'e
t
I
8
}'
UNIVERSITY OF ~ORTfl Ci\ROLINA
Department of Statistics
Chapel Hill l N. C.
ANALYSIS OF IRREGULAR FACTORIAL FRACTIONS
by
R. C. Bose and J. N. Srivastava
August 1963
Grant no. AF-AFOSR-84-63
In this paper we develop the theory for irregular symmetrical and asymmetrical factor:i.al fractions. To aid in the
developnent of the theoryl a 7-.-fuilction has been introduced.
Using this l it has been shown how to compute the matrix
(M I say) to be inverttld for soh-ing the normal equations.
Although in this paper we mainly' discuss analysis of fractions, the theory developed here paves the ".'lay for the construction of good fractions by throwing light directly on
the nature of M and hence on the variance-covariance matrix.
This research was supportefl by the Mathematics Division of the Air
-Force Office 'of Scientific Research.
Institute of Statistics
Mimeo Series No. 373
:;11ALYSIS OF IRREGULAR FACTORIAL FRACTIONSI
by
R. C. Bose and J. N. Srivastava
1.
Introduction and Summary.
Factorial fractions are finding increasing use in agricultural, biological
and industrial experimentation.
(For a general introduction to the various as-
peets of the subject see [2, 11, 13., 18J).
Though in many situations we are
interested only in the main effects of the different factors, in most others a
study of two-factor interactions becomes necessary or desirable.
However,.goOd
fractions are at present nc:t available for most of the aSynJIll.etrical and even synJIll.etrical factorials.
•
By a
good fraction we mean that it should possess each of
the follOWing features to a reasonable degree.
Firs~ly
it should be economic; i.e.
should involve as few observations as desired.
Secondly, the correlations between
the esti:rnates of various effects should be small, particularly those involving min
effects.
Thirdly the variances for the different main effects, or for the inter-
actions should not widely differ from one pair of factors to another, i.e. there
should be some kind of synJIll.etry or balance in the fraction.
This requirement is
connected with a fourth one, viz. that the fraction should be analyzable with
reasonable ease, and should be well interpretable.
A fraction is called irregular if it is not orthogonal, Le. if some of the
effects are correlated.
The large amount of work (see for example [3, 5, 7, 8,
9, 10, 12, 14, 15, 16]) already done in this area shows that only a few orthogonal
q
fractions are economic.
Thus for even the vast majority of cases likely to arise
in practice, good irregular fractions are still unknown.
~his research was supported by the Mathematics Division of the Air Force
Office of Scientific Research.
2
In this paper we develop the theory for irregular symmetrical and asymmetrical
factorial fractions.
been introduced.
To aid in the development of the theory, a
A-function has
Using this, it bas been shawn how to compute the matrix
to be inverted for solving the normal equations.
(M, say)
Although in this paper we mainly
discuss analysis of fractions, the theory developed here paves the way for the
construction of good fractions by throwing light directly on the nature of M
and hence on the variance-covariance matrix.
In a separate paper [6]
we discuss the analysis of a special class of frac-
tions, called partially balanced, where we discuss special methods of inverting M.
Based on these two papers will be several later communications, pUblished elsewhere
in which the construction of good irregular fractions will be considered•
•
Most of the theory contained herein will be found in the unpublished report
[17J, and a rather detailed summary in [4J.
2.
Definition of interactions:
Consider an
Normal equations.
Treatment combinations can be written
sm factorial experiment.
in the lexicographic order a
jl
j2
jm
a 2 ••• am , 0 < j < s-l,
- r-
l
the symbol for a treatment, the exponent
j
I'
of a
r
r
= 1,2, •••m.
If in
is zero, 'ole shall omit this
= 0 for all r, the symbol will be written ep •
r
Exactly similar notation will· be used for interactions except that a's will be
a
I'
from t he symbol.
If j
replaced by At sand ep
by IJ..
It is well known (see, for example [2, 11]) that each interaction degree of
freedom can be expressed as a linear contrast of all treatment combinations:
... Amm
k
•
=
..
"
Let
a
denote the column vector of all assemblies in the above mentioned
lexicographic order:
••• ,a 2.~
m
(2)
s-l s-l
••• , am- 1 am
.
... ams-l J •
•••0.'
Let
.
~
A denote the column vector of A's
in the same order.
Then equations
(1) could be 'Written in matrix notation as:
(3)
=
A
.
where D is an
D
m m
s x s
matrix.
,
a
It is known [2J
that the sum of produc,ts of the
corresponding elements in any two rows of D is zero.
of squares of the elements in the i-th row of D.
dividei.throwby
the cell (i,i).
I).
A
=
(4)
&i'
Let
I).
= I). D
Then C
bean
s
Let
5~J. denote the sum
To make D orthogonal we
m xsm diagona1:matrixwith &i-1 in
is an orthogonal matrix,
Hence using (3) we get
C! 1 so that
a
=
C'
~ A
=
D' 1).2 A •
Let A be partitioned as
A'
where
=
( L'. :' I')
0
'
L is the co1Ulllll vector of all interactions up to and including all the
2-factor interactions (say
higher order interactions,
are assumed to be zero.
v
in number),
and I
is the vector of 3-factor and
o
We shall suppose hence forth that all elements' of I
o
The effect on the estimate of
L of replacing this last
4
ass'\.1lIlIltion on I
by other assumptions, will be studied in a separate paper.
o
can be easily checked that
Under the assumption that elements of
a
-
where
(D'
~2)
o
=
I
o
are zero, we get from
(4)
and
It
(5):
(D' ~2) 0 L I
is the matrix obtained by cutting out from (D'
~2) the (sm - v)
columns which correspond to I
~2o be obtained from ~2 by cutting
out the last
~2 ,and Dof from D by cutting
out the
in (4). Let
o
(sm - v) columns and rows from
(sm - v) columns.
(8)
a
=
D'
o
Then it can be easily checked that
~2
L
0
•
Let T be a fraction, i.e.
a set of assemblies in which any given treatment
combination may not occur or occur once or more times.
Let the expected
of the assemblies in T written in the form of a vector be denoted by
for simplicity, we are not assuming any block effects.
Let
E'
r
val~s
where,
be the matrix ob-
tained from D' by cutting out the last (sm - v) columns corresponding to I ,
o
0
and also by o.m.ttting (or repeating) the rows corresponding to treatment combinations
omitted (or repeated) from !
to get
r.
Note that the rows of E'
in such a way as to correspond to the elements of ;t*.
observations corresponding to
r.
Let X be the vector of
Since we are assuming no block effects, it is
clear that
Exp (X)
=
r
=
E'
~~
are arranged
L = E' R ,
The normal equations for estimating ]
can then be
say.
written
5
(EE t) ~
(10)
1\
.
where ~
= E 1. ,
.
stands for the estimate of ~.
L as the best linear unbiased estimate of
L
(11)
=
~-2 (EEt)-l E Y
o
-
is nonsingular, we get
L, where
~-2 (EEt)-l x
=
(EEt)
Hence if
0
-
say
,
He shall now demonstrate a useful and easy way of calculating x.
(5) and the definition of Dt
we get
o
=
(12)
at
or
D'
o
combination
e
z(e)
e
of a.
D0 a
of length sm in the following way.
Let the element in z
be denoted by z
= o·
= total
=
L
Corresponding to ~, construct a vector z
Take any element
From (3) and
(e).
e r/.
if
yield of
corresponding to the treatment
We define
T
e
(from all repetitions of
e in T)
if
e€
T.
Then from the definition of Et, it is easy to check that
(14)
x
-
=
Ey
-
=
D
0
z
-
Comparing (12) and (14) we find that
L is obtained from
~,
x
is obtained from ~ in the same way as
and that we do not need to write down E explicitly.
Finally, the calculation of
L from
~
or of'
making tviO-iVay tables for ltthe total yields If
~
from
~
can be easily done by
of various level-combinations of each
pair of factors.
Example.
Consider the following
10 assemblies of a
23 factorial:
aia~a~ (10 , 12), a~a~a~ (17, 15) (twice each), and a~a~a3 (5),
e
6
0 (
)
a 10a1
2a; 12,
1
a 10a0
2a;
(
)
The figures in brackets are (artificial) observed
7.
The set of all interactions is
values.
AI
=
The vector a
-
Al;}
(I-l; A1 .. A2 , ~; A1A2 , AlA;'
AIA2~)'
The matrix D in (;)
is defined similarly.
is (usually) defined
as
D
II ...11
1
1
1
1
1
1
1
1-
...1
...1
1
1
...1
1
...1
...1
1
-1
1
...1
1
1
...1
...1
...1
1
...1
1
1
1
1
-1
...1
1
1
...1
-1
1
1
-1
1
-1
-1
1
...1
1
1
1
-1
...1
-1
...1
1
1
...1
1
1
1
...1
...1
-1
=
e
2
Clearly 5
i
for each i.
(A1A~),
+ ; + ;
= 9,
v =1
horizontal line in D.
E
e
=
= 7,
Also I
and Dr
0
From D0
l
1-1
here is the vector with the single element
0
is the transpose of the :matrix above the
we get
1
1
1
1
1
1
1
1
1
-1
1
1
1
1
...1
-1
-1
1
-1
...1
1
1
-1
..1
1
1
-1
-1
1
-1
...1
-1
1
1
1
1
-1
-1
-1
1
1
1
-1
-1
-1
..1
1
-1
-1
1
...1
..1
1
1
-1
-1
1
-1
1
-1
-1
-1
-1
-1
1
1
1
1
-1
-1
..
7
where the columns correspond to the assemblies in the order given above.
calculate x we proceed as explained earlier by calculating
Using the above in (14) and ell), one gets
~,
To
which here equals
L.
Consider (11) again, which gives the solution of the normal equations.
the computation of x is easy, and the diagonal matrix
-
b,
0
Since
can be easily evaluated
the main problem in the analysis of a fraction T reduces therefore to the
evaluation of the matrix E E'
and its inversion.
to a set of assemblies T 'tdll be denoted by
3.
A.-operator and t:he calculation of
The matrix E E'
corresponding
(EE')T.
(EE').
Let 0m,s denote the total set of sm assemblies when we have m factors
jr
each at s levels. Let (a ) denote the sum of 'observed yields' of all treatr
ments of 0
m,s
j
in which the symbol
(a r) occurs.
r
Then we define the main effect
,
r
where the
(16)
d's
I:
d.(K) = 0,
J
.
s-l
I:
••• , m;
0 ~k
t
~
s-l,
are real numbers satisfying
s-l
j=o
= 1,2,
dj(K) dj(K')
o < K < 6-1 ,
= 0,
K ~ K', 0 ~ K, K' ~ s-l ,
j=o
and
d.(O)
J
= 1,
for all
j
K
•
K
2
Then in (1) the interaction ~ l A2
[3]) such that
K
••• Amm is defined (see for example
8
Let
(1
Sr S
jl j2
jr
a
• •• a
~11i2,···,ir
il i2
ir
m) occurs among the treatment combinations in the fraction T that
we have.
where
~J.·11j2,···,jr
I'e
be the number of times the symbol a
So far as the latter set of symbols are concerned we assume that they
~fS
are real numbers.
jr
a
(1 S r S m) such
i
Q
j~
that no term in any polynomial in J contains any symbol a
at a power higher
i
, 2 1
jl:5
r ' j, 5 j4
than unity. Thus for example, terms like (a ) a , (a ).
(a.) a.
l
2
i1
~,~4
If.)
Consider the set <J
of all polynomials in the symbols
do not occur in these polynomials.
w by:
U
Vie define a A-operator on the polynomials in
(18)
j
jl j2
A{a. a
~l i 2
... a i r)r
j
jr
j
f,.[~(a.lai2 ••• a )
i
~l 2
r
jl,j2'·· ·,jr
A..
.
=
=
,
~1'~2'· ··'~r
~
jl,j2' ····'jr
Ai'
. 1,J.2 ,···,J.. r
,
9
•••
+
1 ~ r"u
where
nomial
in
S Tn
(j?
and f3 t S
"A(P)
are real numbers.
Thus if P
denotes any poly-
is a real number which naturally depends on T.
phasize this last fact we sometimes write
is in excess" is in defect,
A(P)
A(P,T)~
as
or is zero in T according as
To em-
We shall say that P
A(P"T)
is
>
0"
= 0.
< 0, or
and T
are two fractions or sets of assemblies. By T + T
2
l
2
we denote the set of assemblies in which a particular assembly e is repeated
Suppose T
l
e
ca times provided that
and
cal + ~ = q.
e
occurs exactly
cal times in T
l
Then from the definition of the
it is obvious that if P E
and
~
times in T
2
A-function, given in (18),
(f , then
It is useful to note that the A-function defines a kind of homomorphism
:t
from the is~t (f of polynomials to the set of real numbers. Studies
:.".'' "...;'; k,· ,'.
,-
f\.
on this will
however be made elsewhere.
Given any fraction T, the matrix EEf
the A-function.
Every row of E
exactly one element of
e
L.
can be directly expressed in terms of
and hence every column of EI
Hence the element
E(i,j)
corresponds to
in the cell (i,j)
of EEl
corresponds to the elements in the i""th and j-th row of L. Consider any two
jl j2
j3 j4
elements of L, say Ai Ai
and Ai Ai. Note that there is no loss of generalit.y
1
2
3 4
in taking the elements in this form, since the main effects A~ and the general
mean 'Il are obtainable by choosing one or both of the j's as zero.
Let
10
j2
j, j4
e(A Ai ,Ai Ai) be the element in EE t which stands at the intersection of
i
1
2
,
4
jl j2
j, j4
the row corresponding to Ai Ai
and column corresponding to A. Ai. For the
1
'2
~,
4
evaluation of the said element one has to consider three different cases.
jl
(20)
(ii) If
jl j2
jl j2
e(ai ai' a i a i )
1
2
1
2
(21)
(iii)
say
i l = i,
If
and
= A[
i2
= i 4,
then
2
s-l
k
(~~ (jr) ~ (j~) a i } ]
r=l k=o
r
(11 , i 2 ) has exactly one factor common with
(i"
i 4)
1 = i , then
1
4
e{a
jl
i
1
a
j'2
i
,a
'2
ji
a
j3 )
1
i
l'
The value of the left hand side of (19) is obtained by multiplying the
jl j2
corresponding elements in the row R against the element ~ Ai
in L, and
l
l
j, j4
fl f'2
'2
in a
the row R'2 against the element Ai, A
a'2. •• am in the vector ~,
1
i4
are respectively
Proof:
lm
,
where, as indicated,
jl' j'2' j, and
i,.tb and i -th places.
4
respectively to
j4
stand respectively at
il-th, i -th,
2
Now using (3.) and (4), these two elements reduce
11
The product ot these two is
6,
where
The number ot times we get the same product
elements ot R
l
is equal to the number ot times the symbol
and R
2
li
a.
& by multiplying two corresp;niling
1
~l
f.~2
li
li
3
4
ai i
a
a
i
2
3
4
occurs in the treatment conta:J.ned in the vector 1:,-
Hence we get,
By definition, this number is
12
4
s-l
= A [JI
r=l
( E
k=o
ak )
i
r
C\(jr)
J,
as desired.
Case II.
il
=i2
and i
3 = i4
Then
jl j2
a
e (a
i1 i2
=
Case III.
,
a
• J 2•
•
J• l
i
a
1
i
)
2
2
A [ TL
r=l
(i , i ) has exactly one factor common with (1 , i ), say i l = i •
4
2
l
3 4
~hen
Proof of cases
II and III follows on lines similar to case I.
4. Further remarks on the A-operator.
From the last result we see that each element of EE'
linear function of the A'S.
is expressible as a
In this section we propose to demonstrate a number of
techniques that could produce further simplification in the calculation of these
elements.
s-l k
Since for all i, we have A( E a i )
k=o
defined by
I.
a
s-l
a
=
E
k=o
k
ai
'
A(a)
= n,
=
n
we shall create a new symbol
•
•
13
s-l k
k
').. {( E a l ) a 2 )
i
i
k=o
1
2
I
Consider
where
i
l
~ i
2
•
From (18) this quantity equals
==
Hence ve observe an important property of a viz:
(24)
a
k
a a 1
i
l
II.
Let
Tl , T2 , ""., T
k
Then using
...
,
k
i
k
k
= ai a = ai
k
k
k
2
l a. 2
etc •
ai = ai
J.
2
1
2
a
be k
= a ,
a a
sets of assemblies and let
T = T + ... + T •
l
k
(19) - (22), one easily ftnds that
e
Let T be a set of assemblies expressible in the form T = U ® U '
2
l
where @ denotes Kronecker product" U is a set of assemblies involving say
l
k factors, and U the remaining m-k factors. Let the first set of k factors
2
III.
be denoted by G
l
belong to Gl "
and the second set by G •
2
and
iI, i
2" •••" i~
j'
j'J
a 2r ••• a ~t ,
i2
i
j
j
').. {a. l a 2
J.
i
l
•
2
The above result has a useful coro1.lary.
where
Pl
~
T )
f
J. 1
=
i " i " ••• "
2
l
belong to G • Then it is easy to check that
2
jW
a. 1r
(26)
Let the factors
Let P
€
is a polynomial involving factors from
@ ." and suppose . P = P1 P2"
G
l
and P
2
from
G " Then
2
•
14
A.{p "T)
IV.
•
Suppose T is a set of assemblies which satisfy a certain set of equations
over a finite field GF(s)
prime power).
with s
elements (we are now assuming s
to be a
VIe shall state a 'Well known result and prove another one" both of
which shall be useful in the computation of the A.(p" T).
Lemma 4.1.
s
Let there be m factors
is a prime power.
taking in EG(m"s)
,-
al" a , ••• , am" each at s levels" 'Where
2
Suppose we obtain a s -k fractional replication T by
(28)
k
independent linear equations
g11xl
+ g12x2 + • ••• + glmxm = c
l
g21xl
+ g22x2 + • ••• + g2mxm = c2
•
•
•
~lxl
•
•
e
•
+ 9:t2x2 + • ••• +
•
•
~xm
•
•
e
=
~
where all symbols represent elements in GF(s).
Suppose all linear combinations
of the above equations contain d
which have non-zero coefficients
g'Se
(i)
or more x's
Then
(for the case
d
= 2t)
assuming interactions of
(ii) (for the case
d
assuming interactions
jl
(iii) the symbol a.
J. l
all interactions up to t-factors can be estimated
(t+l)-factors and higher orders to be negligible.
= 2t+l)
all interactions up to t-factors can be estimated
of (t+2)-th and higher orders to be negligible,
j2
jr
a i ••• a i
' where 1 < r < d and J l "j2" ••• ,j =
r
2
r
jl"j2'·· ·jr
A..
times in the assemblies of the fraction T"
is independent of iI' i 2 " ••• ,i r
for
1 < r < d.
,
15
(iv)
s
m-r-k
•
In the context of the above theorem" let us find the value of
when
r
> d (for the case of the fraction T). Let ex"
o ••• exs- 1 be the s
elements of GF( s)" and
of any factor correspond to the element
jl"j2" ·,··jr
exj • Then for finding the value of
A.. of .
i "we have to find how many
J. l ,ll<2"··· r
solutions of the equations (28) are such that
x.J.
••
=
l
let the level
jl"
jl··· jr
For obtaining A..
i
J.
equations (28).
Case I.
j
l
•••
we therefore substitute the values (29)
r
in
Two cases may arise:
Substitute the values (29) in equations (28).
If the resulting
equations are inconsistent" then
•• , jr
••• i r
Case II.
=
0
•
After substitution we may find k'«
k) equations to be independent.
By
the previous theorem" we shall then have
=
The above gives us
•
number of points on an (m-r-k ~) flat in EG(m" s)
•
16
Lemma 4.2.
equations in GF(s), s~ as at (28), which generate
Let there be k
an orthogonal array T of strength d.
Let S be the set of all equations ob-
tained by taking the linear combinations of equations (28).
Let S I
be that sub-
set of S in which only Xi , ••• ,x.
occur, and let the number of independent
jlj2· u j r
r_k"
Then exactly s
of the i\.i i
. (r > d) are
1
equations in S'
be k".
J. r
1 2'· ·J.r
nonzero, namely those, for which
k
satisfies the equations
leaves k'
S'.
= 1"
Further if the substitution (29) in equations (28)
independent equations, then
s m-r-k '
•
We shall noW describe a
s
~ractical
(30)
(m-k)
(for higher
=2.
Let
(28)
be written
= c
an obvious matriX notation" where G is a
matrix of rank
= 2,
(EE')T' when T is an orthogonal
First consider the case d
G x
•
technique for the case s
the generalisation is obvious) of computing
array of strength d.
in
2" ••• r
kxm matrix,
Let G be an
whose rows are orthogonal to those of G.
(m-k):xm
Then it is well
known (see for example [2J) that G possesses the property P viz that no d
d
columns of G are linearly dependent. (In fact one way to construct equations
(30) for obtaining an array of strength d is to start from
matrix G which is orthogonal to it).
has m columns which we may denote by
correspond to the i-th factor.
•
(§i + §j).
and then get the
Assume now that G is given to us.
~l" ~2".
"§m.
It
Let the i-th column
Form an mxm table, the i-th row or i-th column of
which corresponds to the i-th factor.
vector
G,
In the (i"j) cell of the table, put the
Notice that only the cells above or in the diagonal of the
table need to be computed.
In the process of constructing economic fractions" we
~
17
shall encounter only those cases where m-k is small.
eX~ination
of this table is very revealing.
If d
For such cases an eye
<;,
then if a vector
~f
occurs in any cellaI' the table, say in the (i,j) cell, then it can be easily
checked (using Lemma. ;.2) that we must have
.:!:
m-k
2
vThere
,
The s~e holds for all values of the pair (i,j)
in which cell ~f occurs.
If
d < 4, then we shall similarly have
•
P4 = (ai - a~ ) (a; .. a~ ) (ar .. a~ ) (at .. a~ ) ,
1
1
1
12
provided that the cells (iI' jl) and (i 2, j2)
~r in them.
2
J2
2
of our table have the same vector
An examination of the formulae (20) - (22) shows that if d is at least 2,
then only the expressions of the form (21) and (;2) need to be evaluated, since
in that case we assume ~(l)
If c
the signs
= 1,
do(l)
= -1.
is. the zero vector,
then it can be easily checked that one must use
.
-
and + on the
r.h.s.
of (;1) and (;2) respectively.
cases an examination of i-th, j-th, f~th (for
jg-th (for (;2»
V.
s
and il-th, jl-th, i -th,
2
columns of G is usually quick enough for this purpose.
We shall consider now the :matrix EE'
Case Io
(;1»
In other
= 2.
Here we shall write
in detail for the cases
s=2 and s=;.
•
18
(:33)
(EE' )
where the M's
=
M
l
M'2
M
2
M4
~
M'
5
~
M
5
M6
are submatrices such that the 1st row block (and the first column
block) refer to the general mean
IJ., the second row and column blocks correspond
to the main effects, and the third ones to the interactions, all expressed in their
natural order, as in' L.
is an m x (~)
Thus for example the matrix M
5
matrix
whose rows refer to the main effects and columns to the interactions.
When we wish
to be more expressive, we shall refer to this fact by saying that M
corresponds
to
[Al , A2 , ••• , Am: A12, Al ;, ... , Am_l,m
abbreviate the symbol AiA
j
[IJ.:
to A •
ij
J,
where for
Similarly M
2
Al , A2 , ... , Am ], and the matrix
[M2
n
2
dl(l)
M;]
to
n
2
5
designs we shall
corresponds to
[IJ.: Al , A2,· .. ,AmiA12,· .Am_l,m]
etc.
Since for the
case, we have
= 1,
do(l)
4 are given by e(Ai , Ai) which equals ,,(ai + a~)
of M
= -1,
= n.
the diagonal elements
Similarly M
l
and
the diagonal elements of M
6 are all equal to n. The elements of M2 are of
the type €(IJ., Ai)' which equal "(at - a~), written for short "(l'-o')i or even
"(1 '-0')
in
li
it being understood that i-th factor is under consideration.
or 0
1
or
l'
or 0'
The prime
denotes that they are ordinary algebraic symbols
representing levels of a factor, and that they can be manipulated in the same way
as
a 1i
2 1'.
or a 0 • Thus instead of writing
i
•
type
( ) or even
2,1'
Hence the elements of M are of the type
2
"(It - 0')
where
1 we shall write
2 ai'
i
= "(a - 2 0')
has been omitted from "'~ •
e(IJ.J AiA )
j
which equals
=
n - 2
,,0 ,
Similarly the elements of
"'(1 8 _ 0')(1' _ 0')
= (,,11+
M;
are of the
,,00) _ (,,01 + ,,10),
•
19
in an obvious extension of our notation. Also
(;4)
•
=
e(A , A)
i
j
"-(1' - 0')
(1' ,.... 0.1)
i
j
For the full operational case of this symbol, we must note:
Thus we have
which together with
(;4)
gives
=
e(Ai,Aj )
•
e(Ai~' Aj~)'
for k ~ i,j
•
Elements of M
are of the type e(A , AiA j ) or e(-'\" AiA j ), which respectively
i
equal "-(lJ - O')j and "-(I' - O')i (1' - O')j (1' - O')k. Similarly the ele-
5
-'\,A ) in M6 equal "-(1' - O')i ••• (1' - O·)f}.
f
Let us call a fraction T to be (1,0) symmetric with respect to triplets
ments of type e(AiAj
jl
j2
j;
,
ji
j2 a j3
1
T), where each jJ = l+j (mod 2).
i;
1 i 2 i;
This means for example that for any triplet of factors, the combination (101) occurs
if ,,-(a.
~1
ai
a
,T)
2
as many times in T as
"-100
= "-011
etc.
=
,,-(ai a
(010 ) 1 i. e.
"-101 = "-010 ~
Similarly "-000
= "-11\
Also we have
J
=
"- (1'(1'
+ ot) j (1' + 0')k - 0'i (1' + 0') j (1' + 0')~
i
=
"- [(lSl~l')+(l'O'O')+(l'l'O')+(lrOrlr)_(O'lrl')_(OtorO')_(0'1'0')_
(O'O'l')J
=
0
•
•
20
Hence
1
0
= ')I..
and M
are
5
zero is that the fraction be (1,,0) symmetric with respect to triplets. This con')I..
Thus a sufficient condition that the matrices M
2
dition is necessary also, as can be verified by starting fram the above equations
and solving back.
Case II.
s
=3.
He shall use an obvious extension of the notation for the case
further explanation.
The matrix
(EEf)
=
•
s = 2, without
is broken up as
~
M
4
M
a
M
M
12
M
6
9
M
5
M
10
Mil
M
13
M
14
M
15
M
16
M
17
M
la
M
19
M
20
Sym.
M
2l
where the row blocks (and the column blocks) respectively correspond to
[~]; [B l , B2 , ••• , Bm ]; [B~, B~, ••• , B;]; [B1B2 , B1B , ••• , Bm_1Bm] ;
3
2 2
2-2
2
2
22
2].
[Bl B2 , BIB3' • •• , Bm_1Bm]; and [BJ:B2' B1B2 , ••• , Bm_1Bm WTJ.tten for short
(~), {Bi }, (B~}, {B1Bj }, {BiB~} and (BiB j }. Thus for example M2l is an
m(m-l) x m(m-l) matriX corresponding to [{B~Bj}] and M is m x (~) and
9
corresponds to
[{B } : {BiB }].
j
i
The calculation of various elements of EE'
be illustrated for some of the matrices
(37)
do(l) =
d (2)
0
--
-1,
dl(l) = 0,
1,
d (2) = -2,
l
=
Then we have for example
M.
To apply (20) - (22),
d (1)
2
d (2)
2
=
1
=
1
will
we shall define
21
.-.
=
(a)
M1:
E(~,~)
(b)
M:
E(B1,Bi )
7
22
= ~1j
E( B1,B j )
(c)
n
= ~{2' - 0')i(2 1 - 0 1)1
= ~(a - Ii) = n - ~i1
00
+ ~ij -
02
~ij
E(B1,Bia~) = ~(21
M10 :
= ~(21 - 0 1)1 ( a
= ~(21 + 0 1 )1
20
~1j
-
- 0 1 )1 (2 1 - 2 l' + 0 1)1 (2 1 - 2 I' + Ol)j
- 3
Ij)
2
0)
(21
Dl)
= ( Ai - ~1 - 3 ~ij - ~ij
E(B1,B?~) = ~(21 - or \ (a - 3 Ij) (a - 3
= ~(2'
•
- Ol)i (a - 3
J.k)
Ij - 3 1k + 9(1 ' 1
l
)jk)
_ (~2 _ ~o) _ 3(~21_~01) _ ;(~21 _ ~01)+ 9(~211 _ ~011)
i
i
1j ij
1k
ik
1jk
1jk
-
(d)
M21 :
E(Bis j , BiBj)
= ~(2'
=~ {(2
1 -
2 l' + 0')1}2 {2' _ 01)j}2
+ 4 I' + Ol)i (2' + 0').
= ~(2t-0')1
J
(2' - O')j
E(B~j' B~k) = ~(a
+ ;
= E(Bi ,
= ~(a
+ ;1') (a I!)
1
J
Bj ).
Ii) (2 1 - Ol)j (2'
-
QI)
k
22
€(B~Bj,BiB~)
~
€
A(2 f
=
(~~j' B~i)
€(Biaj,BjB~) = A(a
=n
Ol)i (2 1 - Of)j (a + 3 Ik - 6 Ik)
-
- 6 A [Ik(2 1 - 01)i(2 1 - Ol)j]
- 3 1i)(a - 1j) (a - 3 Ik)
1
1
11
1
(11
11
111
- 3(Ai + Ak) + 9 Aik - Aj + 3 Aij + AJk ) - 9 Aijk
€(BiBj,~B() = A(a - 3 1i)(a - 3 Ik) (2' - 0')j(2 1
-
0 1)( , etc.
A fraction T will be called (2,0) symmetric with respect to triplets if
j1 j2 j3
ji j2 j}
A(a i a. a i ' T) = A(a. a i a i ' T), where each j' = 2 + 2j (mod 3). This
1 ~2
3
~1
2 3
100
meanS that' for example A
= A122, A020 = A202, A012 = A210 etc. Using the above
A-calculus, we can easily check that this is a inecessary and sufficient condition
{B }
i
in order that the matrices M ,
2
matrices. In fact this breaks
22
(m + 1) x (m + 1).
in order that the effects
2
and {B Bj } are uncorre1ated With the rest, i.e.
i
M , M , M , M , M , M
are all zero
and M
8 9 10 6 15 18
20
EEl into two matrices of orders m2 x m2 and
The material in this section is useful not only in the analysis of fractions,
but also in their construction.
In many situations for constructing fractions, one
has a number of sets of assemblies T , T , •••• The formulae in this section
1 2
help one to readily evaluate (EEl) for each T • By examining the nondiagona1
i
elements of EE' for each T, one can then piece together a few of the TIS in
such a way that the resulting fraction T. + T. + ••• + T.
J1
the nondiagona1 elements are small enough.
J2
Jk
Achievement of
symmetry introduces further simplicity in inverting
lations of the estimates, some of them to zero.
say is economic and
(1,0) or (2,0)
(EEl) and decreases the corre-
2;
6. Examples for symmetrical fractions.
To illustrate the preceding theory 'He shall present tllO fractions.
7
A 2 fraction in 44 assemblies.
Example:
The fraction is T
Tl
= T1
T2
1100
rOOOOOOOJ-
Llllllll
•
1
lly
1
100J010
-
001
f
®
1-
01l
101]
110
It is evident that T2 is coltP1etely (1,,0) symmetric. AlsoT4 t: T:; "
Le. T is obtained by interchanging 1 and 0 in T • Hence the whole
4
3
fraction T is (1,0) symmetr:i.c. Let the two sets of factors be denoted by
AI" A2"
'3'
We have for
B1 " B2" B:;.
A4 and
i
I:
j
€
(A."
J.
Consider now the matrix M4
€
For
€ (Ai"
Bj )
in (EE I)
in (33).
"
A.)
J
=
2 + 2 x (2-4) + 3 x (3 - 2) + 3 (3 - 2)
where the different terms correspond to the T
i
=4
Similarly
IS.
(B."B.) = 2 + 6 x (2 - 0) + 5(1 - 2) + 5 (1 - 2) = 4 •
J
J.
we use the property stated in sec. 5. II.
€(A." B)
J.
j
=
E
Tfs
=2 +
tit
1111
0001
0010
0100
1000
r
®
1110
1011
0111
LIIO I
-000
®
1001
0101
0110
0011
= 29)
+ T2 + T; + T where the T are given by
4
i
T;
T4
0000
1010
(v
Hence M4 could be written
(AI. _ AO
Ai
Ai
)
(Al
Bj
He get
- ~ )
j
(0 x 0) + (3-2)(1-2) + (-3+2)(2-1)
=0
,
r
I
1
(0) J 43
40 14 + 4 J 44
40 13 + 4 J 33
(0) J;4
24
J
I-
This can be easily inverted to get
P2
P1
-1
M4 =
I
P'2
1
where P1 To I 4 -
=
~ J 44 , P2 ':=. (0) J 4; ,
P;
and
1
P; = To I; -
1
520 J;;
An examination of M 1 shows that the main effects, which are correlated only
4
among themselves, have a small correlation of the magnitude 0.08 nearly.
e
27 fraction in 48 assemblies
It is interesting to compare this with a
proposed by AddeJ.lnan [lJ.
In the latter :plan, 5 main effects are correlated with
some of the interactions, the correlation being
31
for four of them and
21
for the remaining one" but the other correlations are lesser than in our :plan.
~
Let us now consider
terms of M "
1
These are
~and
and the interactions.
M
6 are a1reaay known except those which involve 4 factors.
_ ,0
( ,1
,\0a
'"a
i
i
=2
Using (;4), we find that all
)
e
(B 1" B2:Z:
B )
+ (;-;)(1-1) + (;-2)(;-0) + (2-;)(0-;)
=2
e(AiAj
" ~B~)
e(AiAj
, ~B~) =
./
=8
;
+ (2-4)(2-0) + (;-2)(1-2) + (;-2)(1-2)
=
-4
2 + (;-;)(1-1) + (1-4)(1-2) + (4-1)(2-1) = 8.
25
Let the submatrix of EE I
'tvhich corresponds to [Il; A A ) Al~' A A , A2~' A A4 ,
I 4
2
I 2
A'!4; B B , B B , B B ; AIB ,· .. ,A B , A B , ... , A B , A B , ••• , A B
2 3
4 2
I 2
I 3
4 l
4 3
I 2
I 3
1
order be denoted by O. then from the above calculations
I
44
1
I
4 J l ,6
4 J l ,3
(0) J 1,12
Q
(-4) J 6,3
Q
J
I
1
I
I
l
Q=
2
Q
%
4
Q
Sym.
where
J
m,n
II
n
=
or I
=
mn
5 .J
(nxn) identity matrix
(mxn) matrix with unity everyvThere
I
6
being a
with unity in the cross diagonal and zero elsewhere,
.....
Q2 = [Q6 Q6 Q6J, Q6 =
I
I
!
i--
~ =
40 I
3
+ 4 J 3 ,3
0
0
6
0
6
0
0
6
6
0
6
0
0
6
6
0
6
0
6
6
0
0
:-1
nxn matrix
in this
e
26
(0) J 1,4
Q4
=
CO)
J l ,4
L 8 J 14
l
(0) J l ,4
8 J l ,4
Co)
8 J 14
(0) J 14
14
(0) J 14
8 I 4 - 4 J 44
8 I 4 - 4 J 44
40 14 + 4 J 44
8 14 - 4 J
44
Sym.
The
J
40 I 4 + 4 J 44
inverse of the matrix (38) which could easily be obtained by using the algo-
rithm for the analysis of multidimensional partially balanced designs is given in
sec. 8 of [ 6] •
Example 2.
4 fraction in 45 assemblies.
A 3
(\)
= 33).
This fraction
T
which is symmetric with respe¢t to the different factors is given belO't".
0001
0021
1002
2120
1120
2110
0010
0012
2001
2021
1102
0112
0100
0120
2010
1022
1210
2011
1000
0210
1020
0122
1012
0211
2221
0102
2201
1220
1201
1111
2122
1200
2102
0212
0121
2222
1222
2100
2012
1202
2101
0000
As indfcated, apart from the assembly (1111),
assemblies of the types
the fraction consists of four
(0001) and (2221) each, and twelve assemblies of the types
(0021), (2201) and (1120) each.
Obviously, the fraction is
(2,0) symmetric.
•
2'7
vle shall. first compute the diagonal elements of'
"'here for convenience, we write
uf using
(BE' )T
3n factorials, in place
(2~ + Ie + 0') = b, for
a.
= ~(b +
e(BI B2 , BI B2 )
3 If)
= ~(2'
= 96
•
+ 0')1 (2' + 0')2
= ~(b
= ~(b - bi - b~ + bi b~) = 48
e(B2B2 , B2B2 )
I 2
I 2
= ~(b
= ~(b
- 32 + 4
= 20
+ 3 b l ) (b + 3 b ~ )
l
2
=
= n + 6 ~. + 9~"
e(BiB2' BiB2)
1
1
- bi)(b -b2)
+ 3 bi)(b - b;)
Next we turn to the nondiag'Onal elements.
separately for each submatrix M of
i
48 + 96 + 36
= 48
= 180
+ 48 - 16 - 12
•
= 68.
The computati<Cilns shall be presented
(EE')T
as at (40).
(1) ~.
€(Bl,BlB~) =~(b
2
€(B ,B B )
l l 2
- bi)(b - 3
b~) = 48
- 16 - 48 + 12
= .4
= e(B1 ,B2 ) = O.
E(Bl,B2B~) = ~(2'
- Ot)(2' - O')(b - 3 hI)
~
0 _ 6(A
122
120
_ A
)
=6
28
e(B1,B2)
=0
e(Bl,B~B3) = 6
e(BIB~, BIB~)
=
A(b -b~)(b - 3 b1)(b - 3 b1)
= A(b
- b1)(b - 3 b1)2,
say,
48.. 112
=
e(Bia2' BiB3 ) = A(b
+
+ 60 ..
18
= .. 22
3 bl )(2' .. 0')(2' .. 0')
= 2 A[b(2' .. 0')(2' - O')J - A(b - 3 b1 )(2' - 0')(2' - 0')
0 - e(B1, B2B~) = -6
=
e(Bis2' B~4) = A(b .. 3 b1)(b - 3 b1 )(2' - 0')(2' .. 0')
= e(B1, B2B~) .. 3 A [1'(2' - 0')(2' - o')J
1122 _ Al120 ) =-6
= 6-6(2-3) + 9 x 2 (A
e(~, Bia~) = A(b - 3 b1)(b .. 3 bl
+
9A[1'1'(2'-Or)(2'-Or)J
)
= A(b .. 6 bI + 9 b11) =48 - 96 + 36 = -12
-12
e(B1B2, B1B )
3
=
=
1
A(b-b+)(2' ..0')(2'-0')
e ( B2,B ) - 2( A122 - A120 )
3
=0
(
- 2 2-3)
=2
29
2 B2B2)
e(B12B2,
3 4
= A(b
- 3 bl )4
=
A [b + 4{-3 b+) + 6(-3 b~)2 + 4(-3 bl )3 + (-3 b1)4J
=
n -
12 Al + 54 All _ 108 Alll
= 48 - 192
+
= 81
Allll
216 - 216 + 162 = 18 •
=0
e(Bi, B1B2 )
=
e(Bl, B2 )
e(Bi, B2B;)
=
e(B2, BiB3 )
= 6
A[b + 3(-3 b~) + 3(-3 b1 )2 + (-3 b1 )3 J
= n - 9 Al + 27 All _ 27 Alll
+
= 48 - 144
+ 108 - 54 = .42
e(B1B2, Il)
•
=
€(B1B2, B~)
2
0
=6
2
€(B1B3, B2B4 )
= -6
·e
30
The two matrices
corres~onding to [[Bi }; [BiBj}J and
[BiB~JJin which (EEf)T is split be~ause of (2,0) symmetry
01 and 02
[[~}; {Bi ]; {BiBj };
can now be ea.f;Jily written dow ..
Ex 3.
m
3
For a more theoretical example, consider a balanced fraction
factorial, i.e. a fraction T
to all the
m factors.
We define
for which
(EE')T
T
from a
is symmetrical with respect
T to be commutative, if for all unequal
i, j
and k, we have
Let us obtain an equivalent condition (used in sec. 5 of [6J) in terms of the Afunction.
4It
The above could be written
A[(b - bi)(b - ,;
=
A
[(b
b~)(b
+ 3
- 3
~) ]
bi)(b~ - b~)(b~
-
b~)
]
This simplifies to
the suffixes being dropped because of symmetry.
7. The asymmetrical case.
The theory for this can be developed on lines parallel to the symmetrical case
given in the preceding sections..
For lack of space only the results will be stated
m2
and the details will be ommited, restricting ourselves to 2~ x 3
factorials.
.
by
•
A general treatment combination (and also its 'true effect') could be denoted
i1 i2
i ml
jl j2
j~
a
b
b ••• b
1
where i!s € (0,1) and
a ••• a
2
2
l
l
m1
m2
Given a set of assemblies T, the number of times the symbol
e (0 , 1,2).
j2
j2
jr ji
j~,
a i ••• a i b
bit ••• bit
1
2
r ~l
2
r'
jl
Q = ai
j's
o ,
occurs among the assemblies in T will be
jl,j2,··~,jrJ jf, j~, ~.~,j:,
denoted by
Ail,i ,···,i J if, i
2
3
of section
r
2, ... ,i:,
31
or by
A(Q), extending the notation
.
°2,3.
The set of all possible assemblies will be denoted by
effects in which one is interested fall in three groups:
factors at
2 levels each, or pure A-effects (i1)
AB effects, i.e. those which involve at least one
(i)
The set of
Pure effects of
Pure B-effects (iii) Mixed
A factor and one
B factor.
We shall adopt an obvious generalization of the notation and terminology of the
previous sections, and define a general interaction of our mixed factorial by
=
t
jl,J2 , .... ,jm J J.,l ,J• 2I ,···,. j m
.
2
l
1::
j's, jl 's
'\1,k2 , ••• ,k~j kl,k
2,••• ,k~2
.
.,
j'
m
(a ... a lb ••• b 2)
m 1
m
l
2
l
jl
Jm J l
,
where
jl' ... ,j m J jl', ..
l
'\1' ...,km j
l
where the
the
·,j'm
2
k ' , ••• ,k I
l
m2
dj(k) refer to the factors
d's defined at
A (at 2 levels each) Qt1 are the same as
U{,) , and the dj(k) refer to the factors B(at 3 levels
each) and are the same as the corresponding
d's defined at ( 37) •
In writing the symbol for the interactions, we shall stick to the earlier
convention that if any k
or k'
is zero, we drop that factor from the symbo1.
The normal equations for the mixed case can be set up by approaching exactly as
for the symmetrical case.
•
It can be seen that corresponding to the matrix EE',
which we had to invert for the symmetrical case, in the mixed case we shall have to
invert another matrix which may be denoted by FF'.
explained below without proofs.
The nature of FF'
will be
e
32
As for EE I, the rows and columns of FF'
(i)
correspond to the effects:
[~},
the general mean, (ii) [Ai}' [AiAj }, the pure A-effects, (iii) [Bi },
(Bi}, {BiB j }, {B~j}' {B~~}, the pure B-effects, and (iv) (AiBi ,}, [AiBit}, the
mixed AB effects. The form of FFt
(40)
=
FF'
°1
n3
nt 3
°2
is exhibited below
where °1 is of the form:
{~}
{Bi }
{Bi2}
{BiBj }
(B?j}
(B?~}
}
Ml
M2
M:;
M4
M
5
M
6
M.r
fBi}
Mt
2
Mll
M12
M
13
M14
M
15
M16
[Bi2}
~
Ml2
M20
M
21
M22
M
23
M24
}
M
Ml3
M~l
M28
M
29
M;o
M
31
(B?j}
Mt
5
Mi4
M22
M
29
M;5
M:;6
M:;7
[B?~}
M6
M~5
M
23
M
30
~6
M41
M42
(AiAj }
M.}
Mi6
M24
~l
~7
M
42
M
46
(~
e
(41)
{BiB j
.
•
4
(AiAj
}
e
;;
°2
is of the form:
{Ai}
2
{AiBj }
{AiB j }
{Ai}
M
50
M
5l
M
52
{AiBj }
M
5l
M5;
M
{AiB~J
M
52
M;4
M
55
(42)
0;
and finally
(4;)
54
,
can be represented as :
{BiBjl
{B?j}
{B?~}
{AiAj }
The order of the vrhole matrix FF'
~
v0
= 1
+
m(~+l)
The whole
is v
o
x v
0
where obviously
+ 2n2 + 2 mn.
matl~:ix
FF'
bas 100 submatrices.
The set or sets of interactions
to which a submatrix corresponds are shown on the left of the rows and above the
34
of size mn x mn,
etc.
M
45
corresponds to
Each element of FF'
[(B?~) i {AiB~}] and is of size (~) x mn,
thus corresponds to a unique pair of interaction.
element in the cell corresponding to the pair of interactions
't'1I'itten e(x,y).
Thus for example
e(B?2' B?4)
. lies in
The
(x,y) 'Will be
~6'
e(B1 ,B2 ,Bi)
in
M
21 etc.
The following lemma can be easily established on the lines of Theorem
3.1.
Lemma 7.1
(44)
=
k , k,j' f II f , , at most two are nonzero;
j
1
1
Where, (i) out of the four symbols
and similarly for krl. k t , frl' ftl ,
(ii)
1'1(A)
is a polynomial in a~
k;
k
e(A - A
j
i
and similarly fiB)
j
k
,
A
r
r
k
At
t
defined by
)
~ (Pl(A)} ,
is defined by
f
f i' j , k r I k t •
e(B i' Bj • , Brl Btt )
(iii)
=
=
~(P2(B»)
after expanding the product P1(A) P (B) algebraically, the value on the
2
r.h.s. of (44) is obtained by using in the usual way, the definition of
.
any assembly
Q•
We shall now illustrate the calculation of elements of FF'.
~(Q)
for
35
=
'\( 1
0)(b2
bO)
~ a l + al
2 - 2
= A(b22
-
bO)
2
.,,2
= A.,,2
.,,0
- A.,,2
..
where the dot shows that no factors of the type are involved•
• ~
(1,,0) and (2,0) symmetry are defined as before. The following lemma is
The
an extension of previous results.
Lemma 7.2.
~
Let T be a set of assemblies from the
2
.
m
x 3 2. factrifLl.
Then
(i) if T is (1,0) symmetric With respect to triplet of the fact~s ~, the
matrix FFt
may be broken up into two orthogonal parts, one corresponding to the
222set of effects (~}" {Bi ), (Bi ), (BiB j ), (B1Bj ) , {BfBj} and (AiAj ) and the other
corresponding to the set {Ai}' (AiB j ) and {AiB~),
(ii)
if T
is
(2,0) symmetric with respect to triplets of factors B, the two
is broken are {~), (Bi)" (BiB j
sets in which FF'
2
{AiBj
(iii)
}
}"
(B~~}, (AiAj }" {Ai)'
2
and (B i }, {Bi Bj l, {AiB j } •
if it has both the above properties" then FF'. is broken into 4 parts
corresponding to
(c) {AiBj
},
2-2
( a ) {f.!}, {B2 }, [BiB ) .. {BfBj}'
{AiAj },
j
i
and (d)
{Ai}
and [AiB~1 •
«(
b) B ),
i
2
[BiB
j
}
;6
All the above results will be amply illustrated in the next section.
8. Examples of the analysis of asymmetrical factorial fractions.
Example 1. Suppose we have an orthogonal array f* of strength 4 in
m(= ml + m2 ) factors" with s =; and N assemblies. From this array" one could
m
m
obtain a fraction for a 2 l x; 2 factorial by making the transformation
of the levels of m of the factors. This means that for these
l
m factors we replace the level symbol 2 by 1. In this example" we shall illusl
trate the above developed method of analysis of asymmetrical fractions by considering the analysis of certain fractions obtained by such transformations.
Let the set of N assemblies obtained after making the above transformation
be denoted by r 1.
Obtain N more assemblies
transformation T = .(01 10 2 ).
2
0
Obviously"
r 2 by making the complementary
Consider the analysis of r
+ r2 •
r is (1 / 0) sylllmetric With respect to the factors A. Hence
by using Lemma 7.2" and equation (40)" we see that 0;
matrix.
=rl
Also 01 in
in
(FFt)r is a zero
(40) can be written as
(EE' );
L
'\ =
where
(EEt)
4 m
represents the matrix EEl the; 2 factorial fraction obtained by
ignoring the factors
I
(.
A in the mixed fraction
of strength 4 in factors
B" the matrix EE'
whose elements can be calculated as in Ex.
must equal
r.
However since
r
is an array
is a diagonal nonsingular matrix
6.2. Also fram (41)" the matrix
LI
37
~
vIe now show that except
all other matrices in
L are null.
It is
sufficient to observe that each element in all such matrices is of the form
A[P l (A)}[P 2 (B)}, where Pl(A)
is of the form
(ar
a~)(a~
-
-
a~)
and P 2(B)
is a product of one or two polynomials, each of which are of the form
bk - bk0)
( 2
(45)
( b 2 - 2b1 + b 0) • Now
k
k
k
or
A[Pl(A) P2(B), rlJ
= A[(a~a~
+
a~a~
-
a~a~
-
= A[ (4 b1 b1j + b 0 a 0 - 2b1 b 0 - 2b1 b 0)P (B) , P*
i
i j
j
i j
i 2
a~a~)P2(B),
]
rlJ
,
by going back on the transformation T " Since I'* is an orthOgonal array of
l
strength 3, the above expression is clearly zero. Similarly for r 2" This com-
.e
pletes the proof of our assertion"
Hence
is split up into a diagonal matrix and the matrix
01
~
(46)
M
46 J
M+
Any element of
since
j' f
A. ~,
~~
~
is
A[(a; -
N
must all equal -9
elements in cells of the type
elements in M
46
involving
I
a~)(a~
in
-
P*.
a~),
which fram (45) equals
Also from this it follows that the
(AiA , A ,,\)
i
j
4
rJ
in M
46 also equal 2N/9.
factors one gets in a similar manner
For
38
4
A[
.rr.
r=l
(a~ - a~ ), r 1
r
+
r
r2 J
4
0
4
21
21
= A[( J:[ (b i + bi - 0i )} + {]I (-bi - b i + b~ )},
r=l
r
r
r
r=l
r
r
r
=
4
4
A[ f. ::If. (b - 2 b~ )} + { ]I (2 b~ - b))" r'* J
=
2A [
r=l
4
JC
r=l
r
(b - 2b~ ),
r
r=l
r'*J
r
I'*J
because of symmetry with respect to quadruplets of factors.
The last
~xpression
eVidently equals
2(N - 8 •
~
+ 24·
~
N
- 32 • -27 + 16 •
8~) = 2N/8l
Formulae by which the inverse of M
and hence of (46) can be quickly ob46
tained are available in sec. 5" [6J" ~nd will not be reproduced here. However
it would be of interest to knoW' the correlations between the various interactions
Thus the main effects of factors
any pair is found to equal
A are correlated and the correlation between
(-l/m + 8).
l
•
where x's
take the three values
fraction of a
3
out of the 243 assemblies of the
0,1,2.
6 factorial that we get this way, we eliminate all the assemblies
that contain the level 2 of any of the factors represented here by xl' x2 '
and x4. vle then get the 48 assemblies T = T + T + T , exhibited below.
2
l
3
Tl
T
2
r1111 -I
0001
0010
0100
,,-0
.-
0000
1110
1101
1011
0111
x
Since each of the
[~J
x
11
10 I
U
22
1000~
T4
!llOe""
1010
1001
0110
0101
L001l
x
X,
[~l
16 combinations of the levels of A-factors occurs exactly 3
times, it is clear that the effects
{Ai}' {AiAj }
are all completely orthogonal,
Hence leaving these two sets of effects aside, and noting that the fraction is also
(2,0) symmetric, we find from lemma. 7.2 that FF'
and (iv)
2-2
(
{Il}, {Bi2}, {BiBj }, {Bj;Bj},ii)
{Bi }, {2
BiBj }, ( iii ){ AiB j
{AiB~}. We shall take up each part separately.
Part (i)
This corresponds to
corresponding to:
( i)
.......
M
~
~
M20 Mm. M
1
M
4
M
6
23
4 M21 M28 ~O
M
6 M23 li~O M4l
M
e
is broken into four parts
I
o
o
-2
96
6
-2
-6
96
-2
-6
22
-2
=
Sym.
6l
198
J
40
To see this, we notice that leaving
matrices are all
M1
=
e{~,~)
M4
=
e(~,BIB2)
M6
=
e{~,B1B2)
(1 x 1),
M;' M21 , M23
20 , the rest of the
and M
and
= 48 ,
22
= A(2'.0·)(2'.O') = A(22
+ 00 - 02 - 20)
= A(b - 3b11)(
b •1
3b2) = 48 -
=
48 - 3 x 2 x 16 + 9 x 6
1
3(~1 +
11
A12) + 9A12
=6 ,
M28
= e(B1B2, B1B2) = A(b - bi)(b - b~) = 48 - 2.x 16 + 6 = 22 ,
M41
= e(Bin~, Bis~) = ~(b
M30 = e(B1B2, BiB~)
+ 3bi)(b +
:;b~) = 48
+
2 x 16 x :; + 6 x 9
= A(2' - 0')(2' - 0') = -2 •
Also clearly
M3 = [0 OJ
since
e(B~, B~) = ~(b
+ :;bi)
= 48 + 16 x 3 = 96
e(B~, B~) = e{~1 Bi B~)
e(~, B~) = A(bi - 2 bi
The inverse of
=
+
6
b~) = 0
* is given by
°1
1
2If"
* -I 1
°1 ;:::; '2
e
= 5 x 2 - 6 x 2 = -2,
0
0
l ri
0
0
f
....
1 I
r2
f:;
f'
f
3
4
=
198 ,
41
where
1
r2 =
17 x
45
..-
2
2
I _!!3 II '32
2
I
136
135 x 9800
[r'1 r'3 r4] =
''"\
4
II
- 22
!
* =
°20
IMll
I.-
M14 !
I
Sym•
M35
J
-2
2
32
--~b
=
I
Sym.
62
-2
"l
_~ I
_ _ _ _M
-2
62.J
by calculations similar to above, and the inverse is
......
~.
68
4
4
2
-2""
68
-2
2
-2
35
1
1
35
I
1
2Ibo
2
Part (iii)
I
:
He have
n;2-j'·
0;3
=
e(AiBl , AiBl )
=
e(Bl,Bl )
e(~Bl' A2Bl )
=
A(l-O)l (1-0)2 (2 + 0)1
= A(l-O)l (1-0)2 (b
= _(All,l + AOO,l) +
=
32
- bi)
= -A(ll +
(A01,1 + A10,l)
= -(3 + 3 + 2) + (2 + 2 + 4)
= 0
:j
I1
-2
l--- -2
99
i
'3
~_
Part (ii) Here we get
i
00 - 01 - 10)a a bi
l 2
42
e(~B1' A1B2 )
=
e(A B , A B )
1 1
2 2
=
=
[(3+3) + 4x2] - [(2+2) + 2x2]
=
6 •
Hence
* = 6 J 44 - 8 I 4 '
'12
and
Part (iv)
Here
'*
1°41
M
55
=
Sym.
!-
*]
Q~2
°43
where
*
°41
*
= °43
= 96 I 4 '
Also
1
1440
*
°42
=
.
24 I 4 - 18 J 44
, 16 I 4 + J 44 i -4 I + ~ J 1
II ---- --------1-----------/
L Sym.
16 I 4 + J 44 J
~
)
•
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1.
1
Addelman, Sidney (1962).
plans.
2.
Bose, R. C. (1947).
Sankhya
;.
Technometrics
Symmetrical and asymmetrical fractional factorial
447-58.
Mathematical theory of the symmetrical factorial design.
8 107-166.
Bose, R. C. and Connor, W. S. (1958).
am
x;n
designs.
Analysis of fractionally replicated
Proceedings of the ;lst session of the International Statis-
, Tti-cn1 Institute, .:ert16selth, ; - 22.
4.
Bose,' R. C.
and Srivastava, J. N. (196;).
factorial designs. I",
Analysis.
Mathematical theory of the
To be published in the 'Proceedings of the
;6th session of the International Statistical Institute, ottawa'",
.e
5.
Bose, R. C. and Srivastava, J. N. (196;).
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Construction.
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session of the International Statistical Institute, ottawa'.
6.
Bose, R. C. and Srivastava, J. N. (196;).
Multidimensional partially balanced
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7.
Connor, H. S. and Young, Shirley.
(195~).
Fractional factorial designs
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Applied Mathenatical
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D. C.
8.
Chakravarti, I. M. (1956).
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9.
Daniell, Cultbert.
(1962).
SankhYa, 17 14;-164.
Sequential fractional replicates in the
#-q
I
series.
80urnal of the American Statistical Association.
10. Finney, D. J. (1945).
Annals of Eugenics. 12
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291-;01.
44
11. Kempthorne, Oscar (1952). The Design and Analysis of EePeriments.,
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12. Kishen, K. (1948). On fractional replication of the general symmetrical
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~d Srivastava, J. N.
(1960). Mathematical theory of confounding
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Society of Agricultural Statistics.
Journal of the Indian
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r
experiments.
Biometrika, 33 305-325.
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Bureau of Soil Science, Technical Communications No. 35 •
•
"