•
TEE THEORY OF THE ASSOCIATION ALGEBRA AND
THE RELATIONSHIP ALGEBRA OF A PARTIALLY
BALANCED INCOMPLETE BLOCK DESIGN
by
Junjiro Ogawa
University of North Carolina
This research was supported by the Office of
Naval RGsasl'U undsr Contract No.' Nonr-855(06)
tor research in probability and statistics at
Chapel Hill. Reproduction in whole or in part
for any purpose of the United States Government
is permitted.
Institute of Statistics
Mimeograph Series No. 224
April, 1959
THE THEORY OF THE ASSOC IATION ALGEBRA AND
THE RELATIONSHIP ALGEBRA OF A PARTIALLY
BALANCED INCOMPLETE BLOCK DESIGNl
by
Junjiro Ogawa
University of North Carolina" Chapel Hill" N.. C.
Introduction and Summary
The concepts of the
I1
rel ationship matrices" and the "relation-
ship algebra" of a design introduced by All To James
L-1J seems
to
offer an interesting mathematical tool to explore the theory of
eX1Jerimental design.
It has been shown by James
L1.J,
that in
the analysis of a balanced incomplete block design the partition
of the total sum of squares corrected by the mean into partial sums
of squares due to treatments» blocks and errors corresponds exactly
to the decomposition of the unit element of its relationship algebra
into mutually orthogonal principal idempotents.
Since a balanced incomplete block design is a special partially
balanced incomplete block design with only one associate class" and
furthermore there are classes of balanced incomplete block designs
whose dual designs are Partially blanced, it is in itself interesting to construct the relationship algebra for a partially balanced
incomplete block design and to analize its algebraic structure in
1
Tb1·s reB~arch" was supported by the Office of .Naval Research_
under Contract No. Nonr-855(06) for research in probability and
statistics at Chapel Hill. Reproduction in whole or in part for
any purpose of the United States Government is 1Jermitted.
2
in connection with the "association algebra" of the partially balanced
incomplete block design" that had been introduced by R. C. Bose
1:2].
It so turned out that the relationship algebra of PBIBD is a natural
generalization of the relationship alsebra of BIBD.
In case a BIBD
has a dual design being partially balanced the relationship algebra
of the original design turns out to be a subalgebra of the relationship algebra of the dual design.
Thus there are some special structual
features of BIBD belonging to this class of designs.
1.
The association algebra of an association.
We say" following Bose ~2~, that an association is defined
in a set of v
element~if
we are given certain relations among elements
of the set, called the association, in such a way that
(a) any tvo elements of the set are either the 1st, or 2nd,
th
••• , or m associates"
(b) every element of the set has n (which is independent of
i
th
the individual element) i
associates" and
(c) i f the two elements a and 13 are i th associates, then the
number of such elements 7 as are j th associates of Q and at the same
th
time k associates d' tHs P~k which is also independent of the individual pair (0,,13)0
An element a can be regarded as the
0
th associate of itself.
Thus it will be consistent to add the following conventions:
n
(1.1)
o
= 1,
3
where 8 ij
=
= k.
1,
if
i
0,
if
i::l k.
With these notational devices, the relations which exist
among the parameters which describe the given association can be
expressed as follows
~=o
n i = v.
k
k
Pij
(1 ..2)
= PJi •
t,;=o P~k
=~
i
j
_
1
k
_
ni P jk - n j Pik -
~Pij
The association matrices are defined as follows
(1.3)
where
l~
{2J:
i f two elements a and 13 are i th associates.
otherwise
01
It will be clear that A is nothing but the unit matrix of degree v.
o
From the very definitions of the association matrices it
follows immediately that they are all symmetric, linearly independent
and
(1.4)
A +A
o
l
+ ••• +A
m =Gv •
where Gv is the matriX of degree v whose elements are all unit;r.
Furthermore, it can be shown
L2] that
1+
By the symmetry of Pk with respect to i and j, it is readily seen
ij
that
(1.6)
i.e. the multiplication within the set
is commutative.
6L of all association matrices
(1.5) expresses the fact that the linear cloSllr2.,LOTJ
of the matrices set
or..
I
i.e., the totality of finite linear combina-
tions of the association matrices Ao ,Al' ••• 1 Am with real coefficients,
is closed with respect to the mUltiplication. In other words, the
linear clogu:re
COO is
a linear associative and commutative algebra,
which we call the association algebra of the given association and
denote it by or..
The matric algebra
.LOLl generated
by the association
matrices is a faithful representation of degree v of the abstract
algebra
Ol. •
Since the constants P~j constitute the so-called "structural
constants" of the association algebra
oc.~
it is well-known that the
mapping
( m)
where
(1,,8)
i
gives a representation of
~
~
= 0,
1, 2, ••• , m
, and it is called the regular representa-
of or and denoted by ( ot )"
Since 0'{ is the so-called "abst,ract counterpart" of the matric
algebra
L7JO generated by symmetric
matrices Ai it is completely re-
ducible in the field of all rational numbers, hence in any field of
5
characteristic 0.
On the other hand, ShuI' , s lemma
[4] shows
us that
any irreducible r-epresentat1on of a conmute:tive algebra in' an
algebraically closed field must be linear.
In particular'any
irrl!idueibJte !'e)ol'esentation of a eOID$utative rnatrie algebra in a
field
containing ell characteristic
rOO~8
muS't be linear. Any representation of a
algebra decomposes into
irreduc~bl€
,·whMb.·
113 {!;t!l1ivs1er.l.t
to
,
-
of the
ma~ric&s
completely reducible
constitutents, 'each of
one of the irreducible constituents of
Hence the regular representation
of the association algebra must decompose into inequivalent m + 1 linear
representations in the field of all complex numbers.
But these linear
constituents are the characteristic roots of the symmetric matrices Ai'
hence they are all real.
(~)
Thus, after all, the regular representation
decomposes into m + 1 inequivalent linear representations in the
field of all real numbers.
Even more, if the characteristic roots of
all association matrices are rational, (od decomposes into m +1
inequivalent linear representations in the field of all rational numbers.
On account of the fact that
i
= 0,
1, ".J m
we can choose a non-singular matrix C of degree m + 1 in the field of
real numbers, being of the form
1
(1 ..10)
c=
1,
Cll C12
•
o
..
•.
•
000
1
Q··
Cm
•
•
o
•
6
in such a way that simultaneously
(1.11)
cpi C- l =
,
. ..
z
i
= 0, 1, •••,
m
mi
where
(1.12)
and, of course
Z
00
=
zl0 = .....
=
z me
= 1.
If m + 1 Pi are actually given, we can determine the transforming matrix
C excepting proportionality factors.
Let
then
(1.14)
Multiplying Ckj to both sides of (1.14) and summing up with respect to
j, we get
(1.15)
7
On the other hand, the commutativity of mUltiplication of association
matrices yields
(1.16)
A.*A*
-It 1
= A*A.*
= (~.1=0 cij z.
i-1t
kJ )A.*
Uk
Since (Ai) is a set of linearly independent matrices and C is non-singular the set {At l must be a set of linearly independent matrices" it
follows that
and hence
(1.18)
for i/= k.
This means that the set of matrices
(1.19)
is a set of m + 1 mutually orthogonal idempotents of the algebra
LI0t7,
and each of which gives rise to a linear representation
(1.20)(0):
Ai ~ za.i'
i=O,l, ... ,m; Ct=O,l, ... ,m.
It should be remarked here that
~",=0
C ••
~J
Z.jFO,
~
i=O,l, .... ,m.
Indeed, suppose that
for some index i, then this together with (1.17) give us .
8
C
io
=c il = ....
= ~ 1m = 0..
But this is eVidently impossible because of the non-singularity of C.
From the relations (1.4) , it follows immediately that
~=o
(1.21)
z0 j =
~j=o zij
~=o
nj = v
i=1,2, •• ~,m
= 0,
We shall determine the multiplicities of linear representations
1n the matric algebra
L-«-.7.
Let
°
0
,
(ll' • o.~ (lm be the respective
multiplicity of linear representation in
L-et7,
then by considering
the trace of G we get
v
(l
o
(1.22)
Comparing the1races of A ; Al ,
o
Since the matrix
II
Zij
II
= 10
000'
Am' we also have
of degree m + 1 is non-singular (because of
the inequivalence of m + 1 linear representations), a ,(l2'
am
l
are determined uniquely by any set of m linearly independent equations
0 •• '
9
2.
EXamples of association algebras.
There is a certain amount of detailed knowledge concerning the
types of association scheme which give rise to associations with two
associate classes
1.:6_7, L7 J.
Now we shall look at some of the types of frequent occurrence
from the point of view of the corresponding association algebra.
Given an association scheme, we can write down the association matrices
Ao ' All and A2 , and from which the association algebra is generated
as their linear closure. The association scheme may be regarded as
the realization of the association expressed by the association matrices.
Thus the problem of existence of the association with given parameters
is reduced to the following:
Given an association algebra with given
parameters, is it possible to select a set of basic matrices with
elements either 0 or 1 and all diagonal elements are O\s uniquely
except for permutations of rows and columns? This problem is out of
the scope of the present paper.
1.
Group divisible type.
The number of elements is v=mn
where m and n are positive integers.
They can be diVided into m
groups of n elements each, such that any two elements in the same
group are 1st associates and two elements in different groups are
2nd associates.
If the elements are numbered in the dictionary
order with respect to groups, i.e., the
j th
element in the i th group
has the number (i-l)+j, then the association matrices have simple
forms:
10
•
G
n
It is easy to see that
nl
= n-l
} n2
=n
(m-l).
The regular representation is given by the mapping
o
o
0
0
1
n-l
n(m-l)
= C· l
n(m-l) n{m-l) n(m-2)
C
-n
o
The multiplicities of irreducible constituents are
(2.4
respectively.
Three principal idempotents of this association algebra are
11
(2.6)
It is known that in this case the association scheme is determined
uniquely by the parameters
II.
[13].
The number of elements is v = n (n-l) /2
Triangular type.
where n is a positive integer. We take an nxn square" and fill the
n(n-l»).2 positions above the main diagonal by the different elements,
taken in order.
The positions in the main diagonal are left blank"
while the positions below the main diagonal are filled so that the
scheme is symmetrical with respect to the main diagonal. Two elements
in the same column are 1st associates, whereas two elements which do
not occur in the same column are 2nd associates.
The regular representation of the association algebra is given
by the following mapping.
o ~ P0
A
Al~
Pl
= 13 •
=
o
2n-4
o
1
n-2
n-3
o
4
2n-8
001
o
n-3
n-B
(n-2)(n-3) (n-3){n-4) (n-4)(n-5)
222
Three linear representations have respective multiplicities
(2.8)
in
I:(}o.
2n-4
n-4
-2
12
2
III. L Tytle: The number of elements in this case is v=n ,
2
2
where n is a positive integer. Th&se n elements are arranged in an
nxn square in such a way that any two elements in the same row or the
same column are 1st associates, whereas any two elements neither in
the same row nor in the same column are second associates.
The regular representation of the corresponding association
algebra is given by the mapping:
A ~ il
o
=1
0
"
0
Al~Pl
=
2n-2
0
1
n-2
n-l
2n-2
0
2
=C- l
-2
0-2
20-4
C,
(2.9)
1
0
2n-4
0-1
2
2
(n-l) (n-l) (n-2) (0-2)
0
-e
A2~
P2
=
0
=C- l
(n-1C.
1
-n+l·
Three linear representations have respective multiplicities
(2.10)
in
L7iO.
......-
IV. Cyclic Tytle:
the v elements.
We shall consider the numbers l,2,3,. ....,v are
The 1st associates of i are
i + d1i i + d21 ••• J i + d~ (mod. v),
where d's satisfy the conditions
(1) the d's are all different and 0 < d j < v J j=1~2,.~.,nl
13
(2) among the n1 (n -1) differences dj-dj_ reduced mod. v, each of the
1
numbers d , d2 , ••• , d occurs 0 times, whereas each of the numbers
1
n1
e , e 2, •• GI e occurs f3 times, where
1
n2
are all different v-1 numbers 1,2,3, ••• , v-1.
EVidently there should hold a relation
(2.11)
The regular representation of the corresponding association
algebra is given by the following mapping:
A
o
'e
~
P
0
Al(~Pl
= 1 ,
3
=
0
1
n1
a
0
n1 ..a-1
0
A2 --;. P2
=
0
0
n -o-1
0
=c-1
f3
n1 -f3
~
Z21
1
f3
1
n2 n 2 -n l - a+1 n2 -n1+f3-1
--1
=C
n2
Three linear representations have respective multiplicities
(V-1)Zll+1'11
Z21-Z1l
in L~7,
C,
Zll
c:.
Z12
Z22
14
3. The relationship algebra of a partially balanced incomplete block
desi@.
We say that a partially balanced incomplete block design (PBIBD) is
given, if along with an association among v treatments, the following
conditions are satisfied:
each of b blocks contains k different treatments,
each treatment occurs in r blocks, and
any two treatments which are 1th associates occur together in Ai
blocks.
The parameters of the association are given as in Section ;1, aad
we qave further .relations
t:.oni Ai
= rk.
where we have put
\,5
r
and
kb = rv
=n (say)
Thus there are n experimental units or plots on the whole. We number
them in Bome way
but once for all, for instance, the i th experimen-
tal unit in the jth block bears the number (j-l)+i.
We shall define the relationship matrices of this PBIBD. To
this end it is convenient to define the incidence vectors and the incidence matrices for treatments and for blocks. These are defined as
follows:
the incidence vector of the treatment ex is
15
if the experimental unit f receives
the treatment a,
, where taf
=
otherY11se'
•
and the incidence matrix of treatment 1s
The incidence vector of the block a is
".
T)al
'l)a2
~=
•
1 1 if the experimental unit f belongs
, where flaf
•
=
to the block a,
[
0, otherwise.
~
Tlan
and the incidence matrix of block is
There are four groups of the relationship matrices corresponding
to four types of the relations among the experimental units
(I) Identify relation:
L1. J
~
Corresponding to this relation we take I=I , the
n
unit matrix of degree n.
(II)
Universal relations:
This relation must be represented by G=G ,
n
where Gn stands for the matrix of degree n whose elements are all
unityo
16
III.
Block relation:
This 1s represented by a zr.a.trix
where
r 1, if the two experimental units f and $ belong to
(3.8)
b
rg
=
i
the same block
0, otherwise.
It will be seen that
IV •..Treatment relations:
These can be described by m+l matrices of
degree n
where
and
if the two experimental units f and g receive treatments
and a and f3 which are i th associates
otherwise.
It can be seen that
(3.12)
17
Since~=o Ti = G = Gn2 it follows immediately that
~i=o T i =G=G.
n
Also,
2
G
=n
G" BG
= GB = kG"
2
B
= kG
and
Since
~ !la. = ~=l tar flaf
== tl-aa, =
r 1, block
if the treatment a occurs in the
a,
1 0" otherwise,
we get a relation
(:5.16)
is the incidence matrix of the design.
It has been well known
["2 J
that
Multiplying to both sides of the above relation
<J>t
~
from the left and
from the right, we get a relation
By making use of (3.17), we obtain more general relations
18
=
J!lt_-0
m
~
t
k PikP11j)Tt .
k.V=O"ll
~
(r..
It is readily seen that
Thus it is easy to see that the linear closure of the matrices set
of 4m+ 3 matrices
I, Gj B, T,
u TuB; BTu , BTuB, u=l,2, ••• ,m.
is multiplicatively closed, i.e., these 4m+3 relationship matrices
generate a linear associative algebra1R, which we call the relationship
algebra of the PBIBD 8
The mUltiplication table of this algebradt can easily be worked
The relationship algebraffi contains a subalgebra
/
.
I.JG:Tu ' u=1,92,9
o. o,m which
OZ*
consisting of
is isomorphic to the association algebra (J[
corresponding to the association defined for treatments.
As a special case, the relationship algebra of a BIBDis the linear
closure
or changing the basis by means of the relation
r
19
the relationship algebra of a BIBD is expressed as
This is the one which was discllssed by James
[1 J
in some detail.
4. The analysis of the relationship algebra of a partially balanced
il:3.complete block design&
The relationship algebra'lR of a PBIBD is non-commutative in
Since tR is generated by symmetric matrices; it is completely
general.
reducible.
Hence all irreducible representatiors of m are obtained by
the reduction of its regular representations of factor algebrasof'eR
J:4J~
L'J.
[GJ,
the totality of multiples of
G,
is a one-dimensional
two-sided ideal of ffi and as we have seen in the preVious section
hence we obtain a linear representation
of~, ioeo~
(4~1)
Next we shall consider the factor algebra'CR mod G.
end" it will be convenient to change the basis of'1R.
(4 0 2)
T*i
= ~A*~t
= ~J=o
1
c 1jTJI
Then it is easy to see that the m sets
rT*;
L"
u
BT*"
BT*BJ,
u T*B"
u
u
u=lJl2, •• "1m
Put
To this
20
.
form m two-sided ideals of eR mod. G and they are annihilating each other •
Hence there must be m inequivalent irreducible representations of the 2nd
degree each of :wh:lcl11s derivable through a one -sided ideal of each of the
two-sided ideal, provided that
(4.4)
Now by direct calculations it follows for each u that
BLT*,BT*,
BT*BJ
u
u T*B,
u
u
= L~,
(4.5)
and
~, T~B, B~BJ
0
0
0
0
1
k
0
0
o
o
0
0
0
0
1
k
TiLT*,M*,T* B,BT*B-J
u
U
(4.6)
U
u-'
= LT*,
u
BT*,
u T*B,
u BT*BJ
u
Z
/JIlf- ~ 0
ui kV=O ZUV-JA
tl
o
0
o
0
o
0
Indeed,
1 u
and
= ~Ai ~t~A*~t
= r~A1A*Q'
= rz ui ~A*~t
u
u
u
= rZ
0
0
0
0
m
fj
r~1lV'=~u~1k\
for i=0,1,2, ••• , m.
T T*
0
T*
ui u
21
Since
CP i C- l
•
>=
, 1=0,1,.
•o
Q
.,m,
it follows that
Thus we obtain m inequivalent irreducible representations of the
2nd degree as follows:
1
m(2):
0
I~
U
1
0
,
(4.7)
rZ
T
i
.z: Z
Ui
0
0
0
0
G'~
kV
u
,B_ ~:
0
1
k
rtf!ik\
, 1=1,2, ••• , m.
~
0
0
The irreducibility of m~2) can be expressed as follows:
Suppose
thatCR(2) is reducible, then 1t 1s possible to find a non-singular matrix
u
c=
,
1n such a way that simultaneously
c
o
o
1
k
* *
o *
22
and
c
* *
o *'
o
These leads to the relations
C Cll+kC C21 = 0
22
22
12
f/
21 _
CllrZ uiC21C + rzuvPik~C21C - 0,
Now since C is non-singular,
.
~t
i=1,2, _. _,me
11
can not be that C
= C21 = O.
If
1
(4.8)
or
(4.8)
.~
V,k
Z liPikf/ ~
u~-
= rk
Z ., i=112, ... ,m.
u~
If C = 0 and C22 .1
r 0, then C21
21
= 0 and
consequently C11
= O.
If
C22 = 0, and C21f 0,. then
cr
(4.10)
&Zuf/ prk~ = 0,
1=1,2, ••• ,m.
Thus the representationffi(2} is irreducible unless (4.8) or (4.10)
u
hold.
2,3
Other irreducible representations of m are obtained by considering the regular representation of the factor
LG,
TuS' u=1,2, .. Qr;,mJ..
algebra~
mod.
These are all linear and given by
Since in 1s a completely reducible algebra. I it must be of the form
13 aiel) + f3 :mel) + a~(l) + ~
(4 ..12)
o
I loG
0
o:~ (2) •
u=l u u
Now by considering the tra.ces of matrices G" To" T ,
l
we get
a*o
(4.1,3)
1
III
=v
(4.14)
•
G
~
Q
a
0
co oS'
0
~
•
~
~
•
0
•
•
- 1
0
•
•
and
(4.16)
(3
o
+ 131 + a*
+ 2atl + .... + 2a*
== n.
0
m
Comparing (4.14) with (1.23), it follows immediately that
(4.17)
0:*
u
=0:,
u
u=1,))2 ,
...,<> •,• m..
T and B, I,
m
24
This means that the irreducible representation of second degree of the
relationship
alaebra~
corresponding to the irreducible constituent
(ZuV) of the association algebra
tR as that (Zuq) has 1n
LV has
the same multiplicity O:u in
Lot7.
From (4.15) and the first equation of (4.14), we get
f3 l = b - v.
Then from
(4~16),
we obtain
f3a =n-b-v+l
Thus finally we can determine the structure of R completely as follows:
(4.20)
In m(2) T == T is represented by
u I
a
r
o
0
where
zu = ~i=0
(4.22)
Zui A1.
In particular for a balanced incomplete block design
Zu
,
= Z uo Aa + ZulA-l = r-A
hence for a BIBD, m(2 ) turns out to be
u
25
which confirms the James' result
/:1_7.
5. Significance of the relationship algebra of a design to its analysis.
We shall denote the observations arranged in the order of experimental
units by an n-dimensional vector";4 Then the treatment totals and
block totals are given by
! = !'!1 where
To
= ~!I
a=l~ ... , v
B = '!f'x
where Ba = Tl'x.
a=l., .. ••, b
-'
a-'
-
respectively. And
1
-2
n-!'G! = nx,
(5.2)
-
1...n
=n
~f=l
xf '
2 ).
!r -x'Tx- =!r (~'x)t
(~tx)
= !(T2+•••+Tv'"
-...1
1
it!' B!
where
1··
= it
(W'!)'('!ff!)
x
1
= it
2
(B1+ou+
2
BbL
.L
T'
1k -
NB -
Let
then it follows that
L
Xl
rk -
TBx =
-
L
(~IX) r~,.lr(.irtx)
rk
'I' 'I' -
.=
.L r.v
T S (Q)
- - rk a=1
Ct Ct
26
We have seen in Section 1 that
(5.3) JlAu
= (~i=0
i
cui Zui)·l ~i=0 aiAi
== ~i c· A.,
U
=0
U
J.
u=O,l, s • • ,m
are mutually orthogonal idempotents, i.e.
f::f
A
=
tf A
u
U
tfA
.5
u
•
uv
or
~i=~.I. a Ui Ai tt Av = i(Au 8uv
I
or
i
!
I:a
i
Z. =8
U
•
uv
UJ.
Thus we get
0
0
+ 01c1 + .. 0 + 0
o
m
0
ccn0
a oo +0:01 + ••• + rvO
'"" =
m,
= 1
1
0 0 0
nl + C¥J. Zll + ••• + am Zml =
00
o
•
•
c
•
•
•
•
•
•
•
•
•
•
•
°
•
••• +cPZ
m mm =0
Comparing (5 s 6) With(4.14), it follows that
and consequently
m
o
D "
1
= 0:o =v
•
27
Let us put
if.
T
u
= tJ1i=o
a Ui Til
u=o,l,. .,m
0
and consider m matrices
Since
it follows that
(5.11)
Hence
2
VU
=r2
=r
=r
(~-
l.A
-
VN')~A
k.·u
r(k-l)-Z \
w
1
... Z
k
1
f...
L_
k
(0f~N
V')
umm r (0-ltVNt) Au(Qt{N'W t )
r(k-l}-Z u··~
lA- eo. Zumm
A
k
A
(rIv - -k NNt) nAu
VU
In other words,
are mutually orthogonal idempotents having the traces
28
Thus there are m corresponding sums of squares which are mutually independent and obey the chi-square distributions with respective degrees
of freedom a l , ••• "Ctmo For each index u, the sum of square is
= r(k-l)-Z
k
Q.?fA Q
u'
u=l, •••
1"- ••• -Z um>..m -
U """1
,m.
or
~~ ••• -Zumm
A
~l R~
r(k-l)-Zu
•
where
R = -+I A Q =
-u
u-
a Ou-Q + a lu A1Q
- +
0.. + cPAum-Q
or
Since ~Au is of rank au' there should be only Ctu non-vanishing components
Rua • Thus the sum of squares due to treatment (eliminating the block
effect)
can be decomposed into v-l independent constituents as
= ~
z*
u=l a
k·
R2
r(k-l}-Zu--~
l"-o •• -Zumm
~
ua
•
29
6. Exact distribution of statistics in the analysis of a randomized
block design_
Given b blocks each of which contains k more or less homogeneous
experimental units and k treatments to be compared by experiment, we
shall say that the design is a randomized block design if we assign k
treatments at random to plots within each block. The important point
in this set up is that the incidence vectors
~,
a=l, ••• ,v of treat-
ments are random vectors, whereas the incidence vectors.!lex' a=l, ••• ,b of
blocks are fixed vectors.
In this case there are only four relationship matrices,
(6 ..1)
1, G, B, T
and they are commutative.
It is easy to see that
222
G =nG, B =kB, T =bT.
(6.2)
i.e.~
n=bke
Hence the relationship algebra m is the linear closure
Indeed, the multiplication table is as follows:
T
G
B kB
G kG
T G bT bG
G kG bG nG
1
(5.4)
B
The decomposition of the unit element into orthogonal idompotents
is given by
30
or
(6,,6)
The underlying model is that the conditicnal distribution of
e
-
= -x
- g 1
-n
~
-
t -Vb
-
-
There will be no harm in assuming
= bt~
tt1c>
= .Q
0
If we denote the nxk matrix whose elements are all unity by
J, we have
G = J
~v
=
~ J~
and since
and
(~t -~J~ )!=glb+~'i~ ~ + ~~V £.
(6 .. 11)
+
1.b
= t +
= g ~ _._~jt~!_*J'iV£
~te_l Jfe
n-
(~t_~Jf)~
31
it follows that
1
x t (?JI
J.-a)x
b
n
-
(~bl
= -e vb
-!J)
n
(~bl f .lJf )e
n-
Take an orthogonal matrix of the form
/b -tt(~tJ~Jt)
b
n
/
/t'b(~,J~Jt)(~_!
J)t
b
n
b
n-
u..2
o
and let
~* == p~
or
~
then we have
= P'~* •
32
Now
1 1
'01 (~--J)
b n
....~
•
o
hence
1
0
o ••
o
o
•
(6.16)
•
•
Q
o
Therefore
(6.17)
Xl
(~T - ~B)! = el2
+ 2
.yb~r~
= (el + Jl~r ~)2 +
+
b~'~ + Q(e~,
Q(e~,
.0.,
.0 0' e~)
The conditional distribution of this statistic divided by
~
e~)
df
given
is a non-central chi-square distribution of degrees of freedom k-l
1 1
(the rank of bT-~)
and with non-centrality parameter b!l! /2
C1.
Hence
consequently the absolute distributions of this quadratic form is also
the same non-central chi-square distribution.
As far as the author is aware, no one took care of ·the
assignment of treatments in his analysis.
by the description given above.
ran~om
This point is taken care of
33
7. The a.ssociation algebra of the dual design of a BIBD.
Suppose that a BIBD with parameters
which satisfy the well-known conditions
rv
= kb~
r(k-l)=A(v-l)
has the dual design being partially"balanced with m associate classes.
Le"1; its incidence matrices be
~,
'if and N, then there hold the
relations
and
Now the association in the dual design will be defined as
th associates if they have exfollows~ Two blocks are said to be 1
actly i treatments in common for i=1~2, .OQ, m-l, and two blocks are
th
said to be m associates if they have no treatment in common. It is
known
caJ
that
A block is regarded as oth associate of itselfc The association matrices
AO' Al ,
·.°1
Am of this association must satisfy the relations
Ao + Al + .... + Am-1 + Am = Gb
(7 ..4)
and
where P~j are the constants of the association.
Let the regular representation of this association algebra be
•
C,
1:0,1, ••• ,m.
·e
Due to the relations (5.,4) and (5 ..5),we get relations
rk+n
l
+2n +· • • + (m-l) n 1 = rk
2
m-
k + Zll + 2 Z12 + • • • + (m-l) Zlm_l
(7.8)
k + Z21 + 2 Z22 + •
•
e
0·.·
k + Zml + 2 Zm2 + • • •
and
= r- ~
+ (m-l) Z2m_l = 0
0.0 •.••...
+ (m-I) Zmm- 1 = 0
35
Furthermore we know that the following m relations to hold:
1 + Zll + Z12 + • • • + ZlITt • 0
(7.10)
\
• • • • •
.. .
• • • •
1 + Zml + Zm2 +
•
0
~
0
0
0
• + Zmm = 0
and by considering the traces of association matrices we have
.+Ct
m =b
• • o+azl=o
mm
{7 . ll}
o
where alJ' a2 "
0 •• '
•
•
•
•
•
• •
0
..
0
(.\
•
•
t!
•
•
am are the multiplicities of m ineqUivalent linear
representations in the association algebra ..
By putting
e.
=
k
rll
:2
1
•
•
•
I
1 =
•
.
m-l
•
•
0
1
these can be written as
i
I
and Z
= II ZUi
"
b
rk
Z~
=
Zl
r-~
(7.12)
=
0
0
•
•
•
•
•
•
0
0
and
Hence it follows that
1
1
a1
(7.13)
'e
=
zt-1z
.
1
•
•
..
•
•
1
•
am
From (7.13), the multiplicities a , ••• , am can be calculated as
l
(7.14)
On the other hand, from the first and third equations of (7.12),
it follows that
(7.15 )
rk + a
hence, if r-~ f. 0, we get
l
(r-~)
= kb,
37
al
==
kb-rk
r-~
= v-1
Thus
zOl == (v-l) /b
Now we shall determine the association algebras of the dual
designs of certain balanced incomplete block designs.
Case Io
1:9 J
Suppose that a balanced incomplete block design
with parameters
v, b, r, k,
is given.
~
= 1
In this case it is easy to see that no two blocks have more
than one treatment in common.
We can introduce an association among the blocks by calling two
blocks 1st associates if they have exactly one treatment in common and
2nd associates if they have none in common.
The fact that this definition of the association satisfies the
conditions
(a)~
(b), and (c) in Section 1 will be proved in the following.
Let the incidence matrix of the original balanced incomplete block
design be N which satisfies the well-known relation
(7.18)
NN'
= (r-~)
I v + Gv
We shall define three matrices A ' A , and A2 by
o l
38
and for i=1,2
if two blocks a and ~ have (2-i)
treatments in common,
otherwise
Then it will be evident that
and
If we can show that these :5 matrices generate a commutative
linear associative algebra, then they constitute a set of association
matrices corresponding to the association defined above.
Now since
where ni denotes the number of blocks which have (2-i) treatments in
common with a given block, it follows from (7.20) that
(7.23
Further
39
whereas
hence we obtain a relation
(7.24)
Consequently we get
A1A2
I:
L-nl-~-lU'Ao + Cnl-k2+2k-r J Al + (nl -k2 )A2
A~
=
Ln2-nl+k(r-l)~Ao + (n2-~+k2-2k+r)~+(n2-nl+k2)A2.
Thus it is readily seen that Ao " Al , and A2 generates a commutative
linear associative algebra. The structural constants of' this algebra are
nl
o
= Pll
P~2
(7027)
o
n2 = P22
= ~-l);
1
222
Pll = r-2+(k-l) ,
Pll = k •
=0
Pi2 = (k-l)(r-k)"
,
=b -1-k( r-1) ,P122
Pi2 = k(r-k-l).
_ (kL.l)(r-k)(r-k-l),p~f') = (r-k)2+2(k-l)
k
~A
r(r-l)
k
The regular representation of' this algebra is given by the
mapping
40
A
~
o
P
0
;;: 1 31
o
1
0
2
2
Al --;> Pl = k(r-1) r-2+(k-l) . k
o
(k-l)(r-k) k(r-k-l)
o
o
o
==e- 1
1
.O!-J)(r-k)
k(r-k-l)
In this case eVidently Ao =k, \=1$
~r;:O~
Hence
/
Zll
= r-l,
k + Z2l ;:: 0
Thus "We have
Zll
= r-k-l
Z21 ;:: - k"
Z12 = -(r-k)
Z22 = k - 1
The multiplicities al and a2 are determined by
al
as "Were expectedo
n
=C-
(k.J)~~k-l) (p-k)2+0<_1}r(i1 )11
-l-k(r-l)
k +
C,
bk-rk
;;: r-l = v-l, a 2 ;: b-v
2
l
, ..
41
Case II.
It is known
C8J that
a balanced block design with
parameters
)(k+2) I r:'k+2" k" .2
v =1
'2lt(k+l),1
b= 2' (k+l
is given, then any two blocks···have one or two treatments in common.
Let the incidence matrix of the design be N, which satisfies
the well-known relation
NN' = (r-2)Iv+2Gv •
Now define three matrices Ao " Al and A2 by
1" if two blocks a and
in common
{
0, otherwise
~
have i treatments
then eVidently
and
Starting from relations (7.36) and (7.37) and proceeding exactly
same as in Case I, we get relations
42
2 2 2 2 2
A + 4A A + 4A = 2k 1\0 + (2k -k)A + (2k -k)A2
2
l
l
1 2
2
212
1 2
12)
A + 2A A + A = 2'(k +3k )A +2'(k +3k - 2 )A + '2 (k +3k-2 A2 ,
2
l
O
l
1 2
(7 .38) ~ + A A
2k A + (2k-l) A + 2k A
--1
=
1 2
"l.A +
2
1
0
2'
212
12
12
A2
2'(k -k)AO + 2' (~ -k) Al + '2 (k -k-2)A21
=
whence we obtain
2
A =2kA
1
o
AA =
I 2
A~ = ~(k-l)AO
+
!tAl
(k-l)A
l
+
4A2
+
2(k-2)A
2
+ !{k-l)(k-2)Al + !(k-2)(k-3)A2
Thus it is readily seen that Ao ' Al and A2 generate a commutative
linear associative algebra, whose structural constants are given as
follows:
This means that if we call two blocks the i th associates when they have
exactly 1 treatments 1n common (i=l,2), the above defined matrices
A ,A1 and A2 are the corresponding association matrices of this
O
associationD
The regular representation of this association algebra is given
by
the mapping
43
e
Ao
Al
~
P0
Pl
~
= 13
=
nl
0
1
0
2k
k
4
0
k-l
2k-4
(7.41)
A2 ~ P2=
-1
Zll
Z2l
0
0
1
0
k-l
2k-4
lk(k_l) ~(k-l)(k-2) ~(k-2)(k.3)
In this case
"'0=k,
'1.=1,
~=2,
C,
n2
=C· l
Z12
C
Z22
hence we have
k + Zll + 2 Z12 == r-2
(7.42)
k + Z21 + 2 Z22 = 0
Thus we obtain
=r+k,
Z2l =k-2,
Zll
Z12 = r-k-l
Z22
= k+l
with multiplicities al=v-l and a 2=b-v respectively.
Case III. A balanced incomplete block design with parameters
v, b, r, k, A is said to be affine resolvable if the b blocks can be
separated into r sets, each forming a complete replication such that
any two blocks of different sets have the same number of treatments in
common.
It has been shown that the parameters of such a design can be
expressed in terms of two positive integers nand t (n ~ 2, t >:"0) in
the
following manner:
44
v::nk~n2 [In... l)t+1J~ b=nrr:n(n2t+n+l) I r=n2t+n+l,
(7.44)
k = n Lln-l)t+1J, A = nt+l
•
It can be easily seen that the dual design 1s partially balanced with
two associate classes.
Let the incidence matrix of the original design be N, which
satisifies the well-known relation
mil
=n
L(n-l)t+1J!v + (nt+l) Gv •
Let the association matrices of the dual design be Ao =
A2 satisfying the relations
~3
All and
(7.46)
and
where
cr
denotes the number of treatments common to two blocks from the
different sets.
Now clearly since
nl
and
we have
= n(r-l)
and n2
= n-l,
On the other hand
{NIN)2=NI (NNI )N=n Lln-l)t+l ]~PN + (nt+l)n2 Lln-l)t+l_rGb
2
n Lln-l)t+l J{nt+2) A + fn{n:It+l) a+(nt+l)n2(n-lt+l)~
o
2
• A +(nt+l)n L(n-l)t+l_rA ,
2
1
=
whereas
hence we obtain a relation
Hence we can see that
o = (n-l) t+l,
and consequently
(n-l) A
o
46
Thus the parameters of the association are
nl
o
= Pll
= n2 (nt+l).
1
Pll
= n(n2t+n-l),
1
P12
I:
n-l,
III
0
1
P22
2
2
Pll = n (nt+l),
,
The regular representation of this association algebra is given by the
mapping
Ao
~
P0
= I~,
'",
'0
Al~ Pl
==
1
2(nt+l)
n
0
n(n2t+n_l)
l
l
n2 (t+l) =C-
o
n-l
0'
o
o
1
o
n-l
0
n-l
0
n-2
C,
Z
n
Z21
n2
=0 ..1
Z12
C,
Z12
In this case it is seen that
hence
2
nLln-l),t+1J + L(n-l)t+l_7 zll=(n t+n+l)nLln-l)t+1J
n[ln-l)t+l] + Lln-l)t+l.7 Z21= n F(n-l)t+1J
Thus
Zll = n (nt+l),
Z12
= - n2 (nt+l)-l
,
Z22
&::I
2
-
1
with mUltiplicities
respectively.
8.
The relationship algebra of the dual design of a balanced incomPlete
block design.
Let
The relationship algebra R of the original balanced incomplete block
t
design that is
turns out to be a subalgebra of the relationship algebra
of the dual design which is partially balanced with m associate classes.
'LR*
is also completely reducible. We want to find out all irre-
ducible representations of lR*.
If we proceed formally as in Section 4" these are following
possibilities:
48
it=l,2, ••• ,m
{8.5)lR~
( 2)
I 1 0
: I ~I 0 1
,(HJ
If the above m 2nd degree
0 0
0 0
"T~
0 0
1 r
,Bi~
f/
kZ ui rJ'zU,JPikPk
0
l( \(~
representationsZR~(2), u=l,2, •• e,m were
irreducible" we would have relations (4 c 14) and (4 ..15) where v and b
1
are interchanged.
Now~!(2)
But since vas; b, this is eVidently impossible.
induces a representation of the 2nd degree of the
sUbalgebraCR of<R*, that is
(8.6)
-(2)
('R
:1 ~
1 0
0 1
~T~
o0
1 r
which is equivalent to
SinceQR(2)is
irreducible,CR~(2) must be irreducible with multiplicity v-l •
.i.
In particular for the case
linear representations
(8,,8)
m=2"ZR~(2)must decompose
into two
where
~~(l): I ~ 1, G ~ 0, T ~ 0, B ---kZ2l , i=1,2, ••• ,m
SinceaR* is completely reducible; it must be of the form
(8010)
Now, by comparing the traces of G, Bo ' Bl' B2 and T, we see that
(8.11)
The fact that
~l
= 0 is not surprising$ because, as a matter of fact,
henceW*(l} does not a.ppear in the regular representation of?R*.,
1
References
James, A.T o, "The relationship algebra of an experimental
design," An~. of Math o Stat., Vol. 28 j 1957.
L2J
Bose, R" C. and Mesner, DoMo, "On linear associative algegras
corresponding to association schemes of partially balanced
designs." Institute of Statistics, University of North
Carolina, Mimeogra.ph Series No 188, 1958.
0
Fisher,9 Re A. and Yates, F., Statist:i.t:!!. Tables !2!. Biological,
Agricultural and Medical Research, 1st ed., 1938, Oliver
and Boyd •
.£4]
Weyl, H., ~ Classical Groups Their Invariants and Representations,
Princeton University Press, 19390
Ogawa, J., "The theory of linear associative algebra. and some of
its applications to the theory of experimental designs."
Lecture note, University of North Carolina, 1958.
50
L-6J
Bose, R C., and Shimamoto" P., "Classification and analysis of
partially balanced designs with two associate classes,,"
~.~.~. ~." Vol. 47, '1952"
{1J
Bose" R.C., and Connor, W.S., "Combinatorial properties of group
divisible incomplete block designs." ~.~. ~.,
Vol. 23, 1952.
9
Connor" W.S., "On the structure of balanced incomplete block
designs." ~. 2!~. ~." Vol 23, 1952.
Shrikhande" S. S., " On the dual of some balanced incomplete block
designs. " Biometrics I Vol 8, 1952 •
•