ENUMERATION OF (sn, sk) CONFOUNDED
SYMMETRICAL FACTORIAL DESIGNS
by
Pierre Robillard
University of North Carolina
Institute of Statistics Mimeo Series No. 559
December 1967
This research was partially supported by the
Universite'" de Montreal under a fellowship of
National Research Council of Canada during
the summer 1967 and the U.S. Army Research
Office-Durham under Grant Number
DA-ARD-D-31-124-G910.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N.C.
.~
1.
Introduction.
A series of papers by Bose (1) and by Bose and Kishen (2) showed how
the theory of symmetrical factorial experiments can be developed when the
treatments of the experiment are identified with the points of a Finite
Euclidean Geometry.
In particular a method of completely enumerating the
various possibilities of confounding for designs of the class (sn,sk) was
n k
n
described; a design of class ( s,s) is a s factorial design with n factors,
each at s levels, these being s k blocks each containing s n-k plots, in a
complete replication.
This report describes a new geometrical method of enumerating designs
n2
(S,8),
03
n4
of class ( s,s),
and (s,s).
of the class
(s6,s3) is carried out.
The actual enumeration for designs
It is shown here that we can estab-
lish a 1:1 correspondence between designs of class (sn,sr) and designs of
class (sn,sn-r) which preserve the equivalence relation defined by two
designs being of the same type.
The number of types of designs in class
(sn,s2) is derived.
This research ,,,as partially supported by the Universite de Montreal
under a fellowship of National Research Council of Canada during the
summer 1967 ~nd the U.S. Army Research Office-Durham under Grant Number
DA-ARD-D-31-124-G910.
2.
Definitions
Consider a sm factorial design involving n factors F , F , ••• F each
2
n
l
at s levels, s a prime number or power of a prime. Any treatment in which
these factors occur at levels x 'x2 , ••• ,xn respectively is denoted by
l
••• , x ).
n
A linear function of the treatments may be written in
L = x ,x E
, •• ,x Cxl ,x2 ,.·,xn T(Xl ,x2 , ••• ,x),
n
n
l 2
Xi taking all the s possible values.
Contrasts, orthogonality relation and degree of freedom are defined as
usual.
The s levels at which a factor occurs are identified within the
elements of a Galois Field GF(s).
The treatments T(X ,x , ••• x ) may be
n
l 2
represented by the points (X 'X ' •• 'Xn ) of the n-dimensional Finite EUClidl 2
ean Space EG(n,s). A 1:1 correspondence is thus established between the
s n treatments and the s n points of the space EG ( n,s).
Splitting and con-
founding of degrees of freedom can be translated in geometrical terms.
A symmetrical factorial design s n is said to be of class (n
s,s k) if
) of s n-k treatments each.
each replication is laid out in s k blocks (
~ n
n k
In paper (1) a way to look at the problem of construction of an ( s, s )
design is to consider k independent parallel pencils P , P2 ,···, Pk in EG(n,s)
l
With:
P
l
= p(all ,a12 , ••• a ln )
a
P
k
.
ij
A
= p(akl ,ak2 , ••• a kn )
€
GF(s)
= (aij )
and think of Pi as the main effect pencil of a general factor
2 -
has rank k
'i'
(i=1,2'.J k ) •
•
Consider another set Pk+ , ""Pn of n-k independent pencils in EG(n,s):
l
Pk +l
= P(bk+l,l'
b k+l ,2,···,b k+l ,n)
Pk+2 ~ P(b k +2 ,1' b k +2 ,2,···,b k +2 ,n)
P
n
b
ij
€
.
= P(b nl ,
b
n2
, •.. ,b
oo
)
GF(s)
and think of Pi as the main effect pencil of a general factor 'i(i=k+l, .. ,n).
The intersection of the k pencils P , P , ""P defines a partition of
2
k
l
n-k
k
EG ( n,s ) in s sets of s
elements each, and correspondingly a partition of
n
k
n-k
the s treatments in s sets of s
treatments. If each set of treatments
n k
is assigned to a block, the design obtained is of the class ( s,s).
P
l
, P , " " P are said to be the generating pencils of the design.
2
k
In such a design each of the general factor '1' '2' ••• ,i k remains at
constant level in each of the blocks.
Thus the s k -1 degrees of freedom
belonging to all possible generalized interactions of i , f , ••• ,i are
2
k
l
confounded.
k
In terms of the original factors the s -1 degrees of freedom confounded
belong to the pencils with coefficients
("'1' "'2""''''k)
all a 12
a
a
21 22
a I
10
a
2n
("'1' "'2' .•• , "'k)
~
(0, 0, •.. ,0)
(2.1)
a kl
~
akn
"'i
€
GF(s)
each of these pencils carrying (s-l) degrees of freedom.
There are (Sk_l)/(S_l) confounded pencils.
••• ,i
It is known that if the
th
coefficients of a pencil are the only non-null coefficients,
e
- 3 -
the (s-l) degrees of freedom carried by this pencil belong to the (e-l)-order
interaction between Fi ' F i ' .•. ,F i factors, (iI' i 2 , •.. ,ie , a subset of
1
2
e
I, 2, .•• ,n). This characterizes the degrees of freedom confounded by a
n
k
design of type ( s,s) completely.
3.
Designs of
~ ~ ~
similar designs
Consider two designs of class (sn,sk) identical if the (sk_l)/(S_l)
n k
pencils confounded in each are the same. Let D be a (s , s ) design generl
ated by the k independent pencils Pi, P
Pit the coefficients of which
2' ...
make up a
k x n matrix A:
A
::r
'e
The elements of the i
th
row of A are the coefficients of Pi.
The rank of
A equals k.
A is the generating matrix of the design P.
n k
( ,
Let D be a s
s ) design generated by k independent pencils
2
2
2
2
PI ' P2 ' ••• ,Pk the coefficients of which make up a
matrix B of rank k:
B •
- 4-
k x n generating
',-
Then we have the obvious:
Lemma 3.1
Two (sn,sk) designs D and D with generating matrix A and B
2
1
respectively, are identical if and only if we can write A
= CB
and
Ic I ~ o.
We will not make a distinction between identical designs as mathematical
objects.
Two designs of class (sn,sk) are defined as of the same type if we can
make them identical by permuting in the same way the coefficients of the
n k ) design
In other words if D is an ( s,s
l
with confounded pencils {P(~l) i :: 1, 2, ••• , m} and D another (sn,sk)
2
design with confounded pencils {p(~j) j :: 1, 2, ••• , m}, m = (sk_l)/(S_l),
confounded pencils of one design.
we say D and D to be of the same type if there exists a permutation
2
l
matrix E such that
n
{p(~j)
;
i
= 1,
... , m}
= {P(~jEn)
; j :: 1, ••. , m}.
We have immediately the following characterization of designs of the same
type in:
Lemma 3.2
n
k
Two ( s,s) designs D and D with generating matrix A and B
2
1
respectively, are of the same type if and only if there exists a
permutation matrix !n and a non-singular matrix kCk
such that
A = CBE
n
Before defining similar designs the following notions must be introduced.
Two n-vectors a and b are said to be of the same structure when their nonnull coefficients occur at the same position.
(2,0,0,4,1,0) are of the same structure.
- 5 -
e.g. (1,0,0,3,2,0) and
Two finite sets of n-vectors are said to be of the same structure when it is
possible to establish a 1:1 correspondence between both sets such that two
corresponding vectors are of the
~
structure.
Two (sn,sk) designs D and D with a set of confounded pencils
2
l
{P(!i) i = 1, 2, •• " m} and {p(~j) j = 1, 2, .,', m} respectively, are
said to be similar when the sets of vectors {!i
{~j
j
= 1,
i
= 1,
2, ••• , m} and
2, .,., m} have the same structure.
The problem of enumeration of (sn,sk) confounded designs is to
divide the (sn,sk) designs into classes of similar designs of the same type,
and produce the confounded pencils in each class.
4. Enumeration of (sn,sk) designs
n k
Let D be a s
( , s ) design With generating matrix A.
By lemma 3.2 we
I
( IC ~ 0 and En a permutation
n
matriX) is possibly a different (sn,sk) design of the same type as D. Thus
know that the design associated with CAE
there is no loss of generality in considering the generating matrix in a
canonical form, i.e., of the form:
1 0
o1
o0
...
0
0
1
Pll P12
P2l P22
Pln-k
P2n-k
Pkl Pk2
Pkn- k
Given {P
= [I:P]
} the design associated with such a matrix will have
ij
(sk_l)/(s_l) confounded pencils, the coeffic:1.ents of which are given by
(sk_l)/(S_l) different vectors of EG(n,s) :
- 6 -
1 0
o
1
(4.2)
o0
1
I
Pkl Pk2
Pkn-k
I
for different choices of (A , A , ••• , A ) ~ (0, 0, ••• , 0).
l
2
k
Consider (A , A , ••. , A ) as coefficients of a (k-2)-flat in PG(k-l,s)
k
2
l
and the non-null columns of [I:P] as points in PG(k-l,s).
The pencil with
coefficients given by (4.2) will belong to the interaction Fi ' F , ••• F
i2
ir
1
th
th
th
if and only if the i l ' i 2 ' ••• ,i r coefficients of the vector (4.2) are
the only ones which are non-null, or if and only if among the n points
] only the i th ' i th ' ••• ,i th belong to
represented by the columns of [ I:P,
l
r
2
the (k-2)-flat considered.
Given a generating matrix of the form (4.1) we associate a graph in
''--
PG(k-l,s), each of the n points on it being given by a column of (4.1) and
thus corresponding to a factor.
I have shown how each (k-2)-flat of
PG(k-l,s) corresponds to a pencil.
Its degrees of freedom belong to the
interaction between the factors whose corresponding points do not lie on
the (k-l)-flat.
This can be illustrated by a design of class (s6,s3) whose generating
matrix is
100
H
=
and
010
001
,
'--
- 7-
~ 0
This gives rise to the following generating graph:
"
o
{b;' )
\
'5
•
1+
~
The i th point represents the i th column of H.
Points 6 and 1 are identical.
Points 2, 3, 5 are collinear.
Let us consider all the possible (s3_ l )/(S_1)
lines in PG(2,s).
The line containing 2 and 4 corresponds to a set of (A , A , A ) = ~'
l
2 3
such that A'H has null coefficients 2 and 4; all the others are non-zero
because this line does not contain points other than 2,4.
coefficient
The pencil with
~'H
belongs to F F F F interaction.
1 3 5 6
The line containing 2 and 3 contains 5 also.
Such a line has
th
nd
rd
as coefficients and A'H is a vector with 2 , 3 , and 5
coefficient nUll; the pencil with such coefficient belongs to the 2nd order
A'
= (Al,A2'~
interaction F1F4F6... and so on for the other lines of PG(2,s).
From such a graph we immediately realize that no main effect factors
are confounded because no lines can meet in 5 points simultaneously. There
st
is no 1 order interaction confounded; no lines pass through any 4 points
of such graph.
It is obvious that any graph with the same structure will correspond to
a similar design.
The number of similar designs is equal to the number of w
ways of placing the points 3, 4, and 5 in PG(2,s) such that the graph
obtained is of the same structure as (4.3), multiplied by a factor of (8_1)3
(each (S-l) representation of a point gives rise to a different design).
- 8 -
The number of designs of the same type is seen to be the number of
permutations of the numbers 1 to 6 which change the structure of the graph.
Permuting 1 and 6 does not change the structure; the same holds for permutation of 2, 3, and 5.
But permuting 1 and 4 gives rise to a different
design of the same type.
Here the number of types is 6!/2!3!
The case of a null column is a degenerate one.
factor is always unconfounded.
= 60.
The corresponding
In the graphic representation we simply omit
such points.
5.
Construction of (sn,sk) design ~ the ~
Case k
= 2:
= 2,3,4
a (sn,s2) design corresponds to a choice of n points in
the geometry PG(l,s).
a-flat.
k
The (S+l) pencils confounded correspond to the (s+l)
A confounded pencil belongs to F
the points iI' i ,- •. ,i r do
2
~
i1
' F
i2
' ••• ,F
interaction if
ir
lie in the corresponding a-flat.
The 23
6 2
types of designs of class (s ,s ) were enumerated and will be produced in
another paper.
Case k
= 3:
a (sn,s3) design corresponds to a choice of n points in
PG(2,s).
The s2+S +l pencils confounded correspond to the (s2+8+l) lines in
PG(2,s).
A confounded pencil belongs to F
' F ' ••• ,F
interaction if
i1
i2
ir
do not lie in the corresponding line. The designs
the points iI' i , ••• i
r
2
of class (s6,s3) are enumerated.
Case k
PG(3,s).
= 4:
They divide in 38 types.
a (sn,s4) design corresponds to a choice of n points in
Here the (s4_ l )/(S_1) pencils confounded correspond to the
(s4_ l )/(S_1) planes of PG(3,s).
A confounded pencil belongs to
- 9 -
F
' F ' ••• ,F
interaction if the points i , i 2 , ••• , i r do not lie in
l
i
i
i
1
2
r
6 4
the corresponding plane. The 23 types of (s ,s ) designs were enumerated
and will be produced later.
In each class of designs, every type is described in the following
synopsis.
The generating matrix is given in a canonical form; the "x"'s
in the matrix represent the non-null element in GF(s) chosen such that all
possibly non-vanishing sub-determinants of the matrix must indeed be nonvanishing.
Special cases of vanishing sub-determinants are indicated.
generating graph is produced.
The
The various pencils confounded are grouped
under the interaction they belong to.
The number of similar designs and
designs of the same types are produced.
6.
Types of similar designs
We will need the following lemmas:
Lemma 1
If a and b are two n-vectors of the same structure then the sets
I
{~ ~'~
= o}
and {II ~'1
::I
o} are of the same structure.
The proof is by induction on the number of non-null elements Of!' and b'.
If a and b have only one non-null element in the p
{~
I!' ~
{II ~'1
::I
oj •
= oj
::I
(x
-
Ixp
[11 yp
= oj
::I
th
position then
is obviously of the same structure as
oj.
Let us suppose the lemma true when a and b are of the same structure
and possess (r-l) or fewer non-null coefficients.
Let us investigate the
case when exactly r coefficients of a and b are non-null.
{~ !'~
I
= oj
=
{II ~'1
= oj
::I
+•.• + a
ir
Xi
oj
::I
oj
r
+••• + b i Yi
r r
- 10 -
::I
In that case:
I
But by the hypothesis of induction the subset l~ a
xi +..• +a x
i1 1
ir ir
= 0, j = 1,2, ••• r} is of the same structure as
at least one x.
lX Ib i
-J. j
= 0,
at least one Y = 0, j = 1,2, •.. r}; in particular
ij
both sets contain the same number of elements. But the number of solutions
Yi + ••• +b
1
of ~'~
j
1
=0
r
and ~'X
= 1,2, ••• r}
structure.
i
Yi
= 0,
r
=0
is the same, hence lX
and (y b'y
=0
~ 0, j
and Y
i
a'x
=0
and x
= 1,2, ••. r}
i.1
~ 0,
are of the same
j
This terminates the proof.
In a similar way we may prove the
Theorem 1
If l~i} and l~j} are two sets of vectors of the same structure
and dimension then the sets
(~I ~1~
=0
all
iJ
and
111 ~jI = 0
all j}
are of the same structure and dimension.
n s k ) design D, with generating matrix H.
( ,
Assume a s
design D* with generating matrix H* such that rank H*
'--~
said to be the dual design of D.
The (n
s,s n-k)
= n-k
and HH*
= 0,
is
In other words D* is the design the
confounded pencils of which have their coefficients orthogonal to H.
Lemma 2
The correspondence of a (sn,sk) design to its (sn,sn-k) dual
establishes a 1:1 correspondence between designs of class (sn,sk) and
n n-k) •
of class ( s,s
Proof:
From the definition, we notice that for every design D of class
(.n,sk) there is only one corresponding dual design.
Conversely every
design D* of class (sn,sn-k) is seen to be the dual of only one (sn,sk)
design, namely the design having the coefficients of their
co~1ounded
orthogonal to the generating matrix D*.
Lemma 3
The dual designs of two similar (sn,sk) designs are similar
( s n ,s n-k) designs.
- 11 -
pencils
Proof: Let D and D be two similar (sn,sk) designs with generating matrix
2
l
H and H respectively. The row vector space of H and H have the same
2
2
l
l
structure and equal dimension.
The orthogonal spaces spanned respectively
by the rows of Hi and H will be of the same structure and the same dimen-
2
sion from theorem 1.
Hence the designs Dt and
D~
associated with Ht and
H~
are similar.
Lemma 4 The dual designs of two (sn,sk) designs of the same tyPe are
( s n ,s n-k) designs of the same type.
Proof:
If D and D are (sn,sk) designs of the same type, their generating
2
l
matrices H and H are such that
2
l
ICI~
Their duals are Dr and
,
2
H2H
= 0;
D~
2
0 and En is a permutation matrix.
with generating matrix Hf and H such that
let us take H En
= Hr·
Then H1Hr
,
"
2
,
..
CH H
= O.
2
2 2
.. CH2EnE H
n
Such a choice of Hr is valid; this shows Dr and D to be the same type of
2
designs.
Corollary The class of designs (sn,sk) and (sn,sn-k) can each be partitioned in an equal number of corresponding type of similar designs;
two corresponding classes have the same number of designs.
- 12 -
7.
The number of different types
~
n
2
similar ( s,s ) designs
In section 5 we saw how the different (sn,s2) designs correspond to the
different ways of choosing n points on PG(l,s) and how to work out the
confounded pencils in each case.
PG(l,s) may be represented as (s+l) points
(X 'X ) on a line, two points (xl' x2 ) and (Yl' Y2 ) being identical when
I 2
X = pX2 (p a non-null element in GF(s)).
2
As example let the design with generating matrix
Xl
= PYl '
lOXXXOX\... H
( Olxooxxf
x being any non-null element in GF(s) such that all the possible non-singular 2 x 2 matrices are non-singular.
Such a design can be represented by
the graph in PG(l,s):
•
(1,4,5)
where the i
th
•
.
•
(2,6)
(7)
point is the one corresponding to the i
th
column of H.
We
saw that the confounded pencils are the ones with coefficients given by
(~l' ~2)(· lOx x x 0 x )
OlxOOxx
Taking (~l' ~2) as coefficients of an O-flat in PG(l,s), the confounded
pencil for this
(~l' ~2)
will belong to interaction Fi ' F. , ••• ,F. if the
1
~2
~r
th
i
, i2th, ••• ,irth points do not lie in the O-flat in question.
l
From this equivalence of a design with a repartition of factors on a
line we have the following observations:
two designs will be similar and
of the same type when the number of points corresponding to k factors are
equal in both graphic representations of the design (k
= 1,2, •.• n-I).
The
number of different types of similar matrices will be the number of ways we
- 13 -
can arrange n objects in at least two different cells (when the n column is
non-null) plus the number of ways we can arrange n-l objects in at least
two different cells (when only one column is null) and so on.
Two arrangements are identical if the number of cells with k objects
is the same in both arrangements for k
= l, •.. ,n-l.
Let V be the number of
n
wa;ys we can arrange n objects in at least two different cells.
n
of different types of similar (8 ,s2) is thus
n
~
=2
V.
n
The number
The quantity (V + 1)
n
is the number of ways we can arrange n objects in one or more cells, two
arrangements being identical when the number of cells with k objects is
the same in both arrangements for k
= 1, ••• ,no
The last quantity is well
computed (see Riordan's book, page 122,(3) as example).
The number of type of (Sn,6 2 ) designs is:
3 for
7 for
13 for
23 for
n
n
n
n
=3
=4
=5
=6
for
n
=7
for
n
87 for
128 for
n
=8
=9
= 10
37
58
n
- 14 -
1
b
Type 1
o
100XXX]
o lOX X X
[ 001 X X X
main
effect
first
order
second
order
fourth
order
third
order
F F F F4
1 2 3
F F2F F
1 3 5
F F F F
1 2 3 6
F F2F4F
1
5
F F F4F
1 2 6
F F F F
1 2 5 6
F F F F
1 3 4 5
F1F F4F6
3
F1F F F6
3 5
F1F4F F6
5
F2F F4F
3 5
F2F F4F6
3
F2F F F6
3 5
F2F4F F6
5
F F4F F6
3 5
1
Fl 2 F4F F6 (s-4)
5
1
F1F3F4F5F6 (s-4)
1
F2F3F4F5F6 (s-4)
1
F1F2F3F4F5 (s-4)
1
F1F2F3F4F6 (s-4)
1
F1F2F3F5F6 (s-4)
1
1
1
1
1
1
1
1
1
Number of similar designs: (S_1)5(s_2)(S_3)2(s_6)
Number of analogous types:
1
15
fifth
order
2
(s -5s+10)
....,
Type 2
~,
~ooxx~
010 X X X
o 0 1 XXX
main
effect
first
order
second
order
F F F
1 4 5
...
fourth
order
third
order
1
" -j
Fl'4 F5F6
1
F2F3F4F5F6 (s-4)
F F4F F6
3 5
F F F F
1 2 4 6
F F F F
1 2 5 6
F F F F
1 3 4 6
F F F F
1 3 5 6
F F F F4
1 2 3
1
FIF3F4F5F6 (s-3)
1
FIF2F4F5F6 (s-3)
1
FIF2F3F5F6 (s-4)
1
FIF2F3F4F6 (s-4)
1
FIF2F3F4F5 (s-3)
1
F1F~l3F5
1
F~l3F5F6
1
F2F F4F 1
3 6
F F F F 1
2 3 4 5
F1F2F F6 1
3
Number of similar designs:
(s-1)5(s-2)(S-3)(s-4)
Number of analogous types:
20
16
fifth
order
2
(s -5s+9)
Type
3
.
[rOOXX~
o 0 1 X X .,
1
OlOXX~
main
effect
first
order
second
order
third
order
F F F
1 2 5
1
Fl 2F 4
1
F F2F4F
1
5
F F2F F6
I
3
F F F F6
I 3 5
F F F4F
I 3 6
F2F F}+F6
3
F F F F
2 3 5 6
F4F F F
3 5 6
F F4F 1
2 5
F F4F }
I
5
fourth
order
(s-3) FIF2F3F5F6 (s-3)
1
F}F2F F4F6 (s-3)
3
1
F2F3F4F5F6 (8-3)
1
FIF3F5F6F6 (s-3)
1
1
1
Number of similar designs:
(8-1)5(S-2)(S-3)
Number of analogous types:
15
'-.._-
17
.....
fifth
order
~.~
Type 4
X 0 ~J
X
U010
o 0 XX
1
0 0 X X
1.
X
main
effect
first
order
'
~
second
order
third
order
F2F6F4 1
F F F4 1
1 5
F1F2F F6
5
F1F F F6
3 5
F2F F F6
3 5
F F F F
1 2 3 6
F F F F
1 2 3 5
F F4F F6
3 5
F F F4F6
1 3
F F F F4
1 2 3
F2F F F
3 4 5
.
fourth
order
1
F1F2F3F5F6 (s-4)
1
F1F3F4F5F6 (8-3)
1
F2F3F4F5F6 (8-3)
1
F1F2F4F5F6 (s-2)
1
F1F2F3F4F6 (s-3)
1
Fl 2F F4F (s-3)
5
3
1
1
1
Number of similar designs:
(s-1)5(s-2)(s-3)
Number of analogous types:
90
18
'j
:)
fifth
order
2
(s -58+8)
"-
'i'
Type 5
liooxx~
010 X X 0
first
order
second
order
.,
third
order
F F2F
1 3
1
FlF2F}.j.
1
F2F F4F (s-3)
3 5
F1F F4F (S-3)
3 5
F F2F4F (S-3)
1
5
F1F2F F (S-3)
3 5
F F2F F4 (S-3)
1 3
F F F 1
1 2 5
F F F4 1
1 3
F F F 1
1 3 5
Fl 4F5 1
F F F 1
2 3 4
F2~"3F5 1
'-
,
¢ ..
OOlXXO
main
effect
t
•
F F F 1
2 4 5
F F4F 1
3 5
Number of similar designs:
4
(s-l) (s-2)(s-3)
Number of analogous types:
6
19
.)
fourth
order
2
F1F2F3F4F5(s -4s+6)
fifth
order
Type 6
[1 0 0 X 0
j
010 0 X 0
o 0 1 XXX
main
ef'f'ect
i.
f'irst
order
second
order
third
order
F2F 1
5
F F4F 1
1 S
F F4F F6
3 5
F2F F4F6
3
F F F F
1 3 5 6
F F F F6
1 2 3
F F F4F
1 3 5
F F2F F4
1 3
Number of similar designs:
(8_1)5(s_2)
Number of' analogous types:
60
20
..
b
f'ourth
order
1
F2F3F4F5F6(s-2)
1
FIF3F4F5F6(s-3)
1
FIF2F4F5F6(s-1)
1
FIF2F3F5F6(s-2)
1
FIF2F3F4F6(s-3)
1
FIF2F3F4F5(s-2)
f'if'tb
order
2
(s -Ss+{))
Type 7
lOOXXOJ
o 0 1 XXX
~
01.0 X 0 0
-
main
effect
first
order
-
third
order
second
order
F F4F F6 1
3 5
F2F F F6 1
3 5
F F F F 1
l 3 4 6
F F2F F6 1
1
3
F F2F4F (S-2)
1
5
F F F F 1
1 3 5 6
Number of similar designs:
(s-1)5(s-2)
Number of analogous types:
90
21
fourth
order
F2F3F4F5F6(s-2)
FlF3F4F5F6(s-3)
FIF2F3F5F6(s-3)
FIF2F3F4F6(s-2)
fifth
order
2
(3 -4s+5)
TlPe 8
[1010
°° X
X 0X0]
X
OOlOXX
main
effect
first
order
second
order
third
order
F F F6 1
3 5
F2F4F6 1
F2F F4F
3 5
F F F4F6
1 3
F F2F F6
1 5
F F2F F6
1 3
F F2F F
1 3 5
F F2F F4
1 3
F1F4F 1
5
Number of similar designs: (S-1)5(s-2)
Number of analogous types:
120
22
fourth
order
1
F2F3F4F5F6(s-2)
1
FIF3F4F5F6(S-2)
1
FIF2F4F5F6(s-2)
1
FIF2F3F5F6(S-3)
1
FIF2F3F4F6(S-3)
1
FIF2F3F4F5(S-3)
fifth
order
2
(s -5s+7)
',,-
Type 9
U
100xrs~rlar
t
010 X X X
0010XX
main
effect
first
order
V't
second
order
third
order
F F2F 1
1 3
F4F F 1
5 6
F F F4F
1 2 5
F F F4F
1 2 6
F F F F
1 2 5 6
F F F F
1 3 4 5
F F F F
1 3 4 6
F F F F
1 3 5 6
F F F F
2 3 4 5
F F F4F
2 3 6
F F F F6
2 3 5
"-
~
Number of similar designs:
(8_1)5(8_2)2
Number of analogous types:
10
23
/"
')
"'>
fourth
order
1
F2F3F4F5F6(s-3)
1
F1F3F4F5F6(s-3)
1
F1F2F4F5F6(s-3)
1
F1F2F3F5F6 (s- 3 )
1
F1F2F3F4F6(s-3)
1
F1F2F3F4F5(s-3)
1
1
1
fifth
order
2
(s -5s+8)
Type 10
lOOXO~O
010XXX
~
)t
o 0 1 XXX
main
effect
first
order
second
order
third
order
F2F F F6 1
3 5
\
5'
..
.L
fourth
order
F2F3F4F5F6(s-4)
F~l4F5F6
1
F1F3F4F5F6(S-2)
F2F F4F
3 5
F2F F4F6
3
F F4F F
3 5 6
F FFF
1 2 5 6
F F F F
1 2 3 6
F FFF
1 2 3 5
F FF F
1 3 5 6
1
F1F2F4F5F6(S-2)
1
F1F2F3F5F6(S-4)
1
F1F2F3F4F6(S-2)
1
F1F2F3F4F5(S-2)
1
1
1
Number of similar designs:
(S-1)5(S-2)(s-3)
Number of analogous types:
15
24
fifth
order
2
(s -5s+7)
Type 11
[1010
00X
0X]
XX0
..
001 X 0 X
main
effect
first
order
second
order
third
order
F F F 1
1 3 6
F F4F6 1
1
F F F4 1
1 3
F F F 1
3 4 6
F~l4F5
-S
fourth
order
F1F2F5F6 1
F2F3F4F5F6(s-2)
F1F2F3F5
1
FIF2F4F5F6(s-2)
F2F3F5F6
1
FIF2F3F5F6(s-3)
F1F F4F6 (8-3) FIF2F3F4F5(s-2)
3
1
Number of similar dws1gns: (s-1)5(s-2)
Number of analogous tYpes:
-----'
.-----'""-
60
25
fifth
order
2
(s -4s+5)
Type 12
100XXX]
~
010 000
o 0 1 XXX
main
effect
first
order
second
order
,/
third
order
FI F4F F6
5
FI F F F6
3 5
FI F F4F6
3
F F F4F
I 3 5
fourth
order
1
F2F3F4F5F6(s-1)
1
FIF3F4F5F6(s-4)
1
FIF2F4F5F6(s-1)
1
FIF2F3F5F6(s-1)
FIF2F3F4F6(s-1)
FIF2F3F4F5(s-1)
Number of similar designs:
4
(S-l) (s-2)(s-3)
Number of analogous types:
6
26
fifth
order
2
(8 -56+5)
:..
Type 13
'i
'/
~OOXX~
010XXO
£)
main
effect
first
order
F F4
3
0 1 X0 0
second
order
Fl 2F
5
F F F
1 2 3
F F F
1 3 5
F F4F
1 5
F F F
2 3 5
F F F
2 4 5
third
order
1
fourth
order
F2F F4F (s-2)
3 5
F F F4F (S-2)
1 3 5
F1F2F4F (S-3)
5
F F2F F (s-3)
1 3 5
F1F2F F4 (s-2)
3
1
1
1
1
(s_1)4(s_2)
Number of analogous types:
60
F1F2F3F4F5(s2_4s+5)
7-
Type 14
;/
~010XXO
0 0 X X~]
o 0 100 X
first
order
second
order
third
order
F F 1 F F4F 1
1 5
3 6
F F F4 1
1 2
F F2F4F (s-3)
1
5
F2F3F4F5F6(s-1)
F1F3F4F5F6(s-1)
FI F2F3F5F6(S-1)
F1F2F3F4F6(s-1)
4
Number of similar designs: (s-l) (s-2)
Number of analogous types:
.l:;, I.)
fourth
order
F F F 1
1 2 5
F F F 1
2 4 5
",-_
fifth
order
1
Number of similar designs:
main
effect
'>
15
21
fifth
order
Type 15
~
100XXO~
010000
o 0 1 XXX
main
effect
first
order
a
1
second
order
third
order
...
.....
I
i
(\,~)
fourth
order
F1F2F4F (S-1)
5
F F F F6 1
3 4 5
F F F4F6 1
1 3
F F FF 1
1 3 5 6
FIF3F4F5F6(s-3)
:fifth
order
(s 2-4s+3)
F2F3F4F5F6(s-1)
FIF2F3F5F6(S-1)
FIF2F3F4F6(s-1)
4
Number of similar designs: (s-l) (s-2)
Number of analogous types:
60
Type 16
Uooxx~
010 X 0 0
o 0 lOX
main
effect
,i.---,_--t~P
first
order
second
order
F2F4 1
F F2F 1
1 5
F F4F 1
1 5
F FF 1
5 3 6
third
order
fourth
order
F F F4F6 1
1 3
F?2F3F6 1
F1F2F4F
5
~-2)
F2F F4F F6(S-1)
3 5
FIF3F4F5F6(S-2)
FIF2F3F5F6(S-2)
FIF2F3F4F6(S-2)
Number of similar designs: (S_1)5
Number of analogous types:
180
28
:fifth
order
2
(s -4s+4)
Type 17
[1010
00X
1]0
X 0X
l'. ~)
o 0 1 X0 X
main
effect
first
order
third
order
second
order
F2F4 1 F F4F6
3
F2F F6
3
F F F
1 2 5
F F F
1 4 5
fourth
order
F1F F F6 1
3 5
F F F4F (S-2)
1 2 5
F2F F4F6 (s-2)
3
1
1
1
fifth
order
F1F3F4F5F6(s-2)
2
(s -35+3)
F1F2F3F5F6(s-2)
1
Number o~ similar designs: (8_1)5
Number of analogous types:
45
Type 18
OlOXOO
fr°OXO~
OOlXXX
main
~irst
e~fect
order
t
•
second
order
third
order
F F 1 F F F4 (s-2)
1 2
1 2
F F4 1
2
F F4 1
1
)5'
, J
~ourth
fif~h
order
order
FJ!3F5F6 1
F2F3F4F5F6(s-2)
F F F F 1
2 3 5 6
F F4F F 1
3 5 6
F1F3F4F5F6(s-2)
Number o~ similar designs: (S_1)5
Number of analogous types:
•
20
29
F1F2F3F5F6(S-2)
2
(s -3s+3)
Type 19
~
OOXO~
OlOOXO
o0
main
eff'ect
f'irst
order
1 XX
second
order
third
order
F F F F6 1
1 2 3
F1F2F4F (S-1)
F2F3F4F5F6(s-2)
F F F F6 1
1 3 5
F2F F4F6 1
3
F F4F F 1
3 5 6
F1F2F3F5F6(s-2)
5
Number of' similar designs: (S_1)5
Number of' analogous types:
f'ourth
order
45
30
F1F3F4F5F6(s-2)
FIF2F3F4F6(S-2)
f'Uth
order
2
(s -4s+4)
Type 20
[1o 00X
X0]
100 X X
OOlXOX
main
effect
first
order
second
order
third
order
F F2F 1
I
3
F F4F 1
3 6
F F4F 1
I
5
F F F4F
2 3 5
F F F F
I 3 5 6
F F2F4F6
I
fourth
order
1
F2F3F4F5F6(s-2)
1
FIF3F4F5F6(s-2)
1
FIF2F4F5F6(s-2)
fifth
order
2
(5 -5s+ 7 )
FIF2F3F5F6(s-2)
FIF2F3F4F6(s-2)
Fl2F F4F (S-2)
3 5
Number of similar de~igns:
(S_1)5
Number of analogous types:
30
f<
Type 21
[1OlOOXX
000XX]
o0 1 0 0 0
main
effect
F
3
first
order
second
order
third
order
F F F 1
I 5 6
F F2F6 1
I
F F F 1
I 2 5
F F F F6 (S-1)
I 3 5
F2F F F6 (S-1)
3 5
F F F F (S-3)
1 2 5 6
F2F F6
5
FI F2F F6 (S-1)
3
F F2F F (S-1)
I
3 5
1
Number of similar designs: (5_1)3(8_2)
Number of analogous types:
30
31
~
fourth
order
2
s -4s+3
fifth
order
;/ ~;
~
Type 22
..
~o 0000XJ
100 X X
010 X 0
main
effect
first
order
second
order
F F 1
3 5
F F6 1
I
F F F 1
I 2 5
F F2F 1
I
3
F F F
2 5 6 1
F F F
2 3 6 1
~]
/
I
/
"
third
order
fourth
order
fifth
order
2
F~l3F5F6(s-2)
(6 -45+4)
FI F F F6 (s-l)
3 5
FIF2F5F6(S-2)
F F2F F6 (S-2)
I
3
FI F2F F (S-2)
3 5
Number of similar designs: (s-l)
Number of analogous types:
4
90
Type 23
[:o1 0000
~
100 X X
l'>:'4'
•
o 0 1 X0 0
main
effect
first
order
second
order
F F 1
3 4
F F 1
I 6
FIF~l5
third
order
F F F4F (S-1)
I 3 6
FIF2F5F6( 8-2)
1
F F F6 1
2 5
Number of similar designs:
(S_1)4
Number of analogous types:
90
32
\l",5)
fourth
order
FIF2F3F4F5(s-1)
F2F3F4F5F6(s-1)
fifth
order
(s2_ 3s +2)
Type 2U
1 0 0 0 0 X]
[
main
effect
o
o
lOX X X
0 1 0 0 0
second
order
first
order
third
order
F F 1 F F F6 (S-1)
1 3
1 6
fourth
order
F1F2F4F
5
1
F2F3F4F5F6(s-1)
F2F1?5F6
1
FIF 2F }l5 F 6(5-2)
fifth
order
2
(6 -36+2)
FIF2F3F4F5(s-1)
Number of similar desiGns:
Number of analogous types:
Type 25
lOOOOX]
oI 0 0 0 X
o 0 I XX0
ma.in
effect
first
order
~
second
order
/
third
order
F F I F F4F
I
I 6
3 5
F2F6 I F F2F6 (s-2)
1
F F
1
1 2
Number of similar designs:
Number of analogous types:
fourth
order
F2F3F4F5F6(s-1)
FIF3F4F5F6(s-1)
FIF2F3F4F5(s-l)
(8-1) 4
20
33
fifth
order
(s 2 ~3s+2)
Type 26
~010
000X 0XX]X
o
main
effect
F
3
first
order
0 1 000
second
order
third
order
F F F
1
l 5 6
F2F4F6 1
1
fourth
order
Fl F F F6 (S-1)
3 5
F2F3F4F6 (S-1)
fifth
order
FlF2F4F5F6(s-2)
2
(s -3s+2)
FlF2F3F4F5(s-1)
F F F4F 1
l 2 5
Number of similar designs:
Number of analogous types:
....
Type 27
/~, ...
~OOOOJ
OlOOOX
first
order
second
order
)
~--L
o 0 lOX X
main
effect
• I>
third
order
F F
1 F F F
1
F F F F6 (S-2)
l 2
2 3 5
1 3 5
F F 1 F F F 1
F F F F6 (S-2)
2 3 5
l 6
l 3 5
F F 1 F F F
1
F F F F (S-2)
2 6
l 2 3 5
3 5 6
F F2F6 (s-2)
l
Number of similar designs:
(S_1)4
Number of analogous types:
60
fourth
order
2
(s -3s+3)
fifth
order
p'l:i .. )
./1'
Type 28
o oooo~
lOX X X
o0 10 0 Q
~
main
effect
first
order
• .?---.-----.-J
second
order
third
order
~
fourth
order
fifth
order
F1F2F4F5F6(s-1)
1
2
(s -2s+l)
F2F3F4F5F6(S-1)
Number of similar designs:
(S_1)3
Number of analogous types:
15
Type 29
[1o 000X
OJ
100 0 X
.0 0 lOX
'--
main
effect
F F
1 3
F F
3 5
: l.'".
_ _~'::.. - - - - ' i .
third
order
1 F F F (S-2)
1 3 5
1
F2F F F6 (S-1)
3 5
F F F F (S-1)
I 2 5 6
1
F F F F (s-1)
1 2 3 6
Number of similar designs: (S_1)3
Number of analogous types:
>
',>
second
order
first
order
F F
1 5
q
60
35
fourth
order
2
(s -3s+3)
fifth
order
Type 30
~
lOOOOX~
OlOOXO
OOlXOO.
main
effect
first
order
third
order
second
order
fourth
order
F F6 1
1
F F F F (S-1)
1 2 5 6
F F
2 5
F F F4F (S-1)
6
1 3
F2F F4F (S-1)
:1
F F4 1
3
3
fifth
order
2
(s -2s+1)
5
Number of similar designs: (s_1)3
Number of analogous types:
15
Type 31
~lOlOOXX
0 0 0 0 :]
o0
main
effect
F
3
1
first
order
F F
1 6
1./
tV ..
1 0 0 0
second
order
/".
-_.....
third
order
1 F F F6 (S-1)
1 3
F F F
1
1 2 5
F F F
2 5 6
F F F F (S-2)
1 2 5 6
F2F F F6 (S-1)
3 5
F F F F (S-1)
1 2 3 5
Number of similar designs:
(S_1)3
Number of analogous types:
180
36
.
....--.
:-.,.~
fourth
order
2
(8 -33+2)
Fl 2 F3F F6
5
fifth
order
',-
Type 32
[:1 0 0 0 X
L/~
o~
010 X 0 X
o 0 1 000
main
effect
F
3
1
second
order
first
order
F F 1
I 5
( ....,Ii,I, )
third
order
FI F F (S-1)
3 5
F F F
2 4 6
~
fourth
order
F2F F4F6 (S-1)
3
FIF2F4F5F6(s-1)
fifth
order
2
(s -2s+1)
1
Number of similar designs: (S_1)3
Number of analogous types:
60
...
~.
Type 33
I
~
OOOOX]
01000 X
o 100 X
main
effect
first
order
second
order
F F F (S-2)
1 2 3
F F 1
1 3
F F2F6 (s-2)
1
F F 1
1 2
F F F (s-2)
1 3 6
F F F (S-2)
2 3 6
F F 1
2 3
l
F F 1
2 6
F F6 1
3
Number of similar designs: (S_1)3
Number of analogous types:
l
,L--third
order
F F 1
1 6
•
15
37
fourth
order
fifth
order
Type 34
r
ooo
OIOOXOXJ
oI 000
main
ef'fect
F
3
I
f'irst
order
F I F6 I
F2F I
5
\.
(i, I.)
third
order
second
order
F I F F6 (S-I)
3
F2F F6 (S-1)
3
fourth
order
fifth
order
fourth
order
f'if'th
order
F I F2F F6 (S-I)
5
Number of' similar designs:
Number of analogous types:
.~
Type 35
6
~o 0I
o
main
ef'fect
F
3
first
order
x~
00 00 00 X
0 I 0 0 0
~
second
order
third
order
F F I F I F2F6 (s-2)
6 I
F F
F F F (S-1)
I 2 3
l 2 I
F2F6 1 F F F6 (S-1)
l 3
Fif3F6(s-1)
2
FIF2F3F6(s -3s+2)
Number of similar designs: (S_1)2
Number of analogous types:
~
60
',-
"., ~
Type 36
~.
[1010000
000X]
"'-
o0 1 0 00
main
effect
F 1
3
F ].
2
tf_
(l.';,~)
third
order
second
order
first
order
F F (S-1) F F F 1
2 3
I 5 6
F F F F6 (S-1)
l 2 5
'- Jfourth
order
fifth
order
2
FlF2F3F5F6(s -2s+1)
F F F F (S-1)
l 3 5 6
Number of similar designs: (S_1)2
Number of analogous types:
60
Type 37
~
OOOO~
010000
o1 0 0 0
main
effect
first
order
F F
1 6
1
third
order
second
order
1 F F F (S-1)
1 2 6
F2F3(s-1)FIF3F6(S-1)
Number of similar designs:
(s-l)
Number of analogous types:
90
,
'-
39
fourth
order
fifth
order
Type 38
[
1 0 0 0 0 0
010 COO-]
.0 0 1 0 0 0
f'irst
order
main
eff'ect
z.
third
order
second
order
1
F F2 (S-1) F F F (S 2 -2s+l)
1 2 3
1
F
2
1
F
3
1
F F (s-1)
1 3
F F (S-1)
2
F
l
fourth
order
f'if'th
order
3
Number of similar designs:
1
Number of' analogous types:
20
Ref'erence :
(1)
Bose, R. C. (1947): Mathematical theory of the symmetrical factorial
design. SANKHYA, 8, 107-166••
(2)
Bose, R. C. and Kishen, K. (1940): On the problem of' confounding in
the general Symmetrical Factorial design. SANKHYA, 5, 21-36.
(3)
Riordan,J. (1958): An introduction to Combinatorial Analysis, John
Wiley and Sons, Inc.
I thank Prof'essor I. M. Chakravarti for suggesting this problem and
f'or his belp and guidance.
40