Math 558 Lecture #25
Incomplete Block Designs
Incomplete block designs have number of experimental units less than the number of treatments in each block. Let k be the block size which is constant for all blocks and t be the total number of treatments. Then
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k < t for an incomplete block design. Also assume that each treatment is replicated the same number of times that is r. Then N = bk = tr Where N is the total number of plots in the design. Let i and j be two distinct treatments. Then the concurrence λij of i and j is the number of blocks which contain both i and j. 1. Consider a design with the parameters a design with t = 6, r = 2, b = 4 and k = 3. The t treatments can be arranged in 4 blocks of size 3 in the following manner. {1,2,3},{1,4,5},{2,4,6},{3,5,6} Hereλ12 =λ13 =λ14 =λ15 =1andλ16 =0 2. Consider another design with unequal replications for rest of the parameters be same as in the previous design t = 6, b = 4 and k = 3. One of the possible designs can be {1,2,3},{1,3,4},{1,4,5},{1,5,6} Balanced Incomplete Block Design Definition 1 An incomplete-block design with equally replicated treatments is balanced if there is an integer such that λij = λ for all distinct treatments i and j. Such a design is called ‘balanced incomplete-block design’ and is often abbreviated to BIBD. Example: {1,2,4},{2,3,5},{3,4,6},{4,5,7},{5,6,1},{6,7,2},{7,1,3} Here b = t = 7, = 1, r = 3 = k BIBD Example As an example, suppose that a chemical engineer thinks that the time of reaction for a chemical process is a function of the type of catalyst employed. Four catalysts are currently being investigated. The experimental procedure consists of selecting a batch of raw material, loading the pilot plant, applying each catalyst in a separate run of the pilot plant, and observing the reaction time. Because variations in the batches of raw mate- rial may affect the performance of the catalysts, the engineer decides to use batches of raw material as blocks. However, each batch is only large enough to permit three catalysts to be run. Therefore, a randomized incomplete block design must be used. The balanced incomplete block design for this experi- ment, along with the observations recorded, is shown in Table 4.22. The order in which the catalysts are run in each block is randomized. (Montgomery pg 168) Balanced Incomplete Block Design Theorem .1 In a balanced incomplete-block design, λ(t − 1) = r(k − 1) and therefore t − 1 divides r(k − 1). Proof: Treatment 1 occurs in r blocks, each of which contains k − 1 other treatments. So the total number of concurrences with treatment 1 is r(k − 1). However, this number is the sum of the λ1j for j ̸= 1, each of which is equal to λ, so the sum is (t − 1)λ. Balanced Incomplete Block Design for Catalyst Experiment Treatment Block (Batch of Raw Material) 1 2 3 4 yi. (Catalyst) 1 73 74 — 71 218 2 — 75 67 72 214 3 73 75 68 — 216 4 75 — 72 75 222 y.j 221 224 207 218 870 y. BIBD Constructions Let T = {1, 2, 3, ..t} to be arranged in b blocks each of size k. Then, t This means that each subset of the size k is considered as a block. Where bk t−1 r=t=k−1 r(k−1) t−2 λ=t−1=k−2 This construction is usually used when k = 2, k = t − 2, ork = t − 1. Let t = 5 and k = 2 {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, {4, 5} letk=t−1.b=( t )=t=5.Therefore, t−1 r=bk =t(t−1) =t−1=4 tt install.packages("crossdes") find.BIB(5, 5,4 ) b1 1 2 3 − 5 b2 1 2 − 4 5 b3 1 2 3 4 − b4 − 2 3 4 5 b5 1 − 3 4 5 find.BIB(7, 7,3 ) b1 1 2 6 b2 2 3 5 b3 2 4 7 b4 5 6 7 b5 1 3 7 b6 1 4 5 b7 3 4 6 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com