UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
INCOlVIPLETE MULTIRESPONSE DESIGNS
by
J. N. Srivastava
November 1962
Contract No. AF 49( 638)-213
This paper discusses in detail a simple example of
an incon~lete mUltiresponse design, defined therein.
General formulae have also been presented to enable
an experimenter to analyze a certain wide class of
such designs; of which the example is a member.
This research was supported by the Air Force Office of Scientific
Research.
Institute of Statistics
Mimeo Series No. 340
INCOMPLETE Iv!ULTIRESPONSE DESIGNS
I
by
=
1.
~
===
J. N. Srivastava
University of North Carolina
======- - -- -- -- -- -- - - - =====
== =; == ; = = ====
Introduction
In an earlier paper
1)..1,
Roy and Srivastava introduced two classes of de-
signs which are specially suited to nmltiresponse experiments.
Here we shall con-
sider general classes of designs which are characterized by the fact that not all
the variates are measured on each experimental unit.
Such a situation arises
when either it is either physically impossible or otherwise inadvisable or inconvenient to measure each response on each unit.
As an example, suppose an anthro-
pologist has unearthed a number of skulls, each of which is in a partly mutilated
condition.
Then the kinds of measurements that he can take on a skull may differ
from one skull to the other.
A sketch of the general theory of incomplete-multiresponse (I-M) designs is
For that the reader is referred to ~~7.
outside the scope of this paper.
general discussion using likelihood ratio tests will be found in ~l7.
follows we shall discuss a very simple example of an
the general theory.
At the end
8;
I-M
Another
In what
design to illustrate
few rerr.arks will be made on the general theory
itself and on the new combinatorial research problems which it poses.
2.
f.:. Simple Example of an I-M Design
Suppose we have
mental units.
p=3
For each
variates, and u=8
i
we choose
on each experimental unit in the set
p.«
1-
S .•
1
Sets S.(i = I, 2, ••• , 8) of experi1
p) variates, each of which is measured
This choice of variates is called the
~his research was supported by the Air Force Office of Scientific Research.
2
variate-Wise design and is denoted by D •
l
as shmm belovi:
In the present example, let
Variates
8
i
a
1,2
2,3
1,3
1,2
is supposed to have a (univariate
fined over it.
be
s6
Set
Each set
Dl
block-treatment design
For simplicity again, we suppose
BIBD with parameters
(v,
r, A)
where
v
2,3
D2i
D2i
the same for all
de-
i, being
is the total number of treatments.
The above then fixes the I-M design of our example, viz. the (u+l)-tuplet
(D l , D21 ,···, D2u )·
Let the
p 'true' response to the different treatments be denoted by the
vxp matrix
s
(1)
.................
=
vxp
~
~vl
e:
v2
•••
\7
,
~VP..i
'
= [?1' ~2' ... , ~pJ
where
Let
say
,
g j£ denotes the true value of the £-th response to the j-th treatment.
~(i)
order) of
be a
~
columns (in
matrix whose
Pi
columns are those
which correspond to the
Pi
variates studied on the set
vXPi
Pi
S .•
~
In
the example; we thus have
(2)
It is easy to see that there exists a
= 1,
i
Let
(s
y!
-lS
denote the
= 1,2, ••• , N.;
1
i
pXPi
matrix
B
i
such that
2, ••• , u.
vector of observations on the s-th unit in
= 1,2, •••
u).
Here we have
N.1 = vr, for all i.
8.
~
Furthermore,
3
assume tbcl.t
1', or
if
p.xp.
is the
and
colu~s
~
~
variates.
p
=i
l
matrix obtained by taking the appropriate
of a covariance matrix
matrix of all the
i
~(pJ~),
p.
~
rows
to be called the population dispersion
In our example, we have, say,
fCTll CJ"12
l.: =
I
0"22
!sym.
.---
0"13 \
1'"0"11
Sym
=
E(:5)
CJ"33
L
etc,
j
It is easy to see that
(6)
E(i)
= B!
E B.~
~
i
= 1,2""
u ,
Also we shall make the usual assumption about treatment effects viz,
J
lv
~ n =
_'"
0,
£ = 1, 2, .• , p
Novl consider a fixed set of units
Tij £
and the design
D2i ,
Define
,
=
number of blocks in
=
total (for the £-th variate) response to the j-th treatment over,all
units in
Bij £
Si
=
Si
S.
to which
~
j-th treatment was allotted by D2i ,
Total (for the £-th variate) response over~ll units in the g-th
block in the set
Si •
4.
=
number of units in the g-th block in
is allotted under
r
ij
j-th treatment
S."
to which
J.'
D2i .
=
ntunber of units in the
g-th block in S]..•
=
number of units in
under treatment
Si
(In our example, let K.. = K, r.. ::: r
b.
.].J
lJ
1
~].
n
B
and
finally
Qij£ ::: T ij £ - K £oJ
'"
n'
g=l
lJg
19~
j.
for all
i, g, j.)
fQi,el-1
:::
i.
!
I·
I
lQ~£v ,:
From the theory of block-treatment designs it follows that for all (jfj') and
all
i,
(8)
:::
where
Ci
is a vxv
C.
J
].
natrix with elements given by
1
= - K'
C.{j,j)
J.
=
=
r(l -
1
R)'
here
here.
Also define for convenience,
G,;-r...
= (Q·1,O·2,···,Q·
-:l..2'i
-- J.p. )
:::
, say,
1
so that
(10)
:::
C.
i
J.
VXV
VXPi
= 1,
2, ' •• J u •
J
5
The general theory is concerned with those I-M design,s for which there exist
known matrices
where
F , F , ••• , F
such that
2
l
m
a.~q are known scalars.
This condition is satisfied in the present illus-
tration and we have
,
m = 2,
(lla)
A
= - Ie '
So far as
als
for all
are concerned, we first observe that using
1.
(7)
(10) in the form
(12)
where
re.
=-
(Q.)
~
0".'
S
J.
J.
+
0".
J.
J
7
vv-
~
(i)
i
are arbitrary constants.
(13 )
••• + F
m
as is the case in the example.
=
= 1,
2, ••• , v
Suppose
J vv
,
Then one can write
,
(14)
where
(15)
a!~q
Define for
q
= a.J.q
= 1,
+
0".
J.
q
pX1:p.
~
i
and
2, ••• ,m ,
= (alf q
L
for all
B , ••• , a' B)
uq u
l
g
,
one can write
6
Now
(Qi)
c.J. g
=
B
i
m
= (!; ai q Fq)
q=l
~
B
i
m
= !; Fq
q=l
S (al q Bi )
J
so that
fL
~\
l
(16)
I \
I~2 I
!"L I
( Q) = (Flg, F2b ••• , F ;)
m
•
r
',.. m.J
LL' (mpXn1p),
Consider the matrix
In our example,
,
mp = 6, and !;p. = 18
J.
so that
(1'" )
mp
< 1:pJ..• Also by direct calculation
LL' =
I
u
2
.1: ail
J.=l
I
U
!;
i=l
Di
U
L
D.
J.
is a
correspond to the
I
i:
i=l
I
,2
a'J.",
0
J
D
i
}
~
_J-
px:p diagonal matrix containing unity at those
p.
J.
variates studied on
(18)
ai2 = -
t
I
U
1: ail ai2Di
I i=l
where
.--""/
i
ail a i2 Di I
I
,
"A.
Ie
+
0-i
say
p.
J.
S., and zero elsewhere.
J.
places which
Now
7
It can be show thdt the matrix
zero.
1L'
will be SinL,'1..11ar if we take all
convenient choice here would be
A
(lS')
=
0, i
= 2, 4, 6, 8 •
Then we get
rv
= k"A.v = v\h,
rvl
""i1
=
"A.(v-1)
= (v-l)a ,
k
I
a i2
= 0
= -0:
c-
oay,
,
i
= 1, 3, 5,
7
,
i
= 2, 4, 6,
8
i
= 2, 4, 6,
Hence
u
1
2
a'l
D..~ ::::
J.
i=l
".
NI
rvt
""il ""i2
r. . a122
"-.
D
i
i.(.
l
-
Di
=
2
30:
1
3
;
so that
I
Lt.'
2
= 3a
2
(2'v -2v+l)
I
I - (v-I)
.....
- (V-I) I
I
1
r
~~.'
1
3
(~~
u.
er's
to be
8
Hence
is nonsingu1ar and
I,L'
-,
+(V-1)
2
I
(20)
+(v-1)
\
!®
1
3 = LH1 ,H2_7, say,
(2v -2v+1) J
Thus we can write
rQ L'
-
(LL'f
1
-
7 = (F1 ~,
,
F2 1;)
or
(21)
Now
L'
HI
= (Li L2) HI
rail
=
B'1
.•
- B:
at
u1 u
-
ai2 B'1
a'u2 B'u
"
\
\.
l(ail + v-I ai2)
=
1
22
3(Xv
II
- I
1
2 2
.,Ia v
o
avB'
\.
!
I
3
0
I
!
I
l
I
I
lavB'5
o
avB+ ,
L
0
I
_J
+(v-1)
B' -\
1 ,
! (a' + v-I a' ) B'
u
u2
L
ul
I
7;
2 2
30: v
-
r-avB'l
=
1
I
\
1
~;
1
3
l
\
-
[.
l~)
1
3
9
Hence
1
(22)
3cN
where
and
(24)
Similarly
Q L' H :
2
=
1
3a:v
2 2
1
3a:v
[+
(V-I)
\'
L__
!
1,3,5,7
=
, say
,
( %Bj)-
")-,
G
2,4,6,8
,.
\ -I
\ Q.B! }
~
~,
J
10
where
Zj2 are defined analogous to Zjl' except that the coefficients
(0) are replaced by
(v-l)
and (-1) respectively.
1 and
It is easy to see that
v
v
'\. :
/
.'.-.•.•. 1
'-·--'1
;'
Zjl-
L__
j=l
Z
j2
j=l
We therefore consider the
2(v-l)x3
matrix
rZ
I 2l
•
I
(25)
Z
=
I Z~l
~
Zl-1
=
Z22
---I
say
~ Z21
kJ
2
and first examine the
2(v-l) vectors for linear independence.
l'1e observe that the
Q.o£
~J
(i=1,2, ••• ,u; j=2,3, •• •v; £=1,2, ••• p), (considered
as linear functions of original observations
and so is
Z22"",Zv2'
Also for any
y1 S ) are all linearly independent,
j, Zjl and Zj2
are linearly independent,
since the matrix
rl
!
(26)
!
I
I(v-l)
is nonsingular.
Thus, no Zjl
could depend on any linear function of Zj2'
which shows that the rows of Z are linearly independent.
Let us now calculate the variances and covariances of the
that
Zj~'
We recall
11
) = - ''''
~
c ovu. r ( n 1.J.t.
° r, Q° , • ,
1. J .t.
",j
for
i
= 1,
f. ,
0
2, ••• ,8; j
= ex
V
=
0
0'£
n,' ...~f
JO~Jot
'f'
i=i',
K.
0'££"
if
i=i', j=jt
,otherwise,
= 1,2, ••• v;
£
= 1,2,3.
3av
=
Thus
1
3a:v
• av.2
' (j'
"
11'
(2
1
3av "3
=
0"12)' etc. ,
Hence
(27)
var(Zjl) =
1
L
3av
where
*,
j = 2, 3,
r
!I
*
=
V,
2
2
'3
0"11
~
...
0"12
'3
0"22
'3
0"13
I
!
2
I
•
0"23
I
I Sym.
0'
33
L.
Similarly an easy calculation shows that
var(zJ 21)
o
=
=
122
2 2 f'3(v- l ) + 3v
9a
_7
av 0"11
V
1
3av
2
(2v -2v+l) 0"11
,
and, in general, that
(29)
=
(2v 2 -2v+l)
3av
.
~
*
Exactly in the same way, we further have
=
(v-l)
3av
~
*
,
_.I
12
.and ;for
J,lj'
1
=
cov(Zjl,Zj'l)
COV(Zj2,Zj'2)
=
cov(Zjl,Zj t 2)
=
3ew
1:
2
,
*
2
(2v -2v+l)
2
-
*
1:
,
3ew
(v-l)
*
1:
3ew 2-
The above can be written in the compact form
rz21 -
II
11
...
iZ 1
(30)
Var
1··::'_· I
IZ22
"
1Z
:~
v2
~I
=
\
\
1
3ew
(v-l)
2
(2v -2v+l)
l(v-l)
I
..i
Iv
-1
-1
v
-1
I
Cb\
v
Sym.
W
*
®
1:
..
-1
-1
-1
v
L.
=
®
2
\
I@
1:
J
(V-l)x(v-l)
, say
•
Clearly the two matrices on the r-h-s are positive definite, and so is
order to reduce
(30)
Z.
*
In
to the usual model for multivariate ana~sis of' variance
(MANOVA), we first factorize as below:
W
which is possible since
*
=
W is
RR'
p.d.
,
Then
13
CF
..
\L F20
_
, 10
-1
R
2(v-l)x2(v-l)
I
I
1
2(v-l)x(v-l)
where
FlO and
and F
2
F
are obtained by cutting out the first row and column of
20
respectively, and So
We then find the estimate of
is obtained from
~
F
1
by cutting out the first row.
as
~
'·0
where the matrix inside the curly brackets can be shown to be positive definite.
NOvT
1
W-1
.3ew
::=
1
Ij(V-l)
2
"\ -1
(v-l)
Iv
1
I
(x1
-.::;:
2
-1
v
-1
..
-1
-1
-1
-1
(2v -2v+l)
'-
I
-
30:
=
2(v+1J
tsYIDo
...,
2
®
I
I
1
F20
and hence a substitution in (33)
~o
where
::=
v:,
::=
J
J
\
~-"
.3
-(v-l)
(2\7 -2v+l)
L -(v-l)
W
v
v-l,v-l
1
1
.3
1
1
...
Sym.
-
I v-l
1
,
and
3J
,
gives
£(1 - l)I
fJ
(1 _ ~)I
+ v"J
2v v- 1 + v v- 1 ,v- l'
~
cV
v- 1
v- 1 ,v- 1_7
VII
are certain constants depending upon the chosen values of
on the value of v etc.
Thus one can take as an estimate of
~:
,
(JI
S ,
and
14
=
1
1
(l 2V)
2v
If one Vlants to test a linear hypothesis regarding
standard multiresponse analysis of variance tests.
~,
one can use the
Thus the above method of
analysis leads one to a valid test procedure.
In the model (31), consider the nature of E* • This matrix
is the same as
E, except that the covariances in E* are exactly two thirds of those in E.
This means that if
*
is a typical correlation obtained from E*, then
P££r
2
< "3
In a situation like the one here, where the absolute values of the correlation
have (knovm) upper limits other than unity, it is not know at present which of
the three procedures (viz. largest root test, trace test, and likelihood ratio
tests) would have greater power.
:3.
We shall now present the general forr,lUlae for the analysis of a multiresponse
design, particularly the general conditions on the I-M design so that this analysis
can go through, and various other crucial points in brief.
here have been reproduced from.
The formulae given
£':::7.
Procedure for Analysis
(36)
(i)
First calculate
••• , F
m
el ,
C2 , ••• , Cu.
such that (11) is satisfied, and such that
ill
Li
q=l
Verify whether there exist matrices
F
q
=
J
vv
15
(ii)
LgL~, (g, g'
Next obtain the matrices
a!~g (i
in terms of
such a way that
(iii)
= 1,2, •••u;
g
= l,2, •••m).
= 1,2, •••m)
Choose
~i (i
and hence
LL'
= 1,2, ••• u)
in
is nonsingular.
LL'
Notice that
LL'
is nonsingular if and only if
~£
(£
= 1,2, •••p)
are so, where
a!la!
~
~m
1:1£
=
•
•
ea.
•
•
e
•
•
0
'1
•
~,
• • '*
/~..1
ieU
where
U£
is the collection of sets (among
8 , 8 , ... , 8 ) on which the
2
1
u
response is measured.
~.
£
1J.l . • •
...
...
Hgg ,
= diag
,
2
l
(ngg rJ re' gq" ... , re~g')
£
,
= 1,2, •••
P ,
(g, g' = 1, 2, ••• m)
and observe that
fHll
(LL' f l
=
\. . • . .
LHml
( i v)
H12
Hm2
H
.•
. . lm•
•
H
mID
= LH1 ,H2 ,···
J=
H , say
say,
g
Hm-7
Calculate
and
•
= 1,2, •••
m •
,
£-th
16
(v)
Calculate. the qua.ntities on the
L .h.s
below and verify whether they
can be factorized in the form shown on the r.h.s.
(a)
\ '-I,
/
L-...
'Yu'
j
(b)
,
£ ~ (
Jr q' q I
\
a!~q ,
m
\---1
\
/
L_,_'
a'iq11: £qq' \!
=
/
.. ,
w·JJ
qq'
•
, q==l
for all
(vi)
= 1,2, •••
£'
I.,
:pj
q,q'
= 1,2, •••m;
j,j' = 1,2, ••• v •
Verify whether the m(v-l)xm(v-l) matrix
w
==
is positive definite.
(vii)
Also verify whether
dependent.
(It
~hould be
= 2, •••v;
E
= 1,2,.~.m)
are linearly in-
noted t.hat this also will depend on the choice of cr.).
~
F
go
I
I
L~m J
~2
=
~o
•
F
'-- mo
I
and ~o are defined as earlier.
) Zl'\
Var
I
=
Z2
Ii. m
where
q
If all the above goes through, 'then the following holds:
-,
lZl II
~F~O
(viii)
where the
Zjq (j
j
W
Q{l
_/
2:
*
,
Also
17
*
0"££/
=
Y£ £ t 0"££ t; 2., /.
t
= 1,2,...
P
From here on the analysis proceeds as in the example above.
Any mUltiresponse design for which the above analysis can go through is
called regular.
It is beyond the scope of this paper to go into the detailed
characterization and properties of regular I-M designs, and the new and interesting
areas of research they lead to.
However, some brief conrraents on some of these as-
pects will be offered.
It turns out that for a regular design m should be large.
above
properties
Henc e we should take
In case the
hold, it can be easily checked that
t'
fPl+P2+' • ·+P
:P
u
m as near
notes the largest integer less than
l
~
as possible, where
f
a_]
de-
a.
Notice that except for (36) (ii), we have not put
an~r
other condition on the
matrices
F , F2 , ••• , F • However in special situations, the proper choice of
l
m
these matrices may be very important. This choice determines how the different
treatments stand relative to each other in the over all design.
It will also
affect the facility with which the matrices involved in the final estinlate of {
can be inverted.
If one chooses
F , F2 , ••• , F as the ill association matrices
l
m
arising out of a (generalized or ordinary) partially balanced association scheme
with (m-l) associate classes (though it is not suggested that this should necessarily be done), then one interesting problem for research would be to study PB
association schemes with large m, and their use in I-M designs.
However the central problem of I-M designs is to have matrices
the design D , and the choice of
l
but also that (ii)
E*
O"i
Fl' F2' • • • ,Fm,
such that (i) not only is (36) satisfied
is (in some sense) close to E, (iii) the variances of
.
18
the treatment contrasts of interest are small and the power' of the MANOVA test
procedure that we use is relatively large.
The author is thankful to Professor S. N. Roy for going through this faper
and for many stim121ating discussions.
REFERENCES
1-~7
Monahan, I. P. (1961).
Incomplete-Variable Designs, Unpublished Thesis,
V.P.I., Blacksburg, Virginia.
~~7
Roy,
s.
N. (1957).
~~7
Roy,
s.
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