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A method for constructing variance balanced designs
Journal
of Statistical
Planning
and Inference
127
23 (1989) 127-131
North-Holland
A METHOD BALANCED James
FOR CONSTRUCTING DESIGNS
A. CALVIN
Department
of Statistics
Kishore
Received
of Statistics, 8 September
Recommended
Abstract:
Science,
University
of Iowa, Iowa City, IA 52242, U.S.A.
Birsa Agricultural
1986; revised
Calvin (1986) provides
(2) permit
technique
fewer experimental
Subject
Key words:
University,
manuscript
received
Ranchi
834006, India
5 July 1988
by A.S. Hedayat
tend his construction
block
and Actuarial
SINHA
Department
AMS
VARIANCE
Classification: Balanced
a technique to (1) produce
for constructing
variance
balanced
designs with more than two distinct
designs.
We ex-
block sizes and
units. Primary
incomplete
62KlO;
block
secondary
designs;
variance
62510. balanced
block
designs;
incomplete
designs.
1. Introduction In many experimental situations it is a severe restriction to require that all blocks in the experiment be the same size. Variance balanced designs (VBDs) form a class of designs that are flexible extensions to balanced incomplete block designs (BIBDs). They provide the experimenter with the ability to design an experiment, with equal precision among all pairwise comparisons, without being restricted to equal block sizes and equal replication of the treatments. The construction of variance balanced designs has been studied by several authors, including Calvin (1986), Sinha & Jones (1988), Mukerjee & Kageyama (1985), Gupta & Jones (1983), Khatri (1982), Tyagi (1979), Kageyama (1976) and Hedayat & Federer (1974). This note extends the work of Calvin (1986) to allow for more than two distinct block sizes and possibly fewer total observations. It has been brought to our attention by the editor that Hedayat and Stufken (1988) have shown an equivalence between VBDs and pairwise balanced designs (PBDs). They show that given a design from either class one can construct a design from the other class. Thus, any VBD can be constructed from a generating PBD. Their example 2.2 demonstrates this relationship with the designs in Calvin (1986). 0378.3758/89/$3.50
0
1989, Elsevier
Science
Publishers
B.V. (North-Holland)
J.A.
128
In what follows, the phrase ‘distinct a different
the term ‘block’ will refer to a subcollection of treatments and blocks’ will refer to a set of blocks in which each block contains
subcollection
2. Generalized
Calvin, K. Sinha / Variance balanced designs
of the treatments.
designs
A technique for constructing variance balanced designs, which is based on the unionizing block principle of Hedayat and Federer (1974), was described in Calvin (1986). To construct a design for u treatments, using blocks of size s andj+ 1, this technique requires that one choose positive integers t and j such that (1) (t- l)/(jl)=s an integer> 1, and (2) t + s= u, the number of treatments. To start the construction divide the o treatments into two groups. Let GO contain the first t treatments and Gi contain the last s treatments. The design will have one distinct block of size s, containing the s elements in Gt.To build the blocks of size j-t 1 consider the BIBD formed by constructing all (5) combinations of the I treatments in GO. For this BIBD r=(fI f), k=j, and A =(;I;). Repeat this BIBD s times, each time adding a different element of G, to each block. This yields sx (f) distinct blocks of size j + 1. The algorithm produces a VBD with a reduced normal equations matrix of C= UN- AsJ when the blocks of size j+ 1 are repeated j + 1 times and the block of size s is repeated As* times. The generating BIBD for the blocks of size j+ 1 can be replaced by any BIBD(t, b, r, A) where r/l =s. Choosing the smallest such BIBD minimizes the total number of experimental units required for the VBD. These results can be extended to allow for more than two distinct block sizes. Letting u represent the number of treatments and t,j and m be positive integers such that o = t + ms a VBD for u treatments can be constructed with blocks of size s,s+ 1, . . . ,s+ m - 1 and j+ m under the following restrictions: (1) (t- l)/(jl)=s an integer> 1, (2) t + ms = o the number of treatments, and (3) (t/j-s)>(l)(s/(sl))+* for m22. (The third restriction is clearly satisfied when t and j are such The algorithm used to construct the design is recursive. We can (1986) that the design for u = t + s treatments is based on a BIBD for In general, the design for u = t + ms treatments is based on u = t + (m - 1)s treatments.
that t/j>s1.) see from Calvin u = t treatments. the design for
VBD algorithm. Let d, be the design for o, = t + ms treatments. Then d, can be the distinct any BIBD(uc,b,r, j,,l) where r/A =s. For rn? 1, we first construct blocks for each block size and then determine how many times each block must be repeated to yield a VBD.
J.A.
Calvin, K. Sinha / Variance balanced designs
into m + 1 groups.
Divide the u, treatments GO, the next s treatments
129
Assign the first 1 treatments
to G,, and so on, until the last s treatments
to group
are assigned
Thus, d, _ , is based on the treatments in GO through G, ~ 1. Repeat each distinct block in d+, s times, each time adding a different element of G, to the block. Since d,_ , contains blocks of size s, . . . , s+ m - 2 and j + m - 1, the new blocks in d, will be of size s+ 1, . . . , s + m - 1 and j + m. Finally, the m distinct blocks of size s are made up of the m sets G,, . . . , G,.
to G,.
Let pO be the number of repetitions of the blocks of size j+m, let pi, i= of the blocks of size s + m - i and let qh, 1, . . . , m - 1, be the number of repetitions m, be the number of repetitions of the block consisting of the s treatments h=l,..., in Gh. The reduced normal equations matrix for this VBD is
C,=v,,,W-W’J po, . . . ,p,,_ ,, q,, . . . , qm take on the following
when the integers
values:
Pe=j+m,
pi=k-(s+m-i)(s-l)‘P1(s-t/j),
i=l,...,m-1,
q,=/ls m+2-h(shP1-(s-l)hP1(s-t/j)),
(1)
h=l,...,m-1,
qm=Pf’. Note that the values in (1) are integers, since by the relationships among the parameters of the BIBD in do, rt = bj implies that b = M/j is an integer. If all the values in (1) have a common divisor a smaller design is possible, The following
is a sketch
of the algebra
required
to prove that d, is a VBD.
Proof (sketch). Let A’, . . . ,AmP ‘,B’ , . . . , B” represents the distinct blocks of size j+m,s+m-l,..., s+ 1,s ,..., s. Then
C,=poAo+
... +pm_,AmP1+qlB1+
C-matrices
for
the
... +qmBM.
Let A:, u represent the submatrix of A’ associated with the treatments Define Bl, in a similar manner. Then it is also true that
in G, and G,,.
m-1 G,u=
c
i=o
piA,,+
f
q/zB:u.
h=l
The submatrices Al, o and B& are pattern matrices, and to prove the result we need only check that when the coefficients take on the values in (I), the resulting C-matrix is
C,,,=o,W’Z-AsmJ. This can be done by considering each element as a polynomial in s and using some combinatorial relationships from Feller (1968) to show that the coefficients of sj, j
130
J.A.
3. Construction Consider
Calvin,
K. Sinha / Variance balanced
example
an experiment
with 5 treatments.
and t +s= o, so m = 1. Thus GO = { 1,2,3} based on the BIBD(t=3,b=3,r=2,k=2,A=l) Table
designs
If we choose t = 3 and j= 2, then s = 2 and
Gi = {4,5}. The resulting is in Table 1.
design
1
Variance
balanced
design for IJ~=5 treatments
Block
# of replications
Block
# of replications
1,2,4
3
2,334
I,2,5
3
2,3,5
3
1,3,4
3
4s
4
193s
3
3
We can expand the experiment to 7 treatments, by letting m = 2. Then G, = {6,7} and the new design with blocks of size 2, 3, and 4 and C=281-4J, is in Table 2.
Table 2 Variance
balanced
design
for IJ~= 7 treatments
# reps
Block
Block
# reps
Block
# reps
I,2,4,6
4
1,3,4,7
4
2,3,5,6
4
1,2,4,7
4
193,596
4
2,3,5,7
4
1,2,5,6
4
I,3,5,7
4
4,596
3
1,2,5,7 1,3,46
4 4
2,3,4,6 233,497
4 4
4,5,7 4,5
3 4
6,7
8
Acknowledgements The authors would like to thank the editor, for pointing out the reference of Hedayat and Stufken, and the referees and Dr. N. Sedransk for their comments which were helpful in improving the presentation of the article.
References Calvin,
J.A. (1986). A new class of variance
balanced
designs.
J. Statist.
Plann. Inference
14, 251-254.
Cochran, W.G. and G.M. Cox (1957). Experimental Designs, 2nd edition. Wiley, New York. Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume I, 3rd edition. Wiley, New York. Gupta, S.C. and B. Jones Biometrika
70, 433-440.
(1983).
Equireplicate
balanced
block
designs
with unequal
block
sizes.
J.A. Calvin, K. Sinha / Variance balanced designs Hedayat,
A. and W.T. Federer
(1974). Pairwise
and variance
balanced
131
incomplete
block designs.
Ann.
Inst. Statist. Math. 26, 331-338. Hedayat, A. and J. Stufken (1988). On a relation between pairwise balanced and variance designs. Stat. Lab. Technical Report No. 88-05, University of Illinois at Chicago. Kageyama, Khatri,
S. (1976). Construction
C.G.
Mukerjee,
of balanced
(1982). A note on variance
R. and S. Kageyama
block
balanced
block
Utilitas Math. 9, 209-229. J. Statist. Plann. Inference 6, 173-177.
designs.
designs.
(1985). On resolvable
balanced
and affine
resolvable
variance-balanced
block designs
with unequal
designs.
Biometrika 72, 165-172. Sinha,
K. and B. Jones
(1988). Further
equireplicate
balanced
block sizes.
Statist. Probab. Lett. 6, 229-230. Tyagi,
B.N.
333-336.
(1979).
On a class of variance
balanced
block
designs.
J. Statist. Plann. Znference 3,
of Statistical
Planning
and Inference
127
23 (1989) 127-131
North-Holland
A METHOD BALANCED James
FOR CONSTRUCTING DESIGNS
A. CALVIN
Department
of Statistics
Kishore
Received
of Statistics, 8 September
Recommended
Abstract:
Science,
University
of Iowa, Iowa City, IA 52242, U.S.A.
Birsa Agricultural
1986; revised
Calvin (1986) provides
(2) permit
technique
fewer experimental
Subject
Key words:
University,
manuscript
received
Ranchi
834006, India
5 July 1988
by A.S. Hedayat
tend his construction
block
and Actuarial
SINHA
Department
AMS
VARIANCE
Classification: Balanced
a technique to (1) produce
for constructing
variance
balanced
designs with more than two distinct
designs.
We ex-
block sizes and
units. Primary
incomplete
62KlO;
block
secondary
designs;
variance
62510. balanced
block
designs;
incomplete
designs.
1. Introduction In many experimental situations it is a severe restriction to require that all blocks in the experiment be the same size. Variance balanced designs (VBDs) form a class of designs that are flexible extensions to balanced incomplete block designs (BIBDs). They provide the experimenter with the ability to design an experiment, with equal precision among all pairwise comparisons, without being restricted to equal block sizes and equal replication of the treatments. The construction of variance balanced designs has been studied by several authors, including Calvin (1986), Sinha & Jones (1988), Mukerjee & Kageyama (1985), Gupta & Jones (1983), Khatri (1982), Tyagi (1979), Kageyama (1976) and Hedayat & Federer (1974). This note extends the work of Calvin (1986) to allow for more than two distinct block sizes and possibly fewer total observations. It has been brought to our attention by the editor that Hedayat and Stufken (1988) have shown an equivalence between VBDs and pairwise balanced designs (PBDs). They show that given a design from either class one can construct a design from the other class. Thus, any VBD can be constructed from a generating PBD. Their example 2.2 demonstrates this relationship with the designs in Calvin (1986). 0378.3758/89/$3.50
0
1989, Elsevier
Science
Publishers
B.V. (North-Holland)
J.A.
128
In what follows, the phrase ‘distinct a different
the term ‘block’ will refer to a subcollection of treatments and blocks’ will refer to a set of blocks in which each block contains
subcollection
2. Generalized
Calvin, K. Sinha / Variance balanced designs
of the treatments.
designs
A technique for constructing variance balanced designs, which is based on the unionizing block principle of Hedayat and Federer (1974), was described in Calvin (1986). To construct a design for u treatments, using blocks of size s andj+ 1, this technique requires that one choose positive integers t and j such that (1) (t- l)/(jl)=s an integer> 1, and (2) t + s= u, the number of treatments. To start the construction divide the o treatments into two groups. Let GO contain the first t treatments and Gi contain the last s treatments. The design will have one distinct block of size s, containing the s elements in Gt.To build the blocks of size j-t 1 consider the BIBD formed by constructing all (5) combinations of the I treatments in GO. For this BIBD r=(fI f), k=j, and A =(;I;). Repeat this BIBD s times, each time adding a different element of G, to each block. This yields sx (f) distinct blocks of size j + 1. The algorithm produces a VBD with a reduced normal equations matrix of C= UN- AsJ when the blocks of size j+ 1 are repeated j + 1 times and the block of size s is repeated As* times. The generating BIBD for the blocks of size j+ 1 can be replaced by any BIBD(t, b, r, A) where r/l =s. Choosing the smallest such BIBD minimizes the total number of experimental units required for the VBD. These results can be extended to allow for more than two distinct block sizes. Letting u represent the number of treatments and t,j and m be positive integers such that o = t + ms a VBD for u treatments can be constructed with blocks of size s,s+ 1, . . . ,s+ m - 1 and j+ m under the following restrictions: (1) (t- l)/(jl)=s an integer> 1, (2) t + ms = o the number of treatments, and (3) (t/j-s)>(l)(s/(sl))+* for m22. (The third restriction is clearly satisfied when t and j are such The algorithm used to construct the design is recursive. We can (1986) that the design for u = t + s treatments is based on a BIBD for In general, the design for u = t + ms treatments is based on u = t + (m - 1)s treatments.
that t/j>s1.) see from Calvin u = t treatments. the design for
VBD algorithm. Let d, be the design for o, = t + ms treatments. Then d, can be the distinct any BIBD(uc,b,r, j,,l) where r/A =s. For rn? 1, we first construct blocks for each block size and then determine how many times each block must be repeated to yield a VBD.
J.A.
Calvin, K. Sinha / Variance balanced designs
into m + 1 groups.
Divide the u, treatments GO, the next s treatments
129
Assign the first 1 treatments
to G,, and so on, until the last s treatments
to group
are assigned
Thus, d, _ , is based on the treatments in GO through G, ~ 1. Repeat each distinct block in d+, s times, each time adding a different element of G, to the block. Since d,_ , contains blocks of size s, . . . , s+ m - 2 and j + m - 1, the new blocks in d, will be of size s+ 1, . . . , s + m - 1 and j + m. Finally, the m distinct blocks of size s are made up of the m sets G,, . . . , G,.
to G,.
Let pO be the number of repetitions of the blocks of size j+m, let pi, i= of the blocks of size s + m - i and let qh, 1, . . . , m - 1, be the number of repetitions m, be the number of repetitions of the block consisting of the s treatments h=l,..., in Gh. The reduced normal equations matrix for this VBD is
C,=v,,,W-W’J po, . . . ,p,,_ ,, q,, . . . , qm take on the following
when the integers
values:
Pe=j+m,
pi=k-(s+m-i)(s-l)‘P1(s-t/j),
i=l,...,m-1,
q,=/ls m+2-h(shP1-(s-l)hP1(s-t/j)),
(1)
h=l,...,m-1,
qm=Pf’. Note that the values in (1) are integers, since by the relationships among the parameters of the BIBD in do, rt = bj implies that b = M/j is an integer. If all the values in (1) have a common divisor a smaller design is possible, The following
is a sketch
of the algebra
required
to prove that d, is a VBD.
Proof (sketch). Let A’, . . . ,AmP ‘,B’ , . . . , B” represents the distinct blocks of size j+m,s+m-l,..., s+ 1,s ,..., s. Then
C,=poAo+
... +pm_,AmP1+qlB1+
C-matrices
for
the
... +qmBM.
Let A:, u represent the submatrix of A’ associated with the treatments Define Bl, in a similar manner. Then it is also true that
in G, and G,,.
m-1 G,u=
c
i=o
piA,,+
f
q/zB:u.
h=l
The submatrices Al, o and B& are pattern matrices, and to prove the result we need only check that when the coefficients take on the values in (I), the resulting C-matrix is
C,,,=o,W’Z-AsmJ. This can be done by considering each element as a polynomial in s and using some combinatorial relationships from Feller (1968) to show that the coefficients of sj, j
130
J.A.
3. Construction Consider
Calvin,
K. Sinha / Variance balanced
example
an experiment
with 5 treatments.
and t +s= o, so m = 1. Thus GO = { 1,2,3} based on the BIBD(t=3,b=3,r=2,k=2,A=l) Table
designs
If we choose t = 3 and j= 2, then s = 2 and
Gi = {4,5}. The resulting is in Table 1.
design
1
Variance
balanced
design for IJ~=5 treatments
Block
# of replications
Block
# of replications
1,2,4
3
2,334
I,2,5
3
2,3,5
3
1,3,4
3
4s
4
193s
3
3
We can expand the experiment to 7 treatments, by letting m = 2. Then G, = {6,7} and the new design with blocks of size 2, 3, and 4 and C=281-4J, is in Table 2.
Table 2 Variance
balanced
design
for IJ~= 7 treatments
# reps
Block
Block
# reps
Block
# reps
I,2,4,6
4
1,3,4,7
4
2,3,5,6
4
1,2,4,7
4
193,596
4
2,3,5,7
4
1,2,5,6
4
I,3,5,7
4
4,596
3
1,2,5,7 1,3,46
4 4
2,3,4,6 233,497
4 4
4,5,7 4,5
3 4
6,7
8
Acknowledgements The authors would like to thank the editor, for pointing out the reference of Hedayat and Stufken, and the referees and Dr. N. Sedransk for their comments which were helpful in improving the presentation of the article.
References Calvin,
J.A. (1986). A new class of variance
balanced
designs.
J. Statist.
Plann. Inference
14, 251-254.
Cochran, W.G. and G.M. Cox (1957). Experimental Designs, 2nd edition. Wiley, New York. Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Volume I, 3rd edition. Wiley, New York. Gupta, S.C. and B. Jones Biometrika
70, 433-440.
(1983).
Equireplicate
balanced
block
designs
with unequal
block
sizes.
J.A. Calvin, K. Sinha / Variance balanced designs Hedayat,
A. and W.T. Federer
(1974). Pairwise
and variance
balanced
131
incomplete
block designs.
Ann.
Inst. Statist. Math. 26, 331-338. Hedayat, A. and J. Stufken (1988). On a relation between pairwise balanced and variance designs. Stat. Lab. Technical Report No. 88-05, University of Illinois at Chicago. Kageyama, Khatri,
S. (1976). Construction
C.G.
Mukerjee,
of balanced
(1982). A note on variance
R. and S. Kageyama
block
balanced
block
Utilitas Math. 9, 209-229. J. Statist. Plann. Inference 6, 173-177.
designs.
designs.
(1985). On resolvable
balanced
and affine
resolvable
variance-balanced
block designs
with unequal
designs.
Biometrika 72, 165-172. Sinha,
K. and B. Jones
(1988). Further
equireplicate
balanced
block sizes.
Statist. Probab. Lett. 6, 229-230. Tyagi,
B.N.
333-336.
(1979).
On a class of variance
balanced
block
designs.
J. Statist. Plann. Znference 3,