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

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IE 410 Lecture 21: Nested and Hierarchical Designs and Rules for Expected Mean Squares

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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:

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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 0 ------------------------------------- -2 0 -2 2 -3 4 0 -4 2 2 Batches ------------------------------------- -2 -1 1 3 0 0 -1 1 -2 2 ------------------------------------- 1 0 3 4 4 3 2

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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:

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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:

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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.

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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.

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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

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SSB(A) = n i j [ Y ij. - Y i.. ] 2 SSE = i j k [Y ijk - Y ij. ] 2
SST = SSA + SSB(A) + SSE df abn-1 a-1 a(b-1) ab(n-1) Hopefully this makes sense. A is at a levels --> a-1 df

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At each level of a we have B at b levels --> b-1 df thus overall a levels of a we have a(b-1) df for factor B
Recall the analysis of the two factor factorial design SST = SSA + SSB + SSAB + SSE df abn-1 a-1 b-1 (a-1)(b-1) ab(n-1) We could easily show that if we take the two factor factorial SS then:

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SSB + SSAB = SSB(A) from factorial From nested
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