9.1 INTRODUCTION

The construction and looking into the existence or nonexistence of designs is as important as providing the analysis for a given class of designs. Finite geometries, methods of differences, patchwork methods, and recursive methods are commonly used procedures for constructing designs. In Section 9.2, a brief review of Galois fields (GF) is provided, and in subsequent sections, some construction methods of the previously introduced designs are considered.

9.2 GALOIS FIELDS

Given a set of elements, F, a system is said to be a field, if the following conditions are satisfied:

1. a, b ∈ F ⇒ a + b ∈ F,  a · b ∈ F.
2. a + (b + c) = (a + b) + c,  a · (b · c) = (a · b) · c.
3. a + b = b + a,  a · b = b · a.
4. ∃0, such that a + 0 = a,   ∃1, such that a · 1 = a.
5. ∃ (-a), such that a + (-a) = 0.
6. ∀ a ≠ 0,   ∃ (a^– 1), such that a · (a^- 1) = 1.
7. a · (b + c) = a · b + a · c.

The number of elements, s, in a field is finite if s is a prime or prime power. These elements are called Galois fields and are written as GF(s). The quantity a is said to be congruent to b modulus s and is written a ≡ b(mod  s), iff, a – b is divisible by s. If s is a prime p, the set of integers {0, 1, …, p - 1} is a field if these are multiplied or added in the usual manner and are reduced by mod p. For example, with s = 5, a field of 5 elements is {0, 1, 2, 3, 4}. Addition and multiplication of these elements are shown by the addition and multiplication tables given in Tables 9.2.1 and 9.2.2.

The entries in Tables 9.2.1 and 9.2.2 are, for example, 3 + 4 = 7 ≡ 2(mod 5) and 4 · 2 = 8 ≡ 3(mod 5). The entry in row 4, column 5 in Table 9.2.1 is 2, and the entry in row 5, column 3 is 3 in Table 9.2.2.

If s = pⁿ, where n is a positive integer and p is a prime, the field elements are given as α₀ = 0,  α₁, …, α_s–1 where for multiplication purposes α_i = x^i–1 and for addition purposes , where a_i’s are not all zero and belong to GF(p), and x is a primitive root satisfying an nth degree minimal function given in Table 9.2.3 that is equated to zero.

The multiplicative and addition form of GF(3²) are

9.3 Generalized Youden Designs

In Definition 5.6.4, a GYD was introduced. From that definition, it is clear that a GYD is an arrangement of v symbols in a k × b array such that

1. Every symbol occurs m or m + 1 times in each row, as well as either n or n + 1 times in each column where m = ⌊b/v⌋ and n = ⌊k/v⌋.
2. Every symbol occurs exactly r times.
3. Every two distinct symbols occur together λ₁ times in the same row and λ₂ times in the same column.

Letting L_ij to be a v × v Latin square for i = 1, 2, …, m and j = 1, 2, …, n, clearly, the following theorem holds:

Again,

9.4 Williams’ Balanced Residual Effects Designs

CODWR were first constructed systematically by Williams (1949) using one or two Latin squares depending on the number of treatments v is even or odd. In this section, construction methods of CODWR with k = v will be given.

Let M be a module of v elements. If n_i is a k-component vector such that n′_i = (a_i1, a_i2, …, a_ik), then denote by n′_i,θ the column vector (a_i1 + θ, a_i2 + θ, …, a_ik + θ)′, where a_i1, a_i2, …, a_ik, θεM. The following theorem provides a method of constructing BRED, using the method of differences.

Specializing Theorem 9.4.1 to the cases of v even and odd with v = k, the following corollaries can be obtained.

Corollary 9.4.1.1 A BRED with parameters v = k = b,  t = 1,  λ = v,  μ = v - 2,  υ = 1 always exists when v is even.

Proof Let M = {0, 1, …, v - 1} and v = 2m. Further, let n′₁ = {0, 2m - 1, 1, 2m - 2, …, m + 1, m - 1, m}. It is easy to verify that n₁ satisfies all the requirements of Theorem 9.4.1, and the columns given by n′_1,θ for θ = 0, 1, …, v - 1 provide the required design.

Corollary 9.4.1.2 A BRED with parameters v = k,  b = 2v,  t = 2,  λ = 2v,  μ = 2(v - 2),  υ = 2 always exists when v is odd.

Proof Let v = 2m + 1 and M = {0, 1, …, v - 1}. Further, let n′₁ = {0, 2m, 1, 2m - 1, …, m + 1, m} and n′₂ = {m, m + 1, …, 2m - 1, 1, 2m, 0}. Then n₁ and n₂ satisfy the requirements of Theorem 9.4.1, and the columns given by n_1,θ, n_2,θ (θ = 0, 1, …, v - 1) provide the required design. Note that n₂ is n₁, written in reverse order.

It is interesting to see whether BREDs with v = k = b, t = 1, λ = v, μ = v - 2, υ = 1 exist for odd v. An exhaustive count shows that such BREDs do not exist for v = 3, 5, and 7. According to Hedayat and Afsarinejad (1978), the design for v = 9 as given in (9.4.1) was found by K.B. Mertz by using a computer, and the design for v = 15 as given in (9.4.2) was found by E. Sonneman by mimicking the design found by Mertz:

(9.4.1)

(9.4.2)

For v = 21, the design (9.4.3) was given by Hedayat and Afsarinejad (1975) based on the work of Mendelsohn (1968):

(9.4.3)

Hedayat and Afsarinejad (1978) gave the design for v = 27 as shown in (9.4.4) based on the work of Keedwell (1974):

(9.4.4)

9.5 Other Balanced Residual Effects Designs

The construction of BREDs with k < v, based on the work of Patterson (1952), will be discussed along with other miscellaneous constructions.

Balanced Arrays (or B-arrays) are widely used in factorial experiments, and a class of them can be used as BREDs. The definition of B-arrays is given as follows.

The idea of B-arrays was originally due to Rao (1946). A detailed treatment of orthogonal arrays can be found in Raghavarao (1971, Chapters 2 and 3).

If a B-array exists of strength 2 with λ-parameters satisfying

then the array can be used as a BRED. Such a BRED can be used in sequential experimentation, and the experiment can be terminated or continued at any period.

9.6 Combinatorially Overall Balanced Residual Effects Designs

Two series of COBREDs will be given in the following theorems.

9.7 Construction of Treatment Balanced Residual Effects Designs

Following Section 8.5, we take ∞ to be the control treatment and 0, 1, …, v - 1 to be the active treatments. Let M be the module of v symbols, {0, 1, …, v - 1}. Let ∞ + θ = ∞, ∞ - θ = ∞, θ - ∞ = - ∞ for every θ ∈ M. Analogous to Theorem 9.4.1 for constructing BRED, we have Theorem 9.7.1.

As an example, let v = 3 active treatments and one control treatment ∞ and let a = 1. The three column vectors n′₁ = (∞, 0, 1), n′₂ = (0, ∞, 1), and n′₃ = (0, 1, ∞) satisfy the requirements of Theorem 9.7.1, and the design is as follows:

If there exists a BRED for v treatments in b units and k(<v) periods exist, then we can get a TBRED in k periods and bk subjects. We repeat the BRED k times and replace row i in the ith repetition by the control treatment ∞ for i = 1, 2, …, k. The parameters of this TBRED can easily be obtained. Using the columns n′₁ = (0, 1, 6), n′₂ = (0, 2, 5), and n′₃ = (0, 4, 3) to get a BRED from Theorem 9.5.1, we can construct a TBRED for 7 active treatments and one control treatment ∞ in 3 periods and 63 units. The sets needed to get the design following Theorem 9.7.1 are (∞, 1, 6), (∞, 2, 5), (∞, 4, 3), (0,∞, 6), (0, ∞, 5), (0, ∞, 3), (0, 1, ∞), (0, 2, ∞), and (0, 4, ∞).

Given a BRED for v treatments in b units and k (<v) periods, we can get a TBRED in k + 1 periods. To construct the design, we replicate the BRED k times and add an extra period in each replication of the control treatment ∞. The extra period comes before the first period of BRED in the first replication and after the last period in the last replication. It is inserted between the (i - 1) and i periods of BRED in the ith replication for i = 2, 3, …, k - 1.

From the following BRED

for 3 treatments in 2 periods and 6 units, we get the TBRED

for 3 active and one control treatment in 3 periods on 18 units.

9.8 Some Construction of PBCOD (m)

We need to use PBCOD (m), when the number of treatments is very large and it is difficult to use many periods for experimentation.

If a PBIB design for v treatments in block size k exists, we can get a PBCOD (m) for v treatments by forming BRED for the treatments in each block of the PBIB design.

If v = n(n - 1)/2, the treatments can be arranged in a n × n array with empty diagonal and writing the v treatments above the diagonal and symmetrically filling the bottom of the diagonal. The treatments θ and ϕ are first associates if they occur in the same row or column of the n × n array, otherwise, second associates. This is called triangular association scheme. For example, with 6 treatments, we arrange them in the array

and understand

and

Using the triangular association scheme, we get a PBCOD (2) by forming n BREDs using the symbols in each row of the n × n array.

If v = n², we get L₂ association scheme by writing the v symbols in a n × n array and defining (θ, ϕ) = 1 if θ, ϕ occur in the same row or column, otherwise, second associates. For v = 9, we form

and get PBCOD (2) by forming six BREDs using the symbols in each row and each column.

Given two arrays

we define the symbolic Kronecker product of D1, D2, denoted by D₁ ⊗ D₂ as

For i = 1, 2, if D_i is two BREDs or PBCOD (m) with v_i treatments in k_i periods with b_i units, it can be verified that D₁ ⊗ D₂ is a BRED or PBCOD (m) for v₁, v₂ treatments in k_1, k₂ periods with b_1, b₂ units.

Let v = 2n and the treatments are arranged in a 2 × n array:

(1, 1) (1, 2) … (1, n).

(2, 1) (2, 2) … (2, n).

We form two BREDs D_i with n treatments (i, 1), (i, 2), …, (i, n) for i = 1, 2. We add an extra period by putting (2, j) if the last period of D₁ is (1, j) and (1, j) if the last period of D₂ is (2, j) for j = 1, 2, …, n. The resulting design is PBCOD (m) with rectangular association scheme

9.9 Construction of Complete Set of Mols and Patterson’s Bred

We defined a complete set of MOLS in Section 5.11. In this section, we will discuss their construction method, and using them, we will give a construction method of Patterson’s BRED, whenever v is a prime or prime power.

Let v be a prime or prime power, and let α₀ = 0, α₁, α₂, ..., α_v–1 be the elements of GF(v). It is known that the v - 1 arrays L_i of v × v dimensionality whose (j, j′) cell is filled by provide a complete set of MOLS of order v (see Raghavarao, 1971, Chapter 1). If we call the columns of L_i to be ℓ_i0, ℓ_i1, ℓ_i2, …, ℓ_i(v-1), the columns of L_i+1 are ℓ_i0, ℓ_i2, ℓ_i3, …, ℓ_i(v–1), ℓ_i1.

For example, let v = 5, and consider the elements of GF(5) given by α₀ = 0, α₁ = 1, α₂ = 2, α₃ = 4, α₄ = 3.

The 4 MOLS of order 5 are

To construct BREDs of Patterson, interchange the rows and columns of a complete set of MOLS, and write them side by side to form a v × v(v - 1) array and take the first k rows to form BRED, for required k. To get a BRED for five treatments in three periods, we use the first three rows of the following array:

9.10 Balanced Circular Arrangements

Given v, we arrange the v symbols in a circular form filling v(v - 1) positions such that every p symbols θ and ϕ occur next to each other once. Such arrangement always exists for any v, and we will discuss them in this section.

First, we assume v is even, say v = 2m, and we construct Williams’ BRED for 2m elements and wrap the columns in a circle without any repetition of a symbol and filling v(v - 1) positions. Proceed wrapping columns i, m + i for i = 1, 2, …, m ignoring repetitions to get the required design.

For v = 6, the BRED

gives the circular arrangement

For odd v = 2m + 1, we first prepare the circular arrangement in 2 m symbols and add (2m + 1)th symbol by replacing any 1 by {(2m + 1),1}, 2 by {(2m + 1), 2}, …, 2m by {(2m + 1),(2m)}.

From the circular arrangement given previously for v = 6, we get the arrangement for 7 symbols as follows:

If we want an arrangement where each treatment gives residual effect on itself, we can repeat each of the treatment i, for i = 1, 2, …, v in the circular arrangement in v(v–1) symbols.

9.11 CONCLUDING REMARKS

Some general construction methods of cross-over designs, discussed in the earlier chapter, were presented in this chapter. A catalogue of cross-over designs is given by Patterson and Lucas (1962). In practical situations, one may construct a suitable cross-over design, not necessarily a BRED or PBCOD (m), and analyze it. Recently, one of the authors has come across a situation in which a placebo (p) and two active treatments (A and B) were used as CODWR in a two sequence, five period design given by

P P

A P

P B

B P

P A

The analysis for such ad hoc designs can be easily constructed by the methods of Section 6.2.

The nonexistence of BREDs is not systematically studied. Isolated cases were discussed in Patterson (1952) and Pigeon and Raghavarao (1987).

There is a need to expand classes of PBCOD (m) by developing designs based on other association schemes. TBREDs were constructed by Pigeon (1984) following analogous methods discussed in Section 9.4. He gave a list of parameter sets whose solutions are unknown. The construction of those unsolved cases will present interesting venues for research in this area.