•
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
HIERARcmCAL AND p-BLOCK IvlULTIRESPONSE DESIGNS
AND THEIR ANALYSIS
by
S. N. Roy and J. N. Srivastava
November 1962
Contract No. AF 49(638)-213
This paper introduces two classes of multiresponse designs: (a) hierarchical designs, and (b) designs with
p block systems. The former will be useful for those
situations where the responses could be arranged in a
descending order of importance, so that with respect to
any two responses it is known which one needs to be
measured on a larger number of experimental units. The
latter class is called for when the nature and the pattern of heterogeneity in the experimental material differs from one response to the other. The analysis of
such designs together with other properties have also
been discussed.
This research was supported by the Air Force Office of Scientific
Research.
"
Institute of Statistics
Mimeo Series No. 341
HIERARCHICAL ~JD p-BLOCK ~IDLTlRESPONSE DESIGNS AND THEIR Al1ALYSIS l
by
S. N. Roy and J. N. Srivastava
University of North Carolina
= = =1.
=
= = = = = =====a: ;: = = = = = = = = :: = = = = = = = = = = =
~
7-
Introductory' Remarks on
M~iresponse
Designs in General
In any planned experiment what we capture first under our sampling scheme
is a set of erperimental units and according as we study each unit on one or
several responses the experiment is called a uniresponse or multiresponse experiment.
As already stated elsewhere, the overall objective of the experi-
ment being what it is, namely the study of the relationship between the set of
responses and the set of factors and also (in a sense) among the responses themselves,
a11"
appropriate design for the allocation of the units has to be made with
an eye to this overall objective.
Till now, while choosing a design not much con-
sideration has been given to the multiresponse aspect of the experimentation,
the choice being usually made as if we had only a single response under study
and we wanted to carry out that study in a reasonably efficient manner (defined
and described in standard books).
Also till now almost all those that have con-
sidered the analysis and interpretation of mUltiresponse experiments have merely
taken over such designs, and then assumed that each experimental unit is studied
on all responses or chs,racteristics and then carried out the analysis accordingly.
(The only exception, known to the authors, is Monahan [)}.)
However, a little
reflection and even a slight acquaintance with experimental situations will indicate that in many such cases it is neither necessary nor even feasible to
study each unit on all characteristics.
This immediately suggests that every
design has two facets, one relative to the treatments or factor-level combina~
tions and the other relative to the variates or responses.
For uniresponse
~his research was supported by the Air Force Office of Scientific Research.
2
problems the latter facet, of
course, is absent.
The first one is concerned
wi th the a llocation of the experimental units aver the treatments or factor-
level combinations but even here the other aspect (relative to variates) may
well play an indirect role.
In other words, even this allocation rr.ay have to be
made in different 'Ways according as the overall objective is one of uniresponse
or
~ultiresponse
study.
This paper discusses two classes of multiresponse
designs, to be designated as the hierarchical and the p-block designs.
For the
analysis of these two classes of designs, a step-down procedure has been
suggested.
2.
General Description of Hierarchical Designs and the problens Under that
Class
Suppose that on consideration of relative cost or difficulty or on other
ccnsiderations based on a priori information, the different variates or responses
have been arranged in a descending order of importance (for operating purposes)
such that for any pair of variates it is known which one needs to be measured on
a larger number of experimental units.
named VI' V2 , ••• , V
p
Suppose there are
U , U , ••• , Up
l
2
joint.
experimental units.
J
variates that are
in the abave order, and suppose these are measured re-
spectively on the sets of units
Let U.
p
consist of
n.
J
The ll1ultiresponse design determined by
that are not necessarily dis-
(U ' U2 ' ••• , Up)
l
will be called
. • •. -; U
= P Consider the set U.•
J
This, as usual, can be divided into blocks through homogeniety or similar conhierarchical if and only if U
l
siderations for the variate
the set of blocks over U.
J
V •
j
~~
U :J
2
The allocation of treatments together with
defines a (univariate) design D.
hierarchical multiresponse design vdll then be denoted by
(1)
J
over U.•
J
The
3
under the condition that
U. ~ Uj l '
J -"
+
(2)
(j = 1,2, ••• , p-l) •
In set notation define
U! = u. - Uj+l'
J
J
(3)
j == 1, 2, ••• , p-l ,
U' = U
P
Then for
(4)
P
j = 1, 2,
...
)
n! = n. - n +
j l
J
J
p,
U!,
u
,
represents the set of
(np+l
= 0) ,
experimental units on which only the variates
V
l,
V ,
2
••• , V
j
are studied.
To delineate the problem and describe the procedure we have to use some f'u.rther
notation as follows.
r-th variate (r
in Uj.
nj
Let
zj
-··r
= 1,2, ••• ,j;
If ~j(j
denote the
j
n!xl vector of observations on the
= 1,2, ••• ,p),
J
that are made on the
n!
J
units
= 1,2, ••• ,p) denotes the njXl vector of observations on the
units studied under the j-th variate, then we can write
r
.
-,
I·
\
:ZJ
-J
Y.
-J
==
-r
I
j j+l!
Z
I·
I
eJ
_J
r~Iz~
Also let z~s (j = 1,2, ••• ,p; r
element of zj.
\
= 1,2, ••• ,j;
On each unit in U!(j
J
s = 1,2, ••• ,nj)
= 1,2, ••• ,p)
the
j
denote the s-th
observed values
(on the different variates) will be supposed to follow a multivariate normal
distribution with a
jXj
dispersion matrix Ejo
It is clear, that for all
j ~ 1',
L.
L. j
is the
(jxj) top left hand submatrix of the total (px:p) matrix
( ::. L., say), corresponding to all the
p
vuriate
Given any single
V., the expected values of the observations on V.
J
m.xl
(S'l" •• ,Sj
such that
J
will be supposed
J
to involve the
vector of
J
mj )
parameters
ill.
J
k'
(6)
--Js
= 1,2, ••• ,nj),
k,
and
j
k
where ~jS
s(i.e.,
is an
= 1,2, ••• ,k;
j
mjxl
k
= 1,2, ••• ,1';
column vector of known constants
D..
given to us by the design
The null hypothesis
S. , 'tVith components
-J
,
a.
for all permissible
s
variates.
l'
.J-.J
H
o
sS
on the
is then given by
f
'e
b
n rL-J<p·
j=l
II :
o
where
C
j
is a
portant points.
s jXlll
j
matrix, and
(say) .
=
~j
-
.2
is an
S jXl
vector.
The first is that the expectation model here is not the same
as the usual multivariate model.
The second is that, as "Till be shovm later,
the attempt to obtain a step-down procedure for testing the
forms
(8)
where
H
o
H*o :
D.
J
ri
L..
is a known
j
-.:.
-
(D.
J
\
s.m.
J
J
vectol' of unknown parameters given by
of
(7),
trans-
0
p
Q.)
-J
rnatrix,
H
o
H* given by
in a natural manner into an
~\
j=l
Notice two im-
=
=
n
H*
j=l
0
an
SjXl vector, and
oj
Q.
-J
(say) ,
an m.xl
J
5
,-~~ -II
2.2
=
Q. 1
-J+
,
j
0, 1, 2, ••• , p-l
==
i
j
.5.j I
I
5.j+~
with
Ql
-
e:
::::
~l
(10)
ro j
·::?'r
where
is the r-th element of the vector 13., which is given by
_ f
A
- -.:.::.r I-'jr '
r :::: 1, 2, '.'J j;
-J
'0'
'e
(11)
~j =
l,j+l
-1
2: j
Further!nore, it also turns out that on any unit on which at least up to (j+l)
variates have been studied, the (j+l)-th variate, given the others is conditionally
distributed as a normal variate with variance given by
ll:j +ll/IE.J I = O'~J+ 1
of
Y. 1
-J+
sc.y, (j
= 0,
1, 2, ••• , p-l), and the conditional expectation
given by
(12)
where
j
Y.
= l, ••• ,p-l
is a matrix to be defined in the next section in terms of
j+l
and where B .+ l , of dimensions (n. l)x{~ mg ), is given by
J
J+
£=1
J
Af+l ~+l
(13)
Bj +l
=
•••
•
A~+l
I
I
.
\
Aj+l
j+l
. ..
1)
A/
j+l
,
,
(yl, ••• ,Y.),
-
-J
6
and the
matrix
,
(1)+ )
For
j=o, i.e.
(£
~
k;
k
= 1,
2, ••• , p)
at the first stage, we get in place of (12 ),
(15)
~1
H* Pn H*. 'of (8)
o
. 1
oJ
J=
under the model given by (15) and (12) and note that the testability of H .
OJ
of (7) is thrown back on that of its equivalent of H* .• Assume that H* . is
oJ
oJ
testable (the conditions for which will be discussed later), denote by ~. the
We now have the problem of testing the hypothesis
-J
usual least squares estimate of
~.
-J
f\
pj
the dispersion matrix of l j ' by
7
J-
rY.:B.
-
J
and by
T.
J
of (8)
under the model
the rank of
the rank of D..
Also let
J
B , by
j
2
sJ'
square in the least squares estimation at this stage.
(12), by W. the
PI
J
the rank of
be the usual error mean
Then it turns out that
under
(16)
F.
J
-
/\ I
(nj-'Pj)
(
)(
~
.
-J
T.
J
tit
has an F-distribution with d.f.
T
H* ) are independently distributed.
o
follOivS:
j
and
(nj-Pj)' and that the
F 's
j
(u!lc1er
The stepdown procedure suggested is.as
7
H, i.e., H* if F. <
o
0
J and reject otherwise, where ~
(17)
~,
Accept
p
II.
(18)
i=l
rF.J -< ~
Prob -
I
.
= 1,
j
2, ••• , p,
is given py
* 7 =1
oJ-
H .
-
a •
In the next section we sketch the mathematical justification for the procedure
proposed.
3. A Sketch of the Mathematical Justification of the Procedure Proposed
(Assuming Testability of H* .'s)
,
OJ
Toward this end we try to find the conditional distribution of' Y. l'
-J+
given Y , ••• , Y.• Put
- l
-J
Z~'
-J
-1
j '+1
Z.
-J
,
=
Y .,
- jJ
j
~
j'
= 1,
2, ••• , P ,
Z~
_-J
whence
Y..
-JJ
= -J
Y.,
j
= 1,
2, ••• , p •
Also we have (in terms of
(20)
r
p.d.f.)
. l'···'Y
11 -y1 ,···,y·
. 1 I -Y1 ,J+
. l'Y
-J-7 = pry.
- -J+ 1 ,J+
- 2 ,J+
- j , j +1- 7
prob. - -Yj +
=
r
. l' -Y2 ,J+
p- -Y 1 ,J+
. 1'·'" -J,J+
Y. . l' -J+
Y. 1 ,J+
. 1- 7
prY
. 1- 7
. I'Y
. 1'···'Y'
L - 1 ,J+
- 2 ,J+
-J,J+
To comp te the right side of (20) we
j+l
j+l)
( zj+l
Is ' Z2s "."Zj+l,s
is
obsel~e
that the p.d.f. of the vector
8
-1
Const. exp ('2
(21)
r'
_ ~l 2.: -1
j+l~l-7)
'
where
(22)
u'
-1
=
_
{ -r zj+l
ls
( j+l)'
r.'
'~ls
'Whence the conditional distribution of'
(-1) ( '
r -2-
(23)
'Where
'e
const. exp _
~2
]
z1:i,s' given
-1.
t
-1
~l 2.: j +l ~1-~2 2.: j
... ,
l
zi: ,
7
is obtained by cutting out the last canponent of'
~i.
2
(J".
J+
It is chech:ed
1 equal to
2.:.1, and (conditional) mean given by
J+ 1 1/1 J
(24)
E(Z~+l
J+1,8
+
~.
-J
( 25)
j +l
Iz ls
'
Zj+l
2s'
)
... , Z~+l
JS
(say)
j+l
j+l
j+l)
( Zl ,Z2 ' ••• , Z.
(j.,
S
s
JS-J
being the vector defined earlier (11).
is
~2 _ ,
that this conditional distribution is normal 'With a variance
12.:.
... ,
rz j + l ( j+l) t F" 7
28 - !:2s
2.~'
2}: ) -
Thus 'We can write
EC(zj+l) - Aj +l ~j
A~+l ~j
+ A~+l ~~
Aj +l ~
'"'j+l - 1
.21 + -"2
2.2 + •••
J
.... J + j+l2.j+l
j+l A
+ Zj
.t:.j
,
9
j+l
and ~~(k=1,2, ••• ,j) have been already defined by (9)-(11)
j+l
and ( 14) ,Z. 1 is the vector defined by
where
l~
-J+
(26)
and where
(j
=
k
Zj
stands for the
z~
J
~
x
== 0, 1, 2,
••• , p-l) •
matriX
j
. . . . . .. . .. . . . . . . .
==
k
Zl
'e
l-
(j=l,2, ••• ,p-l).
k
Z2 ,~'
'
'Ilk
Hence we have the conditional expecte.tion equation for Y. l'
-J+
it turns out that the Y.
given by (12) and
occuring in the right side of (12) is given by
J
(28)
(k==j+l, ••• ,p),
==
r zj+ll
I j
I
I Z~+2
J
·••
From the above it is easy to check the statements made in Section 2 on the conditional distribution of lj+l' given !l' !2' ••• ' ! j
independence of the
Fj'S
and also about the
under H* • For the latter note that although the
o
Fj +l itself involves C~l' ••• '!j)' its distribution under the n'1:l11 hypothesis
(!l' ••• '!j)' and hence of Fl , ••• , Fj •
The question of the testability of H* .'s and eventually of H* and hence
is a central F and is independent of
~
0
of Ho is one of great interest and will indicate the extent to which the Dj 's
10
of (8)
and hence the
C.ls
J
of (7) might differ for different variates.
This
will be discussed at some length in section 5 following section 4 which introduces the notion of p-block designs in general (for p variates) and considers,
in particular, a hierarchical p-block design.
this is the one discussed in section
5, from which follows as a slightly special
case what is wanted for the present section.
point might be helpful.
The testability condition for
One preliminary remark at this
The testability condition for
*
Raj
under the model
(12) is given by
Rank
(29)
=
4. Multiresponse Designs with p-block Systems
To fix our ideas, let us start with an example.
ment on v
varieties of wheat, the characteristics under study being the yield
and the susceptibility to pests.
correlated.
Suppose we want to experi-
These two variates are well known to be
Before we proceed further one might raise the question:
why study
the susceptibility, since the ultimate yield of a variety is all that matters.
This has several answers, one being that pest incidence mayor may not be unifODnly and independently controllable for all varieties. A variety which may be a
poor yielder because of being susceptible to a large number of pests, may become
the highest yielder when under pest control.
Now suppose that we have
b k(:n, say) experimental units, and suppose that
a fertility gradient suggests dividing these into
b
blocks of k
units each.
Suppose further that with respect to soil fertility, etc. the blocks are very
homogeneous and between-block differences are very large, so that a BIB design
11
(say) with these blocks will be -quite efficient.
The degrees of freedom for
error in the ordinary analysis of variance are n-v-b+l.
The power
of the F-
test :for the equality of yields is a decreasing funcii on of b, provided that
the other parameters are kept constant.
However, the introduction of blocks
causes the error variance to drop so much that ordinarily this would offset the
decrease in power due to increase of b, and, in fact, would push the power up.
The soil fertility has, however, no effect on the pest incidence, and, therefore,
the above blocks cannot be expected to decrease the error variance for the pest
susceptibility, and therefore their introduction may actually decrease the power
of the test relative to that variate.
In other words, for the second variate
we may use the completely randomized design (With b=l).
Thus here we should
have two block-systems, the first system consisting of b blocks and the second
having just one block.
Multiresponse designs with p block-systems essentially deal with differential heterogeneity in the experimental material with respect to the various
characteristics under study.
relative to one response is
If the appropriate stratification of this material
ve~J
different from the appropriate stratification
relative to another response, then two separate systems of blocks are called for
corresponding to these two variates.
The analysis of the above multiresponse design with two block-systems (and
in general of designs with p blcok-systems, if there are p
responses) though
not a special case of the usual model of multivariate linear hypotheses and
analysis of variance, can be handled by suitably generalizing the step-down procedure.
Instead of discussing the analysis for the particular situation when,
on each unit, all the
p variables are actually observed, we shall present the
same for the more general case where a hierdrchical design (in the sense of the
last section) is defined over the units.
12
5.
Testability Conditions..3f Hierarchica.l Designs vri ~1 p-Block Systems
Consider the conditions
Rank (B .)
J
==
B
Rank ( j)
,
D~
<J
These fJave been obtained fram
(29)
vThich are stochastic variables.
by cutting out the parts cont::dning
The conditions
(30)
Y.
J-
l'
will hence forth be
called the "controllable part" of the testability conditions.
We first examine how far
ind.ependent columns exist in
B. •
Also it is clear that
(31)
Rank
J
If (30)
Y
j
>
holds for some
an inequality.
(30)
Now Y
j
j, and
will imply
(29).
Notice that
(Pj - Pj)
'which are linearly independent of cols. of
Rank
(29)
does not hold, then in
consists of
n
j
(31) we
vectors which are distributed inde-
pendently according to a nonsingular j-variate nOTIl1al distribution.
probability that the
j
vectors in
Y.
J
(32)
Rank
(B , Y )
Rank
( J
j
j
B.
Y.
D;
tJ
oJ)
=j
w~
must have
j
B.
Dj
( J)
is unity.
Bj '
+ Rank (B ), for all
< Rank
Thus the
will be mutually independent and also
linearly independent of the column vectors of
probability one, i.e. almost everywhere,
m~st have
j.
Y.
+
Rank
( J)
0
Hence with
13
Comparing (32) with (31) we thus find that, almost everywhere, the conditions
(30) imply (29), a.e.
Hence in our further discussion, we need to consider, only the conditions
(30), which represent the controllable part of the testability
conditio~s.
Let us now go back to (13) and write
~
~
(33)
-
,
}\,~+l
=
"'"
~
for
l':::k~j,
= 1,
2,
••• J
B, = (Bi, B2j ,
J
(34)
e
j
p.
Then for all
... , B.J-j l'
From (30), the above partitioning of B.
J
j
D. = (Di, D2,
J
(35)
••• J
j,
B~)
J
induces a partitioning in
Dj , say
D~)
J
Since our discussion is completely general, we may now assume that corresponding to each variate there exists one system of blocks.
'tole
The situation where
have just one system of blocks for all variates is eVidently a particular case
of the above.
Similarly for each variate we Lay have a different set of treat-
ment effects.
The vectors of parameters for the j-th variate
~j
can therefore
be written
(36)
~j
where
~'b (say
-J
, j
= 1,
2, .•• p
m.bxl) is the set of parameters corresponding to the blocks,
J
and S,t (m.txl say) is the set of parameters representing the treatment effects.
-'.J
J
He have then
14
(37)
Corresponding to
(36), the vectors ~~ (15k 5
j, j
= 1,
2, ••• p) can then
be partitioned in the form
j
(38)
5.k :::
5l j
Now relative to the partitioning (36) and (38) of the vectors
~,this partitioning being
have the corresponding partitioning of the matrices
obtained fram
'-
(15). Thus we write, for 1 ~ k ~
This, in turn, induces a partitioning on the
j, j
~
and I~' we
= 1, 2, •••
P
,
(40)
If the hypothesis
H*o'
Hence for
H
o
concerns the treatments only, then so does the hypothesis
1 ~ k.: j,
j
j
D
kb
(41)
= I,
=0
2, ••• p, we have
,
and
j)
nkj = (0 ,Dkt
where
0
'
denotes a zero matrix of proper order.
The controllable part of the testability condition can then be written, for
e
...
j
(42)
Rank. (Bib' B ,
2b
=
Rank.
C~b
j
B2b
0
)
j
j
j
Bjb , BIt' B ,
2t
j
Bjb
0
... , Bjjt
j
BIt
j
D1t
)
... DB1t )
j
jt
15
Two cases now arise
Case I
For
1.1 2.1 ••• P ,
j ==
say
(43)
This is likely to arise in a great majority of experiments, where we have a
fixed set of treatments, the nature of the expected response for each individual
treatment being exactly the same for each of the p variables under consideration.
In fact it appears that up till now in statistical literature, this has been the
model assumed for treatment effects in multivariate experiments.
An examination of (42) shows that for testability a.e. under the step-down
procedure, we must have
e
j
Dlt
(44)
j
= D2t
=
... - Djt
-
j
;c;
Dj
t
say •
Thus we find that under the present model we cannot, at the j-th stage
(j = 2, ••• p) test different hypothesis for the variates VI' V2 , ••• , Vj
•
The fact that under (43), the j-th stage model (12) degenerates in such a way
that for testability the conditions (44) become necessary, will be referred to
as the "intrinsic confounding" of the j
variables in the j-th stage model.
Let us now consider the effect of the phenomenon of "intrinsic confounding"
on
th~
testability of the hypothesis
H •
o
Because of (38) and (39)
to
(45)
j
Dt (cj
2lt+
~j
~) - 0 •
••• + ~j-l,t
+ ~jt
- , J = 1, 2, .•• p
Cjt
~jt
= 0,
j
= 1, 2, •• , p •
H*
o
reduces
16
vlriting in :full)
is
13 21
l
Dt
2
Dt
2
Dt
1331
~
1332 D3t
f-
(46)
Ho
0
i:\I
rS. lt
0
I
0
0
0
D3t
0
0
.
•
I
\ ,?
,2.2t
i
I
\~3t
:::
.
\
I3 p2 DP
t
13 p)p-lDPt
\
t
!
I
\
IF:
\
I
DPt_
i
I
i
DP
t
..
!
!
.
.
,0
t:-pt j
is
r~lt
0
e
0
0
......
~Pl
while
H*0
(47)
0
•
C
•
2t
.
-,
\-
0
o
: f lt
0
\
·0
0
\
•
0
r.
,_.~
Cpt
:::
s.;t
.
l~
I
\
~,
For any rnatrix M let V(M) denote the vector space generated by the rows of M.
Then a sufficient condition that
H
o
implies
H*
0
is that the vector space
generated by the rows of
(48)
be the same as
is that
V(C jt)'
Similarly a sufficient condition that
It~
implies
Ho
17
...
(i)
(ii)
j
= 1,
2, ••• p •
Combining the two, we find that (49) is a sufficient condition that
Since (46) involves 13' s which are unkn01m, we do not possess
are equivalent.
a set of necessary conditions for the equivalence of
Case II.
j
j
j
B , B , ••• , B
lt
2t
pt
H and
o
H~.
are not all equal
An important example of this situation is the following one:
t
Hand H*
o
0
in the above matrices var.J over the total number
N(=Slxs x ••• x:s ) of treatm
2
ment combinations of a factorial experiment in which there are
k-th factor being at level
sk'
N, it may be plausible to assume that cer-
tain higher order interactions are zero for each of the
j
Blt, ••• ,B
j
jt
m factors, the
However though the total number of treatments
actually tried in the experiment is
A situation where
Let the suffix
p
variables involved.
may not be equal will arise, if we assume for
example that for variate VI all 2-factor and higher order interactions are
zero, others being not negligible, for variate
V ' all 4-factor and higher inter2
actions are zero, and others are not negligible, etc. In such cases even
mIt' m2t , ••• mpt
will not be equal.
For testability, condition (42) will have to be verified.
the matrices
In many cases
Bj , ••• , Bjjt may have a part ( as for example, corresponding to
lt
the main effects) say
B~"*
which is common to all of them.
This will lead to
difficulties of a nature similar to those encountered under Case I.
One remark will be needed here.
For certain special cases, the problems
under Case I may be handled (and have been handled) by methods available in the
present statistical literature using the known designs which are essentially
18
meant for univariate problems.
But the problems under Case II are amenable only
under the present development.
.The use of appropriate mUltiresponse designs
is necessary however for all IDultiresponse experiments that we wish to conduct
in an efficient manner.
6. Nature ?f the Nultivariate Designs for an Important Special Case
Consider now the particular case when (47) is satisfied.
Then in
H*, we
o
can tuke
Di
(50)
= Cjt ,
j
= I,
2, ••• P •
For this special case, the conditions (42) reduce to the following} in view of
(1+3) :
(51)
=
Rank
This shows that at the j-th stage, the design behaves as if there are
block effects.
If there is only one viz. the j-th variable, in the picture, then
the blocks entering into the analysis would have been those which correspond to
j
Bjb'
This shovTS that our method of analysis introduces in effect a larger number
of blocks at the j-th stage (for all
sideration of the j-th variable alone.
of latitude now.
j) than are present there from the conThus we have effectively a larger amount
For example, let us consider a w.atrix
linearly independent contrasts.
Cjt having
mjt-l
If we considered the j-th variable only, ignoring
the rest, then the condition for testability would have been
19
. j
(53)
Rank
j
j
Bjb , Bt
Rank
::::
j
C~b c~"
B
)
Ju
or equivalently that the j-th stage design should be connected.
(53)
vl'ith
(51),
we find that the connectedness is needed now in terms of a
much larger number of block effects.
However, we should still expect the over-
--all design to be qUite complicated, since now we have
each unit.
terms of
Comparing
j
blocks passing through
Further the concept of connectedness v1'ill have to be redefined in
j
systems of blocks in contrast to the present definition of connected-
ness in terms of just one system of blocks.
REFERENCES
L'f:7
Monahan, 1. P. (1961).
Incomplete-Variable Designs, Unpublished Thesis,
V. P. I., Blacksburg, Virginia.
1'5.7
Roy, S. N.
(1957).
Some Aspects of Multivariate Analysis, Wiley, New York.