SOME APPROXIMATE PAIRWISE-EFFICIENT MULTIPLE-COMPARISON
PROCEDURES IN GENERAL UNBALANCED DESIGNS
By
Yosef Hochberg
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 916
MARCH 1974
ii
ACKNOWLEDGMENTS
I wish to express my gratitude to Professor P. K. Sen who, as
my adviser, efficiently guided this work.
I also want to thank Professors N. L. Johnson, D. Quade, and
R. G. D. Steel who, as members of my doctoral committee, contributed
their helpful remarks to this work.
A tribute must also go to my talented typist Mrs. Delores
Gold.
Finally, I would like to record that my work was supported by
the Department of Biostatistics in UNC at Chapel-Hill, N. C.,
U. S. A., and by the Department of Statistics in the University
of Tel-Aviv, Israel.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS. •
LIST OF TABLES .
BASIC NOTATION AND LIST OF ABBREVIATIONS
ii
v
vi
Chapter
I.
INTRODUCTION • . • •
1.1
1.2
1.3
II.
2.2
2.3
2.4
2.5
2.6
III.
Presentation of the problem •
1
Review of some relevant literature. •
8
A summary of the main results in Chapters II to V 15
ON THREE GENERALIZED T-PROCEDURES IN
FULL RANK UNIVARIATE MODELS.
2.0
2.1
1
Introduction. . • • • • • • • •
• • • •
Upper bounds on the distribution of the
studentized range and the studentized
augmented range of a zero mean normal
vector with a dispersion matrix known
except for a multiplicative scalar ••
A conservative procedure for simultaneous
interval estimation of contrasts - the
GT1 method. • . . • • • • • • •
The studentized augmented range and its
utilization in the GT1 method • • • •
An alternative generalization of the
T-method - the GT2 procedure. • • • • •
The GT3 method. • • • • • • • • • . • • • • .
On the choice of the matrix Q involved
in the GT3 method • • • • • • • • • •
18
18
20
23
28
33
38
39
EXTENSIONS TO MORE GENERAL MODELS.
44
3.0
3.1
3.2
44
46
3.3
3.4
3.5
Introduction. • • • • • • . • •
Generalized T-procedures in LFR models.
The distribution of the range in a general
balanced setup and the extended GT1 procedure
Some modifications of the procedures. • • . .
Generalized T~ax procedures • • • • • . • • •
Two notes on simultaneous interval estimation
when only some of the linear functions of the
response vector are of interest • • • • • • •
50
55
60
64
iv
TABLE OF CONTENTS (Continued)
Page
Chapter
IV.
ON MODIFICATIONS AND FURTHER EXTENSIONS OF THE
PROCEDURES IN SOME SPECIFIC PROBLEMS • • • •
4.0
4.1
4.2
4.3
4.4
V.
Introduction. • • • • • • • • •
Extended T and T~~ procedures in some
"Semi-Multivariate problems. • • • . • •
Modified T-type procedures for inference
on treatment x condition interactions in
a split-plot design • • • • • • • • . . • • • •.
The multivariate multisample location problem ••
Analogeous procedures in some different
problems with different statistics. • •
SOME EXAMPLES t EVALUATION AND PROPOSALS FOR
FUTURE RELATED RESEARCH. • • • • • •
5.0
Introduction.
5.3
5.4
APPENDIX:
69
70
77
83
92
94
94
94
5.1 Some examples
5.2
69
Examples of a direct approach for
the shortest confidence intervals
GTI method. • • • • • • • • • • •
Some evaluations of the different
Proposals for future research • •
obtaining
in the
• • • • • • • • 103
procedures • . • 108
• • • •
.• 112
....
• • 115
SOME SUPPLEMENTARY TABLES •
BIBLIOGRAPHY • • • • • • • • • • • • •
• 121
LIST OF TABLES
Page
Table
3.3.1
A comparison of
€ with
1m/ k'
(a)
,\)
·...........
59
·.··········
116
APPENDIX
A.l
(.05) I (.05).
The ratio 2~ ImIk'
,\) qk ,\)
A.2
(a ,)
r
Values of qp,120
A.3
Values of
A.4
(a )
11k'
Values of [qk,~ ]2/2, Xp[(l-a)
] and kXp(k_l) [I-a]. 119
A.5
(a)
(aIm)
The ratio (Ap,k-l,k(n-l)/(k-l)Fk-1,k(n-l»
. . . . . . . · . · · · · · · · · · · 117
Iml~;O~)1[(k-l)F~:~5~]~.
,
,
"i
· · · · · · · · · · 118
~
•
·····
120
e '~
Basic Notation and List of Abbreviations
We write x ~ N(~,a2), x ~ ~, x ~ tv to indicate that the random
scalar x is normally distributed with mean ~ and variance
the
02,
scalar x is a central chi-square variable with V degrees of freedom,
the scalar x is a central Student-t variable with v degrees of freedom,
respectively.
l
~ N(~,E)
To indicate that y, say, is a vector, we write l'
is used to indicate that l is a multivariate normal vector
with a vector of means
~
and dispersion matrix E.
by capital letters A,B, ••••
l
Matrices are denoted
When indication of dimensions for a vector
or a matrix A is necessary we use !p or
l:
pXl for a p dimensional
vector ! and A: mXn for an mxn matrix A.
A square matrix of dimension
n is sometimes denoted by A.
~
n
We write S
W (v,E) to indicate that S
p
is a pXp random matrix distributed as a central Wishart with v degrees
of freedom and E as the associated dispersion matrix.
The shorthands
E(X), VeX), C(X,Y) and Cor(X,Y) are used for expectation (of X),
variance or dispersion matrix (of X), covariance or covariance matrix
(of X and Y) and correlation or correlation matrix (of X and Y) when
X and Yare scalars or vectors, respectively.
We use 1 or 1 for the
-n
n vector (1, ••• ,1)', rCA) for the rank of A and A-: nXm for a
generalized inverse of A: mxn, i.e., a matrix satisfying AA-A=A.
The notation X 2 Y is used to indicate that X and Yare identically
distributed, where X and Y may be scalars, vectors or matrices.
d
also use X ~ Y when P(X~x) ~ P(Y<x) for all x.
The notation
We
e{ ... }
is intensively used to indicate the confidence level of the family of
vii
simultaneous confidence intervals { •.• }.
-1
By AI and B
we mean the
transpose and the inverse of the matrices A and B respectively.
The following abbreviations are intensively used in the sequel.
lID - Independent Identically Distributed.
p.d. - positive definite.
p.s.d. - positive semi definite.
d.f. - degrees of freedom.
DF - Distribution Free.
STP - Simultaneous Test Procedure.
LFR - Less than Full Rank.
CDF - Cumulative Distribution Function.
w.p.1 - with probability one.
CHAPTER I
INTRODUCTION
1.1
Presentation of the Problem
Tukey's T-method for multiple comparisons on a given set of
means gives shorter confidence intervals than Scheffe's S-method for
the pairwise contrasts.
(See Scheffe [1959], Ch. 3).
In the
multivariate case, we have the analogous better performance of the
procedure based on the largest Hotelling's pairwise statistics
(subsequently abbreviated as the T 2
procedure) over the largest
max
root procedure, both proposed by Roy and Bose in their 1953 paper.
Due to the complicated distributions of the statistics involved in
these procedures in any unbalanced design, they are restricted to
balanced models.
In this work we study some exact and conservative generalizations of the T and the T2
max
procedures to unbalanced models.
To be
more specific in the presentation of the problem under study in this
work, we now introduce the Scheffe, Largest Root, T and T2
methods
max
contrasting the first two with the last two.
With this aim we first
introduce a general setup which constitutes the domain of extension
for the T and the T2
max
procedures aimed at in this work.
In section
1.2 we summarize some earlier work on generalizations of the T and
T2
methods.
max
Section 1.3 contains a summary of the main results
obtained in this work.
2
Let 0': pXk = (8 : p Xl, ••• ,8 : pXl)
k
1
(p~l, k~2)
be a matrix of
unknown parameters involved in a fixed effects linear model for a given
set of data, where p is the number of variates and k is the number of
cells in the design at hand.
rolling out 0 by rows.
Let
Let 0' =
2: kpxl be the vector obtained when
(8 1 : pXl, ••• ,8 k : pXl) be a statistic
e relate
(based on the given data) such that on letting
to 0 as 8
relates to 0 the following assumptions i to iv are satisfied.
i)
There exists a set {e} of matrices
0£{0} the corresponding
ii)
e is
such that for any
normally distributed.
k
1.e"'"
={R,£'R. :E(5I.'0)=5I.'0}
.........,
The set
0
is a subspace in \1.k of
dimension e<k, invariant to the different e£{0}.
iii)
To each 0 there corresponds a real kXk matrix B=(b
ij
)
I'
,,-)
,
such that, for any &1'&2£ ~e we have C(&10'&2 0 =&lB&2~' (1.1.1)
invariant to the different e£{0}, where ~ is a p.d.
matrix.
iv)
The set\ of matrices B generated by all 8£{8}
includes symmetric p.d. matrices.
v)
An unbiased estimator Sv of ~ is available, independent
-
of any 0, and such that VSv
~ Wp(v,~).
t }
The set {5I.'0: 51.£
is referred to as the set of estimable parametric
e
functions. The setup (1.1.1) constitutes what we hereafter refer to
as the general normal setup.
Note that this setup does not include an
explicit model for the original data and that any general fixed effects
linear model pertains (in an obvious way) to the general normal setup.
3
We now briefly summarize some well known simultaneous inference
procedures from an estimation point of view.
We use this presentation
to introduce terminology and notation to be used throughout this work.
Definition 1.
A square matrix is said to be uniform if all its
diagonal elements are equal and all its off diagonal elements are
equal.
Definition 2.
if for all
A square matrix A: kXk=(a ) is said to be balanced
ij
l~i<j~k, ~ij(A)
= a ii + a jj - a ij - a
is the same.
ji
We
ek
and £=(c ' ••• ,~)' to denote the contrasts subspace in )'?k
l
k
k
k
and a vector in C respectively, Le., £ e: C ¢::::>
c = O.
i
i=l
use
L
Definition 3.
A general normal setup (as given by (1.1.1»
is said
to be uniform if (k ~
t. e
B = aBI + bB(!!'-I).
It can be verified that this definition
and
1l
contains uniform matrices
together with (1.1.1) implies invariance of a
of uniform matrices B
in~.
B
- b
B
over the class
In the literature, such models are
usually referred to as intra class correlation models.
Definition 4.
A general normal setup is said to be balanced if
contains only balanced matrices B such that,
on
~ and C k ~
t e•
~ij(B)
~
Definition 5.
is invariant
Note that every uniform setup is balanced and
that a balanced setup is not uniform only when
unique B in
~
te
=
7{ k and the
is not uniform.
If in (1.1.1)
is of full rank.
~e = 11. k we say that the normal setup
4
Definition 6.
If
~
contains only diagonal matrices we refer to (1.1.1)
as being a diagonal model.
Scheffe's S-method.
Under the setup (1.1.1) when p=l, the S-method is
embodied in the following probability statement
(1.1. 2)
where
d
F~~~v
is the I-a quantile of the F distribution with parameters
* ,v.
Tukey's T-method.
We include under this heading three different pro-
cedures proposed by Tukey [1953].
These methods are restricted to
special cases of the general normal setup.
1.
The studentized range procedure for simultaneous estimation
of all contrasts in a uniform normal setup (see Definition 3) is
embodied in the following probability statement
(1.1. 3)
where
q~~~
is the I-a quantile of the studentized range distribution
with parameters k,v.
2.
The studentized augmented range procedure is designed for
simultaneous estimation of all linear parametric functions
in a uniform diagonal full rank model, 1. e.,
a>O.
Put ~=(~l' ••• '~k)"
+
-
M(~') • max{E~i,E~i}.
+
8' 'V N(6 ,a 2 B)
~
~
~t~
and B
only
=
aI,
-
~i=max{~i'O}, ~i=max{-~i'O} and define
This procedure is then embodied in the follow-
ing probability statement
(1.1.4)
e-
5
where qk(a} is the l-a quantile of the studentized augmented range dis,\I
Tables for qk(a} do not exist, however
,\I
tribution with parameters k,\I.
(a)
(see Scheffe [1959] p. 78) this quantity is well approximated by qk
,\I
when a < .05 and k > 3.
3.
The studentized maximum modulus procedure can be used to
estimate all linear functions of a finite set of estimable parametric
functions, the estimates of which are independent.
a general (univariate) normal setup, let
A
fi=~i~'
Specifically, in
i=l, ••• ,g, g
~
e
_
and f. = ti'e, i=l, ••• ,g such that
1.
--
(l~i<j~)
A
A
(1.1.5)
1
A
and let
Put f=(fl, ••• ,f }', f=(fl, ... ,f }', cr i =(t'Bt.)~
-i-1.
g
g
b=(bl, ••• ,b } I E"f,.g then
-
g
(1.1.6)
where IMI(a} is the I-a quantile of the studentized maximum modulus
g,\I
distribution with parameters g,\I.
For details on the derivations and
the features of these procedures see Miller [1966] Ch. 2 and Scheffe
[1959] Ch. 3.
We now turn to discuss some multivariate procedures for a
setup as in (1.1.1).
The Largest Root Method (Roy and Bose).
This procedure for simulta-
neously estimating all doubly linear functions ~'e~, ~E~k, hElP is
discussed in Roy and Bose [1953] only in the diagonal full rank model.
Its extension to the general normal model is straightforward and is
discussed in Gabriel [1968] and in Krishnaiah [1969] for example.
6
The procedure in a general normal setup is embodied in the following
probability statement
(1.1. 7)
where A(a) Iv is the I-a quantile of the distribution of the
p,e.,v
-1
largest root of SlS2 ' where Sl ~ Wp(e,E), S2 ~ Wp(V,E), Sl and S2
independent.
Scheffe's S-method (given by (1.1.2»
is a special case
of this method when p=l.
The T2
Method (Roy
--max
- and Bose).
-
This procedure is the multivariate
analog of the univariate studentized range procedure.
in Roy and Bose [1953].
It is discussed
These authors considered simultaneous
interval estimation of all functions in the parametric family
(1.1. 8)
and came up with a procedure more efficient (shorter confidence
intervals) than the Largest Root procedure for this parametric family.
This procedure is usually referred to in the literature as the T2
max
method (see Siotani [1959a]) and is summarized in the following for
convenience.
For any XI : pXk =
(
el' •••
)
b
'~ £~
pXk and H: pxp real,
define the function
(1.1. 9)
Let
~:
kpXl be the vector obtained by rolling out by rows the random
matrix Z: kXp and suppose ~ ~
N(Q,
]8[).
Let Sv be independent of Z
e
7
and such that vS
V
~
W (V,L).
P
-1
distribution of D(Z, SV).
Denote by T(a )
the l-a. quantile in the
p, k ,v
For simultaneous estimation of all
parametric functions in the family (1.1.8) Roy and Bose [1953] proved
that in a uniform diagonal setup with B-aI we have
C{h'(6 -6 )e:[h'(8 -8 )±(ah'S hT(a) )~]} = I-a
- -i -j
- -i -j
- v- p,k,v
The distribution of D(Z , S-l)
does not depend on
V
parameters in the uniform diagonal case are p, k and v.
this distribution (for p > 2) is not known explicitly.
exist.
for approximating T(a)
p,k,V
(1.1.10)
r, its only
Nevertheless
Two methods
The first of these two is based on a
modified Bonferroni method for approximating upper percentiles of
statistics expressible as maxima and is due to M. Siotani.
He dis-
cusses this method in his [1959a, 1959b, 1960, 1964] papers with
reference to different statistics.
The second method provides an
upper bound on T(a ) " based on an inequality due to C. G. Khatri in
p, k , v
his [1967] paper. Siotani's method and Khatri's inequality are discussed in section 1.2.
Note that the T2
procedure can be used for simultaneous
max
interval estimation of all s'Bh, Se:Ck, he:~P as follows
(1.1.11)
Tukey's T-method is a special case of theT 2 method for p=l with
max
T(a)
_ (q(a»2
A multivariate analog of the univariate studentized
l,k,\)
k,\)
maximum modulus is discussed in Siotani [1960].
A multivariate analog
of the univariate studentized augmented range procedure has not been
8
considered in the literature.
The main objective of this work can now be summarized as follows.
Extend the T and the T2
methods to the general normal setup (1.1.1).
max
1.2
Some relevant literature
C. Y. Kramer in his [1956, 1957] papers proposed the use of
various range tests (i.e., the T-test and the multiple range tests) for
several unbalanced designs.
The correct variance is used for each pair-
wise statistic, but the test critical points are based on the distribution of the studentized range of homoscedastic and independent
variables.
Specifically, using the T-test for simplicity of presenta-
tion, Kramer's rule amounts to rejecting the hypothesis 8 =6 iff
i j
(1. 2 .1)
where
~i,(B)
J
are such that
B
in
= bi , +
~
~ij(B)
b"
JJ
- 2b ,.
iJ
All the examples discussed by Kramer
is invariant to the different symmetric matrices
li .
Duncan [1957] conjectures, without proof, that tests of this
type are always on the conservative side.
Let
(1.2.2)
Duncan's conjecture is equivalent to the statement
(1.2.3)
for all k,v and any B.
Unfortunately, it seems that the current
existing theory on inequalities in multivariate normal distributions
is not sufficient to either prove or contradict (by an example) this
9
conjecture.
This point requires further research.
It will, probably,
be resolved, when more and finer inequalities on normal distributions
are obtained.
A generalized T procedure for simultaneously estimating all
linear parametric functions
model is discussed in
B = Diag(bll, ••. ,b
kk
~'~
Spj~tvoll
in a full rank univariate diagonal
and Stoline [1973].
Let
) then their method, to be abbreviated as Ext-T,
is embodied in the following result.
Theorem 1.2.1
(Spj~tvoll
and Stoline)
Another extension of the original T-method is discussed in
Sen [1969].
This generalization is useful for simultaneous
inference on interactions in some factorial experiments.
a pxq stochastic matrix Z = (zij)
Sen considers
following a multivariate normal dis-
tribution with E(Z)=O and
2
i=i' and j=j'
P10 2
ifi' and j=j'
02
i=i' and jfj'
0
E( zij Z i ' j , ) =
P2
P30 2
(1.2.5)
i+i' and
i+i'
where 0 2 >0, -l/(p-l)~Pl<l, -1/(q-l)~2<l and l-P -P +P >O.
l 2 3
An un-
biased estimator s~ of 0 2 is available such that vs /0 2 ~ X2 •
v
V
V
Let
t={L: pXq: !;L=Q~ L!q-Q} and put ILI·2i~12j~ll~ijl, then Sen's main
result is embodied in the following probability inequality.
10
Theorem 1.2.2
(Sen)
(1.2.6)
When Pl=P2=P3=0 and interest is confined to the subset
tl
of
t
defined by
(1. 2.7)
Gabriel et al. [1973] proved the following result
(1.2.8)
where a*=2a/[p(p-l)q(q-l)] and
central t distribution with
V
t~a*)
d.f.
is the l-a* quantile of Student's
This procedure can be used for
simultaneous inference on product type interactions in factorial
experiments.
e·
(See Gabriel et al. [1973]).
Various authors have considered the problem of simultaneous
inference on a finite number of pre-determined estimable parametric
functions.
Let tl, ••• ,t
-
-m
be fixed vectors (i.e., determined in a
way which is independent of the given data) in
t e.
Interest is
t'0h V h£"!P i=l, •.• ,m under
confined to simultaneous inference on -i
-'
,
the general normal setup of (1.1.1).
T =
i
Let
t~(e-8)S-1(0-0)'t
/(t~Bt )~ and define T(a)
by the equation
-~
V
-i -~ -i
max
p( max {T } < T(a»
l<i<m
i
-
= I-a
max
(1.2.9)
From (1.2.9) it follows that
O
O'0h£[O'0:h±(O'B
C{ ~i
- ~i - ~i ~i-hIS V-h
T(a»~]
max
'
\1
h£~P
v _.~,
i 1
}
1
• , ••• ,m • -u
(1.2.10)
11
Clearly, if T(a) is known and the m pre-determined parametric
max
functions constitute all that the experimenter wants to study from the
given data, this procedure is preferable to the Largest Root method.
Unfortunately,
T~~ depends on the &i's and the matrix B besides the
parameters m, p,
v. Thus, although multivariate t and multivariate F
distributions have been obtained in the literature in closed forms,
it is unlikely that exact values of T(a) would be available for use
max
in most designs even in the univariate case.
To overcome this problem, approximations to T(a) were conmax
sidered in the literature.
M. Siotani in his [1959a, 1959b, 1960]
papers considered approximations of T(a) in the multivariate case
max
for three sets of &i's (namely, pairwise comparisons, general independent comparisons, and comparisons of treatments with a control) in a
one way layout.
In his 1964 paper, Siotani considers approximating
T(a) in the univariate case under a general normal setup.
max
All of
Siotani's approximations are based on a modified second order
Bonferroni approximation to upper percentiles of statistics expressib1e as maxima.
This method was first introduced in Siotani [1959a],
which we now describe for later reference.
Let u ,u , ••. ,um be m
1 2
random variables with well defined bivariate distributions.
Suppose
we want to approximate u(a) given by the equation
m
(1. 2.11)
The first order Bonferroni approximation based on the inc1usionexclusion principle is ul(a) defined by
,m
12
=
a
m
L
p{u
i-1
> u(a)}
i ,m
i
(1.2.12)
A second order Bonferroni approximation is u (a) defined by
2 ,m
a
=
I
p{u
i=l
> u(a)} -
i
2,m
L
p{U
i<j
i
The evaluation of an approximation as u
and moreover it underestimates u(a).
m
> u (a,m)' U
2
(a)
2,m
j
> u(a)}
2,m
(1.2.13)
is usually very laborious
To avoid these defects M. Siotani
suggested to approximate u(a) by u (a) defined by
m
3 ,m
a +
L
(1.2.14)
i<j
Clearly, we have
(a) < u(a) < (a)
1 ,m
3 ,m u 2 ,m •
(1.2.15)
U
The main difficulty with Siotani's method is that in many cases the
quantities P(u. > u (a), u > u (a»
1.
1 ,m
j
1 ,m
obtain.
are difficult or impossible to
In the general univariate normal setup these quantities can
be evaluated from existing tables of the distribution of a bivariate
t or a bivariate Itl.
(See, for example, Dunnet and Sobel [1954,
1955], Siotani [1964], and Krishnaiah et a1. [1969]).
In the multi-
variate case, due to the complications involved, Siotani only
approximated the bivariate probabilities for large values of v.
If
the bivariate probabilities are difficult to obtain one can always
use a first order Bonferroni approximation, this was considered by
Dunn [1958, 1959, 1961].
An alternative method for approximating T(a) is based on an
max
.
inequality from Khatri [1967], this approximation is actually an
upper bound.
The use of this inequality in the general normal setup
e-
13
for simultaneous inference on a finite number of pre-determined comparisons is suggested in Khatri [1967] and in Krishnaiah [1968].
For
later reference, I now record two results of Khatri [1967] which can
(u)
be used for approximating Tmax in (1.2.9).
Theorem 1.2.3
Let
and
(Khatri)
~j:
j~l, •••
p X1,
= vijE
C(~i' ~j)
,m, be normally distributed with zero means
for some scalars v ij ' i,j=l, ••• ,m and a p.d.
Then, for any real c , ••• ,cm and any p.d. matrix A: pxp we
1
matrix E.
h~e
P(X~Axj
-J -
m
< c., j~l, .•• ,n) > IT P(Xj'Ax. _< c )
j
- J
j-1
- -J
(1.2.16)
In the following theorem (from Khatri) the case when A is a
random matrix is treated, but first we introduce some notation.
Y: pxn
~
6(p,n,V),
V~
We put
to mean that the random matrix Y has the
density
p
TI~pn IT {r[~(v+n-i+1)]/r[~(V-i+1)]}II +YY'I~(v+n)
~1
On letting
B2 (a,b;
Theorem 1.2.4
(1.2.17)
p
x) = [B(a,b)] -1
IX y a-I (l+y) -a-bdy we
o
have
(Khatri)
Let x.: p X1, j m 1, ••• ,n be as in Theorem 1.2.3 and let
-J
Sv: pxp ~ Wp(V,E) be independent of the ~j'S, then
(1.2.18)
where Y: pxm· (Y1""'y ) ~ 6(p,m,V). Since tables for the quantity
-m
on the right hand side of (1.2.18) are not available (even for
14
c
1
=... =cm)
Khatri suggests upper bounds to approximate this quantity.
We will need only the case c = ••• =c =c, say.
m
1
Theorem 1.2.5
(Khatri)
(1.2.19)
We let C(a)
be defined by the equation
p,m,v
[82~P, ~(v-1-p); C(a)
/v]m = 1-a
p,m,v
(1.2.20)
These results from Khatri may be considered as the multivariate
generalizations of Sidak's inequalities for the univariate case discussed in his 1967 paper.
For example, Theorem 1.2.3 is a mu1ti-
variate generalization of Sidak's well known multiplication inequality.
Throughout this work we use terminology and concepts borrowed
from Gabriel [1969a] without redefining any of the terms in this work.
The reader should refer to Gabriel [1969a] for definitions of the
terms:
a testing family of hypotheses, a monotone
testing family,
a simultaneous test procedure (STP), coherence and consonance of STPs,
an STP being more resolvent than another STP and some modifications
of these concepts.
The Bonferroni method for approximating upper percentage points
of statistics expressible as maxima, is referred to in this work
several times.
We refer the reader to David [1970], Chapter 5, where
this method is clearly discussed.
For reference on multiple range tests we refer the reader to
Miller [1966] Chapter 2.
The methods developed in this work can be
utilized to generalize these procedures to unbalanced designs.
e·
15
1.3
A summary of the main results in Chapters 2 to 5
Three generalizations of the T and the T2
methods for
max
simultaneous inference on all estimable parametric functions, or on
all (estimable) contrasts, in a general normal setup are obtained,
discussed and exemplified in this work.
The first generalization considered in this work (abbreviated
as GTl) is conservative.
It is obtained by deriving new upper bounds
on the quantiles of the distribution of
D{e-e,
Sv-1 )
definition) in a general normal setup.
This method actually produces
a family of procedures each depending on the class
normal setup (see (l.l.l»
at hand.
~
(see 1.1.9 for a
involved in the
Much of the discussion on this
method concerns the choice of a specific
GTI procedure.
We will
find this method to be very efficient and convenient to use in various
designs with a low degree of unbalance.
In designs where a large
degree of unbalance exists, the performance of this method is poor
when comparing with the S-method.
The second generalization (abbreviated as GT2) is based on some
known and new approximations to T{a) in (1.2.9) choosing the
max
(1.2.9»
~. 's (in
-~
in a way that enables extension of the original finite number
of confidence intervals to the continuim of confidence intervals on
all estimable parametric functions.
Having chosen an appropriate
approximation, this method is uniquely defined.
It is also robust to
the degree of unbalance of the design, in the sense that, it produces
shorter confidence intervals than the S-method, for an important class
of contrasts (including the pairwise comparisons) in any normal setup
as (l.l.l).
16
The third generalization (abbreviated as GT3) produces exact
simultaneous interval estimation procedures for all estimable parametric functions.
It is a generalization of
Spj~tvoll
and Stoline's
procedure (discussed in 1.2) to the general normal setup in univariate
and multivariate problems.
This method produces a class of possible
procedures and (as with the GTI method) much of the discussion on this
method concerns the choice of the appropriate specific procedure.
The
GT3 method is also very sensitive to the degree of unbalance in the
design.
Aside from these points, the aims and limitations of this work
will best be gleaned from a content outline of the following chapters.
Since the generalization of the T2
procedure goes basically along
max
the same lines as the generalizations of Tukey's T-method in the univariate case we will, for the sake of simplicity, discuss fully, only
the univariate case.
This is done in Chapter 2 where the full rank
case is studied.
Chapter III contains the extensions of the procedures introduced
in Chapter II to general multivariate models of less than full rank
when all the contrasts are estimable.
Some modifications of the pro-
cedures are studied so that the resulting methods are invariant on the
class
is}.
Also, in Chapter III, we consider certain procedures for
simultaneous inference in a general multivariate setup when only some
of the linear functions of the response vector are of interest.
e·
17
Chapter IV contains miscellaneous contributions to the theory
of simultaneous inference in various problems based on the results
obtained in earlier chapters and few modifications.
Specifically,
included in Chapter IV are problems of simultaneous inference based
on generalized T and T2
procedures in:
max
1) growth curve problems;
2) split-plot designs (main effects and interactions); 3) multivariate multisample location problems (rank procedures).
In Chapter V we bring some examples to exemplify the methods
derived in this work.
different methods.
We also give some comparisons among the
Finally, in Chapter V we discuss some proposals
for future related research.
The discussion in Chapter V is supplemented by some tables
given in an Appendix.
CHAPTER II
ON THREE GENERALIZED T-PROCEDURES
IN FULL RANK UNIVARIATE MODELS
2.0
Introduction
Recall the T-method introduced in 1.1 and its limitations.
In
this chapter we consider three different generalizations of Tukey's
studentized range and studentized augmented range methods to a full
rank general normal setup as given by (1.1.1).
The first method is studied in sections 2.1, 2.2 and 2.3.
second method is discussed in section 2.4.
tain the derivation of the third method.
The
Sections 2.5 and 2.6 conTo supplement the theoretic
discussion a simple example from an ANACOVA with one covariable, is
used throughout the chapter.
In the derivation and discussion of the new methods we will
make repeated use of some results from Anderson [1955], Khatri [1967],
Kimball [1951], Kounias [1968] and Sidak [1968].
For convenience we
record these results here, referring the reader to the original papers
for proofs and details.
Theorem 2.0.1.
(Anderson).
-
density g(x) symmetric about
Let ~ be a random vector in ~ with
a
such that the set
- and
1
k
convex for any non-negative u£~ •
about
Q,
Let
EC~
{~: g(~) ~
u} is
be convex and symmetric
k
then, for any ~£~ and any d, 0 < d < 1
- -
p{x + dy£E} > p{x + y£E}
- -
-
(2.0.1)
e·
19
Corollary 2.0.l.A
(Anderson).
Let x
-
~
N(O, E(x», y
-
of dimension k and x and l are independent.
-
If
L(~)
-
~
N(o, E(y»
-
~
both
>:(y) is p.s.d.,
we have
-
-
P{x£E} < P{yEE}
-
(2.0.2)
for any E as in Theorem 2.0.1.
Corollary 2.0.l.B
(Anderson).
If E in Corollary 2.0.I.A is also
closed we get a strict inequality in (2.0.2).
Theorem 2.0.2.
Let
~i
= (xil,···,x ), (ial, ••• ,n) be n
ip
independent observations from a multivariate normal distribution with
zero means.
(Khatri).
Let Dl(Xll' ••• 'Xln) • D1 and D2 (xij , i=2,3, .•. ,p;
j=l, ••. ,n) = D be sectionwise convex regions, symmetric about the
2
origin and put X=(xl, ••• ,x).
-n
Then
(2.0.3)
Theorem 2.0.3
(Kimball).
For an arbitrary random scalar x
~
F(x),
on letting g.(x), i=l, •.• ,k, be non-decreasing, integrable, increasing
1.
functions of x, we get
k
k
EF(.n gi(x»
1.=1
Theorem 2.0.4
(Kounias).
> n EF(gi(x»
(2.0.4)
i=1
Let El, ••• ,Ek be a finite set of events,
then
(2.0.5)
This is an improvement of the Bonferroni first order approximation.
20
'"
,
. ::.T.:.;;h.::;e.::;0.::;re.::;m=-..;2:::.. :. .0::.. :. :.5=--_(;>.:S:.,:i:..;d:.:a:.:k:;.L.) .
Let (Xl" ••
,~)
be a random vector having a
k-variate normal distribution with mean values 0, variances 1, and
with the correlation matrix
~(A) •
a ~ A ~ 1, in the following way:
have Plj(A) = AP lj
j~2,
Pij(A)
some fixed correlation matrix.
(Pij(A»
depending on a scalar A,
Under the probability law PA' we
=
i,j~2,
Pij for
ifj, where (P ij ) is
Then; P(A) = PA{lxll<cl, .•. ,I~I<ck}
is a non-decreasing function of A, 0 < A < 1 for any positive numbers
Corollary 2.0.5.A
v
')
(Sidak.
If (xl""'~) has a normal distribution
with mean values 0 and with an arbitrary covariance matrix, then
k
II
i=l
Corollary 2.0.5.B
(Sidak).
p{lxil < c.}
(2.0.6)
1.
Let (xl""'~) be as in Theorem 2.0.5
and let s be a positive random variable which has under all P
fixed distribution, and is independent of
(xl"'.'~)'
A
a
Then;
PA{lxll/s<cl, ••• ,I~I/s<ck} is a non-decreasing function of A, O<A<l.
2.1
Upper bounds on the distribution of the studentized range and the
studentized augmented range of a zero mean normal vector with a
dispersion matrix known except for a multiplicative scalar.
Let ~E~ and denote by R(~) and R(~) the range and augmented
range of
Let
~,respectively.
R(~)
=
~.(ul"
••
and put
max
~=(tl"."£k)'
{Iti-tjl}; R(~) = max( max {ltil},R(&»
l~i<j~k
'~)'
Thus, on letting
we have
(2.1.1)
l~i<k
be a normal vector of zero mean and dispersion
~(~)
e·
21
Q(v)={u: R(u)<v}; Q(V)-{u: R(u)<v}
--
.....,
Lemma 2.1.1.
(2.1.2)
---
For any v>O, Q(v) and Q(v) are convex sets, symmetric
about Q: k X 1.
Proof.
We prove for Q(v), the proof for Q(v) is similar.
~l' ~2£Q(v)
and 0
~
a
R(a~1+(1-a)~2) =
~ a
a
1, we then have
{lauli+(1-a)u2i-aulj-(1-a)u2.1}
max
l~i<j<k
J
max {Iuli-Uljl} + (I-a) max {lu2i -u 2 , I}
l<i<j<k
l<i<j<k
J
aR(~l)
+
(1-a)R(~2) ~
Since R(p
Thus Q(v) is convex.
follows.
~
Let
(2.1.3)
v
= R( -~)
for all
k
~£l(
, Lemma 2.1. 1
Q.E.D.
From Corollary 2.0.l.A it is now clear, that if we construct a
matrix r* such that r*-r(~) is p.s.d., and define a random vector u*
to be normal with mean Q and dispersion matrix r*, we get
}
Hence, for any O<p<l an upper bound on
P{R(u) <
-
-
~
p
}
=p
~
p
(2.1.4)
defined as
(2.1.5)
is - ~* - determined by the equation
p
P{R(u*) < ~*}=p
-
-
p
(2.1.6)
22
We now restate Corollary 2.0.l.A in a slightly extended form,
the proof of which is straightforward.
Let u and u* be random normal vectors in~k with zero mean
-
Lemma 2.1.2.
-
vectors and dispersion matrices L(~) and L(~*) respectively.
a random scalar, independent of ~ and of ~*.
Let T be
Let g(~,T) be a function
with values in i\l such that for any real h and any value t of T the set
Q(t,h)cR
k
defined by
(2.1.7)
is convex and symmetric about O.
-
-
Then, if L(U*)-L(U) is p.s.d., we
get
P{g(~*,T) ~ h} ~ P{g(~,T) ~ h},
Suppose now that L(~)
= a 2 B,
and a 2 is an unknown scalar.
mator s2 of
V
0
1
V hdt
(2.1.8)
where B is a known p.s.d. matrix
Further suppose that an independent esti-
2 is available, such that, vs 2/a 2 ~ X2 •
V
V
We identify
g(u,s2) with either the studentized range or the studentized augmented
-
V
range of u, i.e., with R(u)/s
-
-
V
or with R(u)/s ,respectively.
-
Based
V
on the above discussion we can now describe a general procedure for
constructing families of random variables stochastically larger than
R(u)/s , (R(u)/s).
-
where
~* ~
V
B is
-
V
This is accomplished by constructing B*
= B+B,
any p.s.d. matrix and defining the random vector
N(Q,a 2 B*), the variable R(u*)/s , (R(u*)/s ) is then
-
V
-
V
stochastically larger than R(u)/s , (R(u)/s).
-
a majorization of B.
V
-
We refer to B* as
V
If we majorize B to a uniform (diagonal
uniform) matrix B*, we immediately get upper bounds on any percentile
in the distribution of R(u)/s , (R(u)/s ) as the corresponding
-
V
-
V
e·
23
percentiles in the distribution of R(~*)/SV' (R(~*)/8V).
This is
utilized to define our first conservative generalization of the
T-method which is hereafter abbreviated as the GTl method.
We now turn to discuss the GTl method based on a specific
majorization procedure.
2.2
A conservative procedure for simultaneous interval estimation of
contrasts - the GTl method.
Consider the full rank univariate model, i.e.,
B=(b ij )i,j=l, ••• ,k is completely known, 0
2
e~
N(~,02B),
is unknown but an indepen-
dent estimator s2 of 0 2 is available such that vs 2 /0 2 ~ X2 •
V
V
V
Following
discussion and terminology of 2.1, let B* be a uniform majorization of
B, i.e., B* =
B+B
where
Bis
a p.s.d. matrix and B*
where a and b are some real scalars.
Theorem 2.2.1.
Ck ,
b(!~'-I)
Based on our discussion in 2.1,
the following result is straightforward.
generic notation for a contrast in
= a1 +
(We use £=(cl, ••• ,c )' as a
k
the contrasts subspace in ftk).
For any real a,b such that
a1 + b(ll'-I) - B
""",
(2.2.1)
is p.s.d. we have
(2.2.2)
Theorem 2.2.1 provides a family of conservative simultaneous
estimation procedures of all contrasts
in~.
This is the GTl method
for constructing conservative generalized T-procedures in unbalanced
designs.
In order to obtain the best GTl procedure, we must find
scalars a and b such that, a-b is minimized under the constraint that
24
the matrix given by (2.2.1) is p.s.d.
is not always feasible.
Such an explicit, direct solution,
Some important cases where optimal GTl pro-
cedures can be determined are discussed in a subsequent chapter.
A more feasible approach in the general setup is, to adopt a
certain structured majorization procedure and optimize within its
structure.
We now describe a majorization procedure which is hereafter
referred to as the primitive method for majorizing a symmetric matrix
to a uniform matrix.
As we illustrate in the sequel, this majorization
procedure is, in many cases, easily constructed and in various cases
produces efficient conservative estimation procedures.
The primitive method of majorizing a symmetric matrix B=(b .. )
~J
to a uniform matrix B* = a1 + b(ll'-I).
Si(w) =
Lj~l:j~i
For any real w let
e·
Iw-bijl, i=l, ••• ,k and define
(2.2.3)
q:> (w,B) =
Lennna 2.2.2.
For any real w the matrix
~(w,B)1
+ w(!!'-I) - B
(2.2.4)
is p.s.d.
Proof.
ij
Define matrices B , (l<i<j~k), B .=(b )
where
ij
iJ
ron m,n=l, ••• ,k
w-b
b
ij
=
ron
IW-b ij
0
IrF/:n and either (i=m,j=n) or (i=n,j=m) •
ij
I
m=n and either (m=i) or (m=j) •
otherwise
(l<i<j<k) are all p.s.d.
Note that the matrices B
ij
Let
+ = Diag(al""'~)'
(i-l, ••• ,k) and put B
25
It is now easily verified that (2.2.4) can be written as
(2.2.5)
L
and thus, since B+ and
l<i<j~k
B are p.s.d., Lemma 2.2.2 follows.
ij
Q.E.D.
Lemma 2.2.2 can now be utilized to construct a GTl procedure
by choosing a and b in (2.2.2) as
w.
~(w,B)-w
The value of
optimal choice for w.
~(w,B)
and w respectively for some
depends on w, we thus next discuss the
With this aim we now frame the problem in a
game theoretic setup and study the existence and characteristics of
an optimal choice for w.
Let
X be
the k dimensional simplex in
Rk
and denote by
Xo
its
set of vertices E , .•• ,E , where E is a vector of k-l zeros with +1
k
i
1
as its ith coordinate.
Y be
Let
the interval [min{b .. },max{b .. }].
i;'j ~J i;'j ~J
It is easily verified that in the game (X,Ft,t) where for
~=(x1, ... ,~)'EX, wEFt
k
L
1=1
x. (b
~
U
+ s. (w)-w)
~
the set of admissible strategies for player II, is contained in
Minimizing
~(w,B)-w
(2.2.6)
Y.
with respect to w is equivalent to finding a
minimax strategy for player II in this game.
From Theorem 2.5.1 and
its Corollary 1 in Blackwell and Girshick [1954] it follows that the
game has a value.
We denote by w* the strategy for player II which
is an equalizer rule with respect to the vertices of positive weight
in the first player's strategy which is a mixture of points in X •
o
These features of w* can now be utilized to describe a convenient
26
search procedure for w* as follows.
Define ~l by the equation
(2.2.7)
Check whether for all
i~v
(2.2.8)
If (Z.2.8) holds, stop and choose w*=W •
l
/(1)
~
={i
(1)
l
. (1)
, •..
,1
k
If not, determine the set
}£{l, .•• ,k} of indices such that ~£~(l) iff
1
(2.2.9)
Now, check if
max
..J..' ,(I)
1r 1 £~
Wz determined
by
[~min{l(Ei ,w)+ l(E i "w)}] = t..(E j
w
for some j,j'£A(l), gives for all
,w2)
= t(E j
"w2 )
(Z.Z.10)
~£~(l), ~"'j, ~"'j'
(Z.Z.ll)
If it does, stop and choose w*=w ' if not, continue (in a way which
Z
should be clear now) to form
for some t<k a value w*=W
t
~(Z), determine w3 ' and so on, until
is chosen.
(Equivalently, )(t) is empty).
This procedure will usually require only the very first few steps.
In many cases a straightforward choice for w exists.
example, suppose a scalar b
o
median of the k rows in B=(b
exists such that b
ij
0
is simultaneously a
), i.e., on using m( ••• ) as a generic
notation for the median of the elements ( ••• ) we have
bo·m(bil, ••• ,b
ik
),
V
For
i ..l,. •• ,k, clearly we have w*=b •
o
e·
27
EXAMPLE 2.2.1.
[1965]).
Analysis of covariance with one covariab1e (see Bose
= Ii~l
Assume yij(j-l, •.. ,n i ; i-l, ••• ,k) are n
n i independent
random variables with a common variance 0 2 and with expectations
The least square estimators
_
A
of the mils are given by mi
- __ I
xi
n
i
j=l xij/ni and b
A
k
-
,,-
= Yi-bxi
= Sxy Is xx ,
where
k
ni
l. (x .. -x.)
S
= l
l (x"-Xi)(Yi'-Yi);
S = L
xY
i=l j=l J.J
J
xx
i=l j=l J.J J.
k
\'
n.
(S
yy
i
(i=l, ••• ,k) where y. = I· 1 y . . /n
i
J.
J=
1J
ni
Let S = ~
~ J.(y _y-)2
Li=l Lj=l ij i .
yy
n
\'
-
2
.
(2.2.12)
The unbiased estimator of 0 2 is given by
-SSbA)/v, where SSb = S2xy Is xx and v = tl.i=l
k n.-k-1.
J.
A
The dispersion
matrix of the ~i's is given by
1 +
o 2 [Diag (1
- , ••• ,-)
n
n
l
k
E E']
(2.2.13)
Let h be such that
b
h
= max{b }.
i
If we apply the search procedure for finding w* we
start here by choosing ~1 = m(b
h1
, ••• ,b
hk ), since
max L·~llb .. -m(b· 1 ,···,b ik )1 = L'~1Ibhj-m(bh1,···,bhk)l.
l<i<k J
J.J
J.
J
A numerical illustration. Suppose k=4, n.=20, x.=12+.2(i-1),
J.
J.
1=1,2,3,4 and S =2500.
xx
In this case v=75 and we get
.10076
.05856
.05952
.06048
.05856
.10954
.06051
.06149
.05952
.06051
.11150
.06249
.06049
.06149
.06249
.11350
(2.2.14)
B=
(.05) ~ 3.7.
We find w*=.06149, ~(w* ,B)=.ll55 and Q4,75
Based on Theorem
2.2.1 and the above outlined search procedure, we get the following
28
confidence intervals for the six pairwise contrasts
(2.2.15)
The corresponding intervals obtained by Scheffe's S-method are:
m -m £[ii-i ±.905lS ] for (i,j)-(1,2), (2.3), (3.4);
i j
V
j
mi-mj£[mi-mj±.9053sv] for (i,j)-(1,3), (2,4) and mi -m.£[rn i -m.±.9057s ]
J
J
v
for (i,j)=(1,4).
Finally, before closing this section, we outline a simple procedure for finding w* by solving a corresponding linear programming
problem.
Minimize V, subject to b
restricted in sign.
ii
+
Si(w)~ ~
V, (i=l, ••• ,k), wand V un-
Since Si(w) is not a linear function of w, the
following technique may be used to linearize it.
(1)
Yij
(2)
= max(w-b ij , 0), and Yij
Iw-b .. 1 = y~~)
~J
~J
= max(bij-w,
0).
+ y~~) where yi~) ~ 0 (h=1,2).
~J
Let
Now,
The above problem is
thus equivalent to the following:
Minimize V, subject to:
(1)
\ k
(1)
b ii + Lj=l,j;i Yij
\ k
(2)
+ Lj=l,j;i Yij
- w ~
Yij
(2)
-w~-bij; Yij +w~bij' (i,j=1,2, ••• ,k, j;i).
2.3
The studentized augmented range and its utilization in the
V;
GTI method.
It is generally accepted that the studentized augmented range
method can be used for simultaneously estimating all linear parametric
functions i'e in the uniform setup, i.e., ~ ~ N(~,a2B) and B is of the
form B • aI + b(!!'-I), for some real scalars a and b.
is wrong, and the statement
However, this
29
(2.3.1)
(see Scheffe [1959] Ch. 3 and Miller [1966] Ch. 2) is in error whenever b;O.
The distribution of the augmented range of k unit normal
deviates with a common correlation p is not the same as that of k
independent normal variates with zero means and a common variance l-p.
This is an immediate consequence of the following result.
p.s.d., then
P{R(~) >
d > P{R(y) >
t},
V t>O
(2.3.2)
Let Qx(t) and ~(t) be defined in terms of ~ and y respectively
Proof.
-
-
Since ~(t) and Qy(t) are
as Q(v) in (2.1.2) is defined in terms of u.
symmetric about
Q,
convex (see Lemma 2.1.1) and bounded sets, Lemma
Q.E.D.
2.3.1 readily follows from Corollary 2.0.l.B
Note the strict inequality in (2.3.2) rather than the 'less than
or equal' type inequality introduced in (2.1.4).
Corollary 2.3.l.A.
If ~=(b+d)I+b(!!'-I) d~O, then P{R(~»t} is a
strictly increasing function of b.
Corollary 2.3.1.B.
If ~=cr2[aI+b(!!'-I)], E=cr 2 (a-b)I, s
V
is independent
of x and of y, and such that \>sv/cr2 ~ X~, then
i)
If b>O; P{R(x)/s
> t} > P{R(y)/s
> t},
it)
If b<O; P{R(x)/s
> t} < P{R(Y)/s
> t}, V t>O
-
V-\>
V
t>O }
(2.3.3)
-
V-\>
These corollaries are straightforward.
On the basis of Corollary 2.3.1.B
it is clear, that a formula as (2.3.1) can never be correct for any bfO.
30
Rather, if b>O the confidence coefficient is less than 1-a, and if b<O
the confidence coefficient is larger than 1-a.
Lemma 2.3.2.
Let
~=(x1' ••• '~)'
be normal with mean 0 and dispersion
a 2 [I+p(!1'-I)] where p>O, then
(2.3.4)
where
~=(Yl' •..
'Yk+l)' is a normal vector of independent coordinates
with zero means and variances; V(Y ) = a 2 (1-p), i=l, ... ,k and
i
V(Y k+l ) = pa 2 •
Proof.
(2.3.5)
-
where ~+1=0.
~
~
Put xi=p zo+(l-p) zi' (i=l, ••• ,k+l) where the z i 's
(i=O, ••. ,k) are lID z. ~ N(0,a 2 ) and zk+l
~
-(~)~
==
z .
o
Now since
R«xl'···'~+1)')=(1-P)~R«Zl'···'Zk+l)')~R«Y1'···'Yk+1)')
the lemma follows.
(2.3.6)
Q.E.D.
We now discuss a modification of the GT1 procedure which in
many cases results in a more efficient estimation of the contrasts.
We transform the original vector of parameters
mators
e- to
~
and vector of esti-
a basis for the contrasts; e -Ae, 8 =A8 where A of order
c
-
(k-l)xk is a full rank matrix of real elements.
c
-
Suppose now that a
majorization procedure is applied to the matrix ABA' to bring it to
a uniform matrix of the form al k _ l + b(!k-l1k-1-Ik-1)' then from
Lemma 2.3.1 and Lemma 2.3.2 we have for any non-uniform matrix B
e·
31
.
C{~'ac£[~'ec±sv(max{a-b.b})~ q~a~M(~')]. ~ ~£Rk-l}
~
~
Lemma 2.3.3.
(2.3.7)
> l-a
It is always possible to choose a basis and a majoriza-
tion procedure such that the resulting confidence intervals based on
(2.3.7) will be no wider than those obtained by any GTI procedure as
given by (2.2.2) for any contrast.
Proof.
To prove this result
~e
indicate a choice of a basis and a
majorization procedure such that (2.3.7) coincides with (2.2.2).
w and
~(w.B)
be as in 2.2 and let
A: (k-l)xk
Note that i f
we obtain
Let
~'a c
= £' ~. V
=
~£1f
1.0 ••••• 0.-1
0.1 ••••• 0.-1
..
..
..
..
where
£=(R.l.~··.R.k_l. - L~:iR.i)'
(2.3.8)
~= (R.I ••••• R.k_lh::~k-l and
and
M(~') = ~Li~llcil·
k
£d\ ,
Also,
since
(2.3.9)
is a uniform majorization of ABA' it follows from Lemma 2.3.2 that
the confidence intervals obtained from (2.3.7) are the same as those
obtained from (2.2.2) for all contrasts.
Q.E.D.
Alternatively, we may majorize ABA' directly using the primitive
majorization procedure.
and
~(w,B)
Let
wand
~(w,ABA')
correspond to B, respectively.
correspond to ABA' as w
Clearly, b and a in (2.3.7)
can be replaced by any wand <p(w,ABA') respectively.
To obtain the
shortest confidence intervals from such a procedure one is willing to
32
use w* and ~(w*,ABA') (for b and a in (2.3.7), respectively) defined by
the equation
m~n max(~(w,AB~)-w,w) • max(q(w*,ABA')-w*,w*)
(2.3.10)
w
This procedure has some drawbacks which may make it undesirable
in various problems.
in obtaining w*.
ii)
These drawbacks are; i)
The procedure is not invariant under different
choices of a basis for the contrasts subspace.
qk(a) is not tabulated.
,v
The difficulty involved
iii)
The quantity
As mentioned in 1.1, qk(a) can be approximated
,V
by q(a) whenever k>3 and a~.05. However, if one wants to be on the
k,v
(a) cannot b e used to approximate qk,v
-(a
) . 11y
sa f e si de, c 1ear 1y, qk,v
espec1a
when a>.05 is used.
Still, in some problems w* is easily obtained and an obvious
choice for a basis exists such that this procedure gives shorter confidence intervals than the estimation procedure discussed in 2.2.
In
some cases an improvement over the original GTI is obtained when using
w.
*
EXAMPLE.
Suppose that in Example 2.2.1 we had
i=1, .•• ,4 and the rest remains the same.
xi =12,13,14,15,
Following the original GTI
procedure we obtain
(2.3.11)
If we use a procedure as in (2.3.7) with an A matrix as in (2.3.8) and
b.a of (2.3.7) replaced by w* and ~(w*.ABA') we get
1.0448 ], l<i<j<k} > .95
V
--
(2.3.12)
•
33
The fact that in some cases this modified GT1 procedure fails to
improve the original GTI can be illustrated by working it out on the
original data in EXAMPLE 2.2.1 and comparing with (2.2.15).
2.4
An alternative (approximate) generalization of the T-method the GT2 procedure.
For any contrast ~.(cl, ••• ,ck)' let Pc={i: ci>O} and Nc={j: c.<O}.
Let ~ij = b ii +
k'=k(k-l)/2.
J
.
k
b j j - 2b ij , (l<L<j<k), put 1~I~Li=llcil and
Let tea)
k',\)
be the value of (T(a»~ in (1.2.9) when p=l
max
and the m &i'S are the k'-m pairwise contrasts.
Theorem 2.4.1.
The probability is l-a that all the contrasts simulta-
neously satisfy
(2.4.1)
Proof.
To prove the theorem, it is sufficient to prove that on
letting Y1' ••• 'Yk and
aij~O,
l<i<j<k, be real scalars, we have the
equivalence
(2.4.2)
To
prove~in
(2.4.2) substitute the pairwise contrasts in the right
hand side of (2.4.2).
To prove at, we first note that
Then on writing
34
(2.4.3)
and multiplying (2.4.3) by 11£1 we get
k
11£11 l
i=l
ciY i =
L
i£P
L
£
j€N
l
c i (-Cj)Y i +
i£P
£
L
£
j£N
cicjY j
S
(2.4.4)
=
From the left hand side of (2.4.2), the right hand side follows
based on (2.4.4).
On identifying Yi with 8i -8 i , a ij with
the theorem follows.
l~i<j~k,
~
• (a)
svtk',V~ij
Q.E.D.
Next we consider a similar approach for simultaneous estimation
of all linear parametric functions.
M(&') as in 1.1.
Let P n and Nn correspond to ,Q, as P
'"
to c.
if-M(~')
Define {M}=P,Q,
and put k"=k(k+1) /2 ~
Let
'"
:
tk(~),v
Li~l ,Q,~
-
and N correspond
C
and {M}-N,Q, i;
C
M(~')-= li~l ,Q,~
be the value of (; (a»~ in (1. 2.9)
max
when p=l and the m &i's are the k'+k=k"=m vectors consisting of the k'
. .
k
ti ces
pa1rw1se
contrasts an d th ever
p(~)
=
L
L ,Q,i(-,Q,j)~~j'
0
f t h e s i mp 1ex 1n
. ~k
~.
Put
Q(&)
i£P,Q, j£N,Q,
-
Theorem 2.4.2.
~'~
The probability is l-a that all parametric functions
simultaneously satisfy
(2.4.5)
Proof.
To prove this theorem, it is sufficient to prove that on
letting Yi , aij>O and bi>O,
l<i<j~k
be real scalars, the condition;
.
35
is necessary and sufficient for all the linear combinations
2i : l
£i Y1
to satisfy
This result is proved along the same lines as the analogous result
in the proof of Theorem 2.4.1 on realizing that ~.k1
~.y.
can be
L~=
~ ~
written as a contrast in cO,cl' ••• ,ck ' where Co = _(~.k1
£.),
c.=£.,
~=
~
~
~
i=l, ••• ,k when putting yo·o.
• (a)
~
• (a)
On identifying Yi with
8i -8.,
~
a .. with
~J
~
svtk" ,V1/Jij and b i with sVtk",Vbii' l<i<j<k, the theorem follows.
Q.E.D.
tea)
m,V'
We now consider some possible approximations of
(a) for
Based on Corollary 2.0.5.B we may substitute Iml m,v
where Im\
tm,v'
(a)
m=k' , k".
m=k' , k"
(a~ is as in (1.1.6). This will provide us with a con-
m,v
servative procedure.
Utilizing Theorem 2.0.3 it can be shown that
these upper bounds are sharper than those obtained by the mu1tip1ication inequality which are sharper than a Bonferroni first order upper
bound.
Whenever the values of Iml
(a~ are not available (for some
m, v
tables see Miller [1966], Armitage and Krishnaiah [1964] and Hahn and
Hendricson [1971]) one may use tea) ;; t(a*) m=k' kIt
m, V
V'
"
where t(a*)
V
is the l-a* quantile of a Student's t variable with V d.f. and
a*=[1-(1-a)l/m]/2.
We now consider some general alternative approximations for
T(a) defined by (1.2.9).
max
We discuss only the univariate case, letting
tea) be the value of (T(a»~ in this case.
m,V
max
The results are readily
36
extended to the multivariate case.
Let
G(a~
be the approximation
m,v
obtained by using Siotani's method (introduced in 1.2).
We have
(2.4.6)
where t(a) is the second order Bonferroni approximation of t(a).
2
m,'"
We now consider some alternative approximations possessing
this feature.
Let H(a) be defined by the equation
m,'"
(2.4.7)
where ti=T~ when p=l, i-l, ••• ,m and the Ti's are as in (1.2.9).
i
It
is easily verified that
(2.4.8)
Another approximation can be derived from Theorem 2.0.4.
Let d be a positive constant and define R(a) (d) by the equation
m,'"
(2.4.9)
Define R(a) as the value of d for which R(a) (d)=d.
m,'"
m,'"
Then we have the
following result.
Lemma 2.4.3.
Under the above setup we have for all d>R(a)
- m,'"
(2.4.10)
Proof.
For d=R(a)_R(a) (R(a», (2.4.10) follows from Theorem (2.0.4)
m,'" m,'" m,v
on letting E be the event \til>d, i-l, ••• ,m. Since R(a) (d) is a
i
m,'"
monotone increasing function of d and R(a) (oo)=t(a!2m) the lemma
m,'"
follows.
Q.E.D.
'"
'
37
We now make the following observations.
i)
R~~~(t~a/2m» is
an upper bound for t(a) while neither G(a) nor H(a) is known to have
m,V
m,V
m,v
( ImI (a»
this feature. Other approximations for t(a) are given by R(a)
m,V
m
m,V
and R(a) (t(a»
both bounded between t(a/2m) and t (a). if) I f the
2
m, V 2
'
V
normal variables constituting the numerators of the tits are independent
we have· Iml (a) < R(a)(d} for any d > R(a}. This is so because
,
'm,v
m,v
- m,V
in this case Iml (a) • t(a~ while R(a} is an upper bound for t(a).
m,v
m,v
m,V
m,V
However, if the numerators of the tits are strongly correlated we may
expect R(a}(d} to be smaller than Iml (a) for values of d sufficiently
m,V
m,V
close to R(a}. In particular when d_t(a/2m) we might expect this to
m,V
V
be so.
Finally, we consider a very convenient approximation for tea)
m,v
as follows.
From Theorem B in Halperin [1967], it follows that, for
any d>O and in any normal setup as (1.1.1) we have
L.t
I i I ~d, It j I ~d } ~ m(m-1)
2
P( t
l~i<j~m
Thus, on letting
mP(lt
I 1 I ~d}] 2
(2.4.11)
[PC t
t~a} be defined by the equation
I
> t(a»
- m(m-1) p[(lt
1-2
2
I > t(a»]2
1-2
=a
(2.4.12)
it is easily verified that
(2.4.13)
Solving the quadratic equation (2.4.12) we get, say, P( It
and hence obtain
t~a)
t~a).
•
give good results when
V
1I
~
-t (a) )=a2
This approximation may be expected to
is large and the correlations among the
38
numerators of the ti's are low.
• (ex.)
Approximating tk"v in (2.4.1) by
..
(ex.)
ImIk',V'
we get the following
confidence intervals for the six pairwise contrasts mi-mj ,
1~i<j~4,
of Example 2.2.1.
'" '" ± .8570
mi-m
j
± .8572
(i,j) ... (1,2) , (2.3), (3,4)
(i ,j) ... (1,3), (2,4)
(i,j) ... (3,4).
± .8576
2.5
(2.4.14)
An extension of Spj_tvoll and Stoline's approach - the GT3 method.
Spj~tvo1l
and Stoline's approach to the diagonal case can be
extended to the general univariate full rank model.
Further extensions
to models of less than full rank are considered in Chapter 3.
This
extension is embodied in the following result.
Theorem 2.5.1.
Let Q: kXk be any matrix of real elements such that
QQ'=B, then we have
C{~'8£[~'e±s q(ex.)M(i'Q)]} ... l-a
-
Proof.
-
~ -
V
k,v
-
(2.5.1)
First we note that since B is p.d. there always exists a real
non-singular matrix Q satisfying QQ'-B and it follows that
Q-1(~_~) ~ N(Q,cr 2 I).
Next note that the linear combination ~'(~-~)
can be expressed as a linear combination of the coordinates of
Q-l(~_~) as ~'QQ-l(~_~). Now, the following equivalence is easily
verified
(2.5.2)
Thus, an equivalent event to the right hand side of (2.5.2) is
(2.5.3)
'
39
On equating the probability of the right hand side of (2.5.2) to 1-a
we get;
-(a)
~·qk,v
and Theorem 2.5.1. follows.
Q.E.D.
We will refer to this procedure as the GT3 method.
Note that
Spj;tvoll and Sto1ine's method is a special case of this general procedure for a diagonal matrix B-Diag(b
~
the matrix Diag(bll,
••• ,b~kk ) for Q.
11
, ••• ,b
kk
).
These authors use
For any given p.d. matrix B, a
matrix Q satisfying QQ'-B exists and Q is not unique.
The GT3 method
is not invariant under the different choices of a Q matrix, this can
easily be verified by constructing an appropriate simple example.
In
the following section we discuss the problem of choosing an appropriate
Q matrix.
2.6
On the choice of the matrix Q involved in the GT3 method.
The class of all matrices Q satisfying QQ'=B for a given p.d.
k
matrix B contains at least 2 elements which constitute the class of
lower triangular matrices T satisfying TT'=B.
easily evaluated by the square root method.
Ch. 8).
Such a T matrix is
(Cf. Graybill [1969],
It is easily verified that the GT3 procedure is not invariant
even in this small class of matrices each of which may serve as the
matrix Q in the GT3 procedure.
From the features of the function
M(·) it is quite obvious that one should not expect satisfactory
interval estimation of contrasts when using any of the matrices T
for one's Q matrix in the GT3 procedure.
A larger class of matrices Q satisfying QQ'=B is generated by
the latent roots and vectors of B.
Let Al, ••• ,A
k
be the k positive
latent roots of B and let !l" •• '!k be a set of corresponding normalized latent vectors.
Put R-(!l'.'.'!k) and D-Diag(Al, ••• ,A k ).
A
40
matrix Q satisfying QQ'o=B is then given by;
Q=RD~.
It is well known
that if Ai is of multiplicity d, say, the space of vectors Ei (nonnormalized !i) satisfying (B-AiI)Ei-O is of dimension d.
Thus, this
fact and the Zk possible choices of the matrix D~ determine the
multiplicity of choices of a matrix Q in this class.
In general, such a choice of a Q matrix has some drawbacks.
i)
In many cases the latent roots or vectors of B cannot be obtained
explicitly.
ii)
This choice of a matrix Q, always necessitates further
research for finding the appropriate one in the whole class.
This
research will not be unified over different dasigns but rather, will
have to be carried out separately in various different designs.
iii)
Following such a line of attack in some designs we could not
obtain any substantial results.
As an illustration of this method and some of the difficulties
involved we now derive the simultaneous intervals for Example 2.2.1.
We refer the reader to the notation used in 2.1.
Example 2.2.1 is of the form B=(1/n)I+22'.
The matrix B in
The latent roots of such
a matrix are (cf. Graybill [1969] p. 187) A1=1/n + Li~l bi and
A =l/n (i=2, ••• ,k).
i
A latent vector corresponding to A is propor1
tional to b and the latent vectors corresponding to A. (i=2, ••• ,k)
~
k
are any set of k-1 orthogonal vectors in R , orthogonal to 2.
If we
confine attention to inference on contrasts, the problem remains invariant under shifts of the concomitant variable.
assume that b is a contrast.
Thus, we may
If we further assume that the ordinates
of the vector 2 are equally spaced (as in the numerical example of
2.2) we have a convenient way of determining the set of normalized
orthogonal latent vectors.
In such a case the set of the k-l vectors
41
of normalized coefficients for the first k-l orthogonal polynomials
1
together with the vector Ik
vectors for B.
constitute a normalized set of latent
~,
In the numerical example of
2.2.~ ~
can be taken as
~ • ;0(-.3,-.1,.1,.3)', the latent roots are A1-.05008 and \=.05,
(i-2, •.• ,k).
The matrix of normalized latent vectors is
-3/126
~
-1/120
~
-lim
~
3/126
~
11m
~
-3/120
~
3/120
~
1/126
~
R=
(2.6.1)
The resulting confidence intervals for the pairwise contrasts are
ml-m2E[ml-m2±1.11Sv]; ml-m3E[ml-m2±1.19sv]; ml-m4E[ml-m4±1.48sv]
(2.6.2)
m2-m3E[m2-m3±1.19sv];m2-m4E[m2-m4±1.S7sv]; m3-m4E[m3-m4±1.93sv]
Comparing (2.6.2) with the results obtained earlier for the GT1, GT2
and the S methods, we see that the GT3's performance for the pairwise
contrasts is inferior to any of these three methods in this example.
This is due to our choice for Q in this case.
We now turn to describe an alternative method for choosing the
Q matrix which has some satisfactory features.
matrix of the same structure as that of B.
Tukey's T-method and
Spj~tvoll
general diagonal case.
We try to find a Q
This is in analogy with
and Sto1ine's Ext-T method for the
This approach has three desired features.
Firstly, it usually produces a unique GT3 procedure, even though there
may exist more than one matrix Q satisfying QQ'=B and being of the
same structure as that of B.
This is demonstrated in the sequel.
Secondly, for modestly unbalanced designs,Q being of the same structure
42
as that of B will be only modestly non-uniform and thus we may expect
satisfactory results from a GT3 procedure in such a setup.
Thirdly,
this method for choosing Q is a generalization of the choice for Q in
Tukey's and
Spj~tvo11
and Sto1ine's special cases.
this method in Example 2.2.1.
Bc(l/n)r +
~~'.
Let us demonstrate
The matrix B has the structure
We look for a matrix Q of the form
(2.6.3)
satisfying QQ'=B.
With this aim we write
(2.6.4)
We get a 2 =1/n and
h·d~~ where d is a scalar satisfying d(2a+d~'~)=1.
In Example 2.2.1 we find a-0.2236, d-1.3087.
.29003
.07664
.07789
.07915
.07664
.30150
.07919
.08047
.07789
.07919
.30410
.08180
.07915
.08047
.08180
.30670
Q=
e-
We get
(2.6.5)
which produces the following approximate (q~:~;) ~ 3.7) confidence
intervals for the pairwise contrasts mi-m ,
j
'" '" ± .839s ,
mi-m
v
j
'" '" ± .856s '
mi-m
v
j
'" '" ± •870s '
mi-m
v
j
l<i<j~k.
(i,j) = (1,2), (2,3), (3,4)
(i,j) "" (1,3), (2,4)
(i,j) = (1,4) •
The improvement from (2.6.2) to (2.6.6) is high.
however note that:
(2.6.6)
We should
firstly, a matrix Q, of the same form as B, such
43
that QQ'-B, does not always exist for every given p.d. B. This is
demonstrated in the sequel.
Secondly, this choice of a matrix Q
does not necessarily provide the optimal intervals for all pairwise
comparisons simultaneously.
In many cases one can find a different
Q which provides shorter intervals for some of the pairwise contrasts.
Spj~tvoll
This is the case also in
and Stoline's simple
diagonal case.
Such an optimality feature exists in the uniform diagonal
case, i.e., B=aI and
Q=a~I. To verify this, note that on letting
gi denote the ith row of Q we have
(2.6.7)
Our conjecture follows on realizing that
min
&E~: &'&=2a
M(~') = a~
-
(2.6.8)
CHAPTER III
EXTENSIONS TO MORE GENERAL MODELS
3.0
Introduction
The generalized T-procedures introduced in Chapter 2, were given
only for the univariate full rank case.
In this chapter, we extend
these procedures, to the general multivariate normal setup, as given by
(1.1.1) and introduce some modifications especially useful in LFR models.
We will, however, confine ourselves, throughout this chapter, to LFR
models where
te
=
k
C.
This extension has two dimensions (i.e., from
full rank to LFR and from univariate to multivariate) which we treat
sequentially.
In sections 3.1 to 3.3 we consider generalized T-type
procedures in univariate LFR models.
Then, in sections 3.4 and 3.5 we
utilize the results of 3.1 to 3.3 in deriving generalized T2
max
and
some other procedures, in a general multivariate normal setup.
Clearly, any LFR model can be reparametrized to a full rank
model and then treated by the procedures of Chapter 2.
However, this
simple minded solution does not necessarily guarantee satisfactory
results, nor will all the methods discussed be invariant to different
reparameterizations.
As discussed in the sequel, better solutions
exist in many LFR models.
A more detailed outline of this chapter now follows.
In section
3.1 we show that all three procedures introduced in Chapter 2 for models
of full rank can be similarly used in LFR models based on any 0E{0}
\
e-
45
(equivalently any
B£~).
8£{8}, is discussed.
Invariance of the procedures to the different
It turns out that the GTl method based on the
primitive majorization procedure is not invariant to the different
choices of B
in~.
To overcome this difficulty we first (Section 3.2)
derive the exact distribution of the range of a normal vector with a
zero mean vector and a general balanced dispersion matrix.
definition 2 in 1.1).
(See
Utilizing this result we define a GTl procedure
based on a majorization of a given symmetric B matrix to a general,
symmetric, balanced matrix.
This extended GTl procedure is invariant
to all the different choices for a symmetric
te = Ck ,
B£~
in any LFR model (when
as assumed).
In Section 3.3 we consider some modifications of the methods.
Firstly, a modified GTl procedure based on a more efficient majorization
method (than the primitive majorization introduced in 2.2) is derived.
This modified GTl procedure is not always uniquely determined, but in
many LFR models (as illustrated in 5.1) it has an obvious structure
which produces better results than the original GT1.
Secondly, a
modified GT1-GT3 type procedure (i.e., a procedure with combined
features of the GTl and the GT3 methods) is introduced.
We will find
this procedure (Section 5.1) very useful in some specific problems.
Thirdly, a conservative modified GT2 type procedure is discussed which
is useful in some specific problems.
In Section 3.4 we briefly summarize the multivariate analogs of
the GTl, the GT2 and the GT3 methods (i.e., generalized T 2
procedures)
max
in the general normal setup (1.1.1).
46
In Section 3.5 we consider the problem of simultaneous interval
estimation of doubly linear parametric functions in a general mu1tivariate normal setup in (some) cases when not all the functions of the
response vector are of interest.
Note that in the univariate case e-0 and
- - (see definitions
~.0
-
in 1.1) we thus use in Sections 3.1 to 3.3, ~, ~ and {~} for 0, 0 and
{0}, respectively.
3.1
Generalized T-procedures in LFR models.
Consider the general univariate setup of (1.1.1).
A given vector
~E{~} corresponding to a certain BE~, is not necessarily an estimator of
e.
The actual dispersion matrix of
matrix E which depends on B.
B
- is
Q
a 2 (B+E ), say, for some real
B
The following result summarizes the
application of the GT1 method to LFR models based on any symmetric matrix
BE~.
(Recall our assumption
Theorem 3.1.1.
Ck = ().
e
For any ~E{~} corresponding to a sYmmetric BE~ with a
dispersion matrix a 2 (B+E ), on majorizing B to a uniform matrix
B
B*
= aI
+ b(!!'-I), we have
(3.1.1)
Proof.
Let ~ ~ N(Q,a 2B*) and ~* ~ N(E(~),a2(B*+EB»' then from (1.1.1)
and the discussion in 2.1, it follows that in models where
1e = Ck , we
have (recall that R(.) stands for the range of (.»
d
i)
R(2*-E(~» ~ R(~-E(~»;
ii)
From i and ii Theorem 3.1.1 follows.
R(~*-E(~» ~ R(~)
Q.E.D.
(3.1.2)
e·
47
Thus, a GTl procedure, based on any symmetric
BE~,
can be used.
However, the resulting procedure is not necessarily invariant to the
different choices of
B£~.
Majorization procedures as described in
Chapter 2 operating on different B matrices may result in different
GTl procedures.
BE~,
This is so since in models where!
e
= Ck ,
on letting
we have that any matrix of the form
l~i<j<k
also belongs
to~.
(3.1.3)
It is clear that different GTl procedures may
result from using B and B+E.
The original model can of course be
reparametrized to a full rank model and then a procedure as discussed
in 2.3 can be used.
This method has its drawbacks (see 2.3) and not
always do we want to use it.
In Section 3.2 we derive an extended
GTl procedure which is invariant to all the different choices of a
matrix B
in~.
Note however that the GTl procedure is invariant
over the class
{B+tll'
--
The GT2 procedure causes no problems in LFR models.
invariant to any choice of
B£~
(3.1.4)
'
It is
We now discuss the use of GT3 type
procedures in LFR models.
Theorem 3.1.2.
Let B be any p.d. matrix in ( and let Q satisfy
B
QBQi=B, then
(3.1.5)
Proof.
The dispersion matrix of ~ is given by
48
(3.1.6)
From the properties of a model in which
l e = Gk
we deduce the relation
(3.1. 7)
l<i<j~k
We now look for a matrix LB
=
! x
k
~i' ~BE~
satisfying
(3.1.8)
Such a matrix always exists since (3.1.8) and the form of L imply
B
E
B
=10
-
(Qh) , + l'
-
~
Qh + 11'
--
~
h 'h
- -
(3.1.9)
From (3.1.7) and (3.1.9) it follows that a matrix L as in (3.1.8)
B
always exists.
Now, note that
(3.1.10)
By an argument similar to the proof of Theorem 2.5.1, Theorem 3.1.2
now follows from (3.1.10) and the fact that eke Rk •
Q.E.D.
Suppose now that B , B are both p.d. matrices in
2
l
ll.
and
Q2Qi = B2 • From (3.1.3) and (3.1.9) it follows that a matrix L: 1o<k
exists such that (Q2+L)(Q +L)'
2
= Bl
k
and L is of the form 1 0 h' , hE~ •
Thus, a GT3 procedure based on B2 with Q2Qi
-
=
-
B is equivalent to a
2
GT3 procedure based on Bl with Ql=Q2+L and QlQi
= Bl •
Alternatively,
one may first reparametrize the original model to a full rank model
by a transformation
e-e-Te,
Since
e
~
-e
--e
-
e -Te; T:(k-l)xk, r(T)-k-l
N(e-e ,a 2 TBT'), we may use the GT3 procedure as in 2.5.
(3.1.11)
e-
!~9
Is the GT3 in this form invariant to the different possible
choices of a transformation matrix T as in (3.1.11)?
~ei=Ti~ ~ N(~ei,a2TiBTi)' i=1,2, and Q2Qi
exists a non-singular matrix A such that
Now
~'8
, - -e2
= ~'A-l8
-el'
Q1Q'1
V ~£~k-l and Q
l
= T2BTi,
then clearly, there
~e2 = A-1~e1 and ~e1 = A~e2'
= ~Q2
= AQ 2Q'A'
= AT 2BT'A'
2
2
Let 8 =T 8 and
-e i i -
give
• T1 BT'1
(3.1.12)
Thus, the confidence interval for the parametric function
-1
in terms of ~el is given by ~'A
~'~e2
which
1
~el' is determined by M(~'A- Ql)'
From (3.1.12) it follows that
(3.1.13)
thus, the GT3 based on B and Q is equivalent to the GT3 based on B
2
2
1
In some important problems, an exact GT3 procedure exists based
on the distribution of the studentized range.
This is given by the
following theorem.
Theorem 3.1.3.
Let q., (i=l, ••• ,k) denote the rows of Q satisfying
-~
QQ'=B for an arbitrary p.d.
B£~
in a given LFR model.
If q.l = q.1
-~-J-
for all l<i,j<k,
(3.1.14)
Proof.
We have the equivalence of the following two events
(3.1.15)
Now, the condition ~1l • ~jl,
V (1<1,
k
j~k) implies that S'Q£C
for
50
k
any £EC.
Thus, since
from (3.1.15).
3.2
-
£'(~-~)
• £'QQ-1 (~-~),
Theorem 3.1.3 follows
Q.E.D.
The distribution of the range in a general balanced setup and
the extended GT1 procedure
Theorem 3.2.1.
and
~ij
= 0ii +
Let
~(X1' ••• '~)' ~
N(O,E) where E=(Oij) is p.s.d.
0jj - 20 ij .. a, say, for all (l<i<j<k), then
(3.2.1)
Proof.
Let!d = (X1-X2'···'X1-~' x2-x3' ••• '~_1-~)' and
~d .. (zl- z 2,··"zl- z k' Z2- Z3'···'Zk_1-Zk), each of dimension k(k-1)/2.
k
To prove the theorem it is sufficient to show that !d and ~a)2~d have
the same dispersion matrices.
C(Zi-Zj , Zm-ZR.) =
The dispersion matrix of
~d
is given as
2
if
i""1l1 and jaR.
1
if
either i=m., j;R. or j=R., i;m
-1
if
either i=R. ,
0
j;m
(3.2.2)
or jam, i;R.
otherwise
We now show that the dispersion matrix of !d is given by
a
C(xi-xj ,Xm-X ) =
if i-m and jaR.
~a
i f either iam, j;R. or j=R., i;m
~a
i f either i=R., j;m or j=m., i;R.
0
Clearly, C(xi-x , xi-xj)-a for all
j
for the case
j~t
(3.2.3)
otherwise
i~j.
Note that if (3.2.3) holds
and i-m it holds also for the cases i+m and j-R.,
e-
51
i-R, and jJ'm, and i;R, and j-m since
To show that (3.2.3) holds for the case i-m and j;R, note that
(3.2.5)
When i'J'j and m;R, we have
(3.2.6)
Thus, the theorem follows.
Q.E.D.
Based on Theorem 3.2.1 we now propose an extended GTl (EGT1)
procedure based on majorizing any symmetric
B£~
to a sYmmetric balanced
matrix rather than to a uniform matrix as in the GTl method.
The EGTl
is embodied in the following theorem.
Theorem 3.2.2.
1
Let B£~ be symmetric and let wi£l , di>O, i=l, ••• ,k,
satisfy
k
e
bii+di+j~ll~ (wi+wj )-b ij I = wi + 2 ' V (i=l, ••• ,k)
(3.2.7)
j'J'i
for some e>O, then we have
(3.2.8)
Proof.
From equation (3.2.7) it is clear that the balanced matrix
B*=(b~j); b: i + wi + e/2; b~j = ~(wi+Wj)' (l<i;j<k), is a majorized
form of B. From this, together with Theorem 3.2.1 Theorem 3.2.2
follows, based on arguments introduced in 2.1.
Q.E.D.
52
Optimizing the EGTl procedure amounts to minimizing e in (3.2.7)
1
by an appropriate choice of numbers wiER ,
di~O,
i=l, ••• ,k.
Note, that an EGTl procedure always exists since the GTl is a
special case of the EGTl when wiDW, V (i-l,2, ••• ,k).
We now show that
the EGTI is invariant to the different choices of symmetric
matrices in models where
BE~
k
t e • C.
Suppose B , B E ~ both symmetric,
l
2
then there exists a real symmetric matrix E-( e
ij
) such that
(3.2.9)
Both B and B are candidates for use in the EGTI procedure.
2
l
Lemma 3.2.3.
An
EGTl procedure given by Theorem 3.2.2, based on B ,
I
is equivalent to an EGTl procedure based on B •
2
Proof.
Since
k
k
Ie = C we have £iBl£2 - £iB2£2'V £l'£2EC which
implies that E satisfies
(3.2.10)
An
EGT1 procedure based on B2 with
1
w2iE~
, di>O (i=l, ••• ,k) and e>O
satisfying
(i=l, ••• ,k)
(3.2.11)
is equivalent to an EGT1 procedure based on B with the same di's the
1
same e and with w1i - w2i + e i , i-l, ••• ,k.
This proves that the best
EGTI procedure based on B is equivalent to the best EGTl procedure
l
based on B •
2
Q.E.D.
e-
53
The EGTI procedure can be used in various cases when a straightforward efficient choice for the wi's and the di's exists.
As an
example we consider the type of design in Example 2.2.1 but we take k=3
rather than k-4 as in the original example.
suppose that n=20
xi =12,13,14
Using the notation of 2.2
(i=1,2,3) and S =2500; we obtain the B
xx
matrix in this full rank design as
B=
[
.1076
.0624
.0624
.1176
.0728
.0672
.0728
.1284
This matrix is unbalanced.
.0672 ]
(3.2.12)
However, by adding to B the p.s.d. matrix
Bl=Diag(O, .0012, 0), the matrix B+Bl=(bi ) is a balanced majorization
j
of B with bii+bjj-2bij = .1016, V (1~i<j~3).
majorization procedure we get B* = .1340 1
uniform majorization of B.
3
Using the primitive
+ .0728(~3~3-I3) as the
In this case the EGTI procedure is strictly
better than the GTI procedure since 2(.1340 - .0728) > .1016.
Some designs possess symmetry features with respect to the treatments under study which guarantee, as we subsequently show, that a best
EGTI procedure in the class
in a certain subclass
to any
BE~,
~
of~.
is no better than the best GTI procedure
Since the best EGTI procedure is invariant
we choose a B in a subclass of
~
and show that the best
EGTI is equivalent to the best GTl in that subclass.
a given design a subclass of
has the following structure.
ii)
~
exists such that any B in that subclass
i)
All diagonal elements in B are equal.
In each row of B all the off diagonal elements can be partitioned
into d, say, exclusive sets with 1
the 1
Suppose that for
a
a
elements in the ath set, such that
elements in the ath set of the ith row are equal to b
a
say, for
54
all iml, ••• ,k, aal, ••• ,d.
We use the term semi-uniform for such a
matrix and designs which produce such a subclass of matrices
denote by
~u
the subclass of semi-uniform matrices
in~.
We
B~
The follow-
ing result justifies the use of a GTI procedure in a semi-uniform
design.
Lemma 3.2.4.
i) The GTI procedure is invariant over the class
semi-uniform design.
~
u
in a
ii) The optimal EGTI is equivalent to the best
GTI based on a matrix BE"-u •
Proof.
i) Since we have assumed that
for any Bl , B2 in
from (3.2.8).
~u;
le
=Ck ,
it can be verified that
1
Bl =B 2+t!!' for some tER.
ii) Follows directly
Q.E.D.
Denote the diagonal element in a given semi-uniform B matrix by
a, and recall our notation and discussion in 2.1 and in 3.1.
If a > max {b }, then the optimal choice w* for w in
ex
l<a<d
the GTI procedure is any-median of the k values in any row of B.
Lemma 3.2.5.
Proof.
First note that
~(w,B) - w m a +
where
~(w,B)
L
Ibij-wl-w, i=l, ••• ,k
(3.2.13)
j:j;'i
is given by (2.2.3).
Since a > max {b } we may write
l<a<d a
(3.2.13) as
k
cr{w,B)-w =
L Ibi.-w\,
j=l
i-l, ••• ,k
which is minimized when w '"' median{bil, ••• ,b
Q.E.D.
(3.2.l4)
J
ik
} for any iE{l, ••• ,k}.
e·
55
Various important unbalanced designs fall in this class.
Further
discussions and examples are postponed to Chapter 5.
3.3
Some modifications of the procedures
We now introduce an alternative procedure to the one discussed
in 2.2 for majorizing a given symmetric matrix to a uniform matrix.
The new procedure is an improved version of the primitive majorization
procedure, however, it is not always uniquely defined.
modified GTl as MGTl.
Let
BS~
be symmetric.
We abbreviate
Denote by Pf the set of
double indices {ij: l<i<j<k}. Suppose that Pl""'Pm are m exclusive
m
sets such that LJ P. = P • Denote the elements of PR, by
f
i=l 1
m
(ij)n , ••• ,(ij)n
where kl, ••• ,k satisfy
k. = k(k-l)/2 and
Nl
NkR,
m
1= l
1
L.
ki~l,
i-l, ••• ,m.
Let DR, be the set of all dR,' say, different indices
which appear in any position in the set PR,'
We further assume that
for any pair of elements (a,B) in PR, we have
(3.3.1)
The following result is the backbone of the MGTI method.
Lemma 3.3.1.
If for all R,=l, ••• ,m, the set PR, is given by
(3.3.2)
then a,b in (3.1.1) can be chosen as
m
a·
max {b + L Ii(DR.)[1+(dR,-2)~(w-bR,)]lw-bR,I}, bew
l<i<k ii R,-1
where Ii(PR,)-l if iSPR, and zero otherwise and
zero otherwise.
~(x)=l
if x<O and
(3.3.3)
56
Let BJ/,: kXk be a symmetric matrix with w-bJ/, as its (i,j)
Proof.
th
element for all (i,j)EPJ/,' [1+(dJ/,-2)~(w-bJ/,)]lw-bJ/,1 as its (i,i)th
element for all iEDJ/, and zeros elsewhere.
all J/,=l, ••• ,m.
Clearly, BJ/, is p.s.d. for
It can be verified that ~~l BJ/,+B gives the matrix
with all off diagonal elements equal to wand the ith diagonal element
given by
m
b ii +
2
J/,=l
Ii(DJ/,)[1+(dJ/,-2)~(w-bJ/,)]lw-bJ/,1
(3.3.4)
Following similar arguments to those involved in the proofs of
Theorem 2.2.1 and Lemma 2.2.1, Lemma 3.3.1 follows.
Q.E.D.
For a given partition of P which satisfies (3.3.1) and (3.3.2)
f
an optimal MGTI is the MGTI procedure which minimizes a-b of (3.3.3).
e·
The optimal value of w can be searched for in a similar way to the
search procedure devised in Chapter 2 for finding the optimal GTI
procedure.
However, more than one partition satisfying (3.3.1) and
(3.3.2) may exist which may provide different MGTI solutions.
Clearly,
one may define the optimal MGTI procedure in a unique form by letting
it be the one providing the smallest value for a-b among all partitions and among all possible values of w.
such a procedure is difficult to determine.
However, in many problems
Note that if all off
diagonal elements in B are different the MGTI reduces to the GTI
procedure and is uniquely defined.
The MGTI procedure can be extended in a way similar to the
extension of the GTI discussed in 3.2 to make it invariant to all
BE~.
Next we discuss a modified GTl-GT3 (abbreviated as MGT(1,3»
type procedure possessing features of both the GTI and the GT3 methods.
57
Consider the univariate general normal setup (1.1.1) with
ing to a p.d. matrix B.
,..,.. ,..
satisfy QQ'-B.
Lemma 3.3.2.
Let B-B+B where
B is
-e correspond-
p.s.d. and let Q=Q:kxk
i) The probability is at least 1-a that all estimable
parametric functions
&'~
simultaneously satisfy
(3.3.5)
ii) If
t e = Ck
and the sums of the elements in the k rows of Q are
(a)
the same, then -(a)
qk,v in ( 3.3.5 ) can be replaced by qk,v'
Proof.
We only prove i, since ii can similarly be proved following
the line of proof of Theorem 3.1.
We may pretend (based on our dis-
cus~,ion in 3.1) that ~ '" N(~,a2B) •
Note that the event
(3.3.6)
implies (3.3.5).
But Q-1(~_~) '" N(O,02Q-1BQ-l,) and since
B=B-B
(where B is p.s.d.) it is clear (from our discussion in 2.1) that
q~~~ is an upper bound of ~ defined by the equation
(3.3.7)
hence the lemma follows.
Q.E.D.
This method (which combines features of the GT1 and the GT3
procedures) can be used in various problems to improve the results
obtained by either a 'pure' GTl procedure or a 'pure' GT3 procedure.
In some problems this procedure may also be more convenient to use
than any of the original procedures.
This is demonstrated in 5.1.
58
Next, in this section, we consider a modification of the GT2
method.
~
A conservative GT2 procedure is based on an upper bound on
defined by the equation
(3.3.8)
where
~ij
From Theorem 2.0.5 we have obtained
is as in 2.4.
as an upper bound on
(a)
Irn Ik',V
In some cases, however, Theorem 2.0.5 can be
~.
utilized more efficiently, as we now discuss.
Suppose that the vector ~ can be partitioned to 8'=(8',8')
-1-2
where ~i: kixl=(8il,···,8ik )', i=1,2, kl >2 and k l +k 2=k, such that on
i
partitioning
(3.3.9)
B- [
where Bi J,
11'
kiXkj , . (b i J, }o-l
;v- ,
••• ,
k i,JtJ0'=1 , ••• , kj ' i,j=1,2 we have
Bll=(al-bl)Ik +bl!k!k for some aI' b l • Similarly partition
1
1 1
11
i 'i '
~'=(~i'~i) and let ~ii,ji' be defined in terms of b ii , b j j
and
ii'
b ij (i,j=1,2) as ~aB is defined (in 2.4) in terms of baa' b
and
BS
baS' respectively.
An upper bound on ~ is then given by ~* satisfying
P{R(~l-~l} ~ [2(al-bl}]~sV~*}P{Mk*~~*}
~
,
where k*=k l k 2+(kf} and
l-a
~*,v stands for a variable distributed as the
studentized (v d.f.) maximum modulus of k* lID normal variates.
(This is easily proved using Theorems 2.0.5 and 2.0.3).
(from Sidak [1967]) that
~*<Imlk(~}'
,V
It is clear
However, since the distribution
of the studentized maximum modulus has not been intensively tabulated,
~* is not easy to obtain.
Alternatively, an upper bound on ~* which
59
depending on k, k 1 and v, may be better or worse than
defined by the equation
this follows from Theorem 2.0.5.
Iml~~~v' is €
Since the distribution of the
studentized range has been intensively tabulated (Harter [1969]) we
can approximate ~" (from above, say) by a simple search procedure.
It is clear that
€ will be
Imlk(~),v for sufficiently
smaller than
large values of k, k and V for any given a. To illustrate the use1
fulness of this method we bring the following table, the (+) entries
in which indicate combinations of values of k, k
€<Iml~~~v
for a-.OS.
1
and V where
The efficiency of this method could however be
much better emphasized in a table containing larger values of k, but
tables for
Iml~~~v
for such values of k do not exist in the present.
Table 3.3.1
Comparisons of
€ with
Iml~~~v
V
k :: k
1
I
I
4
5
6
I
I
I
I
I
I
I
I
I
I
3
5
8
10
15
20
2S
30
40
60
120
3
-
- -
-
-
-
-
?
+
+
+
4
- - - ? ?
?
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
5
I
The (-) entries indicate that
Iml~~~v < ~
indicate cases where the relation between
determined by the search procedure.
while the (1) entries
Iml~~~v
and
€ could
not be
60
Generalized T2
procedures
max
3.4
Recall the general multivariate normal setup (1.1.1), the notak
tion introduced in L 1 and the assumption C ...
to.
e
We now bring a
result which is a generalization of Theorem 3.2.1 and subsequently
utilize it to derive the multivariate analog of the EGT1 procedure in
LFR models.
Lemma 3.4.1.
Let 0E{0} be such that its corresponding matrix B=(b .. )E~
~J
is symmetric and balanced, i.e., b
l<i,
j~k.
ii
+ b jj - 2b ij = e, say, for all
Then, the distribution of any function having
only argument and such that it is a function of
e-e
e-e
as its
only through the
quantities ~i-~j-(§i-~j)' (l<i, j<k) is invariant to the values of
b
ii
, i-1, ••• ,k, as long as e>O is held fixed.
Proof.
The proof is similar to the proof of Theorem 3.2.1.
It follows
on verifying that
- - - C(~i-~j'~m-~R.)
eI:
if
i-m and j-R.
~eL
if
either i-m, j;'R. or j=R., i;'m
(3.4.1)
=
~eL
if either i=R., j;'m or j=m, i;'R.
O:pxp
This is easily verified.
Corollary 3.4.1.A.
otherwise
Q.E.D.
If the function considered in Lemma 3.4.1 is also
a function of random quantities other than
independent of
-e,
-
e which
are statistically
the result of Lemma 3.4.1 still holds.
This is
straightforward.
Corollary 3.4.1.B.
Under the setup of Lemma 3.4.1, we have
e·
61
k
C{c'ete:[c'eU(~et(ak) tis t)~~I Icil],
-
-
-
-
p"
V-
i=l
V-
~e:llk}
\I Se:Ck,
=
I-a
(3.4.2)
The event D(e-e , S-l)<t
where D(.,.) is defined by (1.1.9),
V -'
Proof.
implies
1(2i-2j-(2i-2j»'~1 ~ t~(~ISV~)~' V l~i, j~k, ~e:RP
(3.4.3)
From Lemma 3.4.1 it is clear that we may consider B as being of the
form B =
~eI. Thus, on choosing t
=
~e T(a)
the corollary now
p,k,v'
follows since (3.4.3) implies
Is'(e-0)~1
k
<
(~ISv~t)~ ~.I
which implies the corollary.
Definition.
Icil, V Se:Ck,
1.=1
~e:lP
(3.4.4)
Q.E.D.
In the uniform diagonal setup of full rank, we
define the multivariate analog of the univariate studentized augmented
range as the statistic
-
-1
max(D(e,Sv)'
- -1max {ei's e.})
l<i<k - V-1.
(3.4.5)
-
where e and Sv are as in (1.1.1).
Consider now the (multivariate) uniform diagonal full rank model
with B
= aI,
for some a>O.
We have
~ ~ N(~, aI ~ E).
-(a)
Denote by T
the 1-a quantile of a
p,k,V
(3.4.6)
-1- -
-1
D(0-0,S,,).
v
We get the
following result (the proof of which is straightforward).
(3.4.7)
We now discuss the multivariate analog of the GT1 procedure in
62
the general multivariate normal setup.
(generic notation) with rows
~i:
Let X: kXp be a real matrix
1x p, i-1, ••• ,k.
Let H be any symmetric
p.d. matrix and define the sets
Q(v) = {X:D(X,H) ~ v}, Q(v) = {X:D(X,H) ~ v}
Lemma 3.4.2.
(3.4.8)
Both Q(v) and Q(v) are symmetric about O:kXp and convex
for any v>O.
Proof.
We prove the result only for Q(v), the proof for Q(v) is
similar.
Denote ~ij(X) = (~i-~j)'H(~i-~j)' l<i<j<k.
obvious.
To prove convexity, suppose XiEQ(V),
Symmetry is
Xi=(~i1'
••• '~ik)"
i=1,2.
For any O<a<l we have
~ij(aX1+(1-a)X2) = a2~ij(X1) + (1-a)2~ij(X2)
(3.4.9)
+ 2a(1-a)(~li-~lj)'H(~2i-~2j)
Since H is assumed to be p.d. there is a non-singular matrix T of
real elements such that H=T'T and we have
(~li-~lj)'T'T(~2i-~2j) ~ [(~li-~lj)'T'T(~li-~ij)]~
[(~2i-~2j)'T'T(~2i-~2j)]~
which follows from the Cauchy-Schwartz inequality.
(3.4.10)
Substituting the
right hand side of (3.4.10) in (3.4.9) we get
since this holds for all
i~j
we finally get
(3.4.12)
63
which proves the convexity of Q(v).
Q.E.D.
It is easily verified that any majorization of a matrix B£( also
majorizes the matrix B 0 E.
Based on this observation, the results
contained in Lemma 3.4.1 and its two corollaries, the generalized T 2
max
method which is the multivariate analog of the EGT1 method (Theorem
3.2.2) is now straightforward.
Note that the value of T(a)
is not
p,k,v
known and approximations of the type discussed in 1.2 must be substituted.
The generalization of the GT2 method to the multivariate case
presents no difficulty.
The extension of the GT3 method to the general multivariate
normal setup is embodied in the following two results (recall our
assumption
t "" Ck) •
e
Lemma 3.4.4.
In the multivariate setup (1.1.1), consider a matrix
0£{~} for which the corresponding matrix B£~ is p.d.
Let Q satisfy
B
QBQi = B, then
(3.4.13)
Proof.
Similar to the proof of Theorem 3.1.2.
Lemma 3.4.5.
Under the setup of Lemma 3.4.4 on letting
~i'
i=l, •.. ,k
denote the rows of QB' if gil is the same for all i=l, .•. ,k, we have
&e:~}
= I-a
.... v- p,k,v )~M(c'Q
- B)], V ~e:Ck,
-~
C{c'GR,e:[c'0R,± (R,'S R,T(a)
..
....
Proof.
..
..
Similar to the proof of Theorem 3.1.3.
(3.4.14)
64
3.5
Two notes on simultaneous interval estimation in a multivariate
normal setup when only some of the linear functions of the
response vector are of interest
I.
In many multivariate problems one can specify, without
'looking' at the data, a finite number of linear functions of the
variates which constitute one's only, or main interest in the experiment.
In this section we discuss simultaneous interval estimation of
all doubly linear functions b'0t., j=l, ••• ,m and
-
-J
k
bE~
k
or bEe.
--
Clearly, the Largest-Root method introduced in 1.1 can always be
used, however if m is moderate in size (relative to p) we may obtain
better results (shorter confidence intervals) from the following
procedures.
Lemma 3.5.1
If (1.1.1) is a full rank uniform diagonal model with
B = aI, we have
If (1.1.1) is an LFR model such that t
e
=
Ck and the matrix
corresponding to 0 is balanced with b ll + b 22 - 2b
12
Proof.
= e,
We prove (3.5.2), the proof of (3.5.1) is similar.
fixed t., we have
-J
B=(b
ij
we have
For a
)
65
Put t ij -
~j(~i-~i)/[~e~jSV~jl~,
i-1, ••• ,k, j-1, ••• ,m.
Based on the
equivalence
(3.5.4)
We can find an A satisfying
(3.5.5)
by finding an A satisfying
max
p{
j=l, ••• ,m
(R ) ~ A} < a
j
Using the Bonferroni inequality, we get,
(3.5.6)
A.q~~~m).
The lemma now
follows from the equivalence of (3.5.5) and (3.5.6).
Q.E.D.
Similarly, we can use the Bonferroni inequality to extend the
S, the GTl, the GT2 and the GT3 for simultaneous inference on a finite
number of pre-determined functions of the response vector in unbalanced models.
Let
~j'
j=l, ••• ,m be as in Lemma 3.4.1 and assume
an unbalanced general multivariate setup as (1.1.1).
Let e be defined
as in Theorem 3.2.2, let P c , N c , k, and '''i
' l<i<u<k
be as in 2.4 and
'f'u
--let Q be as in (3.4.13).
B
-We put
-
c_'e~_j=cj' c'e~.=c. and ~~S,,~.=s:.
- -J J
-J v-J J
We then have the following result.
Lennna 3.5.2
C{cjdej±sj «k-1)F~~7~~'B~)~l,'V
~e:Ck,
C{Cjdcj±sj~e)~q~alm)~~
Icill, V ~e:Ck,
,v i-1
j-1, ••• ,m}
~
j-1, ... ,m}
1-a
(3.5.7)
~ 1-a
(3.5.8)
66
L
itP
L
£
utN
(3.5.9)
£
(3.5.10)
As an example of the efficiency of these methods vs the
Largest-Root method in cases when m is moderate in size, we look
at a one way MANOVA with equal sample sizes n, say, from k populations.
Suppose p=m, a case which often arises when linear combina-
tions of the response vector make no sense.
We compare the lengths
of the intervals obtained by the Largest-Root method with those
obtained from (3.5.7) for all contrasts among 'treatments' ('populations').
In this comparison the ratio of the lengths of the
e-
confidence intervals is equal to
(a)
I
(aIm)
~
(Ap,k-l,k(n-l) (k-l)Fk_l,k(n_l»
(3.5.11)
Table A.5 (see Appendix) gives some indications on the value of
(3.5.11) as a function of p, k and n for a=.05.
The values in this
table should be considered as approximations to the actual values
rather than exact since the values of A(a)
were obtained
p,k-l,k(n-l)
by interpolations in the charts of Heck [1960].
II.
Suppose that in a given problem pertaining to (1.1.1),
only linear combinations within subsets of the components of the
response vector make sense.
We now discuss a simple application of
the Bonefrroni inequality, to obtain shorter confidence intervals
than those obtained by the Largest-Root method in a special case of
this sort.
67
Suppose p=2g and the response vector can be partitioned into
two sub-vectors of length g each ~'·(~i'l2), such that the only meaningful combinations
&'l
are those generated by &'6 of the form
,Q,'-(O,O, •.• ,O,,Q,l' ••• ',Q,) or &'-(,Q,1,,Q,2, ••• ,,Q,g'O, ••• ,O).
-
Denote the
g
set of all vectors in ~p of either of these two forms byR P .
The
o
following lemma gives conservative simultaneous confidence intervals
for all the estimable doubly linear functions b'8,Q"
-
-
b£t , ,Q,£RP which
-
e
-
0
are generally shorter than those obtained by the Largest-Root method.
Lemma 3.5.3.
The probability is at least l-a that the following set
of confidence intervals will simultaneously cover all the corresponding true values
[b'0,Q,±(A(a/2)b'Bb,Q,'S ,Q,)~ b£~, ,Q,£~]
- g,k,v- -- v- ' - e - 0
Proof.
We prove the result for the full rank case.
proved by a reparametrization to a full rank model.
p.d. (see assumption iv in (1.1.1».
The LFR case is
We assume B is
Let B=QQ', Q:kXk non-singular.
Denote by @~:pxl the ith row in Q-1 (0-8).
k -*-*,
ee
Li=l
-i-i
(3.5.12)
Form the matrix S*:pxp
*
and partition Sand
S\) with respect to y-l and y-2 in
y '= (y' ,y') as
-1-2
Sv =
[
Sv , 11
S\) ,12
S\) , 21
S
v,22
J;
S* =
S~l
[ S~l
*
S12
S~2 ]
(3.5.13)
The event
sup{&'S*&/(&'Sv&)} ~ A
,Q,£RP
-
is equivalent to
0
(3.5.14)
68
max(sup {~'S~l~/(~'SV,ll~)}' sup {~'S;2~/(~'SV,22~)}) < A
~ERg
~E~g
(3.5.15)
Utilizing the Bonferroni inequality, an A which makes the probability
of (3.5.15) no less than 1-a is A
=
A(a/2).
g,k,v
From the equivalence of
(3.5.14) and (3.5.15), following the same arguments as in Roy and Bose
[1953], we have that the probability is at least 1-a that
(3.5.16)
which implies the lemma.
As
Q.E.D.
an example, consider a one way MANOVA setup with equal sample
sizes n-10.
Let p-4, k=3 and a=.05,
(.025) (.05)_
we get A2,3,10/A4,3,10-.76.
Note that a similar procedure can be obtained with the T2
max
method.
e·
CHAPTER IV
ON MODIFICATIONS AND FURTHER EXTENSIONS OF THE
PROCEDURES IN SOME SPECIFIC PROBLEMS
4.0
Introduction
In this chapter we are concerned with the use of generalized
T-type procedures in some specific problems where modifications of
the methods introduced in earlier chapters are necessary or useful.
We provide T-type methods in some problems where such methods are
not available and improve some procedures proposed elsewhere for
other problems.
In Section 4.1 we are concerned with simultaneous inference on
growth curves and on main effects in split-plot designs.
In Section
4.2 we consider simultaneous inference on interactions in a two way
layout where we improve one of the results of Gabriel et ale [1973].
The design used for this discussion is a split-plot which enables
extracting results more general than results obtained from a similar
discussion in a two factor crossed experiment.
In Section 4.3 we
consider alternative procedures to those proposed by Gabriel and Sen
[1968] for simultaneous inference based on rank statistics in the
multivariate multisample location problem.
Section 4.4 contains
some examples of augmenting the basic premises of the GTI, the GT2
and the GT3 procedures to problems pertaining to (1.1.1), where
inference is based on statistics other than the range or the augmented
70
range and is restricted to special cases of the general setup (1.1.1).
This includes the slippage problem, the problem of comparing several
treatments with a control, bounding a regression curve and simultaneous
prediction of the dependent variable at a finite number of realizations
of the independent variables in a linear regression.
Extended T and T2
4.1
max
procedures in some "Semi-Multivariate"
problems.
Various experimental designs and models for univariate data
exist in which multivariate techniques for inference have been considered in the literature.
Some important examples that fall in this
category are i) analysis of repeated measurements in a general experimental design (refer to Grizzle and Allen [1969]).
approach to the two factor mixed model
ii) Scheffels
(see Scheffe [1959] Ch. 8).
iii) simultaneous inference on all product-type interaction contrasts
in a balanced two factor experiment.
(See Gabriel et ale [1973]).
We use the term "semi-multivariate" to describe a problem in that
general category.
It is clear that the choice between univariate and
multivariate techniques in a given "semi-multivariate" problem is
determined by the basic assumptions incorporated into the underlying
model.
In this section we discuss simultaneous inference based on
generalized T and T2
procedures in problems pertaining to i and iii
max
above.
First we consider the problem of repeated measurements.
Suppose that N individuals have been randomly assigned to the
i
i
th
of the r cells of some design and that the same characteristic has
been measured under p different conditions on each individual.
I
=( (1)
~iY
(p»
x iy , ••• ,xiy
Let
be the vector of the p measurements on the y th
e-
71
individual in the i
th
cell, r-l, ••• ,Ni, i-l, ••• ,r.
Put N
X:pxN=(xll, ••• ,x
} and consider the model
-rn r
x • BtA+E
(4.1.1)
where B:pXq, A:rxN are real known matrices, (:qXr is a matrix of
unknown parameters and the columns of E:pxN are lID with the distributional law N(O,E).
The model (4.1.1), introduced in Potthoff and Roy
[1964], is usually referred to as a growth curve model.
Rao [1965]
proposes a conditional solution to (4.l.l) which leads to a 'regular'
linear model with stochastic predictors as follows.
B2 :pX(p-q} be of full rank such that BiB=I and BiB=O.
Let Bl:pxq and
Let Yl=BiX,
Y2=BiX, then if the model (4.l.l) is right we have E(yl}=tA, E(Y }=0.
2
Conditional on Y2 ' we have the linear model
(4.l.2)
where
y:qx(r+p-q}=[(:~],
D:(r+p-q)xN=[A':Yi]' (n is the matrix of
regression coefficients of Y on Y2 ) and the columns of E are lID
l
2
A
Let y be the least squares estimate of y in the
conditional model (note that since Y has a continuous distribution
2
(DD')
-1
exists w.p.l).
The least squares estimate of ( in the con-
ditional model is the left qXr minor of
A
r
explicitly given by (see
Grizzle and Allen [1969])
(4.1.3)
A
where S=X[l-A'(AA,)-lA]X'.
N(t, R
l
e
The conditional distribution of ( is
E2 ), where R is the rXr leading minor of (D'D)
l
-1
and
72
L =(B'L- l B)-1. The residual error sum of products matrix from fitting
2
the conditional model is S -Y (I-D'(DD,)-lD)Y'=(B'S-lB)-l and its conY 1
1
l
ditional distribution given D is W (N-r-p+q,L ) independent of f.
2
q
All these results follow from Rao [1965].
In the conditional model
2
(4.1.2), the use of generalized Tmax
procedures for simultaneous
inference on all comparisons among the r growth curves is straightforward.
Krishnaiah [1969] discusses the use of T2
procedures for
max
inference on a finite number of comparisons among growth curves based
on Rao's reduction of the original model to the conditional model with
covariables.
An alternative model for
X
X is the usual linear model
= (A+E
(4.1.4)
where (:pxr is a matrix of unknown parameters, A and E as in (4.1.1).
For simplicity, suppose that (4.1.4) is a one way layout, A is an
indicator type design matrix,
t=(~l' •• "~
~
~r
E(eiT)=§i-(~il), ••• ,~ip»,
where
) y=l, ••• ,N i , i=l, ••• ,r, and let us follow the discussion
in Koch [1969].
Two hypotheses of interest in this problem are the
hypothesis of no treatment effects Hot' say, and the hypothesis of no
condition effects H ,say, given respectively by
oc
Ho t:
~l=···=~
~
~r
and Hoc :
~i(l)=···=~i(P), V
(i=l, ••• ,r)
(4.1.5)
The appropriate procedure for testing either of these two hypotheses
depends on the assumptions we are willing to make regarding interaction between treatment and condition effects and the form of L.
e·
73
Case I.
Interaction of unknown form may exist between treat-
ments and conditions and nothing is known about
~.
For simultaneous
inference on all atomic component hypotheses of Hot (i.e.,
r
~'tc=O, ~ ~ERP, ccC ) we may use either Bose and Roy's Largest-Root
-
-
~
~
method or any generalized T2
procedure.
max
For simultaneous inference on all atomic components of H
oc
(1. e.,
!:' ~i' V SECP, i=l, ••. , r) we proceed as follows.
Firs t make the
transformation
u
- ty
=Cx
-iy
' y=l, ••• ,N i , i=l, ••• ,r,
(4.1.6)
where C:(p-l)xp is a matrix whose rows are linearly independent
contrasts.
Next, obtain Ti =
N
i=l, ••• ,r, SN=
_ (11
~i • -
Ni~i.(CSC,)-l~i.'
Li~l y=i(~iY-~i.)(~i -~i.)'
N.
where u =(l/N.)~ l~u. ,
-i •
~ Ly= -1y
and
=(
(1)
(p) )'
Ni ) ~Ly= il~iy
- Xi
. , ••• ,x.~. •
is the matrix of sum of squares and cross products associated with the
hypothesis
H~~): ~~l)= ••• =~~p),
i=l, •.• ,r, and
matrix (.).
ch~(.)
Shi independent ofSe=CSC' for all
denotes the
~
th
largest latent root of the
Following Roy's union-intersection principle, a simulta-
neous test procedure for the family of hypotheses
c ~Cp , .~-l
{Hoi'£i·c'~
·-iZi=0 ' uv -i~
- , ••• , r}
is to accept Hoc iff for all i, T
i
(4.1.7)
~ d(a) , where d(a) is defined by
the equation PH ( max {T } > d(a» = a. This is a special case of
o l<i<r i
testing quasi-independent hypotheses in a general linear (multivariate)
model discussed in Krishnaiah [1965].
(Quasi-independent hypotheses
are defined as hypotheses with corresponding independent sums of
74
max {T } is
l<i<r i
a "SMANOVA" test proposed by Krishnaiah in the more general setup.
squares and cross products matrices).
The test based on
Because of the difficulty involved in obtaining d(a) , Krishnaiah
finally proposes the use of another test.
For our problem, this
alternative test is given by the rule; accept H iff for all i=l, ••• ,r
oc
U
we have Vi = chl(Shi)/chp_l(Se) ~ dl ), where diu) is given by the
equation PH ( max {Vi} > d~U» = u.
o l<i<r
However, based on our discussion in 3.4, conservative SMANOVA
test and estimation procedures are easily obtained by utilizing
Khatri's inequality (see Theorem 1.2.5).
Alternatively, approximate
SMANOVA test and estimation procedures can be obtained from Siotani's
modified Bonferroni second order approximation to d(u) based on the
discussion in Siotani [1960].
Case II.
No interaction exists between treatments and condi-
tions, L is cOmpletelY unknown.
In this case we have the additive
model
t,;~j)=m+Ui+Bj'
r
I
(4.1.8)
ui=o,
Bj=O, j=l, ••• ,p, i=l, ••• ,r.
i=l
j=l
The hypothesis Hot: u1= ••• =u is tested as follows.
r
N
Obtain
(.)-(1/ )'\ p X(j)
(.) -- (1/ N )'\
i
(.)
-1
·-1
x iy
P lj=l iy , xi.
i lY=l x iy , y- , ••• ,N i , 1- , ••• ,r.
We have E(x(·»=m+U V(x(·»=1'L1 /N and C(X(·) x(·»=O for l<i<i'<r.
i .
i'
i.
-p -p
i
i. ' i'.
2
The unbiased estimator of l'Ll
-p -p is l'Sl
-p -p /(N-r) ~ XN-r •
--
Thus, any
generalized T procedure discussed in Chapter 2 can be used for
simultaneous inference on all contrasts in the ui's.
To test H we use a procedure as in Case I with one minor
oc
change, namely, the matrix CSC' is replaced by
e·
75
N
S*
u
m
Li -r l Ly- Ii
(u
-u
- i y - ..
)(u
-u
- i y - ..
distributed as W l(N-l,CrC').
p-
)' where u
= (l/N)L
I
~i
which is
Y
Thus, on letting T~ be defined in
i Y
terms of S~ as T is defined in terms of CSC', we have that T~ follows
i
a central Hotelling T2 distribution with p-l and N-l d.f. rather than
with p-l and N-r d.f. as the case is with T •
i
Case III. Treatment and condition effects are not assumed
additive, r is assumed to be uniform, i.e.,
~ =
a 2 [I+p(!!'-I)], -l/(p-l) < P ~ 1.
(4.1.9)
(This is the assumption of "compound synnnetry" of the error distribution in Koch [1969]'s terminology).
For simultaneous inference on the components of H one may
ot
use the same procedure as in case I.
We are not aware of any alterna-
tive procedure for simultaneous inference for this family of hypotheses which utilizes the information on
~
given by (4.1.9).
For simultaneous testing (estimating) the family of hypotheses
(4.1.7) (the family of corresponding linear parametric functions) we
now propose a conservative procedure based on Roy's union-intersection
principle.
To derive this procedure we utilize the results of
Bhargava and Srivastava [1973] together with inequality (2.0.4).
~i=(cil, ••. ,ciP)'ECP, i=l, ••• ,r, and put ak=l-(l-a)
11k'
Let
where
k'=k(k-l)/2 and k is an integer.
Lemma 4.1.1.
The probability is at least l-a that for all SiEeP,
i=l, ••• ,r, we simultaneously have
(4.1.10)
76
Proof.
Following Bhargava and Srivastava [1973] (where the one sample
case is discussed) we make the transformation
zi~) = xi~) - d xi~) ,
j=l, ..• ,p, y=l, ..• ,N , i-1, ... ,r, where d • 1+(l-P)~[1+(p-1)P]~.
i
This value of d makes all the
and
E(Z~~»
=
zi~)'s independent with a common variance
~ij)-(d/P)Lj~l~ij).
Note that for any contrast £ we have
c'(x i -E(xi »=c'(zi -E(zi », where zi =(Zi(l) ""'Zi(P»" and that
- - y
- y
- - y
- y
- y
y
Y
L.
.(z~j)-z(·)-z(j)+z(·»2 = L
(x~j)-x~')-x(j)+x~'»2, where
iy
1.
1.
i,y,j 1.y 1.y
1.
1..
1.,y,J 1.y
z~·), z(j) and z~·) are defined in terms of the z(j),s as x(·), x(j)
1.y
i.
1..
iy
iy
i.
and xi:) are defined in terms of the
xi~)'s. Thus, for a given
iE{l, .•• ,r} we have
(ex )
r } = (l_N)l/r
Pi = P{Ri -< N ~
i svqp,v
"
(4.1.11)
1
where R = R«xi:) _~i1) , ... , xi:) -~ip) )') and the func tion R(.) : JI'p-+R
i
is defined in 2.1.1.
Let Pi(sv) be the conditional (on sv) probability
of the event { .•• } in (4.1.11).
r
E( II
i=l
Pi(s
»
V
From Theorem (2.0.3) we get
r
r
>
II
i=l
E(Pi(s
IT p. =
» = i=l
V
1.
(4.1.12)
I-a
which together with the identity R = sup {ILC.(X(j)-~(j»I/~Llc.I)}
i
• J i.
i
j
J
CECP J
imply the lemma. Q.E.D.
Case IV.
Treatment and condition effects are additive, L is
uniform as in (4.1.9).
have
x~:) ~ N(m+a i ,
given by s
i
N
V
Here we
(a 2 +p(p-l»/(N P», i-l, ••• ,r, and clearly, the
xi(·),s are independent.
•
(We use the notation of (4.1.8).
An unbiased estimator of (a 2+p(p-l)/p is
- ~ r ~ i(x(')_x('»2/v where v-N-r.
li-lly=l iy
i.
'
The use of a
T procedure (in the balanced case) and generalized T procedures
e·
77
considered in this work (in the unbalanced case) for testing H is
ot
thus straightforward.
To test H we propose the same procedure as in Case III.
oc
We
are not aware of any way which utilizes the additional assumption of
no interaction to improve the procedure discussed in Case III for
testing this hypothesis.
4.?
Modified T-type procedures for inference on treatment x condition
interactions in a simple split plot design.
Recall the split-plot model, the terminology and notation in
4.1.
We now discuss simultaneous inference on treatment x condition
interactions based on T or T2
type procedures when L is completely
max
unknown (Case I) and when L is assumed to be uniform (Case 2).
We
r
p
r
use D:rxp, c =(c , ••• ,c ), and c =(c P, ... ,c )' as generic notations
-r
r
-p
p
1
1
for a matrix and vectors satisfying
r
D1 =0 l'D=O' and c EC
c EC P
-p -' -r - r ' -p
respectively.
We consider two families of parametric functions,
namely, the family
n
o
= {
and the family
(4.2.1)
no
of all atomic (interaction) parametric functions
2 di'~i(j):
V D=(d .. )
J
~
i,J
'
nl
(4.2.2)
satisfying (4.2.l)}
of all product-type (interaction) parametric
functions
n1 = {c'~c
: V c , c satisfying (4.2.l)}
-p"-r
-p-r
Case 1.
L is completely unknown.
considered in this case.
Only inference on
(4.2.3)
nl is
Recall the definitions of T(a)
and of
k,p,v
78
C(a)
from 1.1 and 1.2 respectively.
p,k,v
Lemma 4.2.1.
The probability is at least I-a that simultaneously for
all c
satisfying (4.2.1) c'tc-r falls in the interval
~p
and c
~r
~p
c Xc ±(C(a)
c'S c )~ \
-p ~r
p-l,r~v -p v-p i~P
I
c
je:N
c
-r
~r
where v=N-r, Sv=S/(N-r),
(4.2.4)
x=(x~~»
and
~ij=(Ni+Nj)/(NiNj).
Nl= ... =N ,c(a) ,
can be replaced by
r
p- l ,r ,v
If
~T(a)
1
making (4.2.4) an
r,p- ,v
exact I-a simultaneous estimation procedure.
sUP{~'Sh£/(£'Se£)}= sup {&'C~C'&/(&'CSeC,&)}gChl(S~s:-l)
(4.2.5)
l
&e:1P-
£e:CP
e·
where C:(p-l)Xp is as in (4.1.6) and S~ ~ Wp_l(vh,CLC') and
S* ~ W l(v ,CLC') are independent.
e
pe
From (4.2.5) and from the GT2
method in the multivariate case this lemma follows.
Case 2.
Q.E.D.
L is given by (4.1.9), cr 2 and p unknown.
first, and suppose Nl= ••• =Nr=n.
Consider Q
O
A conservative simultaneous estimation
procedure for Q is given by the following result.
O
Lenuna 4.2.2.
(")
The probability is at least l-a that all Li,jdij~iJ
generated by all D=(d
ij
) satisfying (4.2.1) will simultaneously satisfy
L d ~(j)e:[ L d x(j)±svn~
i ,j ij i i , j ij i.
qr(pa)"~iL,jld1'J.I]
v
where v and sv are as in (4.1.10).
Proof.
Obvious from the proof of Lemma 4.1.1.
(4.2.6)
79
For the unbalanced case any of the following three sets of confidence intervals procedures gives a conservative simultaneous interval
estimation procedure for nO.
are given by
\' d x(j)±s ( max {N })~ q«() 1, \' Id I
l
ij i
V
i
rp V~ l
ij·
i,j
•
l<i<r
' i,j
Conservative GT2 type intervals for
L. diJ.x(j)±s
Iml (~),
l
i.
V
r p,v . j
i ,J
~,
(j)
Ei,jdij~i
(4.2.7)
are given by
~
l d . j (-d i , J. , )1/I i J. i' J. , / fh .LId
.. j)
P
' ., N ~
. ~J
£ D i J £ D
~,J
(4.2.8)
where PD={ij:dij>O}, ND={ij:dij<O}, 1/Iiji'j,=(Ni +N i ,)/(N i Ni ,) and
r'=r(r-1)/2, p'=p(p-1)/2.
The GT3 type intervals for
(j)
Ei,jdij~i
are given by
(4.2.9)
Next we discuss inference on n
design.
1
starting with a balanced
Denote the common sample size by n, and let
~
be as in
4.1.
Lemma 4.2.3.
The probability is at least 1-() that either of the
following two families of statements will simultaneously (for all
r
p
c £C , c £C ) be true
-r
""p
«() )
r
£[c'Xc
±s n ~ ~"Z(c
,c )]
-p -r -p -r V
~,v
-r-p
I:c'~c
(4.2.10)
«() )
II:c'~c
£[c'Xc ±s n~q p ~(c ,c )]
-p -r -p -r V
r,v -r-p
(4.2.11)
80
Proof.
For any matrix Y:pxr={yij) of real elements and any real
scalar t the following equivalence holds (see Gabriel et al. [1973]).
M-
max
l'::i<j.::p
~
{I -p
c' Yc I <
-r -
We now identify Y with
i-~~J'{c-r ,c-p ), "'I c e:C P c e:C r }
p ' -r
x-t and
(4.2.l2)
derive two upper bounds for the I-a
quantile of the distribution of M.
Note that for a given i' and j',
Ri ,.,= max IY"-Y"'-Yij'+Yi,.,1 is distributed as the range of p
J l'::i<j.::p ~J ~ J
J
lID normal variates with a zero mean and variance 2cr 2 {1-p)/n. Using
e-
Theorem 2.0.2, we have
(a )
P{
max
{Ri'j'} <
l<i'<j'<r
(2/n)~ Svqp,~
) > I-a
From (4.2.12) and (4.2.l3) we obtain (4.2.l0) of Lemma 4.2.3.
second half of Lemma 4.2.3 is similarly proved.
(4.2.l3)
The
Q.E.D.
(a )
Table A.2 (see Appendix) gives q
(a)
and a.
(Note that the ratio q
~20 for some values of p,r
r'
~ does not
{a
~ /q r,v
p,v
(ex)
is evident from the table that q
depend on v).
It
(a)
:. /q ~
P,v
r,v
< 1 when p<r for all values
of p,r and a considered.
These procedures can be used in a crossed two way factorial
experiment.
The intervals obtained by (4.2.l0) or by (4.2.ll) are
always shorter than the intervals given by the "Bonferroni modification of Tukey's method" proposed by Gabriel et a1. [1973].
81
Finally in this section we discuss simultaneous inference on
nl
in an unbalanced setup (when E is as in (4.1.10».
The GT1 Method.
Use either (4.2.10) or (4.2.11) replacing n
by min{N }.
i
Let Y be the set of all ~ii,-(Ni+Ni,)/(NiNi')'
The GT2 Method.
l~i<i'~r,
then
(a. )
r
({c'tc £[c'Xc ±s q v K(c ,c ,'f)],
-p -r -p -r V p,
-r -p
V
p
r
c £C
c £C } > 1-0.
- p ' -r
(4.2.14)
where sv is as in (4.1.10) and
p
K(£
r
'£ ,')~
p
l Icil i£Pl
i=l
l
c
i'£N
-r
c
-r
Arguing as in the proof of (4.2.13) we get here
~
P (max
{~i' ,R. i'}
l<i<i'<r
~ ~
~
(a.r )
svqp ,v
(4.2.15)
) 2:,. l-a.
Thus to prove that (4.2.14) provides conservative confidence intervals
for the family
nl it is sufficient to show that for any matrix Y as in
(4.2.12) and scalars a
ii
,2:,.O,
l<i<i'~r,
we have
(4.2.16)
where.A. is related to the a
ii
, , s as V is related to the I/J
ii
, 's.
To
prove that (4.2.16) holds we write
c 'Yc
-p -r
=
e+f+g+h
(4.2.17)
82
where e = \L. ie:P
c
-r
We have the following straightforward equations
c~J= L (-c~,)~
j'e:N
i'e:N
L
i' e:N
(-c:,)
~
c
-r
L
j e:P
c j ' g by
c
-p
and dividing each by ~
(-c~,)
L
Thus on multiplying e by
r
L
~=l
c
-r
r
c.
L
ie:P
Ic~1
~
c
-r
L
J. 'e:N c
-p
L
j 'e:N
J
c
-p
r
~=l
(4.2.18)
Icil.
(-c~ ,) , f by
(-c~,), h by
c
-p
c~
L
ie:P
c
-r
L
j e:P
c
p
j
c
-p
P
L Icil
we obtain (after obvious re-
~=l
arrangements of terms)
c'Yc-r
-p
=4
ie:P
c
,je:P
c
-p
-r
Li 'e:N
,
c ' j 'e:N c
-p
-r
P
r
(Yji-Yji'-Yj'i+Yj'i')/(
L Ic~1 ~=l
L Icil)
~=l
From this we can easily verify that (4.2.16) holds and hence on
identifying y. with
Ji
x(j)-~(j)
1.
i
':>
and a
ii'
with s
V
q(~r)'I~
p, V 'l'ii'
we get that
(4.2.14) provides a procedure as conjectured.
The GT3 method.
The probability is at least
l-~
that all
product interaction contrasts simultaneously satisfy
where
Sv is as in (4.1.10). This can be proved using Theorem 2.0.2
and the main result in Bhargava and Srivastava [1973] discussed earlier.
e·
83
4.3
The multivariate multisample location problem.
In this section we discuss some alternative procedures to those
proposed by Gabriel and Sen in their [1968] paper for simultaneous
inference based on rank statistics in the multivariate, multisample
location problem.
Here we intensively use terminology from Gabriel
[1969a] which is not redefined here.
Terms borrowed from Gabriel
[1969a] are only underlined in this section and we refer the reader
to Gabriel's paper for definitions.
Consider
the N =
y
th
Li:1 n i
observation
k~2
independent samples of sizes nl, •.. ,n , each of
k
observations being made on p variables.
.
~n
the i
th
sample on the j
th
Denote the
(j)
variable by Yiy'
The
_( (1)
(p»,
Yiy , ••• ,Y iy
is denoted by Fi (l)' We assume
p
Fi(l) = F(l+~i)' ~i=(e~l) , ••• ,ei »" i=l, ••• ,k, where F(.) is an
CDF of
~iy-
unknown absolutely continuous CDF.
(j )
{ (j)
Yiy
Let R(j) R(j)
be the ranks of
iy' ii' ,y
(j ) }
among the ordered Yi1 , ••• ,y in
and
(j )
(j ) (j )
(j)
i
{Y i1 ""'Y in 'Y i ' ""'Yi'n.'} respectively.
i
~
corresponding rank scores
Ei~)
and
Eii~,y
Further define
for j=l, •.. ,p based on p
score generating functions as in Puri and Sen [1966).
denote the mean score of the i
th
Let T~~
~~ ~
sample on the jth variate in the
pooled ranking of the pair of samples i,i'.
and VN=(v
,), j,j'=l, ••• ,p, where uii)·O for all j=l, ••• ,p,
Njj
i=l, ••• ,k and
(4.3.1)
We use G and S as generic notations for a group of e out
e
a
of the k populations and a set of a out of the p variables.
Let
84
S
Ha
o,G e
be the hypothesis that specifies the equality of the location
vectors of the variables in S for all samples in G. Let n denote
S
a
e
a
the family of all H G and put N = Li£G n • The null hypothesis
G
i
S
0,
e
e
is H = H P . Let vjj'(a) denote the (j,j')th element of the inverse
o
o,G
N
k
of the principal minor of V corresponding to the variables in Sa.
N
Based on the Chatterjee and Sen permutational (conditional)
approach to the multivariate multisample location problem the statistic
\
nn,
\
u(j)u(j')v jj '(a)/2N
i i
L
ii' ii' N
i,i'cG
j,j'cS
(4.3.2)
L
e
a
is proposed by Gabriel and Sen [1968] as an appropriate conditionally
S
a
DF statistic for the hypothesis H G. These authors consider the
0, e
S
S
S
n} is a joint
f amily of hypotheses n~,and prove that {
H aG,L a : H a GC~'
o , e Ge
o 'e
monotone increasing testing family. Under some regularity conditions
S
S
a
a
they prove that under H G ' the asymptotic distribution of NL /N
o, e
Ge Ge
is a central chi-square with a(e-l) d.f.
The large sample STP of level a proposed by these authors for
the family of hypotheses
S
n is thus, to reject each H a iff
oG
S
L a > ~(k_l)[l-a], where xk[a] is the a th quantile of e a central chiG
e
square distribution with k d.f.
Gabriel and Sen also provide a simultaneous estimation procedure for all the location differences
A(j)' = e(j)-e(j)
i
i'
ii
u
j =1 , ••• ,p, 1~iT
~i' ~k
(4.3.3)
based on the well known 'sliding principle' as follows.
u~i~(a)
as
uii~
computed upon replacing each
y-l, ... ,n , i-l, ••• ,k, j-l, ••• ,p.
i
Let
y~~)
by
Define
y~~)-a,
e·
85
'" (j)
fl ii , ,U
... sup{a:u~i~(a) ~-[X(k_1)p[1-a]vNjj
1
N ]~}
nin i ,
(4.3.4)
'" (j)
flu' , L =
inf{a:u~l~ (a)
1
N
~ [X(k_1)p[1-a]v Njj nin , ]~}
i
They prove that the simultaneous confidence intervals
(4.3.5)
have an asymptotic confidence coefficient at least 1-a.
We now discuss some alternative testing and estimation proDenote by R(j) the rank of y~j) among the
iT,Y
1Y
(j)
(j)
(j)
ordered {Y11 ""'Yk~} and let EiT,y be the corresponding score
cedures for this problem.
determined by the jth score generating function.
of these scores for the i
th
sample.
T~j)
be the mean
1
Let
Let
(. ')
k
ni ( )
k
n i ( ')
E
J
/N-[
L
L
E
j
/N][
L
L E j /N]
VNjj ' =
i=l y=l iT,Y iT,y
i=l y=1 iT,y
i=l y=l iT,y
k
T
n i ()
L LE j
and define the matrix
(4.3.6)
V~ = (V~jj')' l~, j'~. Under some regularity
conditions (cL Puri and Sen [1971] Theorem 5.4.2) the probability
limits of V and of V~ are the same.
N
We denote this limit by V=(v
Thus, we have two consistent estimators of V and we use VN=(V
a generic notation for either of these two.
'"
"'(a)
V as v~J
N
",-\j'
is related to V and put ~
N
set of all p variates.
_( (1)
(1)
,).
,) as
Let v~j'(a) be related to
"'()
for v~J
a
(p),
_( (1)
when Sa is the
Let T..
,- Tii" ••• ,T 1.,) , T.T.1
-11
-1
i
l~i<i'~k, ~ii'=ti-ti,=(W~i~,···,wi~~)'
Njj
jj
(p»'
, .•• ,T.1
and £ii'=!ii'-!i'i=
(p) ,
( Uii' , .•. ,Uii') •
The approximate large sample dispersion matrices of the tits
\
and the tii' 's is now recorded for reference in the sequel (for
,
86
•
details see Puri and Sen [1971] Ch. 5 and Ch. 6).
v
iat, i'=t'
(4.3.7)
ni,n t ,
v
ninii,nit'
i-tj i,i',t' different
O:pxp
i,i' ,t,t' all different
(4.3.8)
where 0ii,=l if i=i' and 0ii'=O otherwise.
A
p
Hereafter we assume that V ~ V and that the asymptotic distriN
bution of the Ii's and of the Iii,'s is normal.
(See Theorems 5.4.2
and 5.5.1 in Puri and Sen [1971] and the proof of Theorem 3.1 in
Gabriel and Sen [1968] for the appropriate regularity conditions).
e·
For simultaneously testing the family of hypotheses Q we now
propose 'the following large sample procedures.
Let ~(a) and ~(a) be
pr
pt
defined by the equations:
p
p
N-l nin i ,
P (max
{l l w(j ) w(j , ) ~ j , } < ~(a»
pt
Ho l<i<i'<k N n ii , j=l j'=l ii' ii' N
-
Procedure 1.
= I-a
(4.3.10)
-
S
H
a
Reject
o,G
iff
e
f
S
nii ,-1 nin i ,
a
U (j ) u (j , ) Qj j , (a) } > ~(a)
{
max
(4.3.11)
Tl G •
pr
j,j
£S
ii' ii' N
nU'
' e
(i"i')e:G e nU'
a
e
87
S
Procedure 2.
Reject H a
o,G
iff
e
S
a
T2 G
, e
(4.3.12)
Lemma 4.3.1.
Based on the Chatterjee and Sen permutational (con-
ditional) approach to the multivariate k-sample location problem,
under the permutational conditional probability measure, we have
S
S
i) the family {HO~Ge' Tl~Ge}' when
joint monotone testing family.
VNjj ,
is taken as
VNjj
"
is a
(See Gabriel [1969a] for definitions).
ii) procedure 2 provides a coherent testing procedure for the family
of hypotheses Q.
Proof.
i) This follows along the line of argument used in Gabriel
S
S
a
and Sen [1968] to prove that {H a ' L } is a joint monotone testing
G
G
0, e
e
family.
ii) From the lemma given in the appendix of Gabriel and Sen
[1968] we get that if Sb
~
Sa' then
(4.3.13)
Thus, clearly, for any Gd
~
Ge and Sb
~
Sa we get
S
S
b
a
T
> T
2 , Gd - 2 'G
e
which proves the coherence of procedure 2.
(4.3.14)
Q.E.D.
Note that both procedures are not always consonant in their
decisions since the statistics involved are not strictly monotone.
However, procedure 1 is an STP and thus possesses the feature of
having a as the upper bound for the probability of rejecting at
88
S
least one true H a
in Q.
o,Ge
Procedure 2 does not possess this feature.
It does however satisfy the basic requirement from a multiple compari-
•
son test, namely, that the probability of rejecting at least one
hypothesis in Q is at most a when Ho is true.
Lemma 4.3.2.
i) If all sample sizes are equal, then both ~(a) and ~(a)
pe
pr
asymptotically converge to
~T(a)
p,k,oo • ii) For the general case, an
upper bound to both ~(a) and ~(a) is given by X [(l-a)l/k'], where
p~
pr '
p
k'=k(k-l)/2.
Proof.
i) Follows from (4.3.8) and (4.3.9).
ii) Follows from
Q.E.D.
Theorem 1.2.3.
S
Procedure 3.
Sa
T3 G max
, e (i;i')EG
Reject H a
iff
o,Ge
e
max
jES
e.
{"~
V
a
I (j )I}> (N-l)
N ~ (l-(l-a)l/p)
~
qk 00
/ [ min {nJ]
Njj ' W",
11
,
l<i<k 1
(4.3.15)
S
Procedure 4.
Reject H a
o,G
iff
e
(4.3.16)
S
Procedure 5.
Reject H a
o,G
iff
e
n n , L n -1 L
( i i )~( ii )~ max{v~ ,lu(j)I}>lml (a)
n ii ,
n ii
'ES
Njj
ii'
k'p,oo
J
a
If nl·n2· ••• ·~·n we may use the following.
(4.3.17)
89
S
Procedure 6.
Reject H a
o,G
max
(i~i' )£G
{ vA~
max
j£S
e
iff
e
Njj
,
lu(j)I}>( 2 )!~ (l-(l"(X)l/p)
,
- - q ex
ii
2n-1
k, l
(C L18)
a
Lemma 4.3.3.
The asymptotic probability of rejecting ultc;:;sc
hypothesis in
n,
OI,E'
under H , is at most a. for any of l:he procedl·.res
a
'j
4, 5, and 6.
Proof.
For any Sand G we have
Procedure 3.
a
e
.
where R(! (j»
denotes the range of coordinates in T (j) .-'
.1)
(
• ",
..
'f']."
k.
Thus it is sufficient to show that asymptoticaHy
.
l/p (N 1:1/. . " ) l.ry
(1-(1-a.)
) N-1) [Hi',l' \'. ) I
( max { vA~° ° R(T(J).1.<
_
')5-Qk 00
l<'<k l
Ho 1~2P NJ J ,
1,
P
To show that this holds, let us write (for Luge samples based on
(4.3.8»
(4.3.21)
where each of the vectors ~O'~l""'~k is
N(Q. V),
the ~i'
(i=l, ••. ,k) are independent and
The resulting dispersion matrix of (~O,~i""'~k) 1s p.s.d. and thus
~
(4.3.21) is valid.
We now note that on letting ~i""(N/[(N-l)ni]) ~i'
~i'" ( qU~· .. ,qip ) , , i
a
1, ... ,k and ~ (j) =(q1j, .. ·,Qkj!\
'
we may write
90
(4.3.20) asymptotically as
(4.3.23)
To show that (4.3.23) is at least I-a we use
where n=min{nil.
Theorem 2.0.2 and the upper bound on the distribution of the range
The proof of Lemma 4.3.3 for procedures
involved in the GTI method.
4 and 5 is relatively simple and thus omitted.
The proof of Lemma
4.3.3 for procedure 6 follows on verifying that the correlation
structure
0f
(j) 's for any j --1 , ••• ,p 1S
. asymptotically the
t h e k ' Uii'
same as that of the k' pairwise differences among k independent unit
normal variables and then following the same line of argument used
in proving Lemma 4.3.3 for procedure 3.
Lemma 4.3.4.
Q.E.D.
For the family of hypotheses
n we
4, 5 and 6 are all coherent and consonant.
VNjj
form STP's when
Proof.
is taken as
V
Njj
have i) Procedures 3,
ii) Procedures 5 and 6
•
Similar to the proof of Lemma 4.3.1.
S
The testing family {H a
Lemma 4.3.5.
S
S
a
' T
1 is strictly narrower
G
l
0, e
'G
e
S
than {H a
' LGal.
0, Ge
e
Proof.
Let
no
S
be the set of all H a
0, Ge
S
a pair of samples.
For all H a
0, G
e
En0
En
such that G contains only
we have
Sa
2 Sa
LG • k Tl G
e
S
and for all H a
o,G
e
En-n0
e
' e
we have with probability one
(4.3.24)
91
S
L a>
G
e
2
k
Sa
T
(4.3.25)
1 G
' e
S
The proper inequality of (4.3.25) holds also for H =H P .
o o,G k
Corollary 4.3.5.
Q.E.D.
Procedure 1 is strictly more resolvent than Gabriel
and Sen's [1968] STP.
Proof.
Follows from Theorem 5 in Gabriel [1969a].
contains the set
n00
of all atomic hypotheses in
Note that
n.
no
Gabriel [1969a]
restricted the discussion on resolution, to resolution over the set
of atomic hypotheses.
However, when comparing two non-consonant STP's
such a restricted definition of resolution, although simpler may be
misleading.
In our case Procedure 1 is more resolvent than the STP of
Gabriel and Sen [1968] over the class n which properly contains n
o
unless p-1, in which case
no =n00
00
Exact comparisons among STPs in
terms of resolution (in the sense of Gabriel [1969a]) are possible
only if they are of the same level.
The rest of the STPs considered
above are all conservative and their exact levels unknown.
We now discuss the asymptotic estimation procedures based on
the above test statistics and the sliding principle.
(j)
Define Uii' (a)
as in (4.3.4) let
(4.3.26)
(j)
(j)
bii"L(~)=inf{a:uii,(a)
<
n ii ,
n ii ,
~
~[vNjj n ,-1 nin , ] }
ii
i
and put
(4.3.27)
92
Lemma 4.3.6.
~
(l-(l-a)
.. qk ,00
i) In the balanced case, on letting
1/
p)
intervals for all
~2 ~ T(a~
p, ,
..
00
~
or
' (4.3.27) provides asymptotic simultaneous confidence
~ii~'s with a confidence coefficient at least I-a.
ii) In the unbalanced case the analogous result holds with either
or ~ ..
Proof.
ImIk'p,oo
(a)
•
We prove ii), the proof for i) is similar.
Under any admissible
l<i<i'<k
hypothesis the common distribution of the {U(j)(~(j»
ii' ii' , -'
(j)
j=l, .•• ,p} is the same as that of {U ii "
For both choices of
~,
l<i<i'<k,
j=l, ••• ,p} under H •
o
it is true under any admissible hypothesis that
asymptotically
(4.3.28)
This follows from the Appendix to Gabriel and Sen [1968] and our
earlier discussion in this section.
u~l~(a)
From (4.3.28) and the fact that
is non-decreasing with a, Lemma 4.3.6 follows.
Lemma 4.3.7.
Q.E.D.
In the balanced case the confidence intervals given by
(4.3.27) and Lemma 4.3.6 are shorter than those obtained by Gabriel
and Sen as given by (4.3.5).
Proof.
From Theorem 5' in Gabriel [1969a] and the proof of Lemma
4.3.5, Lemma 4.3.7 follows.
4.4
Q.E.D.
Analogeous procedures in some different problems with different
statistics.
The slippage problem.
(Refer to David [1971] Ch. 8).
Consider
the general setup (1.1.1) and the problem of testing the hypothesis of
e.
93
slippage, namely; Ho : 2l··.·=~k versus the alternative HI
where Hli is the hypothesis that all
is different from the other k-l
~j
~j
but
~i
k
= Ui=l Hli
are equal and that
Qi
's in any unspecified direction.
When the model is diagonal and uniform, the statistic proposed by
- - max {(~i-~
) '5 -1 (~i-~
)},
l<i<k
v
~.=Ii~l~i/k. The test is: reject Ho iff-P>b(~), accept H1i if
P = (8.-8 )'5- 1 (6 -6 ) where b(~) is a constant fixed by letting the
Siotani [1959a] is proportional to p.
-1 - .
V
-i-.
level of the test =~.
Siotani considers approximations to b(~)
based on his method introduced in 1.2.
If (1.1.1) is a general full
rank model not necessarily balanced one may use procedures similar to
the GTI or the GT2 type procedures.
In any general slippage setup,
the given matrix B may be majorized to any uniform matrix and thus,
similar to our earlier discussions, an upper bound on b(~) obtained.
A GT2 procedure may be used by utilizing (in a straightforward
manner) Theorem 1.2.5.
Similar remarks about the use of the
techniques introduced in this work
with the aid of" the pairwise
comparison statistics, in different problems and statistics can further
be illustrated in the following areas.
i) The problem of 'many-one'
comparisons, i.e., treatments versus control, for which a procedure
exists only in the uniform diagonal case (see Miller, Ch. 2).
ii) The
problems of bounding a regression surface and that of simultaneously
predicting several points in a linear regression based on the distribution of the studentized maximum modulus are discussed in Hahn [1972].
His methods are in analogy with our GT2 method.
It is obvious, however,
how similar procedures to our GTl and GT3 methods may be used in these
problems.
CHAPTER V
SOME EXAMPLES, EVALUATION AND PROPOSALS
FOR FUTURE RELATED RESEARCH
5.0
Introduction
In this chapter we exemplify the methods introduced in this
work, study some comparisons among the different procedures, and
discuss some
prospe~ts
of further related research.
In section 5.1, the GT1, the GT2, the GT3, the MGT1 and the
MGT(1,3) are exemplified in some PBIB(2) designs.
In section 5.2
e-
we give some examples of optimal GT1 type procedures, i.e., least
conservative procedures based on a majorization of the matrix B to a
balanced matrix.
Section 5.3 contains some summary remarks con-
cerned with the evaluation of the procedures introduced in this work.
The last section, 5.4, contains some proposals for future research
in problems related to those considered in this work.
5.1
Some examples
In this section we demonstrate the use of the procedures
studied in this work, in some PBIB(2) designs.
Consider a design with k treatments and b blocks, the i
treatment being tested in r
plots.
th
Let n
ij
i
plots and the j
be the number of times the i
th
th
th
block consisting of d
treatment appears in
Put R-Diag(r 1 , ••• ,rk ), D-Diag(d1' ••• '~)' N-(nij ),
th
th
and let Yijm denote the observation in the m plot of the j
block
the j
block.
\
j
95
for the i
th
treatment.
Clearly, Yijm is defined only when it exists.
We assume
Yijrn
= ~+8i+8j+eijm'
(all i,j,m such that Yijm is defined)
(5.1.1)
where the 8 's and the Sj'S are the treatment and block effects
i
(assumed fixed) respectively and the eijm's are lID normal with a
2
zero mean and variance O.
total and the j
th
Let t i and
~j
denote the i th treatment
block total respectively and put
(5.1.2)
From the normal equations for this design it is not difficult
to obtain the equations in terms of
~,
having eliminated
~
and the
8. 's from the equations. These equations, usually referred to as
J
the adjusted intrablock equations for treatments, are given by (see
John [1971], Ch. 11)
(5.1.3)
Usually, attention is confined to connected designs i.e.,
designs where the rank of A is k-1 and that of D-N'R-~ is b-1.
In
such cases all contrasts in treatments as well as in block effects
are estimable.
The estimator s~ is the residual mean sum of squares
based on N-b-k+l=v d.f.
Consider a proper (all di's are equal to d,
say) equireplicated (all rils are equal to r, say) and binary (all
nij's are either zero or one) incomplete block design.
A relation among the treatments termed as a partially
balanced association scheme with two classes is now defined.
96
1.
Any two treatments are either first or second associates, the
relation being symmetric.
2.
Each treatment has n
3.
If treatments a and
that are both j
independent of
th
first associates and n
l
e are
i
th
second associates.
associates, the number of treatments
associates of a and t
a,e.
2
th
associates of
e is
i
Pjt
A proper, equireplicated, binary block
design with that association scheme and such that all pairs of
treatments which are i
th
associates appear together in Ai blocks,
i=1,2, is called a Partially Balanced Incomplete Block design with
two associate classes and is abbreviated as PBIB(2}.
One possible solution to the normal equations (5.l.3) in this
case, is due to C. R. Rao (see John [1971], Ch. l2).
matrix which we denote by B=(b
ij
Rao's solution
} is given as follows.
e·
Put
(5.l.4)
then B is given by
-1
-1
= c22d~r ; b ij = -dc12~r if (i,j) are first
associates and bij=O otherwise
(5.l.5)
The PBIB(2} designs constitute a subclass of all connected incomplete
block designs.
It is clear from (5.l.5) that PBIB(2) designs belong in the
class of semi-uniform
designs (see 3.2 for a definition).
Thus,
based on our discussion in 3.2 we may use a GTI procedure instead of
an EGTI procedure for this class of problems.
97
The PBlB(2) designs have a structure particularly suitable
for the application of the MGTI procedure which was introduced in 3.3.
We now consider the use of the MGTI method in PBlB(2) designs.
. th
he J
y
Jy
th
of treatment indices that are simultaneously Y
refer to 3.3 for terminology and notation.
class of all sets
Thus,
WI
Let n: , y=1,2, be
jy
associates of each other such that the number of sets in
n:J
t
is the
y
smallest possible under the following restrictions:
(i) The number of elements in ,,)(
Wt
jy
Y the same
is L.,
J
for all t, L: being decreasing with j for both
J
values of y.
(ii) No two sets
?ill.
(5.1.6)
exist in which the same two
Jy
indices of treatments simultaneously appear.
We form the sets ~I. ={(a,S): a<S; aE~I. , SEVI. } for all t. , jy
J
and y.
Jy
y
Jy
Jy
Lemma 5.1.1
(1)
The sets
iiI.Jy are all disjoint and their union gives
the set {(a,S): l<a<S<k}.
(2)
For any (a,S), (a',S') both elements in~. we have
Jy
(1)
Follows from (5.1.6).
(2)
Follows from (5.1.6) and the structure of B in PBlB(2)
Proof.
designs.
Q.E.D.
y
In all the well-known PBlB(Z) designs, on letting e
j i denote
y'
th
the number of the sets in nY in which the i
treatment appears, we
jy
98
find that eY
• eY
- ej, say, V(l~i<i'<k). We then get the
jy,i
jy,i'
y
following values for a and b in (3.3.3):
a - b+I Iej [l+(Lj-2)~(W-by)]lw-byl, b-w
(5.1.7)
Y\
where b
=d
-1
-1
c 22 6 r ' b 1 • -d c 12 6 r ' b 2-O as given by (5.1.5).
An
optimal MGT1 procedure is the one based on the w that minimizes a-b
in (3.3.3).
It is easily verified that the only stationary values of
ware
(i) wI
= max(b l ,b 2)
(ii) w2
= min(b 1 ,b 2)
(iii) Any
value in the interval [w ,w 2 ] if both wI and w
2
l
(5.1. 8)
produce the same value for a-b.
Thus, an optimal MGT1 procedure in PBIB(2) designs is easy to obtain
as we now illustrate.
Group divisible PBIB(2) designs.
Suppose that the k=mn treatments are
divided into m groups, each of size n.
Treatments within the same
group are first associates, treatments in different groups are second
associates.
In this case, the matrix B of (5.1.5) is given by
B
= Im ~
U , U = (b-b )I + b 1 l'
n
n
1 n
l -n-n
In this example there is, for each y, only one class of
(5.1. 9)
n:J y 's
of sets ~I which we denote by nY. It is easily verified that in
y
this design on letting ei =e , we have e l -1 and e =n. We get
2
y y
w* - max(b ,b ) and for the use of (5.1.7) we get a-b • b-b l , if
1 2
e·
99
The Latin Square Scheme.
There are k.n 2 treatments arranged in a
square array, two treatments are first associates if they appear in
the same row or column, otherwise they are second associates.
It
can be verified that the matrix B in this scheme has the form
B • I n @ (An -Dn ) + 1-n-n
l'
~
Dn ,
where
(5.1.10)
In this case, again, we have only one class
2n sets of treatments, each set of size n.
~l
which includes
A typical set is the set
of treatments in the same row or the same column of the square
involved in this scheme.
Similarly, only one class ~2 exists with
sets ~~
of n elements in each. Thus, the number of sets in ~2, x,
2
n
n2
say, satisfies; (Jcl-2n) (2) = (2)·
EXAMPLE.
Let n=3, thus we have the square
1
2
3
4
5
6
7
8
9
(5.1.11)
which gives the association scheme as indicated above.
We get
~l
= {(1,2,3),(4,5,6),(7,8,9),(1,4,7),(2,5,8),(3,6,9)}
Q2
= {(1,5,9),(2,4,9),(3,4,8),(1,6,8),(2,6,7),(3,5,7)}
(5.1.12)
For this design we have e =2 and e 2=n-l and it is easily verified
l
that w*
= max(b l ,b 2 )
in any case.
Thus if bl>O we have
a-b • b+(n-2)b l and if bl<O we have a-b
= b-2b 1 •
100
The Triangular Scheme.
There are k=n(n-l)/2 treatments.
The associa-
tion scheme is obtained by writing the treatments as the elements
above the diagonal of a square array of side n, leaving the diagonal
blank, and repeating the treatments below the diagonal in order to
Thus, if n-4 for example we have
obtain a symmetric array.
*
1
2
3
1
*
4
5
2
4
*
6
3
5
6
*
(5.1.13)
Treatments appearing in the same row or column are first associates.
For this example
n1
= {(1,2,3),(1,4,5),(2.4,6),(3,5,6)};
n2
= {(1,6),(2.5),(3.4)}
(5.1.14)
In
n2 ,
each group is of size m where n=2m or n=2m+1.
The number
of groups in n2 , y say, satisfies
(5.1.15)
We get
e2
= 2ym/[n(n-l)]
The class n 1 contains n groups (of size n-l each) and e =2.
1
(5.1.16)
For
n<6 for example we get a-b • b+2lbll in any case.
We now demonstrate the GT3 procedure in LFR models.
Specifically, we illustrate the use of Theorem 3.1.2 in a group
divisible PBIB(2) design.
We use the solution matrix B given by
(5.1.5) for a general PBIB(2) design and by (5.1.9) for the group
e·
101
divisible case.
such that
QBQ~ =
We look for a matrix QB' of the same form of B and
B.
Thus, on letting
(5.1.17)
for some scalars gB' f B•
We get the equations
(5.1.18)
These equations are easily solved and always possess a solution
since B as given by (5.1.9) is p.d.
With that choice of QB' Theorem
3.1.2 can be used and the resulting confidence intervals for the
pairwise contrasts are given by
are first associates
(5.1.19)
if (i,j) are second
associates
The fact that such an approach cannot always be used is best
illustrated by the Latin Square design where it is easily verified
that no Q matrix of the same structure as that of B given by
B
(5.1.10) exists.
QBQ~
In such cases alternative Q matrices satisfying
B
must be found, or a different procedure used.
For this design we may use the MGT(1,3) procedure introduced
~
in 3.3.
~
~
In this case, the matrix B=Im @ Pn , where Pn
b <0, is a majorization of B.
1
= vI n+b l (l
1'-1n ),
~n~n
~
A matrix Q of the same structure as
~
that of B exists (as discussed above when considering the GT3 method
in a GD PBIB(2) design).
(3.3.2) may be used.
In this case, the second part of Lemma
102
We now give a numerical example comparing all the methods in a
GD PBIB(2) design.
EXAMPLE 5.5.1.
Suppose we have a Group Divisible design for
six treatments, indicated by the numbers 1, 2, 3, 4, 5, 6, with groups
[1,2], [3,4], [5,6] arranged in six blocks as follows
[1,2,3]; [3,4,5]; [2,5,6]; [1,2,4]; [3,4,6]; [1,5,6]
(5.1.20)
The matrix B in this case is given by (5.1.9) where
U = [21/48
3/48
3/48 ]
(5.1.21)
21/48
First we consider the GT3 method.
We obtain a matrix Q as
B
in (5.1.17) where
.65974
P
B
=
[ .04737
.04737 ]
(5.1. 22)
.65974
From (5.1.19) we get (here v=7 and we take a=.05)
_ _
{ 3.30s v if (i,j) are first associates ]
8i -8j£ 8 i -8. ±
J
3.81s if (i,j) are second associates
v
~
Using the GT1 method (based on our discussion in this section)
we get
(.05)
Using the conservative GT2 method, obtaining Im115,7 from Hahn
and Hendricson [1971], we get
e·
103
[_.
8 -8 £ 8 -8 ±
i j
i j
{ 3.51sV if (i,j) are first associates
]
3.79s\l i f (i,j) are second associates
Using Scheffe's S-method we get
[_
_
8 -8 £ 8 -8 ±
i j
i j
{3.86SV i f (i,j) are first associates
]
4.l7s\l i f (i,j) are second associates
Thus, in this example, it seems that the ranking of the methods would
be GT3, GT2, GTl, S.
5.2
Examples of a direct approach for obtaining the shortest confidence intervals in the GTI method.
The efficiency of the GTI procedure (shortness of the con-
fidence intervals) in semi-uniform models (refer to 3.2 for a
definition) depends on the majorization method used to majorize the
given semi-uniform matrix B to a uniform matrix.
The most efficient
GTI procedure is given by (3.1.1) when a,b are such that a-b is
minimized subject to the constraint
(5.2.1)
where c~{'} is the smallest characteristic root of {.}.
Depending on the form of B such a minimization is manageable,
difficult or impossible to
accomplish.
This approach is referred to
as the direct approach (for obtaining the optimal GTl procedure).
In
this section we use the direct approach to obtain optimal GTI procedures
in some designs where this is feasible and compare with the GTI procedures based on the majorization methods of 2.2 and 3.3.
104
The direct approach is feasible in the PB1B(2) designs of the
Group Divisible and the Latin Square type which we now discuss.
The direct approach in a PB1B(2) GD design.
this design is given by (5.1.9).
The matrix B in
We get
(5.2.2)
where A = (a-b-A)1 + (b-b )(1 1'-1 ), E = bl 1'.
l -n-n n
n
n
n
-n-n
The
characteristic equation is given by
(This can be verified by using Theorem 8.9.1 in Graybill [1969]).
From (5.2.3) we get the following three different values of the
e·
characteristic roots
(5.2.4)
From (5.2.4) it can be verified that if b
l
< 0 and we choose
A =A =0 we get A > 0 (if A2 = 0 we have Al < 0).
2
l 3
(5.2.4), i f b l > 0
A =A =0
2 3
~Al
> 0 (Al=O qA
2
Similarly, from
< 0).
a=b-bl+bl/m, b=bl/m, a-bab-b l
This gives
}
(5.2.5)
a-b+(n-1)b , b=O, a-b=b+(n-l)b
l
1
Comparing with the results obtained in 5.1 by using the MGT1 method,
we find that in this particular example the MGTI approach actually
produces the best possible GTl procedure.
The direct approach in a PB1B(2) Latin-Square design.
matrix B=B
k
for this design is given by (5.1.10), we get
The
105
aI +b(l 1'-1 )-B -AI = I
-k-k k
k
k
n
k
~
A*+(l 1'-1 )
-n-n n
~
D
(5.2.6)
where
A* = (a-b-A)1
n
+ (b-b ) (1 1'-1), D • (b-b )1 + b(l'l'-1 ).
1 -n-n
-n-n n
1 n
Again, utilizing Theorem 8.9.1 in Graybill [1969] we get the
characteristic equation
2
(a-b-b+Zb
-A) (n-1) (a-b-b-(n-2)b
-A) 2 (n-1)
1
1
(5.Z.7)
2
(a-b+(n
-1)b-Z(n-1)b -A)=O
1
We get the latent roots
A = a-b-b+2b ,
1
1
A
3
=
AZ = a-b-b-(n-2)b ,
1
a-b+(n 2-1)b-2(n-1)b
l
of multiplicities (n-l)2, 2(n-1) and 1 respectively.
(5.2.8)
Going through
considerations as above we get here
a=b+b-2b , b=O, a-b=b-2b
1
1
a=~+b1/n+(n-Z)bl' b=b 1 /n, a-b-b+(n-2)b 1
(5.2.9)
}
Comparing with the results of the MGT1 method in this case (refer
to 5.1) we again find that the MGT1 is the optimal GT1 procedure in
the PBIB(2) Latin-Square design.
Another situation where an optimal GT1 is easy to obtain is
, ••• ,b ) + b*(!!'-1). In this case also the results
kk
1l
from the direct approach coincide with our earlier results, i.e.,
when B=Diag(b
a
=
max(b
11
, ••• ,b
kk
) and b=b*.
Next we discuss the application of the direct approach to
Example 2.2.1 i.e., a one way ANOCVA with one covariab1e.
The matrix
106
B in this case is given by (2.2.13).
d=l/n.
We assume nl= ••• =~=n and put
We get
aI+b(!!'-I)-B • (a-d)I+b(!!'-I)-bb' - A-bb'
(5.2.10)
From Theorem 8.9.3 in Graybill [1969] we obtain the characteristic
equation
(5.2.11)
where A* = A-AI.
After some algebraic manipulations we bring (5.2.11)
to the form
k
2
\
(a-d-A-b) k-2 {(a-d-A+(k-l)b)-[ \l b.(a-d-A+(k-2)b)-b
l b.b.]}=O
i=l 1.
i;'j 1. J
We find that k-2 of the latent roots have the value A=a-b-d which is
non-negative iff a-b>d.
To obtain the other two latent roots we
divide the characteristic equation by (a_d_A_b)k-2 and we get (skipping
some laborious but straightforward calculations)
k
2(a-d)+(k-2)b-
l:
b~ ±
i=l
When b~ we have in the limit min(A l ,A 2 ) = a-d-b-l:i~l b~/2 and thus,
on choosing
a-b=d+l:i~l b~ /2 we have obtained the optimal GTI pro-
cedure for this case.
This example is instructive for cases when the direct approach
is not feasible and one is willing to do some numerical work to
search for a 'good' GTI procedure.
If b is fixed we may look for an
a-a(b) such that (5.2.1) is satisfied and a-b is minimized by letting
e-
107
a=a(b).
We denote by (a,b) any pair satisfying (5.2.1) and call it
a solution.
If by (a(b), b) we denote a pair satisfying (5.2.1) and
minimizing a-b for b fixed, we say that (a(b), b) is a conditionally
optimal solution.
Lemma 5.2.1.
If (a,b) is a solution, then any conditionally optimal
solution (a(b*), b*) for which b*
~
b satisfies
a(b*)-b*
~
a-b
Proof.
have
(5.2.15)
The matrix (a-b)I + (b+b*-b)!!' - B is p.s.d., hence we must
a(b*)~a-b+b*
Corollary 5.2.1.
which implies the lemma.
Q.E.D.
If (a,b) is the optimal solution, then any con-
ditionally optimal solution (a(b*), b*) such that b* > b satisfies
a(b*)-b* = a-b.
Thus to find a 'good' GTI procedure in problems where a direct
approach is not feasible we may choose a 'very' large b and then
search for the appropriate a(b).
Finally, in this section we note that in some cases, a direct
approach to finding the best majorization of B to a uniform diagonal
matrix may be feasible, even where such an approach is not feasible
when considering majorization of B to a general uniform non-diagonal
matrix.
However, such an approach might be very poor in some cases.
As an illustration consider the use of such an approach in our Example
2.2.1.
Suppose n =n = ••• =n =n.
k
l 2
Based on Theorem 8.5.3 in Graybill
[1969] it can be verified that this approach produces:
a-b - lIn +
Ii~l b~. Note that from the original direct approach we
have obtained a-b - lIn +
~Ii:l b~.
108
5.3
Some evaluations of the different procedures.
In this section we evaluate the performance of the methods
introduced earlier mainly from two points of view, namely, shortness
of confidence intervals and convenience in actual use.
Alternative
criteria may be introduced, some of which are discussed in the
sequel.
We will compare the GT2 with the S-method and the GT2 with
the GTI method.
Having compared these two pairs of methods, a direct
comparison between the GTI and the S-methods seems redundant.
Algebraically convenient expressions for practical comparisons of the
GT3 with any of the other methods cannot be obtained explicitly in
non-diagonal unbalanced models.
In view of this fact and the existing
comparison of this method with the S-method in the general diagonal
model given by
Spj~tvoll
and Stoline [1973], we do not discuss a
detailed comparison of the GT3 with any other method here.
First we discuss a comparison between the GT2 and the S methods.
Consider a general univariate normal setup as (1.1.1) where
t e = Ck .
Let R ,S(£) be the ratio of the length of the confidence interval for
2
£'~
obtained by the GT2 method to that obtained by the S-method.
Let
~
k
-
(I:J) be the set of all contrasts c such that c is a comparison
-
of the average of some I means with the average of some other J means,
I+J < k.
We now have the following result.
(Recall our notation from
1.1 and from 2.4).
Lemma 5.4.1.
If all off-diagonal elements in B are non-positive, then
•
(a)
~
on letting w == t , ,v/«k-l)Fk_l,V) , we have
k
wI~
< 1
~R2,S (£)
< 1, V £ECk(I:I)
(5.3.1)
e-
109
For any
Proof.
k
c~(
RZ S(S)
,
sstk (1:1)
For any
c'Bc
we have
=w
I
(5.3.2)
i~P
c
we have
=
It thus follows from (5.3.2) and our assumption that
I
I
R2 ,S(£)
< wI ~
-
i~Pc
[I
i~P
'"
c
-
j~N£
L
j~N
(bii+b. j-2b ij )~
J
(b' i +bj
c
~
<
~
.-2b .. ) ]
1:
wI 2
(5.3.3)
~J
J
The rightmost inequality in (5.4.3) follows from the Cauchy-Schwartz
inequality.
Thus the lemma follows.
Q.E.D.
Note that we always have w < 1 (this is so since t 2 ,
is the
k ,v
I-a quantile of sup {S'(8-8)(8-8)'c/(s2 c 'Bc)} where t P is the set of
p
- - - - v- !:~$.!
( )
k' pairwise
contrasts; (k-l)Fk~l,v is the l-a quantile of
sup {c'(8-8)(6-8)'c/(s2 c 'Bc)}) hence, for any pairwise contrast c we
k -
c~C
have:
- -
- -
R s(£)<l.
Z,
-
v- -
Thus the GTZ method preserves the well-known
advantage of the T method over the S method in any unbalanced design •
In this sense we say that the GT2 is a robust procedure.
.
Since t ,
k ,v
is usually unknown we may be willing to use an upper bound, for
example
Imlk(~),v • Some values of ~ = Iml k(~),v /[(k-l)F(a)
]~ are given
k-l,v
in Table A.3 (see the Appendix).
Clearly,
wdecreases
sharply with k.
Since values of Iml (a) are tabulated only for limited values of m
m,v
(m
~
15, usually),
the efficiency of the GT2 method with respect to
110
the S method cannot be demonstrated by a tabulation of
wfor
large
values of k'.
Next we compare the GTI and GT2 methods.
Let
Rl,2(~)
the ratio of the length of the confidence interval for
£'~
denote
obtained
when using the GTI procedure to that obtained when using the GT2 procedure.
We can always write down an expression for R ,2(S) in terms
l
of the appropriate functions of B and the majorization procedure
involved by the GTl method.
However, in order to get more explicit
expressions for practical use we restrict ourselves to the diagonal
case.
In this case on letting v = q(a)/t(~) we have
k,v k,v
(5.3.4)
e·
Let
v=
q(a)/Iml(~)
•
k,v
k,v
the Appendix.
Some values of
2~/V
are given in Table A.l in
From this table it is clear that the GTI procedure
may give shorter confidence intervals than the GT2 method only if the
variation among the bii's is low, i.e., there is a low degree of unbalance in the design.
To get some notion of what is meant by 'low'
here, we present the following example.
Consider a one-way layout with four groups.
Let n , i=1,2,3,4
i
be the four sample sizes, and to further simplify put n =n =n =n and
l 2 3
n
4
< n.
trasts.
We want to set confidence intervals on the six pairwise conThe following table gives the lowest possible value of n
4
for some given values of n such that (column 1) all pairwise comparisons are better (shorter confidence intervals) estimated by the
GTI method than by the GT2 method, and such that (column 2) three
111
contrasts are better estimated by the GT1 method, the other three
being better estimated by the GT2 method.
We used 0.=.05, and rounded
the resulting values of "mini.ma1 n " to the nearest upper integer.
4
11
(1)
minimal n
(2)
4
minimal n
20
19
18
30
29
27
40
38
36
60
57
54
4
The table above provides some justification for our following
step, namely, carrying out a detailed comparison between these two
methods only in a restricted setup where all but one of the bii's are
equal.
Thus, suppose we have a diagonal model such that k-1 of the
bii's are equal to b , say, and one of the bii's has a different
1
value,
£2' say.
From (5.3.4) it follows that if
~1 > ~2
we have
(5. 3.5)
(5.3.6)
The basic features of the various procedures may now be
summarized, in crude terms, as follows.
1.
Both the GT1 and the GT3 procedures (including the MGT1 and
the MGT(1,3»
in the design.
are not robust to the degree of unbalance involved
These procedures may be expected to produce satisfactory
112
results only in cases when the matrix B is close (using an
appropriate metric) to a uniform matrix.
2.
The GT2 procedure is robust to the degree of unbalance
in the design.
It always gives shorter confidence intervals than
the S-method for all pairwise contrasts.
3.
The main limitation of the GT2 procedure lies in the
fact that t(a) in (2.4.1) has to be approximated and, unfortunately,
m,V
tables of Iml(a) are not yet available for large values of m, as
m,V
often required by the conservative GT2 procedure.
However, this
limitation will be resolved to a large degree when more detailed
tables for Iml (a) become available.
m,V
(This is a reasonable prospect
based on the results of this work and other works in simultaneous
inference, see for example Hahn and Hendricson [1971]).
It is,
however, clear from Table A.3 that in designs which are more than
'moderately' unbalanced and k=3,usually the S method will be
preferred over any of the methods discussed in this work.
servative GT2 procedure based on
Iml~~~v
The con-
gives shorter confidence
intervals than the S method only for k > 4.
4.
Among the three procedures considered in this work, the
GT2 is the only one which always preserves the order of the lengths
of confidence intervals as obtained in a non-simultaneous approach,
for all pairwise comparisons.
It is easily verified that the GTI
procedure never preserves this order and the GT3 procedure preserves
it in some cases (for example, the diagonal case) but not always.
5.4
Proposals for future research.
This study raises a number of questions inviting further
e·
113
investigation.
First, recall the desirability of resolving Duncan's
conjecture regarding Kramer's procedures recorded in 1.2.
There
are grounds to expect that research in this direction may produce
more efficient generalized T-type procedures, at least in some cases.
Second, note that for simultaneous inference on a finite number of
functions of the response vector in a general multivariate normal
setup (Section 3.5) a multiplication type inequality can not be used.
Rather, in order to obtain conservative procedures, the first order
Bonferroni approximation was used.
This brings us to the well known
and, as yet, unresolved problem of whether the multiplication
inequality holds also in the multivariate student distribution.
problem is discussed but not fully resolved in Sidak [1971].
This
A more
general form of this problem (subsequently discussed) is very relevant
to our work.
Recall the procedures obtained in 4.3 for the multi-
variate multisample location problem based on upper bounds on the
large sample distribution of the maximum variatewise ranges.
In the
derivation of these procedures we have utilized a more general
multiplication type inequality as given by Theorem 2.0.2.
In the
analogous multivariate (studentized) normal version of this problem,
the theory required to assess whether such procedures are conservative is not, as yet, available.
Various other conjectures about inequalities on probabilities
in multivariate normal and related multivariate distributions have
appeared in the literature.
Such inequalities could be used (if
proved to be right) to construct conservative union-intersection
procedures (with finite intersections) for some problems which have
not been discussed here and, perhaps, to improve some of those
114
procedures which have been studied.
For example, we refer the reader
..
..,
to Khatri [1970] and SidAk f1973].
In discussing T-type procedures for interactions in a two way
layout the following problem arise.
Recall the union-intersection
type statistic introduced in 4.2. namely, maxl<i<o<r{Rio}, and suppose
_
for simplicity that 0 is known.
J_
J
The upper quantiles of this statistic
can always be approximated by the Siotani's method introduced in 1.2.
However, this method requires knowledge of the upper quantiles of
statistics of the form max(R(~), R(X»
where (~'l)=«xl'Yl)'.• "
(x ,y »' is a simple random sample of size p from the bivariate
p
p
normal distribution with mean (0,0)' and dispersion matrix L
L
=
!.:
1
(i;<)
~
1
or
L= (
1
~
~)
1
(5.4.1)
This suggests investigation of the bivariate distribution of the
variatewise ranges in samples from bivariate normal distributions.
If
0
2
is unknown, but an independent estimator is available,
the problem is to derive the distribution of the studentized analogs
of the above standardized statistics.
e·
116
The Ratio 2~lmlk(;05)/qk(·05)
Table A.l:
~
,v
,v
3
4
5
6
3
1.06
1.08
1.10
1.11
4
1.05
1.07
1.08
1.09
5
1.04
1.06
1.08
1.08
6
1.04
1.06
1.07
1.07
7
1.04
1.05
1.06
1.07
8
1.04
1.05
1.06
1.06
9
1.03
1.05
1.06
1.06
10
1.03
1.05
1.05
1.06
15
1.03
1.04
1.05
1.05
20
1.03
1.04
1.04
1.04
25
1.02
1.03
1.04
1.04
30
1.02
1.03
1.04
1.04
40
1.02
1.03
1.03
1.04
60
1.02
1.03
1.03
1.03
120
1.02
1.03
1.03
1.03
00
1.02
1.03
1.03
1.03
This ratio (relevant to our discussion in 5.3) slightly
decreases for smaller a's and slightly increases for
larger values of a.
e.
117
(ex
Table A.2:
X
ex=.l
0.=.05
(*)
) (*)
Values of qp, 120
r'
2
3
4
5
6
8
9
2
*
3.06
3.42
3.55
3.75
3.99
4.30
3
2.94
*
3.97
4.17
4.39
4.68
4.77
4
3.29
3.90
*
4.50
4.69
4.98
5.05
5
3.55
4.11
4.45
*
4.90
5.19
5.25
6
3.75
4.29
4.63
4.85
*
5.30
5.40
8
3.99
4.57
4.88
5.10
5.29
*
5.63
9
4.10
4.67
4.98
5.19
5.38
5.62
*
2
*
3.43
3.79
4.02
4.23
4.50
4.61
3
3.37
*
4.30
4.50
4.70
4.95
5.07
4
3.70
4.26
*
4.80
4.98
5.22
5.34
5
3.93
4.48
4.79
*
5.18
5.42
5.53
6
4.10
4.63
4.95
5.17
*
5.58
5.68
8
4.37
4.89
5.19
5.40
5.57
*
5.91
9
4.48
4.99
5.29
5.50
5.67
5.90
*
r'=r(r-1)/2; ex ,=l-(l-ex)
r
l/r'
•
This table is relevant to the discussion in 4.2.
118
Table A.3:
Values of
Imlk(;05)/[(k-1)Fk(:015)]~
,v
,v
.'>z
3
4
5
6
5
1.033
0.995
0.959
0.927
10
1.038
0.990
0.951
0.917
15
1.038
0.989
0.947
0.913
20
1.042
0.988
0.946
0.910
30
1.043
0.987
0.944
0.908
40
1.044
0.987
0.943
0.907
60
1.045
0.987
0.942
0.906
This table is relevant to our discussion in 5.3.
119
(a )
Table A.4:
Values of
[qk,~ ]2/ 2
**
(*)
,Xp [(l-a)l/k']
a=.l
~
[q
(a )
p ]2/2
k,oo
and kX (k_l) [I-a]
p
a=.05
2
6
10
6
10
2
1.98
2.89
3.23
2.26
3.16
3.47
3
2.16
3.04
3.38
2.44
3.29
3.59
4
2.26
3.15
3.47
2.55
3.38
3.68
5
2.37
3.22
3.54
2.62
3.40
3.75
8
2.55
3.39
3.68
2.83
3.61
3.32
10
2.62
3.43
3.75
2.89
3.68
3.96
2
2.15
3.15
3.48
2.45
3.37
3.68
3
2.50
3.48
3.80
2.79
3.69
3.99
4
2.79
3.75
4.06
3.08
3.97
4.27
5
3.04
3.99
4.32
3.33
4.20
4.51
8
3.66
4.59
4.89
3.94
4.79
5.08
10
3.99
4.92
5.23
4.28
5.12
5.41
2
2.15
6.93
11.39
2.45
7.41
12.02
3
2.50
8.18
13.55
2.79
8.66
14.16
4
2.79
9.23
15.36
3.08
9.71
15.97
5
3.04
10.16
16.96
3.33
10.63
17.56
8
3.66
12.47
20.95
3.94
12.93
21.54
10
3.99
13.77
23.19
4.28
14.23
23.78
2
X, [(I-a) l/k' ]
p
k~(k_1)[1-a]
(*) a
p
= l-(l-a)l/ p • (**) k' = k(k-1)/2
This table is relevant to the discussion in 4.3.
120
Table A.5:
(a)
I
(aIm)
(A p ,k-1,k(n-1) (k-1)Fk _ 1 ,k(n_1»
~
p=2
p=4
p=6
p=8
p=10
4
6
8
10
10
1.06
1.05
1.03
1.02
30
1.10
1.08
1.06
1.05
50
1.15
1.12
1.09
1.07
10
1.12
1.10
1.09
1.07
30
1.16
1.14
1.12
1.09
50
1.19
1.17
1.15
1.12
10
1.21
1.19
1.17
1.16
30
1.25
1.22
1.19
1.18
50
1.29
1.25
1.22
1.20
10
1.36
1.32
1.30
1.28
30
1.42
1.37
1.33
1. 31
50
1.47
1.41
1.36
1.33
10
1.54
1.49
1.46
1.41
30
1.63
1.56
1.52
1.47
50
1.72
1.63
1.59
1.53
This table is relevant to our discussion
in 3.5.
~
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