C H A P T E R 5
Construction of Partially Balanced Incomplete Block Designs
In the previous chapter we discussed several types of association schemes for PBIB designs with two or more associate classes. There exists a very large number of PBIB designs with these, and other, association schemes. But as we have pointed out earlier, the association schemes themselves do not constitute or generate the actual plan, that is, the assignment of treatment to blocks. To obtain such plans, different methods of construction have been developed and used. It is impossible to mention all of them as the literature in this area is immense. Rather, we shall discuss in some detail only a few methods with emphasis mainly on three types of PBIB designs, namely PBIB(2) designs, in particular, group-divisible PBIB(2) designs, cyclic PBIB designs, and EGD-PBIB designs. The first class is important because it constitutes the largest class of PBIB designs. The second class also represents a rather large class, and the designs are easy to construct and are widely applicable. The last class is important with respect to the construction of systems of confounding for asymmetrical factorial experiments (see Chapter 12).
has been obtained fromD by interchanging treatments and blocks. This implies that D has parameters t=b, b=t, k=r, r=k. The design D is referred to as thedual of the designD (and, of course, vice versa). If the designD has a certain structure, such as being a BIB or PBIB design, then the dual design D very often also has a particular recognizable structure. It is this relationship between D and D that we shall now utilize to construct certain GD-PBIB(2) designs from existing BIB designs. Consider the following theorem.
Theorem 5.1 LetD be a BIB design with parameterst =s2, b=s(s+1), k=s, r =s+1, λ=1, where s is prime or prime power (see Sections 3.34 and 18.7). Then the dualD is a GD-PBIB(2) design with parameterst=(s+ 1)s, t1=s+1, t2=s, b=s2, k=s+1, r=s, λ1=0, λ2=1.
Proof As can be seen from the construction in Section 3.7,Dis a resolvable (α=1) BIB design, consisting of s+1 replication groups with s blocks each and containing each treatment once. Denote the blocks byBj where j denotes the replication group (j =1,2, . . . , s+1) and denotes the block within the jth replication group (=1,2, . . . , s). Now consider a treatment θ, say, and suppose it occurs in blocksB11, B22, . . . , Bs+1,s+1. Then in D block θ con- tains the treatments 11,22, . . . , s+1s+1. Further, let θ1, θ2, . . . , θs−1 be the treatments that occur together with θ in B11 in D. Since {θ , θ1, θ2, . . . , θs−1} appear all in different blocks in the remainings replication groups (sinceλ=1), it follows then that in D treatment 11 occurs together exactly once with all j (j=2, . . . , s+1;=1,2, . . . , s), but not at all with the s−1 treatments 1(=1). This implies the association scheme
11 12 . . . 1s
21 22 . . . 2s
s+1 1 s+1 2 . . . s+1 s
where treatments in the same row are 1st associates and 2nd associates otherwise, which is, in fact, the GD association scheme, and since the GD association is unique (Shrikhande, 1952), the resulting design is the GD-PBIB(2) design.
Example 5.1 Consider the BIB design D with t =32, b=12, k=3, r= 4, λ=1, given by the plan [denoting the treatments by(x1, x2), x1, x2=0,1,2]:
Block Treatments 11 (0, 0) (0, 1) (0, 2) 12 (1, 0) (1, 1) (1, 2) 13 (2, 0) (2, 1) (2, 2) 21 (0, 0) (1, 0) (2, 0) 22 (0, 1) (1, 1) (2, 1) 23 (0, 2) (1, 2) (2, 2)
Block Treatments 31 (0, 0) (1, 2) (2, 1) 32 (1, 0) (0, 1) (2, 2) 33 (2, 0) (0, 2) (1, 1) 41 (0, 0) (1, 1) (2, 2) 42 (1, 0) (0, 2) (2, 1) 43 (2, 0) (0, 1) (1, 2) The dual D is then given by
Block Treatments (0, 0) 11, 21, 31, 41 (0, 1) 11, 22, 32, 43 (0, 2) 11, 23, 33, 42 (1, 0) 12, 21, 32, 42 (1, 1) 12, 22, 33, 41 (1, 2) 12, 23, 31, 43 (2, 0) 13, 21, 33, 43 (2, 1) 13, 22, 31, 42 (2, 2) 13, 23, 32, 41
The reader can verify easily that this is indeed the plan for a GD-PBIB(2) design with the parameters as indicated. It is isomorphic to plan SR41 given
by Clatworthy (1973).
5.1.2 Method of Differences
We denote and write thet =t1t2 treatments now as follows:
01 02 . . . 0t2
11 12 . . . 1t2
(t1−1)1 (t1−1)2 . . . (t1−1)t2 (5.1) that is, the(i+1)th group in the association scheme for the GD design consists of the treatments(i1, i2, . . . , it2)withi=0,1, . . . , t1−1, these treatments being 1st associates of each other. The treatments in theth column are referred to as treatments of class.
The method of differences as described for BIB designs (Section 3.2) can now be extended for purposes of constructing GD-PBIB(2) designs as stated in the following theorem.
Theorem 5.2 Let it be possible to find s initial blocks B10, B20, . . . , Bs0 such that
i. Each block containsk treatments.
ii. Treatments of class(=1,2, . . . , t2)are representedr times among the s blocks.
iii. Among the differences of type (, ) arising from the s blocks, each nonzero residue mod t1 occurs the same number of times, λ2, say, for all=1,2, . . . , t2.
iv. Among the differences of type (, ) arising from the s blocks, each nonzero residue mod t1 occurs λ2 times and 0 occursλ1 times, say, for all(, )(, =1,2, . . . , t2, =).
Then, developing the initial blocksB10, B20, . . . , Bs0 cyclically, modt1 yields a GD-PBIB(2) design with parameterst =t1t2, b=t1s, k, r =ks/t2, λ1, λ2.
The proof is quite obvious as it is patterned after that of Theorem 3.1. By looking at array (5.1), it is clear that through the cyclic development of the initial blocks, any two treatments in the same row occurλ1 times together in the same block, and any two treatments not in the same row occur λ2 times together in the same block. This then satisfies the association scheme for the GD-PBIB(2) design.
We shall illustrate this with the following example.
Example 5.2 Let t =14, t1=7, t2=2; that is, array (5.1) is 01 02
11 12 21 22 31 32 41 42 51 52 61 62 Let the initial blocks of size 4 be
B10=(01,02,11,12) B20=(01,02,21,22) B30=(01,02,31,32)
To verify conditions 2–4 of Theorem 5.2 we see that treatments of class 1 and 2 occurr=6 times. Further, the differences of type (1, 1) are
B10: 01−11≡6 11−01≡1 B20: 01−21≡5 21−01≡2 B30: 01−31≡4 31−01≡3
and the same holds for differences of type (2, 2); similarly, differences of type (1, 2) are
B10: 01−02≡0 01−12≡6 11−02≡1 11−12≡0 B20: 01−02≡0 01−22≡5 21−02≡2 21−22≡0 B30: 01−02≡0 01−32≡4 31−02≡3 31−32≡0 and the same is true for differences of type (2, 1). Hence λ1=6, λ2=1. The plan for the GD-PBIB(2) design withb=21 blocks is then
(01,02,11,12) (01,02,21,22) (01,02,31,32) (11,12,21,22) (11,12,31,32) (11,12,41,42) (21,22,31,32) (21,22,41,42) (21,22,51,52) (31,32,41,42) (31,32,51,52) (31,32,61,62) (41,42,51,52) (41,42,61,62) (41,42,01,02) (51,52,61,62) (51,52,01,02) (51,52,11,12) (61,62,01,02) (61,62,11,12) (61,62,21,22)
which is the same as plan S13 of Clatworthy (1973). We note that this PBIB design is, of course, resolvable and that it lends itself to two-way elimination of heterogeneity in that each treatment occurs exactly three times in the first two
and the last two positions of the 21 blocks.
5.1.3 Finite Geometries
We shall first consider a method of using a projective geometry, PG(K, pn), to construct GD-PBIB(2) designs. The result can then be applied similarly to a Euclidean geometry, EG(K, pn). (See Appendix B for details about projective and Euclidean geometries). The basic idea in both cases is to omit one point from a finite geometry and all theM-spaces containing that point. The remaining M-spaces are then taken as the blocks of a PBIB design. More specifically we have the following theorem.
Theorem 5.3 Omitting one point from a PG(K, pn) and all the M-spaces containing that point, gives rise to a GD-PBIB(2) design, if one identifies the remaining points with the treatments and the remainingM-spaces with the blocks.
Any two treatments are 1st associates if the line joining them goes through the omitted point, they are 2nd associates otherwise.
Proof The PG(K, pn) has 1+pn+p2n+ · · · +pKn points. Omitting one point leads to
t=pn+p2n+ · · · +pKn
=
1+pn+ · · · +p(K−1)n
pn
≡t1t2 (5.2)
treatments. In the PG(K, pn) there are ψ (M, K, pn) M-spaces, of which ϕ(0, M, K, pn)contain the omitted point. Hence there are
b=ψ (M, K, pn)−ϕ(0, M, K, pn) (5.3) M-spaces left that constitute the blocks, eachM-space (block) containing
k=ψ (0, M, pn)=1+pn+p2n+ · · · +pMn (5.4) points (treatments). Since each point is contained in ϕ(0, M, K, pn) M-spaces and since each line joining a given point and the omitted point is contained in ϕ(1, M, K, pn) M-spaces, it follows that each retained point (treatment) occurs in r=ϕ(0, M, K, pn)−ϕ(1, M, K, pn) (5.5) retainedM-spaces (blocks). Further
n1=ψ (0,1, pn)−2=pn−1=t2−1 (5.6) n2=t−n1−1=
pn+p2n+ · · · +p(K−1)n
pn
=(t1−1)t2 (5.7)
and
λ1 = 0, λ2=ϕ(1, M, K, pn) (5.8)
Thus, we are obviously led to the association scheme for the GD-PBIB(2) design.
Because of the uniqueness of the association scheme, the design constructed in
this manner is a GD-PBIB design.
As an illustration we consider the following example.
Example 5.3 Letp=3, K=2, n=1. The points of the PG(2, 2) are given by triplets(u1, u2, u3)withu1, u2, u3=0, 1 except(u1u2u3)=(000). Suppose we omit the point (100), then the remaining points are: 010, 110, 001, 101, 011, 111, that is,t =6. The lines (i.e.,M=1) not passing through 100 constitute the blocks of the design and are given by
001 010 011
110 001 111
010 101 111
110 101 011 (5.9)
that is, b=4, k=3, r=2. The association scheme is obtained by determining for each point which other points lie on the line going through it and the omitted
point. For example, the only point on the line going through (001) and (100) is obtained from (µ0+ν1, µ0+ν0, µ1+ν0) with µ=ν =1, that is, the point (101). Doing this for all six points leads to the following association scheme:
0th Associate 1st Associates 2nd Associates
010 110 001, 101, 011, 111
110 010 001, 101, 011, 111
001 101 010, 110, 011, 111
101 001 010, 110, 011, 111
011 111 010, 110, 001, 101
111 011 010, 110, 001, 101
This is, of course, the GD-PBIB association scheme if we write the treatments in the following array:
010 110
001 101
011 111
Inspection of the plan (5.9) shows that λ1=0, λ2=1. With suitable labeling
this is plan SR18 of Clatworthy (1973).
5.1.4 Orthogonal Arrays
Orthogonal arrays (for a description see Appendix C) play an important role in the construction of experimental designs (see also Chapters 13 and 14). Of particular interest are orthogonal arrays of strength 2. This is the type we shall employ here to construct GD-PBIB designs. More specifically we shall employ orthogonal arrays OA[N , K, p,2;λ], wherepis prime. Their relationship to the GD-PBIB(2) design is as follows: Replace any integer x appearing in the ith row of the array by the treatment (i−1)p+x. The ith row contains then the treatments
(i−1)p, (i−1)p+1, . . . , (i−1)p+p−1
each occurringr =N /p times since each symbol occurs equally often in each row. The total number of treatments ist =Kpand we havet1=K, t2=p. The columns of this derived scheme form the blocks of the PBIB design, that is, there areb=N blocks, each of sizek=K. Treatments in the same row are 1st associates, that is, each treatment hasn1=p−1 1st associates and since these treatments do not appear in any other row, it follows that λ1=0. Treatments in different rows are 2nd associates, giving n2=(K−1)p. Since we have an OA[b, k, p,2;λ], that is, each possible 2×1 column vector appearingλtimes, we have λ2=λ. This is, of course, the association scheme of a GD-PBIB(2) design with the parameters as stated above.
We illustrate this with the following example.
Example 5.4 Consider the following OA[8, 4, 2, 2, 2]:
A=
0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1
0 0 0 0 1 1 1 1
0 1 1 0 1 0 0 1
This leads to the derived GD-PBIB(2) design (with columns representing blocks):
Block
1 2 3 4 5 6 7 8
0 1 0 1 0 1 0 1
2 2 3 3 2 2 3 3
4 4 4 4 5 5 5 5
6 7 7 6 7 6 6 7
with t =8, b=8, k=4, r=4, λ1=0, λ2=2. This design is isomorphic to
plan SR36 of Clatworthy (1973).
For further specialized methods of constructing GD-PBIB(2) designs the reader is referred to Raghavarao (1971). An essentially complete list of existing plans for GD-PBIB(2) designs is given by Clatworthy (1973) together with references and methods concerning their construction.