,
BU-513-M
MULTIDIMENSIONAL
BALANCED
DESIGNS*
by
D.
A.
Anderson*'~
and
w.
T.
Federer
Cornell
University
Abstract
June,
1974
If
v =
4A
+ 3
is
a prime
or
prime power,
then
the
k =
2A
+ 1
quadratic
residues
in
GF(v) form a
(v,k,A)
difference
set.
A
general
construction
is
given
for
m-way
comPletely
variance
balanced
designs
where each
factor
has v
levels,
m
is
any
integer
less
than
or
equal
to
k,
and N = vk.
The
construction
gives
rise
to
a
variety
of
designs,
easily
enumerated,
with
the
same
parameters
pairwise
but
with
differing
variance
properties.
For
m = 3
there
are
only
two
distinct
designs
possible,
and
their
relative
efficiency
is
shown
to
be
2X
2
/(2~
2
-
1).
\
*Paper
No.
BU-513-M
in
the
Biometrics
Unit
Mimeo
Series,
Cornell
University.
Research
partially
supported
by
Public
Health
Research Grant
5-ROl-GM-05900
from
the
National
Institutes
of
Health.
**
On
leave
from
the
University
of
Wyoming.
AMS
1970
subject
classifications
60C05,
62K05,
62Kl0,
62K15
MULTIDIMENSIONAL
BALA~ICED
DESIGNs*
BU-513-M
by
June,
1974
D.
A.
Anderson~:-
and
'vl.
T.
Federer
Cornell
University
1.
Introduction
~
Background. Consider an experiment
involving
m
factors
at
s
1
, s
2
,
···,
sm
levels,
and suppose
that
all
interactions
between
factors
can
be
neglected.
A
design
T
for
the
experiment
of
size
N
is
the
specification
of
N
combinations.
Denote by
B.
.
the
si
X
sj
incidence
matrix
of
factors
i and j
for
:LJ
the
design
T. That
is,
the
element
in
Bij
corresponding
to
level
x
of
factor
i
and
level
y
of
factor
j
is
the
number
of
combinations which have
factors
i and j
at
levels
x and
y,
respectively.
The
elements
of
Bij
are
integers
greater
than
or
equal
to
zero.
In
an obvious
wa.y
we
let
B
11
=
Diag[r
1
,
information
matrix
for
the
design
T
is
the
block
matrix
r
2
,
···,
r
].
Then,
the
St
(<Bij))
i,j
=
1,
2,
···,
m,
where
we
have
already
adjusted
for
the
mean.
If
the
design
T
is
connected,
M = ((B
..
>)
+
DiagL-J
, J , • •
•,
J J
is
nonsingular
and M-l
is
a
conditional
l.J
sl
sl.
SaBz
sill
sm
inverse
of
(<Bij)),
(<Bij))-
= M-l =
(<vij)).
Definition
1.1.
The
design
T
is
said
to
be
(i)
variance
balanced
with
respect
to
factor
i
if
Vii
= a
..
I +
b.J
, and
J.
st
J.
s1s1
(ii)
completely
variance
balanced
if
it
is
variance
balanced
with
respect
to
every
i =
1,
2,
···,
m.
*
Paper
No.
BU-513-M
in
the
Biometrics
Unit
Mimeo
Series,
Cornell
University.
Research
partially
supported
by
Public
Health
Research Grant
5-ROl-GM-05900
from
the
National
Instj_tutes
of
Health.
**
On
leave
from
the
University
of
W,yoming.
- 2 -
We
illustrate
the
above
"\·lith
the
folloiling
construction.
Suppose
for
a given
n
we
haxe a
set
of
t
orthogonal
Latin
squares
of
order
n,
O(n,t),
which
we
will
assume
to
be
arranged
so
that
each
has
first
row
(0,
1,
2,
···,
n-
1).
Cut
off
the
first
row
from
each
of
the
t
squares,
and
consider
the
t + 1
factors
corre-
spending
to
the
column
effects
and
the
t
treatments
of
the
t
Latin
squares.
It
is
apparent
that
for
any
pair
of
factors
each
level
x
of
one
occurs
exactly
one time
with
each
level
of
the
second
except
x.
The
combination
(x,x)
occurs
zero
times.
Thus
the
incidence
matrix
for
itn
and
jth
factors,
Bij'
is
=
Jnn
I
n
if
J =
1,
2,
···,
t +
1.
(1.1)
Since
the
matrices
I and J - I
are
closed
with
respect
to
multiplication,
com-
n
nn
n
plete
variance
balance
is
obvious
if
the
design
is
connected.
It
is
important
to
observe
that
if
we
have a
design
with
m
factors
each
at
n
levels
in
N
observations,
an obvious
necessary
condition
for
connectedness
is
N
~
1 +
m(n
-
1).
(1.2)
Thus
if
n
is
a prime
or
prime power and t = n -
1,
we
may
use
any t
(not
t +
1)
of
the
factors
to
obtain
a t-way
variance
balanced
design
in
N =
n(n
-
1)
runs.
If
n
is
a prime power
or
not
and t < n -
1,
we
have
(t
+ 1)-way complete
variance
balanced
designs.
The
purpose
of
this
paper
is
to
produce
families
of
m-way
completely
variance
balanced
designs each
with
the
same
parameters
pairwise,
but
which give
rise
to
designs
with
different
efficiencies.
The
constructions
are
based
on
difference
sets
of
the
form
(v,k,~)
=
(4)..
+
3,
2~
+ 1,
).)
where 4). + 3
is
a prime
or
prime
power,
The
complete
variance
balance
follow·s from
the
fact
(Theorem
2.1)
that
the
matrices
I,
B,
B' form a
basis
for
a
linear
associative
commutative
algebra
where B
is
the
incidence
matrix
of
the
corresponding
symmetric
balanced
incomplete
block
design.
- 3 -
Hedayat and Raghavarao (1973) have
obtained
sufficient
conditions
for
the
existence
of
three-way pair.·rlse
balanced
designs,
but
not
necessarily
variance
balanced,
based
on
difference
sets.
Their
construction
when
applied
to
the
(v,k,~)
difference
set
above
is
one
of
the
possible
constructions
in
this
paper
for
m = 3.
Afsarinejad
and Hedayat (1972) have given
constructions
for
multistage
Youden
designs
from
difference
sets
in
a manner
similar
to
the
constructions
here;
however,
the
complete
variance
balance
and
the
differences
in
variance
properties
were
not
observed. Preece (1966) has
given
several
Youden
designs
for
two
sets
of
treatments
obtained
by
cutting
rows from Graeco-Latin
squares.
For
the
case
v
=
4~
+
3,
a prime
or
prime power,
his
designs
are
essentially
those
of
the
pres-
ent
paper
for
m =
3,
and he
has
noted
the
different
variance
properties
of
the
different
possibl~
designs.
The
present
paper
might be
regarded
as
an
extension
of
these
constructions
to
m-way
completely
variance
balance
which
easily
provide
all
possible
combinatorial
configurations
of
the
given
type.
Agrawal (1966}
has
given
constructions
for
three-way
designs
which
are
related
to
those
of
Hedayat and Raghavarao (1973), and
of
this
paper
for
m =
3·
Potthoff
(1962a.,b, 1963) has given a number
of
specific
designs
of
three
and
four
dimensions
which can
be
obtained
by
one
of
the
constructions.
Causey (1968)
with
a
related
construction
produces
some
designs
for
t~e
case
discussed
in
this
paper
of
up
to
five
dimensions,
but
does
not
obtain
a.
construction
for
the
maximwn
number
of
factors.
2.
On
(4~
+ 3,
2A
+
1,
\)
Difference
Sets.
If
v =
4~
+ 3
is
a prime
or
prime
power
and k =
(v
-
1)/2
=
2A
+
1,
then
it
is
well-known
that
the
quadratic
residues
in
GF(v) form a {v,k,A)
differeuce
set.
Let
the
quadratic
residues
be
arranged
in
some
arbitrary
but
fixed
order
(for
convenience
we
take
the
first
element
to
be
the
multiplicative
identity
denoted by
1)
asS=
(1,
d
2
, d
3
,
···,
~).
We
observe
- 4 -
that
the
v-
l
vectors
XS
= (x, xd
2
, xd
3
,
···,
x~),
x € GF(v),
are
(v-
1)/2
permutations
of
the
vector
Q
as
x
ranges
over
the
k
quadratic
residues
of
GF(v),
and
(v
-
1)/2
permutations
of
nonquadratie
residues
as
x ranges over
the
non-
quadratic
residues.
In
particular,
since
if
xis
a
quadratic
residue,
then
(-x)
is
nonquadratic
and
-g,
is
a
permutation
of
the
nonquadratic
residues.
Further,
xg,
-
yg,
=
(x
- y)g,
is
a
permutation
of
quadratic
or
nonquadratic
residues
as
(x -
y)
is
or
is
not
a.
quadratic
residue.
Thus
with
the
ordering
of
~'
we
have
specified
(v
-
1)
ordered
vectors
xg,
x € GF(v), x f
o;
such
that
the
vectors
together
with
(0,
o,
···,
0)
are
closed
with
respect
to
vector
addition
over
GF(v).
Corresponding
to
this
(v,k,A)
difference
set,
we
can
construct
a symmetric
balanced
incomplete
block
design
v =
b,
r =
k,
X,
whose
oth
block
is
~
and
whose
xt
~'~
block
ia
Q + (x x • • •
- ' ' '
x)
=
(1
+
x,
d
2
+
x,
d
3
+
x,
···,
~
+
x),
x
ranging
over
the
nonzero elements
of
GF'(
v).
Let B denote
the
v X v
incidence
matrix
of
this
design
so
that
B'B =
BB'
=
(2X
+
l)I
+ X(J - I
),
v vv v
(2.1)
and
BJ
=
B'J
=
(2A
+
l)J
•
(2.2)
We
prove
the
following
theorem
of
central
importance
in
our
constructions
of
multidimensional
balanced
designs.
Theorem
2.1.
!f. B
~
~
incidence
matr.;.x
of
the
cyclic
balanced
incomplete
~
design~
initial
block
Sl'
then~~
matrices
I,
B, and B'
form~
basis~
a
linear
associative
and
co~~tative
algebra.
Proof.
Since S and
-S
are
permutations
of
the
quadratic
and
nonquadratic
residues,
respectively,
and 0
is
neither
inS
nor
in
-~,
it
follows
that
Iv
+ B + B' = J and
the
three
matrices
are
linearly
independent.
From
(2.1)
B'B =
BB'
=
(2X
+
l)I
+
XB
+
AB'.
v
(2.3)
