•
ON CONSTRUCTION OF BALANCED FACTORIAL EXPERIMENTS
by
Chung-yi Suen
A dissertation submitted to the faculty of
the University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in the Department of Statistics
Chapel Hill
1982
Approved by
•
CHUNG-YI SUEN.
On Construction of Balanced Factorial Experiments.
(Under
the direction of I. M. CHAKRAVARTI.)
Shah (1958) gave the definition of balanced factorial experiments (BFE)
and proved that a BFE is necessarily a partially balanced incomplete block
design with an extended group divisible association scheme.
The purpose of
this thesis is to construct BFE's such that their main effects and lower order interactions are estimated with high efficiencies.
Three types of BFE's are discussed, and their C-matrices are expressed
by using the Kronecker product of matrices.
Efficiencies of treatment con-
trasts are derived from the eigenvalues of the C-matrix.
In an s xs x... xs
m
l 2
BFE, if the main effects of the first factor are estimated with full efficiency, then the block size is shown to be a multiple of sl.
Some efficient two- factor BFE' s are shown to be equivalent to balanced
arrays of strength two with parameters A(x,y)=A
not.
or A2 according as x=y or
l
Transitive arrays of strength two are constructed by using doubly tran-
sitive permutation groups.
In particular, a transitive array of strength two
with six symbols and index two is constructed; this corresponds to a 6x6 BFE
with block size six and all the main effects unconfounded.
Other balanced
arrays are constructed by partly orthogonal arrays.
Multi-factor BFE's are constructed from two-factor BFE's.
construction are given.
Two methods of
The first method is the product of balanced arrays
which is simlar to the product of orthogonal arrays defined by Bush (1952).
The second method was given by Shah (1960), which generates a BFE from two
given BFE's.
•
These methods can provide efficient BFE's if efficient two-fac-
tor BFE's are used.
BFE's with two-way elimination of heterogeneity are studied.
ial conditions of two-way BFE's are derived.
Combinator-
Two-way BFE's are oonstructed
ii
by suitably arranging the treatments in each block of one-way BFE's.
Patch-
work methods, which were used in the construction of generalized Youden squares
by Kiefer (1975), are used in constructing two-way BFE's.
For s a prime power
and 23, a method of constructing an s2 two-way BFE with s(s-l) columns and
s(s-l) rows such that all the main effects are estimated with full efficiency
is given.
•
iii
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to my advisor, Dr. Indra M .
Chakravarti, for his initial suggestion of the topic and more importantly,
for his constant help and encouragement during my graduate career at the
University of North Carolina.
I thank Dr. B. V. Shah for providing me with his Ph.D. dissertation,
which was of invaluable help to this research.
I also wish to thank my
committee members, Dr. Norman L. Johnson, Dr. Barry Margolin, Dr. Douglas G.
Kelly, and Dr. Donald Richards, for their comments.
I am grateful to the entire Statistics Department for the support during
my stay at the University of North Carolina at Chapel Hill.
I particularly would like to thank Ms. Ruth Bahr for many hours of the
difficult task of typing this work.
Last but not least I would like to thank my wife, Lee-jen, for her encouragement and moral support which have made this work possible.
iv
TABLE OF CONTENTS
CHAPTER I:
INTRODUCTION
1
1.1
A review of earlier work
1
1.2
Summary of this thesis
3
PRELIMINARY FORMULATIONS
5
2.1
General notation, definitions
5
2.2
Some results regarding C-matrix
8
2.3
Balanced arrays
10
SOME PROPERTIES OF BFE's
14
3.1
Efficiencies of BFE's
14
3.2
Symmetrical BFE's
19
CHAPTER II:
CHAPTER I II :
24
3.3
CONSTRUCTION OF BALANCED ARRAYS
27
4.1
Construction of transitive arrays
27
4.2
The product of balanced arrays
31
4.3
Construction of some balanced arrays of
CHAPTER IV:
strength two
CHAPTER V:
33
TWO-FACTOR BFE'S
40
5.1
slxs2 BFE's with block size
s1(sl~s2)
40
5.2
slxs2 BFE's with block size s2(sl<s2)
43
5.3
s1xs2 BFE's with block size a common multiple
of sl and s2
5.4
s
2
symmetrical BFE's with block size ns (1<n<s)
46
50
v
CHAPTER VI:
MULTI-FACTOR BFE'S
55
6.1
sm symmetrical BFE's with block size s
55
6.2
Methods of constructing mUlti-factor BfE's
58
6.3
Examples of multi-factor BFE's
62
CHAPTER VII:
BFE'S WITH TWO-WAY ELIMINATION OF HETEROGENEITY 74
7.1
Introduction
74
7.2
Construction of two-way BFE' s
80
7.3
An
algorithm for constructing an s2 two-way
symmetrical BFE
BIBLIOGRAPHY
92
101
e't
01APTER I
INTRODUCTION
1.1
A review of earlier work
Balancing is a nice property in the design of experiments.
In incom-
plete block designs. balancing is defined with respect to normalized treatment contrasts.
If all the normalized treatment contrasts are estimated
with the same variance, the design is said to be balanced.
A balanced in-
complete block design (BIBD) is an arrangement of v treatments into b blocks,
each block containing k plots, such that each treatment occurs r times and
two distinct treatments occur together in A blocks.
Alv-l) = r(k-l), r and A are determined
Since vr = bk and
once v, band k are known.
The
above design is denoted by BIBD(v,b,k).
If there is a factorial structure imposed on design, balancing must be
defined with respect to the main effects and interactions of the factors.
A design is balanced if the following two conditions are satisfied:
(1)
all the normalized treatment contrasts belonging to the same interaction (or main effect) are estimated with the same variance;
(2)
estimates of two treatment contrasts belonging to different interactions (or main effects) are uncorrelated.
Shah (1958, 1960) gave the exact definition of a balanced factorial experiment (BFE) and showed the remarkable result that every BFE is necessarily
a partially balanced incomplete block design (PBIBD) with an association
2
scheme called extended group divisible scheme by Hinkelmann (1964).
Bose (1947) was the first to consider the problem of balancing in factorial designs.
He considered sm symmetrical factorial experiments, where
s is a prime power.
By identifying the sm treatments with the points of
the Finite Euclidean Geometry EG(m,s),
he was able to construct balanced
designs with block size sk such that main effects are unconfounded.
To re-
duce the size of experimental units, he also constructed designs such that
balancing is achieved only in main effects and lower order interactions.
However, when s is not a prime power or the factorial design is asymmetrical, this method is not applicable since finite geometries are not available.
Nair and Rao (1948)
signs.
considered the problems of two-factor block de-
They pointed out that certain PBIBD's are BFE's.
Properties of
two-factor balanced designs were studied extensively and constructions by
using orthogonal arrays and orthogonal Latin squares were given.
Kishen
and Srivastava (1959), and Kishen and Tyagi (1964) have also constructed
many asymmetrical BFE's by using finite geometries and pairwise balanced
designs.
Shah (1958, 1960) derived the eigenvalues of the C-matrix for a BFE
and proposed a very powerful method of constructing a BFE from two known
BFE's.
He has constructed 117 asymmetrical BFE's in his unpublished Ph.D.
thesis (1960).
Kurkjian and Zelen (1963) defined a certain property, called
property A, for designs.
anced factorially.
Designs having property A were proved to be bal-
Kshirsagar (1966) proved the converse of the result,
viz. that balanced factorial designs possess property A.
Puri and Nigam (1976, 1978) developed the theory of balanced factorial
experiments in varying replications and varying block sizes.
They intro-
e"
3
duced the structural property A* as a generalization of property A of Kurkjian and Zelen, and proved that BFE's possess the property A*.
1.2
Summary of this thesis
Since a BFE is equivalent to a certain PBIBD, constructing BFE's is
therefore the same as constructing PBIBD's with a certain association scheme
defined on the treatments.
Although many BFE's have been constructed by
using different methods, the problem. of constructing BFE's is far from
solved except for the symmetrical case in which finite geometries can be
applied.
The purpose of this thesis is to study the methods of constructing
BFE's in which main effects and lower order interactions are estimated
with high efficiencies.
However, we shall restrict designs to those with
the same replication for each treatment and the same number of plots in
each block.
Chapter II gives the definitions of balanced factorial experiments,
partially balanced incomplete block design, the C-matrix of a design, property A, and balanced arrays.
Some useful results are also included.
In Chapter III, properties of three types of BFE's are studied.
C-matrix of a BFE is given by the Kronecker product of matrices.
The
A simple
method is given to find the eigenvalues and eigenvectors of the C-matrix of
a BFE.
Nair and Rao (1948) showed that in a sl x s2 BFE, the block size
must be a multiple of sl if the main effects of the first factor are
mated
with full efficiency.
We extend the result to a sl x s2 x ... xs
estim
BFE.
It is shown that if the main effects and interactions involving the first
q(l~q~m)
factors are estimated with full efficiency, then the block size must
be a multiple of sls2 ... Sq'
Chapter IV deals with the construction of some balanced arrays. Balanced
4
arrays, which are generalizations of orthogonal arrays and transitive arrays,
are very useful in the construction of two-factor balanced designs.
We de-
velop a method of constructing transitive arrays of strength two by using
doubly transitive permutation groups.
Methods of constructing other types
of balanced arrays are given.
Chapter V deals with the construction of two-factor BFE's.
We are inter-
ested in designs in which the main effects are estimated with high efficiencies.
Such designs are constructed by using the balanced arrays constructed
in Chapter IV and balanced incomplete block designs.
Chapter VI deals with the construction of balanced designs with more
than two factors.
We use the product of balanced arrays constructed in
Chapter IV to generate balanced arrays which are equivalent to multi-factor
BFE's.
We also apply the method of Shah (1960) to construct mUlti-factor
BFE's from two-factor BFE's constructed in Chapter V.
When s is a prime
power, we can construct an sm symmetrical BFE with block size s for any specified parameters
~. IS.
1
Chapter VII deals with designs suitable for two-way elimination of
heterogeneity.
are defined.
heterogeneity.
BFE's which permit two-way elimination of heterogeneity
We derive the combinatorial conditions for BFE's with two-way
Some two-way BFE's are constructed from one-way BFE's by
suitably arranging the order of treatments within each block.
For s a
prime power, a method is given to construct an s2 BFE with s(s-l) rows and
s(s-l) columns such that all the main effects are estimated with full efficicncy.
~"
a-IAPTER II
PRELIMINARY FORMULATIONS
General notation, definitions
2.1
Except in Chapter VII
we shall restrict our consideration only to de-
signs satisfying the following conditions:
1)
There are v treatments, each replicated r times.
2)
There are b blocks, each having k plots.
3)
No treatment occurs more than once in a block.
The fixed effect model is assumed:
(2.1.1)
y ..
1J
= )J
+ T. +
1
B.
where y .. is the yield o f t h e
1J
the overall effect,
t he
. th
1
treatment,
B.
J
E ..
1J
+ E ..
J
1J
. th
1
i=l, ... ,v;
j=l, ... ,b
treatment applied to the jth block,
)J
is
is the effect of the jth block, T. is the effect of
1
is the experimental error.
normal distributions with mean 0 and variance
0
2
E ..
1J
's are independent
.
In a factorial experiment, let F , F , ... ,F be the m factors of the
2
l
m
treatments at sl' s2, ... ,sm levels respectively.
Let v(=sls2 ... sm) treat-
ments be denoted by the levels of the factors as (x 1 ,x ' ... ,x ), where Xi
2
m
th
is the level of the i
factor and takes values 0,1, ... ,s.-l. Throughout
1
this thesis the v treatment combinations will always be arranged in lexicographic order (cf. Kurkjian and Zelen (1963)).
Let T(x ,x , ... ,x ) represent the effect of the treatment combination
m
1 2
6
(X l 'x2 '·· .,Xm)·
A treatment contrast Il(X l ,X 2 ,··· ,x m) T(x l ,x 2 ,··· ,x m)
(summation is over all x ,x ' ... ,x ) belongs to the (q_l)th order interm
l 2
action between the factors F. ,F. , ... ,F. , if 1(x ,x , ... ,X ) depends only
l 2
m
J
J
J
l
on X. , x . , ... , x .
)1
J2
Jq
2
q
and
s. -1
1
L
(2.1.2)
x.=o
1
l(X I ,x 2 '··· ,x m) = 0
2
If L1 (x ,x , ... ,x ) = 1, Ll(x ,x , ... ,x ) T(x ,x , ... ;x ) is called a norm
m
l 2
m
l 2
l 2
malized contrast.
Shah (1960)
used the following definition on balanced factorial exper-
iments.
Definition 2.1.1
A factorial experiment is called a balanced factorial
experiment (BPE), if the following conditions are satisfied:
(1)
Each treatment is replicated the same number of times, say r.
(2)
Each block has the same number of plots, say k.
(3)
Estimates of contrasts belonging to different interactions are uncorrelated
(4)
with each other.
All the normalized contrasts belonging to the same interaction are
estimated with the same variance.
Shah (1958) and Kshirsagar (1966) showed that a BFE is necessarily a
partially balanced incomplete block design with an association scheme called
extended group divisible scheme by Hinkelmann (1964).
Partially balanced incomplete block designs were first introduced by
Bose and Nair (1939).
To define PBIBD's, we need the concept of the asso-
ciation scheme for v treatments as given below:
7
Definition 2.1.2
Given v treatments 1,2, ... ,v, a relation satisfying
the following conditions is said to be an association scheme with m classes:
(1)
th
Any two treatments are either 1st, 2nd, ... , or m associates, the
relation of association being symmetrical; that is, if the treatment
th
th
a is the i
associate of the treatment B, then B is the i
associate of a.
(2)
Each treatment a has n.1 i
th
associates, the number n.1 being indepen-
dent of a.
( 3).
I f any two treatments a an d
.
are 1.th assocIates,
t h en t h e number
Q
~
.
treatments t hat are J.th assocIates
0f
th
.
a, an d k
assocIates
0f Q
~,
0f
•
IS
P~k and is independent of the pairs of i th associates a and B.
The numbers v, n
i
i
(i=1,2, ... ,m) and Pjk (i,j,k=1,2, ... ,m) are called the
parameters of the association scheme.
If we have an association scheme for
the v treatments, we can define a PBIBD as follows:
Definition 2.1.3
Given an association scheme with m classes, a PBIBD
with m associate classes is an arrangement of v treatments into b blocks of
size k«v) such that
(1)
Every treatment occurs at most once in a block.
(2)
Every treatment occurs in exactly r blocks.
( 3)
I f two treatments a an d
Q
~
.
are 1.th assocIates,
t h en t h ey occur toget h -
er in A. blocks, the number A. being independent of the particular
1
.
paIr
f
0
1
.th assocIates
.
1
a an d
Q
~.
The numbers v,b,r,k,A.(i=1,2, ... ,m) are called the parameters of the
1
design.
For a PBIBD, the following conditions hold:
8
(2.1.3)
vr
bk
(2.1.4)
r(k-I) =
m
L
n.A.
1
i=l
1
Let the v(=s l s2' .. sm) treatments be denoted
(x.=O,I, ... ,s.-l).
1
1
by (x ,x ,· .. ,x )
m
1 2
An extended group divisible association scheme with
m
m
2 -1 classes (EGD/2 -1) for the v treatments is defined by: two treatments
(x I ,x 2 ,···,xm) and (Y1'Y2"."Ym) are
CI..
1
(Cl.
I ,Cl. 2 ,
=0 or 1 according as X.=y. or not (i=1,2,
1
1
m)
,Cl.
th
associates, if
,m).
The parameters of the extended group divisible association scheme are:
(2.1.5)
(2.1.6)
where f. (0,0,0) = f. (1,0,1) = f. (1,1,0) = 1, f. (0,0,1) = f. (0,1,0) =
1 1 1 1 1
f.(1,0,0) = 0, f.(O,l,l) = s.-I, and f.(1,I,l) = s.-2.
1
2.2
1
1
1
1
Some results regarding C-matrix
Let the vxb matrix N= (n .. ) be the incidence matrix of the design,
1J
where n .. is the number of times the
1J
Il . .
1J
.th
1
treatment occurs in the /h block.
= 0 or 1, since each treatment occurs at most once in each block.
The
reduced normal equation for estimating the treatment effects are known to be:
9
T = the column vector of the v treatment effects;
T = the least square estimate of T.
It is well known that 0 is an eigenvalue of C with corresponding eigen2
vector (1,1, ... ,1)', and the variance-covariance matrix of Q is Co .
We
first prove the following lemma.
Lemma 2.2.1
value
e,
If t is a normalized eigenvector of C with non-zero eigen2
A
then t'T is estimable and Var(t'T) =
Proof:
0
18.
Q,'C = 8t', since Q, is an eigenvector of C with eigenvalue
=
e1 ~'CT,
A
and hence
t'
Var(t'T) = VarCe-o)
=
is estimated by
5(,'T
t'
e-ee
Q,
0
2
= 0
2
18.
5(,'1"=
1
A
e.
1
82' C1" = e5(,'Q.
Q.E.D.
In the complete block design, where none of the interactions or main
effects is confounded, the estimable treatment contrast t'T is estimated with
.
varlance
0
2
1r.
By comparing the variances, we can define the efficiency of
a treatment contrast.
Definition 2.2.1
In a block design in which each treatment occurs
r times, if a treatment contrast t'T is estimated with variance
its efficiency is defined to be 8/r.
When 8/r
=
0
2/8, then
1, t'T is said to be esti-
mated with full efficiency.
Kurkjian and Zelen (1963) introduced the following defintion.
Denni tion 2. 2. 2
If C is the C-matrix of a design in v(=sls2 ... s m)
treatment combinations, then the design is said to possess the property A,
if
(2.2.2)
C=
10
where
x
is the Kronecker product, a.
1
depending on a.
1
IS,
J. is an s .x s. matrix with all elements equal to 1, 1. is
1
1
the s.xs. identity matrix,
1
= 0 or 1 , the h (0. 1 ' ... ,am) are constants
1
and
1
a.
J.
1
1
1
a.
= J.1 if a.1 = 1 and
J.
1
1
=
= O.
1. if a.
1
1
They showed that designs with property A are balanced factorially.
It
is easy to see that (2.2.2) is equivalent to
(2.2.3)
C=
a.
where
are constants depending on a. 's, and (J. - 1. )
if a.
= I.1
1
=.J . - 1.
11111
1
if
0,.
1
=
o.
In fact, we can show the following relations between the coefficients
of (2.2.2) and (2.2.3).
L
(2.2.4)
{(al, ... ,a ) :a.~I3. for all i}
mIl
{ ( 0.
Lemma 2.2.2
2.3
L~_l(I3·-a.)
\( -1)
L
(2.2.5)
1
1-
)
m
"",0.
1
:a.~.
13 1
1
1
f or all
·}
I
h(a l ,a 2 ,···,a)
m
g ( 0. 1 ' 0. 2 ' ... , a
)
m
Property A is equivalent to (2.2.3).
Balanced arrays
The concepts of orthogonal arrays were first introduced by Rao (1946).
They play a vital role in the construction of symmetrical and asymmetrical
confounded factorial experiments and fractionally replicated design.
(1946), Bose
Rao
and Bush (1952), Bush (1952), Plackett and Burman (1946), and
Addelman and Kempthorne (1961) have constrcuted some useful orthogonal arrays.
Chakravarti (1956) introduced the concept of partially balanced arrays, which
generalize the concept
of orthogonal arrays. He (1961) constructed partially
balanced arrays from tactical configuration and pairwise partially balanced
designs.
Srivastava and Chopra have made contributions to the theory and
11
construction of partially balanced arrays, renaming them balanced arrays.
Denni Hon 2.3. 1
Let A be an mxN matrix with elements 0,1,2, ... ,s-1.
Consider the s t ordered t-plets (x"! ,x ' ... ,X ) that can be formed from a
2
t
t-rowed submatrix of A and let there be associated a nonnegative intger
A(X 1,x 2 ' .. · ,x t ) that is invariant under permutations of xl ,x 2 '
,x t'
If
for every t-rowed submatrix of A the stordered t-plets(x ,x '
,x ) each occur
1 2
t
A(x ,x , ... ,x ) times, the matrix A is called a balanced array of strength t
l 2
t
in N assemblies, m constraints, s symbols, and the specified A(x l ,x 2 ,··· ,xt )
parameters.
A is denoted by BA(N,m,s,t).
Definition 2.3.2
for all (x ,x , ... ,x ).
1 2
t
Let A be a BA(N,m,s,t) with parameters
A(~l"
.. ,Xt)=A
A is called an orthogonal array of strenth t in N
assemblies, m constraints, s symbols and index A.
A is denoted by OA(N,m,s,t).
Clearly N = A st in an OA(N,m,s,t) with index A.
in orthogonal arrays of strength 2 only.
We shall be interested
The definition of resolvability
will be useful in this thesis.
Definition 2.3.3
An
2
OACAs ,m,s,2), where A = as, .is said to be S-resolv-
able if it is the juxtaposition of as different OA(Ss,m,s,l)'s. A I-resolvable
array is said to be completely resolvable.
Definition 2.3.4
2
An OA(As ,m,s,2) is said to be partly resolvable if
there exist s assemblies which form a OA(s,m,s,l).
A completely resolvable orthogonal array is certainly partly resolvable.
The following example gives a partly orthogonal array which is not completely
resolvable:
12
Bose and Bush (1952) proved the following theorem:
Theorem 2.3.1
If A and s are both powers of the same prime p, a
2
completely resolvable OA(AS ,As,s,2) can always be constructed.
Adde1man and Kempthorne (1961) gave a method of constructing an
n
n
OA(2s , 2(s -1)/(s-1)-1,s,2).
n
Theorem 2.3.2
n
If s is a prime power, an OA(2s , 2(s -1)/(s-1)-1,s,2)
can always be constructed.
Their method is first to construct an OA(sn,(sn_ 1)/(s-1),s,2) of index unity with the factors represented by x1,x2, ... ,xn,x1+x2, ... ,xl+ ... +xn'
i.e. all the (sn -1) / (s-l) main effects and interactions of n factors . Then
add
more
(sn_ 1)/(s-1)-1 factors, which are obtained by adding x~ to the
n
above (s -l)/(s-l) factors except Xl.
2(sn- l )/(s-1)-1 factors.
The first half contains all the
The second half is suitably constructed to make
it an orthogonal array.
There are 2(s
n-1
-l)/(s-l)-l factors which do not contain x n in the
OA(2s n ,2(sn- l )/(s-1)-1,s,2).
If we delete these factors from the orthogonal
array, the remaining array is completely resolvable.
Corollary 2.3.1
If s is a prime power, a completely resolvable
n
n-1
OA(2s ,2s
,s,2) can always be constructed.
13
Finally we shall give the definition of a transitive array.
Definition 2.3.5
if x. = x. for some
1
J
Let A be a BA(N,m,s,t) with parametersA(x ,··· ,xt)=O
l
i~j,
and A(x , ...
l
,X
t
) = A if all x. 's are distinct. Then
1
A is called a transitive array of strength t in N assemblies, m constraints,
5
symbols, and index A.
In a TA(N,m,s,t), m
A is denoted by TA(N,m,s,t).
~
s since all the symbols in an assembly are dis-
tinct, and also N = As(s-l) ... (s-t+l) where A is the index.
CHAPTER III
SOME PROPERTIES OF BFE'S
3.1
Efficiencies of BFE's
Let F ,F , ... ,F be the m factors at sl,s2" .. ,sm levels respectively
1 2
m
and N be the incidence matrix of a BFE.
treatment contrasts of
.
eIgenvectors
0
To calculate the efficiencies of
a BFE, we need to find out the eigenvalues and
f th e C
'
-rnatrlx
r I -k
_l~TN'.
~~
We assume an extended group divisible association scheme defined on the.
m
treatments since a BFE is an EGD/(2 -l) - PBIBD.
Consider the matrix
a
a
a
(Jl-I ) lX(J -I ) 2x ... X(J -I ) m as defined in Equation (2.2.3). The ele2 2
l
m m
th
th
column of
ment, which is in the (x l ,x 2 ,· .. ,x m)
row and (Yl'Y2"" ,Y m)
the matrix (the treatments are in lexicographic order), is 1 if treatments
(X ,x ,···,x )and (Yl'Y ""'Y ) are (a ,a , ... ,a )
l
2
otherwise.
m
2
m
l
2
th
m
Two treatments which are (a ,a , ... ,am)
l 2
th
associates, and is 0
associates occur to-
gether in A
blocks; hence we have the following lemma:
a l a 2 · •. am
Lemma 3.1.1
(3.1.1)
NN' =
where Aoo ...
O
Further
Let N be the incidence matrix in a BFE;
then
is defined to be r.
let
j.
1
h.
1
= s.I x1 vector with all elements equal to 1;
any s.xl vector orthogonal to j.;
1
1
15
S.
h.1 1 =
t
1
if 8.1 = 1
J.
if B. = 0
1
1
Then
a.
(3.1.2)·
1
B.
8.
h. I =H.(a.,B.) h. I
1
1
(J.,-!.)
1
111
1
where H.(a. ,B.) is given by the following table:
111
~
0
0
1
s. -1
1
1
-1
1
1
Thus,
(3.1.3)
=
Define
(3.1.4)
We have the following theorem:
Theorem 3.1.1
The eigenvalues of NN' of a BFE are g(B ,8 , ... ,8 )'s
l 2
m
with corresponding eigenvectors
m
II (s. -1)
i=l
1
The multiplicity of
B.
1
.
16
1
Bl B2
Bm
Since C = r I - INN', hI Xh x... xh 's are also eigenvectors of C
m
2
1
with corresponding eigenvalues r - k g(B ,B 2 ,.·.,B m)·
l
say
B.
J1
,B. , .. . ,B. ,
J2
If qof Bi's are 1,
then it is easy to check that the treatment contrast
Jq
8 B
Bm
th
(hI 1 xh 2 x... xh m )T belongs to the (q-l)
order interaction between
2
F. ,F. , ... ,F . .
Jl
J2
Let E(B ,B , ... ,B ) denote the efficiency of this treatm
l 2
Jq
ment contrast; then
Corollary 3.1.1
In an s xs BFE, Nair and Rao (1948) showed that the block size k must
l 2
be a multiple of sl if the main effects of the first factor are estimated with
full efficiency.
We generalize
Theorem 3.1.2
the result to an m-factor BFE.
In an s xs x ... xs BFE, if the main effects of F are
l 2
m
i
estimated with full efficiency, then the block size must be a multiple of s ..
1
Proof:
Assume i = 1 without loss of generality.
Let
.R,.
J.
an d
th
,
1
be the numbers of treatments in the jth block which are at the i
ii
th
levels of F respectively.
l
l
.R,.
J.
and
,
1
Counting the number ·of treatments in the
jth block, we have
s -1
1
(3.1.5)
L
i =0
1
.R,.
J.
=k
1}
.
t h e numb er
Count1ng
0
f or dere d pa1rs
.
0
f treatments at the l. th 1eve 1
l
0f
PI in all blocks, including pairs of the same treatment, we have
b
(3.1.6)
L
j=l
Counting the number of ordered pairs (x,y), where treatment x is at the
17
th
i th level and treatment y is at the ii
level of F , in all blocks. we
l
l
have
b
0.
L
(3.1.7)
~.
~.
J.
j=l
1
1
J.,
1
1
= 5 5
2 3
, .. 5
m
0. ,., . ,am
2
A1rv
rv (s2- 1)
~2···~m
2
am
•.• (s -1)
m
Hence
b
(3.1.8)
=
L
j=l
=
2
~.
Ji
I ~~
1
I
L
J' =1
1
~2'
0. 2
-2
J.,
1
j =1
2 s s ... s. [
2 3 m rv
(s2- l )
b
b
+
rv
(A.
. '~m
-A
)
00. , .. a
10. , .. a
2
2
m
m
am
... (sm- l )
]
=
2
5
25 3 " .sm g(l,O, ... ,0)
If main effects of F are estimated with full efficiency,
l
g(l,O, ... ,O)
=
0 by Corollary 3.1.1.
Hence
,b
L.-l(~'
JJ.
1
implies~.
=~.
J.
1
J. ,
1
1
1
But
L.1C
1_
0
~.
J.1
Since i
1
J. ,
1
1
)
2
=
0, which
and iI' are arbitrary,
l
1
~j.
s -1
for all j.
-~.
then
=
constant for all j
=
1,2, ... ,b; i
l
= 0,1, ... ,sl-l.
1
= k;
therefore
k
s
.
k must be a multiple of sl' since
1
Q.E.D
is an integer.
From the proof of Theorem 3.1.2, we know that each level of the factor
F occurs the same number of times in each block if the main effects of F
l
l
are estimated with full efficiency.
We shall consider the case that the
first order interaction of F and F are estimated with full efficiency.
l
2
Let ~.
J. .
1
denote the number treatments in the j th block which are at
1
1 2
th
the i
level of F and at the i th level of F2 ; also let ii
l
l
2
t
i l and i
2t
i 2·
By counting the numbers of certain ordered pairs of treatments in all blocks,
the following equalities hold:
18
(3.1.9)
b
l
51,.
51,.
j=l J i1 i J il i
2
2
= S3"
l
'Sm
0.
3
0.
" .a
1)
Aa1 a ... c t
(s3.
m
3
3
m
am
... (s m-1)
b
I
51,.
51,.
j=l J i1 i Jiii
2
z
Hence,
(3.1.10)
If the first order interactions of F1 and F are estimated with full
2
efficiency, then g(l,l,O,O, ... ,0) = O.
Therefore,
(3.1.11)
The equation (3.1.11) is an identity if i l = ii or i 2 = ii·
Theorem 3.1.3
In an s xs 2x... xs m BFE, if the first order interactions
l
of F1 and F are estimated with full efficiency, then
2
51,.
J. 1.
1
1 2
If we also assume the main effects of F and F are estimated with full
l
2
efficiency in addition to the first order interaction, then summing over
19
all i
2,
we have
since t.
J.
1
s -1
But
I/=o
2
t.
J.
1
.
1
1 2
= toJ.
1
1
,
1
= toJ. =-ksl
1
-k-
Therefore to
J
1
0
1
•
1
5
1 2
s
l 2
for all iI' i 2 and j.
Corollary 3.1.2
between F
l
In an s xs x ... xs m BFE, if main effects and interactions
l 2
and F are estimated with full efficiency, then the block size k
2
must be a multiple of s l s2 and t . .
j
1
112
= k/s l s 2
for all iI' i
2
and j.
Similar arguments to those of Theorem 3.1.3 and Corollary 3.1.2 can be
used to prove the general results involving q factors.
Theorem 3.1.4
In an s l xs x ... xs BFE, if the (q_l)th order interactions
m
2
between F , F , ... , and F are estimated with full efficiency, then
l
2
q
(3.1.12)
.,
1 .•
1
Corollary 3.1.3
In an slxs2x ... xsm BFE, if all main effects and inter-
actions involving F , F , ... ,F are estimated with full efficiency, then the
l
2
q
block size k must be a multiple of sls2" .Sq and t . .
j
for all iI' i , ... ,i and j.
q
2
3.2
~
1
1
1 2
.
, .. 1
k/s l s ... s q
2
q
Symmetrical BFE's
When the factorial experiment is symmetrical, i.e. each factor is exper-
imented with the same number of levels s, we can define a balanced design by
20
replacing (4) in Definition 2.1.1 by
(4)
For all q = 1,2, ... ,m, all normalized (q_l)th order interactions are
estimated with the same variance.
A design satisfying this condition is called a symmetrical BFE, which
is a special case of designs satisfying Defintion 2.1.1.
Shah (1958) showed
that a symmetrical BFE is a PBIBD with a hypercubic association scheme.
Definition 3.2.1
Given sm treatments, a hypercubic association scheme
with m classes is given by: two treatments (x l ,x , .. ·,xm) and (Yl'Y2''''Ym)
2
are jth associates, if j is the number of times x. ~ y ..
1
1
The parameters of the hypercubic association scheme are:
(3.2.1)
n. =
J
(3.2.2)
k
p ..
1J
(~) (S-l)j
J
j=1,2, ... ,m
A hypercubic PBIBD has fewer associate classes than an extended group
divisible PBIBD, and possesses a simpler design structure.
Most of the
balanced designs constructed by Bose (1947) are hypercubic PBIBD's.
We
shall be interested in deriving the eigenvalues of the C-matrix of a hypercubic PBIBD in order to obtain the efficiencies of the treatment contrasts.
Let
N = the incidence matrix of a hypercubic PBIBD;
I
= the sxs identity matrix;
J = the sxs matrix with all elements equal to 1;
(J_I)a =.
{
J-I
I
if a=l
if a=O
21
Fix j (O:::;j:::;m); consider the matrix
La
l
_. (J-1)
+... +am-J
al
x(J-1)
a2
am
x ... x(J-I)
.,
of which the element in the (X ,x , ... ,xm)th row and the (Yl'Y2'''''Ym)th
l 2
column is 1 if treatments (x ,x ,,,,,X ) and (Yl'Y2""'Ym) are jth associ1 2
m
· 0 oth
'
T
.
.
ates, an d IS
eTWlse.
wo 'J th aSSOCIate
treatments occur toget h er In
~.
J
blocks; if
~O
is defined to be r we have
Lemma 3.2.1
Let N be the incidence matrix in asymmetrical BFE; then
(3.2.3)
The eigenvectors of NN' of a symmetrical BFE are the same as those of
a BFE; the eigenvalues can be derived from the equation (3.1.4) by letting
S1 = s 2 =... = s = s, and
m
Theorem 3.2.1
~
a l a 2 · .. am
=
~.
J
if a +a +... +a = j.
l 2
m
The NN' of a symmetrical BFE has (m+l) different eigen-
m
j
values g(j) (j=O,l, ... ,m) with multiplicity (.)(s-l) ,
J
where
m
g(j) =
(3.2.4)
and
L P. (j ;m,s)
~.
. 011
1=
P. (j;m,s) is the Krawtchouk polynomial given by:
1
(3.2.5)
P.(j;m,s)
1
For convenience, we list the Krawtchouk polynomials P. (j;m,s) for
1
m=1,2,3,
and 4 in the following table.
22
TABLE 1
Krawtchouk Polynomials Pi(j;m,s)
x
m=2
m=l
0
1
0
1
5-1
1
1
-1
~
0
0
1
1
1
5-1
2
1
-2
1
2
2(5-1)
(5-1)
2
-(5-1)
1
m=3
I;z
0
0
1
3(5-1)
3(5-1)
1
1
25-3
(5-1)(5-3)
2
1
5-3
-25+3
3
1
-3
3
1
2
3
2
(5 -1)
- (5-1)
3
2
5-1
-1
m=4
I>z
3
0
1
2
0
1
4(5-1)
6(5-1)
1
1
35-4
2
1
25-'4
3(5-1)(5-2)
2
5 -65+6
-2(5-1) (5-2)
3
1
5-4
-3(5-2)
35-4
4
. 1
-4
6
-4
2
3
4(5-1)
2
(5-1) (5-4)
4
(5-1)
- (5-1)
(5-1)
-(5-1)
1
4
.3
2
23
Since C
= r I -
..!.k
NN'
we have
The eigenvalues of the C-matrix of a symmetrical
Corollary 3.2.1
BFE are r-g(j)/k (j=O,l, ... ,m).
The efficiency of a (j_l)th order inter-
action is
(3.2.6)
E. = 1 - g(j) Irk
J
The following example of a symmetrical BFE was given by Shah (1958).
Example 3.2.1
Consider a design with two factors each at three levels
2
v = 3 , b = k = 6, r = 4, n
=
l
gel)
r
+
2
= 4, Al = 2, A = 3.
2
1
2
3
4
5
6
01
00
00
01
00
00
02
02
01
02
02
01
10
10
11
10
11
11
12
11
12
11
12
12
20
21
20
20
20
20
21
22
22
22
21
22
Blocks
Treatments
n
(s-2) Al - (s-l) A = 4
2
g(2) = r - 2A
l
El =1, E
2
+
A = 4 - 2'2 + 3
2
=1
- 3/4'6
1'2 - 2·3 = 0
+
3
= 7/8.
Hence the main effects are estimated with full efficiency, and the first
order interactions are estimated with efficiency 7/8.
It is clear from Theorem 3.1.2 that if all the main effects of a symmetrical BFE are estimated with full efficiency, then the block size k must be
a multiple of s.
By Corollary 3.1.2, if all the main effects and first or-
der interactions are estimated with full efficiency, then k must be a multi-
24
pIe of s
2
In general,
if all the interactions with order less than
q(l$q$m) are estimated with full efficiency, then k must be a multiple
of sq.
3.3
If the factorial experiment contains ml+m2+ ... +~(=m) factors in which
m. factors are at s. levels (i=1,2, ... ,h), a balanced design can be defined
1
1
by replacing (4) in Defintion 2.1.1 with
(4)
All the normalized contrasts belonging to the interaction involving
q. factors at s.levels are estimated with the same variance, for
1
1
qi = 0,1, ... ,m i ; i=1,2, ... ,h.
This design was shown by Shah (1958) to be a PBIBD with (m +l)(m +1) ...
l
2
(~+l)-l
associate classes.
ml m2
~
.
sl s2 ... sh treatments IS:
are (al,a2, ...
x.
J
The association scheme defined
on the
two treatments (x l ,x 2 ,··· ,xm) and (Yl'Y2"" 'Ym)
,~)th associates if ai(i=l, ...•h) is the number of times
f y.J (j=ml+···+m.1- l+l .... ,ml+ ... +m.).
1
The parameters of the association scheme are:
h m.
a.
1
1
n (a 1 ' a ' ... , ~) = II ( ) (s . - 1)
2
i=l a
1
(3.3.1)
Ct.=
1
i
O.l, ... ,m.; i=1,2, ... ,h
1
a.
P 1
i=l 8.y.
1 1
h
II
(3.3.2)
a
where P l
8.y·
1 1
is defined by Equation (3.2.2).
Similar to Lemma 3.1.1 and Lemma 3.2.1, we can prove the following lemma:
Lemma 3.3.1
then
ml m2
~
Let N be the incidence matrix of a sl xs x .. ,xs
BFE.
h
2
2S
(3.3.3)
NN'
•
where AOO ... O = r, 8i
=0
or 1.
The eigenvalues of NN' can be derived from Equation (3.1.4) by letting
Sm +1= 5 +2 =... = 5 +
1
mI'
ml m2 = 52
sml+".+~_l+l =... = sm = sh
if 8
+..• +B
m1 +... +m.1- 1+1
ml+···+m.1
= a. for all i=l, ... ,h.
1
Theorem 3.3.1
There are (ml+l) (m +l) ...
2
m
for the NN' of an 51 l xS
different eigenvalues
~
m2
2
(~+l)
X •••
xs h
BFE.
They are
h
(3.3.4)
g(al,a2'···'~) = I
Aa
a
IT Pa (a.;m. ,5.)
n 1-'1'
a
a 1-'1' ··I-'h . 1 1-"
.. l-'h
1=
1 1 1 1
a.,B.
= O,l, ... ,m.;
111
i=1,2, ... ,h
where Pa (a. ;m. ,5.) is the Krawtchouk polynomial given by Equation (3.2.3).
1-"
1 1 1
1
The multiplicity of
g(al,a2'''''~)
Corollary 3.3.1
are r - g(a l , ...
,~)/k
is n(al,a2' ...
'~)
given by (3.3.1).
m
The eigenvalues of the C-matrix of an 51 l x... xs
(a.=O,l, .. ,m.; i=l, ... ,h).
1
1
~
h
The efficiency of any con-
trast belonging to the interaction involving a. factors at s. levels is
1
(3.3.5)
BFE
1
26
Example 3.3.1
Consider the 22 x3 BFE with v = 12, r = 3, b = k = 6,
•
Blocks
Treatments
Hence,
1
2
3
4
5
6
000
000
000
010
010
010
110
110
110
100
100
100
001
011
011
011
001
001
111
101
101
101
111
111
012
002
012
002
012
002
102
112
102
112
102
112
E(O,l)=E(l,O)=E(l,l)=l,
E(2,0)=
8
9'
E(2,1)=
5
9·
The balanced factorial experiment discussed in this section is a genera1ization of those discussed in Section 3.1 and 3.2.
When m1=m2= ... =~=1, the
h
association scheme defined on the treatments is the EGD/(2 -1) association
scheme and the design is the BFE discussed in Section 3.1.
Theorem 3.3.1
is a generalization of Theorem 3.1.1 for P (8. ;l.s.) = H. (a. ,8.).
a. 1
1
1
1
1
When
1
h=l, the association scheme becomes a hypercubic association scheme and the
design
beco~es
a symmetrical BFE.
CHAPTER IV
CONSTRUCTION OF BALANCED ARRAYS
4.1
Construction of transitive arrays
Transitive arrays are defined in Section 2.3, and transitive arrays
of strength two are useful in the construction of two-factor BFE's. Therefore we are especially interested in constructing transitive arrays of
strength two.
In this section we shall give a method of constructing
arrays of strength t by t-ply transitive permutation groups.
First we
give the defintion of a t-ply transitive group.
Definition 4.1.1
The group consisting of all permutations of n sym-
boIs {O,1,2, ... n-l} is called the symmetric group of degree n, denoted by S .
n
Definition 4.1.2
A subgroup G of S
n
is called a t-ply transitive
group, if G contains a permutation replacing any whatever given ordered
set of t symbols by any whatever other given ordered set of t symbols.
A t-ply transitive group is called transitive if t=l, for t=2,3 we
often use the terms doubly, triply transitive.
Clearly a (t+l)-ply transi-
tive group is t-ply transitive; more properties of t-ply transitive groups
can be found in the
usef~l
boo~
by Carmichael (1937).
The following theorem is
in the construction of transitive arrays.
Theorem 4.1.1
The order of a t-ply transitive group G of degree s is
As(s-l) ... (s-t+l), where A is the order of the largest subgroup H of G, each
element of which leaves fixed a given ordered set of t symbols.
28
From this theorem, we see that the order of a t-ply transitive group
of degree s is equal to the number of assemblies in a transitive array
strength t with s symbols.
of
This suggests a relation between transitive
groups and transitive arrays.
Indeed, we can construct a TA(As(s-I) ...
(s-t+l),s,s,t) from a t-ply transitive group of degree s.
Theorem 4.1. 2
If a t-ply transitive group G of degree s and order
As(s-l) ... (s-t+l) exists, then a TA(As(s-l) ... (s-t+l),s,s,t) can always be
constructed.
Proof:
For each element
g in G, we associate it with an sxl vector
(g(O),g(l), ... ,g(s-l)),where g(i)(i=O,l, ... ,s-l) is the symbol replacing i
by the action of g.
Construct an sxAs(s-I) ... (s-t+l)
matrix
by
using
As(s-l) ... (s-t+l) sxl vectors associated with each element in G as columns.
For any t elements i ,i , ... ,it from {O,l, ... ,s-l}, the set {(g(i ),
l 2
l
g(i ), ... ,g(i )) :geG} contains any ordered set of t symbols exactly A times
2
t
by the property of a t-ply transitive group.
This means that for
any
t-rowed submatrix of the sxAs(s-l) ... (s-t+l) matrix every ordered set of
t symbols occurs'as a column exactly A times.
Hence the sxAs(s-I) ... (s-t+l)
matrix is a TACAs(s-I) ... (s'-t+I),s,s,2) by Definition 2.3.5.
Q.E.D.
We would like to construct TA(As(s-I),s,s,2)'s in this section; hence,
by Theorem 4.1.2, we shall serach for doubly transitive groups.
Usually
A is required to be as small as possible so that the size of the transitive
array won't be too large.
However, if A is not restricted to be small, we
can always construct a TA(As(s-I),s,s,2) for any s
~
2.
For example, the
symmetric group of degrees is doubly transitive with order s!; so we can construct a TA(s! ,s,s,2) with index A = (s-2)!.
Two examples follow which illustrate the application of Theorem 4.1.2.
29
If s is a prime power, then there exists a TA(s(s-l),
Corollary 4.1. 1
s,s,2) .
Let GF(s) denote the Galois field of order s.
Proof:
permutation group ort all the elements of GF(s).
Consider the
Let G be the group consis-
ting of the following permutations:
a#O, a,bEGF(s)
g(x) = ax+b
(3.1.1)
where· and + denote multiplication and addition respectively in the GF(s).
Then G is a doubly transitive group of degree s and order s(s-l);
ther~fore
Q.E.D.
we can construct a TA(s(s-l) ,s,s,2) by Theorem 4.1.2
e
r
Example 4.1.1
For s=3, we can construct a TA(6,3,3,2).
0
1
2
0
1
2
1
2
0
2
0
1
2
0
1
1
2
0
The TA(s(s-1),s,s,2) constructed in Corollary 4.1.1 is completely re2
solvable, and is equivalent to the existence of the well known OA(s ,5+1,5,2)
or (s-l) mutually orthogonal Latin squares of order s.
Corollary 4.1. 2
(5- 2 ),5,5,3).
If 5-1 is a prime power, then there exists a TA(s(s-l)
Moreover, if 5 is even, then we can
always
construct
a
TA(s(s-1)(s-2),s,s,2).
Proof:
element
00.
Let S be the set consisting of all elements of GF(5-l) and the
Consider the symmetric group on the elements of S.
subgroup consisting of the elements
(4.1.2)
g(x) = ax+b
a#O,
a,bEGF(s-l)
Let G be the
30
or
a
g(x) = x+b
(4.1.3)
where
a+ OO =
+
a!O,
C
a,b ,cEGF(s-l)
and + denote multiplication and addition defined in GF(s·-l), and
0
00,
a/ oo = 0, and a oOO =
00
for all a!O.
group of degree s and order s(s-1)(s-2);
Then G is a triply transitive
hence
we
can
construct
a
TA(s (s-l) (s-2) ,s,s ,3) .
If s is even, the subgroup of G of all even permutations is doubly transitive of order s(s-1)(s-2)/2; thus we can construct a TA(s(s-l) (s-2)/2,s,s,2).
Q.E.D.
Example 4.1. 2
o
For s=6, we can construct a TA(60,6,6,2) in the following:
0 1 122 3 344 5 5
1 1 2 2 4 400 5 5 3 3
240 3 354 5 120 1
35540 1 1 2 3 0 4 2
4 2 305 354 2 110
534 5 102 1 0 324
o
0 1 1 2 2 3 3 4 4 5 5
2 2 335 5 440 0 1 1
1 5 250 402 1 334
3 4 4 0 3 1 1 5 2 5 2 0
5 1 5 240 2 0 3 1 4 3
43041351520 2
001 1 2 2 3 3 4 455
4455001 1 332 2
1 334 1 5 250 204
520243045 1 1 3
3 1 435 1 5 2 200 4
2 5 2 0 3 4 4 0 1 5 3 1
o0
112
33001
4 5 240
21534
544 2 3
1 2 355
o0
1 1 2 2 334 4 5
5 5 4 4 3 3 2 2 1 i 0
230 5 1 4 1 405 2
4 1 3 2 500 5 2 3 1
3 2 504 1 4 150 3
14230 550 324
2 3 3
155
3 0 1
5 2 4
0 1 0
4 4 2
4 455
2 244
3 5 1 2
100 3
5 321
0 130
5
0
3
4
2
1
The above TA(60,6,6,2) is 2-reso1vab1e; we can add 12 columns consisting
of each of (i,i,i,i,i,i)' (i=O,l, ... ,5) twice to get a 2-reso1vab1e
OA(72,6,6,2).
Then we can add a row (0 ... 01 ... 1 ... 5 ... 5) in which each sym-
bo1 repeats 12 times to'obtain an OA(72,7,6,2).
2
m~
AS -1
2
[ 5-1 ] in an OACAs ,m,s ,2).
Hence m
~
72-1)
[~
It is
well
known
that
.
] = 14 In the OA(72,m,6,2);
7 is far less than the upper bound 14.
Also note that it is not possible to construct a transitive array of
strength two in 6 constraints, 6 symbols and index 1.
If a TA(30,6,6,2)
31
exists, we can add 6 columns of the forms (i,i,i,i,i,i)'(i=O,l, ... ,5) to
obtain an OA(36,6,6,2).
But this is impossible, since the existence of an
OA(36,6,6,2) implies the existence of 4 mutually orthogonal Latin squares
of order 6 which is not true.
Letting s = 10, 12 and 14 in Corollary 4.1.2, we can construct a
TA(360,10,10,2),
a TA(660,12,12,2) and a TA(1092,14,14,2).
Rao (1956) has considered the construction of transitive arrays of
strength two and index unity.
Theorem 4.1.3
He showed the following results:
The existence of a resolvable TA(s(s-1),m,s,2) is equiv-
alent to the existence of m-l mutually orthogonal Latin squares of order s.
Theorem 4.1.4
The existence of a TA(s(s-1),m,s,2) is equivalent to
the existence of m-2 mutually orthogonal Latin squares of order s which
have all different symbols in the diagonals.
Since a Latin square of any order with different symbols in the diagonal
always exists, we can construct a TA(s(s-l) ,3,s,2) for s12.
Example
4.l.~
A TA(30,3,6,2) is constructed from a Latin square of
order 6 with different symbols in the diagonal.
aaaaa
1 1 1 1 122 2 2 2 3 3 3 3 3 4 4 4 4 4 5 555 5
1 2 345
a
2 3 4 5
214 5 334 5
4.2
a
a
134 5
a
1 2 450 1 2 350 1 2 3 4
250 134 145 2
a
2 5 3
a
1 430 2 1
The product of balanced arrays
Bush (1952) proved the following theorem for the product of orthogonal
arrays.
Theorem 4.2.1
The existence of OA(N.,m. ,so ,t) for i=1,2, ... ,k implies
111
the existence of an OA(N,m,s,t), where N = Nl ,N , ... ,Nk , s = sls2 ... sk' and
2
m = min(m ,m , ... m ).
k
l 2
32
The product of orthogonal arrays can generate orthogonal arrays from
several known orthogonal arrays.
The procedure can be similarly used to de-
fine the product of balanced arrays and to generate new balanced arrays
from known balanced arrays.
Theorem 4.2.2
The existence of BA(N. ,m. ,s. , t) for i=l, 2, ... ,k implies
111
the existence of a BA(N,m,s,t), where N = N N ... N , s = s s2' .. sk and
l 2
k
l
m = min(m ,m , ... ,m ).
k
l 2
If the symbols of the BA(N,m,s,t) are denoted by
ordered k-tuples, then the parameters are ACCaU,a2l, .. ,akl),(a12,a22, .. ,ak2)'
. . . , (a It ' a 2t ' . . . , ~ t) ) =A(all' a12 ' . . , at ) A(an ' a 22 ' . . , a 2t) . . . A(a k 1 ' a k 2' . . , a kt) .
Proof:
Let the BA(Nl,ml,\,t) be denoted by the mlxN l matrix A = (aU)
and the BA(N ,m ,s2,t) be denoted by the m2 xN matrix B = (b ij ). Let
2 2
2
Al and B denote the first m rows of A and B, respectively.
l
Then form
the mXN N matrix:
l 2
...........
. (a 1,b mN ). • . (a
b 1)'
mN l' m
m
2
. (a
mN
b
l'
mN
)
2
which can be shown to be a BA(N N2 ,m,sl,s2,t) with parameters A((al,b ), ... ,
l
l
(at,b )) = A(a , ... at)A(b , ... ,b ).
t
l
l
t
From this array by following the same
procedure with BA(N ,m ,s3,t), we get a BA(NlN2N3,m,sls2s3,t}.
3 3
this procedure, we finally get a BA(N,m,s,t).
Example 4.2.1
Continuing
Q.E.D.
The product of the following two balanced arrays
33
BA(2,2,2,2)
BA( 6,2,3,2)
01
012012
10
120201
A(O,l)=l
A(0,0)=A(1,1)=A(2,2)=0
A(O,O)=A(l,l)=O
A(0,1)=A(0,2)=A(1,2)=1
is a BA(12,2,6,2)
00 01 02 00 01 02 10 11 12 10 11 12
11 12 10 12 10 11 01 02 00 02 00 01
with parameters
4.3
Construction of some balanced arrays of strength two
In this section, we are interested in constructing balanced arrays of
strength two with parameters A(x,y)=A
1
or A according as x=y or not.
2
In
particular, we are especially interested in the BA((ms-l)sA,ms,s,2) with parameters A(X,y) =
(m-I)A or rnA according as x=y or not.
For brevity, we
shall call it the balanced array of type T with index A and denote it by
BA(T) (m,s ,A).
It is clear that a BA(T) (l,s,A) is a TA(As(s-l) ,5,5,2).
In constructing
a BA(T) (m,s,A) for any given m and s, we would like A to be as small as possible so that the size of the balanced array is not too large.
However, if
there is no restriction on A, we can always construct a BA(m,s,A) for any
m and s.
Theorem 4.3.1
some A.
For all m and s, there always exists a BA(T)(m,s,A) for
34
Proof:
For all m and s, there exists a TA((ms-l)msn,ms,ffiS,2) for
some n from the discussion in Section 4.1.
Let the symbols of the transi-
tive array be denoted by {O,l, ... ,ms-l}. If we replace each symbol in the
transitive array by x(mod m),
then
the
transitive
array
becomes a
2
BA((ms-l)ms n,ms,s,2) with parameters A(x,y) = (m-l)mn or m n according
as x=y or not, which is a BA(T) (m,s,mn).
Q.E.D.
The method of construction used in Theorem 4.3.1 does not usually produce balanced arrays with as small a number of assemblies as we desire. In
the following, we'll discuss methods of constructing balanced arrays of
'type T with index unity.
Theorem 4,3.2
The existence of a partly resolvable '(Definition 2.3.3)
2
OA(ms ,ms,s,2) is equivalent to the existence of a BA(T)(m,s,l),
Proof:
2
If a partly resolvable OA(ms ,ms,s,2) exists, then there
exist s assemblies which form an OA(S,ffiS,S,l). We can permute the symbols of
the orthogonal array in each row such that these s assemblies are of the
forms (i.,i, ... ,i)' for i=O,l, ... ,s-l.
Deleting these s assemblies, we ob-
tain a BA(T)(m,s,l).
On the other hand,
if theres exists a BA(T)(m,s,l), we can obtain a
2
partly resolvable OA(ms ,ms,s,2) by adding s assemblies of the forms
(i,i, ... ,i)'(i=O,l, .. "s-l).
Q.E.D.
We first give a method of constructing BA(T)(m,2,1) by using Hadamard
matrices.
A square matrix H of order n is said to be a Hadamard matriX, if
every element of H is either 1 or -1 and HH' = nI, where I is the indentity
matrix of order n.
A necessary condition for a matrix H of order n to be
a Hadamard matrix is that n must be a multiple of 4 except that n can be 2.
35
Corollary 4.3.1
If the Hadamard matrix of order 4m exists, then a
BA(T)(m,2,1) can always be constructed.
Proof:
If the Hadamard matrix of order 4m exists, we can arrange its
elements such that all the elements in the first column and first row are 1.
All other columns must then contain 2m lIs and 2m -l's.
Deleting 2m rows
whose second column is 1, we obtain an OA(4m,2m,2,2) with all the elements
equal to 1 in the first column and equal to -1 in the second column. By
Theorem 4.3.2 we can construct a BA(T) (m,2,1) since the OA(4m,2m,2,2) is
partly resolvable.
Q.E.D.
If the symbols of the BA(T)(m,2,1) are denoted by a and 1, the balanced
array becomes the incidence matrix of a balanced incomplete block design
with 2m treatments, 4m-2 blocks of m plots each, and any two treatments occur together in m-l blocks.
Theorem 4.3.3
Thus we have
The existence of a BA(T) (m,2,1) is equivalent to the
existence of a BIBD(2m,4m-2,m).
Corollary 4.3.2
If the Hadamard matrix of order 4m exists, then a
BIBD(2m,4m-2,m) can always be constructed.
Since it is well known that Hadamard matrices of order 4m exist for
all m
~
25, we can always construct a BA(T)(m,2,1) for m=1,2, ... ,25.
36
Corollary 4.3.3
If m and s are both powers of the same prime p, a BA(T)
(m,s,l) can always be constructed.
Proof:
By Theorem 2.3.1, we can construct a completely resolvable
2
OA(ms ,ms,s,2), then apply Theorem 4.3.2.
Example 4.3.2
For m
Q.E.D.
= 3 and s = 3, we can construct a BA(T)(3,3,1).
012
012
012
012
012
012
012
012
120
012
120
201
201
012
201
120
012
120
201
201
012
201
120
120
120
201
201
012
201
120
120
012
201
201
012
201
120
120
012
120
201
012
201
120
120
012
120
201
012
201
120
120
012
120
201
201
201
120
120
012
120
201
201
012
120
120
012
120
201
201
012
201
37
Corollary 4.3.4
~ ~
If s
n
p
m
= 2s ~ , where p is a prime, n
~
1, and
0, then a BA(T) (m,s,l) can always be constructed.
Proof:
By Corollary 2.3.1, we can construct a completely resolvable
2
OA(ms ,ms,s,2), then apply Theorem 4.3.2.
Example 4.3.3
Q.E.D.
For s = 3 and m = 2, we can construct a BA(T)(2,3,1).
012
012
012
012
012
012
120
201
120
201
120
012
201
201
120
201
201
012
120
120
120
201
120
012
201
201
120
120
201
012
Bose and Bush (1952) introduced the method of differences in the construction of orthogonal arrays of strength two.
The concept can be used to
construct the type of balanced arrays discussed in this section.
Theorem 4.3.4
Let M be a module of s elements.
If it is possible to
choose m rows and N columns (N=A +A (s-1), Al and A are integers)
2
l 2
all a 12 ·
a lN
a 2l a 22 ·
a 2N
with elements belonging to M such that among the differences of the corresponding elements of any two rows, the element 0 occurs Al times and the
other nonzero elements occur A times, then by adding the elements of the
2
module to the elements in the above array and reducing mod s, we can generate
Ns columns; this constitutes a BA(N,m,s,2) with parameters A(x,y) = Al or A2
according as x=y or Xfy.
The balanced arrays that can be constructed by Theorem 4.3.4 are
38
completely resolvable. We shall give two examples to illustrate the application of Theorem 4.3.4.
Example 4.3.4
Let M = {O,I,2}.
Among the differences of the correspond-
ing elements of any two rows of the following array, 0 occurs 6 times whereas
I and 2 each occur 8 times.
o o o o o
2 2 o 1
I
2 2 o
I
1
o
o
o
0
0
0
000
o o
2
1
2
1
2
1
I
0
0
2
1
2
2
2
1
1
0
o
0
o o
2
o
o
0
1
2
1
2
0
2
0
1
2
1
0
1
2
000
2
I
I
2
2
0
1
0
021
I
2
211
o
2
o
o
1
2
1
I
2
2
o
1
o
0
2
2
1
221
1
0
2
0
2
1
2
1
1
2
2
0
1
0
0
1
2
122
1
1
o
2
o 2 I 2 1
o o 2 I 2
I o o 2 1
o 1 o o 2
2 o 1 o o
2 2 o I o
I
2
2
010
212
2
1
1
0
1
I
2
2
121
2
2
1
1
2
1
I
220
o 1
o o
2 o
012
1
2
2
1
o
o o
2 o
o 2
1 o
1
2
1
122
o
2
001
2
1
2
2
1
1
2
1
2
112
1
o
2
0
0
I
2
1
2
2
1
0
2
1
211
1
1
o
2
0
o
1
2
1
2
2
0
1
Hence we can construct a BA(66,12,3,2) with parameters A(X,y) = 6 or 8
according as x = y or not, i. e. a BA(T) (4,3,2).
Let M = {O,1,2,3}. Among the differences of the corre-
Example 4.3.5
sponding elements of any two rows of the following array,
0 occurs 4 times,
whereas 1,2, and 3 occur 6 times.
(1
0
o
0
3
0
I
3
3
2
0
0
0
o
0
o
0
000
2
o
o
2
1
1
3
2
3
1
o
3
2
o
1
2
0
2
1
1
3
2
1
1
0
3
3
3
0
120
2
1
1
3
2
1
1
0
3
2
3
3
o
I
2
0
2
1
1
1
211
1
3
2
3
3
0
I
2
o
2
1
3
1
2
1
1
1
3
2
3
3
0
1
2
0
2
3
3
1
2
2
1
1
3
2
3
3
0
1
2
o
2
331
o
2
1
1
.')
2
3
3
o
1
2
0
233
2
0
2
1
132
3
3
U
1
2
023
o o o o o o
o 2 3 3 1 2
2 o 2 3 3 1
3 2 o 2 3 3
o 3 2 o 2 3
1 o 3 2 o 2
1 1 o 3 2 o
2 1 1 o 3 2
1 2 1 1 o 3
3 1 2 1 1 o
1
2
o
2
1
3
2
3
3
o
3
2
0
2
3
3
1
2
1
o
1
2
0
2
1
3
2
3
3
0
3
2
0
2
3
3
1
2
1
0
1
2
1
3
3
2
0
2
3
(1
1
1
39
Hence we can construct a BA(88,12,4,2) with parameters A(X,y) = 4 or 6
according as x=y or not, i.e. a BA(T) (3,4,2) .
Efforts have been made to reduce the number of assemblies in Example
4.3.4 and 4.3.5 by a half, i.e. to construct a BA(T) (4,3,1) and a BA(T)
(3,4,1), but without success.
Examples 4.3.1, 4.3.2, and 4.3.3 can also
be constructed by the method of Theorem 4.3.4, but certainly there are balanced arrays which can be constructed by Corollary 4.3.1 and cannot be constructed by Theorem 4.3.4.
For example, a BA(T) (3,2,1), which can be con-
structed by Corollary 4.3.1, is not completely resolvable. Therefore, it cannot be constructed by Theorem 4.3.4.
However, all the balanced arrays that
can be constructed by Corollary 4.3.3 can also be constructed by Theorem
4.3.4, since the orthogonal arrays used in Corollary 4.3.3 are constructed
by the method of difference.
CHAPTER V
TWO-FACTOR BFE'S
We shall discuss the construction of two-factor BFE's in this chapter.
The construction of more-than-two-factor BFE's, which can be done by using
two-factor BFE's, will be discussed in the next chapter.
We are only inter-
ested in the BFE's of which the main effects are estimated with high efficiencies.
These designs can usually be constructed by the balanced arrays dis-
cussed in the previous chapter.
Let F and F be the two factors in a BFE at sl and s2 levels respecI
2
tively.
We can assume
sl~s2
incidence matrix of the BFE.
without loss of generality.
Let N denote the
By Equation (3.1.4), the eigenvalues of NN' are:
(5.1.1)
(5.1.2)
(5.1.3)
Corollary 3.1.1, we can derive the efficiencies of the main effects in the
following:
(k-l)SI
(5.1.4)
E(l,O) =
k
41
(k-l)s2
(5.1.5)
E(O,l) =
k
If the main effects of F are estimated with full efficiency, i.e.
l
E(l,O) = I, then the block size k must be a multiple of 51.
k = 51 throughout this section.
(5.1.6)
We shall assume
For k = 5, Equation (5.1.4) becomes
E(l,O)
E(l,O) = 1 if and only if A = OJ that is, two treatments at the same level
OI
of F never occur together in the same block.
l
Theorem 5.1.1
.
e
In an slxs2 BF[ with block size 51' the main effects
are estimated with full efficiency if and only if A = O.
OI
This design is
equivalent to a BA(AlOs2+Alls2(s2-l), 51 ,5 2 ,2) with parameters A(X,y) =
A or All according as x=y or not.
lO
Proof:
The first part of the theorem has been shown; we need only
prove the latter part.
Supposing such a balanced array exists; if we iden-
tify columns, rows, and symbols with blocks, the levels of F , and the levels
l
Q.E.D.
of F respectively, then it is the specified BFE.
2
In proving Theorem 5.1.1, we don't really use the condition
hence it is true for all 51 and 52.
For k=s, and AOl=O in Equation (5.1.5),
(5.1. 7)
E(O,l)
(sl-1)s2 AU
=-.".-.,.-.::.--.-.~=-=:.;-~
sl[(s2-l)All+AlO]
E(O,l) has the maximum value
(5 1-1)5
5
1
(5
2
2
-1)
when A
lO
= o.
sl~s2;
42
Theorem 5.1.2
sl(sl~s2)'
In as slxs2 BFE with block size
if the main
effects of F are estimated with full efficiency and the main effects of F
1
2
(sl-l) s2
are estimated with maximum efficiency
then the BFE has parameters
sl (s2- 1) ,
A10 = A01 = 0 and All
r O.
This design is equivalent to a TA(A11s2(s2-1),
s1' s 2,2) •
Since A = 0 means that two treatments at the same level of F do not
10
2
occur together in the same block, which implies s2
sl
~
~
k = sl' we do not need
s2 in the construction of designs in Theorem 5.1.2.
The construction of a TA(s2(s2-1).AU,s2,s2,2) has been discussed in Sec-
tion 4 01.
Deleting any (s2-s1) constraints from a TA(s2(s2-1)A11,s2,s2,2),
we obtain a TA (s2(s2-1)A11,sl,s2,2).
If we restrict All = I, then the ex-
istence of a TA(s2(s2-1),sl,s2,2) is equivalent to the existence of sl-l
mutually orthogonal Latin squares of order s2 or sl-2 mutually orthogonal
Latin squares of order s2 with different elements in the diagonal.
A 3 x4 BFE with b=12, k=3, r=3, A =\0=O, and A =l,
11
01
can be constructed from a TA(12,3,4,2).
Example 5.l.1
Blocks
1
2
3
Levels
of. F
1
Ij
4
5
6
7
8
9 10 11 12
Levels of F
2
0
0
1
2
3
0
1
2
3
0
1
2
3
1
1
0
3
2
2
3
0
1
3
2
1
0
2
2
3
0
1 .3
2
1. 0
8
1. 0
3 2
5
E(l,O)=l, E(O,l)=.g, and E(l,l)=g in this design.
Example 50102
A 3x6 BFE with b=30, k=3, r=5, A =A , and A =1 can
11
01 10
be constructed from a TA(30,3,6,2) in Example 4.1.3.
The efiiciencies are
43
4
3
E(l,O)=l, E(O,l)=S' and E(l,l)=S.
Let sl=s2=s in Theorem 5.1.2; then we have a symmetrical BFE.
Corollary 5.1.1
In an s2 symmetrical BFE with block size s, if all
the main effects are estimated with full efficiency, then the BFE has parameters Al=O and A ,O.
2
Example 5.1.3
This design is equivalent to a TA(A S(S-1),s,s,2).
2
If s is a prime power, then there exists a TA(s(s-l),
Hence we can always constrcut an s
s,s,2) by Corollary 4.1.1.
2
symmetrical
BFE with r = s-l, b = s(s-l), k=s, Al=O, A =1, El=l, and E = s-2
2
2 s-l·
Exampl e 5.1. 4
A6
2
symmetrical BFE with r=lO, b=60, k=6, Al=O, and
A =2 can be constructed from the TA(60,6,6,2) in Example 4.1.2.
2
encies are E =1 and E =
l
2
The effici-
i.
2
2
2
Similarly we can construct 10 , 12 , 14 BFE's by using TA(360,10,10,2),
TA(660,12,12,2) and TA(1092,14,14,2) respectively.
If the main effects of F are estimated with full efficiency, then the
2
block size k must be a multiple of s2.
Assume k = s2 throughout this section.
By Theorem 5.1.1, E(O,l)=l if and only if AlO=O. Furthermore, the design is
equivalent to a BA(AOlsl+Allsl(sl-1),s2sl,2) with parameters A(x,Y)=A
or
OI
All according as x=y or not, if we identify the columns, rows, and symbols
of the balanced array with the blocks, the levels of F , and the levels of
2
F of the design
l
g
Example 5.2.1
can be constructed from an OA(4,3,2,2).
44
Blocks
1
Levels
of F
2
Levels of F
l
0
I
1
i
2
;
I
2
E(O,l)=l, E(l,O)=E(l,l)="3
2
4
3
0
0
1
1
0
1
0
1
0
1
1
0
in this design.
For A = 0 and k = 52' Equation (5.1.4) becomes
lO
(5.2.1)
Note that
1
E(l,O) =
AOl~O'
since k =
5 >5
2
1 implies that at least two treatments in a
given block are the same level of F ,
l
To maximize E(l,O), it is required
that A01/\11 be as small as possible,
Theorem 5.2.1
the following inequality holds:
(5.2.2)
5
-2
When the equality holds, E(l,O)=l and E(l,l) =.2-..
•
s2- 1
Proof:
g(O,l)=O in this BFE, since the main effects of F are esti2
mated with full efficiency. By (5.1.2), we have
(5.2.3)
Substituting the r in Equation (5.1.1) with Equation (5.2.3)
(5.2.4)
But g(l,O)
~
0, since g(l,O) is an eigenvalue of the nonnegative definite
45
matrix NN'.
Therefore
we have Equation (5.2.2), and the equality holds
if and only if g(l,O)=O, i.e., E(l,O)=l.
Q.E.D.
Since a necessary condition for E(l,O)=l is that block size k must be
a multiple of sl' we must assume s2 = msl(=k) for some integer m in order to
construct a BFE such that all the main effects are estimated with full efficiency.
When s2 = ms , Equation (5.2.2) becomes
l
A
(5.2.5)
...2.!.
> m-l
All -
m
Corollary 5.2.1
fects of F and F2 are estimated with full efficiency, i f and only i f 52 = ms l ,
l
AlO=O, and AOI/A ll = (m-l)/m for some m.
This design is equivalent
to
a
BA((ms l -l)s,A,ms ,sl,2) with parameters A(X,y) = (m-l)A or mA according as
l
x=y or not, i.e., a BA(T)(m,sl,A).
By Theorem 4.3.1, for any given m and sl we can always construct a BA(T)
(m,sl,A) for some A.
Thus we can always construct an mslxs
BFE such that
l
all the main effects are estimated with full efficiency, but a large replication may be needed.
The constructions of BA(T) (m,sl,l) for some m and sl
are discussed in Corollaries 4.3.1, 4.3.3, and 4.3.4. In Examples 4.3.4 and
4.3.5, we also give a BA(T) (4,3,2) and a BA(T)(3,4,2).
Example 5.2.2
A 2x4 BFE with b=6, k=4, r=3, AlO=O, AOl=l, and All =2
can be constructed from a BA(T)(2,2,1) in Example 4.3.1.
Blocks
1
Levels
of F2
2
3
4
5
6
Levels of Fl
0
1
0
1
0
1
0
1
0
1
1
0
0
1
2
1
0
0
1
0
1
3
0
1
0
1
1
0
46
2
E(O,l)=E(l,O)=l, E(l,l)=X in this design.
5.3
slxs2 BFE's with block size a common multiple of sl and s2
In an slxs2 BFE with block size s2' if s2 is not a mUltiple of sl' then
the main effects of F cannot be estimated with full efficiency. To estimate
l
all the main effects with full efficiency, the block size k must be a common
multiple of sl and s2'
Let sl=ps and s2=qs, where s>l.
A method is given
below to construct art slxs2 BFE with block' size pqs such that all the main
effects are estimated with full efficiency.
Theorem 5.3.1
If there exists a resolvable BIBD with qs treatments
and block size q, then there exists a psxqs BFE with block size pqs such
that all the main effects are estimated with full efficiency.
Proof:
Construct a BA(T) (p,s,n) for some integer n by Theorem 4.3.1.
In the resolvable BIBD, there being s blocks within each replication, we can
number the blocks in each replication by O,l, ••• ,s-l.
Now replacing the symbols in the balanced array by a group of symbols,
which represent blocks in the BIBD
for each replication, we obtain a pqs x
(ps-l)snr' matrix, where r' is the number of replications in the BIBD.
As-
sign i th level of F to the rows from the (iq+l)th to the (i+l)qth, where
l
i=O, ••• ,ps-l. Identifying columns and symbols with blocks and the levels of
F , we get a psxqs design with block size pqs.
2
We shall show that all the main effects of the design constructed above
are estimated with full efficiency.
Let A'be the number of blocks two treat-
ments occur together in the BIBD, then (qs-l)A' = (q-l)r'.
and A'=(q-l)m, where m need not be an integer.
Assume r'=(qs-l)m
Let A ' A ' All denote
lO
Ol
the parameters and r denote the number of replications in the psxqs design.
47
Then
A = (ps-l) (q-l)mn = (ps-l)nA'
OI
(5.3.1)
A = (p-l) (qs-l)mn = (p-l)nr'
lO
All = (p-l) (q-l)mn+pq(s-l)mn = (p-l)nA'+pn(r'-A')
r
(ps-l) (qs-l)mn = (ps-l)nr'
Substituting the parameters of Equations (5.1.1), (5.1.2) and (5.1.3)
with Equation (5.3.1) and using Corollary 3.1.1, we have E(O,l)=E(l,O)=l,
s-l
and E(l,l) = 1- (ps-l) (qs-l) •
Q.E.D.
Given any q and s, there always exists a resolvable BIB design with
qs treatments and block size q if the number of replications is allowed to
be large.
For example, the unreduced BIBD of qs treatments with block size q,
in which each of the (qs) possible q-element combinations from a block, is
q
resovable with parameters
(5.3.2)
v
= qs,
b
r =
q, and A
Hence we can always construct a psxqs BFE with block size pqs such that all
the main effects are estimated with full efficiency by Theorem 5.3.1.
But
usually we like a design to have few replications; hence the sizes of the
balanced array and BIBD are required to be small.
Several applications
of Theorem 5.3.1 are given in the following examples.
Example 5.3.1
A 4x6 BFE with block size 12.
resolvable BIBD with 4 treatments and block size 2.
Consider the following
48
Zo
ZI
0
Y
I
1
0
1
2
3
3
2
X Xl
o
0 2
YO
1
3
where XO' Xl' YO' Yl' ZO' and ZI represent the blocks.
BA(T)(3,2,1) given below:
0
0
0
0
0
1
1
I
0
1
0
1
1
0
0
1
1
1
1
0
0
1
1
0
I
0
1
0
0
1
0
0
1
1
0
0
0
1
0
1
1
0
1
1
1
1
0
0
Also consider the
1
1
1
0
0
0
1
1
0
0
1
0
By Theorem 5.3.1, we can construct a 4x6 BFE with k=I2, r=15, b=30, AIO=S,
14
A =6, A =8, E(O,I)=E(I,O)=I, and E(I,I) =15 •
OI
ll
Blocks
I
3
2
Levels
of F
2
4
1
Xo X Xl Xl
o
Xl Xo Xo Xl
Xo Xl Xl Xo
Xl Xl Xl Xo
4
5
10
1l
12
13
14
15
Xl
Xl
Xl
YO
YO
YO
YO
YO
Xo Xo X Xl
o
Xl Xl Xl Xo Xo
Xl Xo Xl Xl Xo
X Xl X X X
o o o
o
X X X Xl Xl
0 o o
Xl
YO
Yo
YI
Xo Y YO
I
Xo YO YI
Xl YI YI
YO
Y
I
Y
I
Y
I
Y
I
YO YO
X Y
o
I
YO
o Xo Xo Xo
Xo X
3
9
6
7
Levels of F
I
0
2
8
5
Xl
Xl
X Xl
o
Xl
Xl
Xl
Y
I
YI
Y
I
YO YI
Y
l
YO
49
Blocks
16
17
18
Levels
of F
2
I
20
19
21
22
23
24
25
26
27
28
29
30
Levels of F
l
0
Y
l
Y
1
Y
l
Y
l
Zo
Zo
Zo
Zo
Zo
Zl
Zl
Zl
Zl
Zl
YO
Y
l
Y
l
1
YO
YO
Y
l
Zo
Zo
Zl
Zl
Zl
Zo
Zo
Zo
Zl
Zl
2
Yl
Y
l
YO
YO
YO
Zl
Zo
Zo
Zl
Zl
Zl
Zl
Zo
Zo
Zo
3
YO
Y
l
Y
l
YO
YO
Zo
Zl
Zl
Zo
Zl
Zo
Zl
Zl
Zo
Zo
4
Yl
YO
YO
YO
Y
l
Zl
Zl
Zl
Zo
Zo
Zl
Zo
Zo
Zo
Zl
5
YO
YO
Y
l
Y
l
YO
Zl
Zl
Zo
Zl
Zo
Zo
Zo
Zl
Zl
Zo
Example 5.3.2
A 6x8 BFE with k=24, r=35, b=70, A =15. A =14, A =18,
Ol
lO
ll
34
E(O.l)=E(l.O)=l, and E(l,l) =3'5. Use the resolvable BIBD with 8 treatments
and block size 4 given below:
o 4 0 I
0 2
o 1
I 5
2 3
I 3
3 2
4 5
2 6
5 4
6 4
5 4
7 6
3 7
7 6
7 5 6 7
0 I
0 2 0 I
2 3
I 3
3 2
4 5
4 6
6 7 5 7
and the BA(T) (3,2.1) in Example 5.3.1.
A 6x9 BFE with k=18, r=20. b=60. A =5, A =4. A =7,
Il
Ol
IO
19
E(O.l)=E(l.O)=l, and E(l,l) = 20. Use the resolvable BIBD with 9 treatments
Example 5.3.3
and block size 3 given below:
0 1 2 0 1 2 0 3 6 o 1 2
5 3 4 4 5 3 1 4 7 3 4 5
7 8 6 8 6 7 2 5 8 678
and the BA(T)(2,3,l) in Example 4.3.3.
50
Example 5.3.4
A 8x12 BFE with k=24, r=77, b=308, A =14, AlO=ll, A =20,
11
Ol
E(O,l)=E(l,O)=l, and E(l,l) = ~i.
Use the resolvable BIBD with 12 treatments
and block size 3 given below:
o
2
4
7
1
358
2
4
6
9
3
5
7 10
4
6
8
0
5
7
9
1
1
6
5
10
2
760
3
8
7
1
4
9
8
5 10
9
3
6
o
10
4
3
8
9 11
4
9 10 11
5
10
o 11
6
o
1 11
7
2 11
8
2
2
1
6
8 10
2
7
9
o
3
8 10
1
4
9
0
2
5
10 1
3
6
7
105
8
2
1
6
9
3
2
7
104
3
8
o
5
4
9
9
3
5 11
o
5
611
1
7 11
2
7
8 11
4 11
10 4
6
3 11
and the BA(T) (2,4,1) which can be constructed by Corollary 4.3.3.
o
5.4
1 2 3 012 3
012 3 012 3 0 123
0 123
012 3
10322301
3 2 1 0
0 123
1 0 3 2
230 1
3 2 1 0
23010123
230 1
1 0 3 2
3 2 1 0
1 0 3 2
3 2 1 0
32102301
1 0 3 2
1 0 3 2
230 1
3 2 1 0
0 123
o
123
1 0 3 2
1 0 3 2
3 2 1 0
3 2 1 0
230 1
230 1
1 0 3 2
3 2 1 0
230 1
3 2 1 0
230 1 0 123
1 0 3 2
23011032
3 2 1 0
230 1 0 123
3 2 1 0
1 0 3 2
32103210
012 3
230 1
1 0 3 2
230 1
1 0 3 2
s2 s~etrical BFE's with block size ns (l<n<s)
The construction of s2 SYmmetrical BFE's with block size s such that the
main effects are estimated with full efficiency has been shown to be equivalent to the construction of a TA(A s(s-1),s,s,2) in Corollary 5.1.1.
2
section
In this
we shall discuss the construction of s2 SYmmetrical BFE's with block
size ns (l<n<s) such that the main effects are estimated with full efficiency.
By using the same argument in Theorem 5.3.1, we can show the following
theorem of constructing s2 SYmmetrical BFE's with block size a mUltiple of s.
Theorem 504.1
Let £,m,n,p,q be positive integers, if there exist a
51
BA(T)(q,p,£) and an n-resolvable BIBD(pq,mp,nq); then we can construct a
pqxpq symmetrical BFE with block size npq2 and parameters
Al = £mn(nq-1)
(5.4.1)
2
A = £(mn q-A)
2
r
= £mn(pq-l)
where A = mn(nq-1)/(pq-l) is the number of blocks two treatments occur together in the BIBD.
The efficiencies of the BFE are El=l and E =1 2
Corollary 5.4.1
p-n 2.
n(pq-l)
If 5 is a prime power, then there exists a symmetrical
2
2
5 BFE with block size 5(5-1), 5(5-1) blocks and parameters Al = 5 -35+2 and
2
A = 5 -35+3.
2
Proof:
For any integer 5, there exists a BIBD(s,s,s-I).
power, there exists a TA(s(s-1),s,s,2).
For 5 a prime
apply Theorem 5.4.1 with £=m=q=l,
No~
p=s, and n=s-l, and we get the 52 BFE.
In the following
Q.E.D.
2
we give examples of 5
can be constructed by Theorem 5.4.1.
Example 5.4.1
(see also Example 3.2.1).
Use the BIBD(3,3,2)
o
o
1
1
2
2
and the TA(6,3,3,2)
o
1
2
o
1
2
1
2
o
2
o
1
2
o
1
1
2
o
symmetrical BFE's with
s~7
which
52
Example 5.4.2
4
2
symmetrical BPE's
8
=9
a)
Use the resolvable BIBD(4,6,2)
o
2
0 1
0 1
1 3
2 3
3 2
and the BA(T)(2,2,1) in Example 4.3.1.
b)
Use the BIBD(4,4,3)
1
0
o
0
2
2
1
1
3
3
3
2
and the TA(12,4,4,2) which can be constructed by letting s=4 in
Coro llary 4.1. 1•
Example 5.4.3
a)
52 symmetrical BPE's
29
k=10, b=40, r=16, ;'\=4, /"'2=7, E1=1, and E2='32.
Use the 2-reso1vab1e BIBD(5,IO,2)
0
1
2
3
4
0
1
2
3
4
1
2
3
4 0
2
3
4 0
1
and the TA(20,5,5,2) which can be constructed by letting 5=5 in
Corollary 4.1.1.
b)
23
k=15, b=40, r=24, /"'1=12, /"'2=15, E1=1, and E =2'4 •
2
Use the 3-reso1vab1e BIBD(5,10,3)
0
1
2
3
4
0
1
2 3
4
1
2
4
2
4
3
0
3
0
1
1
2
3
4
0
3
0
4
1
2
53
and the same TA(20,5,5,2) in (a).
c)
63
k=20, b=20, r=16, 1.. 1=12, 1.. =13, E1=1, and E = 64 •
2
2
Use the BIBD(5,5,4)
1
0
0
0
0
2
2
1
1
1
3
3
3
2
2
4
4
4
4
3
and the same TA(20,5,5,2) in (a) •
2
Example 5.4.4
A 6 BFE with k=12, b=75, r=25, 1.. =5, 1.. =9, E =1,
1
1
2
23
and E =2s. Use the resolvable BIBD(6,15,2)
2
0 2 4 0 1 3
o1
2
0 1 2
0 1 3
1 3 5
3 5 4
4 3 5
5 2 4
2 4 5
and the BA(T)(2,3,1) in Example 4.3.3.
Example 5.4.5
a)
7
2
symmetrical BFE's
67
k=14, b=126, r=36, 1..1=6, 1.. =11, E =1, and E ="72 •
2
1
2
Use the 2-reso1vab1e BIBD(7,21.2)
0 1 2 3 4 5 6
1 2 3 4 5 6 0
0 1 2 3 4 5 6
2 3 4 560 1 3 4 5 6 0 1 2
0 1 2 3 4 5 6
and the TA(42,7,7,2) which can be constructed by letting s=7 in
Corollary 4.1.1
b)
26
k=21, b=42, r=18, 1.. 1=6, 1.. =8, E =1, and E =2"7 •
2
1
2
Use the BIBD(7,7,3)
o 1 234 5 6
1
342
064
2
5
3
5
6
0
1
and the same TA(42,7,7,2) in (a).
S4
(c)
47
k=28, b=42, r=24, 1.. 1=12, 1.. =14, E =1, and E = 48
2
1
2
.
Use the 818D(7,7,4)
0
1
2
3
4
S 6
6
S 1
4
0
3
2
1
4
3
6
2
0
S
3
0
4
S 6
2
1
and the same TA(42,7,7,2) in (a) •
·e
CHAPTER VI
MULTI-FACTOR BFE'S
sm symmetrical BFE's with block size s
6.1
The s
m
symmetrical BFE has been shown by Shah (1958) to be equivalent
to a PBIBD with a hypercubic association scheme.
We shall consider the con-
struction of such designs with block size s in this section.
By Equations (2.1.4) and (3.2.1), we have
(6.1.1)
m
.
\' m
1
r(s-l) = L (.)(s-l) A.•
i=l 1
1
Hence
(6.1.2)
r =
m m
L (.)(s-l) i-I A.•
. I I I
1=
r is completely determined by the values of A. 'sO
1
power, we shall show that there exists an s
s for any given 1. 1 ,1. 2 "'"
Lemma 6.1.1
an s
m
m
When s is a prime
symmetrical BFE with block size
and Am'
If s is a prime power, then given j
(l~j~m)
there exists
symmetrical BFE with block size s and parameters 1..=1, 1..=0 for all ilj.
J
Proof:
1
Given a.(i=l, ••• ,m-l), b.(fO, i=m-j+l, ••• ,m-l) in GF(s), conI
1
sider the block containing elements (x ,x , ••• ,x ) satisfying the following
l 2
m
set of equations:
56
(6.1.3)
x
. = a
m-J
x m-J+
. 1
.
m-J
b . x +a
m-J+lm m-j+l
.........
Any two treatments in this block are at the same levels of the factors
Fl, ••• ,F
. and at different levels of the factors F . 1, ... ,F • By letting
m-J
m-J+
m
a. 's and b. 's run through all possible values in GF(s), we obtain Sm-l(s_l)j-l
1
1
blocks with the above property.
Let {il,i , .. .,i
.} be any m-j integers from {l,2, ••• ,m}. We can simim-J
m-l
j-l
larly construct s
(s-l)
blocks which contain treatments at the same
2
levels of the factors F. ,F. , ••• ,F.
and at different levels of the other
1
1
1.
1
2
m-J
..
.}
f actors. For eac h 0 f { 1 ,1 ,.0.,1
. we canstruct s m-l (s-l) j -1 such bloc k s,
1 2
m-J
and let the design consist of all (~)sm-l(s_l)j-l such blocks. Then in this
J
design any two treatments with exactly j factors at
~ifferent
levels will
occur together in one block, and any two treatments with exactly i(ifj) factors at different levels will not occur in the same block.
metrical BFE with parameters A.=l and A.=O for all ifj.
J
1
This is the symQ.E.D.
The efficiencies of the symmetrical BFE constructed in Lemma 6.1.1 can
be calculated by Equations (3.2.2) and (3.2.4).
(6.1.4 )
E.
1
=1
P.(i;m,s)
1
- -
J
(~) (S_l)j-ls
S
i=1,2, ••• ,m
J
. 1ar, wh en J=m,
.
P (.l;m,s ) = (_l)i(s_l)m-i and Equation (6.1.4)
I n part1cu
m
becomes
57
1
E. = 1 - - -
(6.1.5)
1
S
(s-l)
i-I
i=1,2, ••• ,m
s
This balanced design has been constructed by Bose (1947); the main effects
are estimated with full efficiency since El=l in Equation (6.1.5).
Theorem 6.1.1
there exists an s
••• , A
m
If s is a prime power, then for any given A ,A , ••• ,A
l 2
m
symmetrical BFE with block size s and parameters A ,A ,
l 2
Q
m
Let D. denote the design constructed in Lemma (6.1.1.) The
Proof:
J
symmetrical BFE consists of A Dj'S for j=1,2, ••• ,m has parameters A '\2, •• ,A •
j
l
m
Q.E.D.
Now consider the case when s is not a prime power.
In an s
2
symmetrical
BFE with block size s, if we can construct a design with Al=O and A/O, then
the main effects are estimated with full efficiency.
By Corollary 5.1.1 such
a design is equivalent to a TA(A s(s-1),s,s,2).
2
. 1 BFE with block size s, if we can conIn the case of an s m symmetrlca
struct a design with parameters A #0, and A.=O for i=l, ••• ,m-l, then the main
m
1
effects are estimated with full efficiency.
we can multiply (see Theorem 4.2.2) m-l
([As(s-l)]
if xi
r Yi
m-l
If a TA(As(s-1),s,s,2) exists,
such transitive arrays to get a BA
.
m-l
,s,s,2) wlth parameters A((xl •••• 'xm_l)'(yl' •••• Ym_~) = A
for all i=l, ••• ,m-l, and A((xl, ••• ,xm_l)'(Yl""'Ym_I)) =
° other-
wise.
Identifying rows with the levels of F , symbols with the levels of
I
F , ••• ,F , and columns with the blocks, we obtain an sm symmetrical BFE in
2
m
ml
[As(s_l)]m-l blocks of s plots each with parameters A =A - and A.=O for
m
1
i= 1, ••• ,m-l.
Thus we have the following theorem.
Theorem 6.1.2
of an s
m
The existence of a TA(As(s-l) ,s,s,2) implies the existence
symmetrical BFE with
b = [As(s-l)]
m-l
, k=s, r=[A(s-l)]
m-l
m1
, A =A m
58
and A.=O, for i=l, ••• ,m-l.
1
6.2
Methods of constructing
multi~factor
BFE's
In this section, we shall discuss methods of constructing multi-factor
BFE's by using the known two-factor BFE's or other multi-factor BFE's already
2
constructed.
struct an s
m
We have seen in Theorem 6.1.2 that from m-l s
BFE by using the product of balanced arrays.
.
BFE's we can conIn general, the
product of balanced arrays can generate many efficient BFE's.
Theorem 6.2.1
i
meters Ai (x'Y)=~O or
x ••• xs
1
= ~a
m
If there exists BA(N.,s ,s.,2)(i=1, ••• ,m-l) with para1
m 1
i
~1
according as x=y or not, then there exists an slxs
2
BFE with k=s , b=Nl···N 1
m
m- ,
2
m-l
or 1.
a ••• ~ a _ , where a.=O
1
ml
1 2
~
Proof:
Multiply the given m-l balanced arrays to obtain a BA(N N •••
1 2
Nm_ l ,sm,s l s2 ••• s m_l,2) with parameters A((x l ,x 2 , ••• ,xm_ l )'(Yl'Y2""'Ym_l))
2
m-l
••• ~
where a.=O or 1 according as x.=y.or not. Identifying symbols
a
a _
1
1
1
ml
l 2
with the levels of F ,F , ••• ,F _ , rows with the levels of F , and columns
ml
l 2
m
=
~
1
a
~
with blocks, we obtain an slxs2x ••• xsm BFE with the specified parameters.
Q.E.D.
The method used in Theorem 6.2.1 can usually produce efficient BFE's, if
we use balanced arrays corresponding to efficient two-factor BFE's.
While
applying this method, the block size remains the same but the number of blocks
increases very rapidly.
Hence this method is used when the numbers of assem- .
b lies in the balanced arrays are not too large.
Example 6.2.1
Consider the product of the OA(4,3,2,2) in Example 5.2.1
and the TA(6,3,3,2) in Example 4.1.1.
59
00
01
02
00
01
02
00
01
02
00
01
02
01
02
00
02
00
01
11
12
10
12
10
11
02
00
01
01
02
00
12
10
11
11
12
10
10
11
12
10
11
12
10
11
12
10
11
12
01
02
00
02
00
01
11
12
10
12
10
11
12
10
11
11
12
10
02
00
01
01
02
00
which is a BA(24,3,6,2) with parameters A((x ,x 2), (Yl'Y2)) = o or 1 according
1
as x =Y or not. By Theorem 6.2.1, this corresponds to a 2x3x3 BFE wi th k=3,
2 2
=1, A001=A010=Al00~Al10=A101=O, E(O,l,O)=E(O,O,l)=l,
b=24, r=4, A =A
011 111
E(l,O,O)=E(l,l,O)=E(I,O,l)=E(l,l,l)=~and E(O,l,l)= 21 •
Example 6.2.2
The product of a BA(T)(3,2,1) in Example 5.3.1 and a
BA(T)(2,3,1) in Example 4.3.3 generate a 2x3x6 BFE with r=25, b=150, k=6,
A010=A100=A110=0, A001 =2, A011 =4, A101 =3, and A111 =6.
The efficiencies are
4
21
E(O,Ol):::E(O,I,O)=E(l,O,O)=l, E(O,l,l)=E(l,O,I)=E(l,l,O)="5' and E(l,l,l)= 25.
We can also obtain an efficient 2x3x6 BFE by collapsing the first factor
of the 6
2
symmetrical BFE in Example 5.1.2 into two factors, one at 2 levels
and the other at 3 levels.
The BFE has parameters r=10, b=60, k=6, A001=A010
=A100=A010=0, and A011=A101=A111=2.
The efficiencies are E(O,O,I)=E(O,l,O)
4
=E(l,O,O)=E(l,l,O)=l, and E(O,l,l):::E(l,O,l)=E(l,l,I)=S.
All the main effects
are also estimated with full efficiency like Example 6.2.2, but we only need
10 replications in this design.
The second method of constructing multi-factor BFE's we shall discuss
was suggested by Yates (1957), and employed by Nair and Rao (1941), Li (1934)
and Kishen (1958).
The general form with exact conditions for validity was
proved by Shah (1960).
This method replaces different levels of a factor in
one design by distinct sets of treatment combinations forming the blocks of
another design.
60
Assume that there exists a BFE with m factors F ,F , ••• ,F at s ,s ,
l 2
m
1 2
••• ,sm levels respectively, each of the v*(=sls2 ••• sm) treatments replicated r* times in b* blocks of k* plots each, with the incidence matrix
(6.2.1)
N* = [AiIAzl ••• IA* ]
b*
Further assume that b*=pq, and the pq blocks can be divided into p groups
of q blocks each, such that the design consisting of p blocks formed by
adding together all the blocks of a group is a BFE.
The incidence matrix
is
rA~I r
r
N* = [
A* . I••• I A*
.]
pq
j=l J j=l q+J
j=l pq-q+J
(6.2.2)
For a resolvable design N*, the corresponding N* exists with p=r*.
pq
The following theorem was proven by Shah (1960).
Theorem 6.2.2
Let there be a BFE with incidence matrix N in n+l
factors FO,F m+ 1••• Fm+n at q,s m+ l' ••• 's m+n levels respectively in b blocks
of k plots each.
Also let there be two BFE's with incidence matrices N*
and N* as given by Equations (6.2.1) and (6.2.2) respectively.
pq
If the
level j-l of the factor F is replaced by the block A. .(j=1,2, ••• ,q) in
O
1q+J
.
each of the treatments of N, then the design obtained by adjoining the p designs so formed (for i=O,l, ••• ,p-l) is a BFE in m+n factors in bp blocks of
kk* plots each.
This method generates an m+n factor BFE from an n+lfactor BFE and an
10
fac-
tor BFE. Thus from the two two-factor BFE's we can generate a three-factor RFE.
If the two-factor BFE's are efficient, then the three-factor BFE is also efficient.
We can therefore construct efficient mUlti-factor BFE's step by step
from efficient two-factor BFE's.
While applying this method, the number of
61
blocks does not increase so quickly as the first method, but the block size
does increase.
It can be seen that Theorem 5.3.1 and Theorem 5.4.1 are the consequences
of Theorem 6.2.2 if we let m=n=l in Theorem 6.2.2.
Example 6.2.3
Let N be the incidence matrix of the 3x6 BFE constructed
by identifying rows, columns, and symbols, with the levels of the second factor, the blocks, and the levels of the first
factor respectively in the
o' Xl' X2 , YO' Yl , Y2 represent blocks.
where X
Then by Theorem 6.2.2, we
2
can construct a 3 X6 BFE with r=IO, k=18, b=30, A =5, A =2, A =3, A =4,
20
OI
21
II
9
AIO=O, E(2,1)=iO and all the main effects and first order interactions are
estimated with full efficiency.
Blocks
1
2
3
4
Levels
of F
3
The BFE is given below:
5
6
7
8
9
10
11
12
13
14
15
X
Xl
X
2
Levels of F and F
2
I
a
X
\
X
2
X
o
Xl
X
2
X
o
Xl
X
2
X
o
Xl
X
2
1
X
o
Xl
X
2
Xl
X
2
X
o
X
2
X
Xl
Xl
X
2
X
2
Xl
X
2
X
X
X
2
X
2
X
o
Xl
X
2
X
o
Xl
Xl
3
X
2
X
o
Xl
X
2
X
1
X
Xl
X
o
Xl
X
2
Xl
X
2
X
o
Xl
4
Xl
X
2
X
o
X
2
X
o
Xl
Xl
X
2
X
X
Xl
X
2
X
2
5
X
2
X
Xl
Xl
X
2
X
Xl
X
2
X
X
2
X
Xl
X
o
o
o
o
o
o
o
o
o
0
o
o
X
o X2 o
Xl
X
2
X
2
X
X
o
Xl
Xl
X
2
o
o
X
o
62
Blocks
16
17
18
Levels
of F
3
6.3
19
20
22
21
Levels of F
23
and F
l
24
25
26
27
28
29
30
2
a
Yo
Y
Y
Yo
Y
Y
2
Yo
Y
Y
Yo
Y
Y
2
Yo
Y
Y
1
Y
Y
Y
2
Y
l
Y
2
Yo
Y
2
Yo
Y
Y
Y
2
Yo
Y
2
Yo
Y
2
Yl
Y
2
Yo
Yo
Y
Y
Y
2
Yo
Y
Y
2
Yo
Y
Y
Y
Yo
3
Y
Yo
Y
Y
Yo
Y
l
Yo
Y
Y
Y
Y
Yo
Y
Y
Y
4
Y
Y2 Yo
Y
Y
5
Y Yo
2
Y2 Yo Yl
Y Y Yo
2
l
Y
Y
a
2
l
l
2
l
l
Y
l
2
l
l
2
l
l
2
l
l
2
2 Yo
l
l
2
Yo
l
l
l
2
l
l
l
2
l
2
2
l
a
Y
a Yl Y2 Y2 Yo Y1
Y Yo
2
Yl
Y Y
1 2
Yo
Examples of multi-factor BFE's
In this section
we shall use the methods discussed in the preceding
section and some known BFE's to construct examples of multi-factor BFE's. We
are especially interested in BFE's of which the main effects and lower order
interactions can be estimated with high efficiencies.
Type I.
-
If there exists TA(s.(s.-l), s ,s.,2) for i=1,2, ... ,m-l, then
11m 1
.
m-l
by Theorem 6.2.1 we can construct an s xs x ••• xs BFE wIth k=s , b=IT. IS. (5.-1),
12m
m
1= 1 1
r=IT::~(Si-l), All •• l=l, and other A's being O.
(6.3.1)
m
r + IT
i=l
By Equation (3.1.4)
H.(l,a.)
1
1
= r + (-1)
L~_la. m
I-a.
11 IT (s.-l)
1
i=l 1
Hence
(6.3.2)
= 1 _ g(a l ,0. 2 " . , ,am)
rk
Lma.
::: 1 -
1
s
m
(-1) 1 1 IT~ l(s.-l)
1=
1
m-l
s m IT.1= 1 (s.-l)
1
I-a.
1
63
= 1 - -
(-1)
1
Lm1a.
(s -1)
m
1
s m - - - -m- - - -a.
1
s m II.1= l(s.-l)
1
Let 0..=1 and 0..=0 for i#j; then Equation (6.3.2) becomes
J
1
s.-s
(6.3.3)
E(O'···ho,l,O, ••• ,O)
jt place
= 1
m
J
- s (5.-1)
m J
This is the efficiency of the main effects of the factor F., and is 1 when
J
j=m.
Hence the main effects of F are estimated with full efficiency.
m
general, let a. =1 for i=l, ••• ,q
Ji
(q~m),
In
and other a's be 0; then Equation
(6.3.2) is
.
(6.3.4)
which is the efficiency of the (q-1)
Example 6.3.1
exist.
For any given
th
q
(s -1)
m
order interaction between F. ,F. , ••• ,
J1 J2
(i=1, ..• ,m-1), TA(s. (5.-1),3,5.,2)'5
1 1 1
1
s.~3
Hence we can always construct an slxs2x ••• xsm_1x3
b =IIm.-1
=1 s. (5. -1), r = II.m-1
-1 (5. -1), All
1
(-1)
.
1
= 1 - -s
m
1
1
1-
1
••
BFE
with
k=3,
1=1, and other A' 5 being 0.
Other examples of this type include 5 x 4 x 4, 5 x5 x4, 7 x 5 x 4, 7 x 5x 5, 7 x 7 x 5,
••• ,etc. BFE's.
Type II.
Let sm=n1s1=n2s2= ••• =nm_1sm_1 and there exists BA(T)(n i ,si,1)
for i=1,2, ••• ,m-1.
k=sm' b=(sm- 1)
m-1
sl s 2··· s m_1' r=(sm- 1 )
m-1 a i
1-ai
II i =l n i (n i -1)
(6.3.5)
By Theorem 6.2.1 there exists an slxs2x ••• xsm BFE with
m-1
,
By Equation (3.1.4)
m-1 a.
I-a. m-1
1
r +
2 (II
n. (n.-1)
1)( II H.(a.,B.))H (1,B)
.-1 1
1
·-1 1 1 1 m
m
a ••-a _ 11m1
1
64
r + (-1)
13m
(5 -1)
m
Let 13 =1, 13.=0 for i=1,2, ••• ,m-l
m
(6.3.6)
1
m-l a.
I-a.
1
IT n. 1 (n. -1)
H. (a. ,13.)
1
1
1
1
a •• a _ i=l 1
1 m1
l-S m
\
L.
in Equation (6.3.5),
m-l a.
I-a.
a.
1
IT n. 1
(n.-l)1
(5.-1)
1
1
a a ··a _ 1.=1 1
1 2
m1
\
g(O,O, ••• ,O,l) = r -
L.
m-l
a.
I-a.
IT (5 -n.) l(n._l)
1
. 1 m 1
1
a ••• a _ 1=
1
m1
2
= r -
m-l
= r - IT [(5 -n.)+(n.-l)]
i=l
m 1
1
= (5 _l)m-l
m
=
(5 _l)m-l
m
°
Similarly, we can show that g(l,O, ••• ,O) = g(O,l, ••• ,O) = g(O, ••• ,l,O) = O.
Thus, all the main effects are estimated with full efficiency.
=6 m=1, and 13.=0
for i=1, ••• ,m-2; then (6.3.5) becomes
1
(6.3.7)
g(O, ••• ,0,1,1) = r - L (A a •• a
01Hm_l (0,1)
a 1 •• a _
1
m-2
m2
m-2
+ A
IlH 1(1,1» IT H. (a. ,0)
a 1 ··am_2
mi=l 1 1
r +
\
L.
a • .a _
1
m 2
= (5 -1)
m
In general, when L~=l a i = q
(6.3.8)
m-2 a.
I-a.
a.
1
IT n. 1
(n.-l)1
(5.-1)
i=l 1
1
1
m-I + (5 -1) m- 2
m
Equation (6.3.5) is
Let Sm_I
65
Hence,
(6.3.9)
(5 _l)m-l+(_l)q(s _l)m- q
=
1 -
m
m
-~----...,.--;':-'--
(5
-1)
m-l
m
1
(-l)q
5
m
= 1 - -5- - ------':.....-q---l
m
5 (5 -1)
where [.1=
\~ lct.=q
1
m m
It can be seen that Equation (6.3.9) is the same as Equation (6.1.5)
with 5 =5; hence
m
the efficiencies are equal to those of the sm symmetrical
m
BrE in Lemma 6.1.1 with j=m.
Example 6.3.2
A BA(T)(3,2,1) is given in Example(4.3.3); hence we can
construct a 22 x6 BFE with k=6, b=lOO, r=25, A =9, A =6, A =4, and A
ll
2l
Ol
lO
=A
20
21
=
=0.
4
The efficiencies are E(l,O)=E(O,l)=l, E(1,1)=E(2,0)=S' and E(2,1)
25 .
Example 6.2.2 is also of this type; other examples include 2 x 2 x 4,
2x4x4, 3 x 3 x 6, 2x5 x lO, ••• ,etc. BFE's.
The following example is also a 22 x6 BFE with only 5 replications; .the
main effects are estimated with full efficiency but some interactions are
not
estimable.
Example 6.3.3
A 22 x6 BFE with k=6, r=5, b=20, A =2, A =3, and
Ol
2l
4
All=AlO=A20=0, E(0,1)=E(1,0)=E(2,1)=1, E{l,l)= 5' and E(2,0)=O.
use Theorem 6.2.2 to construct this design.
We shall
Let N be the incidence matrix
of the 2x6 BFE corresponding to a BA(T)(2,3,1) by Corollary 5.2.1.
Let N*
be the incidence matrix of the following 22 design with block size 1,
00
11
01
10
and N be the following 22 BFE with the interaction confounded,
22
66
00
01
11
10
Apply Theorem 6.2.2, and we get the following 22 x6 design:
,'Blocks
.
1
2
Levels
of F
3
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Levels of F and F
1
2
0
00 00 00 00 00 11 111111 11 01 01 01 01 01 10 10 10 10 10
1
00 00 11 11 11 00 00 00 11 11 01 01 10 10 10 01 01 01 10 10
2
11 00 00 11 11 11 11 00 00 00 10 01 01 10 10 10 10 01 01 01
3
00 11 11 00 11 00 1111 00 00 01 10 10 01 10 01 10 10 01 01
4
11 11 11 00 00 11 00 00 00 11 10 10 10 01 01 10 01 01 01 10
5
11 11 00 11 00 00 00 11 11 00 10 10 01 10 01 01 01 10 10 01
Type III.
Let there exist a BA(T)(n ,s,1), by Corollary 5.2.1).
1
This
corresponds to an n s x s BFE with k=n s, b=(n s-1)s, and
1
1
1
A
oo
(6.3.10)
= n l s-1 = r
A01 = 0
A10 = n 1-1
By Equations (5.1.1), (5.1.2), and (5.1.3), the eigenvalues of NN' are
(6.3.11)
g(l,O)
0
g(O,l)
o
If there exists a resolvable BA(T)(n ,s,1), then this corresponds to .
2
a resolvable n s x s BFE.
2
By Theorem 6.2.2, if we replace the levels of the
67
second factor of the nlsxs BFE by the blocks of the nZsxs BFE, we get an
Z
nlsxnZsxs BFE with k=nlnZs , b=(nls-l) (nZs-l)s, and
= AOo(nZs-l) = (nls-l) (n s-1)
A
OOO
2
A
= A (n s-1) = 0
OOI
Ol 2
(6.3.12)
A
= AOO(nZ-l)+AOl(nZs-nz) = (nls-l) (nZ-l)
OIO
A
= AOOn2+AOl(n2s-n2-l) = (n s-1)n
Oll
l
2
A
= A (n s-l)
lOO
lO 2
(nl-l) (n 2s-l)
A
= All (n s-l) = n (nZs-l)
2
lOl
l
A
= AIO(nZ-l)+All(n2s-n2) = (nl-l) (nZ-l)+nl(nZs-n )
110
z
e
A
= AlOn2+All(n2s-nZ-l) = (n -l)+n +n (n s-n -1)
1
2 1 2- Z
lll
where AOO ' AOl ' \0' All are given by Equation (6.3.10).
of NN' are
g U\, S2' (3) = n l n 2s
(6.3.13)
2
The eigenvalues
if S =S =S =1
1 Z 3
Z
= nlnZs (n l s-l)(n 2s-l)
if S =S =S =0
123
= 0
otherwise
1
Hence, E(l,l,l) = 1 - (n s-l)(n s-l) , and all the main effects and first
1
2
order interactions are estimated with full efficiency.
If, further, there exists a resolvable BA(T) (n s,1), we can replace the
3
levels of the third factor of the nlsxnZsxs BFE by the blocks of the Tl3SXS
3
BFE to obtain an nlsxn2sxn3sxs BFE with k=n n n s such that all the main
l 2 3
effects and interactions are estimated with full efficiency except the third
order interactions, which are estimated with efficiency
1-(nls-l)(n2;-1)(n~s:f)o
SI,
Continuing this procedure, we get an nlsxn2sxoooxnSl,sxs BFE . with k=s n n •• nSl,'
l 2
68
b=s(nls-l)(nzs-l) ••• (n£s-l) and r=(nls-l)(nZs-l) ••• (n£s-l).
The A's can
be calculated recursively by the following formulae:
(6.3.14)
We shall prove that E(l,l, ••• ,I)=l - 1/(n 1s-1)(n Zs-1) ••• (n£s-1) and all
either efficiencies are 1.
The proof is given by induction.
Equation (3.1.4)
can be rewritten as
(6.3.15)
=
L
a 1 • .a£_1
£-1
(II
H.(a.,S.)) [Aa ••• a _ 00
. 1 1 1 1
o
1=
1
Jfv
1
Using Equation (6.3.14), we have
£-1
g(SI,SZ, ••• ,So_I'O,O) = L
('-1
II H.(a.,S.))[A
0
Jfv
1
1
l a 1 ·.·a£_1
a 1 •• a£_1 1-
69
(on-I) (Ons-1)+A
Tv
g(SI,S2, ••• ,Sn_l,O,1)
Tv
a 1 ••• a5/,-1
I(Ons-on)(Ons-1)
Tv
Tv
Tv
5/,-1
(.IT 1 H.1 (a.1 ,S.))[A
1
a 1 ••• a n_ 0
a 1 •• a5/,_1 1=
Tv 1
L
=
onTv s (05/,s-l)
+
Aa •••a5/,_1 l(s-I)] = 0nTv s (onTv s - 1) g(SI,S2, •• ,Sn_l'O)
Tv
1
=
5/,-1
(IT H.(a.,S.)) [A
o(on s - 1)
.-1 1 1 1
a l ···a5/, 1 Tv
a 1 ··a5/,_1 1 -
Tv
L
(6.3.16)
A
1(05/,s-05/,-I) (s-I)] = 0
a 1 ···a5/,_1
g(SI,S2, ••• ,Sn_1,1,1)
=
Tv
5/,-1
s 1
(IT
•
1 H.(a.,S.))[A
1
1
1
a 1 ••• a n _ 1 o(onTv - )
a 1 •••0.5/,_1 1=
Tv
L
70
R,-l
L
(IT H. (a.,B.))(A
0
n 1
·-1 1 1 1
al···a Nal ••• aR,_l 1-
By the recursive formula (6.3.16) and the initial values (6.3,13) we have
(6.3.17)
g(B l ,B 2 ,···,BR,+1) = s
= s
R,
R,
R,
IT n.
i=l 1
if B =B =••• =B
=1
l 2
hl
R,
IT n.(n.s-1)
i=l 1 1
if Bl =B 2=•• • =BR,+l = 0
= 0
e
otherwise
Hence the efficiencies are
E(B l ,B 2 ,···,BR,+1) = 1
(6.3.18)
= 0
Theorem 6.3.1
R,
R,
1/ IT (n.s-l)
i=l 1
if Bl =8 2=••• =8R,+1 = 1
otherwise
If there exists a BA(T)(n ,s,1), and a resolvable BA(T)
1
(ni,s,l) for i=2, ••• ,R"
with k = s
-
then we can always construct an nlsxnZsx •• oxnR,sxs BFE
R,
R,
R,
ITi=l n i , b = s ITi=l (nis-l), and r = ITi=l (n i s-1) such that
E(l,l, ... ,l) = l-l/r and all other efficiencies are 1.
Example 6.3.4
~
A BA(T)(3,Z,1) is given in Example 5.3.1, and a resolvable
"
BA(T)(2,2,1) is given in Example 4.3.1 which is equivalent to the following
71
4x2 resolvable BFE:
Xo~
where
Xo
Xl
YO
Yl
Zo
Zl
00
01
00
01
00
01
10
11
11
10
11
10
21
20
20
21
21
20
31
30
31
30
30
31
Xl' Yo' Y , Zo' Zl represent
I
the blocks.
Hence we can construct
a 6x4x2 BFE with k=24, b=30, r=15, AOOl=O, A =5, AOll=lO, A =6, A =9,
OlO
lOO
10l
14
E(l,l,l)=IT' and all other efficiencies are L
The design
can be expressed as the same table in Example 5.3.1; the differences are the
rows representing the levels of the first factor and the X ' Xl' YO' Y , Zo'
l
o
Zl representing the blocks as shown above.
Other examples include BFE's 4x4x2, 6 x6 x3, 6 x3 x 3, 6 x9 x3, 8 x4 x4, etc.
Corollary 6.3.1
~ ~+m-1
BFE with k=2 s
If s is a prime power, then there exists a
~
, r=(2s-l) (s-l)
m-1
~
, b=(2s-l) (s-l)
m-l
s,
E(~,m)
(2s)~xsm(m~1)
= 1 - l/r,
and all other efficiencies are I.
Proof:
BA(T)(2,s~1)
This is a consequence of Theorem 6.3.1 since a resolvable
and BA(T)(l,s,l) (i.e. a TA(s(s-1),s,s,2)) exist for s a prime
power.
Q.E.D.
If in addition to the conditions in Theorem 6.3.1, there exists a resolvable BIBD with
n~+ls
treatments and block size n + , then we can replace
i l
the levels of the last factor of the nlsxn2sx ••• xn~sxs BFE by the blocks of
BIBD to get an nlsx ••• xn~sxni+1s BFE with block size n1 ••• nini+lsi.
All the
main effects and interactions are estimated with full efficiency except the
nth ord
.
er .lnteractlons.
N
72
Type IV.
If there exists a BA(ps 2_ s ,u,s,2) with parameters A(X,y)
= p-l or p according as x=y or not, and a resolvable BA(qs 2- s ,t,s,2) with
parameters A(x,y)=q-l or q according as x=y or not.
Then, similar to
Theorem 6.3.1, we can construct a uxtxs BFE with k=ut, r=(ps-l)(qs-l),
b=s(ps-l)(qs-l), AOOl=O, AOlO=(ps-l)(q-l), AOll=(ps-l)q, AlOO=(p-l)(qs-l),
AlOl=P(qs-l), AllO=pqs-p-q+l, and Alll=pqs-p-q.
The efficiencies can be cal-
culated nelow:
E(l,O,O) = E(O,l,O) = E(l,l,O) = 1
E(O,O,l) = 1
(ps-u) (qs-t)
- (ps-l)
(qs-l)ut
(6.3.19)
E(O,l,l) = 1
qs (ps-u)
(ps-l) (qs-l)ut
ps(qs-t)
E(l,O,l) = 1 (ps-l) (qs-l)ut
2
pqs
E(l,l,l) = 1 (ps-l) (qs-l)ut
Usually the BA(pS 2- s ,u,s,2) can be obtained by deleting ps-u constraints
in a BA(T)(p,s,l) if it exists; similarly, the resolvable BA(qs 2- s ,t,s,2) can
be obtained by deleting qs-t constraints in a resolvable BA(T) (q,s,l).
Other
methods of constructing this type of balanced arrays is still to be developed.
Example 6.3.S
As in Example 6.3.4, if we use the BA(lO,S,3,2) obtained
by deleting a constraint in BA(T)(3,2,1), and other procedures being the same,
then we get a Sx4x2 BFE with k=20, b=30, r=lS, AOOl=O, AOlO=S, AOll=lO,
A =6, A =9, A =8, A =?,
lOo
lll
lOl
llO
E(O,l,l)= ;~, E(l,l,l)= ;~.
E(l,O,O)~E(O,l,O)=E(O,O,l)~E(l,l,O)=E(l,O,l)=l,
The SX4x2 BFE is given on the next page.
73
Blocks
1
2
3
Levels
of Fl
4
5
6
7
9
8
10
11
12
13
14
15
YO
Levels of F and F
3
2
0
Xo
Xl
X
o
X
o
X
o
Xo
X
o
X
o
Xl
Xl
Xl
Xl
Xl
YO
YO
YO
YO
1
X
o
Xl
Xl
Xl
X
o
X
o
Xl
Xl
YO
YO
Y
l
X
o
Xl
Xl
Xl
Xl
X
o
X
o
X
o
Y
l
YO
YO
Xl
Xl
Xl
X
o
Xl
Xl
X
o
Xl
X
o
X
o
Xl
Y
l
Y
l
Y
l
Y
l
Xl
Y
l
Y
l
YO
Xl
X
o
X
o
YO
Xl
X
o
X
o
X
o
X
o
Y Y
l
l
Y Y
1
1
2
3
X
o
4
YO
YO
Blocks
Levels
of F
1
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Levels of F and F
2
3
0
Y
l
Y
l
Y
l
Y
l
1
YO
YO
YO
2
Y
l
3
YO
Y
l
Y
l
4
Y YO
1
Zo
Zo
Zo
Zo
Zo
Zl
Zl
Zl
Zl
2
Y
l
Y
l
Y
l
Zo
Zo
Zl
Zl
Zl
Zo
Zo
Zo
Zl
Zl
YO
YO
YO
Zl
Zo
Zo
Zl
Zl
Zl
Zl
Zo
Zo
Zo
Y
l
Y
0
YO
YO
Zo
Zl
Zl
Zo
Zl
Zo
Zl
Zl
Zo
2
0
YO
Y
l
Zl
Zl
Zl
Zo
Zo
Zl
Zo
2
0
Zo
Zl
1
where XO' Xl' YO' Yl , ZO' Zl represent the blocks of the 4x2 BFE in Example
6.3.4.
Other examples include BFE's 4 X 3 X 2. 6 x 5 x 3, 5 x 3x 3, 6x8x3, 7x 4 x 4, ... etc.
CHAPTER VII
BFE'S WITH TWO-WAY ELIMINATION OF HETEROGENEITY
7.1
Introduction
In the previous chapters, we restricted our consideration of BFE's to
one-way designs only.
We shall consider two-way designs in this chapter,
i.e., designs with rows and columns as blocks.
Assume that the designs
satisfy the following conditions:
1)
There are v treatments, each replicated r times.
2)
There are b columns and k rows.
3)
There are u plots at a given row
and a given column.
Hence, there
are ku plots in each column and bu plots in each row.
We do not need the condition that each treatment occur at most once in
cach row or each column.
this assumption.
(7.1.1)
Yij1
In most cases u=l, but for generality we don't make
The fixed effect model is assumed:
= ~+L.+a.+8
1
J
1
+E .
where Y"o is the yield of the i
iJ1
th
treatment applied to the jth row and 1
IJN
column,
~ is the overall effect,
i=1,2, ••• ,v; j=1,2, ••• ,k; 1=1,2, ••• ,b
T.
1
is the effect of the i
is the effect of the jth row, 8 is the effect of the 1
1
the error.
E.
th
th
treatment, a.
J
row, and
E ..
o is
1JN
'o's are independent normal distributions with mean 0 and var-
1JN
.
th
2
lance a •
Let N X b = (n· o) and M X k = (m .. ) be the column incidence matrix and
V
IN
V
1J
75
the row incidence matrix respectively, where n
ii
is the number of times
th
column and m.. is the number of times treatIJ
.
h
.th
ment i occurs In t e J
rows. Let
treatment i occurs in the i
T = the column vector of the v treatment totals;
A = the column vector of the k row
totals;
B = the column vector of the b column totals;
G = the grand totals of all yields;
j
the vxl vector with all elements equal to 1·,
J = the vxv matrix with all elements equal to 1.
Then the reduced normal equation for the treatment effects is
(7.1.2)
CT =Q
where
1
(7. L 3)
C
=r
(7.1.4)
Q
= T - ..!..
bu
I - -
bu
2
1
r
NN' + bku J
ku
MM' - -
MA
..!..
NB
ku
It is well knwon that E(Q) = C T,
of C with eigenvector j.
this chapter.
+
rG
bku j
Var(Q)
= Ca
2
, and 0 is an eigenvaltle
Also Lemma 2.2.1 is valid under the conditions in
Hence we can calculate the efficiencies of the treatment con-
trasts by Definition 2.2.1.
b
h e m f actors
Let F , F2'.'.' Fmet
1
being experimented with s. levels.
1
0
.
f t h e experIment,
t h e 1.th f actor
We define balanced factorial experiments
with two-way elimination of heterogeneity below.
Definition 7.1.1
An experiment is called a balanced factorial experi-
ment with two-way elimination of heterogeneity
are satisfied:
if the following conditions
76
1)
Each treatment is replicated the same number of times, say r.
2)
There are k rows and b columns.
At a given row and a given column,
there exist u plots.
Estimates of contrast belonging to different interactions are uncor-
3)
related with each other.
All the normalized contrasts belonging to the same interaction are
4)
estimated with the same variance.
Kurkjian and Zelen (1963) and Kshirsagar
(1966) gave the sufficient
and necessary condition for an experiment to be balanced factorially.
The necessary and sufficient condition for an
Theorem 7.1.1
ment to be a BFE
experi-
is that the C-matrix of the experiment possesses property A
in Definition 2.2.2.
We shall give a combinatorial condition for an experiment to be a BFE
based on Theorem 7.1.1.
Let the v(=sls2 ••• sm) treatments possess the extended
m
group divisible association scheme with 2 _l classes and let
k
A.. ,
11
=
L n·n
1)(,
£=1
n ·,n
1
)(,
(7.1.5)
b
8 .. , =
11
I
m..
j=l
1J
m., .
1 J
Then, by Equation (7.1.3), the (i,i,)th element of the C-matrix is
1
(7.1.6)
rO 11
.. '--b
8 11
.. ,
U
2
r
1
-k
, + bku
u A..
11
where 0 .. , = 1 or 0 according as i=i' or not.
11
If the C-matrix possesses property A, then by Equation (2.2.3)
(7.1.7)
C
=
77
where
~(al,a2,
••• ,am)'s are constants depending on ai's.
Suppose that treat-
ments i and i' are (Sl,S2"",Sm)th associates, by Equation (7.1.7) the
. ") th e 1ement
( 1,1
0f
. .IS
t h e C-matr1x
1
(0
0
0)
~ ~1'~2""'~m'
1
r
2
b e i i , + k \ i ' = ruo ii , + bk -
(7.1.8)
u~
(6 l ,6 2 ,· .. ,6m)
0 .. , is a function of 6. 's only, since 0 .. , = 1 or 0
11
1
11
1
Hence
according
as 6 =8
1 2
1
=••• =6 =0 or not. Therefore -b e .. , + -k A.. , is a constant for all treatments
m
11
11
th
i and i' which are (6 ,6 , ••• ,6 )
associate, if and only if the C-matrix
1 Z
m
possesses property A.
Theorem 7.1.2
A two-way experiment is a BFE if and only if given any
two treatments i and i' which are U\,S2, ••• ,6m)th associates,
~eii'
+r\i l
is a function of (6 ,6 , ••• ,6m) only.
l 2
Letting b=k in Theorem 7.1.2, we have the following corollary.
Corollary 7.1.1
If the number of rows equals the number of columns
in a two-way experiment, then it is a BFE if and only if given any two treatments i and i' which are (Sl,6 "",Sm)
2
th
associates; e .. , + A.. , is a func11
11
tion of U\,Sz"",Sm) only.
By Equation (7.1.5) NN' = (A .. ,) and MM' = (8 .. ,), thus MM' + NN' =
11
11
(8 .. ,+A .. ,).
11
11
If we let L = (M:N), then LL' = NM'+NN' and the condition of
Corollary 7.1.1 means LL' possesses property A.
Hence
Corollary 7.1.1 can
be explained in the following equivalent way.
In a two-way experiment with the number of rows equal to the number of
columns, it is a BFE if and only if the one-way experiment with blocks consisting of all columns and rows is a BFE.
Example 7.1.1
and u=2:
2
Consider the following two-way 3 design with r=2, b=k=3,
78
01
02
10
12
10
11
21
22
20
22
00
01
!
20
21
00
02
11
12
The one-way design consisting of all 3 rows and 3 columns as blocks forms a
BFE given in Example 3.2.1
0
Hence this is a two-way BFE with 8 +A = 4,
0 O
8 +A = 2, and 8 +A = 3.
l l
2 2
Consider the following two-way 2
Example 7.1.2
3
design with r=2,
b=k=4, and u=lo
000
100
111
all
110
010
001
101
101
111
000
010
all
001
110
100
The one-way design consisting of all 4 rows and 4 columns as blocks forms
a symmetrical BFE with Al=l, A =2, A =3.
Hence it is a two-way BFE.
3
The condition of Corollary 7.1.1 does not require both 8 .. , and A.. ,
2
11
be
11
functions of CS ,8 , ••• ,Sm) only, which is certainly sufficient for
l 2
1
1
-b 8 11
··'+-k A11
.. ,
A two-way design is a BFE if given any two treatments
Corollary 7.1.2
i and i' which are CS
l
,8 2 , ••• ,Bm) th associates; both 811
.. , and A.. , depend
11
only on CB l ,S2, ••• ,Sm).
Not many BFE's satisfying the condition of Theorem 7.2.1 but not satisfying the condition of Corollary 7.1.2 are known.
are two of them.
Examples 7.1.1 and 7.1.2 nrc
Corollary 7.1.2 means that in a two-way design if the one-
way design formed by considering rows as blocks, and the one-way design
formed by considering columns as blocks are BFE's, then the two-way design
is a BFE.
79
Example 7.1. 3
Consider the following 3x3 design with r=2, b=3, k=6,
and u=l.
11
12
10
21
22
20
22
20
21
00
01
02
00
01
02
12
10
11
If columns are considered as blocks, then it is a one-way BFE with AOl=AIO=O
and AlI=l.
If rows are considered as blocks then it is a one-way BFE with
AIO=AII=I and A =2. Hence it is a two-way BFE.
Ol
If the conditions of Theorem 7.1.2 are satisfied, then
-.!..MM'
bu
(7.1.9)
where
na
1
= b-
,a , •• ,a
l 2
m
associates.
u
e..
,
11
+
1
k- A.. ,
u
11
We can similarly derive the eigenvalues of the C-matrix as in
Theorem 3.1.1.
The eigenvalues of the C-matrix of a two-way BFE with
Theorem 7.1.3
na
parameters
's given in Equation (7.1.9) are
1a 2 • .am
H.(a.,S.)
(7.1.10)
III
where a.,S. = 0 or 1, and H. (a.,S.) are given in Equation (3.1.2).
1
1
III
Example 6.1.4
The two-way BFE in Example 7.1.3 has parameters
. 2 + -13 • 2 = 1
. 2 -31 . 0 = 31
. 1 13 . 0 = 1
1 .
1
1
1 =
nl l = 6" . 1
2
3
1
nOO = -6
1
nOl .- 6
1
n lO = -6
+
+ -
+
6
80
Hence by Equation (7.1.10)
f(O,l) = r - nOO+nOl-2nlO+2nll = 2
3
f(l,O) = r - noo-2nOl+nlO+2nll = "2
3
1
The design has efficiencies E(O,l)=l, E(l,O)='4' and E(l,l)="2.
m
In s
ml m2
designs arid sl x S x
2
xS
mh
h
designs, if the treatments have the
same association schemes as those defined in Sections 3.2 and 3.3 and b~ MM'
+
1
ku NN' can be expressed in the ways as in Equations (3.2.3) and (3.3.3),
then we can have two-way BFE's similar to the one-way BFE's defined there.
The eigenvalues of :u MM'+ :u NN' can be derived similarly to those in Equations (3.2.4) and (3.3.4).
The design given in Example 7.1.1 is a two-way symmetri-
Example 7.1.5
cal BFE
with nO =
2
3'
nl =
1
S'
and n 2 =
1
2·
The eigenvalues of the C-matrix
are
f(l) = r - nO-nl+2n2 = 2
f(2)
=r
Hence E = 1 and E2 =
l
7.2
3
n o +2nl -n 2 = 2
3
i.
Construction of two-way BFE's
In this section we shall construct BFE's satisfying the condition given
in Corollary 7.1.2; that is, it is a one-way BFE when either rows or columns
are considered as blocks.
Since we have constructed some one-way BFE's in
Chapter VII and VI, if we can rearrange the positions of treatments within
each block in a one-way BFE such that it is another one-way BFE when rows
are considered as blocks, then we can obtain a two-way BFE.
81
To achieve balance factorially in one-way designs, the necessary and
sufficient condition is that the C-matrix possesses property A.
It is not
necessary that the design should be incomplete or each element should occur
at most once in each block.
The partially balanced block design (PBBD) de-
fined by Cheng (1978) can be used to obtain BFE's.
Definition 7.2.1
If we are given an association scheme with m classes
for the v treatments, then a PBBD is a design such that
There are k plots in each block, and each treatment occurs in each
1)
block [k/v] or [k/v] + 1 times, where [x] is the largest integer
~
x.
Each treatment is replicated r times.
2)
Let A.. , = l.~ In .. n.,., where n .. is the number of times treatment i
11
0J= 1J 1 J
1J
th
. t h e J. th bloc.
k
occurs 1n
I f treatments i and i' are a
associates,
3)
then the quantity A.. , depends only on a.
11
It is denoted by A •
a
Given a PBIBD with block size k, if each treatment is added to each block
n times, then we obtain a PBBD with block size nv+k.
On the other hand,
given a PBBD with block size k, if each treatment is taken from each block
[k/v] times, then we get a PBIBD with block size k-[k/v]v. If a PBBD has an
extended group. divisible association scheme, then the C-matrix of the design
possesses propoerty A, and therefore is a BFE.
Since we can obtain a PBIBD
from a PBBD and vice versa, it is sufficient to discuss only PBIBD's in
one-way designs.
For a two-way design, if it is an extended group divisible PBBD when
rows or columns are considered as blocks, then the C-matrix possesses property A.
We consider PBBD's instead of PBIBD's, because PBBD's are generali-
zations of PBIBD's and have less restrictions.
82
Example 7.2.1
From an OA(9,4,3,2), we can construct a 4x3 one-way
BFE with block size 4 by Theorem S.LL
00
00
00
01
01
01
02
02
02
10
11
12
10
11
12
10
11
12
20
21
22
21
22
20
22
20
21
30
31
32
32
30
31
31
32
30
If we rearrange the treatments within each block, then we get the following
one-way BFEif rows are considered as blocks.
10
21
22
32
11
12
31
20
30
00
31
32
01
30
20
22
02
21
30
11
00
10
01
31
02
32
12
20 ·00
12
21
22
01
10
11
02
Hence this is a 4x3 two-way BFE with b=9, k=4, u=l, r=3, AOl=O, A =A =1,
lO 11
3
8
A =3, 6 =6 =3, and 6 =6 =2. The efficiencies are E(O,l)='4' E(l,O)=g
00 01
OO
10 11
and E(l,I)=
t.
to the 4x3 two-way BFE in Example 7.2.1, then we obtain another 4x3 two-way
BFE.
When rows are considered as blocks, it is a complete block design. When
columns are considered as blocks, it is a one-way BFE with r=4, b=12, k=4,
AOl=O, A10 =2,and All=l.
9
The efficiencies are E(O,l)='16' E(l,O)= 1 , and
E(l,l)=i
Example 7.2.3
If we add 3 columns of the following forms
83
00
01
02
10
11
12
20
21
22
to the 3x3 two-way BFE in Example 7.1.3, then we obtain another 3x3 two-way
BFE.
It
is a complete block design when rows are considered as blocks, and
a one-way BFE with r=3, b=9, k=3, AIO=All=l, and AOl=O.
The efficiencies
are E(O,l)=E(l,l)=t, and E(l,O)= 1 •
Example 7.2.4
From a TA(6,2,3,2), we can construct a 2x3 one-way BFE
with block size 2 below by Theorem 5.1.1.
00
01
02
00
01
02
11
12
10
12
10
11
If we rearrange the treatments within each block, then the following design
is a complete block design if rows are considered as blocks.
00
01
02
12
10
11
11
12
10
00
01
02
3
1
!-Ience it is a two-way BFE with efficiencies E(l,O)= 1, E(O,l)="4' and E(I,l)="4.
Eaxmples 7.2.2, 7.1.3, 7.2.3 and 7.2.4 are special cases of the following
theorem.
Theorem 7.2.1
If s is a prime power, then there always exist the follow-
ing BFE's.
A01=0, A10 =2, and A11 =1.
s2+ s +2
and E(l,l) =
. 2 •
(s+l)
2)
s
2
The efficiencies are E(O,l)=(s+l) , E(l,O)=l,
2
An sxs BFE with k=3, b=s -s, u=l, r=A
OO =8 00 =s-1, AOl=AlO=O, All=l,
84
8
1O
=8
ll
=s-2, and 8
=s-1.
The efficiencies are E(O,l)= 1, E(l,O)
s-2
and E(1, 1) = - 1
s-
= s(s-2)
(s-l)
01
2
2
An sxs BFE with k=s, b=s , u=l, r=AOO=800=801=810=811=s, A =A =1,
10 11
3)
s-l
The efficiencies are E(O,l)=E(l,l)= and E(l,O)= 1 •
s '
The efficiencies are E(O,l)= S(S-2~, E(l,O)= 1 ,
(5-1)
and An = 1 •
2
and E(l,l)=
Proof:
(1)
s -3s+1
2 •
(s-l)
For s a prime power, there exists an OA(s2,s+1 ,5,2).
We add s assemblies of the following forms
01
• s-l
01 • • • s-l
•
•
0
•
•
01 • • • s-l
2
to obtain a BA(s +s,s+1,s,2) with parameter A(x,y)=2 or 1 according as x=y
or not.
By Theorem 5.1.1 we can construct an (s+l)xs one-way BFE with
k=s+l, A =0, A =2, and A =1.
10
01
11
Arrange the treatments within each block
such that each row contains all the treatments once.
This can be done by
applying the method of Smith and Hartley (1948).
(2)
For s a primer power, there exists a TA(s(s-1),s,s,2).
By Theorem
5.1.1, we can construct an sxs one-way BFE with k=s, A =A =0, and A =1.
11
01 10
Since the TA is resolvable, we can easily arrange the treatments within each
.
11t e
h treatments except
co 1umn sue h t h at the l· th row contaIns
a
(i_1)th level of the first factor.
8
10
=8
11
=s-2, and 8
01
t
hose at t hc
Then it is a one-way BFE with parameters
=s-1 if rows are considered as blocks.
e.
85
(3)
Add s columns of treatments given below,
(s-l) 0
00
01 • • • O(s-l)
10
11.
l(s-l)
......
(s-l) 1 . . . (s-l) (s-l)
to the design constructed in (3), and we obtain the desired design.
(4)
Discard a constraint from a TA(s(s-1),s,s,2), and we obtain a
TA(s(s-1),s-1,s,2).
By Theorem 5.1.1 we can construct an (s-l)xs one-way
BFE with k=s-l, AOI=AlO=O, and A =!'
ll
Arrange the treatments within each
column such that all the treatments occur exactly once in each row.
Then
it is a complete block design when rows are considered as blocks.
Q.E.D
Similarly, we can prove the following theorem.
Theorem 7.2.2
If there exists a resolvable BA(T)(m,s,l), then we can
always construct the following two-way BFE's.
1)
2
An msxs BFE with k=ms, b=ms -s, u=l , r=A 00 =8 00 =ms-l ' 0
A1
=0' A10 =m-l '
A =m, 8 =8 =ms-2, and 8 =ms-l.
1l
l0 ll
0l
1-
2)
1
1
"" '2 ' E(O,l)= 1 , and E(l,l)= 1- ms-l
(ms-l)
2
An msxs BFE with k=ms, b=ms , u=l, r=AOO=800=80l=8l0=8l1=ms, AlO=All=m,
and AOl=O.
3)
The efficiencies are E(l,O) =
The efficiencies are E(O,l)= E (1,1)= 1- :s ' and E(l,O)= 1 •
2
An (ms-l)xs BFE with k=ms-l, b=ms -s, u=l, r=AOO=800=80l=8l0=8ll=ms-l,
AOl=O,
~\O=m-l,
and Al1 =m.
1
The efficiencies are E(O,l)=l - - - - 2 '
(ms-l)
ms
E(l,O)= 1 , and E(l,l)= 1 2 •
(ms-l)
It is easily seen that designs (2) and (3) in Theorem 7.2.2 exist even
if the BA(T) (m,s,l) is not resolvable, because we can apply the method of
Smith and Hartley to obtain them.
We are unable to prove that design (1)
86
also exists without resolvability, but it is conjectured that it does exist.
The following are examples of Theorem 7.2.2 from a nonresolvable BA(T)(3,2,1).
A 6x2 two-way BFE with k=6, b=10, u=l, r=A
Example 7.2.5
A =0, A =2, All =3, 8 =8 =4, and 8 =5.
10 11
01
01
lO
24
E(l,O) = 25
OO
=8
00
=5,
The efficiencies are E(O,l)= 1 ,
4
' and E(l,l) = 5'
21
41
20
30
40
50
10
31
51
11
30
31
41
40
00
21
50
51
20
01
41
51
31
00
50
10
01
40
11
30
10
20
00
51
11
41
21
01
40
50
Sl
00
50
11
21
30
31
10
01
20
00
10
11
21
31
01
40
20
30
41
Example 7.2.6
If we add two columns of treatments given below
00
01
10
11
20
21
30
31
40
41
50
51
to the 6x2 two-way BFE in Example 7.2.5, then we obtain a 6x2 two-way BFE
with k=6, b=12, u=l, r=Aoo=800=801=810=81l=6, A10 =A 11 =3, and A01 =0'
The
efficiencies are E(O,l) = E(l,!) = ~, and E(l,O) = 1.
Example 7.2.7
=8
10
=8
11
A 5x2 two-way BFE with k=5, b=10, u=l, r=AOO=800=801
=5, A =0, A =2, and A =3.
11
10
01
19
=1, and E(l,l) = 25
24
The efficiencies are E(O,l) = 25' E(l,O)
87
00
10
11
21
31
01
40
20
30
41
41
00
20
30
11
21
10
31
40
01
30
31
00
40
21
41
01
10
20
11
10
41
31
00
40
30
21
01
11
20
21
20
41
11
00
10
31
40
01
30
Example 7.2.1 suggests the existence of the following design:
2
(*)An (s+l)xs two-way BFE with k=s+l, b=s , u=l, r=A
=1, 8
1O
=8
11
2
E(l,O) = s
=s-1, and 8
01
=s.
OO
=8
00
=s, A =0, A =A
01
lO 11
The efficiencies are E(O,l) = E(l,l) = S~l' and
;1 .
s
2
This is constructed from an OA(s ,s+l,s,a) where s is a prime power. It
can be easily checked that (*) does not exist fro s=2.
But we have success-
fully constructed (*) for 3=3,4,5; therefore we conjecture that (*) exists
for s a pri::1e power and greater than 2.
The following example is a three-factor two-way BFE, constructed hy
arranging the treatments of each block in a one-way BFE.
Example 7.2.8
A 3 x 3x.3 two-way BFE with k=9, b=12, u=l, r=AOOO=8000
=4, A001=A010=A100=0' A011=A101=A110=2, A =1, 8100=8110=8101=8111=1,
111
8010=8011=3, and 8001 =4.
The efficiencies are E(O,O,l)=E(l,O,l)=E(O,l,l)=l,
E(O,l,O)=E(l,l,O) = ~~, E(l,O,O) =
e
;6 '
and E(l,l,l) =
fo
000
001
002
011
012
010
022
020
021
101
102
100
012
010
011
022
020
021
000
001
002
110
III
112
021
022
020
000
001
002
011
012
010
122
120
121
102
100
101
112
110
111
121
122
120
202
200
201
III
112
110
120
121
122
102
100
101
211
212
210
120
121
122
101
102
100
110
III
112
220
221
222
201
202
200
210
211
212
220
221
222
000
001
002
210
211
212
221
222
220
201
202
200
012
010
all
222
220
221
202
200
201
212
210
211
021
022
020
88
In the remainder of this section, we shall discuss a method of constructing two-way BFE's called "patchwork method".
It was introduced by
Kiefer (1975) in the construction of generalized Youden squares.
applied this method to construct Youden hyperrectangles.
Cheng (1979)
We describe this
method in the following:
Denote the kxb array G of the design to be constructed as
where GIl is an a v x a v array with equal replications of treatments in rows
1
2
and columns, e.g., this can be obtained as an a x a array of Latin squares
l 2
of order v.
(G
Zl
,G
22
The remainder of G is pieced together, when possible as follows:
) is an EGD-PBIBD and each row of G has treatments replicated a
2
2l
times, and similarly for the EGD-PBIBD
(Gi2,G~2).
With these conditions,
we can see that G is an EGO-PBBD if either columns or rows are considered
as blocks.
Example 7.2.9
=A
lO
A
2x4 two-way BFE with k=lO, b=12, u=l, r=15, A =2l, A
Oo
UI
=18, A =19, 8 =25, 8 =22, 8 =20, 8 =23. The efficiericies are E(l,O)
11
10
ll
01
00
89
Example 7.2.10
1..
01
=1..
10
=28, 1..
79
Be1,0) = 80'
e
A 3x5 two-way BFE with k=18, b=20, r=24, 1..
=29, 8
00 24 23 22 21 20 14 13 12 11 10 04 03 02 01
00 01 12 23 14
01 00 24 23 22 21 20 14 13 12 11 10 04 03 02
10 11 22 03 24
02 01 00 24 23 22 21 20 14 13 12 11 10 04 03
20 21 02 13 04
03 02 01 00 24 23 22 21 20 14 13 12 11 10 04
24 00 11 02 23
04 03 02 01 00 24 23 22 21 20 14 13 12 11 10
04 10 21 12 03
10 04 03 02 01 00 24 23 22 21 20 14 13 12 11
14 20 01 22 13
n
10 04 03 02 01 00 24 23 22 21 20 14 13 12
03 14 00 21 22
12 11 10 04 03 02 01 00 24 23 22 21 20 14 13
13 24 10 01 02
13 12 11 10 04 03 02 01 00 24 23 22 21 20 14
23 04 20 11 12
14 13 12 11 10 04 03 02 01 00 24 23 22 21 20
22 13 04 00 11
20 14 13 12 11 10 04 03 02 01 00 24 23 22 21
02 23 14 10 21
21 20 14 13 12 11 10 04 03 02 01 00 24 23 22
12 03 24 20 01
22 21 20 14 13 12 11 10 04 03 02 01 00 24 23
21 12 13 24 00
23 22 21 20 14 13 12 11 10 04 03 02 01 00 24
01 22 23 04 10
24 23 22 21 20 14 13 12 11 10 04 03 02 01 00
11 02 03 14 20
00 01 02 03 04 24 20 21 22 23 14 10 11 12 13
00 01 02 03 04
11 12 13 14 10 00 01 02 03 04 23 24 20 21 22
14 10 11 12 13
22 23 24 20 21 12 13 14 10 11 00 01 02 03 04
23 24 20 21 22
A 2x2 two-way BFE with k=b=6
Example 702.11
-
=8
=
e
10
26
=12, 1..
27'
10
=8
01
=13, and 1..
11
=8
11
=14.
=32,
The efficiencies are
=36, 8 =8 =32, and 8 =30.
01 11
10
00
215
527
E(O,l) = 216' and E(l,l) ='540.
11
00
,
r=9
A =8 =15 A
' 0 0 00
' 01
The efficiencies are E(O,l)=E(l,O)
23
and Ee1,1) = 27 •
00
11
10
01
00
11
01
00
11
10
01
10
10
01
00
11
11
00
11
10
01
00
10
10
00
01
11
10
00
01
11
10
00
01
10
11
90
A3
=8 =24
o
'
2
two-way symmetrical BFE with k=b=12, r=16, A
O
15
A =8 =20, and A =8 =20. The efficiencies are E = 1 , and E = 16·
2 2
1 1
1
2
Example 7.2.12
00
22
21
20
12
11
10
02
01
00
11
22
01
00
22
21
20
12
11
10
02
01
12
20
02
01
00
22
21
20
12
11
10
02
10
21
10
02
01
00
22
21
20
12
11
21
00
12
11
10
02
01
00
22
21
20
12
22
01
10
12
11
10
02
01
00
22
21
20
20
02
11
20
12
11
10
02
01
00
22
21
12
21
00
21
20
12
11
10
02
01
00
22
10
22
01
22
21
20
12
11
10
02
01
00
11
20
02
00
01
02
21
·22
20
11
12
10
00
22
11
11
12
10
00
01
02
22
20
21
12
01
20
22
20
21
12
10
11
00
01
02
21
10
02
Theorem 7.2.3
Ifs is a prime power, then there always exist the fol1ow-
ing two-way BFE's:
1)
2)
2
An sxs two-way BFE with b=k=s(s+l), u=I, r= (s+ 1) , A =8 =5 2+45+3,
OO 00
2
2
2
A01=Al0=A11=811=s +35+3, 8 =5 +35+2, and 8 10 =5 +35+4. The efficienci cs
01
2
3
, E(l,O)=l - 1
are E(0,1)=1 and E(1,1)=13'
. (5+1)3
(5+1)3·
(5+1)
2
An sxs two-way BFE with b=k=s(s+l), u=l, A =8 =s +45+3, A =A =A
OO 00
01 10 11
=801=810=811=52+35+3, i.e.
a generalized Youden square with 52 treat-
ment. All the efficiencies are 1 -
Proof:
(1)
2 3.
(5+1)
Apply the patchwork method by letting
GIl = a Latin square of order 5
2
G = the design (4) in Theorem 7.2.1
21
G =
12
Gil
e
91
00
G =
22
(2)
10
.• . .• ·.
.... ··.
(5-1)0
01 11
•
•
•
•
0(5-1) 1(5-1). •
(s-l)l
. .. . ..• ...
· (5-1) (5-1)
Also apply the patchwork method by letting G , G , G be the
21
22
11
same as in (1) •
G is obtained from the G in (1) by interchanging the
12
12
Q.E.D.
two factors.
For 5=3 in Theorem 7.2.3, we have the following two examples
A 3x3 two-way BFE with k=b=12, r=16, 1..
Example 7.2.13
e
=~~,
and
E(1,l)=~~.
00
22
21
20
12
11
10
02
01
00
11
22
01
00
22
21
20
12
11
10
02
01
12
20
02
01
00
22
21
20
12
11
10
02
10
21
10
02
01
00
22
21
20
12
11
21
00
12
11
10
02
01
00
22
21
20
12
22
01
10
12
11
10
02
01
00
22
21
20
20
02
11
20
12
11
10
02
01
00
22
211 10
20
00
21
20
12
11
10
02
01
00
22
11
21
01
22
21
20
12
11
10
02
01
00
12
22
02
00
01
02
21
22
20
10
11
12
00
10
20
11
12
10
00
01
02
20
21
22
01
11
21
22
20
21
12
10
11
00
01
02
02
12
22
Example 7.2.14
A 3x3 two-way BFE with k=b=12, r=16, 1..
A01=A10=A11=801=~10=811=21.
=8
00
=24, 1..
61
The efficiencies are E(0,1)=64-'
=1..10=1..11=811=21, 801 =20, and 8 10 =22.
E(l,O)
00
All the efficiencies are
~~.
00
=8
00
=24,
01
92
00
22
21
20
12
11
10
02
01
00
11
22
01
00
22
21
20
12
11
10
02
10
21
02
02
01
00
22
21
20
12
11
10
20
01
12
10
02
01
00
22
21
20
12
11
12
00
21
11
10
02
01
00
22
21
20
12
22
10
01
12
11
10
02
01
00
22
21
20
02
20
11
20
12
11
10
02
01
00
22
21
01
02
00
21
20
12
11
10
02
01
00
22
11
12
10
22
21
20
12
11
10
02
01
00
21
22
20
00
01
02
21
22
20
10
11
12
00
10
20
11
12
10
00
01
02
20
21
22
01
11
21
22
20
21
12
10
11
00
01
02
02
12
22
If one design is to be chosen among Examples 7.2.12, 7.2.13, and
7.2.14, we may choose Example 7.2.12, since all the main effects are estimated with full efficiency.
But this design cannot usually be constructed
for s a prime power, while Examples 7.2.13 and 7.2.14 can always be constructed for all such
7.3
SIS.
An algorithm for constructing an s
2
two-way symmetrical BFE
By Corollary 5.4.1, we have seen that there always exists an
S
2
one-
2
. 2 2
way symmetrical BFE with r=(s-l) , b=k=s(s-l), A1=s -3s+2, and A =S -35+3.
2
If we can arrange the treatments within each block in this design so that
it is a one-way BFE with the same parameters when rows are considered as
blocks, then we obtain a two-way symmetrical BFE with all the main effects
estimated with full efficiency.
We shall give an algorithm of this arrangc-
ment in this section.
Lemma 7.3.1
Suppose s~3, and let A = {0,1,2, ••• ,s-1} and X. = {XEAI
xri} for i=1,2, ••• ,s-1.
I
Then there exists an (s-l)x(s-l) matrix such that
.
the l.th column contaIns
al 1 t h e elements
0
f X. an d so d oes the l.th row.
I
93
(1)
Proof:
(2)
If
s~4,
If s=3, then the matrix is
2
0
o
1
then we can construct a Latin square of order s-l with
symbols {1,2, ••• ,s-1} such that the diagonal elements are 1,2, ••• ,s-1.
Re-
placing all the diagonal elements by the symbol 0, we obtain the desired
Q.E.D.
matrix.
For s=4 in Lemma 7.3.1, we obtain the following 3x3
Example 7.3.1
matrix.
032
301
210
If s is a prime power and ~ 3, then there exists an s
Theorem 7.3.1
2
2
two-way symmetrical BFE with b=k=s(s-l), u=l, r=A =8 =(s-1) ,
O 0
2
2
A =8 =s -3s+3. The efficiencies are El=l and E2=1 3 .
2 2
(s-l)
Proof:
We will show this theorem ,for s=4 by giving an algorithm; the
general case for s a prime power is similar.
Let X ' Xl' X ' X represent the following sets of treatments:
O
2
3
02
02
01
01
03
03
03
02
Let Y., Z., W. (i=0,1,2,3) represent the similar treatment sets as X. except
111
1
that the levels of the first factor are 1, 2, 3 respectively.
5.4.1, we can construct a 4
2
From Corollary
one-way BFE in the following, if columns are
94
considered as blocks.
(7.3. 1 )
X
o
X
o
X
o
Xl
Xl
Xl
X
2
X
2
X
2
Y
1
Y
2
Y
3
Yo
Y
3
Y
2
Y
3
YO
Z2
Z3
Zl
Z3
Z2
Zo
Zo
W
3
WI
W
2
W
W
o
W
3
X ] =
o
r
z
Y
l
X
3
Y
2
X
3
Y
l
Yo
Zl
Z3
Zl
Zo
Z2
WI
W
3
W
o
W
o
W
2
WI
l
01
02
02
Ol]
03
03
03
X
3
Now consider the matrix.
(7.3.2)
[X
O
X
o
02
If we arrange the treatments within each column to get a Latin square, we
can write it in the following form:
x*l
o !
X* •I =
O!
X* j
oj
(7.3.3)
01
03
02
03
02
01
02
01
03
where X is the row vector with the same treatments as X ' but the arrangeo
o
ments of treatments are different according to the positions.
Also, by Lemma 7.3.1, we can do the following arrangement:
(7.3.4)
[Y1
Y
2
Y3]
=
[0~~
13
10
11
where Yt's are similarly defined as
flY~
12J [V*]
Y~
11
10
X~.
=
y*
3
In general, we have
Y*
(7 0305)
[Yo
1
Y.
J
Yk ]
=
for all i, j, k distinct.
y*
k
After arranging the treatments within each column as above, we can rewrite
95
Equation (7.3.1) in terms of row vectors as below:
(7.3.6)
x*
0
x*
1
Xu
x*
X*0
y*
1
y*
2
y*
3
Z*
2
Z*
3
x*1
y*
0
y*
3
y*
2
Z*
3
Z*
2
Z*
0
W*
2
x*
2
x*
2
x*
2
y*
3
y*
0
y*
1
Z*
0
Z*
1
Z*
1
W*
3
1
Z*
3
W*
I
W*
3
W*
0
W*
0
W*
3
W*
I
W*
2
x*
3
X*
3
x*
3
y*
2
y*
1
y*
0
Z*
1
Z*
0
Z*
2
W*
0
W*
2
W*
I
Our task is to arrange (7.3.6) within each of the four columns such
that it possesses the same pattern as (7.3.1) when rows are considered as
blocks.
At this moment we should be careful that the orders of the treat-
ments in the row vector
at different positions.
Y~'s
x~
1
(or
y~, Z~, W~)
111
may be different when it occurs
-
Hence, in the first column of (7.3.6), XO's and
are given in (7.3.3)1
and (7.3.4) respectively,
while Z~'s and W~'s
1
1
are given below:
(7.3.7)
Zil
Z*
3
I
Z* 1
1J
=
20
21
23
W31
21
20
22
W*
23
22
20
I
In the second column of (7.3.6),
w~J
=
30
32
31
32
30
33
31
33
30
96
00
xq
Xi
x*1
(7.3.8)
1
J
L02
00
1
I
20
21
Z3
~
02l
= I 03 02 00
I
Il
I
03
Z*2 ;I
Wil
13
ll2
11
10
10
11
31
33
30
I 33
31
32 )
32
31
t
; 13
LY*2JI
22l
12
!-
1
20
21
23
w*0,
22
L
23
2lJ
W3J
I 30
Y3l
1
I
iL.. Z*0 J1
03J
r::l
r
=
11
=
1
l
'-
~
--"
In the third column of (7.3.6),
roo
03
x* 1 = ; 03
01
00 I\
Lx*2JI
I 01
I-
00
03 i
J
YiJ
rz*l
22
23
2ll
Wi
23
22
20
21
20
22
rXil
I
I 2:
t
1
I
\
(7.3.9)
Iz~ I =
I 1
I
I
I
L
011,
-I
In the fourth column of
X·3
x*
1
31
=
x*3_,
=
1
r
w*3
,
z*3 I'
y*0
11
11
12
13
~O
13
12j
32
30
33
30
32
31
33
31
32 :
=
_I W*0
!
I z*
I
02
01
y*2
01
00
y*
Lal
00
02
y*0_
f23
22
20
W*0
22
23
21
w*2
lz~2'-'I
20
21
23
w*1
NO\\T
=
1
1
L
consider all
Y~' s
1
!
r.
•
-
10
10
13
12 ,
Lll
12
13 I
33
31
32
31
33
30
32
30
33
r
1
l~l
13
f
--l
(7.3.10)
Zil
r
...
(7.3.6)
~o02
,
1~
12
=
in (7.3.6), which is
e
97
(7.3.11)
y*
3
y*
y*
y*
1
y*
2
y*
3
0
y*
3
y*
2
y*
2
y*
1
y*
0
y*
1
a
Also consider the Latin square of order 4 with symbol i-I at the C1,1.) th
position.
~
(7.3.12)
o
2
3
1
3
1
a
2
132
0
2
0
1 ,,3
If we delete the diagonal, then the remaining symbols in each column of the
Latin square are the same as the indices of each column in (7.3.11).
we can move
Yi, Y3,
Hence
and Yi (2,3,1 are symbols in the first row of (7.3.12))
from the second, third, and fourth columns respectively to the first row.
3, YO'
Move Y
and
Yi
to the fourth row.
from the first, third and fourth columns respectively
Vi, Y3, and YO
Move
columns repective1y to the seventh row.
from the first, second and fourth
Move
Yi, Yo
and
second and third columns respectively to the tenth row.
dures to move Z~'s and
1
W~, s
1
to the remaining rows.
Yi
from the first,
Use similar proce-
Finally move X*l' sto the
fourth, fifth and sixth rows, Xi's to the seventh, eighth and ninth rows. X 's
3
to the tenth, eleventh and twelfth rows, and we arrange (7.3.6) in the following way:
98
x*
y*
y*
3
y*
1
x*
a
z*
x*
a
W*
y*
3
x*
1
z*
3
W*
3
y*
z*
1
W*
1
y*
z*
3
x*
1
z*
a
z*
W*
3
y*
1
z*
x*
W*
a
W*
y*
3
z*
3
x*
y*
x*
z*
x*
W*
y*
W*
3
y*
y*
z*
z*
z*
W*
W*
W*
x*
3
x*
3
x*
3
a
(7.3013)
1
W*
1
2
2
2
a
1
2
Now we can move Y'i;"s,
1
1
1
W~' s
and
Z~'s
1
a
2
a
2
2
a
2
0
2
a
2
a
2
2
1
1
in (7.3.13) so that it has the
e
same pattern as (7.3.1) when rows are considered as blocks.
W*
3
z*
3
y*
3
y*
y*
1
W*
1
a
W*
x*
W*
a
z*
W*
3
W*
x*
z*
a
y*
1
y*
3
x*
z*
z*
W*
3
x*
y*
y*
x*
W*
y*
z*
3
W*
a
z*
X
W*
z*
a
y*
1
x*
3
z*
y*
W*
x*
3
x*
a
z*
x*
y*
x*
a
W*
z*
3
y*
3
x*
1
a
(7.3 014)
1
1
2
2
2
Finally substituting
2
2
2
1
1
a
z*
1
2
2
2
a
2
a
2
a
2
1
1
3
X~'s, Y~'s,
1
1
e
Z'i;"s and
1
W~'s
1
from (7.3.3), (7.3.4),
99
100
01 04 02 03 32 30 34 31 14 10 13 12 41 40 42 43 23 20 21 24
03 02 04 01 42 40 41 44 34 30 32 33 21 20 23 22 13 10 14 11
04 01 03 02 14 11 10 12 23 22 20 24 32 33 30 31 41 44 40 43
02 03 01 04 21 24 20 22 42 43 40 44 13 12 11 10 34 31 30 32
24 21 22 20 00 04 02 03 33 31 30 32 11 10 14 13 42 41 43 44
14 11 10 12 03 02 04 00 43 41 42 40 30 31 33 34 22 21 24 23
42 40 41 44 04 00 03 02 10 12 11 13 24 23 21 20 33 34 31 32
30 32 31 34 02 03 00 04 22 20 21 23 43 44 41 40 14 13 11 12
43 42 44 40 20 22 23 21 00 04 01 03 34 32 31 33 11 12 10 14
23 22 20 24 10 12 11 13 03 01 04 00 44 42 43 41 31 32 34 30
34 30 32 33 43 41 42 40 04 00 03 01 12 14 1311 20 24 22 21
10 14 12 13 31 33 32 30 01 03 00 04 23 21 22 24 44 40 42 41
12 13 11 10 44 43 40 41 21 23 24 22 00 04 01 02 30 33 32 34
32 33 30 31 24 23 21 20 11 13 12 14 02 01 04 00 40 43 44 42
21 20 23 22 30 31 33 34 44 42 43 41 04 00 02 01 12 14 13 10
40 41 43 42 11 10 13 14 32 34 33 31 01 02 00 04 24 22 23 20
31 34 33 30 13 14 12 11 40 44 41 42 22 24 20 23 00 03 01 02
41 44 40 43 33 34 31 32 20 24 22 21 10 13 12 14 02 01 03 00
13 10 14 11 22 21 24 23 31 32 34 30 40 43 44 42 03 00 02 01
20 23 24 21 41 42 44 43 12 11 14 10 33 30 34 32 01 02 00 03
e
.
101
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---
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•
.31>