-
•
•
ON CONSTRUCTION OF BALANCED FACTORIAL EXPERIMENTS
by
Chung-yi Suen
A dissertation submitted to the faculty ofthe University of
North Carolina at ChapelHill in partial fulfillment of the
require-ments for the degree of Doctor of Philosophyin the
Department of Statistics
Chapel Hill
1982
Approved by
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•
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 or-
der 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
l xs 2x... xsm
BFE, if the main effects of the first factor are estimated with
full effici-
ency, 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 l or A2
according as x=y or
not. 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. Two
methods of
construction are given. 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.
Combinator-
ial conditions of two-way BFE's are derived. Two-way BFE's are
oonstructed
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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.
•
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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 encour-
agement and moral support which have made this work
possible.
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CHAPTER I:
1.1
1.2
CHAPTER II:
2.1
2.2
2.3
CHAPTER I II :
3.1
3.2
3.3
CHAPTER IV:
4.1
4.2
TABLE OF CONTENTS
INTRODUCTION
A review of earlier work
Summary of this thesis
PRELIMINARY FORMULATIONS
General notation, definitions
Some results regarding C-matrix
Balanced arrays
SOME PROPERTIES OF BFE's
Efficiencies of BFE's
Symmetrical BFE's
CONSTRUCTION OF BALANCED ARRAYS
Construction of transitive arrays
The product of balanced arrays
1
1
3
5
5
8
10
14
14
19
24
27
27
31
iv
4.3 Construction of some balanced arrays of
strength two 33
CHAPTER V:
5.1
5.2
5.3
TWO-FACTOR BFE'S
slxs2 BFE's with block size s1(sl~s2)
slxs2 BFE's with block size s2(sl
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BFE'S WITH TWO-WAY ELIMINATION OF HETEROGENEITY 74
CHAPTER VI:
6.1
6.2
6.3
CHAPTER VII:
7.1
7.2
7.3
BIBLIOGRAPHY
MULTI-FACTOR BFE'S
sm symmetrical BFE's with block size s
Methods of constructing mUlti-factor BfE's
Examples of multi-factor BFE's
Introduction
Construction of two-way BFE' s
An algorithm for constructing an s2 two-way
symmetrical BFE
55
55
58
62
74
80
92
101
v
e't
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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 treat-
ment 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. Since vr =
bk and
Alv-l) = r(k-l), r and A are determined 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
inter-
action (or main effect) are estimated with the same
variance;
(2) estimates of two treatment contrasts belonging to different
inter-
actions (or main effects) are uncorrelated.
Shah (1958, 1960) gave the exact definition of a balanced
factorial exper-
iment (BFE) and showed the remarkable result that every BFE is
necessarily
a partially balanced incomplete block design (PBIBD) with an
association
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2
scheme called extended group divisible scheme by Hinkelmann
(1964).
Bose (1947) was the first to consider the problem of balancing
in fac-
torial 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
asymmetri-
cal, this method is not applicable since finite geometries are
not available.
Nair and Rao (1948) considered the problems of two-factor block
de-
signs. 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. Designs having property A were proved
to be bal-
anced factorially. 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"
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3
duced the structural property A* as a generalization of property
A of Kurk-
jian 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, prop-
erty A, and balanced arrays. Some useful results are also
included.
In Chapter III, properties of three types of BFE's are studied.
The
C-matrix of a BFE is given by the Kronecker product of matrices.
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 esti-
mated with full efficiency. We extend the result to a sl x s2
x... xs m 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
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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 efficien-
cies. 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 spec-
ified parameters ~. IS.1
Chapter VII deals with designs suitable for two-way elimination
of
heterogeneity. BFE's which permit two-way elimination of
heterogeneity
are defined. We derive the combinatorial conditions for BFE's
with two-way
heterogeneity. 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 effi-
cicncy.
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a-IAPTER II
PRELIMINARY FORMULATIONS
2.1 General notation, definitions
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 .. = )J + T. + B. + E ..1J 1 J 1J
i=l, ... ,v; j=l, ... ,b
E .. 's are independent1J
applied to the jth block, )J is
jth block, T. is the effect of1
is the effect of the
f h . tho t e 1 treatment
E .. is the experimental error.1J
normal distributions with mean 0 and variance 02 .
where y .. is the yield1J
the overall effect, B.J
h. tht e 1 treatment,
In a factorial experiment, let Fl , F2 , ... ,Fm be the m
factors of the
treatments at sl' s2, ... ,sm levels respectively. Let v(=sls2
... sm) treat-
ments be denoted by the levels of the factors as (x 1 ,x2 ' ...
,xm), where Xi
is the level of the i th factor and takes values 0,1, ... ,s.-l.
Throughout1
this thesis the v treatment combinations will always be arranged
in lexico-
graphic order (cf. Kurkjian and Zelen (1963)).
Let T(x1
,x2 , ... ,xm) represent the effect of the treatment
combination
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6
(X l 'x2 '·· .,Xm)· A treatment contrast Il(X l ,X2 ,··· ,xm)
T(x l ,x2 ,··· ,xm)
(summation is over all x l ,x 2 ' ... ,xm) belongs to the
(q_l)th order inter-
action between the factors F. ,F. , ... ,F. , if 1(xl
,x2
, ... ,Xm
) depends onlyJ l J 2 J q
on X. , x . , ... , x . and)1 J 2 J q
(2.1.2)
s. -11
L l(X I ,x2 '··· ,xm) = 0x.=o1
If L12 (x l ,x 2, ... ,xm) = 1, Ll(x l ,x2 , ... ,xm) T(x l ,x
2, ... ;xm) is called a nor-
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 un-
correlated with each other.
(4) 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:
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Definition 2.1.2
7
Given v treatments 1,2, ... ,v, a relation satisfying
the following conditions is said to be an association scheme
with m classes:
(1) Any two treatments are either 1st, 2nd, ... , or mth
associates, the
relation of association being symmetrical; that is, if the
treatment
a is the i th associate of the treatment B, then B is the i th
associ-
(2)
ate of a.
Each treatment a has n. i th associates, the number n. being
indepen-1 1
dent of a.
( . f d Q .th . h h b f3) I any two treatments a an ~ are 1
assocIates, t en t e num er 0
h .th . f d kth . f Q •treatments t at are J assocIates 0 a, an
assocIates 0 ~, IS
P~k and is independent of the pairs of i th associates a and
B.
iThe numbers v, ni (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.
( f d Q .th . h h h3) I two treatments a an ~ are 1 assocIates,
t en t ey occur toget -
er in A. blocks, the number A. being independent of the
particular1 1
. f .th . d QpaIr 0 1 assocIates a an ~.
The numbers v,b,r,k,A.(i=1,2, ... ,m) are called the parameters
of the1
design. For a PBIBD, the following conditions hold:
-
(2.1.3) vr bk
8
(2.1.4) r(k-I)m
= Li=l
n.A.1 1
Let the v(=s l s2' .. sm) treatments be denoted by (x1,x2,· ..
,xm)
(x.=O,I, ... ,s.-l). An extended group divisible association
scheme with1 1
m m2 -1 classes (EGD/2 -1) for the v treatments is defined by:
two treatments
th(xI ,x2,···,xm) and (Y1'Y2"."Ym) are (Cl. I ,Cl.2 , ,Cl.m)
associates, if
CI.. =0 or 1 according as X.=y. or not (i=1,2, ,m).1 1 1
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) =11111
f.(1,0,0) = 0, f.(O,l,l) = s.-I, and f.(1,I,l) = s.-2.1 1 1 1
1
2.2 Some results regarding C-matrix
Let the vxb matrix N= (n .. ) be the incidence matrix of the
design,1J
where is the number of times the .th treatment in the /h block.n
.. 1 occurs1J
Il .. = 0 or 1, since each treatment occurs at most once in each
block. The1J
reduced normal equation for estimating the treatment effects are
known to be:
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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
eigen-
vector (1,1, ... ,1)', and the variance-covariance matrix of Q
is Co2. We
first prove the following lemma.
Lemma 2.2.1 If t is a normalized eigenvector of C with non-zero
eigen-A 2
value e, then t'T is estimable and Var(t'T) = 0 18.
t'Var(t'T) = VarCe-o) = t' Q, 2 2e-ee 0 = 0 18.
Proof: Q,'C = 8t', since Q, is an eigenvector of C with
eigenvalue e.1 A 1 A 1
= e ~'CT, and hence 5(,'T is estimated by 5(,'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
. 21varlance 0 r. 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
Dennition 2. 2. 2
r times, if a treatment contrast t'T is estimated with variance
0 2/8, then
its efficiency is defined to be 8/r. When 8/r = 1, t'T is said
to be esti-
mated with full efficiency.
Kurkjian and Zelen (1963) introduced the following
defintion.
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 =
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10
where x is the Kronecker product, a. = 0 or 1, the h (0.1 ' ...
,am) are constants1depending on a. IS, J. is an s .x s. matrix with
all elements equal to 1, 1. is
1 1 1 1 1a. a.the identity matrix, and J. 1 J. if a. 1 and J. 1
1. if O.s.xs. = = = a. =
1 1 1 1 1 1 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 =
where
if a.1
a.1are constants depending on a. 's, and (J. - 1. ) =.J . -
1.
11111
= I. if 0,. = o.1 1
In fact, we can show the following relations between the
coefficients
of (2.2.2) and (2.2.3).
(2.2.4)
(2.2.5)
L h(al ,a2,···,a){(al, ... ,a ) :a.~I3. for all i} mmIl
L~_l(I3·-a.)\( -1) 1- 1 1 ( )L g 0.1 ' 0.2 ' ... , a{ ( ) 13 f
all I·} m0.1"",0. :a.~. orm 1 1
Lemma 2.2.2 Property A is equivalent to (2.2.3).
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. Rao
(1946), Bose 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
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11
construction of partially balanced arrays, renaming them
balanced arrays.
DenniHon 2.3. 1 Let A be an mxN matrix with elements 0,1,2, ...
,s-1.
Consider the s t ordered t-plets (x"! ,x2
' ... ,X t ) that can be formed from a
t-rowed submatrix of A and let there be associated a nonnegative
intger
A(X 1,x2 ' .. · ,xt ) that is invariant under permutations of xl
,x2 ' ,xt' If
for every t-rowed submatrix of A the stordered t-plets(x ,x2' ,x
) each occur1 tA(xl ,x2, ... ,xt ) times, the matrix A is called a
balanced array of strength t
in N assemblies, m constraints, s symbols, and the specified
A(xl ,x2,··· ,xt )
parameters. A is denoted by BA(N,m,s,t).
Definition 2.3.2 Let A be a BA(N,m,s,t) with parameters A(~l" ..
,Xt)=A
for all (x1,x2 , ... ,xt ). 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. We shall be
interested
in orthogonal arrays of strength 2 only. The definition of
resolvability
will be useful in this thesis.
Definition 2.3.3 2An 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 An OA(As2,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:
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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
2completely resolvable OA(AS ,As,s,2) can always be
constructed.
Adde1man and Kempthorne (1961) gave a method of constructing
an
n nOA(2s , 2(s -1)/(s-1)-1,s,2).
Theorem 2.3.2 n nIf 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 in-
dex 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
nabove (s -l)/(s-l) factors except Xl. The first half contains
all the
2(sn- l )/(s-1)-1 factors. The second half is suitably
constructed to make
it an orthogonal array.
n-1There are 2(s -l)/(s-l)-l factors which do not contain x in
then
OA(2sn ,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-1OA(2s ,2s ,s,2) can always be constructed.
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13
Finally we shall give the definition of a transitive array.
Definition 2.3.5 Let A be a BA(N,m,s,t) with parametersA(xl,···
,xt)=O
if x. = x. for some i~j, and A(x l , ... ,Xt ) = A if all x. 's
are distinct. Then1 J 1
A is called a transitive array of strength t in N assemblies, m
constraints,
5 symbols, and index A. A is denoted by TA(N,m,s,t).
In a TA(N,m,s,t), m ~ 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.
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CHAPTER III
SOME PROPERTIES OF BFE'S
3.1 Efficiencies of BFE's
Let F1,F 2 , ... ,Fm be the m factors at sl,s2" .. ,sm levels
respectively
and N be the incidence matrix of a BFE. To calculate the
efficiencies of
treatment contrasts of a BFE, we need to find out the
eigenvalues and
. f th Ct' r I -k_l~TN'.eIgenvectors 0 e -rna rlx ~~
We assume an extended group divisible association scheme defined
on the.
treatments since a BFE is an EGD/(2m-l) - PBIBD. Consider the
matrixa a a
(Jl-I l ) lX(J 2-I 2) 2x ... X(Jm-Im) m as defined in Equation
(2.2.3). The ele-
th thment, which is in the (x l ,x 2,· .. ,xm) row and (Yl'Y2""
,Ym) column of
the matrix (the treatments are in lexicographic order), is 1 if
treatments
th(X l ,x2,···,xm)and (Yl'Y2""'Ym) are (al ,a2 , ... ,am)
associates, and is 0
thotherwise. Two treatments which are (al ,a2 , ... ,am)
associates occur to-
gether in A blocks; hence we have the following lemma:a l a2 ·
•. am
Lemma 3.1.1 Let N be the incidence matrix in a BFE; then
(3.1.1) NN' =
where Aoo ... O is defined to be r.
Further let j. = s. x1 vector with all elements equal to 1;1
I
h.1
any s.xl vector orthogonal to j.;1 1
-
15
S. t1 if 8. = 1h. 1 1=1 if B. 0J. =1 1Then
a. B. 8.(3.1.2)· (J.,-!.) 1 h. I =H.(a.,B.) h. I
1 1 1 111 1
where H.(a. ,B.) is given by the following table:111
~ 0 10 1 s. -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 l ,8 , ...
,8 )'s2 m
with corresponding eigenvectorsm B.II (s. -1) 1 .
i=l 1
The multiplicity of
-
16
1 Bl B2 BmSince C = r I - INN', hI Xh2
x... xhm 's are also eigenvectors of C
1with corresponding eigenvalues r - k g(B l ,B 2,.·.,Bm)· If qof
Bi's are 1,
say B. ,B. , .. . ,B. , then it is easy to check that the
treatment contrastJ 1 J 2 J q
81 B2 Bm th(hI xh2
x... xhm )T belongs to the (q-l) order interaction between
Let E(Bl
,B2
, ... ,Bm) denote the efficiency of this treat-F. ,F. , ... ,F
..J l J 2 J qment contrast; then
Corollary 3.1.1
In an s l xs 2 BFE, Nair and Rao (1948) showed that the block
size k must
be a multiple of sl if the main effects of the first factor are
estimated with
full efficiency. We generalize the result to an m-factor
BFE.
Theorem 3.1.2 In an s l xs 2x... xsm BFE, if the main effects of
Fi are
estimated with full efficiency, then the block size must be a
multiple of s ..1
be the numbers of treatments in the jth block which are
Proof: Assume i = 1 without loss of generality. Let .R,. , and
.R,. ,J . J .
1 th 1at the i
land
iith
levels of Fl respectively. Counting the number ·of treatments in
the
jth block, we have
s -11
(3.1.5) L .R,. = ki =0 J.
1 1}
C . h b f d d . f h . th 1 1 fount1ng t e num er 0 or ere pa1rs
0 treatments at tel l eve 0
PI in all blocks, including pairs of the same treatment, we
have
(3.1.6)b
Lj=l
Counting the number of ordered pairs (x,y), where treatment x is
at the
-
17
i lth level and treatment y is at the ii
thlevel of Fl , in all blocks. we
have
(3.1.7)
Hence
b
Lj=l
~. ~. = 52
53
, .. 5mJ . J . ,
1 1 1 1
0. 2 amA1rv rv (s2- 1) •.• (s -1)~2···~m m0.2,., . ,am
(3.1.8)b
= Lj=l
2 b~. + IJi j =1
1
b~~ -2 LJ., J' =11
1
= 2 s s ... s. [ I (A. -A )23m rv rv 00.2 , .. a 10.2 , .. a~2'
. '~m m m
0.2 am(s2- l ) ... (sm- l ) ] = 2 5 25 3" .sm g(l,O, ... ,0)
If main effects of Fl are estimated with full efficiency,
then
,b 2g(l,O, ... ,O) = 0 by Corollary 3.1.1. Hence L.-l(~' -~. ) =
0, which
J- J. J. ,1
11 1
implies~. =~. for all j. Since i l and iI' are arbitrary,J . J .
,1
11
1
~j. = constant for all j = 1,2, ... ,b; i l = 0,1, ... ,sl-l.1
1
thereforeButs -1
L. 1_0
~. = k;1C J.
1 1is an integer.
k .s k must be a multiple of sl' since
Q.E.D
From the proof of Theorem 3.1.2, we know that each level of the
factor
Fl
occurs the same number of times in each block if the main
effects of Fl
are estimated with full efficiency. We shall consider the case
that the
first order interaction of Fl and F2 are estimated with full
efficiency.
Let ~. denote the number treatments in the j th block which are
atJ. .
1 11 2the i
lth level of F
land at the i
2th level of F2 ; also let ii 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)
Hence,
bl 51,. 51,.
j=l J i1 i 2Jil i 2
bI 51,. 51,.
j=l J i1 i 2Jiiiz
0.3 am= S3" 'Sm l A 1 (s3- 1) ... (s -1)a a3...ct. m
0.3
" .am m
(3.1.10)
If the first order interactions of F1 and F2are estimated with
full
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 l xs 2x... xs m BFE, if the first order
interactions
of F1 and F2 are estimated with full efficiency, then 51,.J. .1
11 2
If we also assume the main effects of Fl
and F2
are estimated with full
efficiency in addition to the first order interaction, then
summing over
-
19
all i 2, we have
But
since t. = toJ. J. ,1 1 1 1
s -1 k kI/=o t. =to =- Therefore to --J. . J. sl J 0 • 5 l s 22
1 11 2 1 1 1 11 2
for all iI' i 2 and j.
Corollary 3.1.2 In an s l xs 2x... xsm BFE, if main effects and
interactions
between Fl
and F2 are estimated with full efficiency, then the block size
k
must be a multiple of s l s2 and t j .. = k/s l s 2 for all iI'
i 2 and j.1 112
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 2x... xsm BFE, if the (q_l)th order
interactions
between Fl , F2 , ... , and Fqare estimated with full
efficiency, then
(3.1.12)
. ,1 .•
1
for all iI' i 2 , ... ,iq and j.
Corollary 3.1.3 In an slxs2x ... xsm BFE, if all main effects
and inter-
actions involving Fl , F2 , ... ,Fqare estimated with full
efficiency, then the
block size k must be a multiple of sls2" .Sq and tj.. . k/s l s
2... s q1
11
2, .. 1
q
3.2 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 ,x2, .. ·,xm)
and (Yl'Y2''''Ym)
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. = (~) (S-l)jJ J
j=1,2, ... ,m
(3.2.2)k
p ..1J
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 hyper-
cubic 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(J_I)a =. I
if a=l
if a=O
-
21
a l a 2 amFix j (O:::;j:::;m); consider the matrix L _. (J-1)
x(J-1) x ... x(J-I) .,a l +... +am-J
of which the element in the (Xl ,x2, ... ,xm)th row and the
(Yl'Y2'''''Ym)th
column is 1 if treatments (x1,x2 ,,,,,Xm) and (Yl'Y2""'Ym) are
jth associ-
d · 0 h' T' th . h .ates, an IS ot eTWlse. wo J aSSOCIate
treatments occur toget er In
~. blocks; if ~O is defined to be r we haveJ
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 ~ = ~. if a l +a2+... +a = j.m a l a
2 · .. am J m
Theorem 3.2.1 The NN' of a symmetrical BFE has (m+l) different
eigen-
m jvalues g(j) (j=O,l, ... ,m) with multiplicity (.)(s-l) ,
J
where
(3.2.4)m
g(j) = L P. (j ;m,s) ~.. 0111=
and 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)
for1
m=1,2,3, and 4 in the following table.
-
m=l
TABLE 1
Krawtchouk Polynomials Pi(j;m,s)
m=2
22
x 0 10 1 5-1
1 1 -1
~ 0 1 20 1 2(5-1)
2(5-1)
1 1 5-1 -(5-1)
2 1 -2 1
m=3
I;z 0 1 2 30 1 3(5-1)
2 33(5-1) (5 -1)
1 1 25-3 (5-1)(5-3)2
- (5-1)
2 1 5-3 -25+3 5-1
3 1 -3 3 -1
m=4
I>z 0 1 2 3 41 4(5-1)
2 3 40 6(5-1) 4(5-1) (5-1)
1 35-4 3(5-1)(5-2)2 .3
1 (5-1) (5-4) - (5-1)2
-2(5-1) (5-2)2
2 1 25-'4 5 -65+6 (5-1)
3 1 5-4 -3(5-2) 35-4 -(5-1)
4 . 1 -4 6 -4 1
-
Since C = r I - ..!. NN'k we have
23
Corollary 3.2.1 The eigenvalues of the C-matrix of a
symmetrical
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) IrkJ
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
2v = 3 , b = k = 6, r = 4, n l = n2 = 4, Al = 2, A2 = 3.
Blocks 1 2 3 4 5 6
01 00 00 01 00 00Treat- 02 02 01 02 02 01ments
10 10 11 10 11 11
12 11 12 11 12 12
20 21 20 20 20 20
21 22 22 22 21 22
gel) r + (s-2) Al - (s-l) A2 = 4 + 1'2 - 2·3 = 0
g(2) = r - 2A l + A2 = 4 - 2'2 + 3 3
El =1, E2 = 1 - 3/4'6 = 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
symme-
trical 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-
-
2pIe of s
24
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 defined1 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,
for1 1
qi = 0,1, ... ,mi ; i=1,2, ... ,h.
This design was shown by Shah (1958) to be a PBIBD with (ml
+l)(m2+1) ...
(~+l)-l associate classes. The association scheme defined on
the
ml m2 ~ .sl s2 ... sh treatments IS: two treatments (x l ,x2
,··· ,xm) and (Yl'Y2"" 'Ym)
are (al,a2, ... ,~)th associates if ai(i=l, ...•h) is the number
of times
x. f y. (j=ml+···+m. l+l .... ,ml+ ... +m.).J J 1- 1
The parameters of the association scheme are:
(3.3.1)
(3.3.2)
h m. a.1 1
n (a1 ' a2 ' ... ,~) = II ( ) (s . - 1)i=l a i 1
hII
i=l
Ct.= O.l, ... ,m.; i=1,2, ... ,h1 1
a.P 18.y.
1 1
alwhere P is defined by Equation (3.2.2).
8.y·1 1
Similar to Lemma 3.1.1 and Lemma 3.2.1, we can prove the
following lemma:
then
Lemma 3.3.1ml m ~Let N be the incidence matrix of a sl xs 2
2x .. ,xsh BFE.
-
•
(3.3.3) NN'
2S
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 +..• +Bm +... +m. 1+1 ml+···+m.1 1- 1
= a. for all i=l, ... ,h.1
for
Theorem 3.3.1 There are (ml+l) (m2+l) ... (~+l)
ml m2 ~the NN' of an 51 xS 2 X ••• xsh BFE. They are
different eigenvalues
(3.3.4)h
g(al,a2'···'~) = I Aa a IT Pa (a.;m. ,5.)n a a 1-'1' ··I-'h . 1
1-" 1 1 11-'1' .. l-'h 1= 1a.,B. = O,l, ... ,m.;111i=1,2, ...
,h
where Pa (a. ;m. ,5.) is the Krawtchouk polynomial given by
Equation (3.2.3).1-" 1 1 11
The multiplicity of g(al,a2'''''~) is n(al,a2' ... '~) given by
(3.3.1).
Corollary 3.3.1ml ~The eigenvalues of the C-matrix of an 51 x...
xsh BFE
are r - g(al , ... ,~)/k (a.=O,l, .. ,m.; i=l, ... ,h).1 1 The
efficiency of any con-
trast belonging to the interaction involving a. factors at s.
levels is1 1
(3.3.5)
-
26
Example 3.3.1 Consider the 22 x3 BFE with v = 12, r = 3, b = k =
6,
Blocks 1 2 3 4 5 6
000 000 000 010 010 010
Treat- 110 110 110 100 100 100ments
001 011 011 011 001 001
111 101 101 101 111 111
012 002 012 002 012 002
102 112 102 112 102 112
•
Hence, E(O,l)=E(l,O)=E(l,l)=l,8 5
E(2,0)= 9' E(2,1)= 9·
The balanced factorial experiment discussed in this section is a
genera1-
ization of those discussed in Section 3.1 and 3.2. When m1=m2=
... =~=1, the
association scheme defined on the treatments is the EGD/(2h-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.). Whena. 1 1 1 1 11
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. There-
fore 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 is called a t-ply
transitiven
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 boo~ by Carmichael (1937). The following
theorem is
usef~l 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 of
strength t with s symbols. 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 il,i
2, ... ,it from {O,l, ... ,s-l}, the set {(g(i
l),
g(i 2), ... ,g(i t)) :geG} contains any ordered set of t symbols
exactly A times
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 con-
struct a TA(s! ,s,s,2) with index A = (s-2)!.
Two examples follow which illustrate the application of Theorem
4.1.2.
-
Corollary 4.1. 1
s,s,2) .
29
If s is a prime power, then there exists a TA(s(s-l),
Proof: Let GF(s) denote the Galois field of order s. Consider
the
permutation group ort all the elements of GF(s). Let G be the
group consis-
ting of the following permutations:
(3.1.1) g(x) = ax+b a#O, a,bEGF(s)
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
re
we can construct a TA(s(s-l) ,s,s,2) by Theorem 4.1.2
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
Q.E.D.
The TA(s(s-1),s,s,2) constructed in Corollary 4.1.1 is
completely re-
2solvable, 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 If 5-1 is a prime power, then there exists a
TA(s(s-l)
(5-2),5,5,3). Moreover, if 5 is even, then we can always
construct a
TA(s(s-1)(s-2),s,s,2).
Proof: Let S be the set consisting of all elements of GF(5-l)
and the
element 00. Consider the symmetric group on the elements of S.
Let G be the
subgroup consisting of the elements
(4.1.2) g(x) = ax+b a#O, a,bEGF(s-l)
-
30
or
(4.1.3) g(x)a
= x+b + C a!O, a,b ,cEGF(s-l)
where 0 and + denote multiplication and addition defined in
GF(s·-l), and
a+OO = 00, a/oo = 0, and a oOO = 00 for all a!O. Then G is a
triply transitive
group of degree s and order s(s-1)(s-2); 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 transi-
tive 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 For s=6, we can construct a TA(60,6,6,2) in the
following:
o 0 1 122 3 344 5 51 1 2 2 4 400 5 5 3 3240 3 354 5 120 135540 1
1 2 3 0 4 24 2 305 354 2 110534 5 102 1 0 324
o 0 1 1 2 2 3 3 4 4 5 52 2 335 5 440 0 1 11 5 250 402 1 3343 4 4
0 3 1 1 5 2 5 2 05 1 5 240 2 0 3 1 4 343041351520 2
o 0 112 2 3 3 4 45533001 155 2 2444 5 240 3 0 1 3 5 1 221534 5 2
4 100 3544 2 3 0 1 0 5 3211 2 355 4 4 2 0 130
001 1 2 2 3 3 4 4554455001 1 332 21 334 1 5 250 204520243045 1 1
33 1 435 1 5 2 200 42 5 2 0 3 4 4 0 1 5 3 1
o 0 1 1 2 2 334 4 5 55 5 4 4 3 3 2 2 1 i 0 0230 5 1 4 1 405 2 34
1 3 2 500 5 2 3 1 43 2 504 1 4 150 3 214230 550 324 1
The above TA(60,6,6,2) is 2-reso1vab1e; we can add 12 columns
con-
sisting 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). It is well known
that
m ~2AS -1 2
[ 5-1 ] in an OACAs ,m,s ,2).72-1) .
Hence m ~ [~ ] = 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. He showed the following
results:
Theorem 4.1.3 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.
a a a a a 1 1 1 1 122 2 2 2 3 3 3 3 3 4 4 4 4 4 5 555 51 2 345 a
2 3 4 5 a 134 5 a 1 2 450 1 2 350 1 2 3 4214 5 334 5 a 250 134 145
2 a 2 5 3 a 1 430 2 1
4.2 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 implies111
the existence of an OA(N,m,s,t), where N = Nl ,N2 , ... ,Nk , s
= sls2 ... sk' and
m = min(ml
,m2
, ... mk).
-
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 implies111
the existence of a BA(N,m,s,t), where N = Nl N2... Nk , s = s l
s2' .. sk and
m = min(ml ,m2 , ... ,mk). 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)'
. . . , (aIt ' a2t ' . . . ,~t) ) =A(all' a12 ' . . , at ) A(an
' a 22 ' . . , a2t) . . . A(ak1 ' ak2 ' . . , akt) .
Proof: Let the BA(Nl,ml,\,t) be denoted by the mlxN l matrix A =
(aU)
and the BA(N2 ,m2,s2,t) be denoted by the m2xN2 matrix B = (b ij
). Let
Al and Bl denote the first m rows of A and B, respectively. Then
form
the mXNl N2 matrix:
...........
. (a 1,bmN ). • . (amN b 1)'m 2 l' m. (a
mNb
mN)
l' 2
which can be shown to be a BA(N l N2,m,sl,s2,t) with parameters
A((al,b l ), ... ,
(at,bt )) = A(al , ... at)A(b l, ... ,b
t). From this array by following the same
procedure with BA(N3 ,m3,s3,t), we get a BA(NlN2N3,m,sls2s3,t}.
Continuing
this procedure, we finally get a BA(N,m,s,t). Q.E.D.
Example 4.2.1 The product of the following two balanced
arrays
-
BA(2,2,2,2)
01
10
A(O,l)=l
A(O,O)=A(l,l)=O
is a BA(12,2,6,2)
BA(6,2,3,2)
012012
120201
A(0,0)=A(1,1)=A(2,2)=0
A(0,1)=A(0,2)=A(1,2)=1
33
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)=A1 or A2 according as x=y or
not. In
particular, we are especially interested in the
BA((ms-l)sA,ms,s,2) with para-
meters 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 pos-
sible 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
-
Proof:
34
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
2BA((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 pro-
duce 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)
OA(ms2
,ms,s,2) is equivalent to the existence of a BA(T)(m,s,l),
Proof: If a partly resolvable OA(ms 2 ,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
2partly 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 oc-
cur together in m-l blocks. Thus we have
Theorem 4.3.3 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
2OA(ms ,ms,s,2), then apply Theorem 4.3.2. Q.E.D.
Example 4.3.2 For m = 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
-
Corollary 4.3.4 If s
37
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
2OA(ms ,ms,s,2), then apply Theorem 4.3.2. Q.E.D.
Example 4.3.3 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
con-
struction 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=Al +A2(s-1), Al and A2 are
integers)
all a12 · a lN
a2l a22 · a 2N
with elements belonging to M such that among the differences of
the corre-
sponding elements of any two rows, the element 0 occurs Al times
and the
other nonzero elements occur A2 times, then by adding the
elements of the
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 appli-
cation 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
I
2
1
2
ooI
o2
2
o2
I
I
2
1
2
oo1
o2
o2
2
I
1
2
I
2
oo1
o
oo2
2
I
1
2
I
2
ooI
o 01 0
o 12 0
2 2
1 2
1 I
2 1
1 2
2 1
o 2o 0
o 0o 2o 01 0
o 12 0
2 2
I 2
1 I
2 1
1 2
2 1
000
1 2 1
2 1 2
021
o 0 21 0 0
010
2 0 1
220
122
112
211
o2
2
I
2
1
oo2
o1
1
o1
2
2
1
2
1
oo2
o1
000
I 0 2
1 1 0
211
221
122
212
121
012
001
2 0 0
o 2 0
o 0o 02 0
o 21 0
1 1
2 1
2 2
1 2
2 1
I 2
o 1
o 01 2
o 1o 02 0
o 21 0
1 1
2 1
2 2
1 2
2 1
o1
2
1
oo2
o1
1
2
2
o2
1
2
oo2
o1
1
2
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).
Example 4.3.5 Let M = {O,1,2,3}. Among the differences of the
corre-
sponding elements of any two rows of the following array, 0
occurs 4 times,
whereas 1,2, and 3 occur 6 times.
(1 0
3 0
3 3
2 3
3 2
1 3
1 1
2 1
o 22 0
1 2
o 1
o 0I 2
o 13 0
3 3
2 3
3 2
1 3
1 1
2 1
o 22 0
o 0 0 0o 2 1 12 0 2 1
120 2
o I 2 03 0 I 2
3 3 0 1
2 3 3 0
.') 2 3 3
132 3
1 1 3 2
2 1 3
o 03 2
1 3
1 1
2 1
o 22 0
1 2
o 13 U
3 3
2 3
o 03 1
2 1
3 2
1 1
1 3
2 3
o 22 0
1 2
o 33 0
000
o 3 21 0 3
1 1 0
211
1 2 1
3 1 2
331
233
023
2 0 2
3 2 0
oo2
3
o1
1
2
1
3
3
2
o2
o2
3
o1
1
2
1
3
3
o3
2
o2
3
o1
1
2
1
3
o3
3
2
o2
3
o1
1
2
1
o1
3
3
2
o2
3
o1
1
2
o 02 1
1 2
3 1
3 3
2 3
o 22 0
3 2
o 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 bal-
anced arrays which can be constructed by Corollary 4.3.1 and
cannot be con-
structed 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 can-
not 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 effici-
encies. These designs can usually be constructed by the balanced
arrays dis-
cussed in the previous chapter.
Let FI
and F2 be the two factors in a BFE at sl and s2 levels
respec-
tively. We can assume sl~s2 without loss of generality. Let N
denote the
incidence matrix of the BFE. 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:
(5.1.4) E(l,O) =(k-l)SI
k
-
41
(5.1.5) E(O,l) =(k-l)s2
k
If the main effects of Fl are estimated with full efficiency,
i.e.
E(l,O) = I, then the block size k must be a multiple of 51. We
shall assume
k = 51 throughout this section. For k = 5, Equation (5.1.4)
becomes
(5.1.6) E(l,O)
E(l,O) = 1 if and only if AOI = OJ that is, two treatments at
the same level
of Fl never occur together in the same block.
Theorem 5.1.1
.eIn an slxs2 BF[ with block size 51' the main effects
are estimated with full efficiency if and only if AOI = O. This
design is
equivalent to a BA(AlOs2+Alls2(s2-l), 51,52,2) with parameters
A(X,y) =
AlO or All according as x=y or not.
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 Fl ,
and the levels
of F2
respectively, then it is the specified BFE. Q.E.D.
E(O,l)
In proving Theorem 5.1.1, we don't really use the condition
sl~s2;
hence it is true for all 51 and 52.
For k=s, and AOl=O in Equation (5.1.5),
(sl-1)s2AU=-.".-.,.-.::.--.-.~=-=:.;-~
sl[(s2-l)All+AlO](5.1. 7)
(51-1)52E(O,l) has the maximum value5
1(5
2-1) when AlO = o.
-
Theorem 5.1.2
42
In as slxs2 BFE with block size sl(sl~s2)' if the main
effects of F1 are estimated with full efficiency and the main
effects of F2(sl-l) s2
are estimated with maximum efficiency then the BFE has
parameterssl (s2- 1) ,
A10 = A01 = 0 and All r O. This design is equivalent to a
TA(A11s2(s2-1),s1' s 2,2) •
Since A10 = 0 means that two treatments at the same level of
F2do not
occur together in the same block, which implies s2 ~ k = sl' we
do not need
sl ~ 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.
Example 5.l.1 A 3x4 BFE with b=12, k=3, r=3, A01 =\0=O, and
A11=l,
can be constructed from a TA(12,3,4,2).
Blocks 1 2 3 4 5 6 7 8 9 10 11 12
Levels Levels of F2of. F1
0 0 1 2 3 0 1 2 3 0 1 2 3
I 1 1 0 3 2 2 3 0 1 3 2 1 0j 2 2 3 0 1 .3 2 1. 0 1. 0 3 2
8 5E(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, A01=A10, and A11=1
can
be constructed from a TA(30,3,6,2) in Example 4.1.3. The
efiiciencies are
-
43
4 3E(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 para-
meters Al=O and A2,O. This design is equivalent to a
TA(A2S(S-1),s,s,2).
Example 5.1.3 If s is a prime power, then there exists a
TA(s(s-l),
s,s,2) by Corollary 4.1.1. 2Hence we can always constrcut an s
symmetrical
BFE with r = s-l, b = s(s-l), k=s, Al=O, A2=1, El=l, and E =
s-22 s-l·
Exampl e 5.1. 4 A 62
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.
The effici-2
encies are El =1 and E2= i .
Similarly we can construct 102, 122, 142 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 F2 are estimated with full efficiency,
then the
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)=AOI or
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 F2, and the
levels of
Fl of the design g
Example 5.2.1
can be constructed from an OA(4,3,2,2).
-
44
Blocks 1 2 3 4
Levels Levels of Flof F2
I0
I0 0 1 1
1 0 1 0 1;
i 2 0 1 1 0
2E(O,l)=l, E(l,O)=E(l,l)="3 in this design.
For AlO = 0 and k = 52' Equation (5.1.4) becomes
(5.2.1) E(l,O) =1
Note that AOl~O' since k = 5 2>5 1 implies that at least two
treatments in a
given block are the same level of Fl , 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 -2When the equality holds, E(l,O)=l and E(l,l) =.2-..1
•s2-
Proof: g(O,l)=O in this BFE, since the main effects of F2
are esti-
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 effici-
(5.2.5)
ency. When s2 = ms l , Equation (5.2.2) becomes
A...2.!. > m-lAll - m
Corollary 5.2.1
fects of Fl and F2 are estimated with full efficiency, if and
only if 52 =ms l ,
AlO=O, and AOI/All = (m-l)/m for some m. This design is
equivalent to a
BA((ms l -l)s,A,ms l ,sl,2) with parameters A(X,y) = (m-l)A or
mA according as
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 mslxsl
BFE such that
all the main effects are estimated with full efficiency, but a
large repli-
cation 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 2 3 4 5 6
Levels Levels of Flof F2
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
2E(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 Fl cannot be estimated with full efficiency.
To estimate
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 Fl
to the rows from the (iq+l)th to the (i+l)qth, where
i=O, ••• ,ps-l. Identifying columns and symbols with blocks and
the levels of
F2
, we get a psxqs design with block size pqs.
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'.
Assume r'=(qs-l)m
and A'=(q-l)m, where m need not be an integer. Let AOl ' AlO'
All denote
the parameters and r denote the number of replications in the
psxqs design.
-
47
Then
AOI
= (ps-l) (q-l)mn = (ps-l)nA'
(5.3.1) AlO = (p-l) (qs-l)mn = (p-l)nr'
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-land 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, isq
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. Consider the
following
resolvable BIBD with 4 treatments and block size 2.
-
48
Xo Xl YO YI Zo ZI
0 2 0 1 0 1
1 3 2 3 3 2
where XO' Xl' YO' Yl' ZO'and ZI represent the blocks. Also
consider the
BA(T)(3,2,1) given below:
0 0 0 0 0 1 1 I 1 1
0 0 1 1 1 0 0 0 1 1
1 0 0 1 1 I 1 0 0 0
0 1 1 0 1 0 1 1 0 0
1 1 1 0 0 1 0 0 0 1
1 1 0 1 0 0 0 1 1 0
By Theorem 5.3.1, we can construct a 4x6 BFE with k=I2, r=15,
b=30, AIO=S,
14AOI=6, All=8, E(O,I)=E(I,O)=I, and E(I,I) =15 •
Blocks I 2 3 4 5 6 7 8 9 10 1l 12 13 14 15
Levels Levels of FIof F2
0 Xo Xo Xo Xo Xo Xl Xl Xl Xl Xl YO YO YO YO YO1 Xo Xo Xl Xl Xl
Xo Xo Xo Xl Xl YO Yo YI YI YI2 Xl Xo Xo Xl Xl Xl Xl Xo Xo Xo YI YO
YO YI YI3 Xo Xl Xl Xo Xl Xo Xl Xl Xo Xo YO YI YI YO YI4 Xl Xl Xl Xo
Xo Xl Xo Xo Xo Xl YI YI YI YO YO5 Xl Xl Xo Xl X Xo Xo Xl Xl Xo YI
YI YO Yl YO0
-
49
Blocks 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Levels Levels of Flof F2
0 Yl Y1Yl Yl
Yl Zo Zo Zo Zo Zo Zl Zl Zl Zl Zl
1 YO YO YO Yl Yl Zo Zo Zl Zl Zl Zo Zo Zo Zl Zl
I 2 Yl Yl YO YO YO Zl Zo Zo Zl Zl Zl Zl Zo Zo Zo
3 YO Yl Yl YO YO Zo Zl Zl Zo Zl Zo Zl Zl Zo Zo
4 Yl YO YO YOYl Zl Zl Zl Zo Zo Zl Zo Zo Zo Zl
5 YO YO Yl Yl 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, AOl =15. AlO=14,
All=18,
34E(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:
0 I 0 2 0 I o 4 0 I 0 2 o 1
2 3 I 3 3 2 I 5 2 3 I 3 3 2
4 5 4 6 4 5 2 6 5 4 6 4 5 4
6 7 5 7 7 6 3 7 7 6 7 5 6 7
and the BA(T) (3,2.1) in Example 5.3.1.
Example 5.3.3 A 6x9 BFE with k=18, r=20. b=60. AOl =5, AIO=4.
AIl=7,19E(O.l)=E(l.O)=l, and E(l,l) = 20. Use the resolvable BIBD
with 9 treatments
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, AOl =14,
AlO=ll, A11=20,
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 21 6
3 8
4 7
5 10
9 11
1
2
4
358
760
9 10 11
2 4
3 8
5 10
6 9
7 1
o 11
3
4
6
5
9
o
7 10
8 2
1 11
4 6
5 10
7 1
8 0
9 3
2 11
5
6
8
7 9 1
o 10 42 3 11
6 8 10 2
7 105
9 3 4 11
7 9
8 2
10 4
o 31 6
5 11
8 10
9 3
o 5
1 4
2 7
611
9 0
104
1 6
2 5
3 8
7 11
10 1
o 52 7
3 6
4 9
8 11
and the BA(T) (2,4,1) which can be constructed by Corollary
4.3.3.
o 1 2 3 012 310322301
23010123
32102301
o 123 1 0 3 21 0 3 2 3 2 1 0
23011032
32103210
012 3 012 3 0 123 0 123 012 3
3 2 1 0 0 123 1 0 3 2 230 1 3 2 1 0
230 1 1 0 3 2 3 2 1 0 1 0 3 2 3 2 1 0
1 0 3 2 1 0 3 2 230 1 3 2 1 0 0 123
1 0 3 2 3 2 1 0 3 2 1 0 230 1 230 1
230 1 3 2 1 0 230 1 0 123 1 0 3 2
3 2 1 0 230 1 0 123 3 2 1 0 1 0 3 2
012 3 230 1 1 0 3 2 1 0 3 2 230 1
5.4 s2 s~etrical BFE's with block size ns (l
-
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
(5.4.1)
Al = £mn(nq-1)
2A2 = £(mn q-A)
r = £mn(pq-l)
where A = mn(nq-1)/(pq-l) is the number of blocks two treatments
occur to-
gether in the BIBD. The efficiencies of the BFE are El=l and
E2=1 - p-n 2.n(pq-l)
Corollary 5.4.1 If 5 is a prime power, then there exists a
symmetrical
2 25 BFE with block size 5(5-1), 5(5-1) blocks and parameters Al
= 5 -35+2 and
2A2 = 5 -35+3.
Proof: For any integer 5, there exists a BIBD(s,s,s-I). For 5 a
prime
power, there exists a TA(s(s-1),s,s,2). No~ apply Theorem 5.4.1
with £=m=q=l,
p=s, and n=s-l, and we get the 52 BFE. Q.E.D.
In the following2we give examples of 5 symmetrical BFE's with
s~7 which
can be constructed by Theorem 5.4.1.
Example 5.4.1
(see also Example 3.2.1). Use the BIBD(3,3,2)
and the TA(6,3,3,2)
o1
2
1
2
o
o1
2
o1
o2
o2
1
1
2
1
o2
2
1
o
-
52
Example 5.4.2 42 symmetrical BPE's
a)
Use the resolvable BIBD(4,6,2)
o 2 0 1 0 11 3 2 3 3 2
and the BA(T)(2,2,1) in Example 4.3.1.
8=-
9
b)
Use the BIBD(4,4,3)
1 0
2 2
3 3
o 01 1
3 2
and the TA(12,4,4,2) which can be constructed by letting s=4
in
Corollary 4.1. 1•
Example 5.4.3 52 symmetrical BPE's
a)29k=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) k=15, b=40, r=24, /"'1=12, /"'2=15, E1=1, and23
E2=2'4 •
Use the 3-reso1vab1e BIBD(5,10,3)
0 1 2 3 4 0 1 2 3 4
1 2 3 4 0 1 2 3 4 0
2 3 4 0 1 3 4 0 1 2
-
c)
and the same TA(20,5,5,2) in (a).
63k=20, b=20, r=16, 1..1=12, 1..2=13, E1=1, and E2= 64 •
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) •
53
Example 5.4.4 A 62
BFE with k=12, b=75, r=25, 1..1=5, 1..2=9, E1=1,
23and E2=2s. Use the resolvable BIBD(6,15,2)
0 2 4 0 1 3 o 1 2 0 1 2 0 1 31 3 5 2 4 5 3 5 4 4 3 5 5 2 4
and the BA(T)(2,3,1) in Example 4.3.3.
Example 5.4.5 72 symmetrical BFE's
a) k=14, b=126, r=36, 1..1=6, 1..2=11, E1=1, and67
E2="72 •
Use the 2-reso1vab1e BIBD(7,21.2)
0 1 2 3 4 5 6 0 1 2 3 4 5 6 0 1 2 3 4 5 6
1 2 3 4 5 6 0 2 3 4 560 1 3 4 5 6 0 1 2
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..2=8, E1=1, and E2=2"7 •
Use the BIBD(7,7,3)
o 1 234 5 61 342 064
2 5 5 6 3 0 1
and the same TA(42,7,7,2) in (a).
-
S4
1..2=14, E1=1, and47
(c) k=28, b=42, r=24, 1..1=12, E2= 48 .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
6.1 sm symmetrical BFE's with block size s
mThe s 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)
Hence
(6.1.2)
m .\' m 1r(s-l) = L (.)(s-l) A.•
i=l 1 1
m m i-Ir = L (.)(s-l) A.•
. III1=
r is completely determined by the values of A. 'sO When s is a
prime1
mpower, we shall show that there exists an s symmetrical BFE
with block size
s for any given 1.1,1.2"'" and Am'
Lemma 6.1.1 If s is a prime power, then given j (l~j~m) there
exists
man s symmetrical BFE with block size s and parameters 1..=1,
1..=0 for all ilj.
J 1
Proof: Given a.(i=l, ••• ,m-l), b.(fO, i=m-j+l, ••• ,m-l) in
GF(s), con-I 1
sider the block containing elements (x l ,x2, ••• ,xm)
satisfying the following
set of equations:
-
56
(6.1.3) x . = a .m-J m-J
x . 1m-J+b .x +am-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 lettingm-J m-J+ m
a. 's and b. 's run through all possible values in GF(s), we
obtain Sm-l(s_l)j-l1 1
blocks with the above property.
Let {il,i , .. .,i .} be any m-j integers from {l,2, ••• ,m}. We
can simi-2 m-J
m-l j-llarly construct s (s-l) blocks which contain treatments
at the same
levels of the factors F. ,F. , ••• ,F. and at different levels
of the other11 12 1.m-J
f h f { .. .} m-l j -1 kactors. For eac 0 11,1 2,.0.,1 . we
canstruct s (s-l) such bloc s,m-J
and let the design consist of all (~)sm-l(s_l)j-l such blocks.
Then in thisJ
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) fac-
tors at different levels will not occur in the same block. This
is the sym-
metrical BFE with parameters A.=l and A.=O for all ifj. Q.E.D.J
1
The efficiencies of the symmetrical BFE constructed in Lemma
6.1.1 can
be calculated by Equations (3.2.2) and (3.2.4).
and Equation (6.1.4)
i=1,2, ••• ,mP.(i;m,s)
J1E. = 1 - -1 S (~) (S_l)j-ls
J
I . 1 h . P (. ) (_l)i(s_l)m-in part1cu ar, w en J=m, l;m,s
=m
(6.1.4 )
becomes
-
(6.1.5)1
E. = 1 - - -1 S i-I(s-l) s
i=1,2, ••• ,m
57
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 If s is a prime power, then for any given Al
,A2
, ••• ,Am
Proof:
mthere exists an s symmetrical BFE with block size s and
parameters Al
,A2
,
••• , A Qm
Let D. denote the design constructed in Lemma (6.1.1.) TheJ
symmetrical BFE consists of Aj Dj'S for j=1,2, ••• ,m has
parameters Al '\2, •• ,Am•
Q.E.D.Now consider the case when s is not a prime power.
2In an s 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(A2s(s-1),s,s,2).
m . 1In the case of an s symmetrlca BFE with block size s, if we
can con-
struct a design with parameters A #0, and A.=O for i=l, •••
,m-l, then the mainm 1
effects are estimated with full efficiency. If a
TA(As(s-1),s,s,2) exists,
we can multiply (see Theorem 4.2.2) m-l such transitive arrays
to get a BA
m-l . m-l([As(s-l)] ,s,s,2) wlth parameters A((xl ••••
'xm_l)'(yl' •••• Ym_~) = A
if xi r Yi for all i=l, ••• ,m-l, and A((xl, •••
,xm_l)'(Yl""'Ym_I)) = °other-wise. Identifying rows with the levels
of F
I, symbols with the levels of
F2
, ••• ,Fm
, and columns with the blocks, we obtain an sm symmetrical BFE
in
[As(s_l)]m-l blocks of s plots each with parameters A =Am
- l and A.=O form 1
i= 1, ••• ,m-l. Thus we have the following theorem.
Theorem 6.1.2 The existence of a TA(As(s-l) ,s,s,2) implies the
existence
mof an s symmetrical BFE withm-l m-l m 1
b = [As(s-l)] , k=s, r=[A(s-l)] , 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
constructed.2 .
We have seen in Theorem 6.1.2 that from m-l s BFE's we can
con-
mstruct an s BFE by using the product of balanced arrays. In
general, the
product of balanced arrays can generate many efficient
BFE's.
Theorem 6.2.1 If there exists BA(N.,s ,s.,2)(i=1, ••• ,m-l) with
para-1 m 1
i imeters Ai (x'Y)=~O or ~1 according as x=y or not, then there
exists an slxs 2
x ••• xs BFE with k=s , b=Nl···N 1m m m- ,
Proof:
1 2 m-l= ~ ~ ••• ~ , where a.=O or 1.a
1a
2a
m_
l1
Multiply the given m-l balanced arrays to obtain a BA(N1N
2•••
Nm_l ,sm,s l s2 •••sm_l,2) with parameters A((x l ,x2, ••• ,xm_l
)'(Yl'Y2""'Ym_l))
1 2 m-l= ~ ~ ••• ~ where a.=O or 1 according as x.=y.or not.
Identifying symbols
al
a2
am
_l
1 1 1
with the levels of Fl
,F2
, ••• ,Fm
_l
, rows with the levels of Fm
, and columns
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((x1,x2), (Yl'Y2)) = o
or 1 according
as x2
=Y2
or not. By Theorem 6.2.1, this corresponds to a 2x3x3 BFE wi th
k=3,
b=24, r=4, A011 =A111=1,
A001=A010=Al00~Al10=A101=O,E(O,l,O)=E(O,O,l)=l,
E(l,O,O)=E(l,l,O)=E(I,O,l)=E(l,l,l)=~ 1and E(O,l,l)= 2 •
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 21E(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 62
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 Fl
,F2
, ••• ,F at s ,s ,m 1 2
••• ,sm levels respectively, each of the v*(=sls2 ••• sm)
treatments repli-
cated 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
(6.2.2) N* = [ rA~I rA* . I••• I rA* .]pq j=l J j=l q+J j=l
pq-q+J
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 1••• F at q,s l' ••• 's levels respectively in b
blocksm+ m+n m+ m+n
of k plots each. Also let there be two BFE's with incidence
matrices N*
If the
replaced by the block A. .(j=1,2, ••• ,q) in1q+J .
each of the treatments of N, then the design obtained by
adjoining the p de-
and N* as given by Equations (6.2.1) and (6.2.2)
respectively.pq
level j-l of the factor FO
is
signs 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 effi-
cient. 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 fac-
tor, the blocks, and the levels of the first factor respectively
in the
where Xo' Xl' X2, YO' Yl , Y2 represent blocks. Then by Theorem
6.2.2, we
can construct a 32
X6 BFE with r=IO, k=18, b=30, A20=5, AOI=2, A
21=3, A
II=4,
9AIO=O, E(2,1)=iO and all the main effects and first order
interactions are
estimated with full efficiency. The BFE is given below:
Blocks 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Levels Levels of FI
and F2of F3
a Xo \ X2 Xo Xl X2 Xo Xl X2 Xo Xl X2 Xo Xl X21 Xo Xl X2 Xl X2 Xo
X2 Xo Xl Xl X Xo X2 Xo Xl22 Xl X2
Xo Xo X X2 X2 Xo Xl X2 Xo Xl Xl X2 Xo13 X2
Xo Xl X2 Xo Xl Xo Xl X2 Xl X2 X
