Design A Bib

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Design A Bib

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Since constructions of BIB designs will be discussed thoroughly in Chapter 8, only some combinatorial properties are described here. In a BIB design with parameters v, b, r, k, A, the concurrence (concordance) matrix (see Section 2.4.1) is NN' = (r – A)I, + A191%. Also it can easily be shown that |NN' = rk(r–A)** > 0, (6.0.1) noting that v > k implies r > X. Hence, on account of Corollary 2.3.1(a), Fisher's inequality b > v or, equivalently, r > k holds. When the equalities hold, a special type v,b,r,k and λ are known as the parameters of the BIB design. Counting the number of units used in terms of symbols as well as sets, we get vr = bk. (4.3) Taking all the r sets where a given symbol θ occurs, we

can.form r(k −1) pairs of symbols with θ, and from the definition these pairs must be all pairs of other v − 1 symbols, with θ, each pair occurring in λ sets. Hence r(k − 1) = λ(v − 1). (4.4) By writing the incidence matrix N = (nij) where nij = 1(0) according as the ith symbol occurs (does Advanced Experimental Design Klaus Hinkelmann, Oscar Kempthorne. CHAPTER2 2.1 Consider the BIB design Block Block Block 1 1467 7 2367 13 1249 2 2689 8 2458 14 1569 3 1389 9 3579 15 1368 4 1234 10 1257 16 4678 5 1578 11 2356 17 3458 6 4569 12 3479 18 2789 2.2 2.3 2.4 2.5 (a) Give the parameters of this design. (b) Write out its C matrix. (a) Describe a scenario where you may want to use the design given in Exercise 2.1 but in such a form

that.each treatment From Theorem 3.4.11, a necessary condition for the existence of this design is that 1:2 = 61,12 — z2 has a solution in integers zr, y, z. However, it is not hard to see that no such solution exists and thus the symmetric BIB design with parameters 1; = 43, lc = 7, A = 1 does not exist. We have already seen in this section that a BIB design with parameters n = m2+m.+1= b,r = rn+1 = l<:,,\ = 1 can be constructed provided m is a prime or a prime power. When m is not a prime or a prime power, A block design with identical r, (r< = r) is called equi replicate; a block design with identical kt (k = kt) is called proper. DEFINITION 3.14. A with order t AWbalanced equireplicate, proper and (a,, «2) binary

block.design is called simply a balanced block design of order t (BB(<)). If t = 1, then the BB(1) is a completely balanced block design (CBB). A binary CBB is called a completely balanced incomplete block design (BIB), a (n, n + l)binary CBB is a completely balanced complete block There's a bib for every baby in this collection of 34 cross stitch designs by Linda Gillum.observe that for a B.I.B. design N'N=(r — X) /, + \E„ so that \N'N\ = {r+A(v 1)} (r A)'i = rk(r— A)»» ^ 0. This means that N'N is non singular and hence of rank v. But rank (N'N) = rank (N) < b. This proves that b ^ v. Even a sharper inequality b ^ v + r — k is true. This (v \ w t — 1 ) (/□—&)>() which implies r

—v^r—k,.i.e. 6 > v + r — k. The class of designs in which ft=v is called symmetrical B.I.B. designs. In these designs, therefore, the number of treatments common between any two Nonetheless, a certain degree of balance and resulting simplicity in calculations and interpretation of results can be achieved by using a BIB design. A BIB design for a treatments with b blocks, each of size c < a, has the following properties: (a) No treatment is assigned more than once in any block. If n ij denotes the number of replicates of the ith treatment in the jth block, then each nij = 0, 1. (Such a design is called a binary design.) (b) Each treatment is replicated the same number of (We added the qualification noncrossover to

emphasize.that the incomplete block designs we are referring to are the traditional ones which have been used in agriculture, for example, for many years. In these designs each experimental unit in a block receives a single treatment.) Patterson (1952) noted that balanced crossover designs can be constructed by starting with a balanced incomplete block design (BIB design). In a BIB design each treatment is replicated the same number If the precision of the design does not require as many as bk blocks, there would be no need to use that many. For example, if a practical size difference in recipes could be detected with less than 20 subjects and 10 replicates of each treatment level, in the taste test panel, perhaps a BIB design could

be.found with less than 20 blocks. If r is the number of times a treatment level is replicated in an incomplete block design, λ is the number of times each treatment level occurs with every 

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