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blockdesigns2009

# blockdesigns2009 - Statistics 514 Block Designs Lecture 6...

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Statistics 514: Block Designs Lecture 6: Block Designs Montgomery: Chapter 4 Page 1

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Statistics 514: Block Designs Nuisance Factor (may be present in experiment) Has effect on response but its effect is not of interest If unknown Protecting experiment through randomization If known (measurable) but uncontrollable Analysis of Covariance (Chapter 15 or 14 Section 3) If known and controllable Blocking Page 2
Statistics 514: Block Designs Penicillin Experiment In this experiment, four penicillin manufacturing processes ( A , B , C and D ) were being investigated. Yield was the response. It was known that an important raw material, corn steep liquor, was quite variable. The experiment and its results were given below: blend 1 blend 2 blend 3 blend 4 blend 5 A 89 1 84 4 81 2 87 1 79 3 B 88 3 77 2 87 1 92 3 81 4 C 97 2 92 3 87 4 89 2 80 1 D 94 4 79 1 85 3 84 4 88 2 Blend is a nuisance factor, treated as a block factor; (Complete) Blocking: all the treatments are applied within each block, and they are compared within blocks. Advantage: Eliminate blend-to-blend (between-block) variation from experimental error variance when comparing treatments. Cost: degree of freedom. Page 3

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Statistics 514: Block Designs Randomized Complete Block Design b blocks each consisting of (partitioned into) a experimental units a treatments are randomly assigned to the experimental units within each block Typically after the runs in one block have been conducted, then move to another block. Typical blocking factors: day, batch of raw material etc. Results in restriction on randomization because randomization is only within blocks. Data within a block are related to each other. When a = 2 , randomized complete block design becomes paired two sample case. Page 4
Statistics 514: Block Designs Statistical Model b blocks and a treatments Statistical model is y ij = μ + τ i + β j + ij i = 1 , 2 , . . . , a j = 1 , 2 , . . . , b μ - grand mean τ i - i th treatment effect β j - j th block effect ij N(0 , σ 2 ) The model is additive because within a fixed block, the block effect is fixed; for a fixed treatment, the treatment effect is fixed across blocks. In other words, blocks and treatments do not interact. parameter constraints: a i =1 τ i = 0; b j =1 β j = 0 Page 5

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Statistics 514: Block Designs Estimates for Parameters Rewrite observation y ij as: y ij = y .. + ( y i. - y .. ) + ( y .j - y .. ) + ( y ij - y i. - y .j + y .. ) Compared with the model y ij = μ + τ i + β j + ij , we have ˆ μ = y .. ˆ τ i = y i. - y .. ˆ β j = y .j - y .. ˆ ij = y ij - y i. - y .j + y .. Page 6
Statistics 514: Block Designs Sum of Squares (SS) Can partition SS T = ∑ ∑ ( y ij - y ..

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blockdesigns2009 - Statistics 514 Block Designs Lecture 6...

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