UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. C.
SOME RECENT RESULTS IN NORMAL
MULTIVARIATE CONFIDENCE BOUNDS
by
s.
N. Roy
August, 1961
Contract No. AF 49(638)-213
This paper gives a survey of some recent results, most
of them available in print and the rest still unpublished,
in confidence bounds on parametric f'unctions associated
with multivariate normal distributions, that have been developed with a particular end in view. The general philosophy is discussed in greater detail than heretofore, and
most of the results obtained so far are exhibited in that
light, the buJk of the details of the various derivations
being left out. Each parametric f'unction that is considered is interesting either by itself or as a measure
of departure from some (customary) null hypothesis in the
direction of some alternative that may be either standard
or non-standard. Among the results discussed are those
connected with one or two population dispersion matrices,
multivariate linear hypotheses under a linear model, independence between two sets of variates , multiple independence among a p-set or among k sets of Pl,P2, .•• ,Pk'
and the vector analogues of the ratio of Deans and ratio
of variances for a correlated bivariate normal distribution.
Institute of Statistics
Mimeograph Series No. 303
SOME RECENT RESULTS
IN NORMAL MULTIVARIATE CONFIDENCE BOUNDS*
by
S. N. Roy
Department of Statistics
University of North Carolina
1.
Introduction.
Confidence bounds on parametric f'unctions asso-
ciated with Llultivariate nOI'lilal distributions, set up with a parti.cular objective and under a :particular :philoso:phy, have been discussed in a series of :publications beginning in 1953 and continuing
to date.
Partly for the lack of any semi-expository article that
brings together most of the main results and highlights the characteristic features and the
in~lications,
and :partly for other reasons,
the whole :point and :pllr!'ose of this line of work seeD to have been
com:pletely lost u:pon the general statistical :public so far.
The
:present pa:per is intended, in sQ@e measure, to some as such an expository :paper and, in some measure, to offer certain very recent
results not yet available in print.
Intervals based on (H , H). In many, though by no means in
o
all, problems we start from a pair (H ' H) of composite hypothesis
o
and alternative, disjoint but not necessarily exhaustive, and seek
1.1.
a parametric f'unction that might be regarded as a measure of de-
*This research 't-laS supported by the Uni.ted States Air Force through the
Air Force Office of Scientific Research of the Air Research and Development Command, under Contract No. PlF 49( 638 )-213. Reproduction in whole
or in part is permitted for any purpose of the United States Government.
2
'e
parture
frOl~l
H
o
in the direction of
H, or, alterns.tively, SOLle
kind of a "distance function II betvleen the set
H and the uet
o
H.
In such probleLlS we next seek a confidence interval for this pa.ranetric function, one-sided (one way or the other) or two-sided:, depending upon the nature of the pair (H ' H).
o
No claim is mde at
this stage that the parametric function chosen or the confidence
interval proposed for it is, in some sense, optimal.
As to the
confidence coefficient, it would no doubt be very desirable if I
given any permissible
1 - a, the interval could be defined
that this coefficient v;ras equal to
1 - cx.
su(~h
If it does not turn
out that it;ray, the next best thing would be to have a confidence
coefficient
~
1 - a, given any permissible a, such that the
equality is attained, or, in other words, the probability of the
interval covering this parametric function, for some value of this
function, is equal to
1 - a,
If this does not happen (or, at any
rate, cannot be readily proven), then the next best thing vlould be,
for
any per.rlissible
1 -
ex, to have a confidence coefficient whose
greatest lovler bound > 1 - a (and might, in fact, be greater than
1 -
ex), provided that the interval itself is not trivial, for exam-
ple,
these.
(0,
00)
or
(-00, 00),
etc., but is, in fact, much better than
We shall say tP.at such a confidence coefficient is a conser-
vative one, or, alternatively, that the corresponding confidence
region is a conservative one.
For
really complex problems even
this may be difficult to obtain, to say nothing of intervals of the
first or the second kind, and we would consider even this quite worthwhile, especially in view of the fact that we consider it far more
important to
estimate this IIdistance f'unction ll point wise or inter-
3
val wise, than
to test (and acce:pt or reject) the customary null
hypothesis as such.
paper, that
All confidence intervals discussed in this
emerge as the end products for different problems (as
distinct from the confidence regions that occur at the intermediate
stages of the derivation) are conservative in the sense just indicated.
While we still stay with (H , H) there are three other feao
tures that merit special notice.
1.1.1.
Bounds on the Upartials U.
In most complex problems invol-
ving (H , H) there are reasons, set forth partly in T6, 107 but
o
L
more fully in r17, 187, why we prefer to look upon H
-
-
0
as the
intersection of a number of more primitive (coLlposite) hyJ;>otheses
and H as either the union or the intersection of the same number of corresponding and more :primitive (composite) alternatives.
In symbols this may be expressed as
(1)
Ho
=
n
i €
where
(HOi' Hi)
HOi' and H
=
r
U
i €
or
Hi
r
n
i €
Hi
r
form a natural pair and the subscript
i
belongs
to an index set that might even be a part of the continuum.
iug to this v~y of looking at
(H, H), H.
o
o~
and H.
~
(for
Accordi
'Will be called the partials (or cOJl1I>onents) of Hand H.
o
€
r)
For
any such cOJl1I>onent or partial pair (II i' H.) we would be naturally
o
~
interested in a corresponding Lleasure of departure from H.
o~
the direction of
Hi
or a
bounds on it, that goes with
in
"distance functionl~ with appropriate
(H., H.).
o~
~
The question is whether
it is possible to prOVide, vTith a joint conservative confidence
coefficient
> 1 - ex, simultaneous confidence bounds on the "total
4
distance function ll that goes with
(H, H) and on the different
o
llpartial or component distance functions I! that go ",ith the different
(Hoi' Hi)'s.
For the cases discussed in this paper not only has
this been possible, but the bounds on the "partial or component
parametric (distance) functions" are seen to vary 1'11th the components and furthermore, for certain classes of components, the
bounds are also sharper than for the lItotal paranetric function ll ,
which is a very satisf'ying feature wherever it occurs.
This will
be amply illustrated in the following pages.
1.1.2.
The nature of the interval estimation in terms of the per-
centage points of the distribution function used.
The parametric
or the socalled "distance function" set up in the following pages
as also the bounds on it is, in every cuse, such that the percentage points used are those of a distribution function associated
with the customary null hypothesis, and never those of a distribution
function associated VTith the non null hypothesis.
Not only is the
latter kind of distribution functions (mostlY called non-central
distributions), in general, more complicated than the former type,
but in multivariate problems of the types considered these are so
complicated as to be practically unusable.
(
ll
However, paranetric
distance tl ) functions that require for confidence interval esti-
mation the percentage points of the "non-central distributions II
are also, physically, equally meaningful.
For purposes of our
present work the only reason they have not been used is because of
the complexity of the distribution used.
Moreover, as one little
illustration would indicate, they are by no means more meaningful
or significant than the ones being actually used.
Consider, for
5
exaJl1..ple" the parameter p ( correlation coefficient) a.nd 13 (the
regression coefficient of" say"
popula.tion.
y
or x) for a bivariate normal
While both are equally meaningful and useful" it is
well knO'wn that to put bounds on {3
tribution of r
put bounds on
one needs only the central dis-
(the sample correlation coefficient) whereas" to
p" one needs the non-central distribution of r,
which is more complicated but not intractable as in the multivariate
problems discussed here.
For
p the clam is made that" unlike 13,
it is scale invariant, and much is made of that by many people.
How¢ver" my
Ovill
experience vdth nonparametric generalizations of
multivariate analysis has more or less convinced me that in nonparametric situations the analogues of the regression line or lines
are far more meaningful than the analogue of a single measure like
p"
and that is Why I believe that even for the bivariate normal
population 13, in that it builds up the regression line" is at
least as meaningful as
variance or lack of it.
p, notvrlthstanding the issue of scale inGiven the scales in a particular situation,
13 builds up the appropriate regression line. The parametric functions actually used in the following pages should be looked at from
this vie-wpoint.
1.1.3.
(Ho , H).
Additional requirement beyond What is suggested by the pair
In problems based on a pair (li0 , H) it happens qUite often
that while a one-sided interval (vrlth an upper or a lower bound)
for the parametric function is indicated in a natural manner by the
pair
(no , n),
there are other physical considerations that seem to
6
call for one step further to complete the one-sided into a two-sided
interval, by adj oining a l01rer or an upper bound to an upper or a
levIer bound.
In other words, while a pair (H , H) night set in moo
tion the search for a suitable parametric function it may not al'Ways completely determine whether the confidence interval for this
function is to be one-sided or two-sided.
1. 2.
Intervals not prima facie based on any (H , H).
o
Starting
from an appropriate test procedure for any (H ' H) ~re can, under
o
certain broad conditions, obtain a suitable confidence statement
by "inversion tr •
Such statetlents (on some kind of a "distance
function") are almost al'WaYs meaningfUl, though we may not al'Ways happen to be interested in such stateraents.
seems to be, to some extent, a one--way affair.
Hovrever, this
In other \-Tords,
there are situations where a confidence statement of a certain
nature seems to be of direct physical interest in its own right
and can also be obtained without any conscious attempt to associate it with a pair (Ho ' H), although, with some effort" a pair
(Ho " H) to tag it on to can be al'WaYs obtained, which , in many
cases" may not be too interesting or meaningful in itself.
Some-
times the confidence statetlents to be looked for are suggested by
an Ho and its decomposition without explicit introduction of an
H or any deliberate attempt to link the statements 'With a pair
(H , H) and its decomposition.
o
subsections 1.1.1. and 1.1. 2.
Host of the other rerJarks
moo
in
'Will also apply to the intervals
under the heading of this section, and hence such remarks need not
be repeated here.
7
2.
Bounds related to the dispersion matrix
2::
of an N(S,
!:)
pxJ.
f8,
10, lJ:.7.
lJ.'he case belongs to the category of section 1.2.
rather than to thai:; of section 1.1.
s ,
matrix
pxp
With a salll];lle dispersion
let us start from the statement
pxp
at
a
S
a
f!: a -< '"/1. (for all non null a) ,
2
with a joint probability 1 - ex, such that
'"/1.
1
and
'"/1.
2
are
based on the joint distribution of the largest and smallest root
of
a
S !:-l (where 2::
is, in fact, the population disp~rsion matrix).
is a vector of weights to the different variates, and it is to
px1
be noted that (2) is eventually invariant under the choice of this
weight system.
(3)
Let us rewrite this as
at S a
1/'"/1.
2
at a
=
at !: a
at a
<
-
1/'"/1.
1
a' S a
a ' a (for all non null a).
We are, of course, interested in bounds on (a' !: a)/{a'a) for all
non null a, that is, for all values of the weight system a.
Notice
that, for any a, the actual bounds on (a' 2:: a)/{ata) and the width
of the interval depend, among other things, on a
as well.
Among
the infinity of (a I 2:: a) / (a' a ) (for varying a), we are especially
interested in the subset fCh (2::),
max
~
i it)
i'i l
»'
~
i
i')
cll (2::>-7, fCh (2::~~~), ch (2::H~17,
min
max
min
7, and so on (i ~ i' ~ ••• = 1,2, .•• ,p);
min
'
max
'
or, in other words, in the largest and smallest roots of the total
~ch (!:
ch (!: ~.~
.' Of»
-
matrix, and in the largest and snallest roots of the partial Ira trices
obtained by cutting out any one variate, any two variates, and so on.
With a corresponding notation for the partial sample dispersion ma-
8
trices it is shown in
that (3) implies the 2P- 1
f8, 10, 117
pairs of bounds
(4) (i)
1
/1.
ch
I:1aX
2
(S) < ch
(E)
max
ch in(S)
ill
(ii)
<
1
< chnun
, (E) < ~
- ' v1
(i)
chmax(S{i»
( (i)
-< chmax E(,»
J.
Ch~oln(S)
;
.UJ..I.
1
( (i»
< ~
' v1 chmax S(')'
J.
-
(1
= 1,
(i ~ i
l
2, ••. , p-1);
= 1,
and so on until one bas cut out any (p - 1) variates,
2, ••• ,
p -
1);
He have thus
a simultaneous confidence statement ,nth a joint (conservative) confidence coefficient
~
1 -
0:.
Notice here that the largest and
smallest rOots of E are, each, given one pair of bounds, the
largest and smallest roots of any partial are, each, given another
pair of tighter bounds, and the bounds get tighter and tighter as
we cut out more and more variates.
desirable feature.
It is obvious that this is a
It may be further remarked that (ii) and (iii)
do not follow directly from (i) but that all of them follow from
(3) by specializing the vector a.
Another renark is also in order,
The upper bound in the first line of (i) and the lower bound in the
9
second line of (i) are autoluatically the upper and lower bounds
of all roots of E and also of the partials.
But the point is
that (ii), (iii), etc. are providing much tighter bounds to the
partials than these.
These features 'Will be repeated again and
again in the other cases discussed in the following pages, and
hence 'Will not be discussed in as much detail as in this first
case.
Another s et of confidence intervals (for a different set
of parametric functions) based on a different principle of decomposition has been given in
here.
f'27,
As a very special case of (4) my be mentioned that of
~, (,.2), the two-sided bounds on (j2
P=l, 'lThere we have, for N(
~l
(5 )
where
x2_
but this 'Will not be discussed
~l
s
2
2
~ (j ~ ~2
and
~2
s
2
,
are obtained fron the percentage points of the
distribution 'With appropriate
d.f.
3· Bounds related to the dispersion matrices (El , E2)Of N(Sl' E1 ),
N(~2' E )·
2
VIe shall discuss the problem under two heads, namely,
confidence statements in the category of section 1.2., not explicitly
based on any (Eo, E), and those in the category of section 1.1, based
explicitly on pairs of (Eo, H)
3.1.
Intervals not based explicitly on any (H , H).
a
He start from
an analogue of (3) 'With an exact probability 1 - ex £8, 9, 10, 11_7,
and obtain, rlith a joint (conservative) confidence coefficient
~
1 - ex, simultaneous bounds on
Chm;El E2 -"!:.7,
r
(i,i')
chmin _ E1(i,if)
ChJZl E2 -"!:.7 ;
(1,1,)-1
E2 {i,i,)-7;
etc.
10
Denoting the corresponding sample dispersion matrices by 8
1
8
2
and
(as in section 2), the simultaneous bounds on the whole set,
total and partials,
pact form
2P- 1
(i ,i , •.. ,
1
(6)
in number, can be expressed in the com-
2
"-I chmin 181(il,i2,···,
]
'\'There
il
< i 2 < ••• < i (0
PI
~ PI ~ P - 1)
is any oidered subset from
(1, 2, .•. , p) and the right hand upper and lower subscripts indicate
that the corresponding rows and columns have been cut out.
Notice
that the bounds as also the width of the interval depend upon
(iI' i 2 , •.• , i ), getting tighter and tighter as we increase Pl'
PI
.
Another point worth mention is that, unlike the case Oil' section
2, here we don't have separate pairs of bounds for the largest and
the smallest root.
we obtain for
(7)
Al
As a very special case of (6), putting p = 1,
N(~i~ CTi)
(i
= 1,
2 2
2 2
sl/s2 ~ CTl /CT2 ~ ""2
2), the confidence interval
2/ 2
sl s 2
'
11
where
~l
(central)
and
~2
are based on the percentage points of' the
F-distribution with al?l?ropriate
d.f..
For another
set of confidence statements (on a different set of parametric
fu...'1ctions) based on a different principle of decomposition, the
reader is referred to 1'2.7.
,.2.
Intervals based explicitly on pairs of (H , H) 1l§.7.
a
sider an H ~
o
Con-
I: l = I: 2 as expressed in the form. of a decOtlposition
and three different alternatives, given respectively by
starting from suitable analogues of (3) (different for the three
different pairs (H , H(i»
o
i
= 1,
2, 3», vdth an exact probability
1 - a, we have, with a joint (conservative) confidence coefficient
~
1 -
a, the following three sets of confidence intervals for the
three cases 1l§.7.
(i , ... , i
(10)
.l
Pl )
ch.
;-I:1C'
)
mn J . , .... , i
l
Pl
> ~l
(iI' ... , i p )
ch.
mn -;-S1 ( i , ..• , i I )
l
PI
12
(iI' ... , i
(11)
chmax
LL. l (.
~l'
PI
... , ~. )
PI
(iI'
< A2 ch
max
(12)
(10)
U
)
... , i PI )
LSl (.
i
~l"'"
(ll),
(iI'
... ,
i
l
r
PI
S2(i ,· .. , i ) ]
l
PI
)
PI
with different values for
,
Al and A ,
2
The right hand side upper and lower subscripts have the same interpretaticn as in subsection 3.1.
H(i)
For other interesting alternatives
the reader is referred to Ll§.7.
order.
Two other connnents are in
First, for the very special case of
analogues of (7) appropriate
p
= 1,
vre easily obtain
to (10)" (ll), and (12).
Second,
applying tm principle of section 1.1.3., we can extend (10) or (11)
and obtain a set of two-sided bounds of the form (6).
4.
Linear hypothesis under a nmltivariate linear model.
to the category of section 1.1.
here is called model I of
This belongs
For simplicity, the model we consider
PJ10VA and MANOVA.
This, together wi th-
the associated tests and confidence bounds is discussed in flO, 11,
14, 15_7.
For Model II and mixed models for MANOVA (with the tests
and the associated confidence intervals) the reader is referred to
L15,
I§}.
X :
L~
pxn
1
The model I for MANOVA
x2 .•. xn-7 p
1
vectors aucl1 that
1
consists of
is the following.
n
independently distributed
13
r
N
-
'& (X')
(ii)
=
~
meters"
_7
= 1"
(i
2" ••• " n) ,
A
~
nxm mxr
nx.P
where
~
PXJ?
lIZ
if) (x.)"
J.
~
stands for expectation"
unkno~m para-
is a matrix of
A is a matrix of given constants (that might include some
observed concomitant variables as well) depending upon the design
and the model" of rank
C
~
s:xm
IllX.P
r < m < n - p.
= o
U
pxu
Under this model we set up
against
H:
C
sxu
~
ufO
where
C and U·· are given mtrices of constants of ranks
u
and where it is furthermore assumed that
p
~
=ikf~7 =
bility of
Ho
Q
and
error
L14,
19_7 as the condition for testa-
in the strong sense.
Let us denote by
~"
o
*
the symmetric positive serni-definite matrices
Q__
li
o
given in terms of
(A" C)"
A and
(A, C) in
L10,
11" l4" l'2.7.
They are of course available in other forms as well from the extensive literature of ANOVA" and they might be conveniently called
the matrices due to the hypothesis" the error and the alternative.
As distinct from these let us define
(15)
e
SH
o
uxu
= ~s -rU'
Serror
uxu
UXJ?
=
1
n-r
X
Q
H
Xl
U _7,
0
pm
fut
UXJ?
nxn
X
pm
nx.P
pxu
Qerror
nxn
Xl
nx.P
U _7,
pxu
, say),
s < rand
nV '
milk
which is characterized in
(~
sxu
14
where the left side matrices are called respectively the dispersion
matrices due to the hypothesis and, to the error.
Starting from a
statement analogoUD to (3) (except that it is one-sided) vr.tth an
exact probability 1 -
Q:,
L9,
it is possible
10, 14,
1.27
vr.tth a joint (conservative) confidence coefficient ~ 1 -
r'rt
l/2
Chmax
-
taneous confidence bounds on
'I
J
to put,
Q:,
simul-
Q*-l
Tl7 and on partials
H ·....
o
obtained by cutting out (i) one or more columns of
T}
more rows of
(i. e., one or
T) t)
more columns of
and
T) t)
(ii)
one or more rows of
T}
(
i . e., one or
together with the corresponding ro"iV's (and
the same columns of the s;ynunetric matrix
* '
%
which, as can be
o
checked fron Llo, 11, 14, 15_7 means cutting out one or more rows
of C) •
If, as in many problems,
C
stands for a set of treatment
contrasts, then this means cutting out one or more contrasts.
cutting out one or more columns of'
cutting out one or more
C01UInnS
T}
(i. e ., rows of
T) t)
Also,
means
of U, i.e., certain linear combi-
nations out of the total set of linear combinations of the variates
in which we are interested, or , in case U is an identity matriX,
tial of
H
o
in the direction of its alternative which is its comple-
15
ment.
To indicate the nature of these simultaneous confidence bounds
we first note that the exact probability statement analogous to (3)
which is our starting point in this case implies
<
where
~
l
ch / 2
max
is obtained
(8
H
o
)
+
~
Chl / 2
max
(8
)
error'
from the percentage point of the distribution
of the central largest root
f!:J:.7.
But the analog of' (3) also implies all the partials, and :bile l7hOJ..e
u
s
set of partials together with (:16)" (2 _ 1 x 2 _1) in number, can be
expressed compactly as
16
(0 :5 u :5 u - 1) is any ordered subset from (1, 2, ... , u) and
l
il < i
2
< .•• < is (0:5 sl
(1, 2, .•• , s).
~ s -
1)
is any ordered subset from
1
For the symmetric matrices
SHO' Serror and
*
~o'
the right hand super and subscripts indicate that the corresponding
rows (and the sane columns) have been cut out.
For the matrix
T}
(in general rectangular and asymmetriC) the right hand super and
subscripts indicate that the corresponding rows and columns (not
of course now
and
Inatching
the ro't'1s) have been cut out.
So far as
T}
*
QH
are concerned the relationship of the partials to the
o
columns of U and rows of C has been already explained. So
far as
~o
and
Serror are c0D-cerpe.cl tlle riaht band
subscripts go with just the corresponding columns of U.
~t1.l?e:r ~nd
For the
symmetric matrix SH ' the left hand subscripts indicate that the
o
corresponding rows of C llave been cut out, resulting in a change
in the structure (or composition) of
~
that does not alter its
o
order and finally in a change in the structure of
does not alter its order.
SH that also
o
For the actual fonns of these partials
(and the total) involved in (9), especially the forms of the parametric "distance" :f'unctions that occur in the middle, in the case
of treatment contrasts under BIB and PBIB designs the reader is
17
referred to
.£'57.
It is clear frora. (9) the.t the bounds depend upon
(i , i , •.• , i ) and (jl' j2' .•• , ju )' while the
l
2
sl
1
width of the bound depends upon (jl' j2' ••• , ju ) alone. A set
l
of bounds for which even the width should depend upon (i , :1. , ••. , i )
2
l
sl
both sets
as rlell have also been investigated in
here.
£117,
but will not be discussed
It is also obvious that for the partials we are getting tighter
and tighter bounds, especially for the partials rdth respect to
(jl' j2' .•• , j
). A further remark is called for here. The pair
ul
(Ho ' H) or the partial pairs would seem to suggest a lower bound
rather than the two-sided bounds.
However, we still seek the two-
sided bounds on the principle of section
1.1.,.
For a set of con-
fidence statements different from (9) and based on a different principle of decomposition of H
the reader is referred to
r57.
0 - -
(17)
Three special cases of
4.1.
The case
tion.
8.8.
Here
u
deserve special, rlf;ution here.
= 1.
This is the univariate or the liliOVA situa-
8
are scalar, usually referred to the mean
8 ,
Ho
error
due to the hypothesis and the error respectively, 'While
is an
1 x s
row vector.
This means that we can drop the
l / 211 take
"Chmax
'
u l = 0, and express ( 17 ) in the form
(18)
r,
L
~
11 , .•• ,1 s
)
1.
s H-7-
1
.L
'Ar- s error-7 -<
1..
0
;l.,
(i , ... ,1 )-1
*
l
81
Q H (1 , ••• ,i )
o l
sl
1.
J..,
'A
£ s error-7 ,
T}'
18
where
is based on the percentage point of F-distribution with
i\
suitable d.f.
4.2.
The problem of two mean vectors in N(Si'
~)
(i = 1, 2)
pxl PXJ?
17,
117·
Here Ho : Sl = S2 against H: Sl ~ S2 ' and this
case is easily handled by a proper choice of A, C and U. Notice here
10,
that U = I.
nl
and n
To fix A, we assume that two random samples of' sizes
have been drawn from the two ]?OJ?u1ations.
2
X2 the two sample mean vectors and by
S ,
2
PXJ?
px1
sample dispersion matrices and set S
= (nlsl
Denote by
the two
+ n2s )/(nl + n2 ).
2
Then the set (17) reduces to
(19)
f\xl
-
x2 )
( j 1,···,J. ) I
p
1
Jl,···,J.p ) 1/2
( •
(~_
x2 )
1 ]
1/2
-
i\
chmax
1/2
+
i\
]
,
19
•
jl < j2 < ..• < jp (0 ~ Pl ~ P - 1)
"'-There
is any ordered subset
1
from (1, 2, .•. , p) and A is based on the percentage point of the
distribution of Hotelllng t s T2 v.i th appropriate
F-distribution ",,11th proper d.f.
d. f., 1. e. of the
The number of intervals in (19) is
'&!-1.
4.;.
.
The problem of two comparable mean vectors in a
r-
p
~l
L
ll
t
P
12
N
flO, l}}.
~2
P
,
L
12
1
f
L
P
P
Here vTe have a random sample of size
H :
~l
0
= S2
aga.inst
H:
P
22
n, which fixes the A.
~l ~ ~2' which easily fixes
C
vie have
and sets
p
(20)
U
2pxp
=
[.: J
p
p
As in the preceding section, let Xl and x
vectors for the two different
2
stand for the sa.Ill]?le mean
p-sets and, unlike the preceding section,
- 8
set nov 8* = 811 + 8
- 8k. Then, 'With u = p, (17) now
22
12
~
* is
reduces to a form which looks exactly like (20), except that 8
now differently constituted and A is different and is based on the
percentage point of the distribution of a Hotelling's
different
5.
2
T 'With
d.f.
Independence between tvTO sets of variates
to the category of section 1.1.
£8,
We have here a
10, ~7.
population
This belongs
p
q
p
q
Under this model 'We consider
(21)
H0 :
t
12
pxq
=
°
pxq
against
H:
t
12
~
°
Let us now set
(22)
f3
=
pxq
Notice that
t
12
pxq
-1
22
qxq
t
,
B
8
812
pxq
=
pxq
117
can be regarded as a measure of' departure f'rom
H
o
of
and
f3 and B are the population and the sample "regression"
rlO,
matrices.
-1
22
qxq
Then it has been explained in
H, and similarly the partials of
r~,1S and columns of
f3
-
-
that
l 2
Ch / (f3 f3')
max
in the direction
obtained by cutting out the
f3 (i.e., cutting out from the p-set and the
q-set) can be also regarded as measures of departures from the
corresponding partial bypothesis in the direction of the appropriate
21
(complementary) alternatives.
As in section 3, starting from the
one sided analogue of (3) with an exact probability 1 - ex, we can
:put,
with a joint (conservative) confidence coefficient ~ 1 - ex,
simultaneous bOllllds on
l 2
Ch /
max
(t3 t3') and the :Partials, the whole
set, total and the partials, being expressed., as in section
3, in
the compact form
,
'Where the symbols have the same general explanation as for (9).
It
will be instructive here to drop the superscripts and subscripts and
write the leading interval (involving just the "total") in a much
22
simpler form.
Here
However, certain additional remarks are called for.
i l < i 2 < ..• < ip~ (0 ~ Pl ~ P - 1)
is any ordered subset
..I.
from (1, 2, ... , p) and
j 1 < j 2 < .•• < j q (p ...
ii ~
q - ~)
is
1
an ordered subset ±'rom (1, 2, .•• , q).
Furthermore, in keeping
q, we also arrange that P - P ~ q .. Ql1mch neans that the
l
q
total number of confidence statements is less than (~- 1) x (2 - 1),
with p
~
and can be easily calculated.
In conclusion, as in section 3, while
(HO ' H) and the partials would seem to suggest just the lower bounds
we still seek the upper bounds as well, i.e., the two-sided bounds
on the principle of section 1.1.3.
As in section 3 so also here,
the bounds depend on both sets (i ,i ,· •• ,i ) and (jl,j2, ••• ,j ),
l 2
Pl
ql
but the width depends only on
(i , i , .•• , i ) .
l 2
P
If vle require the
l
width a.lso to depend on (jl,j2, .•• ,j
) as well then we have to use
ql
a different procedure altogether which will not be discussed here.
Bounds tighter than those in (12) but more or less on the same principles have been offered by other workers, but these will not be
discussed here.
5.1.
a
A special case of (23) deserves particular mention.
The case p = 1, i.e., of independence between one variate and
q-set.
Here
p
=1
and Pl
= O.
Furthermore, (1::
scalars, (Zl~9L2)' (lxq) row vectors, (1::
and, in consequence, 13
and B become
,8 )
11 11
become
,8 ) stay (qxq) matrices,
22 22
two (lxq) row vectors while
The 13 and B are usually called the populxq
lxq
lation and the sample regression vector of one variate on the q-set,
8 .
1 2
while
becomes scalar.
2
81. 2
is called the residual variance.
(23) now reduces to
23
(24)
'\
-
I~
1/2 Chl/2
sl.2
max
1/2
1/2
-<[b
1 XCJ. (.Jl,···,J.)
CJ.
l
+
where X.
is based on the percentage point of an F-distribution with
appropriate
d.f.
The well known interval for the regression coeffi-
cient in a bivariate normal distribution (p
= CJ. = 1)
is, of course,
a special case of (24).
6. Multiple independence between a p-set or between sets of Pl'
P2""'~
1 1g7.
VIe shall consider the two cases separately, a,ltI1O\.1.gh
the first one is a special case of the second.
6.1.
Multiple independence between a
p-set.
For an NL~' '£_7
px1 pxl
we express H: '£ = D
(a diagonal matriX) as the intersection
o
O"ii
of the independence of the p-th variate with variates (1,2, ••• ,p-l),
of the
(p-l)-th variate with variates (1,2, .•• ,p-2) and so on till
we get to variate 2 and its independence with 1.
The associated
24
alternatives are the respective complements.
This is called a
step-dow procedure and the results that we ultimately obtain are
not of course invariant under the order in 'Which we take the
variates 0 This is supposed to be a blemish on the technique, but
I feel rather differently for two reasons.
First, there are problems
and situations when the experimenter has some order in mind in which
he is more interested.
The second, and by far the more important
reason is the following.
priate
nonpa~etric
awn experience in the search for appro-
My
analogues of MANOVA and association has more
or less persuaded me that characterizing a nonparametric multivariate
distribution as the product of a chain of conditional probabilities
(in the sense of a step-down procedure) right from the beginning
may, in most situations, be the onl-Y feasible line of attack.
Be
that as it may, let us come back to the concrete problem here ["
_7.
With
0"11
(26)
l:
=
pxp
0"12
O"lp
CT
12
• •
0"22 .
· O"lp
· • 0"2p
.·• .
0"2p
S
,
=
pxp
sll
s12 •
slp
s12
s22 •
s2p
•
• 0"pp
slp
s2p
....
• s
pp
let
(27)
f3 ·i .:\2 ••• i-1 = ["O"li 0"2i' • • O"i_l,i-7
2
1x(i-l'
-1
CT
12
0"22
CT
1,1-1
...
..
,
25
b' i.12 ••• i-1
lx(i-1)
=~s 11
s2i'"
Si_1,~_7
~
sl,i_~
-
-1
s2,i_1
s
s
1,i-1
2,i-1
(for i = p, p-1, ••. , 2) denote respectively the population t.nd the
sample regression vectors of (i) on (1,2, ••. ,i-1), and also Jet
8 _
i 1
(i-l)x(i-1)
=
sll
s12
s
s12
s22
s
•
s
(for
i
=p,
1,i-1
s
..
s
2,i-1
1,i-1
2,i-1
,
i-1,i-1
p-1, ••• , 2).
Then, we have with a joint (conservative) confidence coefficient
~
1 -
c/"
the simultaneous set of intervals
0
(
f3 i
j)
J 1 ,···, i 1
J
1/2
26
ji (0 ~ i ~ i - 2'; i = p, P - 1, ••. , 2)
is
1
1
1/2
any ordered subset from (1,2, ••• ,i-1) and where "A. = + .L1J./(1-E:.7
where
and
ri
j1
IJ.
<
j2
< .•. <
is obtained from the following equation
having an incomplete
for i
= p,
p-1, ••• , 2
t3-distribution with
= sample
and n
d.f. (i-1)/2
and (n-i)/2,
It is to be noticed
p
i 1
that the total number of confidence intervals is ~ (2 - - 1).
i=2
Also, for any i, the bounds depend i and on the set (j1,j2'" ·,ji )'
1
but the 1ddth depends only on
(29) is obtained by putting
(31) (b!J. b.)
J.
1/2
< (t3 i
6.2.
-
"A.
r-s ••
L J.J.
t3 )
i
1/2
i.
i
The leading interval in the set
= 0,
1
1
- si
size -2.
whence we have
-1
S.1 .1- . Js~
7
..-
~ (bi b )
i
1/2
1/2
r-
1/2
chrex
-1
(S.J.- 1 )
I
-1
71 /2
+ AL sii ... si Si_1 si_
Multiple inde:pendence between sets of
P1':P ", "I1r.
2
'tole have here
27
IV
Sl
Pl
Ell
E12
Elk
Pl
~2
P2
Ei2
E
22
E
2k
P2
E'
lk
E'
2k
~
Pk
Pl
P2
Pk
,
•
~k
Pk
1
and
'We
assume, without any loss of generality that P
i
~
,
i-l
E P .
j::l j
The null hypothesis that the population dispersion matrix has all
the off diagonal submatrices zero is expressed as the intersection
of the independence of the
i-set with sets (1,2, ••• ,i-l) (for
i = k, k - 1, ••• , 2), with the associative alternatives as the
respective cOIll]?lements.
'\-lith a corresponding partition of the total
sample dispersion matrix, let
(33)
{3i
=
LEU' E,!,....
E!J.- l,J.' 7
c;J.
P.x(p.
l+ •••+Pl)
J.
J.-1
i
Ell
E
12
E
12
E
E'
l,i-l
E2, J.. l'
22
.El,i_l
.
I
• .E2 ,i_l
. .Ei _l , i-l
,
28
-1
)<
S
1,i-1
Sll
S12
Si2
S22
. ..
S
2,1-1
st
st
...
61- 1 : 1_1
1,i-1
2,i-1
l
J
,
and
(:54)
S2i
P2
=
Si_1
(P1+",+P1-1)x(P1+"'+P1-1)
S1_1,i - 7Pi
Pi - 1
• •.
Sll
S12
Si2
S22
st
1,i-1
I
S2,i_1
'
Sl,i_1
·
• ··
···
·
•
Then, 'Wtth a joint (conservative) confidence coefficient
~
S2,i-1
Si-1,i-1
1 - a,
we have the sets of simultaneous confidence intervals, of which the
leading set is
(35)
1/2
eh....."....
rBJ.' B~_7 - ~
~ L.1;
<
1 2
max -rf3.J.
Ch /
1/2
()t
-1 C.)
1/2
-1
chmax Lrsii- SiSi-1 S J.- 7 chmx (s1-1 )
f3~1.-7
29
and the whole set of sets can be expressed in a compact form by
replacing
(jl,j2,···,jU.)
,
Bi
by
Bi
(fl ,f2 , .•. ,f:.) ,
~
(i)l(jl,j2,···,ju.)
8
where
ii
and
by
jl < j2 < .•• < j
u
(0 < u
-
i
from the set (1, 2, ••• , p.)
~
8(i)
< p. - 1)
i - ~
and
(1 <
K2
by
~
8
is any ordered subset
< ... < (,
~
(0< v.< Pl+P2+
-
~
',,:Pi~l-Pi+ui
is any ordered subset from the set (1,2, ••. , Pl +P
2
+ .•• + Pi-I)'
")..
= + .L1J./(1-IJ.L71 / 2
and
IJ.
is obtained from the
equation
= I-a.
7.
The problem of vector analogue of the ratio of the tlvO variances
and two means for a bivariate normal distribution ["1"27.
iie start
from the same model as in subsection 4.3, with two comparable p-vectors
following a
2p-variate normal distribution.
two cases s eparately
7.1.
We shall consider the
.L13 _7.
The vector analogue of the ratio of two variances.
To fix our
ideas we shall first consider the case of the ratio of two variances
(ji/(j'~ for a (correlated) bivariate normal distribution. It is easy
,
30
to see that if
then
xl
xl - (0"1/0"2) x
and x
2
and
2
has a bivariate normal distribution,
xl + (0"1/0"2) x2
are uncorrelated in
the population (and hence independent in the case of a normal distribution).
It is possible by utilizing this to use the
and obtain for
t-table
O"i/O"~, 'With a confidence coefficient 1 - a, the
confidence interval
{l
<
where
n
+ ~
n-2
2
t a/2
is the sample size and
point of the t-distribution.
,
2 \\
~l-r ~ +
L-(1 + n-22 t 2/
2
is the upper 100 a/2
a 2
t /
We notice that
2
a 2 (1-r » -1]
0"1/0"2
=1
0/0
if and only
if the customary null hypothesis is true.
For two vectors
having the
distribution it can be checked that the set
set
(Xl +
f,.
X )
2
2p-variate normal
(Xl -
f,.
and
where
v
D
and the
I
and the
are uncorrelated in the population (and hence also
indepenaent because of the normal distribution), where
~
~)
f,.
=~
v-
1
and
are given by
is a diagonal matrix idth diagonal elements
Its
-1
-1 I
are the roots of 1::
1::
1::
1::
22
12
11
12
Il,/2,,,·,/p
By utilizing
this fact and using the distribution of the square of the largest
1/2
,
31
canonical correlation coefficient it is possible to obtain confidence
bounds on
cr?/2
max
(r-. r-.') and Ch1~2
(r-. r-. J) and on the "partials".
mJ.Il
Before we go on to that, some remarks are in order as to the physical
role of the mtrix r-..
becomes
i.e.,
0"1/0"2.
= 2,
i. e., for the bivariate case
Furthermore, it can be easily shown that
r-. is an orthogonal matrix, if and
rna. trices, i. e., Ell
E
12
For p
= r-.
E
= X
12
if Ell = E
where
a symmetric
22
and E
12
fidence coefficient
~
matrix.
= Ei2.
r-. r-.' = I,
if Elland E a.re sitlilar
22
r-. is an orthogonal matrix and
r-. (with the same orthogonal
Eb
x
= r-. E22 r-. J,
only
r-.
X ), er QqU:ivaJ.eo.tly
This orthogonal r-. = I, if and only
With a joint (conservative) con-
1 - a, a set of simultaneous confidence inter-
val pairs (including the "partials 11) of which the leading one (involving the "total") is
~~rnax Lchmin(S11
- S12S2;lSk)/Chm1n(S22)'
chmax(S11-
S12S2;lSi2)/c~X(S22)
(Snn- Sl'",Sll-l
Sl"')
-< chmax ()...~) -< ea chmax (Sll)/ch.
JnJ.Il c.c.
c.
c.
,
e = 4/(1-C ), and c
is the upper 100 a 0/ 0 point of the
a
a
a
distribution of the square largest canonical correlation coeffieient
where
32
with
d.f. (p, p, n-l).
The partials obtained by cutting out one
or more variate pairs are not displayed; the form of such partials
(38) of course follows as a very special case of
is by now clear.
(40) .
7·2.
The vector analogue of the ratio of tyro means.
~
Let
i
=( ~ li )
J.xp
and
~2 =(S2i) (i = 1,2, •.• ,p) and
~ = ~li/~2i'
ql'~".' ,~.
for a diagonal matrix with diagonal elements
xl - Dq x
2
is distributed as a
Also let Dq .
stand
Then
p-variate nomal 'With zero mean
vector and !:* = Ell - DqEi2 - El~ q + Dq E22 Dq' l-le utilize this
and use the distribution of Hotelling i s T2 to obtain, with a joint
(conservative) confidence coefficient ~ 1 -
c/"
a set of simultaneous
confidence intervals (including the Upart:ials t1) of which the leading
interval (involving the "total") is
-1n
(41)
2
2
T
ex
p
E
i=l
ch
(8) max ["1 + qi, 1 +
max
IXli X2i I
-,
xl xl -
where
T; is of course the
'tdth appropriate
[" 2
ql'
2
%'
X2 X2
min
max
d.t.,
8
100
ex
2
~,
2 7
... , 1 + ~-
+
2 1/2
... , ~7
0; 0
is the total
2
('11 '
2
... ,
~)
> 0;
point of Hotelling f s
2p x 2p
2
T
sample dispersion
matrix (partitioned in the same manner as the total population matrix),
x
2 (=
x
2i
) (i
= 1,
2, .•• , p)
are the sample mean
lxp
vectors of the t1'1O
p-sets.
The partials (whose torIilS are by now
33
obvious) are not separately displayed.
For p = 1, 1.
e.,
1 - 0,
bivariate case, this reduces, With an exact coefficient
( -2
x2 - k 2
( 42 )
where
8
2) 2
(- 2 q - 2 ~ x2 - k sl
2
xl' x ' sl'
2
2
s2'
8
for tbe
)
(-2
2 r q + ~ - k
to
sl2) ~ 0,
are the sample means, variances and corre2
0
lation coefficient and k = \42/ n, \x/2 is the upper 100 a/2 0
point of the
8.
r
t-distribution with proper d.f. and n
is the sample size.
Concluding remarks.
It has been already stated that we are excluding the problems,
treated elsewhere, involving model II and mixed models of MANOVA.
We are also excluding problems involving other patterns on
r: I)},
including, in particular, the problem of factor analysis and also
problems involVing different
r: I s
for more than two populations [)}.
Nor are we including some other interesting results, somewhat on the
same lines as the ones discussed here, that have been published in
recent years by workers from groups other than the one I have worked
with.
_
REFERENCES
["~7
Bargmann, R., "A study of independence and dependence in multivariate normal analysis," Institute of Statistics, University of
North Carolina Miemograph Series No. 186.
['"2_7
Bhapkar,
v.,
"Confidence bounds connected with ANOVA and MANOVA for
balanced and partially balanced incomplete block designs," Ann. Math.
Stat., Vol. 3 (1960) pp. 741-748.
['"3_7
Gnanadesikan, R., "Equality of more than two variances and of more
than two dispersion matrices against certain alternatives," Ann. Math.
Stat., Vol. 30 (1959) pp. 177-184.
['"4_7
Heck, David, "Charts of some upper percentage points of the distribution of the largest characteristic root," Ann. Math. Stat., Vol. 31
(1960) pp. 625-642.
['"5_7
Roy, J., "Step..down procedures in multivariate analysis," Ann. Math.
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['"6_7
Roy, S. N., "On a heuristic method of test construction and its use
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['"7_7
Roy,
s.
N. and Bose, R. C., "Simultaneous confidence interval esti-
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['"8_7
Roy,
s.
N., "Some further results in simultaneous confidence interval
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19_7
Roy, S. N., "A note on some further results in simultaneous confidence
interval estimation," Ann. Math. Stat., Vol. 27 (1956) pp.856-858.
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e
s.
N., Some Aspects of Multivariate Analysis, New York: John
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1 11_7 Roy, S. N. and Gnanadesikan, R., "Further contributions to multivariate confidence bounds," Biometrika, Vol. XLIV (1957) pp.399-410.
35
L 12_7
Roy, S. N. and Bargmann, R., IITests of multiple independence and the
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["13_7 Roy, S. N. and Potthoff, R. F., IIConfidence bounds on vector analogues
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["14_7 Roy, S. N. and Roy, J. , IIAnalysis of variance with univariate or
multivariate, fixed or mixed classical models,1I Institute of Statis!ics, University of North Carolina Mimeograph Series No. 20§ (1952 ).
["15_7 Roy, S. N. and Gnanadesikan, R., IISome contributions to ANOVA in one
or more dimensions, II Ann. Math. Stat., Vol. 30 (1959) pp. 3°4-340.
["16_7 Roy, S. N. and Cobb, Whitfield, IlMixed model variance analysis with
normal error and possibly non- normal other effects.
Univariate and
multivariate cases, II Ann. Math. Stat., Vol", 31 (1960) pp.939-968.
£17_7 Roy, S. N. and Srivastava, J. N., IIInference on treatment effects and
design of experiments in relation to such inferer:ce," Institute of
Statistics, University of North Carolina Mimeogra~h Series No.274
(1961) .
£ 18_7 Roy, S. N. and Gnanadesikan, R., "Equality of two dispersion matrices
against alternatives of intermediate spec1ficity,1I Institute of
Statistics, University of North Carolina Mimeograph Series No.282
(1961) •
["19_7 Roy, S. N. and Roy, J., "On testability in nO!I:iI!Al ANOVA and MANOVA
•
".
with all 'fixed effects'," Calcutta Mathematical Society Golden
Jubilee Commemoration Volum~ (1958-59) (issued 1961).