•
UNIV£ltSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
Mathematical Sciences Directorate
Air Force Office of Scientific Research
Washington 25, D. C.
AFOSR Report
TN - 591241
ON SOME METHODS OF CONSTRUCTION OF PARTIALLY BALANCED ARltAYS
by
I. M. Chakravarti
May, 1960
Contract No. 49-(638)-213
Methods of construction of partially balanced arrays are
considered in this report. Two methods of construction
are given. One of them derives partially balanced arrays
from (A - ~ - v) configurations and the other is an extension of Bose-Shrikhande method of construction of orthogonal arrays.
Qualified requestors may obtain copies of this report from the ASTIA
Document Service Center, Arlington Hall Station, Arlington 12, Virginia.
Department of Defense contractors must be established for ASTIA services,
or have their "need-to-know" certified by the cognizant military agency
of their project or contract.
Institute of Statistics
Mimeograph Series No. 260
•
ON" SOME jYiETHODS OF CONSTRUCTION OF PAB.'l'IAj~LY BALANCED ARRAYS*
by
I. M. Chakravarti
University of North Carolina
O.
Introduction
Suppose A =
(~ij»
and the elements a..
J.J
is a matrix i
= 1,
2, ••• , m,
j
= 1,
2, ••• N
of the matrix are symbols 0, 1, 2, ""
Consider the st matrices
xl = (xl'
giving different values to x.,
6,
J..
s-l.
x 2 ' ••• Xt ) that can be formed by
X.J. = 0, 1, 2, ••• , s-l; i = 1, 2, ••• t.
Suppose associated with each matrix X there is a positive integer
~
(xl' x 2 ' ... , x t ) which is invariant under permutations of
(xl' xs ' "', xt ). If for every t-rowed submatrix of A, the st matrices
X occur as columns A(x ,X2,.e., xt ) times, then the matrix A is called
l
a partially balanced array of strength t in N assemblies~ m constraints
(or factors), s symbols (or levels) and the specified A (Xl' x 2' ••• , x t )
parameters. When A(xl ,x 2 , ••• x t ) = A for all (Xl' x 2 , ••• , xt ), the
array is called an Orthogonal array.
Orthogonal arrays were definec1in <[,4_7, ['5_7) and construction of
orthogonal arrays were considered in
(1:1_7,
f:2J,
1:3_7, L4J,
["5_7). Partially balanced arrays were defined in ['6J where
their use as multifactorial designs is also discussed.
•
r
.",
*This research was supported in part by the United states Air Force
through the Air Force Office of Scientific Research of the Air Research
and Development Command, under Contract No. AF49(638)-2l3. Reproduction
in whole or in part is permitted for any purpose of the United states Government.
•
2
In this paper, some methods of construction of partially balanced arrays are considered.
One of the methods is applicable when s
::t
2
and derives partially balanced arrays from the well-known (A - lJ. - v)
configurations.
['2_7
1.
method of construction of orthogonal arrays.
An example:
s
The other method is an extension of Bose-Shrikhande
= 3,
An example of a partially balanced array of strength 2,
m = 6 constraints in N = 15 assemblies is given below
Partially balanced array (15, 6, 3, 2)
assemblies
13 14 15
1 2 3.4
5
6
7 8 9 10
1
0 0 0
G
0
1
1
1
1
1
2
2
2
2
2
2
0
2 1 1
2
0
0
1
2
2
0
0
2
1
1
3
1
0
2 2 1
0
2
0
1
2
1
2
0
0
1
4
1
~
0
2
2
2
0
2 0
1
2
1
0
1
0
5
2
2
1
0
1
1
2 2 0
0
0
1
1
0
2
6
2 1
2 1
0
2 1
0
1
0
1
2
0
constraints
0
2
11 12
This array has the A(x , x 2 ) parameters
1
A(Xl'x2)
=2
=1
if Xl and x 2 are unlike
otherwise
This array was constructed by cutting out 3 assemblies and omitting
one row from the orthogonal array A (18, 7, 3, 2)
(
•
2. Construction of partially balanced arrays for s
figurations.
=2
from A -
lJ. -
v con-
•
3
Definition:
["7_7
A A - IJ. -
configuration, of m elements is defined
\I
as the configuration of m elements taken v at a time so that
each set of IJ. elements shall occur together in just A of the sets.
Suppose there are N sets of
e:}.ements each'in the configuration.
v
o
Let Nt denote the number of sets each containing a fixed subset of
elements.
Then it is easily seen that
r-
Nt .. A f m-t) /
'IJ.-t
(2.1)
Consider the matrix A
A - IJ. - v
manner:
t
t
)
t .. 0, 1, 2,
IJ.
$"
\IJ.-t
= « aJ.J
.. »
of form (mx No') derived from a
configuration of m elements in N·' sets in the following
o
Let al' a2' .'"
am denote the
denote the No sets of the configuration.
a ij
=
1
..
0
Consider a IJ.-rowed submatrix of
m elements and sl' s2' ... s~o
Then let
if a
occurs in the set
i
otherwise.
Sj
A with elements a .. as defined above.
J.J
Amongst the No columns of the submatrix, a column matrix X 1 where
IJ.,
its transpose Xl'
.. (Xl' x2 ' ••• , X ), x. = 0 or 1 i = 1, 2, ••• , IJ.
,IJ.
IJ.
J.
occurs A. (xl' x 2 , ... , xlJ.) times. Specifically, let Xi = 1 for
i
= 1,
2, '0', r;
Xi
=0
for i
easy to show that for such an X
(2.2)
= r+l, ••• , IJ.
in X.
Then it is
4
where n stands for the symbol of finite difference, viz.,
Value of A(xl , x 2' ••• , x~) depends only on the count r of unities
in its argument and hence it is invariant under permutation of its
arguments.
Now provided A(xl'~' ••• , x~)
Theorem 2:1,
>
0 for all s~ sets of X, we have
The existence of a (A -
~
- v) of m elements With
A (xl' x 2 , ••• , x~) all positive, implies the existence of a partially
balanced array of strength ~ with parameters s = 2 and A(xl,x2' ••• '~)
as defined in (2.2) •
•
Well-known examples of A -
~
- v configurations are the triple systems,
quadruple systems, etc _, which are defined in
3.
An
£:7J.
extension of Bose-Shrikhande method of construction of orthogonal
arrays and its use in the construction of partially balanced arrays.
Definition:
A pairwise partially balanced design with parameters
(v, kl , k2, ••• , km; bl , b 2, ••• , bm; Al , A2 , ••• , At; nl , n2' ••• , ~)
is defined as an arrangement of v varieties in blocks of m different
sizes kl , k , •• _, km, there being b i blocks of size ki ,
2
m
Zbi = b, satisfying the following conditions:
i=l
•
(i)
No block contains a single variety more than once.
(ii) With respect to any variety, the remro.n.ing (v-l)
varieties fall
•
5
into t
varieties in the i th category,
categories, there being n
i
called the i th associates of the variety;
Two varieties which are i th associates, occur together in A;
(iii)
blocks,
= 1,
i
2, ••• , t.
Then the following relations
~ong
t
l:b.k.(k.-l)
=
l:nivA i
J. J.
J.
i=l
i=l
the parameters hold,
t
m
•
=-
'iJ
l: n.A.
J. J.
i=l
Suppose there exist the orthogonal arrays
i
= 1, 2, "" m
of strength two and index A and in ki symbols. Consider the pairwise partially balanced design defined earlier. There are b. blocks
J.
each of size ki • These bi blocks provide bi sets of ki symbols each.
Using each set of k.J. symbols once in the orthogonal array A.,
one
J.
gets b i such orthogonal arrays. If all such orthogonal arrays are
arranged side by side, then one gets a matrix A with number of columns
N
•
I
=A
m
2
l:bik i
i=l
and number of rows
q
=
min (ql' q2' ""
qm)'
In any
t
two-rowed submatrix of matrix
A,
every ordered pair \ t u) of two
v
.
distinct symbols of varieties which are ith associates will occur AA
i
•
6
times snd every ordered pair
(:~ \
of two like symbols occur Ar j times,
if the variety t
j
occurs in r j blocks of the pairwise partially ba-
lanced design.
Hence the
Theorem 3.1:
The existence of a pairwise partially balanced design
with parameters ( V; kl' k2, ...km; b , b 2, ••• bm, AI' A2' ••• At ,
l
2 q. , k , 2)
, nt ) and of the orthogonal arrays Ai(Aki'
~, n ,
2
J.
i
...
i
= 1,
2,
... , m,
of strength two in
•
imply the existence of the partially balanced array
V
symbols and q = min (ql' Q2' ••• , qm)
and A (xl' x 2 ) = AAi
constraints
where Xl' x 2 stand for two varieties which are
ith associates and A(X,X) = Ar j
where the variety X occurs r j
times
in the pairwise partially balanced design.
As an illustration, a partially balanced array which has been constructed
using the method described above, is given below.
balanced array in v
= 6
symbols,
111is is a partially
N = 48 assemblies m = 5 con-
straints and
=1
i f (x ,x )
I
2
are second associates
= 2 if Xl and x 2 are like
•
I
where Xi'
i = 1, 2, ••• ,
6
are the variety symbols.
''\
f
•
7
In constructing this array, the partially balanced design
(v
= 6,
r
= 2, b = 3, k = 4, nl = 1, n 2 = 4, Al = 2, A2 = 1) in
three blocks
Xl'
x4'
x 2'
x'!2
x2'
xs' Xy
x6
~y
x6'
xl' ~
and the orthogonal array A (16, 5, 4, 2) pave been used.
e,
Orthogonal Array A (16, 5, 4, 2)
R
C
1:1.
0 0 0 0
2
0 1 t t
0 1 t t2
L2 0 1 t
1
0 1 t
3
t
2
t2
1 1 1 1
0 1 t
1
t
t2
0 t2 t
2
t 0 1
t2 t
1
0
t
t
0
t
2
t
t
2
1 t t
t2 0 1
1 0
1 0 t2 t
t
t
t2 t2 t2 t 2
t2
0 1 t
t2 t
1
0
1
t
0 t
t2 0
2
t
1
Making successively the identifications
1
=
Xl
1= x 2
t
::
x2
t=x
t 2 = x4
•
. j
(3)
(2)
(1)
0
=- x
s
t
2
3
= xs
o = x6
1
= Xl
t =x
3
2
t =x
4
o = x6
2
and using them on the above array in place of (0, 1, t, t ) one gets
•
8
is the desired partially balanced array in 6 symbols and 48 assemblies.
Let A* denote the array derived from A by truncating the first row
and the first four columns (as indicated by the horizontal and vertical
* A*2 and A* are obtamed
. from A*
AI'
lines) • Then the arrays
3
using the three identifications of variety -symbols given above.
Let
E denote the array
•
Xl x2 • ••
• •• x 6
xl x 2 • ••
• •• x6
xl x2 •• •
••• x 6
xl x2 '
• •• x6
E
Then the array A~
...array in
~ =-
=-
LE
6 symbols,
A(X,X) = 1, A(x.,x.) =2
1.
A(Xi,x j )
4.
=1
J
xl x2 • ••
• •• x 6
xl x2 • ••
• •• x6
A* 7
3 -
A~
N = 42
..
assemblies,
and x.
J
is a partially balanced
m = 4 constraints and
are first associates and
if they are second associates.
AcknOWledgement:
Thanks are due to Professor R. C. Bose for
kindly going through the manuscript and for his helpful comments.
•
9
-/-1-7
Bose, R. C. and Bush, K. A. (1952):
two and three.
["2_7
Orthogonal arrays of strength
Ann. Math. Stat. 23, 508.
On the com1=C~ition·o.f
Bose, R. C. and Shrikhande, S. S. (1960):
balanced incomplete block designs.
Canad. J. Math. Vol 12 (2),
177.
~3_7
Bush, K. A. (1952)
Orthogonal arrays of index unity.
Ann. Hath.
stat., 23, 426.
~4-7
Rao, C. R. (1946):
On hypercubes of strength d and
of confeundiIlg in factorial experiments.
/:5_7
Rao, C. R.
L6J
Bull. Cal. Math. Soc. 38,
(1947): Factorial experiments
torial arrangements of arrays.
Chakravarti, 1. M. (1956):
J. Roy.
d~ri vable
Carmichael, R. D. (1937)
-(; ...
._h
~
J.
•
9, 128.
Fractional replication in asymmetrical
Sankhya 17,
143.
Introduction to the Theory of Groups of
Finite Order, Dover Bublications, Inc.
,I
from combina-
Stat. Soc. (Suppl)
factorial designs and partially balanced arrays.
L7~
a system