lect21 - IE 410 Lecture 21: Nested and Hierarchical Designs...

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Unformatted text preview: IE 410 Lecture 21: Nested and Hierarchical Designs and Rules for Expected Mean Squares Two Stage Nested Designs Consider the following: Example: A Company deals with 3 suppliers of chemicals. Each Company ships chemicals in Batches. We suspect variation in chemicals is messing up our production and thus do an experiment with two factors: Factor A: Supplier (levels = 1, 2, and 3) Factor B: Batch (levels 1, 2, 3, and 4) In this experiment it seems that Factor A is fixed while Factor B is random. For each combination of batch and supplier we test for purity of the chemical three times (in order to have replicates). Thus the data looks like this: Supplier 1 Supplier 2 Supplier 3------------------------------------- 1 1 2 1 -1 -2 4 0 -3 --------------------------------------2 0 -2 2 -3 4 -4 2 2 Batches --------------------------------------2 -1 1 3 0 0 -1 1 -2 2------------------------------------- 1 0 3 4 4 3 2 0 2 1------------------------------------- so a=3 b=4 n=3 This design looks like a two factor factorial experiment, BUT IT IS NOT Another way to display the data is like this: Supplier 1| Supplier 2| Supplier 3----------|-----------|----------- Bch 1 2 3 4| 1 2 3 4| 1 2 3 4----------|-----------|----------- 1 -2 -2 1| 1 0 -1 0| 2 -2 1 3 Rp -1 -3 0 4|-2 4 0 3| 4 0 -1 2 0 -4 1 0|-3 2 -2 2| 0 2 2 1 Why isn't this a factorial design? Because there are really 12 batches involved. That is, batch 1 under supplier 1 is not the same as batch 1 under supplier 2. The data could just as well be written this way: Supplier 1| Supplier 2| Supplier 3----------|-----------|----------- Bch 1 2 3 4| 5 6 7 8| 9 10 11 12----------|-----------|----------- 1 -2 -2 1| 1 0 -1 0| 2 -2 1 3 Rp -1 -3 0 4|-2 4 0 3| 4 0 -1 2 0 -4 1 0|-3 2 -2 2| 0 2 2 1 This is a "Nested Design " We say that factor B is "nested within factor A" The example above is a "balanced: nested design” which means that Factor B is at b levels UNDER EACH level of factor A. We shall deal only with balanced nested designs. Linear Statistical Model Y ijk = μ + τ i + β j(i) + ε (ij)k where the indexing notation is important to understand: j(i) means "j within i" or "j nested under i" Note also that the replicates index k is nested within combinations of A and B. In factorial designs, replicates are also nested in this way. In fact replicates are, in truth, always nested like this. For the fixed effects model we add the following constraints: i ∑ τ i = 0 j ∑ β j(i) = 0 for all i Decomposition of Sums of Squares SST can be decomposed as follows: SST = i ∑ j ∑ k[ Y ijk- Y ...] 2 SSA = bn i ∑ [ Y i..- Y ...] 2 SSB(A) = n i ∑ j ∑ [ Y ij.- Y i.. ] 2 SSE = i ∑ j ∑ k ∑ [Y ijk- Y ij....
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This note was uploaded on 08/06/2008 for the course IE 410 taught by Professor Storer during the Fall '04 term at Lehigh University .

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lect21 - IE 410 Lecture 21: Nested and Hierarchical Designs...

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