2.7 SOME SPECIAL ARRANGEMENTS
2.7.2 Grouped Blocks
Designs of this type are characterized by the fact that the firstb blocks form a group ofα replicates of the t treatments, so do the nextb blocks, and so on, these groups of blocks being thereby orthogonal to treatments. If there areq of such groups, thenqb=bandqα =r.
Following Shrikhande and Raghavarao (1963) we formalize this in the fol- lowing definition.
Definition 2.4 A BIB design with parameterst,b,k,r,λthat can be arranged in groups of blocks, each group containing α replicates of the t treatments, is said to be α-resolvable.
This concept ofα-resolvability is an extension of the concept of resolvability given by Bose (1942), which refers to the caseα=1.
The following design is an example of a 3-resolvable design.
Example 2.4 Consider the following design with parameterst =t, b=10, k=3,4=6, λ=3:
Replication
Block Treatment Group
1 1 2 3
2 1 2 5
3 1 4 5 I
4 2 3 4
5 3 4 5
6 1 2 4
7 1 3 4
8 1 3 5 II
9 2 3 5
10 2 4 5
We findb=5, q =2, α=3.
Forα-resolvable BIB designs inequality (2.6) can be improved as follows.
Theorem 2.2 If a BIB design with parameterst,b,k,r,λis α-resolvable, then the following inequality must hold:
b≥t +q−1 (2.24)
Proof Since for an α-resolvable BIB design there existq−1 linearly inde- pendent relationships between the columns ofN, we have
t =rank(N N)=rank(N)≤b−q+1
which implies (2.24).
Definition 2.5 An α-resolvable BIB design is called affine α-resolvable if any pair of blocks in the same replication group haveq1 treatments in common and if any pair of blocks from two different replication groups haveq2treatments
in common.
For affineα-resolvable BIB designs the following results are due to Shrikhande and Raghavarao (1963).
Theorem 2.3 In an affineα-resolvable BIB designk2/t is an integer.
Proof Without loss of generality consider the first block in the first replication group,B11say. By definition,B11hasq2treatments in common with every block in the remainingq−1 replication groups, each group consisting ofb/q blocks.
Hence
(q−1) b
q q2=k(r−α)
which, sinceαq=r, yieldsq2=k2/t. Sinceq2is an integer,k2/t is an integer.
Theorem 2.4 For an affineα-resolvable BIB design, the equality
b=t+q−1 (2.25)
holds.
Proof We shall give here the proof for α=1 following Bose (1942) and refer to Shrikhande and Raghavarao (1963) for the general case.
Consider a resolvable BIB design. We then have
t =bk b=br (2.26)
since thebblocks are divisible intor sets ofb blocks each, each set containing each treatment exactly once. Let the blocks belonging to theith set,Si, be denoted by Bi1, Bi2, . . . , Bib for i=1,2, . . . , r. Consider now a particular block,B11
say. Let ij be the number of treatments common to blocks B11 and Bij(i = 2, . . . , r;j =1,2, . . . , b). Further, letmdenote the average ands2the variance of theb(r−1)numbersij.
Now each of the k treatments in B11 is replicated r times, and since the design is resolvable, that is, 1-resolvable, a given treatment inB11will occur in exactly one block each in ther−1 other sets,S2, S3, . . . , Sr. This is true for all k treatments inB11, and hence
ij
ij =k(r −1) (2.27)
and therefore
m= k b = k2
t (2.28)
using (2.26) for the last expression.
Further, the k(k−1)/2 pairs of treatments in B11 each occur λ−1 times together in the setsS2, S3, . . . , Sr. Hence
1 2
ij
ij(ij−1)= 1
2(λ−1)k(k−1) and therefore, using (2.27)
ij
2ij =k[r−1+(λ−1)(k−1)] (2.29) Since
λ= r(k−1)
t−1 = r(k−1) bk−1 we rewrite (2.29) as
ij
2ij = k[(bk−1)(r−k)+r(k−1)2] bk−1
Then we find after some algebra, using (2.28), (2.29), and (2.26),
s2=
ij
(ij−m)2 b(r−1)
=
ij
2ij b(r−1)−m2
=k(t−k)(b−t−r+1)
b2(r−1)(t−1) (2.30)
Since s2≥0, (2.30) implies, of course, the earlier result (2.24), with q=r for α=1.
We now consider an affine resolvable BIB design. Then, by definition,ij =q2 for all i, j. Hences2 of (2.30) equals zero, which implies (2.25) withq=r for
a 1-resolvable BIB design.
Corollary 2.1 If the parameters of a BIB design satisfy (2.25) andk2/t is not an integer, then the design is not resolvable.
Example 2.5 An example of an affine resolvable BIB design is given by the following design with parameters t =8, b=14, k=4, r=7, λ=3, b= 2, q =7, q1=0, q2=2, α=1:
Replication
Block Treatments Group
1 1 2 3 4
2 5 6 7 8 I
3 1 2 7 8
4 3 4 5 6 II
5 1 3 6 8
6 2 4 5 7 III
7 1 4 6 7
8 2 3 5 8 IV
9 1 2 5 6
10 3 4 7 8 V
11 1 3 5 7
12 2 4 6 8 VI
13 1 4 5 8
14 2 3 6 7 VII
A natural model for the analysis of such a design is
y =µI+Xττ+Xηη+Xβ∗β∗+e (2.31) or
yij =µ+τi+ηj+βj ∗ +eij
whereηj is the (fixed) effect of thejth replication group(j =1,2, . . . , q)and βj ∗ is the effect of theth block in thejth group (=1,2, . . . , b). This brings out the point that the blocks are nested in the replication groups and, hence, in the case of theβj ∗ being random variables (as assumed for the combined analysis), thatσβ2∗ now measures the variability of the blocks within groups. Sinceb< b,
it can then usually be concluded thatσβ2∗ for this design will be smaller than the correspondingσβ2 in the BIB design without grouping of blocks. Consequently, only the part of the analysis of variance that deals with the estimation of σβ2∗ (to be used for the combined analysis, in particular, using the Yates estimator) is affected when using this design and model (2.31). This is shown in Table 2.3, where Rj refers to the total of the jth replication group, and Bj refers to the total of theth block within the jth replication group.
2.7.3 α-Resolvable BIB Designs with Replication Groups Across Blocks The designs of this type combine the properties of the designs discussed in Sections 2.7.1 and 2.7.2 and hence lead to a possible reduction in bothσe2andσβ2. Example 2.6 As an example we consider the following BIB design with parameterst =6,b=15,k=4, r=10, λ=6:
Replication Groups Across Blocks
Replication
Block I II Group
1 1 2 3 4
2 1 5 4 6 I
3 2 3 5 6
4 1 2 3 5
5 1 2 4 6 II
6 3 5 4 6
7 1 3 2 6
8 4 5 1 3 III
9 4 6 2 5
10 2 4 1 5
11 5 6 1 3 IV
12 3 6 2 4
13 5 6 1 2
14 4 6 1 3 V
15 3 4 2 5
We find that r=5, s=2, α=2, q=5 in the notation of Sections 2.7.1
and 2.7.2.
A natural model for this type of design is
y=µI+Xττ +Xγγ +Xηη+Xβ∗β∗+e (2.32)
Table 2.3 Analysis of Variance for Resolvable BIB Design
Source d.f. SS E(MS)
Xτ|I,Xη t−1 1 r
i
Ti2−G2 n
Xη|I,Xτ q−1 1 bk
j
Rj2−G2 n
Xβ∗|I,Xτ,Xη b−q Difference σe2+n−t−k(q−1) b−q σβ2∗
I|I,Xτ,Xη,Xβ∗ n−t−b+1
ij
yij 2 −
i
τiQi
−1 k
j
Bj 2 σe2
Total n−1
ij
yij 2 −G2 n
or
yij u=µ+τi+γj+η+βu∗ +eij u
where all the parameters are as previously defined.
The analysis of variance associated with (2.32) follows easily from those presented in Tables 2.2 and 2.3 with the partitioning of the total d.f. as given in Table 2.4.
Table 2.4 Outline of Analysis of Variance for Model (2.32)
Source d.f.
Xτ|I,Xγ,Xη t−1 Xγ|I,Xτ,Xη s−1 Xη|I,Xτ,Xγ q−1 Xβ∗|I,Xτ,Xγ,Xη b−q I|I,Xτ,Xγ,Xη,Xβ∗ n−t−b−s+2
Total n−1
2.8 RESISTANT AND SUSCEPTIBLE BIB DESIGNS