the equivalent
106
1
1-1\ ~
1
=
31\ 1;
1
1\1
=
Under H ' F has an F distribution with 6 and 3 d.f.
lNhen
O 1
0.000002579, F1 = 207.231 and the corresponding p-va1ue = .0048.
Thus at a level .05, we reject the hypothesis of a common mean vector
for the three species of iris.
With all 3 responses measured on each flower, we have a corresponding
CM model:
(3.5.7)
where
E(Y)
= XS,
V(Y)
= 130
Y(30 x 3) =
and
L,
0
[~I
QIO
I,
X (30 x 3) =
IS 1I 3
0
J (10X1)] .
The estimates of interest are:
A
(3.5.8)
SCM
=
V(SCM)
62
[1.4
4.260
5.006
5.936
2.770
5.552
6.588
2.974
= ~ (21 3) -1
0
30428 J,
L
1
= 50
13
and
0[.
In testing the same null hypothesis of species effect, we have
A
(3.5.9)
e=
SH =
[-20798
-0.930
00658J
-4.090
-1. 582
0.454
[37 0103
165.2480
63.2121
,
-57023961
-19.9527
11. 3449
SE =
r09386
4.4986
7.6002
008028J
2.6960
2.4300
.
107
The Wilks LR statistic is
(3.5.10)
The corresponding F statistic is
F
=
2
25
=
3
152.623.
Under H ' the statistic F2 has an F distribution with 6 and 50 d.f.
O
The
corresponsing p-va1ue is less than .0001, and we reject the hypothesis H
O
at a level .05.
Assuming intraclass covariance structure for Land 0 = 0.6, we have
the following comparions:
(1)
Generalized variance
The comparison is given by R in (3.4.8).
(I-rID)
p-l
(I_p)p-l
[1+(p-l)r P]
1
[l+(p-l)P]
[1-(0.75) (0.6) ]2
[1_0.6]2
=
Here
1+2(0.95) (0.6)
1+2(0.6)
3.8704 > 1.
The generalized variance for B
lS larger than the generalized variance
1M
(2)
Asymptotic relative efficiency for flo: CB.J = 0
The comparison is given by ARE in (3.4.19) where
against
108
ARE(0.6)
=
kO
u
(l+(p-l)p]
[1+(p-l)r P]
1
=
8
[1+2(0.6)]
10 [1+2(0.95)(0.6)]
= 0.9263 < 1.
(3)
Cost with C = C =
1
2
••. =
Cp'
We estimate the cost comparison given by ARC in (3.4.29) to be
ARC(0.6)
=
l+(p-l)r l P
l+(p-l)p
CA
cP
1+2(0.95)(0.6)
=
1+2(0.6)
=
0.8636 < 1.
The cost for the 1M model is less than the cost for the corresponding CM
model.
(4)
Minimum risk with s
~rom
=
0.6.
(3.4.42) we have
1+(p-l)r 1s p ~ = {1+2(0.9S)(O.6)(0.6)}~
• 0.9462 < 1,
RR = { 1+(p-l)sp }
1+2(0.6)(0.6)
1+(p-l)r s p ~
l
RN -- ~
{
} = 1.1828 > 1.
k
1+(p-l)sp
O
The 1M model has a sma iler minimum risk (RR < 1), but a larger sample
size is needed (RN> 1) .
CHAPTER IV
ON COVARIANCE STRUCTURES
4.1
Introduction
In multiresponse designs, if the observations arise in a structured
form, such as with the same p responses at each of t di fferent time points,
then the resulting covariance matrix may contain special patterns.
In
this chapter, we study hypothesis testing of some particular patterns
of covariance structures in the model.
A standard MGLM model with tp responses is given by
(4.1.1)
where
YO(NXtp) is an observed data matrix,
X lNxq)
is a design matrix of rank q,
f3 (qxtp) is a matrix of unknown parameters, and
EO (tpXtp)
=
(E .. ), is an unknown posi ti ve definite
LJ
covariance matrix with E.. (pxp), i,j
lJ
=
1,2, ... ,t.
The null hypotheses considered in this chapter are
HI:
~
""0
:::
I 6a L
+
1
(.1 - I) 6a
E ,
2
H : wEI in HI'
2
H :
3
E
l
H : P
4
z
+
PI (.1-1)
::
°1[1
::
1 in 11 •
I, E2
::
°2[1
+
PZ(.1-I)] in Ill' and
3
In each case, the LR test statistic is derived.
For the test
statistic expressed in closed-form, its moments are used to find the
110
asymptotic distributions.
For the test statistic without a closed-
form expression, the Wald test statistic is used.
The asymptotic non-null distribution of a test statistic depends
in general upon the type of alternative being considered.
For the
statistic with closed-form expression, we study asymptotic non-null
distribution of the statistic under a fixed alternative.
As the number
of observations approaches infinity, the power of the test will approach
one against any fixed alternative.
Thus, for each test statistic, we
also consider the asymptotic non-null distributions under a sequence of
local alternatives.
The log likelihood function under H is
O
(4.1.2)
= ~Ntp
L(B,E )
O
The MLE of
B does
-1
£n(21T) -~N £n(IEol) -~trace[EO
(Y-XB)'(Y-X(3)].
not depend on the pattern structure of EO' and is
(4.1.3)
With the MLE of EO'
1
A
(4.1.4)
EO
where
S
=
N S,
= Y' [I-X(XIX)-lXI1Y,
the resulting maximum log-likelihood under H is then
O
Asymptotic distributions of certain LR statistics satisfying a
set of specific conditions are contained in Section 4.1.1.
Asymptotic
distributions of the Wa1d statistic are contained in Section 4.1.2.
The hypothesis HI of block intraclass covariance structure in a
given standard MGLM model is examined in Section 4.2.
With suitable
transformations on the parameter space and the observed data, the MLE's
111
of E and L can be solved directly from the system of partial equations.
2
l
The LR statistic for testing HI can then be derived.
An asymptotic
expansion for the distribution of the LR statistic under HI is contained
in Section 4.2.1 and asymptotic non-null distribution of the LR statistic
when t = 2 under both the fixed alternative and a sequence of local
alternatives are contained in Section 4.Z.Z.
The hypothesis Hz: E = wEI in HI is examined in Section 4.3.
z
Under
A
Hz' the MLE of E can be derived as an explicit function of w, the MLE
l
of w.
But a closed form sOlution for
of partial equations.
wcannot
be solved from the system
However, by using numerical methods, the unique
solution for ~ can be estimated.
Thus, th~ LR statistic for testing
HZ vs. HI can then be evaluated.
The above LR statistic does not have a closed-form expression.
value depends on the numerical estimate of the parameter w.
Its
We can test
the hypothesis HZ vs. HI without using the MLE of the parameters under
H .
2
A test which is asymptotically equivalent to the LR test is the
Wald type test.
The Wald statistic for testing H vs. HI when p
2
contained in Section 4.3.1.
=
2 is
Asymptotic null distribution and asymptotic
non-null distributions under a sequence of local alternatives for the
test statistic are contained in Section 4.3.2.
The hypothesis H of intraclass covariance structures for the two
3
matrices E and [,2 in HI is examined in Section 4.4.
l
The MLE's of E
l
and L under H can be solved directly from the system of partial
2
3
equations. The LR statistic for testing H vs. HI can then be derived.
3
An asymptotic expansion for the distribution of the LR statistic under
the null hypothesis H is contained in Section 4.4.1.
3
Asymptotic non-
null distributions of the LR statistic when p = 2 under both the fixed
112
alternative and a sequence of local alternatives are contained in
Section 4.4.2.
The hypothesis H of equality of elements in the matrix L in H
2
3
4
is examined in Section 4.5.
As in Section 4.4, the MLE's of L
1
and L
2
under H can be solved directly from the system of partial equations.
4
The modified LR statistic for testing H vs. H can then be derived.
4
3
Asymptotic null and non-null distributions of the LR statistic are contained in Section 4.5.1 and 4.5.2, respectively.
Numerical examples based on a data set from the Lipid Research Clinics
Program are given in Section 4.6.
4.1.1
Asymptotic Null Distributions of Certain Likelihood Ratio Statistics
The distribution of a LR statistic can be obtained from its moments.
Let W (0 $ W S 1) be a random variable whose moments are
(4.1.6)
e
a
b
)' Yk h IT r[x. (l+h) + e.]
kn l k
j=l
J
J
E(W h ) = K
xj
b
O a
IT r[y k (l +h) + f k ]
IT x·
k=l
j==l J
b
where
K ==
O
IT
k== 1 r[Yk +X k ]
a
IT r[x .+e.]
J J
j==l
a
I
b
x. ==
J
j=l
L
k=l
Yk'
Define Z == -2p £n W, where 0 < P $ 1.
Then the log characteristic
function of Z is
a
L {-~n
(4.1.7)
f[px.+g.+e.] + 2ispx. ~n x.
j=l
+
~n
J
J
J
J
f[px.(1-2is) + g.
J
.
J
+
e.]}
J
J
113
h
L
+
k=l
{£n f[Pyk+hk+f k ] - 2isPY k £n Yk - £n r[Py (1-2is) + h + f ]}
k
k
k
where
Assume that x
and Y are of the same order, i.e., O(X )
j
k
j
Also assume that P depends on n so that I-p = O(n
-1
).
=
O(Y )
k
=
O(n).
Then by using
an asymptotic expansion of the log gamma function according to Barnes
[see Muirhead (1982), page 305], we have the following result.
Theorem 4.1.1.
Let W be a random variable satisfying the conditions
specified in (4.1.6).
Then the distribution of -2p £n W can be expanded
for large n as
(4.1.8)
P (-2p £n W
< 0 (n
-2 ),
where
a
(4.1.9)
= -2[ L e.
f
(a-b)] and
j =1 J
4.1.2.
a
B (e. )
2 J
x.
f . 1
J=
J
-
.!.[ L
=
d
=1
P
2
1
- d + 6 is the Bernoulli polynomial of degree 2.
Asymptotic Distributions of the Wald Statistic
Let Yl'YZ""'YN be a random sample from a population with density
function f(y;e), where e'
(el •... ,e ) is a q-dimensiontal parameter.
q
=
Then the log-likelihood of the N observations is given by
N
(4.1.10)
L(e)
=
L
j=l
(4.1.11)
Oefine
L.J (e),
where
.
L. (e) = £n fey. ,e) is the log-likelihood from the jth observation.
J
J
2
a L. (e)
(4.1.12)
Io(e) = -E[ae ~e' ]. the information matrix in Yj'
114
~
Then under certain regularity conditions, the MLE of 8, eN' is
asymptotically distributed as a Normal variate with mean
1
e
and variance
-1
N 10 (8).
In testing the hypothesis H: ce
=
YO' where C(rxq) is of rank r.
The Wald statistic is given by
(4.1.13)
Under the null hypothesis H, C8"
N
is asymptoti cally di st ributed as
a Normal variate with mean YO and variance ~[Clo-l(e)c']. Thus, the Wald
2
statistic W is asymptotically distributed X with r d.f.
N
We reject H
for large values of W .
N
When the hypothesis H is not true, we consider a sequence of local
alternatives HN: ce - YO
N.
1
IN"
6, where
=
I::.
is a fixed constant not involving
Under the alternative H , the statistic W is asymptotically disN
N
I
tributed as a noncentral X- with r d.f. with noncentrality parameter
(4.1.14)
The above information matrix 1 (8) depends on the unknown parameter 8.
0
A consistent estimator for IO(B) can be derived by rcpJ;Lcing 8 with its
A
MLE
8N"
That is
-1
-1 "
The estimate for 1 (B), 1 (eN)' may not be unbiased.
0
0
To reduce
the possible bias in the estimator, we apply the jackknife method.
For a random sample of size N, there are N different subsamples for
size N-l.
The jth subsample can he constructed by deleting the jth
us
"-
observation from the original sample.
Let 10 (6 N(j)) be the consistent
estimator for 10(6) from the jth subsample.
That is,
(4.1.16)
=-
1
{
E[
N-l
2
d [L(6)-L j (6)]}
d6 d6'
]
=
6
A
where 6 (j) is the MLE of 6 from the ).th su b samp 1e.
N
Then the jackknike estimator of 1~1(6) is given by
(4.1.17)
1~~(e) = N l~l(eN)
N
- (N-l)
~ j~l 1~1(eN(j))·
-1
-1
A
We replace 1 (6) with its jackknife estimator 1 (6) in the Wald
0J
0
test procedure above.
.e
4.2.
Testing Block Intraclass Covariance Structure for a Covariance Matrix:
HI: l:0 = I i l:l
+
(J-I) i !:2·
Under the hypothesis HI' the covariance matrix can be written as
(4.2.1)
E
0
where
I(txt) is an identity matrix,
=
I 6Il !:l
+
(J-I) 6Il E2 ,
J(txt) is a matrix of 1'5,
l:l (pXp) E2 (pxp) are symmetric.
Since the matrix J-I is symmetric, there exists an orthogonal matrix
f(txt), such that f' (J-I)p
=
O. where O(tXt)
=
Oiag(t-l ,-1,-1, ... ,-1),
with elements being the eigenvalues of the matrix J-I.
Then we have
(4.2.2)
= (f 6Il Ip)(I
=
(f i
Ip)(I
t
i El)(f' 6Il I p )
t
i El
+
+
(f i [p) (0 i E2 )(f' 6Il I p )
0 i E2 )(f' 6Il I p )
116
where BIOck(A1,A "" ,At) is a block diagonal matrix with the
Z
matrices AI,A Z"" ,At on the main diagonal, n l = LI
+
(t-l)L Z' and
From Section 4.1, we know that '"e = eX'X) -1 X'Y is the MLE of
when L is positive definite.
e
Thus, under HI' the log-likelihood function,
O
after being maximized with respect to
S, is
(4.2.3)
L(S, n l ,n 2) = -~Ntp ~n(2rr) - ~N ~n[1
=
~ I p ) BIOckenl,nZ"" ,n z)cr' x I p ) I]
-1
-1
-1
-~trace [(f ~ I ) Block (n
' n , ... ,n ) ([" ~ 11') S]
Z
l
2
p
-~Ntp ~n(2TI) - ~ ~n(lnll) - ~N(t-l) ~n(lnzl)
-~trace [Block
-1
(r
-1
(n l ' n 2 , ...
,nZ-1 )T]
= -~Ntp ~ne2TI) - ~N ~nelnll) - ~Net-l) ~n(ln21)
-1
-~trace(nl
where
S = (Y-XS)'(Y-XS)
T
= er'
~
I ) S (f
P
-1
~trace(nZ
TI ) -
t
T2),
= Y'[I-X(X'X)-lX']Y,
~
I )
P
= (T .. ),
i, j = 1,2, ... , t ,
1)
Tl = TIl' TZ = L T ...
j=2 ))
In the above log-likelihood, the two matrices n
in the same term.
Thus, the MLE's of n
(4.2.4)
I
nl = N
Tl
(4.2.5)
n
,.,
2
and
nz
and n
and
1
=
With n 1 = L
1
results.
l
do not appear
2
can be derived separately as
l
N(t-l) T2 ·
+
(t-l)L , n = L - L , we then have the following
2
2
l
2
e-
117
Theorem 4.2.l.
(4.2.6)
Under HI' the MLE's of L1 and L2 are given by
1
"
L = .!.[n + (t-l) 51" ] = Nt(T + T ) and
l
t 1
2
l
2
A
(4.2.7)
L
2
=
1 "
"
t[51 l - 51 2 ]
=
1
-(T
Nt 1
1
t-l T2 )·
The resulting maximum log-likelihood function is
(4.2.8)
Ll
Corollary 4.2.2.
"A
=
"
L(a,51 l ,51 2 )
= -~Ntp ~n(2t) + ~Ntp ~n(N) + ~N(t-l)p
in(t-l)
The LR statistic used in testing HI vs. H is
O
(4.2.9)
--
When t = 2, the above LR test statistic reduces to
(4.2.10)
which is equivalent to a LR statistic for testing the independence of
2 sets of variables.
The maximum log-likelihood under HI can also be derived without the
use of above orthogonal transformations because the inverse of an intraclass correlation matrix also has an intraclass correlation structure.
In this case the resulting LR statistic is
(4.2.11)
* we have
Equating Al and AI'
(4.2.12)
We then have the following useful result.
118
Let SCtpxtp) = CSiHpXp)) be a positive definite sym-
Theorem 4.2.3.
.. C )) = Crr 6a I ) $ Cf 6a I ), where
metric matrix and TCtpxtp) = (T 1J
pxp
p
p
1'
Ctxt
) is an orthogonal matrix such that
1" [J Ctxt) - It] r = Diag(t-1 ,-1, ... ,-1).
Then we have
(4.2.13)
=
ItCSl+SZ)!
and
=
jl[Ct-1)
S 1 -S 2 ] I
t
t
, where T
L T ...
1
j=2
J)
Proof:
Denoting [(txt)
l' 1 --
T
1
7f
= C1' 1 (t x l)
J
(t x 1)
an
1' 2Ctx [t-1]))' then we have
d
I ) ([2 6a 1 )].
=
p
,
Thus, T = C[1 6a I ) S C[l 6a I )
1
p
P
1
= -t
IT11
L
i,j
1
= t[(J(lxp)
p
e-
6a I ) S (J ( 1) 6a I )]
px
P
P
1
CS S
lJ = t 1+ Z),
S..
= ItCSl +$2) I·
Together with C4.2.12), we have
It
4.2.1
The Asymptotic Null Distribution of the LR Statistic
In this section we derive an asymptotic expansion for the LR
statistic Al as sample size N increases, beginning with a lemma regarding
the moments of the LR statistic when the hypothesis HI is true.
119
Lemma 4.2.4.
When the hypothesis HI is true, the h
th
moment of Al is
given by
(4.2.14)
E(A~) = k
O
(t_l)~(t-l)ph
tp
j~l r[~(l+h)
+ ~(l-q-j)]
p
P
k~l r{~N(l+h) + ~(l-q-k)}k~l r{~N(t-l)(l+h) + ~[l-(t-l)q-k]} ,
where kO is a constant not involving h.
th
The h
moment of Al has the same form as (4.1.6) with
(4.2.15)
--
a = tp, x. = N e. =
J
2'
J
~(l-q-j)
, j = 1,2, ... ,a,
b = 2p, Yk = N t = ~(l-q-k) , k = 1,2, ... ,p,
2' k
N
Yk = I(t -1) , t k = ~[l-(t-l)q+p-k] , k = p+l, ... ,2p.
Thus, analogous to Theorem 4.1.1, we have
Theorem 4.2.5.
When the null hypothesis H is true, the distribution
l
function of -2p ~n ~, where p is given by (4.1.9), can be expanded for
large M = pN as
~n Al
(4.2.16)
p(-2p
where
f = -2[
$ z)
a
4.2.2.
L e.
j=l J
= p(X~
$ z) +
2
0 (M- ),
b
-
2
- ~(a-b)] = ~[(t -2)p + (t-2)].
k=l k
2
Lf
Asymptotic Non-null Distributions of the LR Statistic When t=2
When t=2, the LR statistic, AI' is equal to
T
{~}N
IT111 ITnl
.
The random matrix, T, is distributed as a Wishart distribution W2p (n,n),
where n = N-q and
Q= [Q Il
n I2 ].
n22
When the hypothesis HI is true, n 12 =
o.
120
The power function of the LR test of level
where k
is the upper 100 CI.
CI.
go
> k(x)'
point of the distribution of - 2p Q.n fl., when
the hypothesis HI (rl 12 = 0) is true.
WI" .. ,w
1\
is P(-2p £n
CI.
Let p2
-1
=
Diag(w , ...
l
where
,W )'
p
)
-1
p are the eigenvalues of nllrl12nZZnZl'
Then, under fI
l
P- = O.
We consider two different alternatives
(A)
(B)
A fixed alternative Hk : n 12 ~ O.
A sequence of local alternatives
where
~
~:
is a fixed diagonal matrix.
Define the random variable Z by
-2p Q.n fl.
Z=
(4.Z.l7)
l
M~
Then from Muirhead (1982), we have the following results.
Theorem 4.2.6.
Under the fixed alternative 1\: n
12
~
0, the distribution
function of the random variable Z given by (4.2.17), can be expanded as
Z
-~ n 2
P(-T) S z) = ~(z) - M [I.:- <P(z)
T
(4.2.18)
where 0.
2i
trace P .,
=
J
Z
=
T
40
1'
and
+
(2)·-1
4
:-r(01-0))<P
(z) 1+ OeM ),
T
~
and'" denote the standard normal
~
'v
distribution and density functions, respectively.
Theorem 4.2.7.
Under the sequence of local a1 ternati ves 11 : p
M
The distribution function of -2p Q.n fl.
(4.2.19)
p ( - 2P £n fl.
l
4.3.
=
p2 , p
Test~ng
HZ:
=
1 -
1
=
can be expanded as
2
P(X ::: z)
::: z)
f
+ OeM
where f
1
2
i(p+~+q),
-2
),
and 01
= trace(~).
Proportionality for the Two Matrices, E and [Z' in HI:
1
Z WL I
When HI is true, the log-likelihood function, maximized with respect
L.
=
to S, is given by (4.2.3).
That is
lZl
(4.3.1)
"
L((3,n 1 ,n Z)
= -~Ntp ~n(Z1T) -~N ~n(lnll)
where
n1
~
trace(n
-1
T ) 1 l
~
- ~N(t-l) x.n(!nzi)
trace(n
= El + (t-l)E z' Ez = n 1
- n
-1
T )'
Z Z
z·
Thus, under HZ' the log-likelihood function, maximized with respect to (3, is
(4.3.Z)
'"
L ((3,w,E l )
= -~Ntp ~n(Z1T)
-
~Np ~n[l
+ (t-1)w] -
~N(t-l)p ~n[l-w]
-~Nt ~n[IEll] -~trace {E~1[1+(~_1)WT1 +
The matrix
1
T + __1__ T is symmetric.
1+(t-1)w 1
l-w z
l=W TZ]}'
Thus, the function
L(B,w,E ) can be maximized with respect to E at
1
l
-e
(4.3.3)
The result ing 10g-1 ikel ihood function is
(4.3.4)
"
"
L(S.w.E
)
1
=
+
_l...iNtpUn(21T) - 9,n(Nt) + l} +
~Np Q.n
~N(t-1)p ~n[l+(t-l)w]
[l-w]
- ~N t ~n { I [T 1+TZ] - wI [T 1- (t - 1)TZ]
I}.
To simplify the above expression, we use the following result from
Rao (1973).
Lemma 4.3.1.
Let A and B be pxp symmetric matrices of which B is positive
Then there exists a matrix R such that A = R'AR and B = R'R,
th
l
eigenvalue of AB- .
where A = Diag(A •... ,A ) with A being the k
1
k
p
Replacing A by T - (t-l)T ' B by T + T in the above lemma, we
Z
l
1
Z
definite.
have
122
I [Tl+T Z]
(4.3.5)
- Wl [T l -(t-l)T 2 ] I = !R'R - wR'i\RI
p
=
IR 'I I I - wi\ i IR I = IR' R 1kD1 (1- w\ )
=
ITl+TZlkDl (l-WA k ),
p
Then, the log-likelihood is
"
"
L(S,w,L
) =
l
(4.3.6)
-~Ntp{~n(2rr)
+
~Np ~n[l-w]
- £n(Nt) + I} +
-
~Np(t-l)
~Nt ~n{ITl+Tzl}
-
~Nt
£n[l+(t-l)w]
I
k=l
Q,n[l-{))A k ],
and the MLE of w, w, can be solved from the partial equation
dL(S,W,El)
aw
=
o.
That is
(4.3.7)
(t_l)2
N (-1)
= ~Np l+(t-l)w+ ~
Pl-w
=
=
Since
2[1+(t~~)W][1-W]
Nt
2(1+(t-l)w](1-w]
I
{P[(t-Z) - w(t-l)] +
k=l
{r
(t-2) - w(t-l) + \} .
k=l
1- WA k
1
t-1
< w < 1, [l-w] [l+(t-1)w]
~
"
0, w can be solved from g(w) = 0,
where
(4.3.8)
g(w)
= ~
(t-Z) - (t-1)w + \
k=l
1-A w
k
The function g(w) is a continuous function of w with
g(and
t~l) =p(t-l)
gel) = -p < O.
~Nt
> 0
e-
123
There exists at least one solution of w between - t:l and 1.
To
show the uniqueness of the solution, we evaluate
(4.3.9)
I
dg(W) =
dW
k=l
and use the following lemma.
Lemma 4.3.2.
If matrices T (pxp) and T (pxp) are positive definite, then
2
1
the eigenvalues of [T
l
- (t-l)T ] [T +T ]
2
l 2
-1
are between -(t-l) and 1.
Proof;
Since [T +T ]
l 2
(t-l)[T +T 1 + [T - (t-l)T ] = t T are positive definite.
1
l 2
l
2
We know that both I - [T
-e
(t-1) 1 + [T
l
- (t-l)T ] [T +T ]
1 2
2
-1
and
- (t-l)T ] [T +T ]-1 are positive definite.
Z
1 Z
1
Then the eigenvalues of [T
- (t-1)T ] [T +T ]
2
l 2
l
-1
are
less than 1 and greater than -(t-l).
#
Thus,
(4.3.10)
3g(w)
dW
=
I
(-1) 2 (l-\)[(t-l) +
\1
< O.
k=l [l-\W]
Since g(w) is a strictly decreasing function of w, there exists a
1
unique solution for g(w) = 0 between - (t-l) and 1.
"-
The value of w can be
evaluated through numerical methods, e.g. the havling-interval method.
Summarizing these results, we have
A
Theorem 4.3.3.
Under HZ' the MLE of El is
124
(4.3.11)
"
~l
1 {
:: Nt
T1
l+(t-l)~
+
TZ.. . }
l-w
where w is the unique solution of g(w) :: 0, and g(w) is given by (4.3.8).
The resulting maximum log-likelihood function is
(4.3.12)
L
2
=
.....
"
.....
L(B,w,L )
l
"-
:: -~Ntp{R,n(2TI) - R.n(Nt) + I} + ~N(t-l)p R.n[l+(t-l)~] + ~Np 9-n[l-w]
-~Nt
Corollary 4.3.4.
9-n { IT 1 +T Z\} -
~N t
I
k=l
9-n [l-~\] .
The LR statistic used in testing HZ vs. HI is given by
(4.3.13)
Under HZ' the statistic -Z R-n I\Z is asymptotically distributed as
2
1) _ [~) + 1 ]
a X variable with f degrees of freedom, were f:: Z p (p+
2
2
::
~(p-l)(p+Z).
4.3.1
We reject HZ for large values of -2 9-n A. J
.
•
The Wald statistic when P :: 2
Under the hypothesis HI' the log-likelihood maximized with respect to
13, is given by (4.3.1).
Since
~l
is a positive definite symmetric matrix,
. t s a symmetrIc
.matrlx
. £..1
'" -~ suc h t hat £..1
"'~ I-I
"'~ :: {~l'
\'
th ere eX1S
_!~
-~
~l ~2~1 ' where L
-~
1
l~
:: (~l')-
1
Under HI' the MLE's of L and L are
l
2
.....
1
(4.3.14)
L ::
(T +T ) and
l
Nt
l 2
"-
1
~2 = Nt (T 1
~
~.
, then we have ~2 :: ~lR~2.
1
- t-l T2 )·
Lettlng
.
('\ =
e-
125
With a one-to-one transformation between {Z1,E } and {Z1,R}, the
2
MLE of R is
(4.3.15)
When p = 2, we can express R =
[:~ ::J. L~~ = [:~ ::], T
j
=
[~:~ ~~~]. and
the log-likelihood can be written as a function of e and d.
(4 ..~.16)
L(e,d)=-~Ntp ~n(21T)
+Nt
B~(d,e)
B} (d,e)
-~
A} (e)
~n(D(d)) -~N~n(Al(e)) -~N(t-l) ~n(A2(e))
-
~ ~2(e)
where
-e
(4.3.17)
Al (8)
=
=
II+(t-l)RI
[l+(t-l)e ][l+t-l)e ]
1
3
2 2
(t-l) e ,
2
2
Z
B (d,e) = [1+(t-l)e ] [dlu
+ Zd d u
+d u ]
1
3
2 22
ll
l 2 12
2
- 2(t-l)e [d d u
+d u
+d d u
+d d u ]
Z 3 22
2 12
2 l 2 ll
I 3 lZ
+
ll+(t-l)ell[d~ull+2dzd3u1Z+d~UZ2]'
Z
2
= (1-8 3 ) [dlvll+ZdldZvlZ+dZv22]
and
2
+ 282[dld2vll+d2v12+dld3v12
2
2
+ d 2d 3u 22 ] + (l-81)[d2vll+Zd2d3vIZ+d3vZZ]'
Under HZ: L
Z
=
wEI' we have R = wI.
lZ6
Thus, the hypothesis HZ is equivalent to the hypohtesis that:
C[ :
0, where C(2 X6 ) =
]=
[~
0 -1
0
0
0
0
0
1
:]
As in Section 4.1.2, a Wa1d statistic can be defined
(4.3.18)
where I~~(e,d) is a jackknife estimator of I~l (e,d) given by (4.1.17).
Omitting the lengthy derivation, we have the following matrix
expectations.
(4.3.19)
e[b. ,0]
[6 D] [b.dO] ,
E[b. , B ]
dd 1
dd
d
[ ad ed' ] = Nt -[Oed)] - Nt ~ . - ~ [AI (8)]
aZL(e,d)
and
where
(t-1)
(d~
.)
,
(l+(t-l)8 ,
3
J )'
1
-Z(t-l)8 ,
2
127
2
o
o
-2
-2
o
o
[1+(t-l)e3)UZZ-Z(t-l)eZUlZ+(1+(t-l)elUll
-(t-l)e 2u 22
+[1+(t-l)8 )u
l 12
(Sym. )
()
2(t-l)
o
(Sym. )
The matrices ~8A2' ~dB2' ~ee,A2' ~dd,B2' and ~de,B2 can be derived from
~eAl' ~dBl' ~ee,Al' ~dd,Bl' and Llde,B 1 be replacing (t-l) with -1 and
u""
with v.",
i)
1)
" " T 1 h as a W'IS h art d'IS t rl"b Ulan.
t"
W2 ( n, 6':'~l [l+(t-l)R] ':'~l)
The stat1stlc
6
with expectation:
That is.
128
(4.3.20)
E(uU)
=
.,
n
{dO. [1+(t-1)8 ] - 2(t-l)d d 8 + d 2 [l+(t-l)8_1},
:>
l
Z 3 Z Z
3
[O(d)]Z
E(u
=
-n
{d 2d 3 [1+(t-l)8 l ]-(t-l) (d;+d1d3)8z+dldz[l+(t-l)83]}'
[0(d)]2
E(u
lZ
ZZ
)
) =
n
d
[0(d)]2
2 [1+(t-l)8 ] - Z(t-l)d d 8 -+- di[1-+-(t-l)8 ]L
3
l Z Z
2
l
Similar expectations for vII' vIZ' and v Z2 can also be obtained from
E(u
ll
), E(u
lZ
)' and E(u
zz )
by replacing n with n(t-I) and (t-l) with -1.
The expected values of ~dBl' ~dBZ' ~dd,Bl' ~dd,B2' l'i d8 ,B 1 , and ~d8,BZ
are obtained by replacing {ulI' u lZ ' u 2Z ' vII' v 12 ' v Z2 } with corresponding
expectations in the expression.
With the above expected matrices, the matrix IocgN,dN) can then be
exaluated.
The matrices I o (8 N(j),d N(j))' j = 1,Z, ... ,N, can also be
evaluated in a similar manner.
Thus, we have a jackknife estimator of the
I
inverse of the information matrix, I O
-.
(8,d), and the Wald statistic.
J
4.3.Z.
Asymptotic Distributions of the Wald Statistic when p=2
When p=Z, the Wald statistic W can be obtained from (4.3.18). Under
N
the hypothesis HZ: E = wEI' the statistic W is asymptotically distributed
Z
N
Z
as a X variate with Z degrees of freedom.
We reject HZ for large values
of W .
N
The power function of the Wald test of level a is p(W
N
> k ), where
C1.
k is the upper 100 a% point of the distribution of WN when HZ is true.
a
= (8 -8 ,8 ), under the hypothesis H , we have ~8 = O. Now
1 3 2
2
1
consider a sequence of lo~l alternatives H : ~8 = v'N" ~O' where ~O is a
N
vector of fixed constants. When H is true, the WaId statistic W is
N
N
Letting
~e
e-
129
2
asymptotically distributed as a non-central X variate with 2 degrees
of freedom and non-centrality parameter
(4.3.21)
2
I
-1
-1
A = llO [C 1 (e,d) C I ]
6 ,
0
0J
A
A
2
2
Since the non-central X distribution is a mixture of X distributions
with Poisson weights, we have the power function
00
(4.3.22)
where
2 2
2
P (W N > kal WN - X2 (A )) = L Pk CA) P(X 2+2k > ka ),
k=O
2
2
P (A) = (A / e-~A /k!
2
k
2
Many approximations to the non-central X distribution, X~(A2), have
been suggested.
Among them, Pearson (1959) used a linear function of a
2
central X distribution,
-e
and CX
2
f
+ b agree.
b
=-
c =
CX~ + b,
50
that the first three moments of
X~(A2)
The appropriate values of b, c and fare
4
A
~+3A2
~+3A2
~+2A2
and
4
2
f = ]J + A (3]J +8A )
(]J+3A2) 2
4.4.
Testing Intraclass Covariance Structures for the Two Matrices,
El and E2 , in HI:
H3 : El = all + 0lPl(J-I), E2 = 0 2 1 + 02P2(J-I).
Under the hypothesis HI of block intraclass covariance structure,
the log-likelihood function, maximized with respect to S, is given by
(4.2 .3) .
That i
5
130
(4.4.1)
where
Under H , we have
3
(4.4.2)
and
·Both
nl
and
of generality,
n2
have intraclass covariance structures.
Without loss
-1
n-1
and n can be written as
l
2
e-
(4.4.3)
and
The resulting log-likelihood is
(4.4.4)
"
L(S, d l ,d 2 'Yl'Y2)
= -~Ntp ~n(2'IT) +~Np ~n(dl) +~N(p-1) ~n(l-Yl) + ~N Q,n[l+(p-l)Y ]
1
where
u
+
~N(t-1)p
-
~ dlu l - ~dlYlu2
1
VI
=
Q,n(d ) +
2
r
TlU,j),
r
T (j,j),
2
u
j=l
=
j=l
v
2
2
~N(t-l)
(p-1) Q,n(l-y ) + l~N(t-l) Q,n[l+(p-1JY ]
Z
2
~ d 2v1 - ~ d 2Y2v2'
L
=
i~j
T (i,j)
1
= .~. T (i,j).
2
1
J
131
Solving the partial equations of the above log-likelihood yields
the estimates
(4.4.5)
d
l
Np[(p-l)ul+(p-Z)u ]
Z
=
d
[(p-l)ul-u ] [u +u ]
Z
1 2
U
~
Yl -
-
2
N(t-l)p[(p-l)v +(p-2)v ]
l
2
=
[(p-l)v -v ] [v +v ]
l 2
l 2
V
z
Y2
(p-l)u +(p-Z)u
l
Z
z
- -
(p-l)v +(p-2)v
l
2
We then have the following results.
Theorem 4.4.1.
Under the hypothesis
H ,
3
the MLE's of
L
and L are
2
l
(4.4.6)
where
~
(4.4.7)
=
d 21
+
dzyz(J-I).
The resulting maximum log-likelihood is
(4.4.8)
= -~Ntp~n(27T) +~Ntp~n(Np) +~Nt(p-l)
L
3
Corollary 4.4.2.
~n(p-l) +~N(t-l)p
The LR test statistic used in testing
H
3
Q,n(t-l)
vs. HI is
(4.4.9)
(4.4.10)
(p-l)
A31 = { (p-l) (p-l)u -u
r
.
1
P
ITll
(p-l) u +u
2)
[I
P
(p-l) (p-l)
f
2]
H is rejected for small values of A ,
3
3
1.
[
(p-l)v -v
l 2
P
IT I
2
(p-l)
v
]
l
+v
}2fi(t-l)
2
Ip -]
4.4.1.
The Asymptotic Null Distribution of the LR Statistics
The statistics, A and A , given by (4.4.10) have the same form
3l
32
as the LR statistic Al in Section 4.2.
Applying Lemma 4.2.4 to A31 and
A , we have
32
Under H , the h
3
Lemma 4.4.3.
th
moment of A is
3
p
r[~N(l+h)+~(l-q-J')]
IT
j=l
(4.4.11)
r[~N(l+h)-~l r[~N(p-1)(1+h)-~(p-1)qJ
j
~ 1 r [~ (t - 1) (l +h) + ~(l- q - j ) 1
r[~N(t-1)(1+h)-~q]
where k
o
r[~N(t-1)(p-1)(1+h)-~(p-1)q]
is a constant not involving h.
The above moment has the same form as (4.1.6) with
(4.4.12)
a = 2p,
x. =
J
x. =
J
b
::
4,
Y1
f
::
~N,
e. =
J
~(l-q-j),
~N(t-1),
e. =
~N,
~N(p-1),
1 = f3 =
J
Y2 =
-~q
,
f
2
~ (l
j
= 1,2, ... ,p,
-q - p- j ) ,
Y3 =
= f4 =
j = p+ 1, ... , 2p,
~N(t-l),
-~(p-l)q
Y4
:::
~N
.
The asymptotic distribution of A can then be derived by using
3
Theorem 4.4.1.
Theroem 4.4.4.
of -2p
~nA3
When the null hypothesis H is true, the distribution
3
can be expanded for large value of M = pNt as
2
A <- z) = p(X <- z) + O(M- 2 ) ,
3
f
(4.4.13)
p(-2p
where
f = P2 +p-4,
p = 1
~n
1
-I
[Y
j-1
-1 2
x. (e.-e.+
J
J
J
.!.)
6
4
1
-1 f2 .
Yk (k-fk+ 6)] .
k=l
L
(t - 1) (p - 1),
e
.
133
4.4.2.
Sympototic Non-npll Distributions of the LR Statistic when p=2
When p=2,
(4.4.14)
f = 2, andp= 1 -
2N(~-1)
Applying Theorem 4.2.3 to T
l
(q+t)·
and T ,
2
(4.4.15)
--
If we denote '1 =
matrices under H .
3
r~01f2
and '2 = f 2 '02 f 2' then '1 and '2 are diagonal
We also express
?
p~
as
Diag(~1'~2)'
where
(4.4.16)
'2(1,1)'2(2,2)
Two types of alternatives are considered here.
(A)
A sequence of local alternatives ~: p2 = ~ Diag(8 ,8 )' where 8 1
l Z
M
and 8
(B)
Z
are fixed constants.
A fixed alternative H : pZ #
k
o.
Under H , the characteristic function of -Zp Q,n A is
3
3
(4.4.17)
¢(M,s'~l'~Z) = E {exp[is (-2p
~n
A )I
3
= E {A-2is P }
3
= E
{A- Zis p} E {A: 2iS p}
.)2
31
134
where A and A are statistics for testing independence between two
31
32
variables.
The following result from Sugiura and Fujikoshi (1969) was
used to derive the above characteristic function.
Lemma 4.4.5.
Given A(2 x 2), a random matrix with a Wishart distribution,
(4.4.18)
then the h
th
moment of A is
n
h
(4.4.19)
E(A ) =
In (4.4.19)
Y=
°12 2
[1_y]2
,and
/1
(~,!~n; ~h+h;y).
2F1 is a generalized hypergeometric
°Il °22
function defined by
00
(4.4.20)
with
II (a 1 ,a 2 ;b 1 ;y) = L
k=O
(a)k =
(a 1)k (a 2)k
(b 1 ) k
y
k
IT ,
(a+k-1)!
(a-I) !
Applying Lemma 4.4.5 to the statistics A and A ,
31
32
(4.4.21)
¢(M,s,!/J1,!/J2) = E {A-
2isp
31
} E {A- 2isP }
32
where
(4.4.22)
and
¢(M,s,O,O) is the characteristic function of -2p £n A ,
3
135
G2(M,s'~2)
•
(1-~2)
=
(t-l)M+(t-l) cr
2t
2
(t-I) M+ (t-l)
(t-l) M (t-l) cr _ . (t-l) MOll, )
F ((t-I)M + (t-l)
2
cr, 2t
2 cr, 2t
+
2
lS
t
''¥2'
2 1 2t
2
Under H3, -2p 2n A has an asymptotic X distribution with 2 d.f.
3
Thus,
f
-"2
(4.4.23)
¢ (M ,S ,0,0) = (1-2is)
+ OeM
-2
)
=
(1-2is)
-1-2
+ OeM ).
Using the asymptotic expansion of the generali zed hypergeometric
function 2Ft given by Muirhead (1982, page 724), we have the following
expansions.
8
~1 S ,,¥",)
1)( - _I)}
[1 + 0(M- 2 )]
G1 ( I',
1 = exp {1
2t (1 - 2is
M
(4.4.24 )
8 1•
1
-1
-2
= 1 - 2t (1-1-2is) M + OeM ),
-e
,
G2 (M,s '~2)
and
=1
-
(t-l)8 2
-1
-2
1
2t
(l - 1-2is)M
+ OeM ).
Combining the above results,
f
(4.4.25)
¢(M,S
- 7"
'~t '~2) =
(i-2is)
[ (1- 2is)
f
2
(l - 2 is)
(f+2)
- -y- ]
-2
+ 0 (M
).
Inverting this expansion, we have the following result:
Theorem 4.4.6.
~; p2
..
= JI
M
Under the sequence of local alternatives
Diag(8 ,8 ), the distribution function of -2p 2n A can be
3
l 2
expanded as
(4.4.26)
2
=P(Xf$z)
+
OeM
-2
).
136
= Np =
Let M
l
-1
-2p M
i M,
then from Muirhead (1982),the statistic
~n
1\.31 has the asymptotic mean - !/,n(l-W ), and the asymptotic
l
l
variance of order O(Mi ) = O(M-l). Similarly, for M = N(t-l)p = t~l M,
2
the statistic -Zp M;l ~n 1\.32 has the asymptotic mean - ~n(l-WZ)' and the
asymptotic variance of order O(M;l) = O(M-l).
Thus, the statistic
(4.4.27)
has the asymptotic mean
-1
1
M {M [- R.n(l-W )] + M [- R.n(l-W )]} = - t{R.nO-W ) + (t-l)(1-W )},
l
2
2
l
l
2
and the asymptotic variance of order OeM
-1
).
The variable, Z, defined by
(4.4.28)
has the asymptotic mean 0, and the asymptotic variance of order 0(1).
The characteristic function of Z is
(4.4.29)
~z(s) = E
{exp(isZ)}
= E
{exP[-2isPM-~R.nI\.3]}· eXP{isM~i[~n(l-ljJl)
= ~
(M,sM
= ~
(M, sM
(l-W
where the functions
~,
-~
-~
,W l 'W 2 ) . [(l-W l ) (l-W Z)
,0,0)' (T1 (M, sM
. 1
1
)IS
t
~
M . (l-W
.
2
)IS
1!.:..U
t
(t-l)
~n(1-tjJ2)]}
(t-l) is.!.. M~
] t
-~,-~
,WI)
+
G2 (M , sM
,W 2)
~
M
G , and G are given by (4.4.21) - (4.4.22).
1
Z
To simplify (4.4.29), we use the following two lemmas.
Lemma 4.4.7.
(4.4.30)
The hypergeometric function, 2Fl (a ,a ;b ;y) is equal to
l 2 l
b -a -a
l 1 2
(l-y)
2Fl(bl-al,b1-a2;bl;Y)'
137
Lemma 4.4.8.
[From Sugiura and Fujikoshi (1969)]
The hypergeometric function 2Fl
(-iSM~, -iSM~; ~(M+a) - iSM~,y) has
the asymptotic expansion
(4.4.31)
The expansion of the characteristic function of Z is therefore:
(4.4.32)
<t>Z(s) :: exp{-2s
t [1/J1+(t-I)1/J2]}
2
[ 1+M - ~ {2is + ~ [I/J 1 (1 -I/J 1) + (t - 1)I/J 2 (1- l~ 2) ] (is ) 3}] + 0 (M - 1) .
Letting
(4.4.33)
we have
(4.4.34)
<t>Z (s)
::
s
<t>Z (T)
::
exp[-~s
T
-e
where
a
1
::
2, a
2
::
2
]{l+M
-~
[a
1
.!..T (is)
+
3
1 (is) 3] } + O(M- 1 ),
2 3"
T
1
4 - [I/Jl (1-l/Jl) + (t-l)IjJ2 C1 -1/J2)] .
t
Inverting (4.4.34), and using the fact that
5
e
where <t>(k) (x) is the k
Theorem 4.4.9.
th
-2· .
for k :: 1,3,
derivative of the standard normal density, we have
Under the fixed alternative Ilk: p2 1 0, the distribution
function of the random variable Z given by (4.4.28) can be expanded as
Z
p (:r:S z)
(4.4.35 )
where
<l>
and <t> denote the standard normal distribution and density functions,
respectively.
T
2 is given by (4.4.33), a
l
and a
2
are given by (4.4.34).
138
4.5.
Testing Equality of Elements of Z2 in H3 :
H : Zl =
[1 + PI (.1-1)], L 2 = 1 wJ.
1
4
°
°
Under the hypothesis
H ,
4
(4.5.1)
and
Denoting
(4.5.2)
d
[° 1 (I-PI)]
=
-1
,
l
Yl = [Pl+(t-l)W] [l+P(t-l)w + (P-l)P J- ,
1
Y2 = [Pl-wJ [1-pw+(p-l)P 1 1
-1
-1
,1\]
-1
and 1\-,
anu
are
(4.5.3)
e-
and
The log-likelihood, maximized with respect to S, is therefore
(4.5.4)
"-
LC/3,d,y l ,y Z) =
where
ul =
-~Ntp
R.n(2n) +
I TIC"
j=l
")'u 2 =
J,J
~Ntp
R.n(d)
+~N
L Tl (·1,J')'
i~j
R.n(l-PY )
1
v1 =
+~NCt-l)
In(1-PY )
2
I
T ' ")' v)= L T2 (· ").
j=l 2C ],]
- ifj
I,J
Solving the partial equations of the above log-likelihood yields
A
(4.5.5)
d
=
Ntr(P-l)
,
+ [(p-l)v -v ]
1(p-l)u -u
l 2
1 2
1
Yl = P
- u N+u 1 =
l 2 d
/0'\-
k{
I -
[(p-l)u -u ] + [(P-l)V -V ]}
1 2
1 2
,
t(p-l) (u +u )
l 2
139
I
I
I {
d= P
P
[fp-l)u l -u 2 ] + [(p~l)vl-v21 }
(t-l)
1 - t(p-l)
v +v
1
2
We then have the following results.
Theorem 4.5.1.
Under the hypothesis H , the MLE's of L and L are
4
l
Z
(4.5.6)
(4.5.7)
The resulting maximum log-likelihood is
(4.5.8)
L =
4
-e
-~Ntp
Corollary 4.5.2.
£n(2lT)
+!~Ntr
+~Nt(p-l)
£n[np]
Q,n[(t-l))
N
N
?(p-l) -2(t-l) (p-l)
W~
W
N
t~
(4.5.9)
•1
+~N(t-l)
The LR test statistic for testing H vs. H is given by
4
3
? t(p-l)
H~
£n[t(p-l)]
1
(t-l)~
2
(t-l) (1'-1)
is rejected for small values of A .
4
Under the hypothesis 11 , both
3
structures.
~2l
and
~22
have intradass covariance
Thus, there exists an orthogonal transformation fp(pxp),
such that
(4.5.10)
=
f'Q r
p 2 P
140
Under H , T has a Wishart Wp(n,SG ) distribution.
1
3
1
Denoting Q
- f 'T f ' and applying Theorem 4.2.3,
l(pxp) - p 1 p
p
(4.5.11)
=
has a Wishart WI (n(p-l),W
12
j~2
Q1(j ,j)
) distribution.
Similarly,
(4.5.12)
di s tri but ion.
Further applying Theorem 4.2.3 to SG.
J
(4.5.13)
W12
=
lJ!22
= a\
,~.,
J
j = 1,2, yields
(l-P 1 ) under H4 ·
Since WI and W have different degrees of freedom, the LR test
2
statistic A of Lemma 4.5.2 is biased.
4
That is, the prohability of
rejecting H when H is false can be smaller than the probabi I ity of
4
4
rejecting H when H is true.
4
4
We therefore use the following modified LR statistic:
n
- t(p-l)
2
t
(4.5.14)
n
-(t-l) (p-l)
(t_l)2
4.5.1.
The Asymptotic Null Distribution of the Modified LR Statistic
·
Wh en t h e h ypot h eSIS
LJ
11
4
•
IS
th
true, t h e h
moment
t~nt(p-l)h
(4.5.15)
(t_l)~n (t-l) (p-l)h
0
f'
~ * IS
.
1\4
e-
141
t~nt(p-l)h
=---:---:---:-:---0:---0-:-:--(t_l)~(t-l) (p-l)h
(1+h)]
r[~n(t-l) (1'-1)
(1+h)]
r[~(p-l)] f[~n(t-l)(p-l)]
t~nt(p-l)h
= kO
rp~n(p-l)
( 21lJ ) ~n t (p - 1 ) h
r[~(p-l) (1+h)]
(t_l)~n(t-l)(P-l)h
r[~n(t-l) (1'-1)
(1+h)]
r[~t(p-l) (l+h)]
where k is a constant not involving h.
O
This h th moment has the same form as (4.1.6) with
a = 2, Xl =
(4.5.16)
b
=
L, Yl
!~(p-l),
x
2
= ~(t-l)(p-l), e l = e 2 =
= ~nt(p-l), f l =
0,
O.
Applying Theorem 4.4.1, we have
-e
Theorem 4.5.3.
When the hypothesis H is true. the distribution function
4
of -2p £n A* can be expanded for large M = pntp as
4
(4. 5. 1 7)
p(-2p Q.n A* <- z)
where
P = 1
4
-
=
2
p(Xl$z) + OeM
-,~).
2
(t -t-l)
3n(p-l)
t(t-l)
1
Asymptotic Non-null Distributions of the Modified LR Statistic
4.5.2.
If we denote
(4.5.18)
where
1-1
4
,
~
12
and
~l'
--
are given by (4.5.10), then A = 1 under the hypothesis
In this section, the following alternatives are consiJereJ:
(A)
A general fixed alternative H : t:,. I 1.
k
(B)
A sequence of local alternatives
constant and M = pNtp.
r~:
t:,. =
I +
1
M
e, where e is a fixed
142
Define the random variable Z by
(4.5.19)
k
were
n(p-l)
n(p-l)+n(t-l)(p-l)
=
Z
k2
=1
=
1
t'
t-l
= --t--
- kl
and p is given by (4.5.17).
Then, from Muirhead (1982), we have the following theorems.
Theorem 4.5.4.
Under the fixed alternative, H : 6 .,. I, the distribution
k
function of the random variable Z can be expandcu asymptotically as
(4.5.20)
where ¢ and
denote the stanuard normal distribution and density
</>
e- .
functions respectively, anu
(4.5.21)
?
?
'[- = ~kl k 2a'" ,
a
l
=
a2 =
a
Theorem 4.5.5.
=
1
2T
?
2
1
[klkz(O-+o )-Z] = T [k1kz-lj
2k l k Z
2
'[2
[a
+
2
3
4
I(k l -k 2 )a -k l kZo J,
1-6
k Mk '
l
2
Under the sequence of local alternatives
1~1:
!1 = 1 +
e
M'
the distribution function of -2p tn 1\4* can be expanded as
(4.5.22)
?
p ( - 2p
~n
1\4* $ z)
+
(t-l)e-
2
4Mt
Z
[p(X 3 $x)
?
P (X
i $ z)]
_ 1
+ () (M -).
143
4.6.
Numerical Examples
The data set used in this section is from the Prevalence Study of
the Lipid Research Clinics Program.
This data set consists of 57 white
participants (36 males, 21 females) between ages 50 and 59 from one of the
clinic centers.
For each participant, the variables used are sex, plasma
total cholesterol levels (mg/dl) at Visit 1 (CHOLI) and Visit 2 (CHOL2),
and plasma triglyceride levels (mg/dl) at Visit 1 (TRIG1) and Visit 2
(TRIG2).
Assuming standard MGLM for the data, let
where Y (57 X 4) is the observed data on CHOLl, TRIGl, CHOL2, TRIG2,
O
X(57 X 4) "
-e
[~36 ~J is
the design matrix,
Bl2 x 4) is the matrix of unknown parameters, and
L (4 X 4) is the unknown positive definite covariance matrix.
O
Under H ' then log-likelihood is
O
where N
= 57,
t
= P = 2.
The MLE's of Band L are
O
(4.6.2)
B
=
(228.667
235.048
-797.472
159.238
)~
=
1153.56
-797.472
1024.38
-768.868
- 797.472
7503.29
-1272.68
6068.37
232.083
226. 19
1024.38
-1272.68
1357.54
-917.09
The resulting maximum log-likelihood is LO = -1167.72.
180.75 )
152.47 , and
_768.868]
6068.37
-917.09
7306.74
144
4.6.1.
Testing Block Intraclass Covariance St ructure for
L 2 ] where
L1 '
Under the hypothesis H., EO .[::
matrices, the MLE's of L
and L
1
2
~O
L and L,
1
are
A
Ll
(4.6.3)
=
[ 1255.55
-857.281
-857.281)
7405.01
[ 1024.38
1020.77
-1020.77)
6068.37 .
and
A
L
=
2
The resulting maximum log-likelihood is L
l
= -1169.08,
and the LR
statistic for testing HI vs. H is
O
(4.6.4)
From Section 4.2.1. with p
function of -2p
(4.6.5)
For Al
~n
=
52.5, the distribution
A can be expanded for large M as
~n
p(-2p
=
.921053, M = np
=
Al
~z)
.2555565, -2p
=
~n
2
P(Xf
~ z)
+
OeM
-2
), where f = P
2
= 4.
Al = 2.51315, and the asymptotic p-va1ue
Thus, we accept the hypothesis HI of block intra-
of the test is .642283.
class covariance structure for L '
O
When the hypothesis "l is not true, we first consider the fixed
alternati ve H : L
k
°
=
12
-8
10
-8
72
-12
60
14
-9
(Sym. )
70
According to Section 4.4.2.,
p2
=
[0.0241015
\
the statistic Z
=
0
° ]
t
=
.334199, and
0.0038207
0.14234.
145
= ~Z = 0.4Z5914,
Thus, at the observed z
(4.4.6)
p(~)
> z)
1
=
1- <I>(z) +
M~[r.\(z)
1
Now consider the local alternative
+
the asymptotic power is
~
13
l\1:
p
Z
1
..
=
~:2
asymptotic power at the observed statistic z = -Zp
(4.4.7)
p(-Zp
1\ > z) = 1 -
Q,n
2
{P(Xf ~ z) +
°1
M
= 0.716401.
(0 _o')¢(Z) (z)]
1 '
Z
Q,n
1\
Under
~1'
the
= 2.51315 is
Z
2
[p(X + ~ z) - p(X < z)]}
f
f Z
= 0.650844,
which is slightly larger than the p-value of the statistic -Zp
Q,n
J\
under
the nu 11 hypothes is HI.
Testing Proportionality for E and L in HI
Z
1
4.6.2.
Under the hypothesis H : l.z = Wl. , the MLE' s of ul and E are estimated
l
l
2
numerica 11y as
(4.6.8)
=
w
.815071
and
"-
E = ( 1253.08
1
-75.31Z8
-75.31Z8)
7325.45
The resulting maximum log-likelihood is L')
The MLE's of eN and d are
N
A
8
(4.6.9)
N
=
aN =
For C
=
[~
0.770236]
-0.11284Z
[ 0.801107 ,
I
0.0293072 ]
0.00243191
0.0118626
.
0
-1
0
0
1
0
0
0
(4.6.10)
~]
- .0308711 ]
=
C[ :: ]
[ -.112842
=
-1175.11.
146
The jackknife estimate of the variance of C [
(4.6.11)
l
N
C
1~~(e,d)
C' ::: .0001 [16.3036
6.72915
eN]
dN
is
6.72915]
7.53537
The Wald Statistic has the value
(4.6.12)
Under H , W is asymptotically distributed as a
2
N
2 d.f.
x-?
variate with
The asymptotic p-value of W is
N
(4.6.13)
P(X; >21.(441) ::: 0.00001995,
the hypothesis H is rejected at level
2
a.::: 0.05.
When HZ is not true, we consider the local alternative
Under H , the statistic W is asymptotically distributed as a nonN
N
7
central
x-
(4.6.14)
variate with 2 d.f. and non-centrality parameter
-0.05]'
[ -0.1
[c
rl(e,d)
Thus, the asymptotic power at W
N
(4.6.15)
C,]-1
0.1
[-0.05] ::: .9H0587.
-0. 1
=
21.6441 is
A2
?
2
= ~ [~) k e -7
P(X -(\
) > 21.6441)
l.
2
kl
Z
k=O
.
')
P(X
2+ 2k >21.(441).
We can estimate the above power by summing terms with Poisson weights
k.
A
2)
k
e
2
2
k!
f
(4.6.16)
>.000001.
p(X~(A2)
In this case, we have
> 21.6441)
= .000286.
147
4
A.
---:;:- = -
With Pearson's approximation, we use b =
c =
2+3A.
2+3A. 2
2
1.24755, and f
=
")
.194577,
2.54512.
2+2>.. ..
Also,
2
= P(cXf+b>
(4.6.17)
4.6.3.
21.6441)
=
Testing Intraclass Covariance Structures for L
.000326.
l
and L
2
in HI
Under the hypothesis "3' both L and L have intraclass covariance
l
2
The MLE's of L and L are
2
l
structures.
(4.6.18)
2::
1
[ 4330.28
-857.281
-857.281 )
4330.28
( 3546.36
-1020.77
-1020.77)
3546.38
and
The resulting maximum log-likelihood is L
3
statistic for testing 1I
3
=
-1211.83, and the LR
vs. "l is
(4.6.19)
According to Section 4.4.1, the distribution function of -2p
with p
=
~n
A
3
0.938596, M = Ntp = 107, can be expanded for large M as
(4.6.20)
For A
3
p ( - 2 p In A3 :; z)
=
2.7242 x 10-
of the LR test is O.
19
")
p (Xf:; z)
, -2P In A
3
- ")
2
0 (M .. ), where f = P
+
= 80.2443,
Thus, the hypothesis "
3
+
P - 4 = 2.
the asymptotic p-value
of intraclass covariance
structures for L and L , is rejected.
1
2
When the hypothesis H is not true, we first consider the fixed
3
12
alternative H : L = 100
k
l
[ -8
8
10 -10]
- ] • 1: = 100
2
[
-10 60
72
According to Section 4.4.2., we have
148
p
For z
(4.6.21)
2
0.539697
=
=
0],
= 1.48004, a
T
[
f
=
o
0.555556
1
= 2, a
Z
-.303698, the asymptotic power is therefore
Z
P(-> z) = I
T
-
~(z) +M
-~
[a
1
-1 <jl(z)
T
+
a
.l.
Z T3
<jl(2) (z)] = 0.659049.
1[810
Next, we consider the local alternative ~: p2
= MZ
Under
~~,
(4.6.22)
0.973558,
the asymptotic power at z
=
-2p
~n
~J
8 =8 =1.
2
1
A = 80.2443 is
3
2
8 1+(t-1)8?
2
?
p(-2p ~n A3 >z) = I - P(Xf~ z) +
2tM ~[P(XfS z) - P(X +2 :S z)]
f
=
I-P(X;S80.2443) +
=
o.
2~[P(X;S80.2443)
- P(X;S80.2443J]
CHAPTER V
SUGGESTIONS FOR FUTURE RESEARCH
In this dissertation, we derive optimal confidence procedures for
generalized multivariate linear models.
In an HM model with p responses,
we assume that the covariance matrix of the responses is known or that a
good estimate is available from the experiment with a large sample size.
Considering the correlation information among the responses, a step-down
procedure is used to construct the confidence set for the unknown parameters.
The confidence procedure involves p quadratic forms of chi-
square statistics.
We optimize the confidence procedure by minimizing
suitable norms of the confidence set.
These norms include the weighted
sum of the P quadratic form lengths and the
form length.
~aximum
weighted quadratic
In each case, the weight is the inverse of the degrees of
freedom for the corresponding chi-square statistic.
For the weighted
sum criterion, when all the degrees of freedom are less than 5, the optimal
confidence procedure gives a larger confidence coefficient to the quadratic
form with larger degrees of freedom.
The usual induction method is not
applicable to extend this result to a large degrees of freedom.
The
development of some alternative procedures to accommodate all possihle
values of the degrees of freedom is another area of future research.
When the sample size is small, the individual F statistics should be used
in place of the chi-square statistics to construct the confidence set.
In this case, the confidence procedure involves complicated F distributions.
150
It would-be helpful to develop similar optimal criteria and, hence,
the corresponding optimal procedures related to these F statistics.
In Chapter III, we examine the relative performance of the proposed
special 1M model and its corresponding complete multiresponse model.
These comparisons include generalized variance, asymptotic relative
efficiency, cost, and minimum risk.
The result of each comparison depends
on the covariance structure of the p responses.
We assume a covariance
matrix with an intraclass correlation structure or an autocorrelation
structure with p=3.
Extension of these comparisons to autocorrelation-
structured covariance matrix with a general value of p as well as other
structured patterns of the covariance matrix is desirable.
comparison depends on the alternative hypothesis.
The power
We consider linear
hypotheses of the form CBU = 0 versus the alternative CBU = 6 , where U
0
is a column vector of lIs and 8
0
is a column vector with equal elements.
Future developments in the power comparison include an extension to a
more general setting of linear hypotheses with different values of the
matrices C, U, and 8 ,
0
Finally, we consider the situation when the same p responses are
measured at each of t distinct time points.
We examine the hypotheses of
the block intraclass correlation model as well as some special cases in
the standard MGLM model.
In each case, the LR test is derived.
When
the LR statistic has a closed form expression, its moments are used to
find asymptotic null and non-null distributions.
When a closed form
expression for the LR statistic does not exist, we construct the Wald
statistic to test the hypothesis and then derive the asymptotic null and
non-null distributions.
After testing of block covariance structures,
the next step in this research effort is the testing of the linear
151
hypothesis
cau = 0
when the covariance matrix has one of the specified
block intraclass correlation structures.
used to test the hypothesis.
Suppose the LR procedure were
Then, the LR statistic would be the ratio
of the maximum likelihood under the specified block covariance structure
and the linear hypothesis CBU
=0
to the maximum likelihood under tile
specified block covariance structure.
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