... ~'
~'~
•
UNIVERSITY OF NORTH CAROLINA
DepartQent of Statistics
Chapel Hill" N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D•. C •
AliDSR Report No.
CONTRIBUTIONS TO Tim CONSTRUCTION
.AND
ANALYSIS OF DESIGNS
by
Jagdish Narain Srivastava
July" 1961
•
Contract No. A:F 49(638)-213
A generalized partially balanced association schene
has been defined and has been shawn to follow So
linear associative algebra. The use of tIns association scheme in many directions has been pointed
out. Multidimensional partially balanced designs
have been introduced. The theory of analysis of
fractional replications has been developed and
several methods of construction of fractions have
been given. Particular reference has been mde to
....m n
~
x3
factorials.
Qualified requestors rJay obtain copies of this report from the
ASTIA DocUtlent Service Center, Arlington Hall Station" Arlington
12, Virginia. Department of Defense contractors DUst be established for ASTIA services" or have their "need-to-knowlt certified
by the cognizant llrl.litary agency of their project or contract.
Institute of Statistics
Mimeo Series No. 301
i
.
ACKNOWLEDGMENTS
•
It is a
l~tter
of great privilege and pleasure for me to ex-
press my deep gratitude and heartfelt thanks to Professor Raj
Chandra Bose for his encouragement and guidance during the course
of this work, particularly for the inspiration that I get in
working intimately with him.
To Professors W. J. Hall, and S. N. Roy, I
am
thankful for
going through the manuscript of the thesis and for their suggestions, and again to the latter for his constant
inspirati~
for
my research work.
I am also thankful to the United States Air Force for financial help, and to Professor G. E. Nicholson, Jr.
and the Faculty
for agreeing to offer me a research assistantship fram September
1959 though I had applied too late for the same.
To Mrs. Doris Gardner I am very thankful for her skillful
typing of my manuscript.
To Miss Martha Jordan I extend thanks
for various forms of aid and advice during the past two years.
i1
TABLE OF CONTENTS
•
Chapter
I
II
Page
ACKNOWLEDGMENTS
1
INTRODUCTION .AND SUMMARY
1
GENERALIZED PARTIALLY BALANCED ASSOCIATION
SCHEMES AND THE CORRESPONDING LINEAR
ASSOCIATIVE AlGEBRAS • • • .. • • •
5
1.
Preliminary remarks . • • . • • •
5
2.
Generalized partially balanced
association schemes • • • • • •
5
3. Properties of certain matrices arising
in connection with factorial experiments.
4. Multidimensional partially balanced
designs · . . . . . · . . . . . . . . . . . .
.
5. Product association schemes and the
corresponding product algebras. • •
III
20
39
.....
.. .
FRACTIONAL BEPLICATIONS OF FACTORIAL
EXPERIMENTS • • • • • •
51
1.
Preliminary remarks •
51
2.
Properties desired in fractional
replications • • • • • • • • • • •
... . ..
51
3. Connection between the classical
approach to factorial experiments
and the response surface approach
4.
53
General remarks on fractional
replications . • • • • • • • .
.....
60
5. Analysis of fractionally replicated
designs for symmetrical factorials.
66
6. Fractional designs of the classes
..
m
2
and'
n
. . . . . . . . . . .
......
76
iii
Chapter
..
Page
7.
IV
Analysis of 2m x 3n aSYLJIlletrical
factorial fractions . • • • • • • •
CONSTRUCTION OF FRACTIONAL REPLICATIONS
OF THE ASYMMETRICAL FACTORIAL EXPERIMENTS. • • • • 107
1.
Preliminary remarks .• • • • • • • • •
· • 107
2.
The methods of associated vectors
and truncated geometries - Methods I,
II and III • • • • • • • . • • • • • •
· . . . 109
3.
Use of various types of arrays Methods IV and V .
•
· · · · · · · · · · · · · 117
4. Application of Methods IV and V
to the construction of fractional
replicates of the general asymmetrical
factorials •
• •
..
e
·····
5.
Use of quadrics - Method VI
6.
An
7·
Formation of blocks.
8.
earlier method
·
····
•
···•
· · · · 129
•
··
· 130
134
·n · • · · • · 134
Illustrations from the 'tfU x 3
factorials . • • .
· · · · · · · · · · · • · • 135
•
CHAPTER I
Introduction and S1.Uill1lary
•
The subject of design of experiments, founded by Fisher ["217*
stands out as one of the important branches of statistics.
At the
initial stages, Yates ["59~6g7 contributed to the theory of incomplete
.
block designs and the theory of factorial designs.
The subject may be broadly divided into two :parts, vlh1ch are
ever very intimately connected 'With each other.
how~
These are (i) construc-
tion of designs, (ii) analysis of designs.
Bose ["3-127, pioneered in the development of systematic methods
for the construction of designs and 'Was the first to systematically
attack them.
Bose and Nair ["27
defined :partially balanced incoIlIp;Lete
block (PBIB) designs vThich cover almost all know designs except the
intra and inter group balanced (11GB) designs, which were later defined
by Nair and Rao ["4];7.
The analysis has generally been supplied by the authors 0.10:13 with
the designs.
The analysis in the general case, which is a part of the
theory of linear estimation and testing of linear hypothesis has been
treated by Bose in llg7.
As the list of referencES at the end of the
thesis shows (although it is a vety incomplete list) various other
workers, for example, Tang
*
1 527,
Rotelling £28, 22.7, Rsu £32.7, Rao
Square brackets refer to the list of references, presented at the end
of the thesis.
in the list.
Figures inside the brackets refers to the serial numbers
2
.L46 , 4§J,
Tukey
.L5§l,
Scheffe
.L5}!7
and Duncan
.L25.7,
made signi-
ficant contributions to the subject.
It is a cammon experience that in most of the agricultural, biological and industrial experiments for which designs are constructed,
more than one measurement is taken on each experimental unit.
is to say, the response is in general multivariate.
That
S. N. Roy.L55.7
was the first to establish the needed links between the theory of
multivariate analysis and the theory of design of experiments.
wi th
the help of Roy's union intersection principle
.L5]},
Further,
and the
theory of intersection tests, it has been possible to give the analysis
of designs in cases where the hypothesis regarding the treatments is
kno'WIl to have some physical structure behind it ["5'2.7.
A very ioportant advance in the theory of factorial experiments
was made by Box and his collaborators
.L14-l17
in their work on the
theory of response surfaces.
The theory of construction of symmetrical factorial design was
developed by Fisher ["24-2§}, Bose and Kishen
L}!7,
and Bose
L5,
§J
Sufficient conditions leading to the construction of the general
asymmetrical designs were given by Rao and Nair
["'9,
40, 45.7.
The
generalization of Bose's theory of construction of symmetrical factorial designs to the asymmetrical case 'Was made by Kishen and
Srivastava
L,4,
'~7,
who also gave optimum solution for almost all
practical cases.
Finney L~7 started the theory of fractional replications,
which was later deve loped by many authors as pointed out in the introduction to Cooper IV.
3
Rao £47, 49,
59.7
introduced orthogonal arrays J which have be-
come an integral part of the theory of construction of designs.
These
arrays are useful in many branches of the subject including the theory
of construction of fractional replications.
The theory of partially balanced designs and the corresponding
association schemes has been studied 111' dcta.il by Bose p,p.d
Shinmoto
["27,
etc.
£§.7 ,
Bose and Clatworthy
["19.7
and Bose and Shrikhande
The linear associative algebras connected with these
schemes were studied by Bose and Mesner
["1"27,
who obtained many other
important results in that connections.
References to some other workers in construction of designs will
be found at the end £2, 18,
3§.7.
In this thesis, in each chapter
some discussion bas been made by way of background material.
This is
followed by a development that is basically original, as far as the
author is aware.
Here we shall consider the broad features of the
thesis, the motivation behind it and the usefulness of the results
contained in it.
The second chapter discusses lIGeneralized Partially Balanced
Association Schemes".
All previously known association schemes in-
cluding those of PBIB designs and IIGB designs are particular cases
of the generalized scheme.
The factorial designs too are shown to
follow the generalized association scheme.
uni-ce and cover all the known designs.
the construction of designs for the
multivariate.
This scheme appears to
It may also be helpful for
case where the response is
It has also been used to define a class of multidi-
4
mensional partially balanced designs of '\lhich both PBIBD and IIGBD
are particular cases.
In other words, a development is given that
subsumes all main previous developments as special cases, and furthermore, goes beyond these.
Further study of the generalized partially
balanced scheme appears to have great potentialities in the development of Combinatorics.
The third chapter is devoted to the analysis of fractional replications.
One section deals with the connection between the theory of
fractional replications and the theory of response surfaces.
In the
other sections, the problem of evaluating the matrix (M say) to be
inverted for solving the normal equations has been solved. A near
orthogonal series of fractions' for the 2n factorial has been given.
Properties which the fractions should possess, in order that the matrix
M may be simplified, have been obtained. Analysis for balanced fracm
n
tions from the 2 and 3 series have been given. Apart from the
above, many other results bave been presented which will be found useful in any discussion of the construction or analysis of fractions of
factorial designs.
The last chapter gives a brief account of a number of methods
for the construction of fractions of the
general asymmetrical facm
n
torial design with a special reference to 2 x 3 factorials. A
full investigation of these methods is beyond the scope of this thesis,
and will be produced later.
In developing these methods, use bas been
made of the results obtained in the previous chapters.
CHAPTER II
GENERALIZED PARTIALLY BALANCED ASSOCIATION SCHENES
.AND THE CORRESPONDING LINEAR ASSOCIATIVE ALGEBRAS
2.1.
Preliminary remarks.
In this chapter we shall consider certain association schemes
which are generalizations of the association schemes of partially
balanced designs, studied in
[""l"d.7
by Bose and Mesner.
These new
generalized association schemes and the linear algebras (for a treatment of linear algebras, one may see, for example, Macduffe
[""317 )
corresponding to these schemes are connected in various ways with confounded factorial designs, fractional replications, PBIB designs and
MDIB (multidimensional) designs introduced by Pothoff
["427.
are also connected with the inversion of patterned matrices.
They
In this
chapter, these connections shall be pointed out with varying amounts
of detail, and certain necessary conditions for the existence of these
association schemes will be obtained.
2.2.
Generalized partially balanced association schemes.
2.2.1.
Corresponding to a PBIB design, Bose and Mesner have introduced
the following association scheme
Given a set of v
objects
1,2, ••• v, a relation satis-
fying the following conditions is said to be an association scheme with
ill
classes: --
(a)
Any two objects are either the first, second, or •••
mth asso-
ciates, the relation of association being symmetrical, i.e. if the ob-
6
ject a
is the
ith associate of the object
~,then
~
is the
ith
associate of a.
(b)
Each object
independent of
(c)
0:
has
n
ith associates, the number n.
i
~
being
0:.
If any two objects
0:
are
ith associates, then the
number of objects
~hich
ciate of a
P~k and is independent of the pair of ith asso-
ciates
0:
is
and
are
and ~
jth associates
of
~
and kth asso-
~.
Suppose now that instead of one set of objects there are m
sets of
objects
Sl' S2' ••• Sm'
X
Definition 202.1.1.
i2
the objects in the
ith set being
, ••• , x.
The class
~ni
jj
of
...
the sets
be said to have a partially balanced association if the following conditions are satisfied:
C (i)
vath respect to any x
€ Si'
the objects of Sj can be
ia
disjoint classes (n .. > 0), where each element of
divided into n .
iJ
~J
the ath class is the ath associate of x
objects in the ath class is
.
in Sj' The number of
ia
n~. (i and j may take any value between
~J
1 and n, and may in particular be identical).
The number nij
is
independent of xiao
C (ii)
The relation of association is symmetrical, i.e. if x jb
is the kth associate of x
ciate of x jb
i and j.
in Si'
in Sj' then x ia is the kth assoia
It follows from this that nij = n ji for all
7
Let 5 , 5" and 5 be any three sets in JJ , 'Where
i
k
J
.
i, j, k are not necessarily distinct. Let x
€ 5 , and x
jb € 5 j •
i
ia
Let x"b be the ex- th associate of xi
in 5 oJ so that xi
is the
J
.
a
J
a
cx..th associate of x jb in 5 • Consider the class
(x ' 13, k) of
ia
i
the ~-th associates of x
in 5k~ and the class
jb , r, k) of
ia
the r~th associates of x
is ~. Then' the number of objects comjb
mon to these two classes in a constant
c{iii)
1-1
1-1(x
~, r)
P(i, j, ex, k,
i, j, k, ex, 13 and r
dependent only on
and independent. of the pair
x.~a and x jb with which we start so long as Xia € 51 and x jb
and x
ia and x jb are ex-th associates.
It is clear from the above definition that
p(i, j, ex, k, 13, r)
Definition 2.2.1.2.
=
€
5j
p(j, i, ex, k, r, 13).
A class
:Jj of m sets 51' 52' ••• 5m will be
said to have a partially balanced association, if conditions Care
satisfied, and if so, '\'1e shall express this by writing
.Lp~7
jlj
e .Lp~7 .
is therefore the aggregate of all classes which are par-
tially balanced.
This also includes all classes which have only one
set as a member.
In case a class
C contains only one set
C e .Lp~7, then we shall also (loosely) write
Definition 2.2.1.3.
Let
eB
€
5
€
5, and
.Lp~7.
Induction matrix
LP~7, and contain the m sets 51' 5:2.' ••• 5m• Then
if corresponding to any member in the set
8 (i=1,2, ••• m), all memi
bers in the set5 j (j=1,2, ••• m) are divided into nij associate
8
classes, we shall say that the set
8.
.
on the set
8
j
The m x m matrix (n .. )
•
~J
matrix of the class
\'le
induces
~
n..
~J
associate classes
will be called the induction
cJj •
shall also denote by n., the number of elements in the set
J
S.(j=1,2, ••• m).
J
The above will be illustrated by the example which follows:
EXAMPLE:
Let the symbols N, P, K, and L denote the four fertilizers
Nitrogen, Phosphorus, Potash and Lime (which is sometimes used for acidic soils).
Combining them in twos we get six mixtures NP, NK, NL, PK,
PL, and KL.
Let A,B,C and D respectively denote the best times of
al?]?lication of the fertilizers
8 denote the
1
denote the sets of mixtures NP, ••• , KL.
sets
N, P, K and L.
Let
A,B, C and D and 8
2
In the set Sl let each eleme~t be the first associate of itself and
second associate of all others.
This may be conveniently represented
by the following matrix:
A
B
C
D
A
1
2
2
2
Similarly, in the set
B
2
1
2
2
8 ,
2
C
2
D
2
1
2
2
2
1
2
....
(1)
let each element be the first asso-
ciate of itself, second associate of each element which has exactly
one symbol in common and third associate of each element which has no
symbol in common with it.
The association scheme may then be con-
viently represented by the matrix below:
. -e
9
NP
!ilK
NL
PK
PL
KL
NP
NK
NL
PI<:
PL
KL
1
2
2
2
2
2
1
2
2
2
2
2
3
2
2
2
1
3
1
2
2
3
2
3
3
2
2
3
2
2
1
2
2
2
1
2
• • • (2)
Finally, we may introduce an association scheme between the sets
81 and 8 as follows. Each time of application of fertilizers may
2
be called the first associate of all mixtures vmich contain one component
fertilizer for which this time of application is optimum, and the second
associate of the
e
A
B
C
D
rest~
The association matrix corresponding to this is:
NP
l\1K
NL
PI<:
PL
KL
1
1
2
2
1
1
2
1
2
2
2
1
2
1
1
2
2
1
2
2
1
1
2
1
0 • • (3)
From the first matrix, we find n = 2, n2 = 30 From (3), we have
1
n12 = n21 = 2. Again, from (1), we easily get p(l,l,l,l,l,l) = 1,
p(l,1,2,1,1,2)
= 1, p(l,1,2,1,2,2) = 2, etc.
Similarly from (2), we find p(2,2,2,2,1,2)
p(2,2,3,2,3,2)
= 0,
p(2,2,2,2,1,3)
= 0,
= 1,
p(2,2,2,2,2,3)
= 1,
etc.
For the mixed cases, where both the sets come into the picture, we
obtain from (3), the values:
p(l,2,1,2,1,2)
•
=
2,
p(1,2,2,2,1,3)
= 1
p(1,1,2,2,2,1)
= 2, etc.
Thus we observe that the sets
8 and 8 possess a partially
1
2
10
balanced association scheme.
It is interesting to note that in this
example, the association arises in a natural 'Way.
2~2.2.
Properties of class .Lp~7.
Definition 2.2.2.1.
Association matrices:
Define the
n x n
i
j
matrix
• • • • • • • • •
=
•
•
•
•
• • • (1)
•
rn.l
,
~
b ij
where
e
J:'Of3
bi~v
=
if a
1,
(3
=
Si' f:3
€
€
S. and a
J
is the r·th associate of
in Sj'
otherivise,
0,
and i,j, and r
•
take all permissible values.
The matrices
'Will be called association matrices.
Lemma 2.202.1.
n:.
(b)
n nijo
i
and r.
~J
braiJ'~' for all permissible a,
= L
(a)
f:3
r
= nj
n
r
ji , for all permissible
i, j, r.
i, j
= J n.n. (the n.~ x n.J matrix of all
~ J
(c)
unities)
(d)
~
r
c r B~j
=0
permissible
(e)
(n i x n j ) implies
r.
1
The linear functions of B ,
ij
form a vector space 'With basis
c
r
= 0 for a.ll
11
(a)
Proof:
For fixed a, we have (if a € S ,
i
numbe~
of elements in
Sj
s.),
~ €
J
which are
r~th
associates
of a.
n~j (by definition).
;
(b)
For each a
r-th associates of a.
€
n~j
Si' we have
elements in SjwhiCh are
Thus from (a), we get
"
"
a
b r C\f3
which gives the
;
ij
reQuired result.
(c)
the pair a,
This relation holds, since for all a
~
and n
(d)
This result is obvious.
(e)
This hold;; by virtue of
B~k =
The matrices on the
j
x~
ral
,
ij
=
bro2
.. ,
~J
Sj'
E
2:
( d) •
p(i, k, t, j, r, s)
:t
B~
n.~ x n.
are of dimensions
loh.s.
J
respectively, so that the product exists. . The element
in the ath rOt'T,
(b
Si' ~
are r-th associates for one and only one value of r.
Lemma.
Proof:
€
... ,
column of the product matrix is
~th
b
ron j
ij
s2f3
b jk ' •••
6~
) x
(b jk '
,
b
sn.f3
J )
jk
nj
L:
••••
t=l
Suppose now that a
associates.
Then (3)
€
Si
and
~
€
Sj
and
~j~
are the
is eQual to the number of elements in Sj
t-th
which
,e
12
a.re common to the class of r,.th associates of a
and the class of
s.th associates of ~ and is by definition equal to p(i, k, (, j, r, s)
a,~
since
are fwth associates.
On the other hand, the element- in the a-th row,
of the matrix in the
r.h.s.
one member
~
and
are the
column
is
E p(i, k, t, j, r, s)
t
8ince a
~ ..th
f~th
•
associates of each other, only
in this last sum is nonzero, and the sum reduces to
p(i, k, 1, j, r, s).
This completes the proof of the lemma.
2.2.3.
vle
Linear Associative Algebras.
shall now consider certain algebras connected with our asso-
ciat10n schemes.
Let
8 , where
m
J)
8
i
lle shall begin by proving an important lemma.
be a class in .Lp~7 and contain the m sets
has
n
i
81 , 8 , •••
2
elements (1=1,2, ••• m).
m
Let
n. = L:
. 1
J.=
n.
J.
Consider the
n. x n.
Each matrix D~k has
ri
matrices
defined as follows:
submatrices, the matrix in the i-th
row
block and j-th column block being of order n x n j • For ~ixed
i
j, k and r, the matrix D~k has zero submatrices everywhere except
in the j-th rOvT, block and k...th
column block, where it contains the
n j x ~ matrix B~k. ' The matrices
D~k will be called the com-
ponent association matrices of the class
CAM.
1:;
LeIJTJa 2.2.:;.1.
= z p(j, f,
t, k,
t
if k
for all values of
j, (, k, rand
where
denotes the zero rJatrix of order
Z{n. x n.)
Proof:
= k',
s, which are per.oissible, and
n. x n.
Consider the product D~k D~,! obtained by tlUltiplying the
two r.w.trices blockv1ise.
The eletlent in the
ql-th row block and
%-th colurm block of the product will be a zero matrix if either
the
r
ql-th row block of D
jk
or the
both.
consists entirely of zero subt~trices,
. s
%-th colutm block of D ,( has zero submatrices only, or
k
Since all the row blocks of D~k
consist of zero r.w. trices
except the j-th row block, and all the colutm blocks (except f-th) cantain zero natrices only, the only possible non-zero :r.w.trix, say M,
stands in the
j-th row block and
t-th column block of the product.
However, if k ~ k', the two non-zero matrices will get multiplied
with zero mtrices, and M will be zero.
If k = k', then obviously,
p{j, (; t; k; r, s)
by the last lemma.
This co:m.pletes the proof.
The above lez:ma shows that the product of any two CAM's can be
expressed as a linear function of the various CAM's in-b •
Thus
we shall write
p{j, k, r; k',
!,
t
s; u, v, t) Duv
,
14
Where the
are integers such that
pIS
p(j, k, r; k', (, s; u, v, t)
=
p(j,
t;
t; k; r, s),
if and only if Ie = k', j = u,
v
=
r
Consider the matrices D
~
jk
n jk
=
n •. , say.
jk
= f,
and,
0, otherwise.
• The number of such matrices
The total number of symbols P is
is
n3 ••
•
The set of all linear functions of the component
Dr
form a linear associative algebra with the
jk
D~k as a basis.
association matrices
n •• matrices
.
Proof:
Suppose constants
~
jkr
where
J
C
cr
jk
exist such that
r
ok
J
Z is a zero matrix of order n •• x n ••
~
r
C
r
jk Djk
jkr
which in turn implies that
Using lemma
k, and r.
• Then we get
r
E ( ~ (.jk
Dr)
jk ,
jk r
=
E
r
C1~k
B~k = z(n j
2.2.2.1 (e) , we then get C~k
Hence the matrices
x
=
~).
0, for all
j,
D~k are linearly independent, and
therefore form a basis of the vector space V consisting of linear
functions of these matrices.
Further, from the preceding lemma, it
follows that V is closed under multiplication of elements of V.
Hence V is a linear algebra.
Again, since the members of V are
matrices, and since matrix multiplication follows the associative law,
the algebra V is associative too.
This completes the proof.
15
2.2.4.
In this section, we shall derive various relations among the parameters
p(j) It, t, '{, r, s),
n..
etc.
J.J
In the last lemma, we proved that V is a linear associative alger
s
t
bra. If Djk , D , D
are three matrices in V, then
kK Ki
....
(1)
of
=
=
=
~i
r
( E
u=l
Djk x
~i
E
p(k, i,
uj
(j
D~k
t)
5,
u=l
~
(1)
p(k) i, u,
K,
v
p(j, ij v, k, r, u)
t)
6,
Dji
u
n
ik
= ~
u=l
n
ji
~
p(k, i) u) (,
Similarly, r.h.s.
nj {
=
~
5,
t)
of
D;i ••• (2)
p(j,
ul=l
(1)
f,
Ul,
k,
r, s)
n
p(j,
f)
ul
,
k, r) s)
Since expressions (2) and (3)
ji
E
vl=l
p(j, i,
Vi,
f,
ul
,
t)
D;~.C~)
are equal, we can, by using the last
lemma, equate the coefficients of D~.
JJ.
of r
p(j, i, v, k, r, u)
v=l
from 1 to n ji • We then get
,
in (2) and (3)
for each value
16
~(k, i,
u,
I,
s, t) p(j, i, v, k, r, u)
(4)
njK
=
~
p(j,
I, u,
k, r, s)
p(j, i,v,
I, u,
t)
u=l
for all v
K,
j, k,
2.2.5.
Let
= 1,
2,
... , . nJJ.... ,
and for all per.missible values of
i, r, s, and t.
Further relations between parameters.
J
denote a matrix with x rows and y
xy
unity everywhere.
columns and
Then
.....
(1)
Also, then,
r) B'S
( ~ B..
= J
k nin
r
3.J
J
j
=
S
t
~. ~ B' k
J t 3.
However, first expression in (2) equals
n
i:
S
(B:. B )
J.J j k
r
ik
~
Bt
p(i, k, t, j, r, s)
ik
t=l
Hence
n ..
J.J
~
r=l
t
t
Hence, equating coefficients of B ,
ik
n ..
J.J
r~l
=
P( i, k, t, j, r, S)Bik
for all permissible
s
=
p(i, k, t, j, r, s)
~j
i, k, t,
j
s
~j
~
k
t
Bik
•••
we get
(4)
,
and
s.
The total number of such relations is
=
~
dl , where
,
•••
e
17
where
n
= i:
i
n· k
ik
,
As special cases, we get,
n
= i:
j
~.
J
j = k,
when
jj
p(i, j, t, j, r, s)
i:
=
n
r=l
For
n
s
jj
(7)
we have,
i = k,
ij
p(i, i, t, j, r, s)
i:
(6)
~.
=
r=l
s
n ij
(8) •
By definition, we get
L:
r
r
n ij
= nj
Also,
n ..
r
B
jk
.
Agi
a n, sJ.nce
JJ
~j
p(j, j, t, k, r, s)
t=l
__
Rr
-:it
i:
=
(B~k)' ,
j
t
B..
JJ
(10)
we have considering the terms in the
diagonal,
n~k = p(j, j,
if each term in class
2.2.6.
r
Since l\j
r
D
jk
Define,
r
Ajk =
He call A's,
j
K,
k, r, r),
is the
= (B~k) f
,
(11)
Kth associate of itself.
we get
= (D~j)'.
~j
r
r
= Djk + Dkj
(1)
the symmetric incidence matrices.
Now,
(2)
(18)
[P(j, k, r; i,
I:
==
u,v,t
I,
s; u, v, t) + l?(j, k, r; [, i, s; u, v,t)
But
P(j,k,r;i,K,sju,v,t)
==
p(j, [i t; k; r, s),
==
0, otherwise.
Consider the case, when k
t
is zero.
D!j
=1,k i I,
D~K in (2) is p(j,
coefficient of
cient of
if and only if
i, j
f
v, k
K.
i, and
==
Then the
I,
t, k, r, s), whereas the".coeffir
s
Thus the product. Ajk Ai [ can not alvre.ys be
expressed as a linear function of the
r
A
f
j
K ==
j=u,
A's.
This shows that the matrices
do not form a linear algebra.
jk
2.2.7
The inversion of matrices
Let
'X.
D~k. Let D
D
D~k and their linear functions.
be the linear associative algebra generated by matrices
€
X.
Then we can write
(1)
I:
=:
r,i,j
Suppose
D is nonsingular, and let
-1
D
=
v1.
If w
,
€
then we can write
H = I:
r,i,j
J:: is such that the identity matrix
X ' and that
Suppose
belongs to
I
r
J.J
w..
n •• x n ••
==
I:
i,l,r
I(n •• x
n •• )
(19)
Now
= (
11D
r,i,j
wr
ij
L:
=
r,ij
=
wrij
L:
L:
skl
r
w
ij
L:
r,i,j
D~j)
(
s,k,l
akl
s
r
akl D..
J.J
s
D
kl
s
ajl
p(i,
L:
s,l
L:
s
s
L:
t
l,
Dkn
...
t
t, j, r, S)D
il
and (4), we get
From (3)
t
cil
=
L:
L:
s
r,j
t
cil
=
L:
r
w..
s
ajl
p(i,
wrij
L:
s
ajl
p(i, {, t, j, r, s)
...
The equations (5) are
n••
in number, obtained by varying
i,
or
t
(4)
r,j
J.J
s
over all penrlssible values.
l,
t, j, r, s)
r
The number of unkno'WIB w..
.
J.J
(5)
l,
and
is also
n.o, and obviously all of them enterinto the equations.
The necessary and sufficient condition that H
€
X
is there-
fore that the equations (5) are all independent and consistent.
Now in equations (5), keeping
i
fixed, vary K and t.
This
gives (if m is the total number of sets),
m
L:
K:=l
equations.
V{X
These
in which
Thus the n..
i
n . equations involve only the
i
n.
J..
unknO'wns
is fixed, and x, y take all permissible values.
equations can be broken up into m sets, the
set containing n i •
unknowns.
ith
20
Consider now the i-th set of equations.
the coefficient of wr
ij
t,
t,
j, r
s
we get
a
and
t
J.
r, s).
t, j,
over the
ni~
permissible values,
matrix, which may be denoted by
n .• x n .•
J. .
and t,
is
~ a~t p(i,
Varying
t
For fixed
.Cli
.
It is then easy to see that the necessary and sufficient condition that 11
is, that the matrices
€
.ell , .Cl 2'
••• ,
.rlm
are all nonsingular.
The problem of inversion of IT is then reduced to the inversion
2.3 Propertiea of certain matrices arising in connection with factorial
experiments.
2.3.10
~lctor-vectors
and factor-matrices.
Ccnstder a set of' n symbols
AI' A ,
2
these sy.t11t..iols
r
at e. time for any
the symbol .~
correspond to the case
the set
= o.
Sr' (O~ r ~ n)
Let
in the natural order (Al A
2
Sr
0'..
Definition 2.3.1.1.
r
r
< r < n. Let
0
r
r
A +1 A r+2
A ),
n-r
nn .
our future discussion.
elements of T
such that
(u) symbols obtained by taking Als
denote the set of
Arrange
r
.0., An. lie can cOLlbine
which
A , ••• ,
r
shall suppose to have been done in
r
be a set such that T
T
.0
•
=
Tl
·T.r•
·
Tn
C S.
r. -
will be written as a column .vector.
T
0 ••
N(Z) = number of elements in the set Z.
Let
Let T
1'1e
~
at a time.
r
Let
The
21
n
Then '1 'Will be called a factor vector of the type 2 , (written
n
n
F V(2
since it arises in 2 factorial experiments.
»,
EXAMPLE:
Consider a
,
=
,
~A2A;
~A2A4
= A A;A
l
4
A2A;A4
Al
~
A4
If we take
T
=
~A2
A A
2 4
~A4
~A2A4
A2A;A4
then T
l
=
,
22
and To and T
Ti C
S1'
Let
4 do not contain any element. It is clear that
i = 0, 1, 2, 3, 4. Thus T is a factor vector.
\.[
denote the
are any two factor vectors
[th element in '\.0 Suppose ~ and !!
n
(2 ). Define a matrix Z satisfying the
following conditions:-(i)
the order of Z is
(11) Let
T and
r , r
j
i
ri
N' (x)
t.
~
x N(!!)
and u. respectively be the i-th element of
.J
jth element of !!. Then the element in the 1-th row and
jth column of
on
N(!)
t.
~
Z
is a real number
and P1j'
= N'(t i ), r j
d(r., r., P'j)' depending only
~
J
~
where
= N'(u.),
Pij = N'(t i
J
nu
j
),
= number of symbols in an element x, and
n u.J =
the element formed by the symbols common to
tot
...
and
A matrix Z which satisfies the above conditions
n
will be called a factor matrix (~I). lie will then write Z E F M(2 ).
Definition 2.3.1.2.
If further, lTe have
d(m, n, p)
=
den, m, p),
for all m, n, and p, then Z will be a symmetric factor matrix,
n
written Z E S F M (2 ).
2.3.2.
In a similar manner, we shall define factor vectors and matrices
corresponding to
s
sm factorial experiments ..
Suppose we have m sets of symbols, the i-th set containing t~
o
1
s-l
symbols Ai' Ai' ••• , Ai
' i = 1, 2, ••• m. In the IDS symbols
we have, there are in all m
subscripts and s
superscripts.
Vie
23
an element by combining m symbols, one taken from each set.
get
sm elements.
A factor vector
(SID), written FV(sm)
Thus we
is simply
a set of such elements, the elements being supposed to be arranged in
numerically ascending order, both with respect to subscripts and superscripts. The FV(2m) are easily seen to be particular cases of these.
Definition 2.3.2.1.
An
FV(sm)
will be said to be invariant if it
remains unchanged (except for a permutation of its elements) by any
nonsingular permutation of the subscripts of the symbols constituting
its elements.
EXAMPLE:
The factor vector
o
O
2
~ AO A3.
A1 A12 A3.)' A1 A2 ~
)' -~ 2 )'
,
is an invariant FV(33) •
Consider an s
m
factorial.
Let the
belong to the set of integers (1, 2, •
by
~(x1' x ' ••• ,
2
x.tt)
GO'
Xl' x2 ' ••• x.tt
Then we shall denote
supe~scripts
s-l)
0
an invariant factor vector all of whose ele-
ments have in some order the non-zero superscripts
Xl' x2 ' ••• ,
x.tt'
the other (m-k) superscripts being o.
It can be shown that an invariant FV(sm)
can a1'WaYs be broken
up into sets of the form ~ (xl' ••• , ~), where k may take a number
of values.
The sets
~
will generally be expressed as column vectors.
Also, while writing the sets
~,
symbols with superscripts zero.
write
we shall ignore in each element, all
For the element
00
Ai0
A2 ••• Am'
~. Thus the FV(33) in the example above can be written
we shall
24
"Te shall
assume that factor vectors are written in this form only, i.e. ignoring symbols with superscript
~,
set
x2 ' •••
scriPtsvare
(with L:
.
~
are distinct, and the values of these super-
zl' z2' ••• Zv
~i
=:
If V superscripts out of the
o.
such that
~.
is repeated
zi
k), then the number of elements in the set
Sk(xl , x2 '
1
• •• , i1t)
times
~
is
=
(1)
i
• •• IJ. v •
2.3.3. We shall now define an association scheme for factor vectors.
~ (i l , i 2 , ••• ,~)
Consider the two sets
and Sf (jl' j2' ••• , jf)
where k mayor may not be equal to f.
Suppose there are
distinct integers respectively among the
i' sand
one from each set, we get v v' pairs
Let
0
y an element of Sf'
Suppose there are p
common to x and Yo
Let these be Ag
1
script of Ag .
in x be
~i
~
ordered pair
(~i' ~1)·
Ag , we thus get p
~
••
2
Ag • Let the superP
I
~i •
This gives us an
From the set of common factors
ordered pairs of superscripts.
Ag , Ag , •••
1
2
These p
pairs
may not all be distinct.
(s-l)
2
distinct pairs of nonzero superscripts.
Call these pairs
i
= 1,
2, ••• p,
Sy , ••• Sy ,
'2
where
are not all distinct.
and
subscripts (or factors)
p
Now, there are
v'
Combining them,
x be an element of
Ag
and in y be
j IS.
v and
Suppose now that yl e
P
Sf
25
Then y'
in
..
(i)
and y will be said to belong to the same associate class
SK induced by the element
The element y'
with x, say AQ, ,
1
(11)
x
in Sk
has exactly p
AQ , ,
,AQ ,
p
•••
2
if the superscript of A
Q
,
if and only if
subscripts or factors common
and
in x
and the superscript
is
i
of AQ ,
in yf
is
i
w~,
J.
then the set of ordered pairs (w.,
J.
w~),
J.
i=1,2, ••• P should be the same as the set of ordered pairs
••• , s."
except for a permutation •
P
Since any invariant factor vector V consists of sets of the
type· Sk(xl , x ' ••• ~) only, as pointed out earlier, we can now
2
look upon V as a class of sets and inqui re whether V € LP~7
where the various sets of V are
Sk's.
Definition 2.3.4.1.
• •• Am be
.k < m.
symbols
A ,A , ••• , A taken
CT
CT
CT
l
k
2
out of the total set of m symbols, and let Y be the set of sym-
bols
Let X be the set of k
m symbols, and let
A ,A
P1
P2
, ••• , A
symbols in the set
"k
drawn in a similar way.
X are distinct.
·...
·...
Suppose the k
Then
A
O"k
1'1ill be said to be a nonsingular transformation of the set Z of symbols
Al , A2 , ••• Am onto itself if
26
the k
(i)
symbols in the set Y are distinct,
is changed to A , i=1,2, ••• k.
(ii) A
O"i
Pi
(iii) the symbols in the set Z - X are transformed one to one to
symbols in the set Z - Y.
Theorem 2.3.4.1.
~:
Let
D be an invariant FV(gm). Then D € ~p~7.
Dl , D2 , ••• , Dk , each D. being
J
of the form Sq(i , i , ••• , i ), say in particular D = S (iI' i ,
2
q
l
j
2
qj
(a)
Let D contain the sets'
... ,
Then for any
pose;d into say v
ciate classes in
=
x
j
and k, if x
=
D
j
,
the set
~
can be decom-
disjoint subsets which will be the different assoinduced by x.
i
...
qj
~
i
y
€
A
•••
v
qj
q.
J
qj
Let y
€
D , and let
j
1
f
(1)
Suppose
Q and Q are the same associates of x in D •
l
k
2
Then both Q and 9 have the same number (say I-t) of factor syml
2
bols (i.e. sUbscripts) in cammon with x. Denote these by
A"
zl
••• ,A,
zl-t
and A
,
II
zl
... ,
respectively be the superscripts
respectively.
A,
zi
similarly let
'be the pair for
Then the
3f
I-t
pairs
Q
l
i
)
Let
in x and in
in x and in
Q2.
are, in some order, the same as the
Consider the nonsingular transformation of
27
factor symbols given by
T
=
••••
The transformation T changes
will be changed into elements
Gi,
x
into y, and Q
l
Since, however,
Q~o
Q'
and
Q~
belong to Dko
nonsingular, both g'
and
Q~
have
invariant vector,
1
1
Jl
and Q
2
is an
I1t
Also since T is
subscripts in connnon i'1ith y.
Furthermore, it is clear that y and Gi will generate the set of
Jl
Q
pairs
(1(~,
1(,1) which will be the same (in some order) as the set
~
~
Q
2
wi)
possessed by y and Q~o
(w~,
Thus
2
Qi. and Q
belong to the
same associate class in D induced by y E Dj •
k
On the other hand, suppose Q and Q ~elong to different
l
2
associate classes in D generated by x € Djo Then either (i)
k
Ql and Q have different number of factor symbols (subscripts) in
2
cammon with x, or (ii) the sets of,pairs possessed by (x and Ql)
and
(x and G ) are not the same G In case (i), it is clear that
2
Qi and Q~ belong to different associate classes in D induced by
k
y € Djo Same holds for case (ii) also since the set of pairs possessed
by
(y, Qi)
possessed by
is the same as that possessed by (x, Ql)' and the set
(y,
Q~)
is the same as that of
(x, Q2)o
Thus any associate class in D induced by x € D corresponds
j
k
uniquely to an associate class in 1\ induced by y € D. and vice
.
versa.
J
There exists therefore a one-one correspondence between the
28
1\
associate classes in
induced by x
and by y.
number of associate classes induced by x
induced by y.
Therefore the
is equal to the number
This number "Will be denoted by n
qj~
.
Further, the above argument also shows that if' X is an associate class in
~
induced by x e: Dj , and Y is the class in
induced by y e: D , then the
j
transfo~tion
pondence between the members of
ber of elenents in
J
sets a one-one corres-
X and the members of Y.
X and Y is therefore equal.
ments in the
in D.
T
r-th associate class in D
k
will be denoted by nr
~
The num-
The number of ele-
induced by any elenent
qjqk
Note that the above results hold both when
and k
j
are and
are not equal.
(b)
Finally, we shall prove that if x e: Dj ' y e: D
k
and Y respectively are the
(3-th associate class of y
a-th associate class of x
Dr'
in
and if
x
and
and
in
X
Dr and
yare the
t-th
associates of each other in D
and D , then the number of elements
k
X and Y is a constant p( j, k, t, f, ex, (3), dependent
j
common to
only on the quantities
j,
k, t,
f,
ex and (3, and independent of' the
(x, y) pair with which we start so long as (x, y) are
To show this, take any particular pair (x, y),
of elements cot'lIilon to the classes
11'
j
are also t-th associates,
ponding associate classes in
Dr'
Let the number
X and Y induced by this pair be
Take now some other pair (x', y'), x' e: D , y'
x' and y'
t-th associates,
Let
X'
€
~, such that
and Y'
be the corres-
29
We claim that there exists a transf'ormation
x
into
x
and
x' and
yare
y
into
y'.
T* which changes
To see this, we f'irst observe that
associates, and so have, say A f'actor sym-
tth
bo1s or subscripts in common, and possess as bef'ore, a certain set
of'
ordered pairs as superscripts.
"A.
also have
~ = set of' all f'actor symbols which are in
= set of' all f'actor symbols which are in
2, D.;
Def'ine D.i, D.
H' (z)
If'
W'(y) =
2),
N'(Y'),
A , ••• , A"
and the superscript of'
... "A.
N ' (D.
so that the
y,
i=l,2, •.• p.
Let
si)
1
AE'i'
into
be
SiX
in
x
··.,
x'
Now arrange the
2
and
SYi
into D.~..
• As mentioned
and
"A.
y'
symbols
also are
A,
E
in a new
j
such that the pair corresponding
x
"A.
is the same as that f'or
x
ei
Hence
Let D. be
1
Ai
"trnnsf'ormation (of f'actor symbols)
changes
A
N' (6i).
x
AEIt , ••• , A "
e
(1=1,2, ... ), i.e.
= N' (~).
=
y'.
z, then we
associated ordered pairs are ( Si'
in some order.
order say Aelt ,
to
and
Xl
p
above, the associated ordered pairs of'
(~~,
1
N' (D.1 )
and
and N' (6 )
3
=
in
but not in D.
1
y but not in D. •
x
similarly, corresponding to
we get N' (6 )
2
e2
"A.
x and y,
denotes the number of' f'actors in an element
N'(x) = N'(x'),
have
y'
Let
set of' all f'actor symbols common between
l =
~
and
SUbscripts in common and possess the same set of'
"A.
ordered pairs.
D.
x'
Similarly, then
Ae .
J.*
'1'1
Y
viz (~i' ~i)o
which changes
A
E
i
-> AeIt
i
T be a transformation which
2
x', while changing D.
into D.
Obviously, this
l
D.
1
Let
Now define a
2•
30
can always be done, since both
x
Define a similar transformation
TI"~' ~
N(1-1)
Now let
let
=
U
x'
from
3
~
are in
y
= X - (X (l Y)
W
=
How since
=
Y
n
X
1-1.
U', V'
and
X is the
ath
Wi corresponding to
associate cla.ss of
n(u}
X and
T*
Y'
and
N(Y)
H(X)
=
and
= N(lJ').
X'
into
is
x'"
U is however common
changes
T*
U to
U*.
is obviously nonsingular, it changes distinct factor sym-
bOls into distinct ones only.
associates of
and
Y'.
and
Y and
and
Xl
Now suppose
8
Thus the elements of
13th associates of
Uf
is a member of
"
y!"
then
U* must be
Hence
yt
"
there must be an
are transformed by T*
into
8.
x*
in
8 h:iJ.longs both to
X and a
However, since
y*
ath
'lJ* C U'.
Since there is a one-one correspondence between
Y*
and
Similarly Y is mapped
The set
N(Y')o
x
X'
X is transformed one
= N(X').
Suppose the transformation
Yo
x,
T* transforms
x', and
X', and also that
one-one onto
and
AlSO,
y .. U.
associate class of
to one onto
x*
4'l1.
Y, say.
we conclude from part (a) of this proof that
Since
3
0
agrees with our claim. (It is
Then the theorem will be proved if we show that
ath
T
= X .. U,
Define" similarly"
the
= ~TI.
T2
y' and let
= number of elements in a set
n Y) =
y.. (X
•
Let
number of elements common between the sets
v
to
to
Dj"
operate over disjoint sets of factor symbols).
X and
yl
T
TI ~
Then the transformation T* =
obvious that
and
in
X and
X'
X',
Y which
T*
is nonsingular"
must represent the same element, say w,
which therefore
31
belongs to
.=:>
U'
Hence
U.
is the transform ofw,
= U*.
Uf
= N(U*).
=
N(U)
8
Thus
U* •
N(U')
finite,
Hence
But
But
T*
8ince
and
both U'
so
and
8
0*.
€
U*
are
n(u) = N(U*).
being nonsingular,
N(U V ).
This completes the proof of the theorem.
2.3.,
Let
sets
8
X be an invariant
,
8
ql
~
, •
factor symbols.
is the
x
Aith
~
€
,
where
i = ql'
suppose
X contains
8 " contains elements with
J
%' •.. ,
~,
m
j
we shall henceforth
8 , then there exists an integer
i
Ai
such that
associate of itself and of no other element in 8 •
i
Corresponding to
theorem)
8
For each
assume that if
x
u,
FV(sm), and
X we can now define (since
the matrices
D~k' where
j,
X e LP~7 by the above
k = ql' ~, ••• ~.
Further
we shall have
~
!:
i=ql
where
n. "
= number of elements in
Corollary 2.3.5.1
Let
X.
Thus we get the
X be an invariant
the connected association matrices.
a linear associative algebra
FV(sm).
Let
D~k be
Then their linear functions form
J: , which contains
the identity matrix.
As will be seen later, this result has many connections with the
analysis of fractionally replicated factorial experiments.
2.3.6.
In order to make the ideas c~earer about the calculation
of actual values of the parameters in the association schemes of in-
32
variant factor vectors, we shall now consider an example.
Let
m
FV{2 )
X be an invariant
containing sets
S
Sq ,
We shall suppose that
S •
~
(l)
m > max (qj + qlf:' qj + qt' Cl.r + qK) ,
-
for any three sets
S
qj
necessarily distinct.
tains the element
meters
n
qjqk
, S
qk
,
S
qK
If' any
,
where these sets
is zero" we
q.
J
ass~
are not
that
and the
can have
~
con-
pIS.
0, 1, 2, ... , qj
q.
<
J -
elements
=
+
qk.
~ommon
This fact is ensured by condition (1).
Then any element
with an element
Hence in this case
1
Suppose we define two elements to be t-th associates
t
S
¢ only. Consider the problem of obtaining the para-
Without loss of generality" suppose
have
•••
~'
1
factors in common.
(2)
when they
Thus
Further we shall have
(4)
Let us find
Case I.
Let
qj
n
<
r
qjql~
qk.
for
Then
r
< min (qj" qk). Two cases arise.
33
Then
Case I.
= (m)
number of elements
Clk
... A A 1 ... A inS
r r+
Cl j
Cl j
Take the element
Consider the number of elements in S
Clk
Ar , but do not contain Ar+l' .•• A
Cl j
'
which contain Al , An'
c:.
This number is clearly
m- Clj + r)
(
- r
(6)
~
Cl
elements in x could be chosen in j
r'
Since the set of r
'Ways, we get
r
n jk =
Case II.
(m - qj +r)
(:j)
(7)
qit - r
Bya similar argument, we get
Cl j > CJx .
(m - qj +r)
(;j)
r
n jk =
Further, let y e
Clk - r
and write y
= Al
A2 ..• At Bt +l .•.
B~, where (Bt +l · .. BCJx)
(At+.l .•• A ) = nil, and B's are
Clj
certain A's. Then x and yare tth associates if exactly t
~
n
of the A's, say
of x
in
S( are
St
n:t
~,A2'
.•• At are connnon.
rth associates
r
are njt in number and sth associates of y
in
in
number and we 'Want to find p(:j, k, t, (, r, s).
Suppose z e Sf and z is the
sth associate of y.
s
The
in connnon with y.
Then
z has
Now z has
rth associate of x and
r
factors connnon with x and
Clt factors.
Also t
of the
A's are con:mon between
x
and y.
Out of' these
u (there will be certain lindts in which u
in
caj
varies) factors con:mon.
caf - u f'actors, we have to take r - u out of At +l ,
For the other
..• , A
t, we can choose
x
and s - u
,B~
out of' Bt+1 ' ...
This
in y.
can be done in
t
)
cak (s - u
'Ways.
in
Now we have
z.
u + (r-u) + (s-u)
The rest of' the f'actors,
ca(/ -
= I'
+
S
(I'
+
S -
-
u
diff'erent f'actors
u) in number are to
1\
be selected such that none belongs to A ..• At
l
Bt+l .••
'Ways.
B~
which are
caj + cak - t
in
number.
A._Ll ... A
\,IT
ca j
This can be done in
Hence
p(j, k, t,
f,
r, s)
m - ca· - cak + t )
( cal ~ I' - S + U
(8)
where
j-
sho'Wtl that
consists of' cert.am non-negative integers.
§ is the set of' non-negative integers
It can be
u
such that
each expression in each bracket in (8) is non-negative, and upper
expressions are greater than or ecaual to the corresponding lower ex-
.35
presssions.
We get also the result:
If q.
J
= j,
then
Such expressions 'Will be useful in finding the matrices
~l i
of section 2.2.7,
2. .3,7
As an example, we shall now consider the inversion of the
matrix D given by
.,
;
X
I
11
I
D
i
I
=
x
2
~
I
I
, ,
x
4
x.3 x4 ,··
,
x
2
. x
2
x
2
x 4· x 5
' x x
5 5
x x x '·· x x
4 4 5
4 5
(1)
Symmetric
I
~
I
I
This matrix is obtained by considering the invariant factor
vector
(S"0' Sf)
2' in the notation of previous sections.
The number of elements in X is
=
2
m - m+ 2
2
We can write
r
where as defined earlier D ; j, k = 0, 2 are association
jk
matrices for the sets 8
and 8 • Suppose, D is nonsingular,
o
2
and let W be its inverse and be obtained from (1) by replacing
5.
so that
n .
=
O'
He find here
= 4.
2,
Thus we ge~ two matrices, one
to invert.
and the other
2 x 2
4x 4
Transferring our symbols to the notation of section
2.2."(,·ue find
and similarly for
WI
I n .. x n .•
s.
=
Also we find from (3) that
DO + D2
00
22
.,
so that (section 2.2.7)
c
0
00
=
c2
22
FOr
i
0
= 1, cO2 =
= 0,
c
0
20
=
c
1
22
=
c
0
22
the equation (5), section 2.2.7
z:
z:
j=0,2
s
a~K pC 0,
or
K,
s=o
=
(6) .
0
reduces
to,
t, j, 'O, s)
z:
1
=
a~o p(o,o,o,2,o,s)
x
z: a~2
s=
P(O,2,O,O,0,s)
z:
s::-O,l,2
a~2 p( 0,2, 0,2, (hs)
37
From equation
P( 0,2,0,2,0,0 )
(m-2)
2
=
p(0,2,O,2,o ,2)
(9), section 2.3.6, we get
=
'21 (2
m... 5m + 6) ,
=
1.
Thus we obtain,
1
=
12·
'2 (m - 5m
+ 6) x
5
+ 2(m - 2) x4 + ~
(10)
For i = 2, we get the equations
E
. 'E
r
j=o,2
E a~t
s
p(2, (, t, j, r, s),
... (11)
or
0
0
0
1
=
I
£1::,.I
xl
Sl
I
·---------s-x2
: s2
6
x2
I s3
£7
I
x2
S8
I s4
2l
w J
5
x
2
----x5
x4
w
~-
w
w~3
(12)
by
the help of equations
1
(9), section ;.6, where
Sl
= '2
2
(m - 5m + 6) x2 '
S2
1
= '2
2
(m •. 9m + 20) x + 2 (m - 4) x4 +
5
£6
=
g;
1
= '2
£7
=
[;4
1
= '2
2(m - 4)
g5
=2
(m-2) x2
X;
x5 + 4x4 '
2
x (m - 7m + 12) + x4 (m - ;)
5
(m - ;)
x + (m - 2) x4 +
5
2
(m - 5m + 6)
x5 '
£8
X; ,
=2
(m - 2) x4 .
=
x
and.
X, +
2
4x4 +
X;
0
xl
x
2
x
2
'W2
x
2
X;
x
5
'W
5
2
x4
x4
'W4
0
2
x
X;
'W;
1
x
x
5
0
0
=
(13)
39
It can be easily checked that both equations give the same value for
w2 'which is
x2
L6x~ -
xl
(~
+ x
5
+
4xJ)7 -1
2.3.8 We shall now discuss certain matrices 'Which arise in the analysis of fractionally replicated factorial experiments.
Let
X E: fp~7.
X
be an invariant factor vector.
In section 2.3.5, we have defined mat~ices D~k
ponding to the vector
D~k
X.
Further, corollary 2.3.5.1
form a linear associative algebra
Definition 2.3.8.1.
be
caJ~ed
Then we know that
If D
€
corres-
says that
Xl'
Xl' mentioned above, then D will
an invariant factor matrix.
If also D is synmetric, we
shall call it an invariant synmetric factor matrix (IFSM).
The class
of all invariant symmetric factor matrices will be denoted by
LIID!;7·
It will be seen later that in a factorial exrcriment, if we
assume all intemctions of certain orders to be
nefJ~ligible,
and if
the design is balanced, then the matrix to be inverted for solving
the normal equations belongs to the class £IF~7.
2.4.
2.4.1.
Multi-dimensional partially balanced designs (MDPBD).
Multi-dimensional designs have been defined. by Pothof'f
£4'2.7.
In continuation of his work, we shall define here the MDPB designs
and consider their analysis.
Following Pothoff, suppose lore have
m factors
F , F ,· •. F .
m
l
2
Suppose the ith factor has si levels F , F , .•. , F
il
i2
iSi
Here the word level does not necessarily imply that the levels can
40
be arranged or ordered according to any quantitative criterion.
Thus if F
1
denotes v-arieties of wheat, then
may stand for
sl
There are
levels.
different varieties of 'Wheat.
sl x s2 x .•• x sm
l
Suppose we try N combinations.
hl, 2, ... , m
i , i , ..• , i
2
~,
sl combina-
Let
denote the number of times the combination
m
(il , i , ... i )
2
m
level
= N , say, combinations of
In many cases we my not like to try all the
tions of levels.
l
F , F , ..• ,F
12
11
1Sl
in which kth
is tried.
factor
Thus, if N
(k
= 1,2,
.•. m)
is at
< Nl , and no combination is
repeated, then it is clear that some of the
h's must be zero.
In
what follows we shall
free~
Definition 2.4.1.1.
A multi-dimensional design will be called a
employ the notation of section 1 and 2.
MDPB design of type I, i f the following conditions are satisfied:-. .• i
( ii)
If there are
(F ,
il
,
m
m veC'tors
i
of the form
F , .•• , F
)'
i2
iSi
then the class
c1J
containing the
= 81
say,
m sets
8 , 8 ,
1
possess a partially 'balanced association scheme.
(iii)
Let
h~,s,t
-1.
r'
1
s'
~
1
t
=
=
m
~
i =1
k
2
0
or 1.
41
h
r,s
i r Ii s
=
~
E
k~r,s
'\
hi
E
k~r
=
r
hl,2, ..• m
1 ,1 ", 1
1 2
m
E
~=1
h1,2, .•. m
i ,1 , ..• 1
m
1 2
E
~=1
,
,
etc.
Then we must have
hi
a
(iv)
U:9()~l r~ th
If
Ii
r
(r
= 1,
... m),
inoependent of the levels of the r--ch
guaT'.ti~y
pending
=
r
factor al one.
€ Sand i
x
x
y
of each other, then
i
".x,y
~li
x'
factor, but de..
i
€
Sy
and
i, 1
are a-th associates
x Y
=
y
a constant depending upon
x, y and
a
only, and independent of the
pair (i , i ) 'With Which we start.
x y
2.4.2.
Analysis of MDPB designs of type I, assuming no interactions
present.
Let
X(i , i , ..• i ) denote any combination.
l
2
m
The response
to this combination of factors will be denoted by the symbol
y( i , i 2, .•• , i ).
1
m
The true response to the level
factor will be denoted by T( r, i ).
r
>-7 = T(l,
ELy(il' i 2 , ... i m
...
1
r
of the r -th
l'le shall take as model:
i 1 ) + T(2 , i 2 ) + .•• + T(m, im) .. ·(l)
The number of parameters T to be estimated is
m
E s
= n, say.
r
r=l
42
Let
~ denote the set of N factor combinations on which observa-
tions are taken.
Let 1. denote the set of all factor combinations
in ~ , arranged in the form of a column vector.
FUrther, we de-
fine the (n x 1) vector,
,E' = (T(l,l), .•• , T{l,sl)i T(2,1), T{2,2), .•. , T(2,s2)i
i
T(m,l), T(m,2), ... , T(m,sm) )
.•. (2)
Then we can write
(3)
'Where A is a certain matrix with elements
tained by using equations (l).
normal equations for obtaining
A A' ~
0
and 1, and is ob-
Then it is well kno'Wl'l that the
~
can be written
(4)
= A 1.
The main problem is to obtain (M') and invert it.
We shall
now indicate how this can be done.
The element in the (i, j) cell of M' is obtained by taking
the sum of the product of the corresponding elements in the
column and
ith
row and
jth
jth
Then the product
of A'
column of matrix A'.
ith
Let the elements in the
row of p be respectively T{x, i ) and T{y, j ).
-
x
y
of the elements in the cell (k, i) and cell (k, j)
will be unity, if and only if, the kth element in the vec-
tor 1. contains both
i
x and
j.
y
Hence the (i, j) element of
M' is equal to the number of times the symbols
gether in the various treatment combinations
ix, j
y
occur to-
1n ~ , and is there-
fore equal... to hX'Y.
ix,Jy
however is eqwa1 to
da
x,y
The mat.rix
of each other.
AA'
a
where D
x,y
From the det':lnitioo ot' the design, this
=
,
x and
x,y,a
= 1,
j
are a ..th associates
y
can then be expressed as
,M'
a
dx,y
!:
x, y
if i
m, a
a
Dx,y
= 1,
(5)
2,
ciati en me.trices connec'ted witl1 'the sets
••• J
nxy are the asso-
8 , 8 , ... 8 ,
2
m
1
The matrix M' can then be inverted by the methods of section2.2.7.
It can be easily seen that since
=~,
hX'x
ix,jx
for a11 x,
the matrices in the diagonal of (M') are identity matrices.
It can be show that for the case m
reduce to ordinary PBIB designs.
= 2,
the above designs
They also contain the intra and
inter group balanced designs as a particular case.
2.5
Product association schemes and the corresponding product algebras.
For the development of this theory, we shall borrow the nota-
tioo from section 1 and 2.
2.5.1.
In this section, we consider product sets and their asso-
ciation schemes.
sets, such that for
contains the m
i
Jt!t
S l' ofr2 , · •• ,
Let
i = 1, 2, ..•
sets
8
i1
, 8
i2
t,
be
the class
, .•. , 8
im
i
.
oBi
t
classes of
E
LP~7 and
Further, suppose
that
44
for all permissible (i" j)" the set
denoted by the symbols
8
ij
contains the
V
ij
objects
1
2
v ij
Gij " Gij , ••• " Gij • This will be expressed
in vector notation by writing
(1)
::::
The Kronecker product or (for this Mscussion) s:t.m;ply the product of two sets
8
ij
and.
\ ( will be defined by the corresponding
Kronecker product of the vectors
and 'Will be writtan
r~l
ij
e
8 ij
~.)
f\.f =
where
ith
class
-'~K
(2)
v ij
Gij
In the
-1
vk (
_.~fJ
_J
(['I
o/...)i" take any two sets
jl mayor may not be equal to
j2'
Sij"
1
and 8 ..
~J2
AB in earlier cases" let
B~. . be the association matrix between these two sets" giving
JJ 1 "J 2
o:th associates.
We shall develop the theory for the case
theory can be developed along similar lines.
f = 2.
The general
The sets in class
05
08
will be denoted by 81 , 82 " .•. " 8m" and in class
2 by Tl " T2 "
.•• "T . The u
objects in 8.; 'Will be represented by the symm
...
i
2
boIs
Gl " G2 " .•. " G
Ui
and the
vj
objects in Tj
by HI' H2 , .• ·"
1
H '
The product
Vj
® Tj
8i
is defined as in (2).
ciation matrix between the sets
8
and
i
and T
j
i
by
Also, we shall write
(0:) (f3)
0:
f3
Bij ® C
kK
=
= 1,
=
R
...
(3)
,
(ijXkK)
~;
k,
K
1, 2, .•• , n.J.J'I' f3
=
I, 2, ... , n ,
i, j
for
0:
ath asso-
will be denoted by
8j
B~j , and the f3th association matrix between T
cf3 .
ij
The
2,
,.
1, 2, .•• m
2
=
i J2
,
in an obvious notation.
Write
=
Q
ik
Dcfinitiol1 2,5 ,1.l~
Tk
,Bet\lCen any tno cets
(4)
Si (J)
~k
and
Q
j
K'
'We shall
define a partially balanced association scheme by the associationnatrices
(0;) (f3 )
R
Since Ct tokes nijl values and f3 takes ~K2
(ij) (kK)
values, the number of associate classes in
is equal to
n ijl ~K2
Definition 2·5,1.2
!f) =
The class
06 1
tains all the product sets
8
= 1,2,
~
i
®
and
oBI
®
j) 2'
T , for i
k
08 2
will be called
r!l.J
if the class
= 1,
2, ' ••
ID
l
con-
and
, •• m .
2
Theorem 2.5.1.1.
£p~7, then
J:)
induced by
.
the product of the classes
k
QjK
If the classes
Jj 1 and
0&2
both belong to
of}- E fp~7, and the association matrices of the class
generate a linear associative algebra.
46
Proof: He will first prove the second part.
and
Xl
£}2'
As in section
D~j"
by
1: 2
Let
and
by Lemma
1"
X2
a matrix in
~l
be the algebras corresponding to
we shall denote any matrix in
by
~t.
Since
ell
ell
is an algebra"
2.3.1" there must exist constants p such that
ni"j' ,,1
=
!:
p(i" j'" r"" j" r" r')
r"=l
if
=
r"
Dij , ,
j=i',
(6)
Zero matrix, otherwise
Similarly" there e:dst constants. q such that
s
s'
Fit Ek't'
n
!:
=
sll=l
q(k"
i' ,
s" ,
if
,.. '
s" s')
i =
s"
Fit, ,
k' ,
(8)
= Zero mtrix, otherwise
B~j
Further, the matrices
satisfy exactly some relation as (5)" if
the symbol D is replaced by the
E in (7) is replaced by
The sets
jects.
Hence
Si
B.
Same remarks hold if
C.
and T
j
B~j
symbol
is a
contain respectively u
ui x uj
matrix,
i
and v
C~f is vk x vf
j
obFrom
(3)" we find that the matrix
(a)
«(3)
R
(Ui Vk )
is a
(ij) (kt)
that the product
certainly exists.
~
=
But
(a)
«(3)
R
(ij) (kt)
x
(Uj
Vg)
x
(a' )
R
( j j ')
matrix.
«(3 , )
«( (:')
Thus we find
47
J
by the properties of Kronecker product of matrices.
.
3t
nij'l
=[
!:
a"=l
p(i, j', a", j, a, a')
~i'2
!:
:::
B~J'~'
Hence
..
p(i, jt, a", j a, a')
q(k,
i',
13", (, 13, 13')
13"=1
(a")
x R
(13")
......
(10)
(ij') (ki')
Now consider equation (4).
is
~m2'
The set
The number of vectors of the form
~
8
has u
objects, and T has V objects.
j
j
i
i
Hence, the number of elements in Q
is u v ' and the total
i k
ik
number of elements in all the sets in
is therefore equal to
The order of the matrices
of
in
E's in
tl
X2
is
and E in
v. x v.
:12 ,
D in
£1
is
u. xu. and those
Corresponding to matrices
consider matrices
F such that
.D
48
(i)
the dimension of each matrix F is
u.v. x u.v.
,
F(a) (t3)
(ii) any matrix F is written in full form as
(ij) (kf)
(iii) any matrix F contains ~m
2
blocks,
(iV)
row blocks and ~m2 column
the row blocks are represented by the pair
(i, k), i=l, .•. mlj
(v)
k = 1, 2, ..• m ,
2
the column blocks are represented by the pair
(j,f), j = 1, ...
(vi)
the matrix
~;
F(a) (t3)
is such that it contains zero every-
(ij) (kf)
where except at the juncture of
(i, k) - th row block and
{j, f)-th column block. At this juncture, it contains the
(uiv ) x (U v,,) matrix
j
k
x
R(a) (t3)
defined earlier.
(ij) (kf)
In section 2.2.3, we defined component association matrices.
Thus for example the matrices D and E respectively are component
association matrices for the classes
/)1
and
08'2'
It is then
easily seen that the matrices F are component association matrices
of the class
ciJ .
vle will show that they form an algebra say
L .
To show this, we have to prove that the product of any two matrices
J
in
is in
cl
any two matrices
However it is easy to see that the product of
F(C:) (t3)
(ij)(kf)
a zero matrix of order u.v.
!
=
k'
do not hold.
and
F(a') (t3')
exists and is
(i'j'),(k'f')
x u.v., if the conditions
j
= it,
llhen however, these hold, it can be shown
F(a) (~)
as in Lemma 2.2.3.1, that the product
F
(ij )(kf)
(a')
(~I)
(jj I
Hft- )
_
- p,
can be expressed 'With the help of (9) and (10) in the form
nij'l
p =
L.
a" =1
(a") (~")
(12) .
F
X
(ij I )(kf' )
This completes the proof of the second part of the theoreD.
elY € ["p~7,
To show that
definition 2.2.2.1
we must show that the condition of
is satisfied.
For this it is sufficient to show
that any three sets
Q in ~ have a mutually partially balanced
association scheme.
This however can be verified by observing the
(~)
R(a)
property of the matrices
noted at (10).
(ij )(kf)
This completes the proof of the theorem.
m.
Corollary 2.5.1.1.
x =
FV(s.~), i
is a
Xl
tg)
X
2
Since
Xi
is a
®. . .
~
®
Xn , then
= 1,
X.
2, .•. n, and
€
L p~7 .
m
Proof:
Xi
LP~7·
€
FV( s i i ), we have by theorem. 2.3.4.1., that
Hence by the last theorem,
®
and hence in general (Xl
X2
® ...
JS. ®
®Xn )
~
€
is in LP~7,
.Lp~7 .
Definition 2.5.1.3. Let us refer to definition 2.3.8.1, and let
m
.
be a FV(si i), i = 1,2, ... n. Let X = Xl ® ... ®~. Since
X
€
r-pB7,
_
L
Xi
suppose A~
xy are the component association matrices
corresponding to the class
Then if the matrix A
€
X.
eX ,
Suppose
L'f
c:l. .
form an algebra
"JC"y
then following definition 2.3.7.1,
.50
we say that
GIFM.
~
is a generalized invariant factor matrix, written
The class of all such matrices is denoted by {GIiM} .
It will be seen later, that if in any ba.lanced aSyrn:letrical
fractional factorial experiment we assume all interactions of certain order to be zero, then the matrix to be inverted for solving
the normal equations belongs to
{GIPM}.
The results of this section used jointly with the results of
section '2.'2.7, therefore give a valuable method of solving the normal equations arising in the analysis of balanced asymmetrical
fractional factorial experiments.
Same results will be useful for
the analysis of MDPB designs, when interactions are supposed to be
present.
CHAPTER III
FRACTIONAL REPLICATIONS OF FACTORIAL EXPERIMENTS.
3.1.
Preliminary remarks.
In this chapter we shall mainly deal with the nature and
analysis of fractional replications of symmetrical factorial exn
m
periments (of the types 2 and 3 in particular) and of asymmetrical
m
factorial experiments of the type 2 x 3 n •
A factorial experiment (FE) in which m factors are tested,
r
= 1,
each at sr levels, r
••• x
s~,
2, .'., k, will be written as s:l,
s~
x
and will be called an asymmetrical factorial experiment
(AFE) unless all s. 's are equal to say s, when we call it a symmetri~
cal factorial experiment (SFE).
The symbol FR will be used to denote
lIfractional replication ll •
3.2.
Properties desired in fractional replications.
In a FE with
ill
factors, we can sometimes assume that for some
. t < m, t-factor interactions and those of lower order are not negligible, while those of higher order are relatively small in magnitude
and can be neglected.
In what follows we shall be concerned with the
following three types of investigations:
CI.
Only main effects are present and all two factor and
higher order interactions are assumed negligible.
CII. Main effects and 2-factor interactions are present, and
interactions of higher orders are negligible.
CIII.Main effects and two factor interactions are present and
higher order interactions are assumed negligible; interest
lying however in the estimation of main effects only.
52
In eo.t:!h of the above cases we mayor may not be required to
have a sufficient number of d.f. for errQr.
Under the above conditions, we will try to obtain FR's having
the following prope1:'ties (described briefly).
P- I.
The FR should be economic, i. e., the number of
treatment combinations or
assemb~ies
to be used in
in the FR should be as small as possible while allowing us to estimate all the effects in which we
are interested, and providing only as many d. f. for
error as we desire.
P-II.
The FR should be balanced.
This means that the
variance of the estimate of all main effects should
be the same, under cases C-I and C-III.
Under the
case C-II, it will further imply that the variances
of the estimates of all the two factor interactions
also remain invariant under permutation of factor
symbols.
In general, when the
esti~ates
are corre-
lated, we require that the variance-covariance
matrix of the estimates is symmetrical with respect
to~all
the factors which have the same
numoer of levels.
P-III.
The FR should be orthogonal or near orthogonal.
If
the variance-covariance matrix mentioned in P-II
above is a diagonal matrix, then the corresponding
FR is said to be orthogonal.
On the other hand, if
53
x and y represent any two effects and the correaponding element in the variance- covariance matrix
is cov(x,y), and if this matrix is such that
cov(x,y)
where 0
to be
S€
=
r(x,y) <
€
,
< 1, for all x and y, then the FR is said
€- orthogonal.
If
€
is small, the FR may be
called near orthogonal.
Our main drive will be to achieve P-I and P-III, while preserving
P-II, as far as possible.
3.3. Connections between the classical approach to factorial
experiments and the response surface approach.
3.3.1. Consider a set of n factors.
tive.
Two cases may arise:
crete variable.
Suppose each factor is quantita-
a factor may be a continuous or a dis-
In both the cases we can assume some quantitative
law for the response to any treatment combination.
We shall in-
vestigate the connection between such quantitative laws and the FR
of factorial designs.
The following theorems can be deduced on the lines of Carter
£19_7,
though none of them have been demonstrated by him.
Because
of their importance, it appears worthwhile to discuss them explicitly and with requisite detail.
Theorem 3.3.1.1.
different levels.
(Xlu , x2u '
response.
"., x
Let there be n factors, the i-th factor having si
Let the u-th treatment combination be denoted by
nu ) and let y(x1u ' x2u ' .•• , xnu ) be its expected
Suppose the levels have been selected in such a way that
y satisfies the t- th degree polynomial equation
n
n
n
+ I:
1:: ••• I: c..
. x.
x. . .x.
i =1 i =1
i =l ~1~2"·~t ~lu ~2u
~tU
t
1
2
(1)
where
t < n.
Then in the corresponding
all interactions
Proof:
v4
.....
Consider an interaction
of the
a set
S
I
with (t+1) or more factors.
Take any two levels
i-th factor, i=1,2, .•. n.
of any k
vi'
C\)
Let
u.
and
~
n ~ k ~ (t+1).
Take
factors
dimensional rectangle
(al , a2 , •••
... x sn ),
with (t+l) or more factors, are zero.
involves (t+1) distinct factors.
I
AFE ( sl x s2 x
tt
!it
where a
i = 1, 2, ... k.
i , i , .•• ~ and consider the k2
1
'Whose
coordinates are given by
i
may take any of the two values
Corresponding to the rectangle
1\,
u
i
and
we can
define a contrast:
in symbolical form, the
contl~st
being obtained by opening the pro-
duct on the right symbolically, using the ordinary rules of algebra, and attaching all the combinations of levels of the remaining
n-k
factors to each of the combination of levels of k
factors
formed by the above nRlltiplication, and finally taking their expected value
i), i l
= 1,
y.
Thus, for example, if
in = 3, Vi
~
1
= Vi = 2,
2
u
i1
n = 3, k
= u.= 0,
~2
= 2,
si = 3(all
then (2) reduces
55
to
= y
=
r a 2 c2 -
a 0 c2 - a 2 c 0 + a 0 c)
(b2 + bl + b.0)-7
0
-
y (a , c ' b ) + Y (a , c ' b )
2
2 2
2
2 l
+ yea , b , co)
0
2
+ y (a , b , c )
0
0
l
+ Y (a , b , c )
2
0
2
+
+ y (a , co' b ) 0
0
-
y(a o ' b2 , c 2 ) - y{a o ' b l , c 2 ) - y(a o ' bo' c 2 ) - y(a 2 , b2 , c 0 )
-
y(a , b , c ) - y( a , b , c ).
l
2
0
o 0
2
The contZ'ast z(~) obviously belongs to the k-factor ini , i ,
l
2
teraction between the factors
... \
say I i , i ,
2
l
...
carrying in all
(s~ degrees of freedom.
of any two levels
If for
i
IS
1)
S, and given i, for the choice
and v. of the factor, we can show that
2
i
the corresponding value of Z(~) is zero, then we shall have
shown that
I.
J.
l
u
, i , ...
2
~
z(~)
only.
~
is zero, since the various
consist of the components of the form
Let
y(X1u ' .•. , xi_l,u' 6(vi , u i ), xi+l,u' .•. , XnU )
= y(Xl U' ..• , x.2- 1 ,U ' vi' x i +l ,U ' .•• , xnu )
Similarly, let
d.f.
~,
...
Y(X1u.' .•• , 6(V i , 1.1..1. ),
1
1
=
y(x , .•• ,
vi '
l1.1.
...,
1
x
ru
.., ., xn1.1. )
x ru ' .•• , 6(v.1. , 1.1.i 2 ),
2
...
, .•• , t:::.(v. , u },
1.
i
2
2
x )
n1.1.
(4)
etc., and in: general
xr'1.1.' .•• , A(Yi
,1.1..
), •..
It-l ~-l
x
't
, ... , A(V. ,
ru
=
1.1.
i
),
tt
Y(X ' ... , t:::.(vi ' 1.1.1 ), ... ,
1u
1
1
...
y(x
l1.1.
x
ru
x
.•• ,
, ••• , v:l.:.' ••• , x
tt
n1.1.
X
I
x ru ' ... , 1.1.~, ...
,
)
A(vi
,1.1.i _ ), ...
tt-l
tt-l
••• ,
r 1.1.
)--
... , A(Vi
, ••• , A(v , 1.1.. ), .•• , x , ,
1.
r 1.1.
1l
1
...
nu
k-1
' 1.1..
)
~-l'
...........
, ~~)
It can then be easily checked that
=
Z(R)
-k
Y(X1 , •. 0' A(V ' 1.1.. ),
1.1.
i1
1.
1
... , 6(Vitt ,
u. ), ... , x
~
n1.1.
)
(6)
Now, since by (1), Y is a polynomial in the
n
.•• , xn1.1. , the value of the quantity
on the 1. h. S
.
tained simp~ by
to
x
i1.1.
diff<:reo.eing
and substituting (vi
expression.
the
- 1.1. )
i
variables
0
~1.1.'
of (3) is ob-
r. h. s. of (1) with respect
for
x
iu
in the derived
Since the original polynomial is of degree
t, this
derived expression will be a polynomial of degree at nost (t-l)
57
in the (n-l) variables
x' '
Lu
Similarly the value of
(4)
••• , xi - 1 ,u ' xi+,u
1 ' ... , xnu .
is a polynomial of degree at most
(t-2) and contains all the variables except
vIe proceed on similar lines.
k
If k
~.
?t
= t,
and
l
The value of
polynomial of degree (t-k) in all but the
. •.
i
(6)
i .
2
then is a
k variables
i , i ,
l
2
the polynomial is a constant, and for
+ 1, it reduces to zero.
Hence Z{lk)
=0,
for
k?
t + l.
This completes the proof.
3.3.2. The above theorem can be further extended.
that
(1)
It can be show
implies not only that all interactions with (t+l) or
more factors are zero, but also some
involving t
d.f. belonging to interactions
factors or less are also zero.
We shall however not
go into this any further.
3.3.3
Next we shall consider the converse of the above results .
..;;;Th~e.;;..o.;;.;re;;..;;.;;;m;...3.3.3.l.
In the AFE (6 1 x s2 x ... x sn)' the assumption
that all interactions involving (t+l)-factors or more are zero,
is equivalent
xnu )
to assuming that the expected response
can be represented as a mixed polynomial of the form
w
2
r
= 1,
Proof:
S is the set of all w' s
2, •.• n, and at least
< 'Wr -< s r - 1, for
such that 0
(n-t) of the 'Wls are zero.
Consider any interaction betvreen k
k > t + 1.
wn
X2u .•• Xnu' .... (1)
••• w
n
where
Y(Xlu ' ..• ,
Let these factors be
i , i , .••
l
2
factors where
~
and the inter-
jl j2
jk
action be A=s. A
A~. Then the interaction*
i2
pressed as a linear function of terms of the for.m
Y(X1u ' .•• , x.
l'
:1.
- ,u
1
where
j'
(i) A r(
v i
x, r'
jk
ru ' .,.., 6. (
·be ex-
),
, v2 i
VI i ),
, r'
, r
x
r
= I,
= 1,2,
2, .•• k,
... , j'+l denote any
r
levels of the factor i , for
r j'
(iii)
), ••• , x
)
for
(ii)
ji (
6.
COD
Y(~u' ... , A l(
r
= 1,2,
j'+l distinct
r
..• k, and
j'
r(
), ... , A
), ... ,
),
j' -1
), .•• , A
r
(V.'+l
Jr
'
i ' ... , v 2 i ),
r
' r
j'
... A
k
j
... , A
(
), ••.••• x
nu
)--
I
k
( ),
xnu ) .
It is clear however, that the expression (2) involves at
least k
difference operations over the polynomial (1) involving
k
different variables.
Since no ter.m in (1) contains more than
t
different variable s, the value of (2) reduces to zero, as soon
as k? t + 1. Hence if the response law is given by the poly* For definition, see section 3.5.2.
nomial (I), then all interactions involving (t+l) or more factors
are identically zero, in the AFE (sl x s2 x .•• x
To justify the word
I
S
n
).
equivalent I , underlined in the statement
of the theorem, we must show that any k-factor interaction (k < t)
jl j2
jk
represented by Ai A. . .. A.
where 0 < j < s - 1 (r = 1,2,
r -
Jk
1:1.2
r
... k)
is not necessarily identically zero. To see this, we note
w w
w
that there exists a term
c
xl x 2
x n in the
wl ,'W2 , ... wn l:u 2u
nu
polynomial (1) such that
w.:1. ->
j
r
r
, r
= I,
wi' wi ' .•. , wi
are
12k
non-zero, and
2, .•• k.
Since we have not taken the coefficient of this last term to
be identically zero, this term will n.ot necessarily vanish after
jr
difference operations on the polynomial (1) with respect to the
variable
r
= 1,2, ...
k.
This implies that all interactions
J.renot necessarily identically zero.
This completes the proof.
As a consequence of the above theorem, we get the following
result which can be looked upon us
0.
DltlItttJli.'bution. If'rao the. theory
of (classical) factorial eA":P6riments to the the ory of interpolation
in n-dimensions.
Corollary 3.3.3.1.
Suppose that the function
y(x ' x , .•. xn ) can be expanded
l
2
as a Taylor series, and can be well approximated by the polynomial
(1) in a certain region
such that
~
R
l
of the
n-dimensional Euclidean space
contains a non-degenerate
n-dimensional rectangle
60
R within it.
Then there exists a set of points
T determined
by the assemblies of an FR of the AFE (Sl x s2x •.. X Sn)
which
(t+l)-.factor and higher order interactions are assumed zero,
such that by knowing the values of the function
T, we can calculate the value of y
3.3.4
in
at the points
y
at other points in
~•
The above results have been discussed here to justifY the
very study of the theory of fractional replications.
Frequently,
the experimenters come across situations in classical factorial
experiments where high order interactions are zero.
theorems seem to explain this phenomenon.
The last two
For example, suppose
the experimenter happens to be working in a region of the factor
space in which the res:90nse can be represented by a third degree
polynomial.
Then na.turally, he will find the interactions involv ing
four or more factors to be negligible.
3.4. General remarks on fractional replications.
3.4.1. We first consider FR's of SFE's.
Suppose there are
a.
n
n
factors.
We shall denote them by a ,
l
A treatment combination (or assembly) in which the
factor a
occurs at level j (r = 1,2,
n) will be denoted
j
r
jl ~2
by a
a
a n
Consider a set of T assemblies. Then we
l
2
n
define
jl
number of times the symbol a
i
j2
a
1
i2
occurs among the assemblies in the set
T.
(1)
61
Su:ppose we have an AFE(slxs2X'"
X6).
j1
j
<
r -
r
i -th, i -th,
1
jk
A . •• A~ , where
ie
s -l(r = 1,2, .••k); this being an intemction among t1:e
action "Will be 'Wrltten in the form Ail
o<
Then any inter-
n je
"0'
~-th
2
Consider a SFE(sm}.
factors.
Suppose we assume all (t+1)-factor and
higher order interactions are zero (a k-factor interaction means
Then let
a (k-1)-th order interaction).
m
NtCa )
=
number of e:r:t'ectsto be estimated.
It is obvious that
m
NtCa ) = 1 +
(~)
(s-l) +
(~)
(S_1)2 + .•• +
(~)
(s-l)t ..... (:;)
Also, if N is the number of assemblies used, then we must have
N ~ Nt(Sm)
(4)
.••.
When N is of the form
sr (we are considering an SFE( sm) ,
we have the well kncrwn Rao's inequalities:
(i)
t
even:
sr
(ii)
> 1 + (~) (s-l) + ... + (~) (s-l) ~
todd:
sr > 1 + (~) (a-1) + .•. + (~) (s_l)t +
(n~l) (s_1)t+1
(5).
In other words, we have the important result that the maximum number of factors
tion with
s
r
n, which can be accomodated in an orthogonal fracassemblies, each factor being at
the above inequalities.
s
levels, obeys
We shall denote this number by n (r,s)
2t
and net+l(r,s) respectively in the first and second cases.
62
For cases where
s
is a prime power, Bose has shown that
r
s -1
=
s - 1
=
8+2, if s
is a power of
=
s+l, if
is a power of odd. prime
=
2-1
n {4,s)
=
s2+l , if
n4(5,3)
=
l:t
n (3,s)
3
3
s
r
s
i
2.
m
3
experiment,
The above shows for example, that for a
where
6~
2
m ~ 11, the minimum number of assemblies required
to estimate orthogonally all interact:i.ons up to two·-:factors assuming
the higher order on~s to be negligibJe, is
we have to eot,::'ma:t,e
in
6
SFE (3 ).
243 effects in
35
.,.,
= 243.
Now by (3),
SFE (3~·..i..) and. only
Hence the compl(3tely orthogor.\'3.l
FR
73 effects
in the first
case is very economic, while in the second case, it is very uneconomic.
The problem in later sections will be to cut d.own the
number of assemblies while allowing a little correlation among
the estimates.
3.4.2.
We shall now state a theorem, which is well
kn01-1ll
in the
language of orthogonal arrays, and which 'Will be frequently useful
in later discussion.
Theorem 3.4.2.1:
at
1
~
S
s
Let there be m factors
a , a 2 , ..• am
l
each
levels, where s is a prime pewer. Suppose we obtain a
.
fractional replication T by taking in EG(m,s), k inde-
pendent.
~inea..r
equat.ions:
(1)
where all symbols represent element.s in GF(s ) •
combinations of the above equat.ions contain d
have non-zero coefficient.s
(i)
(for the case
d
gls.
= 2t.)
SUppose all linear
or more
Xl s
which
Then
all interactions up t.o
t-factors can
be estimat.ed assuming interact.ions of {1:.+1)-factors and higher orders to be negligible.
(ii) (for the case
d
= 21:.+1)
all interactions up t.o
t-factors
can be estimated assuming interactions of (t+2)-th and higher orders
to be negligible,
jl j2
jr
a
where 1 < r< d and
(iii) the symbol a. a.
J.
J.
1
l
2
r
>,.jl,j;' ...
times in
jl' 32 , . .. j r = 0,1,2, ... , 6-1 occur
>,.jl,j2'·· ·jr
the assemblies of the fraction T, where
is 1nde-
...
pendent of
(iv)
i , 1 , ..• ,i
l
2
r
~ jl,j2,···jr
".
=
s
,
for
1
j;
< r < d.
m-r-k
In the context of the above theorem, let us find t.he value of
.•• jr
...
i
r
64
r > d (for the case of the fraction
when
be the
s
elements of
GF( s ),
Let a:, .•. c:e 1
o
s-
T) .
and let the level
correspond to the element a:. .
of any factor
j
Then for finding the value of
J
jl,j2' .• ·jr
Ai.
i ' we have to find how many solutions of the equations
1'J.2 ,··· r
(l) are such that
jl .•• j
For obtaining
Ai
i
we therefore substitute the' values
r
1
(1).
(2) in equations
r
Two cases may arise:
Substitute the values (2) in equations (1).
Case I.
If the result-
ing equations are inconsistent, then
jl ... jr
A
•
i
... J.
r
l
Case II.
O.
After substitution we may find k'«
independent.
By
l
ir
...
k) equations to be
the previous theorem, we shall then have
jl .•• j
Ai
=
=
number of points on an (m-r-k') flat in m(m,s)
=
s
r
m-r-k'
..........
(;)
GF(s),
(1),
.
The above gives us
Lemma
;.4.2.1.
Let there be
k
equations in
which generate an orthogonal array of strength
d.
say as at
Let
S be the
set of all equations obtained by taking the linear combinations of
equa~ions
(1).
X. , .•• , Xi
J. l
S'
be k II •
r
Let
S'
be that subset of Sin which only
occur, and let the number of independent equations in
Then exactly s
r-k"
65
nonzero, namely those" for which
st.
satisfies the equations
equations (1) leaves k t
Further if the substitution (2) in
independent equations, then
=
3.4·3.
m-r-k t
.An important probleJJl which arises in the above connec-t1on
is the following.
Let k
s
SUpposethereare
n
factors each at
be the largest integer such that
tained by taking k
s
n-k
=sr
s
levels.
assemblies ob-
equations as at (1) in section 3.4.2, form an
orthogonal array of strength
d.
13 i , i • 1,2, ..• k denote
Let
m-vectors such that
13 1 = (gi' 8 i ' .•• , gi ).
12m
...
Let Vd
be the vector space generated by the vectors
131 , 13 2 ,
••• ,
13k . Two cases will be considered.
Case I.
d
= 2.
Let W
2
denote the set of all vectors in V which
2
have either 3 or 4 nonzero coordinates, and such that no two vectors
in W2 are dependent.
= (131 ,
~
Let
13 2 , .••
l~ (ai' a ), where
j
13k ); ai' a j
€
OF(s),
be the number of vectors in H whose i-th and j-th coordinates are
2
a
and a
respectively. Let
i
j
l~
Further, let
=
Max
i,j,ai,a j
l
= Min
~
f~ (ai' a j ).
l~
, keeping
d
and k
fixed.
66
Case II.
d
= ,.
such that exactly It.
V,
i~(ai,aj)' and
it
VI,
Here let
«~
denote the set of all vectors in
coordinates are nonzero.
exactly as above, replacing W'2
be the minimum value of this new f~
keeping
by
vI,'
Let
wer variations of {3,
d and k fixed.
The problem is to find
a.
Define
land
ft.
We give an outline of
f
method to find a rough upper bound for
first the case
d
=,.
Take a vector v ij 1..11 W,
of its nonzero elements are
j-th places.
and
a
and a j
i
Vectors of the type v ij
ft.
Consider
such that two
respectively at
i-th and
can t t have any other common
nonzero coordinate, and all of these must be independent and so less
than k in number.
f' ~
This S1ves us the inequality
For the case
i1
.L ~7
(k,
min
d
;:
2, define
........
-1 )
vij (in
w2 )
similarly.
(1)
Let
denote the number of vectors which do not have more than two fixed
nonzero coordinates.
Then
......
If we put
f;: f l
+ (2' we need a bound for f 2 ,
(2)
which is not
know however.
A knowledge of
t
and
i'
structing FR's (assuming up to
would be found useful in con2-factors interactions present)
with the help of orthogonal arrays of strength less than
'.5.
4.
Analysis of fractionally replicated designs for the symmetrical factorials.
67
3·5·1.
game general results.
Treatment c~mbinationscan be written,
jm
a , 0 < j < s-l" r=l,2, .•• m.
Consider an FE( sm) .
jl
j2
in the order a
a
1
2
If in the symbol for
we shall omit this
a
a,
r
m
-
treatment, the exponent
from the synbol.
used for interactions except that
¢
by
If
j
j
r
r
of a
= 0,
r
is zero,
for all
r,
¢. Exactly similar notation will be
the symbol will be written
and
r-
a's will be replaced by A's,
Il.
It is well known that each interaction of
(1. f.
can be ex-
pressed as a linear contrast of all treatment combinations:
kl k2
A A
l
2
Let
Akm
m
jl,j2" . ·jm jl j2
jm
k
k a . a ... a
... (1)
.,
j
-kl , 2""
1· 2
m
J l ,J 2 ,··· m
m
=!:
d-
A denote the column vector of all interactions in the
natural order
,. Ais-lA2s-l ...
Let
a
denote the column vector of
a's in the same order.
Then equations (1) could be written
A = D~ ,
in matrix notation, where
.•••.••••
D is an
sm x sm matrix.
(2)
It is
know that the sum of products of the corresponding elements in
any two rows of
D is zero.
Let
ei
denote the sum of squares
68
of the elements in the
divide
at
i-th row of D.
5 ..
i-th row by
J.
Let 6
=
6 D
is an orthogonal matrix.
6 A
= C' 6 A
a
Then
• • • • • • • • • • • ,11
Thus we have
~D ~
=
D orthogonal, we
-1
m
m
maii.d.x having 5
s x s
i
be an
(i, 1) place and zero .elsewhere.
C
To make
= C ~
=
(~D )
.6 A
I
= DI 6 I 6
A
. • • • •• ( 4)
A be partitioned as:
Let
=
A
(~}
o
where
the
L is the vector of all interactions up to and including
v
2·· factor intera.ctions (say
in all in number),
is the V8ct-::C of hig:1.er order interactions.
If all
higher order inteTactions are assumed zero, then
=
a
where
(D' 6' 6)
o
Now 6' 6 = 6 , and so write
rows e,nd D'
o
3-factor and
get
L,
o
is the matrix obtained by cutti..ng out the
2
2
o
(D' 6' 6)
requisite ntL~ber of columns fram
obtained from 6
W~
and I
(D' 6' 6)
(D' 6.6)
corresponding to
= D' 6
2
000
, where
6
2
0
I .
o
is
by cutting out requisite number of coltwms and
is the same as
'With last (sm - v) columns cut
D'
out.
Hence
= D'o
Let
if!
6,2
0
L
be obtained from
~
by om.:ittL'1g certain treatments
and E' the matrix obtained from D'
o
by omitting 'With the help of
(2), the rows corresponding to the treatment combinations omitted.
Then we get
............
E' ~2 L
= E' -P say
o
It is easy to see that a knowledge of
that of
L.
~
is equivalent to
Let if. be the vector of observations such that
...........
= 1.*
(X)
Exp •
(6' )
Then the normal equations can be written (Bose
E
and if
(6)
E E'
E'
R =
E
["12_7),
X
is nonsingular, we get
A
~
as the solution.
1\
L
(
=
E E')
-1
E X
........
(8)
From this we sha.ll finally get
=
-2
~o
1\
R
Hence the main problem in obtaining
L, the vector of interac-
tions in which we are interested, from. X the vector of observations, is to invert
(E E').
't-le will repeat that
E is the matrix
obtained from. D by omitting the last (sm. -v) rOl'TS (which correspond to
3-factor and higher order interactions in A)
and also
omitting all the columns which correspond to treatments on which
observations are not 'taken.
Then l
v x v.
is
Suppose N observations are taken.
N x 1, and E is v x N, so that E E' is always
We 'Will write
B
=
E
X
...........
(10)
70
It is interesting to see that
1? may be interpreted as the
value of the vector of interactions in which we are interested if
those treatments on which observations are not taken are omitted
from calculations.
vle can then 'Write
(11)
:5.5.2.
We will next calculate the elements of E E'.
we need to know the coefficients
d in
For this
(1), section 3.5.1. We
denote by ~ls the set of all possible assemblies when each
factor is at
levels.
s
j
Let Carr)
.r1s
denote the
sum
of yields of all treatments in
j
in which
a r
occurs, for
r
r
= 1,2, ...
m,
0
< j r -< s-l .
-
Then we define the main effect
for all r
where
s-1
.E
dj(k)
= 1,2,
=
0,
kt
~
< k < s -
1,
... m,
0
0 ~
s-l,
.•... (1)
j=o
and
8-1
.E
dj(k) dj(k')
j=o
Further, we define
d.(o)
J
=
1,
= 0,
for all
k ~ k',
0 ~
k, k' < s-l.
(2)
j.
k
1
k
k
Then in (1), section 3.5.1., the interaction Al ~2 ..• ~ is
defined such that
71
(4)
We shall su:p:pose that in the vector of observations
r,
the
comHe
shall also write
......
=
where
13 I S
A. (13
o
a. + 13
o l.l
are real numbers.
1
a
1
+
i1
Denote also,
Also, we adopt the usual multiplication rule:
x2
) ( 13 2 a i )
i1
2
Xl
(13 1 a
=
f3l~2
x2
Xl
ai
a.
1
,.......
J.2
the multiplication being commutative w.r.t. symbols
a~J.
However, we assume the distributive law to hold, i.e. for
Xl
Xl
x
~
2
(13 a
f3i a 1) (132 a
+
+ 13 21 a i 2)
1 i
i
i
1
1
2
2
=
I
x
x 1 x2
Xl
2
a
+
+ f3i 13 2 a i
131132 a i
ai
i
1
2
1
2
I
1\ f3~
Xl X 2
a
a
i1 i2
(6)
72
X'
13i?2
X'
•
.
...
,. "
ai 1 a 2
i
1
2
We al?Sume the function A is addit.ive with respect to
+
Example.
Let
'(2
~
a l0
3,
s =
+ ~~ a 2)
l
A.
2
1
(al - 2al +
=
'"
m =
4.
Then
2
,0
~l
+ ~~ ~
.~
=
a~) (a~
-
a~)
o 2
{ 2 2
1
2
. a l a 2 - 2 a l a2 + a l a 2 -
= ",22
12
-
-
2A.l~
",02
12 + 12
Now, every raw of
ai a~ + 2 a11a °0 - a 1o·a 01
0
20
2",10
A.lo +
10
-
",00
10
E and hence every column of
ponds to exactly one element of L.
Hence the element
cell (i, j) of E E' corresponds to the elements in the
j .. th row of
E' corresE
ij
in
i-th and
L.
Consider any two elements of
Let
j3 j4
a.
a.
~3
be the element in
~4
)
jl j2
j3 j4
L, say Ai· Ai
and A. A.
1 2
~3 ~4
(8)
E E' which stands at the intersection of the
j
row corresponding to
A. l
~1
j
Ai 2 and column corresponding to
2
For the evaluation of the said element, many cases
73
arise, lihich are presented in the following Lennna.
Lemma. 3.5.2.lo
Case I. ( i , i )
l 2
has no factor connnon 'With
jl j2
j3 j4
(a. a i ' a i a. )
~l
2
3 ~4
€
Proof:
::: "-
The value of the
[4
1I
r:::l
loh.s.
s-l
(z
k:::o
j
j
1
L.
2
2
in
L, and the row R
2
r
k
Then
l
ai )
r .~
of (9) is obtained by multiplying
the corresponding elements in the row
Ail Ai
\(j)
( i , i ).
4
3
Rr against
the element
against the element
j3 j4
Ai Ai
3
in
4
The elements in these two rows corresponding to any element
~,
are respectively
,···K
and
where, as indicated,
jl' j2' j3 and j4
11,···,li ,···f
m
i4
3
d
o,0, ... o,j3'O, j4'o, ..• o '
stand respectively at
il-th,
i 2-th, i -th and i~.-th places. Now using (3) and (4), these two
3
elements reduce respectively to
The prod.uct of these two is 8, where
8
= d f . (jl)
~1
d
ti 2 (j2)
di. (j3)
~3
d
ti4(j4)'
The number of times we ge-t the same product
two corresponding elements of R
l
of times the symbol
and R
2
8 by multiplying
is equal to the number
Occurs in the tree.tment contained in the vector ¥..
By definition,
this number is
li
1
l(
4
:::
A,r1t
'r=1
as desired.
Then
s-1
( !:
k:::o
~(jr)
a~
a~
r
)
a
ti
i
_7,
2
2
a
'i
3 a (i4 ]
1
4
3
1
75
..
J
2 .
= Arn
(10)
- hl
Case III.
say 11
=
(1 , i ) has exactly one factor cOlllIl1on nth
1
2
=i4
(1 , i ),
4
3
Then
s-l
A
F( }:
k=o
Proof of cases II and III follows on lines similar to case I.
Given any fraction T, the results of the above lemma nil be very
effective in evaluating the matrix EE') which is to be inverted
for the analysis of T.
3.5.~.
Some further remarks.
From (8) and (9) of section 3.5.1., we get
1\
Var (L)
I'Trite
V
Then V is a
=
=
(1)
.........
1\
Var (L)
v x v matrix.
If any element in V is zero, the
corresponding element in (E E,)-l is zero and vice versa.
Let us agree to call a symbol of the form
jl,j2'
jr
A1l ,i , .•• i
2
r
76
a
A ot' order r.
Then it is easy to see that
is a linear contrast among A's ot' order
(9), section ;.5.2,
4. Similarly (10), section
;.5.2., is a linear contrast among A's of order 2, and (11), section ;.5. 2., among A's of order;.
In this light, the frs,etion
T can belong to two classes:
(i)
Fractions in which for all r ~ 4,
jl,j2'
jr
A..
i
~1'~2' ••.
r
for all possible
=
Ai l' i
ir '
2' .••
jl' j2' .•• jr,the right hand side depending on
iI' i 2 , ••. i
only. For such cases, the matrix E E' shall, in
r
view of the above remarks, be a. diagonal matrix. The fraction T
will in this case be called orthogonal.
(ii)
Fractions in which for all r < 4,
jl,j2'
A. i
jr
i
~l' 2'
r
=
jl,j2'
A
jr
,
for all possible
iI' i , .•• i , the right hand side depending on
2
r
jl,j2' .•. , jr only. In this case, tha matrix E E' will be an
invariant symmetric factor matrix, and will be analyzable by the
methods of chapter II.
The fraction T in this case, will be
called balanced.
;.6. Fractional designs of the class 2m and ;n
;.6.1. Evaluation of matrix E E' •
We refer to section ;.5.2.
the values
dO(l)
0, 1 only.
=
-1,
dl(l)
Then, lemma 5.2.1.
We have
s
= 2.
Thus k
Define
=
1
(1)
gives, for the elements of E E',
takes
77
N being the total number of observations.
.
.
=
€
(A A
f
A A)
Kj
i'
€ (Ai' Ai)
= ~~ijJ~ij
11 + ~oo
= €(~Aj'
=
AiAj )
=
~ 01
..
~ij
..
A10
ij
......
........
N
(4)
€
(Aj , AiAj )
€
100
olo
001
110 ~lol _ ~oll ..
( Ai' A ~ ) = A111 + "'ijk
+ "'ijk + "'ijk - Aijk .. ~ijk
~ijk
j
ijk
€(f.,L, Ai)
..
~ooo
'~ijk
.•..•.••.
(5)
Finally
1
0
)
€ (AiA ., -K
A A(/) = '" (IE
(a
..
a
X
• • k (/
r
r
J
r=J.,J,
,x
(6)
).
Given the fraction T, we obtain first the
",IS,
and then
with the aid of the above equations, evaluate E E' ,
3.6.2.
d,6.2.l.
Analysis of balanced nonsingular designs.
We refer to section 3.5.1.
Consider equations (7) and (10).
It is easy to see that we can write 12 and
12'
~
in the form
= (p ; Pl,P2' .. . ,p ; P12,P13"",Pln,P23"",Pn-l,n)
0
n
B' = (B ; B ,B , .. . ,B ; B12,B13,···,~,B23,···,Bn_l,n)
0 l 2
n
(1)
We shall solve the normal equations at (5), section 3.5.1.,. 'Which
can be
in the form
~ut
(2)
.....
",here
Po
=
:p. , P = . E
.E
i
1
:p., P
all i(tj) iJ
iO
=.E
all i,j(itj)
00
(2')
Pi .... (3)
J
Define similarly
Q
The
=
o
.E
i
B., Q.
1
10
=.E
j
B ., Q
=.E
iJ
00
ij
(4)
B-fJ'
...
e's in (2) are given by
=
£1
=
£ (~, Ai)
£2
=
€
E
=
E(A., A.A,),
3
€(Ai',Aj )= €(l\:Ai , ~Aj)
=
(IJ., AiA)
1
£ (Aj , AiAj ),
J A
€4
=
€(AiA.,.A.. A I/)
J -K x
Actually, the matrix E E' is of the form
N
€l
€l
£1
£2
E
......
N
€2
€2
£1
€1
(;1
€3
... €2
£1
£3
£1
N
£2
N-
S 1l!lDletric
2
£3
~
...
€2
€2
£3
£1
€3
- - - - c. -- - .- eI. - - -€l.-
2
2
4
4
N
N
79
FrOlU (2'), we get on adding over all
1J
€1 Po+ n €2P o+(N.-€2) Po+ n €3 Poo + 2(€1-€3) Poo
n
= go
or
n €lPo + (N+(n-l) €2) Po+(2€il-(n-2) £,) Poo
= Qet
Similarly, adding (2") over all 1 (h), we get
(n-1) €2 Po +(n-l) €3Po+{€1-€3)(Po-P j ) + (n-1){€1-£3) Pj
+ (n-l) £4 Poo +(£2-£4)(2 Poo~Pjo)+(n-l)(€2-€4)PjO
+ (N-2€~ + £4) P.
.:::;
or
JO
=
Q.
JO
,
(n-l) €2 Po + {€l+ (n-2) €3} Po+ (n-~)(€1-€3) Pj
+ (N-2€2 + €4 +(n-2)(€2-€4»
PjO +
{(n-3 ) €4 + 2€2} P oo = Qjo' .(8)
Adding (8) over all j, we get
n(n-1) £2 Po + n{€l +(n-2) €3} Po+n 1(n-3 ) £4+ 2€2} P oo
+ (n-2)(€1-€3) Po + 2 (N-2€2 + €4 + (n-2)(€2-€4»
Poo
= 2Qoo
or
2
n(n-l) €~o + Po f £1(2n-2) + €3(n -2n-n+2>-7
+ PooLr£4(n2-3n+2 - 2n+4) + E (2n-4+2n-4) + 2N7
2
or
2
2
(n _n) £2 Po + Po f(2n-2) €l + (n -3n+2) €}-7
2
+ Poo 1€4(n - 5n + 6) + £2
I
4(n-2) +
'2N_7 =
2Q oo
= 2Q00
80
Thus frOLl (2)" (7) and (9)" we get
N
Po
=
Po
2
(2n-2)€1+(n -3n+2)€3
go
2N+4(n-2)€2
+(n2-5n+6)€
(10)
N-2€2+€4
+(n-2)( €2-€4)1
Q
i
€2
€l
€3
·pol-\
P
x
Q
io
(n-l)
€2
€1+(n-~)E3
(n- 3 )€4+
P
+2€2
.......
I
0
00
(11)
Thus the problem of inverting (E E') reduces to that af inverting t.he
:; x 3 rna trix in (10) and the
2 x 2 rna trix in (11).
Having obtained Pi' Pio' Po' Po and P00' we obtain Pij from
(2").
The method of inverting (E E') given here is an alternative
to the methods of chapter II.
).6.2.2.
Near-orthogonal fractions.
VIe will now consider balanced fractions which are very
I
_2Q09I
......
Similarly" from (2') and (8)" we get
I
130
2€1+(n-2)€,
N+(n-l)€2
n€l
2
(n -n)€2
Poe
-1
€l
81
economic and Which approach orthogonality when the number of facFrom section .3.4.1., we
tors becomes large (say above seven).
find that the number of effects to be estimated is
n
2
N2 (2 ) = N2 say = ~ (n + n + 2)
In what follows, the notation will be
(12)
¢ = all factors at level zero,
a i = only i-th factor at levell,
a .. = only
i-th and j-th factors at level
~J
0,
others at level 1.
The response to treatment shall be denoted by the same symbol
as the treatmeniB ihenselves.
Further, the sums over the re.sponses
of certain treatmenis will be written as
n
= .E1 a.,
S.
o
~
~o
S
~=
..
n
= E
j=l,jfl
ai'
~
, S00 =
E a : . . . •• (1.3 )
~J
all i,j(ifj)
It can be checked that
2
Soo =
n
E
i=l
............
S
io
(14)
The estimated general mean, main effects and two-factor interA
A
A
actions will be denoted by fJ, A., A..A . The series of fractions
J.
J. j
presented in the next theorem may be called U •
l
Theorem .3.6.2.1. Suppose, for our F'R, we take the treatments ¢,
a i (i=1,2, .•• n) and
aij' (ifj) = 1,2, ... n. Then all desired effects
can be estimated, and
(i) A
1
Ai = 4(n-.3) ..[(n-4) a i +
2
n.. 2
82
.......
(ii)
A
n
V(~) =
(iii)
~
If
2
_ 12n3 + 610 - 138n + 120
16(n-3)2 (n_2)2
.....
(18)
2
n* - 12n3 + 610 - 146n + 136 2
2
2
~
,
16(n-3) (n-2)
=
where
4
(16)
2
= variance
r
(20)
per observation
stands for correlations, then the follo'Wing corre-
lations tend to zero for large
n:
(21)
This implies that for large n, the design is near-orthogonal.
Proof:
In this case since N
= N2 ,
we invert E'
instead of E E',
since E is a square matrix, i.e. we solve the ordinary equations
of ex,pectation which in this case is found easier.
Now, we have, from (8), section 3.. 5.1. ,
A
l
,-1
l' = E,-l E- EX = E
'l.
or
'l.
=
1\
E' l'
.......
(22)
.......
(23) .
Written in full (23) becomes
¢ =
ai
=
-
~
~
To + T00
-
(To - 2Ai ) + (Too - 2T io '
(23.1)
(23.2)
8;
aij = \. L + (To-2Ai
- 2A) + (T oo- 2Tio- 2T jo+ 4Ai Aj
.•.•.
)
(2;.;)
where
T
o
=E
i
Ai' T
00
Adding (2;.;) over
=
E
(i~j )
j,
(AiA.),
J
T. o
~
=
(24)
E A.A.
j]. J
we obtain on rearranging terms,
810
= (n-1)\..L
+ (n-;)0
T + (n-5)
T - 2(n-2)
0
0
Ai - 2(n-4) T10 .... (25)
Adding over 1 in (2;.2) and in (25), these two equations together
with (2;.1) give, in matrix notation,
1
-1
n
-(n-2)
1
.... (26)
=
(n-4)
2
n -n
2800
If the ; x; matrix on the
loh.s.
then we can verifY,that;
2
-n +lon -29n+24
-1
1
2
r
='2
2n -6n
4(n -5n+6)
of (26) is denoted by
2
n -7n+10
-4n+8
2
-n +;n-2
r,
1
2
.... (27)
1
From (26) and (27), we get the value of \. L as at (17), and also
of To and Too.
From (2;.1) and (2;.2) we get
(28)
so that using (25) and eliminating T , we get
io
.
which using (27) and (26) gives the result (15).
84
Again, using (2;}Jwe have
4 AiAj
=
aij -
a i - a j + 4(Ai +Aj
)
+
I.l. -
which using (15), (26) and (27) gives (16).
(;0)
;To + Too'
This completes part (i).
For part (ii), the variances can be verified directly
by
using
(15), (16) and (17) and noting that
V(¢) = v(ai) = v(aij ) = rr 2 = variance for observation
=n
v(so)
rr 2 , V(soo)
= n(~-l)
rr2 and V(SiO)
= (n_1)rr 2
(;1).
To verify part (iii), let m denote any number which is of
d
d
the form k. (n ), where k is a constant, i.e. md is of order
o(n
d
).
Then (15), (16), (17) can be expressed i~ terms of order &3,
"Ai
= m- 1
"I.l.
=
¢ + moJ.
a. + m- 1 (s 0 - a i ) +m- 1 Si0 + m- 2 (8 00 - S10 )
m1 ¢+m S +m 2 S
o 0
00
In terms of order, the variances at (;1) become
Veai)
= V(¢) =
V(a J.J
.. )
=
m,
V(S 0 )
0
=
~,
.1
Then we get from (;2),
=
m 2 +m +m 1 +m 1 +m 2 = m .
0
0
85
=
=
m 2 +m 2 +m l +m
•
•
0
m
0
A
V(\-l)
= m2mo + moml + m.?2
= m2
Similarly,
h "Ai) = m 1 m.. + m m + m,m + m 1 m 2 = m
cov (Io!,
~
0
0
.~ 0
••
0
" "
cov (Il, A..A.)
= m 1 ml + m• 1 m0 + m-~.
,m 2 + m.~..m• 2
~ J .
"
1\
cov (Ai' Aj )
=
m. l m. l +
= m0
o m. l + m. l mo + m. l m. 2 +
In
In.
2 m. l
=In.
(by similar calculations)
Comparing the covarial1ces 'With corresponding variances, we
find that all correlations are of the order m.
l
or smaller.
This
completes the proof.
In the series of fractions presented in the a.bove theorem, no
degrees of freedom are apparently available for the estimation of
error mean square.
However, this ca.n always be done by introducing
one or two dunnny factors, (each at two levels) as explained in sec·
tion 4.7., and taking a fraction from the same series (U ), with
l
(n+l) or (n+2) factors instead of only n
3.6.3.
•
factors.
Estimation of main effects when interactions up to two
factors are present .
l
86
f""l
€ 11,
Let
Let
'W
denote all the assemblies in the
'Where
Xl
i
Then
w
assembly of
= (~,
=
0
if x
=
1
if xi
we have that
Lemma
=
1
=
0
factorial.
Let
for i = 1, ..• n.
'Will be called the complementing
For
2
n
w€
T, and if
designs, a set T of assemblies 'Will
Ll,:?,7
symmetric) if for every
occurs
\1
t
'W €
T,
w also
times in T, then
times in T.
3.6.3.1.
n
Consider an SFE(2 ).
containing n assemblies or more.
l
~
••• , x ).
n
2'
n
'W.
be sa.id to be symmetric (or
t
i
x
Definition 3.6.3.1.
occurs
X
2' ..., x~)
(xi,
=
'W
2
Let T be a symmetric set
Then by using T, any set of
d.f. (n ~ n ) belonging to interactions involving an odd
l
2
number (even number) of factors can be estiDJa.ted (assuming
E El
to be nonsingular) orthogonally 'With respect to any set of inter-
actions involving an even number (odd number) of factors.
Proof:
We refer to section 3.5.2.
Consider the matrix E E l
•
Let
A A ·· A~ and A A ... A
be any two interactions, where
jl j2
il i2
jl
k is odd and l is even. 't'Je must show that the element in E E l
in the row corresponding to
~
Ai
. .• Ai
and the column corres-
12K
ponding to A. A ... A
is zero. Using a generalized form of
j
Jl j 2
K
equations (9), (10), (11) in section 3.5.2., and equation (1), section 3.6.1., we can write
A
j
l
)=~f II
r
(1)
87
where
I'
runs over all be
Ar s
",hieh
arc
oither
in
A. , both not in both.
Ai Ai . .. Ai.. or in A. Aj
Jf
12K.
Jl 2
of· Als. Hence
I' shall run over an odd number
Obviously
(2)
The lemma will be proved if we show that
A
r
-
neal - a
I'
I'
O
)
1'-
7
=
A
rn:I'
L
(a
o
I'
l
- a )
7
1'-
This is true, however, since the set T is symmetric with respect to
o and 1.
Corollary 3.6.3.1.
of assemblies.
n
Consider a 8FE(2 ).
Then if we assume
Let T be a synJIIletric set
3-factor and higher order inter-
actions to be negligible, we can estimate the main effects in the
presence of two factor interactions.
3.6.3.2.
The series U
of fractionally replicated designs for
2
n
8FE (2 ).
These are meant for estimating main effect and the general
mee.n wen interactions up to two factors are present.
of construction of these designs is as follows.
(v = n, b,
1',
k
"
.
1. 2
n-k varities say V , V
j
j
1
2
Then corresponding to this we define an assembly
does not contain the remaining
n-k
Suppose a
varieties say Vi ' V. , ••• , V.
1
Vj
Take any BIBD
A) 'With equal or unequal block sizes.
certain block contains
The method
and
~
, .•• ,
88
Thus we get a set T of b assemblies.
such that T is symmetric.
We choose the BIBD
Since v7e have to estimate n+l effects
we must have
b> n=1.
vie choose the BlBD suchthat b
Since a
BlBD with v
is suitably near to n+1.
can be constructed from BIBD with
vQ,riet~es
v + v' varieties by cutting out v' varieties, designs belonging to
the series U can be obtained for any value of' n by using the
2
known BlBD's.
n
3.6.4. Fractional replications from the 3 series.
n
Consider a fraction T of the 3
Definition 3.6.4.1.
factorial.
The fraction T will be said to be (2,0)
symmetric, i.e. symmetric with respect to the levels
if (i) x e T, implies
xe
T, 'tmere
X is obtained from x by in-
terchanging the levels (superscripts) 2 and
and (i1) if x occurs
t
2 and 0,
° in the assembly x,
times in T, then
x
also occurs
t
times in T.
Lemma 3.6.4.1.
Suppose from a
SFE(3n ), we take a set T of assem-
blies which is (2,0) symmetric.
Then in the matrix E E' to be in-
verted for solving the normal equations, we have
€(~, Ai)
= €(Ai , Aj~)
= e(Ai ,
222
= €(A , Aj Ak) = €(~, AiA )
i
j
2
=
e(Ai Aj , l\..A,)
=
0,
= e ( Ai2
Aj
,
A~)
2
Ak2 At)
Proof:
Follows on the lines of Lennna. 3.6.3.1-
3.6.5.
Analysis of balanced and (2,0) symnetric fractions from the
n
SFE (3 ).
Suppose we have a set of assemblies
'With respect to the levels
2
and O.
T which are symmetric
From Lemma 3.6.4.1, it is
easy to see that the normal equations are now broken into two parts,
one corresponding to the two sets
corresponding to the four sets
1A~ A~}.
iAi} and
~A~ Aj
11J.} , {A~}
,
}
and the other
\ AiAj ]
and
Thus the two mtrices to be inverted are
(1)
M n{n-l)
3
M'2
n(n-1)
n
and
1
n
}.i'
5
M'6
(2)
M'
7
1
n(n-1)
M'
9
2
12
M13
n(n-1)
n(n-1)
2
2
M
n
n(n-1)
2
90
;.6.5.1.
First, consider the matrix (1).
denoted by Mis.
The matrix M;
It has two submat:i'ices
corresponds to the effects
2 / 2 }
{AiAj } ' ~ relates to {Ai} and lAiAj
J
etc., which are
indicated above and to toe left of the matrix.
The orders are
indicated below and to the right of the matrix.
The elements in the cells of these matrices are
earlier.
Here we note that since
T
is balanced, we can write:
e(Ai , Ai)
= Yl ,
2 , AiA
2 )
€ ( AiA
j
j
€(Ai , Aj )
= Y2 ,
€(Ai ,
2
e(Ai , AiA j }
=
2
2
€(AiAj , Ai~)
= Y5'
2
2
€(AiAj , AiAk)
=
( 2
2
€ AiAj , Aj~)
= Y6
the case
s
= ;,
A~Aj)
= Y2'
=
2
2
€(AiAj , AiAj )
Y6'
=
Y ,
2
( 2
2
Y6' € Ai Aj , AjAk)
=
Y
€(A~Aj'
=
Y8'
and
for all permissible i, j, k and
The values of
= Y4;
2
€(Ai , AjAk)
Y;,
e's defined
{Af)
7
f.
y' s are obtained by
l.:.G :1ng
lemma ;.5.2.1
for
taking
(4)
do(2)
It 'Will be noted that certain
in the case of
Y
2
and Y6.
=1
€'s are equal, as for example,
That this is so can be verified, by
91
by using lemma 3.5.2.1.
Both the matrices (1) and (2) can be inverted by solving
equations (7), section 3.5.1.
We shall present the solutions that
are obtained through the use of this method.
For matrix (1), these equations can be written as
-..,
~
:j
M'2
{Bi
{Pi}
1
n
(5)
=
{B~Bj}
1piPj
n(n-l)
where the curly brackets denote vectors (of orders as indicated
and of form as those of
[A~Aj ~
{Ai}'
) etc.
In order to so:Lve these equations, we first write:
Ri
=!:
all j(~i)
2
p.p.
!:
=
all i,j,(i~j) ~ J
and
!:
all k(~j,i)
2
PiP. ,
T
=
Pi
=
i
J
~
G,
all i
P.
~
1\2 =
!:
P p~
all j(~i) i J
2
U,
~
p.
all k(~j,i) ~
1\ = R.. ,
i
=
(6)
Tij .
Let primes denote the corresponding sums over B' s.
R'
~J
!:
all j(~i)
u'
=!:
all i(h)
B.
~
Thus:
etc.
We first obtain U and G from
(8)
where
811
=
812
=
Yl + (n-l)Y2
Y2 + Y + (n-2) Y6
3
and
2
Y4 + Y2 + (n-2)(Y + 2Y6 + Y ) + (n _ 5n + 6) Y8
7
5
vie then obtain :Pi" Ri and Ti from
022
""
where
P2l.
= (Y3
- Y6)+ (n-l) y 6
~ = (Y6'" Y7 0"31
==
?YS)
+ ~.n-l) Y8';
(lV'-1) (Y3- Y6)-(¥e • y6!) ,
0"3e =(!2 .,;. 2:16 +Ya) .. (Y; .. Y8}+ (n-l)' (Y6 - Y8)
,0":53
133+
= (Y4
= eYe
- Y5 - Y7 + Y8) - (Y6 - Y8) + (n-1) (Y - Y8)
7
-Y6) + (n::-1)Y6
and
~2 F (Y5 + Y6 - ~8) + (n-1) Y8 '
(11)
93
from
Pi (Y2 - Y6) + Pj (Y3 - Y6) + Y6 U + Ys G
(12)
+ .(Y5 - YS) Ri + (Y6 - YS) Ti + (Y7 - YS) Tj
P.P~=B~j'
J. J
J.
2
+ (Y6 - YS) Rj + (Y4- Y5- Y7+YS) PiPj +(Y2-2y~yS)
and the equation obtained by interchanging i
and
j .
2
The 2 x 2 matrix involving coefficient of Pi Pj
and
in (12) can be written
'Where
(14)
and
On the basis of the above results, the condition for the non-
singularity of matrix (1) can be summarized as below:
Lemma 3.6.5.1.
In order that the matrix (1) be non-singular, it
is necessary and sufficient that the three matrices
be nonsingular.
3.6.5.2.
Consider now, the matrix (2).
Here we write
22
e(Ai , Ai) = u 2 '
2 2 2 2
e(AiAj , AiAj ) = u ' e(Ai Aj , Ai Aj ) = u4 ;
3
2 2 2
e(~'Ai) = vl ' e(~,AiAj) =v2 ' e(~,AiAj) = v3 '
= ul
e ( ~,~)
'
2 2 ) = v ' e ( Ai'
2 AiA ) = v ' e ( Ai,Aj~
2
)
e(Ai,A
j
j
2
3
222
222
e(Ai , AiAj ) = v5' e(Ai , Aj:Ak) = v 6 '
e{AiAj
,
Ai~)
~Ai)
= v7 ' e(AiAj ,
e(~Aj' AiAk
2 2)
= v4'
e;
(
2 2 2 2
e(AiAj , AiAk)
= vlO
( 2 2
' e AiAj ,
AiAj ,
2 2)
Pili
= v4
'
(17)
= va '
= v9
'
.Ak2 A2)
f = vll
•
The equations to be solved. can be written as
e
M4
M
5
M6
~
{~}
{¢}
M'
5
Ma
M
9
M
{pi}
tBi1
M'6
M'
9
M
M
{BiB
M.7
Mio
M
M
13
{B?~
ll
12
10
12
a notation similar to the previous one being used.
Define
(18)
jJ
,
95
with primes for sums over B's,e.g.
V'
=~
B~,
i
etc.
J.
We first obtain
I-L, V,
81 and 82 from the equations
¢
I-L
V
81
6
(20)
=
,
2
where
Q
= n(n-1)
v2
Q
= 2(n-I)
(V2 - v4) + n(n-1) v4
Q
= 2(U3
Q
= n(n-1)
V
3
Q
= 2(n-I)
(v - v6) + n{n-1) v 6
5
Q
= 2(V2 - 2V4 + v9 )
Q
34
= 2(u4
-
9
42
= n v3
+ (u
43
=n
2I
22
23
31
32
33
9
and
- 2V + va) + 4(n-I)(V -VS) + n{n-1) va .
7
7
+ 4(n-1} (v4 - v ) + n(n-I) V
9
9
10 + v11 ) + 4{n-1) (v10- vll ) + n(n-1) v11
2v
2
- v )
3
v4 + 2(V2 - v 4 )
Having obtained
81 , 82 , V, we substitute them in the
2
following equations and get the values of p.,
wi and z..
These
~
~
~,
equations are:
11:
11
11:
11:
12
13
2
Pi
sll ..... S14
B~~
~
V
21
11:
31
11:
11:
11:
+
23
1'1
33
Zi
22
11:
32
11:
i
;31 ..... [;34
8
1
8
2
=
't-l'
i
Z'1
(22)
where
11:
21
= (n-2) (v 2
-
v4)'
11:
22
= (u3
- 2v7 + va) + (n-2)
11:
23
= (v2
- 2v
11:
31
11:
32
11:
33
va)
4 + v9 ) + (n-2) (v4 - v9 )
= (n-2) (v 5 - v6)
= (v 2 - 2v4 + v9 ) +
= (u4
(v7 -
(n-2) (v4 - v )
9
- 2 v10 + v 11 ) + (n-2) (v10 - v 11 );
£11 = v 1 ' s12 = v3' £13 = v4' s14
s21
= (n-1) v 2 '
£23
= 2(V7
[;24
= 2{V4 -
s31
= (n-1) v3'
£22
= (v 2
-
v4)
= v6'
+ (n-1) v4
- va) + (n-1) va'
v ) + (n-1) v
9
s32
9 '
= (v 5 - v6)
+ (n-1)
v6'
97
and
Finally the values of PiP j and P~ P~ are obtained by
substituting the values of the guantit1es obtained earlier in the
following equations:
2 2
~ll PiPj + ~12 PiP j + v2~ + v4 V +
2
2
+ va 81 + v 82 + (v2- v4) (Pi + Pj )
9
7 - va) (l~ + I'S) + (v4 - v 9) (zi +
+ (v
~21
PiP j +
~22 P~ P~
2
+ (v4 - v ) (-'1'1 +
9
rJj
) +
= Bi
Bj
+ v31J. + v6 V +
2
v 8 1 + v ll 8 2 + (v - v6) (Pi + Pj
9
5
+
Zj)
(24)
) +
2 2
(vl0 - v ll ) (Zi + Z) = Bi Bj
,
where
~ll
= U3 -
~12
= V2
s21 =
V
7 + va
2v
- 2v4 + V 9
2 - 2v4 +
V
~22 = U4 - 2vlO +
9
V ll
As in the previous case, the condition for nonsingularity can
be summarized on the basis of the above results for matrix
(2).
The necessary and sufficient condition that the matrix (2) be
nonsingular is that the three matrices
98
(26)
are all nonsingular.
The results of this section give us the complete analysis for
any balanced fraction from the 3n series, which is (2,0) symmetric.
3.7. Analysis of 2m x 3n asymmetrical factorial fractions.
The theory for this can be developed on lines parallel to
the symmetrical case given in the preceding sections.
To con-
serve space the details will be omitted and only the results will
be stated.
3·7·l.
b
A general treatment combination will be denoted by
jl
j2
b
l
2
j's
€
b
(0,1,2).
jn 11 i 2
a
a
n
l
2
a
im
m
where i's
Given a set of assemblies
€
(0,1) and
T, the number
of
times this treatment combination occurs in T will be denoted
by
~jl' j2'
···,jn j i l ,i2 , ... i m
1,2, .....
nj 1,2,
m
Extending the notation of section ,.5, we sroil 'Write this·also as
... ai m7 1
m- (
"
the smaller brackets containing the factors at
th1.~ee
levels and
the square brackets containing those at two levels.
The set of all possible assemblies will be denoted by
1-1 .
He now refer to section 3.5.2. The set of effects in which we are
interested fall in three groups:
99
(1) Pure effects of factors at
2 levels or pure A-effects.
(2) Similarly, Pure B-effects
(3) Mixed AB effects.
These effects will be defined as follows:
The pure A-effects are obtained by taking
1-1 2
tion ;.5.2, where
B-factors, and where
s=2
is obtained from
in (1), (2), (;), sec-
1-1
by
omitting the
d's are defined as
d0(0)
1
=n
do(l)
=
,
=
dl(o)
3
1
n
,
(1)
3
1
-;n
,
~(l)
=
-31n
d's for the case s = 3, fram those for the case
To distinguish the
s = 2, we shall denote the former by dr's.
Then the pure B-effects
are defined in the same 'Way as pure A- effects, with
- d (O) = 2- n
d~(O) = di (0)
2
-2 -n , di(l)
d~(l) =
db(2) = 2
-n
, dj,(2)
= 0,
=
2
d (l)
= 2-n
_2n+l , d (2) = 2-n
2
Then, for k l , k , ..• , k
m e (0,1) and ki' k
2
e (0,1,2), we define a mixed interaction as
k
.A... l
n~
=
k
k
2'"
E
j's,
k
k'
Am B ' 1 B 2
m
1
2
A2
I' s '1t
x
a
jl j2
a
l
2
a
J"
jm
b l
l
m
2 ' ... k'n
k'
B n
n
jl,j2"" ,jm; j'l' j'2,···,J..n
l ,k2 , .•• , km; ki,k;, .•• ,k~
j
(2)
j'
b 2
2
x
j'
b n
n
,
(3)
100
where
= d. (k ) d. (k ) ... d. (k ) d l ., (k ') d"1 (k 1) ... dl.,{k t )
l
2
J1 l
J2 2
Jm m
Jl
J2
In n
(4)
In writing the symbol for the interactions we make the conor k t is zero" we drop that factor from
vention that if any k
the symbol.
Thus if m=;, n=4,then
be written as
also and write
Ai1 A20~1 2
Bl
121.
AJ:1
~ B B .
Sometm.es
l 4
this as ~ A; BiB4'
0 01.
B2 B; B4 mll
we may drop the suffix 1
The normal equations can be s'et up for the mixed case by
approaching exactly as for the symmetrical case in section
•
It can be seen that corresponding to the matrix E E t
;.5.1.
which we had
to invert for the symmetrical case, in the mixed case we shall have
to invert another
IDa trix
which may be denoted by FF'.
The nature
of F F' will be explained below without proofs.
We know that the matrix E E' corresponds to the set of effects
for the case
~BiBj}'
lB?j1 and
In the same 'Way F F' corresponds to
tB?j
1,
s = 2, and to the ,effects
tBiB~},
tlJ.} ,
{Ai
{Ai Bi
lBiB~
1for
1,
{AiAj
t) ,
F Ft
is exhibited below:
l ' fBi 1'
{AiBi,},
is the set of effects to be determined in the mixed case.
of
s=3·
which
The form
where
r) is
- -1
{Ill
.--
of the form:
fB i1 tBi} tBiBj} {B~ Bj } {B?~l
} M1
M2
j
M'2
t
(~)
{AiAj ]
-
~
M4
M
5
M6
M
M11
M12
14
13
M14
M
M16
M3
M12
M20
M21
M22
M
23
M24
{B:rB~
M'4
M13
M~h
M28
M
~O
M31
2(~)
{B?j
M'
5
M14
M22
M29
M
35
~6
~7
(~)
{B?.~j
M'6
M
15
M
23
M30
M36
M41
M42
M'
7
Mi6
M24
M31
M37
M42
M46
1
n
n
(n)
2
2(n)
2
(n)
(m)
"
j
~.
r)-2
is
29
2
-
2
of the form:
[Ai 1
fAiB j
}
[AiB~ ~
{Ai}
M50
M
M52l
mn
{AiBj J
M
51
M
53
M54
mn
{AiB~ }
M52
M54
M
III
on
tm
m
15
7
51
55
(5.3)
1
102
. and finally
..r23
can be represented as:
{~1
1
1
fBi!
lB B1
{B~j 1
n
lB i
n
i j
lB~~}
iAiAj
~
m
mn
mn
The order of the whole matrix
F F'.
is
1)
x
1),
where
(6)
The whole matrix contains 100 submatrices; the order of these
submatrices are indicated at the farthest left and below the matrix.
The set of effects which correspond to
each submatrix are indi-
cated in curly brackets above and to the left of the matrix.
Each
set of effects in any curly bvacket is assumed to be arr.anged in
lexicographic order as in Chapter II.
. Al , A2 , ..• Am
... , BIB
n-
n
;
,etc.
{BiBjJ
means
Thus {Ai} means
BI B2 ,···, BIBn , B2 B ,
3
Take any two curly brackets (i. e. set of
10;
effects)
X
X and Y.
X and y
€
€
Consider two interactions
x and
Y)
at most two out of the
x
and y
where
y being interactions each involving
m +- n
factors.
Suppose
k'
k'
r ' B s'
sI
r
x=
B ,
Then in the submatrix which corresponds to the curly brackets
X and
Y, the element standing in the row corresponding to
column corresponding to
y
x
and
will be denoted by
k
k
k , k ,
A r A s B rl' B s' )
r
s
r
Sl
It is to be noted that at nest two of the k's in x and two of the
k I S in yare nonzero.
To illustrate, we see that when all k IS
are zero, the element reduces to
e('1.'~) stands in the
ment
(Bi B2 , B~ B4)
matrix M50 , e
2
e{AIB1, A2 B;)
M21 ,
;.7·2.
e( l-h lJ.), wl1ieh is M .
The ele-
l
1st row and 2nd column of the sublies in M;6'
54 ,
lies in M
e(Bl B2 ) Bi)
lies in
etc.
Some results regarding analYsis of asymmetrical designs.
The matrix
F F ' is know, when the value of
e's
is know.
For this we state, without proof, the following lemma :
Lemma ;.7.2.1ki kj
e (Ai Aj
fil
Bi,
fjl
Bjl ,
k
k
kIf,
Bs )
Ar r As s Br'r
Sl
=
where, (i)
out of the four symbols
(1)
k., k .,
1
J
f.,
1
and
l.,)
J
at most
104
two are nonzero; and similarly for k,
k s , f r " Ks , .
r
(ii) the symbol on the r.h.s. denotes a linear function of
the
A'S defined in the beginning of section 3.7.1; these
following similar symbolical rules as the
(iii)
£*:7
A'S
A'S in section 3.5.2,
denotes a linear function of a certain set
of symbols for the A-factors, such that lemma 3.5.2.1 gives for
s
= 2, the relation
k.
€
and
k.
(A.~ A. J ,
~
J
(* * *)
(iv)
k
k
Ar
=
AS)
r
s
A
r-* _*7
. • . . • .• (2)
L
denotes a linear function of a certain set
of symbols for the B-factors, such that lemma 3.5.2.1 gives for
s
= 3, the relation
=
Illustration:
Let m = 2, n
(i)
€(A , A A ),
2
I 2
(tv)
(* * *)
....... (3) .
= 2, and let us find
€(B , BI B ),
2
l
(ii)
(iii)
A
€(B
2
l
2
B , A A )
l 2
2
(i) From lemma. 3.5.2.1,
€
and
(Bl
€(Il,
B~,
B2 )
= A {(bi -
b~)
~) = A £ai - a~_7 .
Hence,
€
2
(B l B2 , B2 A )
l
= A
=
/1.22,1
12,1
(bi -
b~) (b~
-
b~)
(ai -
A20,.1 +
12,1
'\. 02,1
("12,1
+
A02 ,0
12,1
-r
105
+
~oo,l
_
12,1
=
(iii)
€
12,1
~ 22, .
+
12, .
~20,
~ 02,.
.
12,.
12, .
~oo,
.
12,.
=
(A , ~ A )
2
2
= ~
~oo,o
a~) =
(ai -
~.,l
.,1
_
~.,o
.,1
,
in an obvious manner.
~'s
This can be expressed in terms of
as
a linear function of 36 terms.
The omissions (dots) after connnas in (ii), and before them
in (iii) shows that the factors at other levels are not under consideration.
m
(~
~)
for short.
n
Let an assembly of a . 2 x 3
design be
Let T be a set of assemblies.
Then T
Definition 3.7.2.1.
will be said to be symmetric with respect to the levels
° of the factors at two levels,
(~
~)
of
~,
e
(~
T implies
~)
1
and
or simply 1l,9} symmetric, if
€
T,
where
a
is the complement
Le. is obtained by interchanging the levels
0 and 1
in
a
Similarly T will be said to be symmetric with respect to the
levels
2 and 0
symmetric, if (,E~)
of the factors at three levels, or simply (2,0)
€
T implies
~J~) €
T, where b
is ob-
106
.
tained from .!? by interchanging the levels
2 and
o.
We now state another important lemma without proof.
. from an
Let T be a set of assemblies
Lemma. 3.7.2.2.
AFE (m
2 x.3 n) .
Then
if T is ~1,~7 symmetric, the matrix F F' which is to
(i)
be inverted for solving the normal equations, maybe broken into
two orthogonal parts, one corresponding to the set of effects
{~}, {Bi~
lB~i
,
'
\BiBjl
'
,
\B?j1
~B?~} and
{AiAj } and the other corresponding to the set fAil,
{AiB~,
{AiB~'1
and
(ii)
T is (2,0) symmetric, the two sets in which F F'
if
t
l'
{~}, ZB~}, Bi Bj1' 1B?~
LAiB~7; and {Bi l' {B~Bj}' {BjAiJ
broken are
and
(iii)
if
T is both (2,0) symmetric and
["Ai Aj _7,
Ll,0_7
["Ai _7
symmetric, then
F F' is broken into four groups, viz
(a)
{~ 1,
{B~}' ~BiBj
(b)
tBi1'
(c)
{AiB j
(d)
{Ai
~
1,
tB?~},
tAiAj
1
tB?j}
,
1and
and
iAiB~l
This lemma will be frequently used in the construction of
designs discussed in the next chapter.
is
CHAPTER III
CONSTRUCTION OF FRACTIONAL REPLICATIONS OF
ASYMMETRICAL FACTORIAL EXPERIMENTS
4.1 Preliminary remarks.
The problem of construction of confounded designs for SFE's
n
n
m
of the type 2 and 3 and AFE's of the type 2m x 3 'Was
first discussed by Yates ~58, ~7.
However, a complete attack on the problem of· construction of
SFE( sm), where
s
is a prime number or a prime power 'Was first
made by Bose and Kishen ~!!7,
and Bose ~27, who utilized for
this purpose linear spaces in finite geometries associated with
G F( s ) .
In the paper by Bose ["27, the problems of balancing
were also discussed.
For the asynmetrical case, Li ~3§J, using methods similar
to Yates', presented seven additional designs for AFE of the
types:
2 x 2 X 4,
3 x 4x 4,
2 x 2 x
5, 2
and 2 x 2 x 2 x 4.
x 3 X 4,
2 x 4 X 4,
Nair and Rao ~4g,7,
3 x 3 X
4,
developed
a set of sufficient conditions, Which, if fulfilled, imply the
m
existence of balanced confounded designs for the AFE(sll x
lIJt
x sk
).
A complete attack on the problem of construction of confounded
ml
designs for the AFE( sl
Srivastava ~34, 3:2.7.
x. • .
lIlt
sk
) 'Was made by Kishen and
They first generalized Bose's methods
of using linear spaces, and used curvilinear spaces in truncated
108
finite geometries.
In ["32,7, the last method was further developed by
them leading to the .use of vectors in Galois fields.
The optimum
solutions for designs for almost all AFE( s{ x .•• x
a:)
likely
to arise in practice were given by them along with the general
theory.
The fractional replicationa were started by Finney £ 2d.7, and
later developed by Plackett and Berman .L4~7, Kempthorne ["3]},
Kishen ["32.7, Banerjee £1:.7 and Rao 1527.
Bose and Connor
1l"J:.7
gave methods for dealing 'With FR's of
AFE's.
(zn
Connor and Young 1227 presented designs for FRls of AFE's
n
x 3 ) for all 5 ~ m +. n ~ 10. Many of these designs are ex-
cellent. Last~, .Patel £4'2.7, utilized the same method (of using
orthogonal arrays) and gave some new designs requiring lesser
numbers of assemblies.
The method of construction used by Connor and Young ["227, and
by PatelI4~7, consisted mainly of associating not necessarily
distinct fractions
JS.'
X , .•• ,
2
Xk
from the
wi th not necessarily distinct fractions
if!!
complete factorial
Y , Y , ••• , Y
k
2
l
:5n complete factorial. The fractions Xi and Yi
orthogonal arrays of strength
2 or 3 or even 1.
from the
are generally
No method, how-
ever, appears to have been fonnulated 'Which tells us which of the
Xi
and Yj
have to be paired together.
An absence of this know-
ledge usually results in a large number of assemblies in the
fraction.
109
In this chapter, several general techniques of construction
of fractional replications will be described and a brief study of
m
n
their properties will 'be made. The fractions for the .AFE (2 x 3 )
which we can get by the use of one or more of these techniques will
also be given, and their nature explained.
For the
ana~sis
of the
designs obtained by the methods discussed in this chapter, the
development in the preceding two chapters and the re$ults obtained
therein will be found greatly useful, and in many cases almost in
dispensable.
4.2.
The method of associated vectors and truncated geometries. Methods I, II and III.
4.2.1.
We consider the problem of construction of FR of .AFE
~
m
IlJt
(Sl x s22 x... x ~ ) . We first define the basic terminology.
Definition 4.2.1.1.
Then
(i)
Any
Let t
be a prime number or a prime power.
ordered set of n elements in GF(t) will be called an
n-vector in GF(t)
(ii) Corresponding to a factor A at k levels, the vector
(0,1,2, •.. ,k-l) in the real field will be called the level
vector of A.
(iii) Corresponding to the level vector of' A, there is an
"Associated vector" (13 0 ,13 ,13 , ... , 13 _ ) of A, where l3's
1 2
k l
are elements of' GF(t), not necessarily distinct.
Let m =
~
+ m + .•• +
2
IIJt.
Let the total number of assem-
110
~
b1ies possible in AFE (sl x
11.
ill2
~
s2 x ... x Sk)
be denoted by
Let us define, corresponding to each factor, an associated
vector in GF(t ) .
Let
w
..r1.
€
We can think. of w as an
ill-
vector in the real field which gives us some particua1r level of
each of the m factors.
level
(
in w.
Suppose
Suppose that a certain factor A is at
(* is the element in the associated
vector of A which corresponds to the element
vector of A.
(
in the level
Let the set of all factors be denoted by F.
we construct a vector w*
(called a transformed assembly) fram
the vector w by replacing, for each A
A in w by the symbol
Then
(*.
Then w*
€
F, the level (
of
is an m-vector in GF{t).
Corresponding to the different assemblies in 1-1, we get obvious~
m
~
1y Sl x S22 x ..• x ~
= v, say, vectors W*. The set of
all vectors
w* may be denoted by
Definition 4.2.1.2.
'-1* .
.1
Any vector in GF(t) used to generate a frac-
tion will be called a generator.
If we have m factors in all,
then the generator will be an m-vector.
The inner product of two vectors a=(a1 ,a2 ,··am)
m
r2 , ••• , r)
is ~ a r, and will be denoted by
m
r=1 r r
Definition 4.2.1.3.
and
r = (r 1 ,
a. r.
Method I
as follows:
of construction of a fraction may then be described
For each factor A e F, define suitable associated
vectors VA in GF( t ), 'Where
the set
•
.11*
t
is a suitable prime power.
of all transformed assemblies w* .
Get
Choose a suit-
111
able generator y
and a constant
tain the set of all
x*
E:
['"2*,
c, such that
c
y.
x*
such that.
GF(t).
E:
=
c.
in
X*
corresponds to one or more assemblies
x.
x*.
Let
the set of all assemblies which correspond to a vector in
x*
Let
be the set of all such vectors or transformed assemblies
x*
Ob-
Each
X be
X* .
Then
X is the fraction produced by Method I, and will sometimes be denoted as
X(t; VA' A
E:
F; y, c).
The properties of the fraction will depend on the Q.uantities
given in the brackets.
The method is obviously very flexible and
covers a large number of cases.
FR's of
SFE(sm) by taking
It covers the theory of obtaining
(m-k) flats in
EG(m,s).
A stU<l¥ of
the general properties of the method will not be made here, but
many important particular series of designs obtainable will be
given.
IIJt
ml
FR's of AFE( sl x... x
sk) when only main effects are
present.
4.2.2.1.
;1
-th fraction of AFE (si x s2)'
We apply method I, with
t
= sl'
For factors with
the associated vector consists of all the
sl
in some suitable order, and for the factor at
vector has
able order.
(al , al , C¥1)
s2
sl prime power.
elements of
sl levels,
GF(Sl)
s2 levels, this
distinct elements of GF(sl)' again in some suit-
If
~
is the unit element of
GF(sl)' the 3-vector
can be taken as a generator.
In the fraction obtained, the main effects of the two factors
at
sl
levels will be correlated.
This series will be called V .
l
112
This method can obviously be generalized to the case When there
are more than one factors with number of levels less than sl' 1.e.
2
for the AFE (sl x s2 x ••• ).
We call this series V2 ·
Some useful designs in these series are given below:
(i) s2 x 2 for all s,
s
prime power, With no d.f. for error,
(ii) in general all s 2 x q designs,
s ~ q > 2,
s
prime paver
such that
(number of effects
to be estimated)
(number of assemblies
in the fraction )
< maximum number of degrees of freedom desired for error
(say n).
e
This gives
Sq - (2 s-2 + q-l + 1) < ne
or s
Since
q < s,
<
1
ne
+
q-2
we get
--=n
q< 1+
q-2
or (q-1)( q-2) < ne
This gives an upper bound for
if we take
q for these designs.
ne = 30, the maximum value of'
For example,
q is 1.
(iii) Similar calculations can be made for fractions of AFE(s2Xqlx ••.
x
~).
Thus for exam;ple a fraction with 30 assemblies can be ob-
tained for
5 x 5 x 3 x 2 design, estimating 12 effects, and
providing 18 d. f .
for error.
ll3
4.2.2.2.
Series
V for FRAFE (Sl x ••• x
3
power and sl ~ max (s2' s3' .•. , sk)'
This series gives a
4.2.2.1.
~),
Sl prime
...2:...
-th fraction. 'VIe proceed as in section
sl
The generator may be chosen as (~, a , •.. , ~). ive choose
l
the associated vector Vi
for the i-th factor such that if
Z :: Z2 + Z3 + ••• + ~, then Z ranges CNer all elements of GF(sl)
as
Zi varies over Vi' i=2, ... k.
involves all the
sl
This ensures that the fraction
levels of the first factor.
In the design for this series also, the normal equations will
be found to be easily solvable.
In case
s2s3 .•• sk' the design could be made
4.2.3.
4.2.3.1.
;
8
is
a factor
of
orthogonal.
qv.1tc
Method of cutting out or method II.
SUppose we have an AFE (sl x
sl ~ max (s2' ..• ,~)
a
sl
-th
s2 x ..• x
and is a pr:tme power.
fraction (r
< k) of the SFE(S~),
~)
where
Then 'We first take
by the standard
1
methods of finite geometries.
Finally, this method says that we cut
our all assemblies in our fraction which have any of the
given levels of the
a fraction having
i-th factor, i = 2, ..• k.
ThUS, for example,
6 assemblies from the AFE(5 x 3 x 2)
obtained by first taking
(sl- si)
may be
~- -th fraction of the 5 x 5 x 5 fac-
torial, and then cutting out two levels say 3, 4
factor, and three levels say 2,3,4
of the second
of the third factor.
The frac-
tion on which cutting out is done may be obtained by taking the 25
assemblies Which lie on the flat
114
in EG(3" 5).
The method of cutting out can be shown to be a partic'l\1ar case
of method I.
Due to lack of s:pace" the :properties of this method will not
be discussed here.
4.2.3.2..
We shall however give an illustration here.
Series V :
4
1 -th
2
s
fraction of
8
3 x· ql x ... x
~
design" only main effects assumed present.
Let the first three factors (at
s
levels) be denoted by
~"
x2 " x
and the rest by Yl " Y2 " .•• Yk . The levels of Yi in
3
k
SFE (s3+ ) which are not to be cut out" form the associated vector
Vi
of Yi " i = 1" .•• k.
We first take a
1
2
s
fraction of the
SFE(s3+k), by choosing the flat in 00(3 + k" s) represented by
the two equations
~ +
al
where
al
and where
a 2 x 2 + a 3 X, +
1:'2 Y2 +
•.. + f3
k Yk
=
c2 "
is the unit element of GF(s)" and a 2 ~ a ~ ~ ~ a 2 '
3
f3"s and f31's are so chosen" that no linear combina-
tions of these equations contains only two of the
than two of the
yls.
For
XIS
and less
s = 3, the method will have to be
changed.
The design of this series can be made completely orthogonal if
one of the
qls is equal to
s.
will be found to be correlated.
Otherwise" the first three effects
•
115
4. 2.4.
Method. III.
(A generalj.zation of method. I).
4.2.4.1.
~
SUppose for the AFE (sl x
lIJ..:
m2
2 x .•• x sk ), we construct a fraction JS. (tl j v;." A € F; Yl' cl ) by the use of method
I. We can further reduce the size of the fraction by taking (i) a
8
number t
Which should be a prime power, (ii) associated vectors
2
~ for each A € F, the element of the associated vectors being
members
of GF(t2 )" (iii) an m-vector Y2 in GF{t2 ), and (iv)
a constant e in GF( t 2 ), and then proceeding as in method I.
2
For this purpose" each assembly Q € X 'Will be converted into
2
a transformed assembly Q* by using the associated vectors VA'
Then each
Q*
by
Q*
® *.
is an m-vector in GF(t 2 ). Denote the set of all
Then our new fraction denoted by X2(t2 j V~, A € F;
Y2; c 2; JS.) consists of all assemblies. ~ such that (i) z e Xl"
(ii) if z* is the m vector representing the transformed assemY2 = c 2 • The above procedure may be repeated,
if we 'Want to cut out some assemblies from X2 also.
bly of z" then z*.
It can be shown that if t l = t 2 " then the fraction X2 can
also be obtained directly by Method I by taking t = ti" and using
suitable associated vectors and generators.
~ t ,
2
l
The situations Where
Thus method III . is chiefly useful in those Cf)ses where t
so that different Galois fields are to be used.
method III is very useful arise in AFE(s{ x .•• x
a:)
'Where the
si are not necessarily prime powers, as for example in 6 x 4 x 3,
6 x 4 x 4" 6 x 3 x 3 x 2" 10 x 5 x 3 x 2 etc.
An
illustration is
U6
presented below.
The success of methods I and III depends heavily on the choice
of appropriate associated vectors.
4.2.4.2.
Consider an example from a
1
1
6 x 6 x 2 factorial.
We ob-
1
'2 and a "0 -th fraction.
tain a ;"
First we divide the
72 assemblies into two sets of 36 each
by using GF(2) te~ber with (i) the generator (1,,1,,1) (ii) the
associated vector (0,,0,,0,,1,,1,,1) for both A and Band
c.
This gives a ~ fraction
X:t. •
To get a ~ -th fraction" we now use
(0,,1,,2,,0,,1,,2)
tions.
~
GF(3) and take (i)
as the associated vector of A and Band (0,,1) as
the associated vector of
A
(0,,1) for
CJ
and (ii) a genem'bor" say· (1,1,2).
-rd fraction can be obtained by taking two
One such fraction
is exhibited below.
~
-th frac1
For this" a "0 -th
fraction L1 is obtained by taking (1,,1,,1) as generator" and
by taking (1,2,,1) as the generator at the
another fraction L
2
second stage. These are eL'libited below.
~ Fraction of 6x6x2 Factorial.
210" 33O"
2Ol" 351,
450"
540
021"
120"
111"
441"
531
030,
140,
250,
30°"
~1O,
520
041"
151"
231, 311"
421"
501
000"
4.2.5.
Further remarks on methods I, II, and III.
These methods may be found useful for construction of fractions
117
which preserve not only main effacts but also all two factor interactions.
Due to lack of space, any detailed discussion of these
methods could not be done here, but a few examples from the
m
n
2 x 3
series
are given at the end.
ple which is not of this type, viz.
a
Here, we consider an exam-
21 -
fraction of the
5x3x2x2 factorial.
We apply method I, with t
in
= 2,
1. e. we use
GF(2).
lIe take,
GF(2), the associated vectors (1,0,1,0,1), (0,1,0), (0,1) and
(0,1) respectively for the four factors say
we use
y
= (1,1,1,1)
A,B,C,
and
D.
Also,
as the generator, and include all assemblies
x in our fraction, such that
transformed assembly of
x
x*. Y
usL~
= '0 E
GF(2), where
x* is the
the given associated vectors.
If only main effects are assumed present, then C and D will
be correlated.
If
2-factor interactions are present, then the Bet
of correlated effects will be
Be} , where a
{AD,
{A,B,
AB,
CD} , {AC, BD} ,
single letter den otes a main effect
a pair of letters gives an interaction
and
d. f ., and
d.f.
The :;0 assemblies in the fraction are displayed below:
1000, 3000, 1200, 3200, 0100, 2100, 4100, 1011, 3011, 1211, 3211,
0111, 2111, 4111, 0010, 0001, 0210, 0201, 2010, 2001, 2210, 2201, .
4010, 400~, 4210, 4201, 1101, 1110, 3101, 3110.
Use of various types of arrays. - Methods IV and V.
4.3.
4.;.1-
Method IV.
Magnified arrays.
m
We shall first explain the various methods for the .AFE(2 x :;n),
and later give indications of their generalization for the general
118
asymmetrical factorial.
Consider the
SFE(3IIH-n).
We take the maximum number k
of
linear equations in EG(IIH-n, 3) such that the assemblies lying on
the flat represented by them t'om an orthogonal array of strength
d.
Let these equations be:
.•• + d2 , m+nxm+n = g2
. ..... . . . . . ... . ... .... .. . . . ....
If we write:
and
xIIH-i = Yi
,
~
X
=
mxl
x
i = 1,2,
... n
.........
.........
,
Y =
x
m
mxl
-'
(2)
gl
Yl
2
(1)
,
Y2
~
Yn
,
g2
=
••• (3)
~
kxl
then the equations (1) can be written in matrix notation as
[D
X
where
D
x
DyJ
['~.J
=
~
••••••
fl
(4)
and D are respectively kxm and kxn matrices.
y
Suppose we 'Want to construct a fraction of a
m
2 x 3
n
fac-
torial, under the assumption that all interactions involving up to
two factors are present.
Many approaches are then open.
•
119
Method IV • 1.
We take
d
= 4.
3mtn·k assem.-
Then in the
blies (which we may denote by @ d) given by (1), we get using
results of section
~~2,
chapter III, that for any set of
r
fac·
........
for j's
E (0,1,2).
To obtain a fraction for the
the level
2
boll, Le.
of each of the
m·factors
factorial, we replace
JS.'~'
...
Xm by the sym·
we make a transformation on the levels represented by
(g i i)·
The set of assemblies resulting from this transforma·
®t)
fraction ®
tion (call then
The
m
n
2 x 3
t
to the two levels
may be taken as a fraction.
obviously is not synmletrical with respect
0 and 1
of the m factors
JS.' ... ,
X .
m
To
achieve synmetry with respect to these two levels and also to make
the matriX
F F' (see section 3.7.2) simpler to invert, we take
another fraction ®~.
This fraction is obtained from
making the transformation
factors
(~
~
g)
Q!:\
by
on the levels of the
Xl'~'
2x3nH-n-k
.•• Xm· Then for our design, we take
assemblies given by
.......
In order to understand the nature of the fraction
consider the matrix F F'.
Since
r4
is £1,
£7
We use lemma 3.7.2.2.
sy.annetric, we can write
(6)
r 4, we
m
e·
120
o
=
F F'
... " " ....
,
,-)
o
.1-2
.111 corresponds to set of
{BiB j J '
B~j J' {Bi B~} and
where
t
1AiBi }
ponds "to the set { Ai}'
effects
{IJ.} , {Bi }
{AiAj } '
and
1-1
,
2
lBi 1'
corres-
and {AiBi}
Using the represent.ation (5), sec"tion
3.7.1,
of
F F1 ,
'We
can wri"te
L
(E E')3
....... (8)
L
1
.
e
where (E E')3
n
represen"ts "the ma"trix to be inver"ted when we have
3
factors each a"t
levels, and
..........
L =
Since
(E E 1 )3
r4
is an orthogona.l a.rray of strength
is nonsingular and is a diagonal matrix.
also "that except
~,
all other matrices in
4 in B' s, the
vTe shall show
L are null.
From
equation (5), we get as a particular case:
jl;
ji, j2
Ai • i'
l'
l'
i
l
2
=
3m+n-k-3
,
....... (10)
ratr.:ix
121
e
for all
and
i
(Y:t.'
e
l
(0,1,2).
e
jl' ji, j'2
•.• y n )
~,
c;;;
and
(i'l' i')
2 e C~
.~' "
.~,
...
,~ )
tin '
Now if we make the transfo:rmation
(: :)
1
=
T
l
1
or
(11)
(: :)
1
.-
~2
.......
0
on the levels of the
Xl' X2 , ••. , Xm ' we get
m factors
,
.
jli J l , j'2
Ai • ii, i'
1"
2
for all
and
e
i
l
e
=
.........
,m+n-k-,
,
... ,
(Yl , ~,
jl e(O,l,2)i ji, j'2
~)
and
ii, i
2
v· ,
e (xl' "2
(12)
...
)~)
(0,1).
e
Similarly
jl' j2 i ji, j'2
Ai , i ;
i'
2 ii, 2
l
for all i l , i 2
jl' j2
€ (
,m+n-k-4
Yl , Y2' .•• , Yn);
e (0,1,2) and
The matrices
=
ji~
j2
M
and M
24
16
e
ii, i
2
........
e (xl'
all three matrices
of the form as at
version of
11 1
X2' ~ ..
Xm);
(0,1).
have elements (see section 3.7.2,)
all of which can be expressed as linear contrasts
form as at (10).
(1')
of A's of the
Hence M 6 and M are null matrices.
1
24
Similarly
M:;l' M'7 and M42 are linear contrasts of A'S
(1'), and are also therefore zero.
reduces to the inversion of
Hence the in-
122
.......
(14)
which can be easily inverted by various methods, including the
formula developed in the example in section 2.3.7.
Next, we consider the nature of
.r12
=
M
50
M
51
M
52
M
51
M
53
M
54
M
52
M'
M
55
54
'(15)
By using Lemma 3.7.2.1, coupled 'With the above argument, we
shall find that M , M and M
are also null matrices for the
54
5l 52
fraction r . By the same argument it will be found that M
and
4
53
~5 are diagonal matrices.
Further, M
'Will be seen to be an
50
invariant factor matrix and can be very easily inverted.
On inverting M , it will be found that the correlation be-
50
tween any two main effects of the factors X (at two levels) is
( - ~8 ), and the variance of any main effect is
~
times the
variance in the completely orthogonal case.
The fraction obtained by this method is of the size
(16)
=
The method is useful for small m and
large
k.
Method IV - 2.
Sometimes one may desire to reduce the size of the fraction
below that given at (16).
l'1e then consider the case
d ~ 4,
i.e. choose the equations (1), so that an orthogonal array of
strength less than
4 is obtained. We then proceed emctly as in
r4 .
Method IV - 1, and get a fraction
Then, since as before,
F F'
breaks up as at
r4 is £1,
equation (7),
_7
s:YIllID.etric, the matrix
(to which we constantly refer).
We therefore turn our attention to the problem of choosing equations
(1) (while keeping
d = ,) such that
1-11
and
.1"1,
are nonsingu-
lar.
A look at equation (8) shows that preferably (E E'), should be
a diagonal matrix, in which case the problem of inversion of
1-11
would be inunediately reduced to that of inverting a matrix of the
For this purpose, however, the fraction r
M .
same order as
46
should form an orthogonal array of strength
Y , Y , ••• Y .
l
2
n
in the factors
This means (see equations (1) and (4»
linear combination of the
k
that no
linear expressions
.............
Dy-Y
should contain
4
4
four or a lesser number of the
(17)
XIS.
Now, since equations (1) give an array of strength less than
4, there will be a set H
of linear combinations of these equations,
such that each equation in the set H involves four or a lesser number
of factors say X
V
1
,
.•• ~
r
(r ~ 4)
out of the set
F
of
124
m+ n
the
factors
which factors out of
Xl' X , .•• , Xm+ .
n
2
F
We will now investigate
should appear in the set of equationa
HI
and in what way.
A look at equation (7) reveals that the £1,27
symmetry of
a fraction implies that the 21 matrices reproduced below are zero:
1
{ ~}
n
{Bi ',
n
£B~}
(18)
{BiBj ?
lB~Bj1
tB?~ 1
mn
m
mn
This has been plucked out from (5), section 3.7.1as explained there.
denoted by
F
1
shows that the
Let the set of rJ factors
and Y , Y , ..• Y
n
1
2
by
F .
2
is.'
The notation is
X , .•. X be
2
m
Then 1eLJma 3·7·2.1-
A's which are involved in the expressions for the
of one of the fo11ovdng types:
(using notation as in the i11us-
125
.~
j1' j2' j:;
(11)
£
.L0 ,
]}.
.,
j1'; ji, J 2
~i .
i'
l' ii' 2
j1,;ji
~i 'i' ,;
l' 1
J• ,••J t , j2,j3
1 1
~.
., l'
~1,J.1' J. 2 ' :;
..,
j
Hence even if the values of the above
~I S
,
are not equal to
the respective constant values obtained by using equations (5)
(taking into account the transformation (11), the matrix (18) is
unchanged and is zero, so long as
r4
is £1,
We now refer to the discussion in section
to Lemma
2.7
symmetric.
:;,4.2
specially
;.4.2.1.
If in the set H, there is an equation which
jl,;ji,j2,j3
involves say four va.riables -'"].
x.., Yl , Y2 , Y~,
then
the
~l
.;
j 1,2,3
are disturbed.
However since we can allow the
~'s
in (18) to be
disturbed, we arrive at the conclusion C :
l
The set H may contain equations which involve either (i)
F , and three out of F , ~ (ii) exactl
2
ly three fa.ctors out of F and none or one from F ,
2
l
ive now consider how far this disturbance of the A,' s at (18)
exactly one factor out of
will affect the symmetr'J of
112
and
volves exactly two factors out of F
l
M46 involves up to four
conclusion C :
2
~ctors
1-11 ,
The matrix M
7
in-
and none out of F , Also
2
out of Fl' Thus we arrive at the
126
In order that ~ll may remain symmetrical with respect to
.,
various factors, we find that the set
H should not contain equations
which involve exactly four factors out of
Fl
F , or exactly two from
l
and two or leas frooF .
2
It can also be seen that if conditions
tained, then
1-12
C
l
and C are main2
r 4 resulting
remains symmetric as in the case of
from method IV - 1.
Method IV - 2 therefore says that we use ~quations (1) giving
4, but maintaining the conditions Co'
array of strength less than
C and C •
l
2
If the condition C
3
is
contradicted, it may not necessarily
:Imply the singularity of the matrices ..e ll and
.r12
•
However,
the matrices to be inverted in such cases may become more complicated.
The assemblies obtained by these methods are called magnified
arrays, since we use a fraction from 3m+n
a fraction for the
factorial for obtaining
m x 3n factorial.
2
4.3.2. Method V.
Another 'Way of getting fractions is as follows.
an example for the
We will give
3n x 2 2m factorial.
Method V - 1.
As in method IV - 1, we start with an orthogonal array
of strength d
from a
sets of factors
JS.'
set of n
3m+n factorial.
X , ••• X and
2
m
®d
As before, we have two
Y , Y , .•• Y . The latter
l
n
2
factors can be put into (1, 1) correspondence with the
127
n
factors at
3
levels each in our design °
The
2m
factors
0. t
each at two levels may be divided into m pairs, each pair having
two factorso
Consider a pair (A _ , A ) °
2r
2r l
pondence with the factor
Xro
This is put into corres-
The four combinations of levels in a
®
pair are (11, 10, 01, 00) ° From
d ' we obtain a fraction @ ~
n
2m
for 3 x 2
faetc:ri!;l.1 b~wnk:1ng (instead of ell) in the last section), the transformation
Tt· (~1 ~o ~1)
from the levels of
for
r
= 1,
X
r
2, ... m.
(1)
to three levels of the pair
A second fraction
H~
(An
cr-
l' An ),
cr
my be similarly
obtained by making the transformation
1
01
Finally, we obtain the required fraction as
Such a method will be powerful especially when m and
ably large.
n
are suit-
In the general case, the size of the fraction is
3m- k
=
2
2m-l
Method V - 2.
This is a variant of the earlier method, and will be explained
first by an example for the
m
2
x 3
n
factorial.
In the last method, we started with a
3m+n
factorial.
In this
128
method we start with a
paver,
sf factorial" where
# ,,,
s
and is a prime
and {is suitab1¥ chosen.
For illustration, consider the
with a
2
2m
x ,n case.
4m+n factorial, obtaining fram ita fraction
is of strength
d.
m and Yl , Y2' .•. , Yn .
pairs,
111
® d'
As before, we have two sets of factors
We divide the
••• , X
levels into
We can start
r-th pair. being
before, we obtain a fraction
which
X1." X ,
2
2m factors at two
(A _ ,
2r l
~r)'
Then as
from ®d' by :making the trans-
Jt
d
formation:
1
2
(4)
from the levels of
for
by
r
= 1,2,
l1Uld.pg
01
10
to the pairs of levels of
X
r
.•• m.
the
From
Jt
d
(A _ , A )
2r l
2r
, we obtain another fraction
1(~
trans~tion
T*
l
2
1
(:
=
'\
(5)
1
2
2/
from the levels of Y to the levels of B " r = 1,2, ..• n.
r
r
2
lar1¥ we obtain a fraction Jtd by using the transformation
T*
2
1
=
(:
and a third fraction
*
T, =
2
1
by using
0
~
(01
1
2
2
o
Finally we take our fraction as
:)
Simi-
(6)
Sometimes
to avoid a large number of assemblies, we may work only with
2
d·
3t
•
129
The method will be useful, if we can take
4.4.
d
< 4.
Application of Methods IV and V to the construction of
fractional replications of the general asymnetrical factorials.
Methods IV-l, IV-2,Vw l.
n~d V~2
can be immediately generalized
8:
2 x .•• x
in an obvious manner for the case of . APE (s( x
s;) .
Due to lack of space, any theoretical discussion of the 'Way of
generalization can not be made here.
We will be illustrating it by
an example, however.
at x
Consider
3
2m
x 2 3n factorial.
We then start 'With a
®
8t+m+n factorial and get an array H d of strength d.
three sets of factors
Y1'Y2 ' .•. Yn •
2, 3 and
the Af S
a
Zl' Z2' .••
Zt j XI'
X2 , .•. Xm,
We have
and
In the original case, we denote the factors at
levels respectively by A, Band Cf s.
into n triplets.
a
Each triplet has
We divide
level combinations
(000, 001, 010, 011, 100, 101, 110, 111), which we put into corres.
pondence with a factor
we divide the
2m
vle thus obtain a fraction say
a transformation T* from the
l
S1m1larly, we obtain
Now
9 level combinations (00, 01, 02, 10, 11, 12,
Then we obtain a fraction
levels of the pair
®d .
B into m pairs, the r-th pair
factors
(B _ , B ) having
2r l
2r
20, 21, 22).
Y.
f
~
from
8 levels of Xr
(B2r_l , B2r ) for r
7t~
7t
@~,
by making
to a subset of
= 1,2,
m, where
1
2
3
4
5
6
02
10
11
12
20
21
by using the transformation
7)
22
(1)
130
1
2
3
4
20
12
11
10
5
6
02
01
.....
and
~~
by using
1
2
22
02
00
4
5
6
12
01
10
:J
(3)
We may then use as a fraction
which ever is more convenient.
for factors at
assemblies.
3
~~
or
123
~d U ~d U ~d'
The former one is (2,0) s~etric
levels, and also requires lesser number of
The value of
d, and the actual choice of equations
@d
for getting the fraction
from the
be guided by considerations of method
4.5.
U ~~
Use of quadrics.
-
S!+nrl-n factorial will
Dl - 2.
Method VI.
n
We sha11 obtain fractions for the
AF'E (8
1
n
1
x
s22
X...
x
such that these fractions are analyzab1e by the methods of Chapter
II.
Consider GF(s),
s
being a prime number or a prime power.
Consider EG(n,s), ~ere
(1)
Let
X be a vector containing n
into m sets, the
2, ••. m.
i-th
set Xi
elements, which are grouped
containing n
i
elements
Let a partially balanced association scheme
fined among and in the sets
~
i
= 1,
be de-
Xi' as in Chapter II, section 2.2.2.
131
That is, we assume that
X e:
£PB_7
(2)
vIe shall follow the notation of Chapter II.
there exist association matrices
d~k
£
D~k' 1ve take a set of constants
GF(S), for all :permissible
~
j,k
D =
~
j,
d:
Jk
r
Because of (2),
k
and
Define
r.
(3)
D:
Jk
. -wheTe it is assumed that the zeros and unities in the matrices
r
D
jk
represent respectively the zero element and the unit element
of GF( s ) • Also -write,
yr
••• , Y2n ;
• ••
.
J
2
... , y
mn
m
),
(4)
m
a vector containing
n.
~
i=l
n
=
~
elements.
Consider the equation
-y'
D
Y
=
Z
'Where Z is a constant and is in
The element Yij
s.1
levels.
GF(s).
!
in the vector
For all factors
at
.
s.1
levels
associated vector Vi' i = 1,2, ... m.
'We
choose the same
Let'the set of all assemw
.r1 ' let
'w*
denote the
transformed assembly after the associated vectors
Vi
are used.
Let A*
w* 'Which satisw
blies
be denoted by
.r1.
stands for a factor at
If
''W
e:
be the set of all transformed assemblies
ty equation
(5).
Denote by A, the set of all assemblies
that its transformed assembly
6* belongs to A*.
6, such
Then we get a
132
fraction b. which may be denoted by
'.
It can be sho'WIl that the matriX, say F F', which is to be
inverted for solving the normal equations, when the design is symmetrical with respect to all factors, belongs to the cla,ss of in~
variant factor matrices of the type (sl x
•. • x
n
smm ).
Thus
from the point of view of F F', the vector X is a factor vector
of the asymmetrical factorial type, and possesses a generalized
partially balanced association scheme which we may denote by ~ .
The superimposition of scheme
cA
on
03
implies that
association scheme, which may be denoted by
cA
is chosen such that
cA·
8
~
£
'x'
has an
cit G 03
The scheme
£PB_7.
By the results of the last section of Chapter II,
'We
then find
that the matrix (F F')b. to be inverted for analyzing the fraction
b. belongs to a linear associative algebra and hence can be inverted
by the formulae developed in Chapter II.
In amy cases it may be found that the number of assemblies in
b. is too large.
In these cases, we may take assemblies lying on
the intersection of two quadrics of the form
=
(5), say
(6)
=
where Dl
tion
cA ·
and D have at their back, the same scheme of associa2
133
A simpler special cases arises when we take
d~k
=
(7)
0,
for j ~ k , and all permissible
.1, k and r.
In this case (5) may be expressed in the form
!:
j
where Dj
Y'
-j
Dj
Y.
-J
=
(8)
Z
are matrices involving D~j
\ole shall illustrate this from the
only.
n
AFE(~ x 3 ).
vie take
In EG(m + n, 3), consider the quadric
s = 3.
D
Xl
x + X' E X = c.
From lemma 3.7.2.2, we may like to have our fraction Ll,27
In (9), we suppose ~J = (~'X2,,,,,xm)
synnnetric and (2,0) synnnetric.
represents factors at two levels and Xl = (Y ,Y ,""
l 2
Yn )
those
at three levels.
For (2,0) synnnetry, it can be easily checked that if
(Xl' x2 ' ••• , xm; Yl , Y2' ••• Yn ) satisfies
(9), then so does
(Xl' x2 ' .•• , Xm; 2yl + 2, 2y2 + 2, ... , 2yn + 2).
Thus for (2,0)
synnnetry, we have together with (9),
X'
where
=
Z
D
jt
+
ZI
E
z
=
(zl' z2' ••• , Zn) and
(10)
c
z1
= 2y1
+ 2, 1=1,2, ••• n.
Let
a
=
(1,1, ..• 1)
(11)
Then
(12) .
Hence sUb.tracting (9) from (10), we get
(2z' +
2 ~') E (2
X + 2 ~) - X' E l.
=
0
S:lJ.n.ilarly, if we desire to have ["1,
2.,7
or ~' E
X + X'
E .~
+ ~' E ~ = 0
symmetry, it can be
(Xl' x2 ' .•. , x ' Yl' .•• Y ) satisfies (9), then
m
n
so does (2X + 1, 2x + 1, ••. , 2x + 1, Yl' •.. , yn ), where XIS
l
2
m
are allowed to vary over 'the 'two elements 0 and 1 only of GF(3).
seen that if
We then get
(2~'+
9:') D (2
~ + ~) +
X'E 1. = c,
and hence
2 a' D x + 2 x' D a
+ a' D a
J\~l the assemblies which satlsf'y
(9),
=
(14)
0
(13) and (14) may con-
stitute our fraction
An earlier method.
4.6.
Method i,TJ:r:
£22.7
This is the name of the method used by Connor and Young
and by Patel
f4'd.7.
As explained earlj.er, this method consis'ts
XI'
Xk
mainly of associating not necessarily distinct arrays
X "",
2
m
fram the 2
factorial with not necessarily di~tinct ar~ays Y , Y ,
l
2
n
.•• , Y from the 3
factorial. For the analysis of t1;.e desigiLs
k
obtained by this method, the results of sec".:;ion 3.4 will be f:mnd
useful.
4.7.
Formation of blocks
Due to lack of space, this problem has not been discussed in
detail in this thesis.
However we may make some remarks here.
135
For all designs in which only main effects are present, we
may form blocks by introducing a block factor.
,za
n
x 3
In a fraction of a
factorial, this d1.lIml1Y factor may have two or three leVJells
according as two or three blocks are being introduced.
suppose three blocks are to be made.
They are represented, as say
rfl
Bo ' B , B , and then we take a fraction of a
1
2
In this fraction, all assemblies in which B
o
same block, those with B
l
In designs
~ere
Thus,
x 3n+l factorial.
occurs belong to the
lie in a separate block, etc.
two factor interactions are also preserved,
we may, by introducing one or two factors,
form
2,3,4, 6 or 9
blocks in the same manner as above.
4.8.
Illustrations from the
'if"
n
x 3
factorials.
We shall now present some examples from the
m
n
2 x 3
series.
In the designs given here, stress has been laid on reducing the
number of assemblies in a fraction without making the analysis too
complicated.
However, it appears that in most of the cases where
a reduction has been made over the number of assemblies in the
fraction proposed by earlier authors, the analysis has also become
slightly more complicated.
This. is however to be expected.
In all
cases, we shall suppose that all interactions involVing up to two
factors are present, their total number being denoted by
24 x 32
Example 1.
Here
v
= 35.
~
design in 48 assemblies.
We use method II.
+ x2 + 2
v.
In GF(3), we take the equation
X:;
+ 2 x4 + 2 x + 2 x6 = 0
5
This represents a flat in EG(6,3). We first get the
35 assemblies
lying on it and then cut out all assemblies containing the level
x,"
2 of any of the factors
xlj." x
5
and x6.
The design can 'be
represented as below:
( ~~) @ (H H, (H)
2 1/
1 0 1 1
o 111
21 x.,5
Example 2.
v
We have"
assemblies
= 62.
®
1100
1 0 1 0
100 1
010 1
o 11 0
0011
1111
1 0 0 0
010 0
o 010
000 1
design in 81 assemblies
We first consider all B-factors and take all
® which lie on the flat
+ Xlj. + X = 2
(1)
5
Let (~" b 2 " b.,,, blj." b'5) be any assembly of the B - factors.
We denote by X(b1 " b2 , b " blj." b ) the set of all distinct assem3
5
blies obtained by. permuting the b' s • We write
~ + x2 +
X,
=
X (2,,0,0,0,0)
X (12200)
X(lllll)
= X3
=
Xl (5)
= X3i
("0)
=
x,,;
X( 02222)
=
X, (1)
The figures in brackets
ments in those sets.
blies lying on (1).
The
X(llOOO)
= X2
(10)
= Xlj.
(20)
=
X(11120)
X (11222)
x.,
(5)
= Xl UX3
Y2
=
= X6
(10)
=
X2
Xlj.j
= X6
and
= Xrr·
after Xi'S denote the number of ele-
Xi'S together exhaust the
We now make two sets:
Y1
X2
= Xl;
u X, u x.,
UXlj. U X6
81 assem-
137
The sets
Y1
and Y
have respectively
2
ments, and each of them is (2,0) symmetric.
where a's
41 and
40
ele-
We take as our design
denote the level of the factor at two levels.
25 x 33
Example :;.
Here, v = 64.
design in 96 assemblies.
We use Method II.
In
GF{ 3), 'We take two equations:
(1)
where
y's refer to factors at
:;
levels or the
B-factors, and
similarly x's refer to A-factors.
The I'm ction is obtained by allowing
x' s to vary over (0,1)
and y's over (0,1,2) in GF(3), and observing the assemblies which
satisfy (1).
0 0 0\
111/®
( 2 2 2)
The design is as below:
(o
. (0 02) ®
1 1 1 1 1);
0 000
1 1 0
221
0 0 1)
( 112
220
(
1 0 0 )
211
022
®
(1 0 1 1 1)
11011 ;
01100
(1 1 1 10)
fxl11101
~ 00011
200)
(o
122
11
(
1 1 1 0 0) .
o 0 0 10;
(
00001
1210
0 2)
021
,.
® (;~g;;);
o 0 100
1 0 1 0
® (110110
1 100 1
1 0 101
o 1 111
138
(~\0 1~~)0 !X\(i0 gg
~~) ;
1 0 1 1
~/
00111
0)
1 1 0 0
10100
( 01110
o 1 101
tX"
101)
212
(
020
~
(012)
120
201
•
'
00° 0)
°
1 °1 0
® (o 0 11 0
11
01001
\ o 0 101
The levels in the first brackets are those of B-factors, and
in the second bracket, of A-factors.
Example 4.
Here, v
2
4 x 35 design in 162 assemblies.
= 101.
We use methods IV and V.
Let
2 and 3 levels respectively.
factors at
X and
In
Y refer to
GF(3), consider the
3 equations:
3 + is. = 1
+ Y + Y4 + Y + X =
2
3
5
2Y1 + Y2 + 2Y4 + X2 =
Y1 + Y2 +
Y
1
Y
(1)
2
°
The set of linear functions of the above equations contain
three more equations which have four or lesser number of variables:
2Y
3
+ 2Y4 + Y5 +
Y
1
+
2Y
5
+
is.
=
is. +
X2 =
Only the last equation contains two
0
2
y l s and two
XI s, and will
therefore produce some deviation from the standard analysis.
The design is obtained as follows:
assemblies
®
lying on (1).
First we get the
The five factors
34
Y correspond to
139
The three levels of X:J. may be mde to correspond to
B-factors.
the 3 level combinations of A
l
tion
(0
00
*
T 1 =
SimilarJ.y
and A
2
given by the transforma-
1
(3)
10
corresponds to the levels of
X
2
~
and A
4 given
T~. Thus, the application of the transformation T~ an
by
gives us
assemblies ®l.
81
We obtain another fraction
®
®2
by
using another transformation
:J
1
=
01
(4)
Then the 162 assemblies of the design are given by
®l
Example 5.
2
=
Here, v
6
102.
U@2
x 34 design in 162 assemblies.
,7
We start from a
factorial and get 81 assem-
blies by taking the following equationa:
Xl + 2X2 +
and
~A4
+ Y1 + Y4
X,
and
=
0
~
+ Yl + 2Y, + Y4 = 2
X,
+ 2Yl + Y2 + Y4 =
y' 6 correspond to
The
~
X,
2
B-factors.
correspond respectively to
¥6
(1)
X:J.'
Let the levels of
three levels of A A ,
I 2
given by the transOomation
u
u
_
140
1
(2)
We then get a fraction
a fraction ®2
®1
in a similar
of 81 assemblies.
'Way-
1
11
®U
1
He obtain
by using t1:le trans.fo.r.mation
:J
®2'
There is only one linear
combination of equations (1).1 which has two
X's
and
two Y' 5,
viz
This lorill cause some deviation tram the regular analysis.
iJ. x:}
Example 6.
Here v • 183.
JS.'
~,
design in 486 assemblies.
We apply method IV-I directly.
We take 11 factors
x"
Yl , Y ' .•• , Y8 each at three levels. In these·fac2
tors we get an array H of strength 4, by using six suitably chosen
equations •
Thus we get 243 assemblies
@.
Then as explained ear-
lier, we use transformations, and get the final design in
2 x 243
=
486
assemblies.
Example 7.
We have
2
6
x 3
v = 246.
8
design in 486 assemblies.
This is a design of the type
directly apply method IV(b) to the fraction
,m+n
factorial.
assemblies.
Here m+tl = 11 and
®
22m x " n and we
® obtained from the
contains (at least) 243
The design, which is ["1,0_7 symmetric is obtained in
141
486 assemblies by proceeding as in method rv(b).
10
6
~le 8.
2
x 3 design in 486 assemblies.
Here,
v
example.
=248.
The design is obtained exactly as in the last
This is a
Miscellaneous
Rernark~:
1
1536
-th fraction.
It may be possible to reduce the number of
assemblies even below that presented in the above examples.
All the
'lfl x .,n designs which could be constructed by using one or more
of the seven methods discussed in this chapter, cannot be presented
here;
it is planned to present them (together with a full ana~sis
of each individual design) in a subsequent report.
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.
~9_7
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Connor, W. S., "Construction of the fractional factorial
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£26_7
Fisher, R. A., IIA system of confounding for factors with
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