We shall now turn to the rather large class of cyclic PBIB designs we introduced in Section 4.6.12. These designs are quite useful since (1) they are easy to construct, namely as the name suggests through cyclic development of initial blocks, (2) they exist for various combinations of design parameters, and (3) they are easy to analyze, that is, their analytical structure is easy to derive.
5.3.1 Construction of Cyclic Designs
As mentioned earlier, the construction of cyclic designs is based on the cyclic development of a set of initial blocks. We can distinguish basically between four types of cyclic designs according to whether the number of blocksbis (1)b=t, (2)b=st, (3) b=t/d, or (4) b=t (s+1/d), whered is a divisor of t. It is then obvious that
For (1) we need one initial block of sizek;
For (2) we needs distinct (nonisomorphic) initial blocks of sizek;
For (3) we need one initial block of sizek such that, if cyclically developed, block b+1=t/d+1 is the same as the initial block and each treatment is replicated r times. This means that if the initial block contains treat- menti, it also must contain treatmentsi+b, i+2b, . . . , i+(d−1)b. This implies thatd is also a divisor of k, sayk/d=k. Then anyk treatments i1, i2, . . . , ik can be chosen and the remaining treatments in the initial block are determined;
For (4) we combine initial blocks from type (1) ifs=1 or (2) if s >1 with an initial block from type (3).
The choice of initial blocks for (1), (2), and hence (4) above is quite arbitrary since any choice will lead to an appropriate design. There is, however, one other consideration that plays an important role in choosing initial blocks and that is the efficiency E of the resulting design. Efficiency here is defined in terms of the average variance of all treatment comparisons (see Section 1.10), and high efficiency is closely related to a small number of associate classes. Whereas the number of associate classes for a cyclic design is, in general,m=t/2 fort even andm=(t−1)/2 fort odd, this number can sometimes be reduced by a proper choice of initial blocks, in some cases even to m=1, that is, BIB designs, or m=2, that is, PBIB(2) designs. For example, considert =6, k=3, r=3, b= 6: If the initial block is (1, 2, 3), then the plan is
1 2 3
2 3 4
3 4 5
4 5 6
5 6 1
6 1 2
and by inspection we can establish the following association scheme:
0th Associate 1st Associates 2nd Associates 3rd Associates
1 2, 6 3, 5 4
2 3, 1 4, 6 5
3 4, 2 5, 1 6
4 5, 3 6, 2 1
5 6, 4 1, 3 2
6 1, 5 2, 4 3
with λ1=2, λ2=1, λ3=0, and
P1=
0 1 0 0
1 0 1 0
0 1 0 1
0 0 1 0
P2=
0 0 1 0
0 1 0 1
1 0 1 0
0 1 0 0
P3=
0 0 0 1
0 0 2 0
0 2 0 0
1 0 0 0
Hence we have a PBIB(3) design. If, on the other hand, the initial block is (1, 2, 4), then the resulting plan is
1 2 4
2 3 5
3 4 6
4 5 1
5 6 2
6 1 3
and, by inspection, we can speculate on the association scheme as being 0th Associate 1st Associates 2nd Associates
1 2, 3, 5, 6 4
2 3, 4, 6, 1 5
3 4, 5, 1, 2 6
4 5, 6, 2, 3 1
5 6, 1, 3, 4 2
6 1, 2, 4, 5 3
withλ1=1, λ2=2, and P1=
0 1 0
1 2 1
0 1 0
P2=
0 0 1
0 4 0
1 0 0
Hence we have in fact a PBIB(2) design. For the first design we obtainE= 0.743 [see (5.14)], whereas for the second design we findE=0.784. Hence the second design is slightly preferable from an efficiency point of view, and this design is therefore listed by John, Wolock, and David (1972). One can convince oneself that no better cyclic design for this combination of parameters exists.
The association scheme for the second design is, indeed, the association scheme for a cyclic PBIB(2) design as discussed in Section 4.6.4, except that we have to relabel the treatments as 0,1, . . . ,5. We then have d1=1, d2=2, d3= 4, d4=5, e1=3, α=2, β=4.
An extensive list of initial blocks for cyclic designs for various parameters with 6≤t≤30, k≤10, r≤10, and fractional cyclic designs for 10≤t ≤60,3≤ k≤10 is given by John, Wolock, and David (1972). A special group in this collection of plans are those with k=2, which are often referred to as paired comparison designs.
5.3.2 Analysis of Cyclic Designs
The general methods of analyzing incomplete block designs (see Chapter 1) apply, of course, also to the cyclic designs just described, or even more specif- ically, the methods of analyzing BIB designs (where appropriate) or PBIB(m)
designs can be used. However, as pointed out earlier, for many cyclic designs the number of associate classes,m, is rather large and hence the system of equations (4.34) becomes rather large. Instead, another method of solving the RNE
Cτˆ =Q
for cyclic designs can be used. This method is interesting in itself as it makes use of the particular construction process that leads to a circulant matrixC and hence to a circulant matrixC−, the elements of which can be written out explicitly (Kempthorne, 1953).
Let us denote the first row of C by (c1, c2, . . . , ct). The second row is then obtained by shifting each element of the first row one position to the right (in a circulant manner), and so forth so thatC is determined entirely by its first row.
The same is true for C−, so all we need to know in order to solve the RNE and to obtain expressions for variances of estimable functions λτ, is the first row of C−, which we denote by c1, c2, . . . , ct. Let dj(j =1,2, . . . , t) denote the eigenvalues ofC where (Kempthorne, 1953)
dj = t =1
c(−1)(j−1)θ (5.10)
andθ =2π/t. Note thatd1=0; henced2, d3, . . . , dt are the nonzero eigenvalues of C. Then, as shown by Kempthorne (1953), the elements ci(i=1,2, . . . , t) are given as
c1= 1 t
t j=2
1 dj
ci= 1 t
t j=2
cos(j−1)(i−1)θ dj
(i=2,3, . . . , t)
(5.11)
Expressions (5.10) and (5.11) can actually be simplified somewhat. Because of the construction of the designs and the resulting association scheme, we have for theλ1i inN N
λ12=λ1t, λ13=λ1,t−1, λ14=λ1,t−2, . . . ,
λ1,t /2=λ1,t /2+2 fort even λ1,(t+1)/2=λ1,(t+3)/2 fort odd and hence we have forC
c1=rk−1 k c2=ct = −λ12
k
c3=ct−1= −λ13 k c4=ct−2= −λ14
k
c(t+1)/2=c(t+3)/2= −λ1,(t+1)/2
k fort odd
or
c(t+2)/2= −λ1,(t+2)/2
k fort even
As a consequence, also the maximum number of differentci values is (t+1)/2 fort odd or(t+2)/2 fort even. With theci values given in (5.11) we can find the variances of treatment comparisons in general via comparisons with treatment 1 as follows:
var(τi−τi)=var(τ1−τi−i+1)
=2(c1−ci−i+1)σe2 fori> i (5.12) This is, of course, a consequence of the circulant form of C−. The average variance of all such treatment comparisons then is
av. var= 1 t (t−1)
ii i=i
var(τi−τi)
= 1 t−1
i=1
var(τ1−τi)
and using (5.12),
av. var= 2 t−1
i=1
(c1−ci)σe2
= 2 t−1
tc1−
t i=1
ci
σe2
= 2t
t−1c1σe2 (5.13)
since t
i=1ci=0 as can be verified directly by using (5.11). From (5.12) it follows then that the efficiency factor for the treatment comparison between the ith and the ith treatment is
Eii = 2σe2/r
2(c1−ci−i+1)σe2 = 1 r(c1−ci−i+1)
and from (5.13) that the overall efficiency factor is E= 2σe2/r
2tc1σe2/(t−1) = t−1
rtc1 (5.14)