I
&
I
I
I
I
I
I
I
It
I
I
I
I
I
I
..I
I
ON THE ANALYSIS OF PARTIALLY BAIANCED nrCOMPLETE
BLOCIC DESIGNS IN THE REGUIAR CASE
by
Junjiro Oea.wa and Goro Ishii
Nillon University, Tokyo and Osa.ka
Cit~l
University, Osaka
Institute of Statistics l-umeo Series No.
December
412
1964
This research was supported by Air Force Office
of Scientific Research Grant No. 84-63_
DEPARTMENT OF STATISTICS
UHIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
I
,
I
I
I
I
I
I
I
It
I
I
I
I
I
I
..I
I
On The Analysis of Partially Balanced Incomplete
Block Designs In The Regular Case
By
Junjiro Ogawa and Goro Ishii
Nihon University, Tokyo and Osaka City
University, Osaka
Contents
Summary
§l.
Partially balanced incomplete block design.
~.
Properties of the association algebra.
§3.
Examples of associations of certain types.
§4..
The relationship algebra of a PHIBD.
~.
Analysis of PBIBD
§(;.
Analysis of variance of PBIBD's of certain types.
§or. A numerical illustration.
References.
Swmnary
Analysis of a. regular partially balanced incomplete block design is
investigated in connection With its association algebra and relationship
algebra.
Properties of the association algebra and the relationship algebra
are summarized in so far as they are useful for the later discussions.
Although the properties of the association algebra have been known already
in another expression,* those of the relationship algebra of a PBIBD are
*Bose,
R. C. and D. M. Mesner (1959) On linear associative algebra corresponding to association schemes of partially balanced designs, Ann. Math.
Stat. 30 21-38.
1
I
believed to be new.
Partitions of the sum. of squares due to treatments adjusted
pertinent to the association being considered are given, and these are supposed
to be new.
As examples, explicit expressions of the partitions are given for
PBIBD's of certain types.
Finally a numerical example is presented for illus-
tration.
§l.
Partially balanced incomplete block design.
For the sake of reader's
convenience, we gave a brief description of a partially balanced incomplete
Reference should be made to [1], [2], [3J, and [7J.
block design.
Given v treatments CPl' CP2' ... , CPv' a relation among them satisfying the
following 3 conditions is called an association with m associate classes:
(a) any two treatments are either 1st, or 2nd, ... , or m-th associates,
(b) each treatment has n i i-th associates, i
= 1,
2, ••• , m, and
(c) for each pair of treatments which are i-th associates, there are
i
P jk (i, j, k
= 1,
2, ••• , m) treatments which are j-th associates of the one
treatment of the pair and at the same time k-th associates of the other.
We have a partially balanced incomplete block design--PBIBD in short, if
there are b blocks each containing k experimental units in such a way that
(1) each block contains k (~v) different treatments,
(2) each treatment occurs in r blocks, and
(3)
blocks, i
any two treatments which are i-th associates occur together in "'i
= 1,
2, ••• , m.
In a degenerate case when m = 1, a PBIBD reduces to a BIBD.
Certain cases in
which m = 2 have been useful in practical applications.
Parameters describing an association are
v, n
i
i
(i = 1, 2, ••• , m), P jk (i, j, k
and additional design parameters are
2
= 1,
,
I
I
I
I
I
I
I
tI
I
I
I
I
I
I
2, ••• , m)
J
I
I
I
,
I
I
I
I
I
I
I
II
I
I
I
I
I
I
..I
I
b,
r, k,
i
i
~i (i
= 1,
2, ••• ,
m).
It should be that
(1.1)
and
P jk = Pkj (symmetry with respect to subscripts).
Further it can be shown that
m
(1.3)
i
E P jk
k=l
=nj
- 6ij ,
and
(1.4)
where 6
stands for the Kronecker delta.
ij
There are r(k-l) treatments occuring in the blocks in which a fixed
treatment
respect
~
occurs and they are classified into m associate classes With
to~.
occuring in
~i
On the other hand, since there are n
i
i-th associates of
blocks, it follows that
A treatment may be regarded as the O-th associate of its own.
add the following conventional notations:
~
o
= r ,
(1.6)
Under these notations, we have the following relations
m
(1. 7)
(1.8)
~
E
i=o
m
n
i
= v,
i
E P jk = n j ,
k=o
3
Thus we
I
,
and
Let A be the unit matrix of order v.
o
of order v such that its element
a~
in the a-th row and in the t3-th column is
1 if epa and cp t3 are i-th associates, and is
Ai
° otherwise,
..... aliv
1
2
a
a
..... av2i
2i
2i
..................
1
ali
(1.10)
Also let Ai be a symmetric matrix
=
i.e. ,
2
ali
, i
= 1,
2, ••• , m,
where
t3
aai
1, if epa and CPt3 are i-th associates
={
0, otherwise.
A ' A , A , ••• , Am are called the association matrices.
o
l
2
It can be seen that
(1.11)
where G stands for the square matrix of order v whose elements are all unity.
v
Hence Ao ' Al , A2 ,
... , A
m
are linearly independent with respect to the field of
all real numbers.
Furthemore we have
(1.12)
Thus the linear closure of the matriX set lAo' A , A , ... , Am} with respect
2
l
to the field of all real numbers is a linear associative and commutative algebra
m,
called the association algebra.
algebra
m is
The abstract counterpart of the matrix
I
I
I
I
I
I
I
--I
I
I
I
I
I
J
denoted by c< •
Let
4
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
..I
I
o
Pok
1
m
Pok ••••• P
ok
o
(' k =
Plk
·
m
1
P lk ••••• Plk
.
k
1
= 0,
1 , 2, ••• , m,
then (1.12) may be rewritten as follows:
A
A
A
1
•
•
A
1
•
•
•
A
0
(1.14 )
·
A
0
~ = Pk
m...
k • 0 , 1, 2 , ••• , m.
1
m
Thus it follows immediately that.
A
A
A
1
••
•
A
A
1
•
•
•
A
0
(1.15 )
0
(Ai+A ) =
k
r iiPk)
m
m
and
A
A
(1.16)
A
1
•
•
A
1
•
•
•
A
0
·
1
0
Ai~
=(> iPk
A
m
m
In other words , the mapping of ~t into the ring of all matrices of order (m+l)
generated by
(1.17)
A
i
->ri
= 0,
i
1, 2, ••• , m
gives the regular representation (n) of the association algebra J1.
5
I
§2.
Properties of the association algebra.
The abstract algebra .it is
completely reducible in the field of all rational numbers [10], hence it is
completely reducible in any number field.
On the other hand, Shur's lemma [11] shows us that any irreducible representation of a commutative algebra in an algebraically closed number field
must be linear.
Hence any irreducible representation of a commutative matrix
algebra in a field containing all characteristic roots of the matrices must
be linear.
From the general theory of algebra [12], we know that any representation
of a completely reducible algebra decomposes into irreducible representations,
each of which is equivalent to one of the irreducible constituents of the
regular representation of the algebra.
Since the rank of
V'{
is m+l, the regular representation (.tr) decomposes
into m+1 inequivalent and linear representations in the field of all complex
numbers.
Since these linear representations are the characteristic roots of
symmetric matrices, they must be all real.
Thus the regular representation
(dl) decomposed into m+l inequivalent and linear representations in the field
of all real numbers.
Even more, if the characteristic roots of all association
matrices are rational as in the cases which will be considered in the next
section, then (ur) decomposes into m+l inequivalent and linear representations
in the field of all rational numbers.
On account of the fact that
i
A-G =GA =nG,
-It v
vii v
= 1, 2, ••• , m
we can choose a non-singular matrix C of order m+l in the field of all real
numbers, being of the form
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
J
6
I
I
I
,
•I
I
I
I
I
I
..I
I
C
c
cll
00
I
I
I
I
I
I
I
C
01
lo
c
c
mo
c00
..... c
...............
C=
with
••••• Com
ml
lm
..... cmm
= c0 1 = ....• = c om = 1,
in such a way that we have simultaneously
A
A
A
A
0
0
l
(2.3)
···
A
c
A
u
l
= Cfl c-l.C
, u = 0, 1, 2, ••• , m,
.•
u
A
m
m
where
o
z
ou
(2.4)
,
=
•
o
z
mu
Let
*
A
u
m
=~
t=O
CtAt ,
U
then
m
=
~
k=o
m
=
~
[=0
7
zou = nu ; u = 0, 1, 2, ••• , m
I
~
I
I
I
I
I
I
I
where we have put
Multiplying (2.6) by c
A* A*
(2.7)
w
and summing up wi"th respect to i, we get
wi
U
~
m
E c i
i:lO
Z i
W
U
A* ,
U
and similarly
(2.8)
Thus we get
m
E c i zwi ~ 0
i:olO
if u " w ,
U
•I
and
#
m
Au = (E c i
i:olO u
(2.10)
Z
u
i)
*
-1
• A
,
U
are orthogonal system of idempotent matrices.
,,#
(2.11)
~
= v
1
=
o
-1
• G
v
u
= 0,
1, 2, ••• , m
It is clear that
,
and
v
if0
• •• + Ifm
Now, since the matrix algebra
~
•
is also a representation of
we are
cJ{ ,
going to find out the multiplicities of linear representations in
~
•
Let a , a , ... a be the respective multiplicities of m+1 linear repreo
1
m
sentations in
~.
First, by considering "the trace of G , we find that
v
8
I
I
I
I
I
J
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
a o = l.
Next, by considering traces of A , A , ••• , A , we get the following system
o 1
m
of linear equations.
a
o
+a
1
+ •••••••• + a
= v,
(2.14)
•••••••••••••••••••••••••••••••••••
an + a z
+ •••••••• + a z
om
m urn
1 ml
=o.
Since m+1 vectors of dimension (m+1)
(1,n1,~,···,nm),(1,Zl1,Z12,···,zlm),····,(1'Zml'Zm2,
••• ,zmm)
are linearly independent, the coefficient matrix of (2.14) is non-singular,
and consequently the multiplicities a , a , ••• , a are detennined uniquely
o
m
1
by the equations (2.14).
§B.
Examples of associations of certain types.
(1) Group divisible type:
and n are positive integers.
The number of treatments is v = mn, where m
They can be divided into m groups of n elements
each, such that any two treatments in the same group are 1st associates and
two treatments in different groups are 2nd associates.
If the whole treatments are numbered leXicographically with respect to
the order of groups and then with respect to the order of treatments in groups,
i.e., the j-th treatment in the i-th group bears the number (i-1)n + j, then
the association matrices have simple forms as follows:
..I
I
m
9
I
,
and
n2
= (m-l)n.
The regular representation (q) of the association algebra is generated
by the mappings
A
->
P
• I ,
003
~ - > PI = II n~1 n~2 n~J,
00
1
- > Po. II 00
n-l
1
1
f (m-l)n (m-l)n (m-2)n
A
2
E c z
.1=0 1J 1J
2
= 1.1
1
~
+ 1. (n-l) • -l(-n) =----1
mm1
J~ C2.1 Z2j = 1.1 • n:r(-1) + 0.0
10
n
=n:r
•
I
I
I
I
I
I
I
•I
I
I
I
I
I
J
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
..I
I
Thus the three orthogonal idempotents are given by
A
o
l
- A J,
2
- A
J.
(mn)-l [(m-l)Ao+(m-l)A
n -1 [(n-l)A
o
1
The multiplicities of the linear representations induced by those idempotents in the matrix association algebra are
respectively, and they are nothing but the ranks of those idempotents.
(2) Triangular type:
is a positive integer.
The number of treatments is v = n(n-l) /2, where n
We take an n/n square, and fill the n(n-l)/2 positions
above the main diagonal by different n(n-l)/2 treatments, 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
Fig.
1
1
n
= 5,
= 10
v
main diagonal--see the following figure.
2
3
4
Two treatments in the same column are
5
6
7
1st associates, whereas two treatments
8
9
which do not occur in the same column
2
5
3
6
8
4
7
9
are 2nd associates.
10
10
Hence
n
l
= 2n - 4,
n
2
= (n-2)(n-3)/2 •
The regular representation of the association algebra in this case is
given by
11
I
A
->p
~
=1,
003
010
A1 - > P.1
=
2n-4 n-2
o n-3
4
2n-8
0 0 0
o
n-3
2n-8
(n-2)(n-3)/2 (n-3)(n-4)/2 (n-4) (n-5)/2
Transforrn1ng by a non-singular matrix
c
=" ~-4
~
~-4-~
n-3 -2
-(n-2)(n-3)
we get
2n-4
o
o
0
(n-2)(n-3)/2
0
-1
, (1:12 C
n-4 0
0
=
j::o
2
~
j::o
0
001
-2
Therefore
2
~
I
0
0
(
- n- 3) 0
CljZlj
=
(2n-4).1 + (n-4).(n-4) + (-4).[-(n-3)]
C2j Z2j
= -(n-2)(n-3).1
+
(n-3).(-2) + (-2).1
Hence we get the three orthogonal idempotents as follows:
[
A
[(2n-4)
o
=
=
n(n-2)
1(n-l)(n-2).
I
I
I
I
I
I
I
•I
I
I
I
I
I
J
with respective ranks
12
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
..I
I
(3) G type: The number of treatments is v = Inn, where m and n are
3
positi ve integers. They can be arranged in the form of m x n treatments in
the same column and the remaining (m-1)(n-1) treatments are the 3rd associates
[8J.
Hence
n
1
= n-1,
n
2
~ = (m-1) (n-1).
... m-1,
The regular representation of the association algebra is generated by
A
0
-> r0
... 1
A ->r =
1
1
(3.17)
A1 ->P2
A ->'3=
3
=
4
0
1
0
0
n-1
n-2
0
0
0
0
0
1
0
0
n-1
n-2.
0
0
1
0
0
0
0
1
m-1
0
m-2
0
0
m-1
0
m-2
0
0
0
1
0
0
n-1
n-2
0
m-1
0
m-2
(n-1) (m-1)
(n-2)(m-1)
(n-1)(m-2)
(n-2)(m-2)
Transforming by a non-singular matrix
(3.18)
C
=
1
1
1
n-1
-1
n-1
1
-1
m-1
m-1
-1
-1
(n-1) (m-1)
- (m-1)
- (n-1)
we get
13
1
I
00
= 1,
zol
= n-1,
z02
= m-1,
z03
= (n-1)(m-1),
zlo
= 1,
zll
= -1,
z12
= m-1,
z13
= - (m-1),
z22
= -1,
z23
= - (n-1),
Z
(3.19)
Z21 = n-1,
z20 = 1,
z30
= 1,
z31
= -1,
z32 = -1,
z33 • l.
Hence we have four orthogonal idempotents
A#
0
= 1:..[
mn
+ A + A J,
2
3
A
0
+ A
1
A1 = ~[(n-1)
Ao
- A1 +(n-1)A2 -
A# = 1:..[ (m-1)
2
mn
A +(m-1)A
- A - A J,
0
123
~J,
~ = 1L[(n-1) (m-1)A -(m-1)A -(n-1)A + A J,
)
mn
0
1
2
3
with respective ranks
0
0
= tr A# = 1,
0
o1
= tr If1
(3.21)
= m-1,
2 = tr A1
O
~.
0) = tr
'i = (n-1) (m-1).
The relationship algebra of' a PBIBD.
relationship matrices of a PBIBD.
= n-1,
We now define the so-called
There are W = bk = vr experimental units
in the whole and they are numbered from 1 through W in any way but once for
all.
(1)
I
= I w,
(2)
G
= Gw,
Identity relation:
Corresponding to this relation, we take
Le., the unit matrix of order W.
Universal relation:
Corresponding to this relation, we take
where G stands for the matrix of order W whose elements are all
w
unity.
(3)
Block relation:
Let the incidence matrix of blocks be
14
~
I
I
I
I
I
I
I
•I
I
I
I
I
I
J
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
..I
I
where
with 11
af
.. {l, if the f-th unit belongs to the
a--th block,
0, otherwise,
then the block relation is represented by a WxW matrix
(4.1)
.,1
B =
(4)
Treatment relation:
""1
where
'I
a =
2
Let the incidence matrix of treatments be
----,v II,
1, if the
treatment occurs
(''''1 <:',..,n ----- '",,) with i"..- "" (0, at the f-th unit,
~
uc
~ft
~
otherwise,
~th
then the treatment relation is represented by the following m+l matrices of
order W:
(4.2)
where
T
U
-= IItfU g II
= 4?A 4? I
,
U
U
= 0, 1, ••• , m.
It is seen immediately that
(4.4)
m
E
u=o
T
u
= G1
•
Also
2
G
= wG,
BG "" GB
= kG,
B
2
= kB,
and
(4.6)
GTU "" TuG
= rnUG,
U
15
= 0, 1, ... , m.
I
Let N
=
II na.all
~
be the incidence matrix of the design, then, since
N
I
I
I
I
I
I
I
= ~'+,
and
(4.7)
NN'
=
m
r: A.uAu ,
u=O
it follows that
m
NN' = r:
u=e
(4.8)
where
Po
=r
+ nlA.l + ••• + n A.
mm
= rk
m
,P
u
p
u
u
= j=a
r:
= 1,
Zuj A..,
J
2, ••• , m.
are the characteristic roots of the matrix NN' with multiplicities a
respectively.
(4.8) is the spectral decomposition of NN'.
said to be regular, if all p's are positive.
u
The design is
We shall be mainly concerned
with regular PBIBDs.
It can be shown that
(4.10)
(4.11)
m
TBT
= iNN'I' = r:
T BT
u w
m
u=o
m
t=e
k,!=e
= r: ( r:
A. T,
u U
A. ! t ) T ,
kPukP!w
t
and
(4.12)
m
t
T T = r: p
Tto
u w t=e uw
Hence the linear closure with respect to the field of all real numbers
of the set of the following 4m+3 matrices
16
•I
I
I
I
I
I
J
I
I
I
,
I
I
I
I
I
I
I
•I
I
I
I
I
I
..I
I
(4.13)
I, G, B, T , T B, BT , BT B,
u
u
u
u
u
= 1,
2, ••. , m,
is a I near associative algebra !R , which is called the relationship algebra
of the PBIBD.
The relationship algebra
T , u
u
= 0,
~
contains a subalgebra ffi* generated by
1, ••• , m, which is isomorphic to the association algebra
m.
As a special case, we get the relationship algebra of a balanced incomplete block design--BIBD in short--as the linear closure [I, G, B, T, TB, BT,
BTBJ.
This algebra was investigated by A. T. James [4J in detail.
H. B. Mann
[6J exploited more general algebra which is associated with linear hypotheses.
The relationship algebra
~
of a PBIBD is located in between James' algebra and
Mann's algebra, so to speak.
The relationship algebra
~
Since
~
of a PBIBD is not commutative in general.
is generated by symmetric matrices, it is completely reducible.
Hence
all irreducible representations of TI are obtained by reducing its regular
representation.
[GJ, the totality of multiples of G, is a one-dimensional two-sided ideal
. of
~
and
2
G
= wG,
BG
= GB = kG,
TuG
Hence we obtain a linear
representation~~l)
(4.14)
I
= GTu = rnuG •
induced by [GJ as follows:
-> 1, G -> w, B -> k, Tu -> rnu •
Next we shall consider the factor algebra ~/[GJ, i.e., consider the
algebra
~
mod G.
To this end, it is convenient to change the basis of
[I, G, B, Tu*, I u*B, BTu* , BTu*B, u
= 1,
2, ••• , mJ •
u
(4.15)
17
= 1,
••• , m,
~
into
I
T**
T
u w
(4.16)
,
*
= IA*uiliAw*I' = riA**
A I' = r( m
E c i Z i)6 T,
u W
i:lO U U uw u
and
*
T*BT* = IA*I'", IA*i' = IA*NN' At'
u W
u
w
u
W
**
m
= (E
i:lO
(4.17)
= pU
Z i~i)tA A i'
U
U W
*
m
(E c i Z i)6 T.
i:lO U U
uw u
The following m subalgebras
[T*, BT*, T*B, BT*B]
u
u u
u
(4.18)
are two-sided ideals of
~
mod. G, u
= 1,
2, ••• , m
mod G, and they are annihilating each other.
Indeed,
for instance,
T*B • T*B
w
u
If
= 0,
Pu
= T*uBTW*
• B
= PU
m c i Z i)6
• (E
T*B •
i:lO U U uw u
•I
then
BTu* = Tu*B = BTu*B = 0,
and consequently the Bubalgebra reduces to [T*] mod. G.
u
Thus in the regular case, there are m inequivalent irreducible representations
of the 2nd degree, each of which being induced by a one-sided ideal of the
above two-sided ideals.
Now by direct calculations, we obtain
Ti[T*,BT*,T*B,BT*B]
u u u
u
= [Tu* ,BT**
T B,BT*B]
uu
u
(4.19)
18
rZ
Puzui
0
0
0
0
0
0
0
0
0
0
ui
rZ
I
I
I
I
I
I
I
ui Puzui
0
0
I
I
I
I
I
J
I
I
I
..I
I
I
I
I
I
I
P
I
I
I
I
I
I
..I
I
B[T* ,BT *,T*B,BT*BJ = [T*,BT* ,T*B,BT*BJ
u
u u
u
u
u u
u
0
0
0
0
1
k
0
0
0
0
0
0
0
0
1
k
; u
= 1,
Thus we get m irreducible representations of the 2nd degree:
(4.20)
2, ••• , m
i = 1, 2, ••. , m
other irreducible representations of !R are obtained by considering the factor
algebra
* u* u*
*
ffi /[aJ/[T ,BT ,T B,BT B, u = 0, 2, •.. , mJ/ raJ.
u
u
They are given by
(4.21)
!R~l):
I->1,a->0,B->0,T ->0,i=1,2, •.. ,m
i
and
(4.22)
~
i
l
):
I -> 1, a -> 0, B -> k, T -> 0,
i
i = 1, 2, ... , m .
Since
12 + 1 2 + 1 2 + m 2 2
=4 m +
3,
there can be no other irreducible representations of ~ •
We shall show that
if the design is regular and therefore v < b.
19
I f v > b, then NN' must be
I
J
singular and hence at least one of p should vanish.
u
Let
!i\ "'''1
~(1)
o
0
+ "1 !R(1) + "1 !R(1) +
1 1
GG
~ ~
u=l
!'It(1)
u u
'
then we get
m
tr I == W == "1
+ "11 + 'YG + 2 t
o
u=l
tr G .. W ...
~
u
,
W'Y G,
m
trB ..
W=
~
k'Y 1 + k'Y G + k
tr T ... 0 ...
u==l
m
rn1 'Y G + r
i
:
U
1
~u'
~uzui' i == 1, 2, ••• , m.
From the first three equations of (4.24), we get
m
'Yo=W-b-
~
u=l
~
u
•
Comparing the last m equations of (4.24) with those of (2.14), we get
~
u
u = 1, 2, ••• , m.
... ex ,
u
Consequently it follows that
"1
o
=W -
b - v + 1 •
Let
(4.25 )
T
#
u
... fA
#
U
u
i!,
= 1,
2, ••• , m,
then it is clear that
# #
(4.26)
Now, let
#
u w == r6 uwTu
T T
US
consider the following m matrices:
20
I
I
I
I
I
I
I
--I
I
I
I
I
I
eJ
I
I
I
..I
I
I
I
I
I
I
P
I
I
I
I
I
I
..I
I
(4.27)
u
= 1,
2, •.• , m.
It can be shown that
V V
u w
= 6uw r(r
Pu
- ---k).Vu ,
and
trY
u
1
#
1
= r . tr(I - - B)~A ~1(I - -k B)
k
u
1
= r • trhu#~rI - -k mi' )
1:k B)~
= r . trA '\' (I u
a .
u
In other words, m matrices
(4.29)
V# =
u
k
r(rk-p)
u
(T# _
u
1:k
BT#)(T# -
u
u
1:k
T#.B)
u
u = 1, 2, ... , m
'
are mutually orthogonal idempotents of rank a , u = 1, 2, ••• , m respectively.
u
Thus the following expression
I
= (I - 1: B k
m
~
u=l
V#) +
u
(1:
k
B -
1:
w
G) +
m
G + ~ V#
w
u=l u
1:
is the decomposition of the unit oflR into mutually orthogonal idempotents,
which will be shown to be useful from the point of view of analysis of variance.
§5.
Analysis of PBIBD.
We are concerned with the linear model which is
often called the intra-block model.
x = 71 + t
where Xl = (Xl' ••• ,
mean,
'T l
=
('T
1
Xw)
T
+
~f3
+ e,
stands for the observation vector, 7 is the general
, ••• , 'Tv) and 13 1 = (13 , ••• , f3 ) are treatment and block effects
b
1
21
I
J
being subjected to the restrictions
v
b
T
respectively, and finally
as N(O] ~ I).
I
I
I
I
I
I
I
= 1: t3 =0
0: ~ a=l a
1:
CX=1
e'
= (e l ,
••• ,
evt
is the error being distributed
We have the adjusted normal equation [2]
or
m rk-p
(5.2)
1:
u=l
k u
A~
=~ •
Let O:u linearly independent column vectors of
Ag
be
a (u)
(u)
-v +1' ~v +2 ' ••• ,
u
,
--I
u
then
( )
a u'o=
-v
-+a .:tr.
u
rk-p
k
()'
uau
t
-v -+a - ,
u
0:-12
-"
••• ,
0:
u
Hence
Balance is achieved over the set of normalized contrasts
(a~u~/ja~Ulo: v
u' u
U
0:
= 1,
••• , 0:u )
Since
X 'v#.x
-
u-
-- -L
g'~# 9. -rk-p
u
t IA#
-
u
and
22
9. ,
... ,
,
u -"
- 12m
+0:
'
I
I
I
I
I
..I
I
I
..I
I
I
I
I
I
I
P
I
I
I
I
I
I
..I
I
it follows that
sum of squares due to treatments
adjusted.
Under this present model, we have
and therefore
2
X
U
= XlV#UX/(J 2
is distributed as the non-central chi-square distribution of degrees of freedom
a and with the non-centrality parameter
u
rk-p
6
u
=
kD"
2
u
The sum of squares due to error s2 is given by
e
m
el(I -
~ B - ~ ~)e
u=l
and therefore
is distributed as the central chi-square distribution of degrees of freedom
2
2
The variates X2 , ••• , X , X are mutually independent in the
m e
l
stochastic sense. Hence under the null-hypothesis
W-b-v+l.
#
A ,.
u
= 0,
the test statistic
#
F
u
=
W-b-v+l
au
XlV X
U
~
s
e
23
I
is distributed as the central F-distribution of degrees of freedom (0 , W-b-v+l).
u
§ 6.
Analysis of variance of PBIBDs of certain types.
In this section, we
present explicit expressions of the partition of the sum of squares due to
treatments given by (5.6) and describe their statistical meanings for PBIBDS
of certain types.
(1)
PBIBD of group divisible type (see §3).
A# .. .1..
o mn
[
In this case we have
+ A ],
2
A +A
0
l
.1..
[(m-l)A0 + (m-l) A - A2 ],
mn
l
A1
:I
A#
2
. -n
1
]
A
l
[(n-l)A0 •
0
:I
0
1
C\ =m-l
O
2
= m(n-l).
Now, we assume the following inner structure of the treatment-effects, which
seems pertinent to the association under consideration.
0=1, 2, ••• , nj i = 1, 2, •.. , m,
(6.1)
which are subjected to the restrictions
(6.2)
m
E
i=l
ei
= 0,
n
E
a=l
i
1t
= 0,
i = 1, ••• , m,
0
so that there are just mn-l independent parameters.
e' = (a , ••• , em) may be
l
regarded as the group-effects and 1t~ may be regarded as the interaction between
the i-th group and the (i-l)n+o -th treatment.
The structural model (6.1) may
be called the inner-parametric representation of the treatment-effects.
By direct calculations we obtain the following expressions:
J
I
I
I
I
I
I
I
--I
I
I
I
I
I
..I
24
I
I
..I
I
I
I
I
I
I
- 8m
- 83 · · ·
· · · . .8 . · · ·8 · · · · · · 8
·
I (m-1)8 1 - 2 - 3
- m
- 8 1 +(m-1)92 - 83 · · · · · - 8m
'l(m-1)8
I
(6.3)
* =
A 'T
1
..I
I
\
\
i
····...········
)
8
8
+(m-1)8
8
- 3 ····· - m
- 1
2
····· ··..
····
- 8 1 - 82 - 83 · · · · · +(m-1)8m
n
········
····
+(m-1)8
- 8 1 - 82 - 83 ·
m
m
\
I
J
and
(n-1)1£
1
1
- 1£11 +
P
I
I
I
I
I
I
- 82
1
(6.4)
*
A 'T
2
1
- 1£ 2
(n-1)1£
- 1£1n
···
1
2
····
- 1£
1
n
n
1
- 1£
1
1
1
- 1£ 2
· · · · · · · +(n-1)1£ n
(n-1)1£
2
1
- 1£22
·····
·····
10:
....
2
2
1£1 +(n-1) 1£2
- 1£
2
n
I
I
;
\
2
1£2
2
+(n-1)1£
n
m
2
...
- 1£mn
- 1£12
........···
(n-1)1£~
- 1£
m (
m
1£1 + n-1)1£
2
-
m
1£1
·······
m
- 1£ 2
·······
m
\
- 1£mn j\
./
,
in
m:
+(n-1)1£ n ~: .
* represents the contrasts between group-effects, whereas A2*'T represents
A1'T
25
I
J
the contrasts between interactions.
Let
n
i
a:1 Q(i-1)n+a = Gi ,
= 1,
I
I
I
I
I
I
I
••• , m,
then since
and
n
1
Q(m-l)n+l - Ii lim
,
t
n
1
- -n Gom
it follows that
(6.6)
x'vI!x
1
=~
n 2v
m
2
~ Qi
1=1
'
and
(6.7)
m n
1
2
x'V#x == (
)
X ~ ~ (Q(i-l)n+a - Ii 0i)
2
r k-l + 1 i=l Q-l
m
~
i=l
26
(j. 2
i
--I
I
I
I
I
I
..I
I
I
..I
I
I
I
I
I
I
In terms or t, these turn out to be
A.V
(6.8)
..I
I
=~
k
xI-...f.x
2
=
tl£t
l'
r(k-l) + A.
I
1 t A#t
k
2
'
which were obtained by C. Y. Kramer and R. A. Bradley [5] by another method.
(2)
PBIBD or triangular type (see §3).
pfI _ 2
o - n(n-l)
In this case, we have
[
A
o
1
1 - n(n-l)
A# _
We assume the rollowing inner structure of the treatment-errects:
P
I
I
I
I
I
I
xlIx
1
n =
8
1
5,
+ 8
2
= n(n-l)
2
v
+ 1f
12
82 + 81 + 1f 2l
8 + 8 + 1f
1
31
3
8
+
1f
8 4 + 8 1 + 1f 4l
84 + 8 2 +
1f
42
8 + 8 1 + 1f
51
5
8 + 8 +
2
5
1f
52
3
+ 8
2
=
10
•
8
+ 8 + 1f
1
13
3
8
+ 8 + 1f
23
3
2
8 1 + 8 4 + 1fll~
8
2
+ 8
4
+
1f
24
8 + 8 + 1f
4
34
3
32
8
8
+ 8 + 1f
43
3
4
+ 8
5
3
+
1f
53
8
8
1f
2
5 + 1i 15 I\
+ 8
~ + 8
5
8
8 + 8 +
4
5
1
4
54
n
i =1
8 i =0,
1f.,
]. J
=1f'i'
J
L:
j ( Ii)
1f' j =0, i =1,2, ... , n,
].
so that there are just v-l independent parameters.
By direct calculations, we obtain the rollowing expressions:
27
+
5
+
1f
+ 8 +
5
The inner parameters are subjected to the restrictions
L:
,
+ 8
1f
25
35
1f
45
I
- 283 -
0
- 282 + (n-2)83 -
0
(n-2)9 1 + (n-2)9 2
(n-2)9
1
0
0
0
n-l
-
28
- 28n-l
-
28
-
0
0
28
J
n
n
. .· • ·. . .
·
(n-2)8 1
. - 28n- 1 + (n-2)9n
- 282 - 283 - 28n
- 28 n _l
- 291 + (n-2)8 2 + (n-2)83 - ·
..• ....... ·• ... ·.. ·.. .
- 29 1 + (n-2)8 2
- 28 3
· . - 29n- 1 + (n-2)8n
..... ... ·..
..· . .
•
- 281
- 282 - 263 - · . . . . - 28n- 1 + (n-2)en
0
0
0
0
0
0
0
0
0
0
0
0
n-l
0
0
(6.10) A#T= 1
1 n
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
./
and
(6.11)
a-th component of A2 T
- (n-3) E interactions corresponding to
1st associates of
associates of T
o
NiT
0
represents the contrasts between main effects e and A;T represents
the contrasts between interactions nij •
§ 70
Numerical illustration.
There are n kinds of ingredients
II' 1 , '0" In which are known to be efficient in gaining weights of hogs
2
if added in the feed stuff. We are interested to know whether there are
interactions between any two of the ingredients when the mixtures of the two
are added in the feed stuff.
28
1
--I
TO
+ 2 E interactions corresponding to 2nd
Thus
n-2
0
I
I
I
I
I
I
I
I
I
I
I
I
,}
I
I
I
..I
I
I
I
I
I
I
P
I
I
I
I
I
I
We make v = n(n-l)/2 mixtures of the possible pairs (Ii' I j ), i ~ j.
The main effects of the n original ingredients are denoted bye., i
l.
and the interaction between Ii and I j is denoted by
2, •.• , n
Then the inner-
parametric representations of the mixtures are given by
if the a..th treatment is the mixture of Ii and I
j
for
Q
= 1, 2, ••• , v =
n(n-l)
2
•
Hence in this situation, the association scheme of triangular type is naturally
defined among the v treatments.
Suppose by taking ten litters of 4 hogs each as blocks, a PBIB of triangular type with parameters
n = 5, v
= 10,
b
= 10,
r
is adopted yielding the following results.
=k
=
4, Al
= 1,
A = 2
2
Observations are the gains of
weights of hogs in pounds after feeding the mixtures of ingredients for 3
months.
This experiment is a hypothetical one and the data are borrowed from
R. C. Bose and T. Shimamoto [3J and therefere this example should be regarded
as a purely illustrative one.
..I
I
~ij.
= 1,
29
Table 1
A Design of' Triangular Type
Blocks
Treatments
1
= (1,2)
2
= (1,3)
3 = (1,4)
4
o
2
3
2.89
= (3,5)
10 = (4,5)
2.54
_e..
6
2.86
1.65
2.29
1.90
2.09
10
9.40
3.06
8.88
2.36
2.04
9.16
2.03
9.02
2.07
9.36
3.02
2.29
2.99
2.87
8.11
9.53 10.47 10.95
9.59
1.57
2.60
9.18
2.28
2.44
10.52
2.23
1.54
Treatment
Total
2.58
2.20
2.49 2.31
2.44
9
1.95
2.81
10.43
8
2.81
1. 77
2.72
7
1.41
2.54
2.28
Block Totals
5
2.51
= (2,3)
6 = (2,4)
7 = (2,5)
8 = (3,4)
9
4
2.31
= (1,5)
5
\.N
1
2.12
7.59
8.91
.-
9.33
2.77
2.09
9.27
8.89 10.61
8.22
93.71
-'-
I
..I
Table 2
Association
Treatment
I
I
I
I
I
I
..I
I
2nd Associates
1
2,
3, 4, 5, 6, 7
8, 9, 10
2
1, 3, 4, 5, 8, 9
6, 7, 10
3
1, 2, 4, 6, 8, 10
5, 7, 9
4
1, 2, 3, 7, 9, 10
5, 6, 8
5
1, 2, 6, 7, 8, 9
3, 4, 10
6
1, 3, 5, 7, 8, 10
2, 4,
7
1, 4, 5, 6, 9, 10
2,
8
2,
9
2, 4, 5,
7, 8, 10
1, 3, 6
1, 2, 5
and
At = i5 ( 6A0 + A1 - 4A2 ),
=
12
(6Ao -
1 +
2A
1 = 4,
0:
2A ), 0: =
2
2
it follows that
1
#
A1Q = 15
A~Q
=
3, 8
1, 4, 7
Now in this case, since
A~
9
3, 5, 6, 9, 10
3, 4, 6, 7, 8, 9
10
P
I
I
I
I
I
I
1st Associates
(6Q + A1Q = 4A2Q) ,
i- (3Q - A1Q + A Q)
2
31
•
5.
I
J
There is a relation
Finally the sums of squares due to main-effects and interactions are given
by
and
x'v#x=
2
k
rk -
p
Q,t!!.Q=.!Q,lA
3
-2
2
2'
respectively satisfying the relation
Thus we get the following table of the analysis of variance (Table 4) by
use of aUXiliary Table 3.
Table 3
Adjusted Treatment Tables and Related Sums
!
I
Q
1
2
9
10
-0.2700
-0.2500
-0.1075
0.2225
-0.5725
0.3350
0.4250
1.0450
-0.6250
-0.2025
Total
0.0000
3
4
5
6
7
8
AQ
1
0.0525
-0.3075
0.8800
-1.0300
0.6600
0.3175
-1.1125
-1.4225
0.6675
1.2950
,
A Q
2
0.2175
0.5575
-0.7725
0.8075
-0.0875
-0.6525
0.6875
0.3775
-0.0425
-1.0925
32
AQ
1
-0.1625
-0.2692
0.2217
-0.1950
-0.1617
0.3292
-0.0875
-0.2225
-0.1992
0.2967
A2Q
-0.1075
0.0192
-0.3292
0.4175
-0.4108
0.0058
0.5125
0.8225
-0.4308
-0.4992
I
I
I
I
I
I
I
--I
I
I
I
I
I
,J
I
I
I
..I
I
I
I
I
I
I
P
I
I
I
I
I
I
..I
I
Table 4
Analysis of Variance
Sources of Variations
Sum of Square s
d.f.
m.s.s.
Variance Ratie
Blocks
3.2284
9
Treatment
0.7467
9
0.08295
main effect
0.13 43)
0.230
interactions
0.6124
:}
0.03357
0.12248
0.841
eliminating blocks
Errors
Total
3.0585
21
7.0336
39
0.14550
References
[1]
Bose, R. C. and K. R. Nair. (1939), Partially balanced incomplete block
designs, Sankhya, 4, 337-372.
[2]
Bose, R. C. (1949), The least square aspects of analysis of variance,
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