ON A MANOVA MODEL AFPLIED TO PROBLl!MS IN GROwrH CURVE
C. G. Khatri
Institute of Statistics
Mimeograph Series No. 399
1964
UNIVERSITY OF NORl'H CAROLINA
Department of Statistics
Chapel Hill, N. C.
ON A MANOVA K>DEL APPLIED TO PROBLEM:! IN GROWTH CURVE
by
C. G. Khatri
University of North Carolina and GUjarat University
Ar~,OSR-Grant No.
84-63
This research was supported by the Mathematics Division of the
Air Force Office of Scientific Research.
Institute of Statistics
Mimeo Series No. 399
ON A MANOVA MODEL APPLIED TO PROBLEMS IN GROWTH CURVE 1
by C· G, Kk.a.~l
University of North Carolina and GUjarat University
=====================================================================================
1.
Introduction and summary.
The usual MANOVA model [11, p. 83]
wa.s generalised by Potthoff and Roy [8],
keeping in view its importance to growth curve problems and they discuss some applications of the generalised MANOVA model.
Rao [9]
In a more special case of one population,
has solved the problem from a different view point and considered some appli-
cations to growth curve.
We give below the generalised MANOVA model and hypothesis
as studied by Potthoff and Roy [8J.
Let
X: pm
tv
(1)
be a random matrix such that
=
E(X)
tV
B
§ A
Ni'JN
and the columns of t"J
X are independent multivariate normals with unknown covariance
matrix
,E :
The matrices
pxp.
B: pxq and A: mxn
,...
N
are assumed to be known and
further, for our purpose and vri thout any loss of theoretical generaJ.ity, they are
assumed to have ranks
r(~
equal to
B and for
,....
q)
and
q and m respectively.
s(~
m) respectively.
A, and we can write, without any loss of genereJ.ity,
1? = '.~3
~)
=
and
~l:
nxs
is
~ ~ ~ = ~
~2)
(A'
,..,1
is
J3'
",5; !l'
"'~ = ~r
~l).! (~s
rs
parameters of
jY~:
rxs.
the estimate of g sa.tisfies the relation
it is evident that the function
~
=~i ~2
and the
With this, we can write
s.
determined in terms of
,
r, ~2: nx(m-s)
~2) r.
From this it is easy to see that the unknown parameters
of
N
Then we can always find the basis for
where ~2: px( q-r) = ~3 ~l' rank of ~3: pxr
(3)
~
~
(2)
rank of
Suppose the ranks of B and A are
C
...N.....
s
~
q,m of
g: q:xm are
""
Hence, if we know the estimate
(3).
From this consideration,
V will be estimable if and only if
('OJ
~his research was supported by the Mathematics Division of the Air Force Office
of Scientific Research.
e
2
(4)
C = (C
C = Cl Ll ' rJ
",1
"'2
r-J f'o/
wle re ~l: cxr, S2: cx(p-r ),
rank of
c
:t
is
(4)
v.
V' =
£2)' ,v2
11 :
Ji
V' = (V'
",1
and
J'2
vxs, J2: vx(n-s), rank of
is true even when the ranks of
£
,..,C
,
V' )
"'2
('¥
is
and the
c
and yare less than
and v, but, for the analysis, the dependent rows of ,..v
C and the dependent
columns of V are redundant.
r-J
Thus"
C g V is
if we assume that the function
~~~
estimable (Which we shall always assume), then we can, without any loss of generality,
assume that the ranks of B: pxq and A: mxn
'"
q(~ p.)
are
and m«
n) respectively.
N
With these remarks, under model (1), we are interested in testing
where
H
o
roJ,v
C: cxq and V:
mxv are of ranks
,..,
was tested by Potthoff and Roy [8]
x =
,3:
singular.
pxp
and
v
(),
~
respectively.
N
N
N
(~~~)
is any arbitrary non-singular real matrix such that
Then the results of Roy [11, p. 83]
has all unit elements,
Rae
were used.
They give same sugges-
When IIl:=v=c=l
IV
and rJ
A: lxn
E by its estimate based on
tv
S = X[I - AI(AAf)-lA] Xf, which is independently distributed of
H , . J ,..
is non-
[9] solved the problem by using the least squares esti-
mates obtained by replacing the unlcnown
"',..,
The hypothesis
by using the transformation
tiona as to the choice of the arbitrary matrix rJG.
i'V
given by
C(Bf G
B)-l B' tv
G X
,
rJ,..,
,.,
",,0
where
c
rV
(6)
~
g V = rJ0) against H( C,.,~N
g V
H0",
(C
H
o
Z = X A'(AAt)-l.
tV
N
t"'J
In section (2.1), we use the likelihood ratio method tor
H
o
(V
tV ('"
under model (1) and
give two other associated test procedures on trace and maximum rootp.
In section (2.2), it is shown that the test procedures derived, in section
(2.1), are applicable to te sting
H'
o
defined by (8) when the model matrix B is
N
completed as
E(X)
,..,
where
B,
tV
!'
~
=
B
g A
rJ ,v roJ
+
A
N
are the same as defined in (1),
,
~ : (p- q)xp
matrix and!l: px(p-q) is called a completion matrix such that
singular.
is an UnkJlown
(~
~l)
is non-
The completion can be done in many ways Which we shall not discuss here.
3
In
e~
BIG B
",,1"" ,..,
h case of completion, these exists a symmetric na trix Q: px:p
such tha.t
tr-J
= ()
and
j(.
(B
B ) I f'IG (B~,.I
B )
.-.J
rJ l
l
is non-singular.
B
N
The hypothesis HI0
is given
When
(8)
by
(8)
Tl X f 5)
aga.inst HI (C -v
-1
)
E- l B)-l (B'
H'(CTlV=O)
,..,
01'J,..,~
where
~
S + (BI
Tl =
,."
"'"
exactly equal to
,."
,..."
(5) .
r'
,..,
~
The hypothesis
~l
(8)
,
k·
~ =
M
0, then
,."
is
can be looked upon as the general
linear hypothesis due to multicollinearity.
The test procedures for
HI
o
are
derived by applying the union intersection principle, and they turn out to be
exactly the same as those derived for
ratio
(5)
under model (1) by using the likelihood
method.
In section (2.3), it is pointed out that
section of two hypotheses
~
procedures for testing
H~
H* = (H , model (1»
is the intero
0
= 0.) under model (7), and the step-down
HI and H( 3 ) ( ~
o
0
""'"1.
rare suggested instead of those derived in section (2.1).
The following consideration brings out the deeper physical implications of the stepdown procedure.
~'incomplete"
Unlike ,.J
A, we clluld not be sure, whether
s
;;J
and NB are, in fact,
as in model (1) rather than "complete" as in model (7).
ThUS, it would
be reasonable and safe to test first the adequacy of model (1) itself by H(3), and
o
then go on to test the hypothesis
H.
Moreover, the main purpose of changing model
o
(1) to model (7) is that if we are uncertain about the order of
the maximum value of
s
N
and NB (because
q can be p), then the test procedures carried out in section
H~
If we want to test H under model
o
(7), then the test procedures are identical with those given by Potthoff and Roy [8],
(2.1) will still hold for
and not for
when rJ
G is defined as in the model (7).
H.
o
Hence, we have an entirely different
interpretation of an arbitrary matrix G introduced by them, and, if the model (1)
rJ
is not true, then the tests with this arbitrary ~ will not be natural and proper
unless G = I or G- l is the covariance matrix based on the similar previous data.
tv
f:'J
tv
This is because H has different implications in two different models.
o
4
In section 3, the distr.ibution and the property of monotonicity of the test
procedures derived in section 2 are established under model (7) (and so they are
also true under model (1».
The simultaneous confidence bounds on th e parametric
function of ~ 1).. Y, under model (7)
section
or
£..t x.
under model (1)
are given in
4, and in section 5 a numerical example illustrates the types of computations
involved in the test procedures.
2.
Derivation of the test procedures:
(2.1)
Likelihood ratio method.
For this purpose, we shall consider model
(5).
(1) and the hypothesis Ho given by
We give below two lemmas which are used in the derivation of the test statistic.
Lemma 1:
B*l' B
,...,
".,
Let
='"O.
!:
!t:
pxq and
px(p-q) be of ranks
q and
(p-q) such that
Then if S: pxp is a symmetric positive definite (p.d.) matrix, then
"'"
S·l _ S·lB(B' S-l B)-lB' S-l
,..
f'I
tV,..
~
tv
('J
tv
=",,1
B*(B*'
1
rJ
S B*)-l B*'
,oJ
('11
.-v
1
.1.
If a matrix T is symmetric and p. d., we shall write it as rv
T = (':tr-J2 )2
Proof:
tV
.1.
where T 2
'"
is a symmetric matrix.
=
rv
6.
=
then
I
1
[8 -2" B(B' Stoj
roJ,...
a.nd so
,.J
1
vTith this notation it .is easy to verify that if
1
1
'R) -2"
f4;:J
I
IV
=
6. 6.'
N
t>J
1
S2 B*(B*' S B*)-2 ]
tv 1"'1 rJ 1 (V ,..,1
= S-~ B(B'
,.J
'"
tV
S·l B)-l B'
IV....,
,
S-~+
1'oJ"""
sl B*(B*'SB*)-:a*,s'i.
,...J,...)
1 ,., 1 tvr'l
From this, the lemma. 2 is obvious.
Lemma. 2:
Denoting
8 (Y)
a matrix whose elements are the differentials of the
""
corresponding elements of Y matrix, we ha.ve
'"
(i) 5°(!1 }2) = 5 (;1)3"2 + !l 5 'J2) and (ii)8 (Log 12'31
if
1.!3'
)=
tr
J3l8 (!3)
> O.
Lemma 2 can be verified directly from the definition.
For the likelihood ratio statistic, we are required to obtain ~ L (i.e.
maxinnnn value of L under H) and
max L where
Ho
,
rv
IN
5
First, we shall derive
(ma; L).
Let us maximize
L with respect to E (assuming
rV
s
tV
to be known).
We find that
(7 L) = (21t n) - 2' pn
1
(10)
1
I ~ 1- 2' n
rw
where
(11)
and S
,oJ
=X
- A'(AA,)-lA ) ~
{I
",n
-.)
rJ
N
,.J
N
! .
I~I with respect to
if we minimize
(max L) will be obtained
H
Taking differential of (Log I~I), we
Now it is apparent that
'"
get after some adjustments as
I!/)
~
B(Log
and so
I~I/
0
0
l'
tV
tv,.,.
IV
s
~
= 1,2, ... ,m)
give us the maximum
as
....
= rv0
S-~
B( s)J ,
IV
N
1 AA'(Z-B s)'
wi:"
"" tv'"
(i = 1,2, ... , q and j
of
tv
(Z - B 1)' S-lB
N
,..,
;";'j = 0
likelihood estimate
(12)
- 2 tr [ 41-
=
g=
or
(B' S,-l B)-l B' S-l Z.
tv,..J
N
N
(V
,.;7
iJ
Putting (12) in (11) and using the lemma. 1, we get the minimum value of
I~N lover
as
=.;
where
I + AA'
Z· ",1
B*(B*'
S B*)-l """B*' Z J
Nm
tV
<V 11"J ",,1
and
(H L) =
(14)
(oJ
!t:
tv
(V
px(p-q) is of rank (p-q).
1
(21t nr2' pn
1..
I~ 1- 2 n (~n
-
.1.
It' )- ~n
,
Hence, we get
exp(- ~ pn)
s
(maHx L), we ha.ve to minimize 1~.1
subject to rvNtV
C V c ,...,
O. For
,_
o
this purpose, let ,sl:(q-c)Xq and ~l:m x (m-v) be of ranks (q-c) and em-v)
Now, to obta.in
and NV' r.J
V = r>I
O. Then, if r~.J.
911 ~~
l
n_: cx(m-v), we get from C
V= 0 ,
respectively such that
and
(15)
C
-.J
s Vl
s
rJ
=
r-J ""
=
C tVC' = r.J
0
l
(V
s
tv-~
C' (C C' )-1
l rJ l (oJ l
~
.,v ,..} rJ
n.. + C' (C C' )-1
rJ·J.
rV
t'V
tV
t'V
n_(V '
l
tVP~,.J
Vl)-l V'
l
(V
,..,
•
=",,1
11
: (q-c)xm
s
....
6
Putting this value of ~ in (10), we find that we have to minimize
~
~
respect to
and similar to
(16)
1C
:J2'
and
First, we shall minimize I~I
(12), the equation for
= (C;J.B'
~
C,)-l
,,~ ,..,1
~'J.
f'I
).
for given
~
with
with resPect to
~
is
s-lBC ,)-l C B' S-l[Z_B C'(C C,)-l n_(V' V )-1 V'].
'" ",1
tvl r J , v
"' ..... f'I
,., ' "
,.:t!. ,., 1 ,..;1
,.., 1
,.J
Now letting ~l:px(P-q+c), a matrix of rank (p-q+c) such that
we get similar to
1,1
2'i ~ E1 =E'
(13 )
(17)
[Z_BG:(cq~)-l n_(V1'V
~
By
lemma 1,
T (T ' S T )-1 T'
,vl
l N 1 r.J ~ 1
,.J
rank
~l =
.-
'" ""'
~
,.J't!. ,..,
rt
f'I.L.
)-lvl,] I('41
remains the same for any -31:px(P-q+c)
of
(p-q+c) satisfying tVT ' B C' =,...O. Hence, let us take
l
1
[B*l S-~(B' S-~)-lC']' Then, we find that
~N
,..J
N
'"
tV
'"
tv
'"
written as
(18)
(min',I)
11, tV
=
I AA' I
tV.,)
rv
Now the minimum veJ.ue of
(18) over
.....~ will be obtained by using
~ = ~l ~3 (~3 ~3)-1
(19)
rv
.
With the help of (15), (16), (19) and the lemma. 1, the value of
I, I
,.J
under
C £V = 0
N,..JAJ,...
is
s
which minimizes
7
e
(20)
=; - (B'
N0,Y
8- 1 B)-l C' Q-l C
tV
('OJ
IV
N
tv
('OJ
gV(V' R V)-l V'
tv N
N
N,.v
~
R
N
'
and
,
(21)
Hence the
where
maximum
is
value of Lunder Ho
-i pn)
(22)
.
Using
(23)
= I ....,JC
I
A
+ Q-l P
rJ
N
I -1
C(B'
8- 1 tvB)-lC'
RVr l (C~V),
Z=XA'(AA,)·l
rV N , . J
tV' p=
'" (C~V'(V'
tv tv:'; N
rJrJ tv
,
tv
'" tv N;:';
1
1
= (B'
8. B)-l B' 8- Z R = (AA,)-l + Z' ~(B*' ~ ~*).l B*' Z and
~
N,.v
N
.-v tV
tv '
,.J
rV
N
tV 1 I'll
;;J A11
f'I 1,..,'
1
1
1
B*(B*' 8 B*)-l B*' = 8- _ 8-~(B' 8- B)-l B' 8- .
where
Q.
tV
tv
g
",1
tv
,
N
1
rv
tV
"'1
N
1
('OJ
I'J
/'oJ
~
IV
N
'"
N
Now with the help of (23), we suggest the following three criteria for testing
H0
against H under model (1):
\ ,
(24)
Reject H
0
if
(25)
reject H
0
if tr P Q-l > ~,
reject H
0
if c~ P Q-l >
N
(26)
where
/\~
""
or
or
..;J
N
-
":?
Ch.(.) = j.th maximum characteristic root of (.), and
J
Pr(A
~ \ I Ho ) = Pr(tr J f l ~
Nmv, when m = v
= c = 1,
"2 mHo)
it is easy to verify that A = tr NP Q-l
,.J
reduces to the test procedure given by Rao [9J
test procedure.
= Pr(Chl ! ~l 2: ~I Ho ) = a
= ch_
.
P Q-l
-~ '" nJ
and so it is a likelihood ratio
In the next section, we shall derive• the test
~rocedures
-
for H'0
under model (7) and they are shown to be (24) or (25) or (26) by union-intersection
principle.
8
(2.2)
Union-intersection principle for testing H~ under model (7):
Lemma-l:
Let M:(r+s)x(r+s) be a symmetric positive definite matrix and
tV
N = NM + yyt, where rv
Y: (r+s)xv.
,..,
Let them be partitioned as
N('oI
~l
r
rY:L2
,
M =
rJ
M'
rv12
~
(~ll - E12 ~~Ei2)
- %1 -
and Y =
N
""
J
Then
Proof:
s
~22
r
Jll
('J
s
!i2
-\.
~2!~ m.2)
We note that
-1
(
)-1
-1
;322 = ~22 + ~2~2
= J122
-1
!12~2
-1
= ~~22
-1
(
-1
)-1
... ~2 J2 ~~ + ~2 ~22!2
J2
-1
!122
'
(
-1)(
~
-1
)-1
-1
+ ~l - rY:t2 ~2 ~ ]~ ~2~2 12
~2 ~2 ' and so .)
~12~;~'yi2 =~2~;~,Yi2 + ~l~
-
~1-~2~;~2){;V+~2 c!1;~2)-1(~1 -l\~;~ ~2)"
From this the lemma 3 is obvious.
With the help of model (7), it is easy to verify that the joint distribution
of
~1 = ~(~rg,~) -lEt2~
means
and
-h~'
= ~i ~ ~l )-~i <1
and .E2
and covariance matrix
C(BrGB)-~'G
,...,; ,.w
fVN
N
N
=
c
=
p-q
c
p-q
e is normal with respective
9
~12 ~~ = - (~ ~ -~r·l~,
Then noting
E"J3l)
III -
and
~~
!l2
C(B' ~ -~)-l C', it is easy to verify that the conditional distribution of
rV
'"
~2
when
S~!:i
~:
(~' ~ -J3rl(~, ~-J3l)~2
E2
=
and covariance matrix
a'C(BI ~ -~)-l C' a
<v""'"
;-.ltV
N
"'"
£' ~2
~'J}. +
a' = a' C"l
where
N""
""
N
CI(1.
N
H' (a' C "l V
oaN
N
N
-
=
V
0(.1
f'J
(V
N
and
f3'
rJ
~
tv
""
f.
0)
and H (a 1
o~
""
V
tv
= 0)
/"OJ
~'Bl'
= 0)
= _ Na' (B'" " ,~-lB)-l~
-...N.--/
given by
(8)
is equiva-
N
=,.;r0)
For this purpose, it is easy to verify that the estimate of
H' ( a 1 V
Now let
and variance
(VN
Now, the problem of testing H'(C
"l V
O,w-vN
lent to testing
X,<~' ,; -~)-l~,.
Then the conditional distribution of
is fixed, is normal with mean
~z:~l).
(V
Al '
,~
,v
cxl be any non-null vector.
2
a 2 under
are respectively given by
,...
(27)
where
,
)-lJ
GB, ( B1GB
1
(28)
(V
........
,...
N""
= (~
»:L2)
c
~J.2
~2
p-q
c
p-q
~
and
(29)
N - M = "{Y' =
N
f'J
"'...,
aC(~'9~-~'~)
)-~'G
(B'GB
""l",;;a.
It is easy to verify that
,....1 rJ
aH2a, I. rP
are independently distributed a's
(Cf H
2, - Ci H
2,) I a2 under H' (alV
a
01'1· tv
o~
7V
and
lV
(;0)
to reject H
oa
if
X2
with n-m-p+q and
rJ
f
= (~,
o~
("VNt"tI
(oj
degrees of freedom.
-
verify that
C(B' 8- 1 B)
v
is
a. With the help of (28)
~l - ~~;~ ~2 =
= 0)
;>C
Hence, the test procedure for testing H
oa
(;1)
=
is fixed, is multivariate normal with mean
-
men
a
rv
rV
/"\(1'{2
-1
C'
f""IJ
=
Q
""
,
and (29), it is easy to
10
-~l
(32)
- ;;il
~2M;}2D-:n
=
("c;. r-c;.
"..
C(BtGB)-~IG[I
)-~~~]X
,v
r-Jp - '"SGB1(Bl!GSGB~
11/"" tv
rJ
rY ",,,,,
","'(01"'.1..
tV
C(BIS-~)-~Is-lx
=
NjVNN
tv J..(V
N
,
NtvnJ
!l - ~~;~2 = ~(~I~-~r~I~-~ ![!'{~tr)r] -~
yl M- l y
(34)
tv'
2 rv22
~2
[VI(AAt)-l.vr~ V'Z'GB
(B'GSGB )-~IGZVrVI(AA')-lvrt
",,,,1 ",,1"'11/"'1"'1 '" IN,,,iJ"
=
tv
N
N
'V
'" N
GB1(B~qSB1)-~ltG
=rJB*l(B*ll
S BiE'l)-l~lt where ~*l is the
rJ.J.tV~rJ
rv
""
tv '"
","l
.~
and on account of lemma 1,
.
'-.I ""
tV
Hence using (31) , (32), (33) and (34) in (35), we have
same as defined in (23).
(35)
nJ tV
N
t'U
f = (a....l rJP ,va)/(a,...,' NQ Na)
whe re
P and Q are the same as defined in (23).
~
tv
interpreted as
The matrices
s.p. (sum of squares and product) matrices due to
.
and
P
~
HI
0
Q can be
~
and
Ht
under
0
Now for testing Ht =
HI, we get the te st procedure by using
o
"" oa
union-intersection procedure! to (30) with the help of (35) as follows:
IOCldel (7).
,.,
(36)
Reject Ho
where Pr[ch..
Q-l ->
--1. P
tv.v
ch.. rV
P Q-l
>
--.J
if
A..
.,
tV
1-'3
--~
I H']
0
= a.
Note that (36) is the same as (26)
suggested in (2.1) and we shall see in section 3, that their distributions under
the null hypotheses are identical and so the same constant is used.
It may now be
noted that (26) or (36) implies (24) and (25), and hence we have the same three test
procedures for testing H~ under model (7).
(2.3)
Connection between H
o
and
H'
0
It is easy to see that ,
H*o
=
[H (C ~ V = 0), model (1)] = H'I( gl = 0, C g V = 0)
o
N
tv'"
=
ThUS, the hy'pothesis
theses
H~3)
and
H~
0
tV
H(3) (
o
s
Nl
= 0) (\
N
I J
H under model (1)
o
N
IV
N
<'J N
H'(C 11 V = 0)
0
rv tV,v
rv
N
under model (7)
under model (7)
is in fact the intersection of two hy,po-
under wider model (7).
We have seen in section (2.2) the
11
tit
test procedures for H'.
o
Now, the test procedures for testing H(3) under model (7)
0
can be obtained as particular cases of (24), (25) or (26) replacing NB by
V
(§ !'l),
.,J
and NC = (0
= tVn
I
N
I
), we get,
",p-q
(38)
Reject H(3)
0
if
P I
I
+ rJg;:l ",1
~ = I rvp-q
(39)
reject H(3)
0
if
tr (,3i1 !l) >
(40)
reject H(3)
if
c~(~lJl) > A'3
0
where nonzero
-1
,
- "2
or
or
-
Chj(Q-l Pl ) = nonzero ch.(AA' Z' B*l(~l' 3 B*)-l ~*l' Z](j=l,2, ••• ) ,
1"11 rV
J roJN N ,.... ~ " "
tV ~
tV
(41)
,
-< AI
P = B'G X A'(AA,,-l A X' G B and Q._ = B' G 3GB,
1 tVl""
('OJ'"
'"V,v,-.l ..... 1
.......'1. ",,1 N"'''"'r.Jl
tV
N
'"
and
tv
= Pr[ch.. (Q-1 p)
--1 t.J'1
Nl
> N..5 /H(3)]
=
0
-
Hence the step down procedures according to Roy and Bargman (12]
ct
or J. Roy [10]
for testing H* in the sense of (37) are suggested as
0
-< \ n ~:5
(42)
Accept H
0
under model (1)
if
(43)
accept H
under model (1)
if (tr P Q-l ~
(44)
accept H
under model (1)
if
where
(45)
0
0
\,
~,
Pre A ~ \
( A
( Ch
l
PQ-1
->
\),
or
~n tr(Pl~l) >
'5)
or
~(\ Ch1(Pl~1) ~ \)
~ satisfy
IH~)
Pre
'i ~ \ I H~3»
C;;
! ~-l S ~ IH~)pr(tr~§l '!J
~ I H~3» = Pr( Ch~l S It I~)pr(C~!:l~
= Pr(tr
~ IH~3);-=-
1 - ct
,
-
12
for the statistics are independent under
suggested in
(42), (43)
H(3)
o
that if
(44)
and
H
o
in place of
(24), (25)
is rejected, then
wi th model
and
(25), (26),
are no
(26)
lon~r
reveal one fact
H0
valid for
Hence, to carry
H under model (7), we obtain, with necessary changes
o
out the test procedure for
(24),
(26)
and
The test procedures
H' under model (7).
o
under model (1), but they are only valid for
in
(24), (25)
(1).
the same test procedures as considered by Potthoff and Roy [8]
when G is any non-singular symmetric matrix as defined in the model (7).
N
we have a different interpretation of an arbitrary
Hence
G matrix introduced by them,
N
and if model (1) is true, then test procedures 'With this arbitrary
G will not be
~
natural and proper unless
similar previous data.
otA G- l
G= I
r-J
N
is the covariance IlRtrix based on the
fV
This is because
H
o
has different
implication~ two
dif-
II
ferent models.
2.1, 2.2
It is of particular interest to note that the procedure in sections
and
2.3
turn out eventually to be invariant under the choice of Jl
far completion
of B into a non-singular matrix, or, in other words, independent of G in the
N
~
sense in which
G occurs up to equation (41).
t'U
Ho~ver,
as has been remarked in the
previous paragraph in a slightly different language, the choice of
~
affects
[8], if a test in that set up were sought
the test procedures of Potthoff and Roy
to be obtained from t he complete model.
3.
Distribution and monotonicity of the test procedures under model (7):Let
r2 =
r
Nl
1..
= Al (A A' )-1 V (V' (A A' )-Iv] -2:
('IJ
A'V[V'A A'
"V
""I ""''''
IV
f'\,J"'-J
,.....,
rv,..",.,~
v'r~ :nx(m-v) '
Then
.-vl
n x v
and
"""
I
N
n
- (
r r,
"'1 1""'1
+
r r')
""2 0'2
is an orthonormal matrix, i. e.
nxn
is an orthogonal matrix.
1..
= I-A'(A A,)-lA
'" tv ,..., -v ,-v
,!':;
r =I
,...'"x ""n-m
Moreover, let
.
1
L+ 2 B*(M' L B*)-2"' ]:
,...
rV l,.Jl
rJ ",,1
pxp be an orthogonal matrix
13
Now, we use the transformation
(46)
1
f"'#
=
r,-2 X r
6.'
N
=
Y: pxn
('xl
s
]4
tV
"',.J
p-q
~'{r
v
The jacobian of the transformation is
q
,vyY 2
1
n
1r,12
, and. with the help of (46),
,..J
(47)
i ~[':; (~~)-~r!
(48)
=
(~ :-~) -! [~1
- !3
!6(~6 !l) -1 ";4]
1
,•
1
[V'(AA,)-l V]-2V' Z' B* (B*' S B*)-l B*' Z V[V'(AA,)-lv]-2= Y'(Y Y6,)-ly4
4 6
'" '" N
N
rv
,v1 rJ1 N,-.I1
r-' 1
(V N
rv,.....N
N
,.J,..J,..J,v
tV
and
(50)
the nonzero ch. roots of
-1
~lSl
are the nonzero ch. roots of
C!4;4 + 35 J'5] .
';6,!6)-1
Q--1 and the matrices
Moreover, since the ch. roots of ("JP1~~
ani
~= C~[Y3 N./
Y~ - ~./f"
y~Y.6'(Y6
y ,)-1 Y6 Y~] rJC'3
rJ rJ 6
r"
,.J
...,./""
""./
1
"'here
C -_ C(B' L
~-1 B)-2
· ~Y2 ' we may ~n
. te grat e out
vv,
I , d0 not contaJ,.n
tV 3
tV
tv
...,
,the joint density function of J1' 13' J4',!5 and J6 is given by
,.J
N[Y1 ;
(51)
(V
~2'
,..,
I ] N [(Y
4 Y5);
,..,,..,
rJq
N
~4'
I
iVP-q
]
N[Y~;
"'./
Then,
0, I ] N [Y
; 0, ,.,p-q
I
]
tv 6
N.-Jq
N
where
1
1
1
g = (B' G r, G B )-2 BI G B £ {V[V'(AA,)-lvr 2 AA'V (V' AA'V '-2) and
",,4
,..J1rJ"" "'"'",1
,...1 tV,...l rJ1
","'''''1 "'1 ..... "',..,1
N
J2
N
"",,-,
<"I
= (~' ~1 ])! ~! [~'~I)-Jrr!, ~ = ~ +(~I E-~)-l(~,
,E-1,,-,B1 ) )1·
14
Let us consider the orthogonal matrices
~]: qxq
(52)
and
Now we use the orthogonal transformations
~:h
(54)
=
(~~):C
~':3 ~
and
=(
1r
§~
~6 I
p-q
n-m-p+q
W4 \
c
q-c
Then, it is easy to verify that the nonzero ch. roots of PQ-l are the nonzero
tv'"
ch. roots of
(55)
Since (55) does not contain
!!J2'
35 and :16' we can integrate out
:J2'
~5
and :16'
and obtain the joint density function of J4' ,15' ;!6' :11,)13 and ,vW4 as
(56)
N[Wl ; 1;5' I c ] N [W3 ;
<:V
rJ
rJ
",
0,
~
Ie] N [W4 ; 0, I c ] N [(Y4' Y5h ~4' I p n]N[Y6;0,I n]'
,..
'" '"
rJ".J
rJ
",
- '2.
",
tV,J p- '2.
""
where
where
G=
...,
(G •• ): cx.v,
J.J
G•• =
"'J.J
are the possibly ch. roots of
° for
i 1= j
J5 J5
or
and
~i' (i = 1,2, .... )
J.
-1
[£(~' ~-l j3)-1£,] (~~Y:)[.Y'(~tJ~f~~ ~
Let us use the orthogonal transformation
(58)
)h
=
~5 ~ ~6'
34
=
~5
23
and
23
=
t-5 YJ8
.
Then the nonzero ch. roots of P Q-l are the nonzero ch. roots of
".., N
15
and the joint density function of ~7' :f8' J3' }4':!5
(60)
N[W ;
tV 7
;
e } I C ]N[W
'" 8
tV
,.J
Now, since )]7
H=
N
~7 -
h.
(i
"'~
.!.
= 1,2, ••• ,c)
i ~ j
[I
tV
+
v
e .
Moreover E(H)
=
f"
is normal when
is the i-th row of
is
6,,.
,...;b
.J6
I
]N[Y ; 0, I
].
tVp-q N 6 rJ N p-q
........
Note that if
are fixed.
and J4
H, the covariance matrix of
~
6-: cxc
°
~: vxv
and
, ..J'(
h.
~~
and. cov (h., h j ) =
for i ~ j.
r-'~,.J,.J
Y4' (Y y ,)-ly4 6J:.]
N 6 '" 6
/V
"'0
N
Now find
N
such that
1
~ J4(!6 ,!6)-1}4 ~6]-2 = .vA, ,: Js where ~ = (r ij ),
and r~i (i = 1,2, ••• ) are possibly the ch. roots of
+
a' a [I
t-",,. y' (Y Y'r
,.,,.,
",v + ,.JO,..,
4 N 6 ,....6
1
N
r 1•
Y6..:Jb
6!
U
(\J
4
= H[I
,.J
,.JV
+ 6
(\J
Y4'(Y
6N
N
Y ')
°
for
1
Y
t-",q-2
TV' 4.~
6 f'I 6
r ij =
Making use of the transformation
-1
(61)
~I"
are given by
are independent normal variates, then the distribution of
28 (!.6 ! 6)- 2!4 J6
(i = 1,2, ••• ,c)
~ ~v
0, T ]N[U~; 0, I ]N(Y4'Y5);
"",.Tc N J .... ,.oJ C
N
N
Y8
and
and 36
=
!:L U
D.- U
t'" 2 ~, N 3
r:-r(
6- u
= N'T,..71
'
we get the joint density function of ("JY4: (p-q) x v, 35: (p-q) x (m-v),
)6: (p-q) x (n-m), NUl: c x(n-m-p+q)
(62)
N[E 2 ;
~, Jc]N[~l;
the nonzero ch. roots of
the nonzero ch. roots of
ch. roots of
91 Jc]N[(!4J5);
PQ-l
N~
as
,.}4'
J p- q]
-1
are the nonzero ch. roots of
C tV....
1) V
f'J
=
° i. e.
~1,Yi)-1~U22P, and.
~6J6)-1
CJLJ4+!5!5)'
-1
v:: 0, the ch. roots of PQ,
,..J....
J'V
N
A,
tr ,.,PQ-l
,y
and
ch_
(PQ-1)
-~ (\J""
__
and the
can be obtained respectively by
using the methods given in Anderson [1], Pi11ai [7]
Koichi Ito [5, 6], and Roy [11, Ch. 8]
and n*
('J
J'~1 are independently distributed whatever rv~ may be. Hence the
distributions of
m*
0, N I p-q] ;
N [Y6;
~
are the nonzero ch. roots of
~P1Sl
We may note that when
and. ,.92: cxv
instead of
s, m,
and
or Heck [2].
n).
or Minoro Siotani [14]
In Heck's notation (taking
H' [or under
H
(I.c-v I -
and
o
0
be
n*
= '21
(n -
ill -
m* =
s*,
The parameters for the distribution of
wunder
.(
s * = mn
c, )
v,
or
'21
P + q - c - 1)
1)
under model (1)]
will
16
,,11
while for the distribution of
= Ch
l
(~l~l)]
under
H~3)( ~l = ,9),
the parameters
are
(64)
= min
s*
m* = ~ (
(p-q, m),
Ip-q-m I -1)
and
1
n* = "2 (n-m-p+q-l).
Now with the help of (62), the results on the monotonicity or restricted monotonicity are the same as given by Khatri [4]
and hence they are not repeated here.
Moreover, instead of model (7) and the hypothesis
the hypothesis
4.
H', if we consider model (1) and
o
H,
= 0 •
o we get the same results as mentioned above, by taking ..../;"'l
..v
2, "Y:
Simultaneous confidence bounds on
[or ~.~
yJ:
For this purpose, vFe may note that if vTe consider P* and Q* given by
rv
(V
(65)
P* = (c
Q
tv
~V-
i"'IJNI"V
('V
~
C" V)(V' R V)-l{c
11J"''''''
,..,.-.",...,
V - C " V) t ,
""',..J'-'"
and tV
R are the same as defined in (23).
and
""'-I""'J""
Q*
rv
=Q
~
,
where
Then using the same method employed
in section (3), it is easy to show that the distribution of the nonzero ch. roots
of P*Q*-l under model (7) is the same as that of the nonzero ch. roots of
tv
....,
under
H~.
Hence, we can find
w,-l
tv"'"
A such that
(66)
where
A will depend on
equal to
c,
Y,
and
n-m-p+q.
Hence (66) implies wi. th prombillty
(1 -ex)
A
at(C
~ V)b
N,...,
N
'V
,-.,.J
(" (atQ
a){b t V' R V b»)
tV,yt'J
I"J
fI""ItJ
t""tJ
~ t-v
1\
< a' (C ~ V)b +
f""'-J
N~"""""
for all non-null vectors ,....
a: cxl
and b:
~
vxl.
same for the simultaneous confidence bounds on
(Le.
N
Sl = I».
~
C
,.....
g V if the model (1) is correct
,y""
s
Hence, we shall not treat the case of C V separately.
If we put some elements of
NN~
a
or ,...,
b
N
dence bounds on the partials of C
~
as zero, we get the simultaneous conti-
V with probability greater than or equal to
~,...;~
(1- ex ).
We may note that (67) will be the
17
Now, let us consider the left hand side of (67).
maximization over
(68)(
b
as
'"
""
.!..!.
[a'(CeV)b][b'V'R V br 2
tV
f'I
N'"
tV
,.",
We rewrite it for the
~
I'V
t'\,J
1
[A(a'Q a)r 2
-
I'V
""
< (a' CTlV
b)[b'V' R V bf2 ,
f¥""'" t-'
~ ~
f"'V
,.",,,,,,
tV tv
JI'J
f"ItJ
and, it is easy to see that maximizing left of (68) with respect to
all non-null vector
b
implies for
N
a: cxl,
[~'(~~~)(S~!)'
<
[Note that if we want to have confidence bounds on
Chj(~'~!)
we have to replace
by
1 + Chj
,!-]t
c{(!'
J,}r
l
a'(C TlV)[V'(AA,)-lvrl(CTl V)'a ,
".J
N"V...,
'"
-,..,"'"
N
(V,-N
'""'"
[{~'(~')-Jr } -l(~f~tQ3r~!3t)-~f,g;)].
Hence, we shall not give here the explicit expressions for the confidence bounds on
the parametric functions of
(C
Tl V)[V' (M'
)
NN"'"
tV"'"
,...I
-J"r l ( CTl V) •• ].
tv
IV ...... N
Similarly, if we maximize (69) with respect to
1
(70)
[chi~l -
1
..[A
1
](Ch~ 3)[Ch;{Y:~!)] ~
a, we shall get
e-J
1
Chi
[(£t,tXH,91,Y)'].
Now, for the right hand side of (67), we may keep the same arguments as
applied by Roy [11, ch. 14]
and finally get the confidence bounds on
Chl(~~~(S~y)t] with gRater confidence greater than or equal to
1
(71)
(chi
~-l
1
~
1
[Ch~ S](ch; (~t ,vR J)] ~
....fA)
ChI
[(~l'!)(£l!)']
(I-a) as
chI (S~YH£:!'y)'
+ { A (Chl 's)(Ch1
X' ~;y)}t
[Note that right hand side of (71) can be improved, but it is not given here.
note that when
H'
Also,
is rejected (71) is always positive.].
o
Now instead of maxinrlz'ing (69) with respect to Na, we minimize the right of
(69) with respect to a, and then (69) will imply
tv
18
ch PQ-l > ", then
We may note that if
c rJ ~
('s~~)(S~.Y)t isp.d. We shall get
'\I ,y 'V
is positive and so
' " " " .....
Chc~l>O if (v~ c). For v<c,we
(67) with respect to a, and then minimize with respect
first maximize the left of
to~.
Chc(C T}V)(CT} V) t
~
(72) the following expression when
By this way, we shall get instead of
v < c:
We give here lower bounds explicitly, for we think that they are important when
H~
is rej ected.
It may also be noted that if we maximi ze (69) [or minimi ze (69)]
to
a
tJ
with respect
subject to some linear constraints, and then minimize [or maximize]
with
respect to these constraints, we shall get
1
1.
1
1
C N
V
[ch~ (PQ- ) - "{A] (ch2' Q)( ch2' V'RV) <
(74)
J
rv""
N
",,-v
1.
ch~ (C T) V)(C T}V)' .
J
N......... ""' ....
The similar results for the independence can be given.
<V
The result
(74) can be
utilized for framing the indecision procedures far MANOVA (and independence) on the
lines of Roy and Gne.nadesikan [13].
the upper bound for
ch
t (C
J
T) V)
(VN'"
(C
T)
This will not be considered here.
Including
V) " we get with confidence greater than or equal
-v""v
to (l-a),
1.
1.
2 Q)(ch 2
(chCN
v
(75)
<
V'RV\
.... : . /
<
.1.
A
"
ch~ [(C sV\(Cs V\']
J
_ ... ;..1 ""'" 'J
+
N
The similar bounds on the partials can be written down easily.
5.
Numerical example:
From the data [8b]
ages (8,
of measurements on
11 girls and 16 boys at 4 different
10, 12, 14), we get A: 2 x 27 to be a matrix composed of 11 (1, 0)
tv
columns followed by
16 (0, 1) columns, and other statistics are given in table 1.
19
Examine the question that the growth curves due to girls and boys are linear.
(i)
(:i i -i -i)
,1}1' =
In this case,
and
J3' = (-~
For this case, we use the test procedures given by (40).
t~)·
-i i
Using the values given
in table 1, we find
.33
zt B*
...... 1
rJ
=( -1.13
Z'B*{B*'
S
'" ",,1 ....1
N
1
"
Nl.
= 2,
with
s*
2010
level.
Bif" S
B*
",1 '" ,..,1
(1263.40
=
13.52
.00010677
M)-l B*' Z =
",1
1V1 ""'
( -.00064836
=
Ch1 (P1 Q.:- )
Hence
.05) ,
-.81
ch1 [M t Z tB*l(B*l' S
tV
N
'" N
~ and n*
m* = -
= 11,
Hence, in the latter
N
""
13.52 \
105.12
J
and
-.00064836 \
.00707551 )
B*l~IcJl8."Z:::, o. 113865
and
w'l = 0.10223
N
tV
which is insignificant even at more than
part, we shall assume that the growth curves due
to boys and girls are linear.
(ii)
Now, we examine the question whether the linear growth rate due to girls and
boys are equal or not?
Here ~ = (0
or (25) or (26).
rv
A.
j
Then
at
=
-
.01130476
(-1 1), and we shall use test procedure (24)
= (22.6628
level.
-.00664795) ,
( -.00664795
.36588157
24.9340)
.474366.829833'
Chl:~l= 0.28262
210
~'
For this purpose, we get from table 1,
(B'S-~) tV.v
1) and
and F 1,23
= 23
B'S
N
-L
N
-Z
(.2530441
.2763562)
.0229005
. 1378823 .
=
-.J
R = (.09101586
tv
x 0.28262
-.00064836
= 6.5
-.00064836 ) .
.06957555
which is significant
Hence, we conclude that the grovrth rates for girls and boys are
different and the
two growth rates is
9510
confidence interval on
S22 -
S21
= difference
between
.
,
20
This shows that the girls' and boys' growth curve s are different, but th ey ar e
linear on account of
(iii)
(i).
Obtain simultaneous confidence bounds on the girls' and boys' growth cruves.
For this purp:>se, we take A =
(2/22)F •
(2,22) = 0.401.
O 025
Then using
= (1
t)
where
,.,b' = (0
1)
for boys' curve in (67), we get the simultaneous confi.dence bounds
a'
N
= age
t
minus eleven, and IV
b'
with probability greater than or equal to
= (1
0) for girls' curve while
951> (refer Potthoff and Roy [8b]
as
follows:
for girls I growth curve
e
~
1.
(24.934 + 0.82983t) i
for boys I growth curve
(iv)
g21t, while
+
0.16703 (89.414 + 3.2492t
g12 +
+
2.7731t 2 )2
£22t.
;fJz-""""'"
The results given in (ii) and (iii) are different tBaa those given by
A
Potthoff and Roy [8b]
by taking a special type of matrix IV
G using previous data.
Here, we give below for comparison the results based on the iE st procedures given
by Potthoff and Roy [8]
when
G = I.
N
(a)
tv
The 95~ confidence interVal on
~
o. 024
g22 -
£22 -
~l
£21 .:s o. 588
while by using the calculations [8b] with their special
(b)
The
95~
is
/i'
~2-
~l
1s insignificant.
simultaneous interval on the girls I and boys I growth curves
respectively are
(22.648 + .4795t)
i
0.1609 (94.415 + 3.145t + 3.0445t2 )
and
e
(24.968 + 0.7855t) i
0.1500 (94.415 + 3.145t + 3.o445t2 )
These results are practically similar to ones we have given in (ii) and (iii).
6.
AcknOWledgement:
The author wishes to thank Professor S. N. Roy for his valua-
ble suggestions and help in preparation of this paper.
TABLE 1
,...;
Matrix ..a
(S. P. matrix)
(\j
68.20
104.91
72.93
02.68
137.64
68.20
97.51
67.0l
Matrix
(means)
97.51
72.93
161.11
102.99
67.01
82.68
102.99
1.24.34
~
21.18
22.23
23.09
24.09
22.87
23.81
25.72
27.47
Matrix M
Matrix 12(model matrix)
.<:att;e) :;1 0 )
1
1
1.
1.
-1
3
-3
1.
-3
-1
1.
3
-1
1
1
-1.
\
Doo-Little method
21.18
67.01
97.51
.7084423 .4868498 .1538797
68.20
0.4954955
137.64
1
22.87
.1661581
71.1172069 24.6142338 49.4768465 11.7354053 12.4780179
.1754571
1
.3432386 .6957085 .1650150
·
·
1
.00726533
-3
-.02179599
.5045045
.007093986
.4864865
.00684063
-
·
·
·
•
•
·
·
83.581.2362 38.5349180 4.0571480
1
.4610475 .0485414
..
·
39.5283046 3.7435621
1
.0945050
5.2349872
.0626335
.1l83923
2.9583460
.001416494 .03539486
5.2411042
.1325912
2.7581586
.1075777
.002721536 .06977680
From tbe above, we get
B'S-~
--.J
,..J
,-.J
0107577~ ~ .00726533
'.007093986
.00684063
.001416h9'+ '.002721536]
.03539486 .06977680
010757,\'] [01538797
'.1650150
.1754571
.0485414
.0626335
=
[-;
.5045045
.4864865
.1.183923
2.9583460
2.7581586 -.02179599
=
[-;
.5045045
.4864865
.1l83923
2.9583460
2.7581586
I
and
B'S-1,.....
Z
,..,t-J
e
e
.1661581
.0945050
.1325912
e
]~
,
f l .
22
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