RELATIOn BETt'JEEII CERTAIll IUCOIlPLETE BLOCK DESIGnS
by
S. S. Shrilthande
University of north Carolina
This research was supported by the United States
Air Force through the Air Force Office of Scientific
Researchpf the Air Research and Development Command,
under Contract Ho. AF 49(638)-213. Reproduction in
.whole or in part is permitted for any purpose of the
Un1ted States Government. .
'
•.
u
-of Stati$t1cs
Uimeograph Series 170.. 207
-u~1-&ute
July, 1958
RELATION BETVIEEN CERTAIN INCOMPLETE BLOCK DESIGNS
l
by
s. S. Shrikhande
University of North Carolina
1.
Summary. Relations between balanced incomplete block designs de-
rived from the symmetrical balanced incomplete block designs with
~
= 1 and 2 and certain partially balanced incomplete block designs
are investigated in this paper.
2.
Introduction.
Let v, b, r, k,
incomplete block design
(2.1)
be the parameters of a balanced
L-l_7 (b.i.b.d.).
vr
(2.2)
~
Then we have
= bk
~(v-l) =
r(k-l)
For a symmetrical b.i.b.d. it is known that any two blocks have
exactly
~
treatments in common.
v
Hence from the design with parameters
=b, r = k,
~
we get by the method of block section
l1._7 another
b.i.b.d. with
parameters.
(2.4)
v'
= v-k,
b'
= v-l,
k'
= k-~,
r'
= k,
~I
= ~.
Thus the existence of a b.i.b.d. with parameters (2.3) implies the exist- ..
tence of another b.i.b.d. with parameters (2.4).
true in general L-2~. However, when ~
= 1,
The converse 1s not
(2.4) corresponds to a
finite Euclidean Plane and it is well known that such a plane can be
L
This research was supported by the United states Air Force through
the Air Force Office of Scientific Research of the Air Research and
Development Command, under Contract No. AF 49(638)-213. Reproduction
1n whole or 1n part is permitted for any purpose of the United States
Government.
embedded into a Finite Projective Plane which is the symmetrical design with
parameters (2.,) for A=l. A s'imilar result has been proved by Hall and Connor
L"3 J
for the case A.
==
'2. We thus have the f'ollowing result.
Let C, C ' D, D be b.i.b. designs with parameters indicated below.
l
l
(2.5)
v
C:
= b = s2., +
s + 1, r
=k = s
+ 1, A.
=1
2
2
v = s , b 'II:: S + s, r == s + 1, k = s, i\ = 1
_ n2 + n + 2
V -- b 2
' r = k == n + 1, A. = 2
v = n(2- l ), b == n(;+l), r = n + 1, k = n - 1, i\
(2.6)
(2.7) D:
(2.8)
=2
Then either C and C (D and D ) are both existent or both nonexi,stent and any
l
l
solution of one yields a solution of the other.
J.:4,5J it follows
From
that the dual of C, is the group divieible de-
signs Cl *.
v
= s2 +
s, b
2
s , r
==
p
=s
== 6,
k
=s
~
+ 1,
= 1,
~
=0
+ 1, q == s
where
p
number of groups of treatments
==
= size of each group
\= number of times any
q
two treatments from
different groups occur together.
A,2
= number
of times any two treatments from any
group occur together.
Again from [""'4 J it follows that the dual D * of Dl is a partially balanced
l
incomplete block design (b.i.b.d.) with two associate classes with parameters
(2.10)
Dl *:
v ==
n{n+l)
n(n-l)
2 ' b = '.
2
' r
= n-l,
k
=n
+ 1,
{n-l (n-2)
n-2n
-2
1-2, n 2 -2
' A-l
1- , A.21
2
Pll = n - 1, Pll = 4.
Conversely using the results of Roy and Laha
L:6J it
can be seen that the dew'
signs with parameters given by (2.9) and (2.10) are linked block designs and
hence the dual of a design with parameters (2.9) is the b.i-b.d. with parameters (2.6).
Similarly the dual of a design with parameters {2-10) is the
3
b.i.b.d. with parameters (2.8).
~ence
the designsC
1
and Cl * (D l and Dl *) are
either both existent or both nonexistent.
A p.b.i.b.d. with two associate classes is said to be triangular
if the number of treatments is n (n
2 1)
['7 J
and the association scheme is an
array of n rows and n columns with the following properties.
(a)
The positions in the principal diagonal are blank.
(b)
The n (n ; 1)
positions above the prinicpal diagonal are filled by
the numbers 1, 2, • •• , n (n-l)
2
corresponding to the treatments.
(c)
The positions below the principal diagonal are filled so that the
array 1s sj1IllID.etrical about the principal diagonal.
(d)
For any treatment i, the first associates are exactly those treatments
which lie in the same row and the same column as treatment i.
The following relations obviously hold.
(1) The number of first associates of any treatment is nl
(2)
With respect to any two treatments 9
1
= 2n
• 4.
and g2 which are first asso-
ciates, the number of treatments which are first associates of both 9 and
1
e2
is
1
1'11
=n
- 2
and 9 4 .which are second asso3
ciates, the number of treatments which are first associates of both 9 and Q
r
3
is
(3)
With respect to any two treatments 9
2
Pn
= 4.
Conversely connor [BJhas show.n that for a p.i.b.d. with two associate
classes if v
=n
(n;l), n
~ 9, (1), (2) and (3) above imply that the association
scheme is triangular, denoted by (T ).
n
In a paper
L"97
submitted
--
to the Annals
of Mathematical Statistics, the author has shown that the same result is true
4
for n
= 5,6.
The cases n = 7 and 8 are as yet undecided.
3. Results for b.i.b.d. with A = 1. In this section we derive some
results for des igns obtained from the symmetrical b .i.b .d. with A = 1. They
provide the motivation for the results obtained in sections 4 and 5.
We have already seen that a design C with paramenters (2.6) is dual of
l
a design Cl * with parameters (2.9) and conversely. Now since Cl * is a group
divisible design, the blocks of C can be divided into (s + 1) sets of s
l
each such that any two blocks of the same set have no treatment in common,
whereas any two blocks ft'QIu:llfferent sets have exactly one treatment in
common.
Hence if we omit one set of blocks we get a group divisible design
Cll given by parameters.
(3.1) Cll:
u ~ b
2
= s,
r
= k = s,
~
= 1,
~2
=0
Further the omitted set of blocks gives rise to another disconnected group
Hence, obviously the s treatments corresPQnding to the blocks of C must
12
all belong to the same group. omitting these trea.tments from Cl, we are left
with Cl l. The association scheme of Cli is obviously group divisible with
- the remaining s groups of size s each. Thus Cl~ is again a. group diVisible design with exactly the same parameters as C •
l1
5
4. Results connected with b.i.b.d. with A = 2.
prove
f~il1
In this section we
theorems for designs connected with b.1.b.d. with
Theorem (1).
~
= 2.
If for any value of n, a design Dl * with parameters (2.10)
exists, then its association scheme is of the type (Tn + 1)'
Proof. We note that the design D1 given by parameters (2.8) does not
exist for n = 6 and 7 ~~o_7. Bence Dl * does not exist for these values
of n.
For n>
n
= 4 and
8, the theorem follows from Connor's result
5, it follows from the result of the author
L-a.J and
for
L~~referred to
previously.
Theorem (2).
For any value of n, the existence of the design Dl with
parameters (2.8), implies the existence of two design Dll and D12 given
respectively by the follow parameters.
v = b = n(n-l)
- 2 -,
(4 .1 ) Dll :
n1
(4.2) D12 :
=2
v
nl
n - 4, n = (n-2) (n-3)/2, ~ = 1, ~ =. 2.
2
1
2
Pll = n - 2, Pll = 4.
=n
= 20
r = k = n - 1,
(n-l)/2, b
- 4, n2
1
= n,
= (n-2)
k
= n-l,
r
(n-3)/2, ~l
2
Pll = n - 2, Pll
=2
= 1,
~2
=0
= 4.
D = Dll + D12 •
l
Proof. Instead of Dl we consider its dual Dl * given by (2.10). Let
Dll* and D12* denote the duals of D1l and D12 respectively. It is sufficient to prove that the treatments of Dl * can be divided into two distinct
Such that
sets Tl and T2 say that by retaining treatments of the set T1 in the blocks
of Dl * we get the design Dl1*, whereas by omitting these treatments (and
hence retaining treatments of the set T2 ) in the blocks of Dl * we get the
design D12*. We note that D12* is nO'\ihing but the unreduced b.i.b.d. with
6
;v
= n,
k :: 2, A.
= 1.
From Theorem 1, we know that the association scheme
of D * is triangular (T +J) which may be written without loss of generality
n
l
as follows.
3
..
n+l
n+2
•
n+l
x
•
n+2
·
. x
1
1
., x
.
•
•
n
,
•
·2n-l
•
•
•
•
•
•
x
•
•
2
n +n
~
2
n
'
2n-l
•
x
n +n'.
•
~
We show that any block of D * contains exactly two treatments from any
l
row of
(4.3). Consider (n+l) blocks si of size {n+l), each containing a
new treatment a and all the treatments of row i of
(4.3), i
= 1,
2, ••• ,
Adding these (n+l) blocks si to the blocks of D * we obviously get
I
2
n +n+2
.
2' r = k = n + 1. The treatment Ct occurs with
a design With v = b =
(n+l).
any treatments of D
"*
in exactly two of these blocks.
Any two treatments of
1
Dl * say
Vand
m. occuring in the same row or column of
a block of D *.
l
They also occur once in a block s1.
not lie in the same row or same column of
(4.3) occured once in
Again i f
Vand
m,do
(4.3), they occur together in two
blocks of D *, but they do not occur together at all in the new bloclt s~s1 •
l
Thus any two treatments
design giving A.
= 2.
f
and m of D * occur together twiee in the. new
l
Hence the design so obtained is nothing but the b .1.b.d.
With parameters (2.7) and hence any two blocks of this design have two
treatments in common.
Let 9
belonging to same row of
exactly one block of D~.
1
ande~e
two different treatments of D *
l
(4.3). Then 9 1 and 9 occur together in
2
7
block of Dl * cannot contain any other treatment ,Q} of the same
r·owof (4.}). For otherwise the design (2.7) obtained above will have two
Further
thi~
blocks containing three treatments in common which is impossible. Consider
in particular the treatments 1, 2, ••• , n occuring in the first row of (4.}).
Then each of the n(n~l) pairs of these treatments uniquely determines a
block of Dl *. Noticing that each pair determines only one block and that
the number of blocks in Dl * is exactly n(n-l)/2, this correspondence can
be setup.. in 1- 1 manner. This proves the statement made at the beginning
of this paragraph.
If we now retain only the treatments of the set T2
=
(1, 2, ••• , n) in the blocks of Dl *, we get the design D12* which is the unre~
duced b.i.b.d. for V = nand k = 2. We now write the nc combination of the
2
n treatments of T
2
(4.4)
in the following array of n rows and n columns.
x
(1,2)
(l,})
(1,4)
(2,1)
x
(2,3)
(2,4)
(3,1)
(3,2)
x
(3,4)
(4,1)
(4,2)
(4,3)
x
•
•
•
(n,l)
(n,2)
(n,3)
As mentioned before each combination determines a unique block of Dl * to
which the combination belongs. By removing these treatments from the corres-
= b = n{n-l)/2, r = k = n - 1,
of the set T ' = (n+l, n+2, ... ,
l
ponding blocks we are left with a design with u
the n (n-l)/2 treatments being the treatments
2
n +n
,
- ) • Now in D * any two blocks have two treatments in common. Hence in
l
this design for treatments of the set T , any two blocks corresponding to
l
two combinations which are either in the same row or in the same column
have exactly one treatment in common, since the corresponding combinations
8
have one treatment in common and which are now omitted.
Similarly, since
any two comibnations not in the same row and not in the same column have no
treatment in common, the corresponding blocks for the above design for the
treatment of
1.
have two treatments in common.
This implies that the blocks
of the above design can be written in a triangular array like (4.4) such
that any two blocks which are either in the same row or the same column
have exactly one treatment in common, whereas any other pair of blocks
have two treatments in common. We may sa.y.that the block structure for this
design is triangular (Tn ).
It is now obvious that this design is nothing
* we see that the n(n+l)/2 blocks
Now considering the dual of Dl' Dll, D12
of Dl can be diVided into sets of ~(n-l)/2 and n respectively suc~ that the
first set of blocks gives the design D and the second set gives the design
1l
D12 • We thus have D = D + D • This completes the proof of the theorem.
l
ll
12
We now prove a partial converse of the above theorem
Theorem 3.
Existence of the design D with parameters (4.1), having
1l
triangular association scheme (T ) for any value of n implies the existence
n
of the corresponding design Dl with parameters (2-~).
Proof Without loss of generality we can represerve the association
(n-l)
(2n-3)
•
•
•
x
2
.
n -n
~
2
n -n
2
x
Consider the n blocks of size (n-l) given by the rows of (4.5). They
constitute the design D12 given by (4.2).
Adding the blocks of D12 to the
9
blocks of D , we get a design with v = n(n-l)/2, b = n(n+l)/2, r = n+l,
ll
k = n-l. Further any two treatments of Dll occuring d.n the same row or
the same column of (4.5) occur together once more in these new blocks.
Also two treatments which are. neither in the same now nor in the same
column of (4.5) already occur together twice in blocks of Dll but they do
not occur together at all in these new block. Hence in the new design
A
= 2,
for all hours of treatments.
Thus the new design is exactly the
design Dl • ~ther we have Dl = DlfPIZ This completes the proof.
The design D is known to be impossible £11_7 for n = 6 so is the
ll
design Dl for the same value of n !:JfI. The design Dl for n = 7 is also
known to be impossible
/..1l0:7,
but the impossibility of Dll for this value
of n has not been proved. If, however, D exists for n = 7, the association
ll
scheme cannot be triangular, as from the above theorem this would imply the
existence of Dl for n
= 7.
From ~8,9~jWe know that the association scheme
for Dll is triangular (Tn) for n2:9 and n = 5. We thus have the following
corollary.
Corollary.
If n
=5
or n
~
9, the eXistence of DIl with parameters (4.1)
implies the existence of Dl with parameters (2.8).
We now state and prove another theorem for the design Dll •
Theorem 4.
If n
=5
or
n~
9, the dual of the design Dll given by (4.1)
is another p.b.i.b.d. with the same parameters as (4.1).
From Theorem 3 and its corollary we have Dll + D12 = D • Consider
l
the dual Dt of D • Then Dlt is Obtained by omitting from D!, the treatments
l
corresponding to the blocks of D12 • We note that in each block of Di' there are
Proof.
two treatments correaponding
to the blocks. of D12 and (n-l) treatments corresa
.
ponding to the blocks of Dll • Further since the dual of D is D1~ which is
12
a b.1.b .d. with parameters v = nj b= n (n-1)/2, k = 2, t = n-l, A. = 1, any
10
two of the n treatments a a
1
2
••• O:n of
D1~
are first associates in
the association scheme of D! is triangular (Tn+ 1)' from
6 or n + 1
DI.
L-a, 9-:1 since
Now
n + 1 =
> 10.
We now show that 0: , 0:2 , ••• , O:n must all be in the same
1
row of the association scheme (Tn+i). For consider any two treatments 0:1
and
0:
2 , say.
Since they are first associates, they must lie either in the
same row or the same column of the association scheme.
Suppose they lie in
in the same row, then since the association scheme is unchanged if we interchange any two rows and the same two columns of the array of (T n+ 1)' we can,
without loss of generality, assume that
and 0:2 are in the first row in
1
, 04' ••• , an all lie in the
positions (1,2) and(1,3) respectively. If
3
first row weare through. Otherwise there is at least one treatme!lt, say,
°
°
which does not lie in this row. Since the first associates of 0: 1 must
3
lie in the first and the second rows,
lies in the second row in position
3
(2,.:) say. Similarly since
is first associate of ct2 , it must be
3
in the third row in position (3,jl),say. Since these two positions of 0:
3
must be symmetrical with respect to the main diagonal we have j = 3 and jl =
0:
°
°
2, i.e., 0:30ccurs in the array of (T n+l ) in positions (2,3) and (3,2). Now
°
Since it is first associate of 1 , it must occur either in the
first row or in the second row in column position V~ 4. Suppose it occurs in
consider 0:4'
position (1, V), ~:;:4·then
C\
does not occur in the same row or same column as
and hence is second associate of 0: which is a contradiction. Suppose 0:4
3
3
occurs in position (2,V), V~ 4, then it is not in the same row or the same
0:
column as
92
and hence is second associate of
°1 and
0:
2 which is
again a contri-
°
2 l1e in a row, all other treatments 3 , .:.. , an
must lie in the same row. We get a similar result if 0: and a lie -...in the
1
2
same column. This proves the result that all the treatments Ct l ,ct2 ••• , an
diction.
Thus if
0:
11
must lie in the same row and hence the same column.
menta
0:
0:
1
2
••• ,
We now omit the treat-
n from the blocks of D'i:. This leaves the designs Dl'i:.
Ct
Further the association scheme of Dli is obviously obtained by omitting
the row and column in which the treatments
scheme is therefore, triangular (Tn).
Ct
l'
0: ,
2
••• , O:n lie.
The
It is now easy to see that the
parameters of D1i are exactly those of D • This completes the proof the
ll
theorem.
5. A constructive proof for embedding the design D1 into the design
D.
As mentioned earlier Hall and Connor
of the design D given by
l
by (2.7).
L3_7 proved
that the existence
(208) implies the existence of the design D given
Their proof is not constructive. We give in this section a
constructive proof of their result.
Suppose the design D eXists, then Di
l
also exists and by Theorem 1,
*
the association scheme of D1is
triangular (Tn+l).
Bn ,
(;.
B 2,,:' of D can be exhibited in the scheme (4.3) when i stands for
n, +'n
l
••• ,
Bi , such
(4.,)
Hence, the blocks B ,
l
t~any two
blocks of D 1n the same
l
r~~,or
in the same column of
have exactly one treatment in common, whereas any two blocks neither
in the same row nor in the same column of (4.3) have two treatments in
common.
Take a new block BI consisting of new treatments up u2 ' ••• , un':" t
Let~he
n(n+l)j2 combinations of these treatments be listed as shown below.
x
(u p u )
2
(ul,u,)
(u ,u )
2 l
x
(u ,u )
2 3
(u"U1 )
(U ,u )
3 2
x
•
(5.1)
•
•
(u"un+l)
•
•
(u2 ' iiri+ 1)
•
•
(u1 ,un+1 )
•
•
•
•
•
•
(up+l'U1 ) (un+l'u2 )
•
(Un+l'u,)
•
•
x
(un+l'un )
(un,un+l)
x
-
12
We now assign the above combinations to the respective blocksB i in the
corresponding positions of (4.3) to get new blocks B~. Thus Bl consists
of treatments of B besides the treatments u l and u ' It is now easy to
l
2
2
2
n +n+2
verify that B' and B~, i=1,2 .••• , n ~n give a design with v = b =~~­
2
'
r = k = n+l. Further i f i and j belong to the same row or same column of
(4.3), then Bi and Bj have one treatment in common and the corresponding
,
,
combinations added to these blocks to give B and B have again one treati
J
.
"
ment 1n common. Thus B and Bj have two treatments in common. If on the
i
other hand i and j neither belong to the same row nor to the same column
of (4.3), then B. and B. have two treatments in common; but now the
J
1
corresponding combinations to be added to them have no treatment in common.
I
I
Thus again Bi and Bj have two treatments in common. Again B' has obviously
two treatments in common with any B~. Thus we get a symmetric design in
which every two blocks have two treatments in common.
This implies that
A = 2. The design thus obtained 1s exactly the design D.
13
REFERENCES
L1 JR.
C. Bose, "On the construction of balanced incomplete block designs ,I,
Annals of Eugenics, Vol.
/..2_7
K. N. Bhattacharya, " A new balanced incomplete block design," Science
~ CUlture, Vol.
["37
9 (1939), pp. 353-399.
9 (1944) pp. 508.
Marshall Hall Jr. and W. S. Connor, "An embedding theorem for balanced
incomplete block designs ," Canadian Journal of Mathematic's, Vol. 6
(1953), pp. 35-41.
{4_7
S. S. Shrikhande, "On the dual of some balanced incomplete block
designs,"
L-5_1
Biometrics, Vol.
8 (1952) pp. 66-72.
R. C. Bose and W. S. Connor, "Combinational properties of group
divisible incomplete block designs," Annals .2! Mathematical
23 (1952), pp. 367 - 383.
Vol.
fo7
sta.!!.~,
J. Roy and R. G. Laha, "Classification and analysis of linked block
designs," Sankhya, Vol.
[7 JR.
17 (1956). pp. 115-132.
C. Bose and T. Sh1m.amoto, "Classification and analysis of partially
balanced designs with two associate classes," Journal of American
Statistical Associatio~, Vol.
[8_7 w.
S. Connor, "The uniqueness of the triangular association scheme,"
Annals
D] s.
£! MathematicalStatistics,
Vol.
29 (1958), pp. 262-266.
S. Shrikhande, "On a characterisation of the triangular association
scheme,"
[10_7 w. s.
designs,"
L-llJ w.
submitted to the Annals of Mathematical Statistics.
Connor, Jr., "On the structure of balanced incomplete block.:
Annals ~ Mathematica.l Statistics, Vol.
23 (1952), pp. 57-71.
S. Connor and W. H. Clarworthy, "Some theorems for partially
balanced designs,"
pp.
47 (1952), pp. 151-190.
100-112.
Annals.2! Mathematical Statistics, Vol.
25 (1954)