1 Chapter 7 Blocking and Confounding in the 2 k Factorial Design
Jan 03, 2016
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Chapter 7 Blocking and Confounding in the 2k Factorial Design
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7.2 Blocking a Replicated 2k Factorial Design • Blocking is a technique for dealing with
controllable nuisance variables• A 2k factorial design with n replicates.• This is the same scenario discussed previously
(Chapter 5, Section 5-6)• If there are n replicates of the design, then each
replicate is a block• Each replicate is run in one of the blocks (time
periods, batches of raw material, etc.)• Runs within the block are randomized
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• Example 7.1
Consider the example from Section 6-2; k = 2 factors, n = 3 replicates
This is the “usual” method for calculating a block sum of squares
2 23...
1 4 12
6.50
iBlocks
i
B ySS
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• The ANOVA table of Example 7.1
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7.3 Confounding in the 2k Factorial Design • Confounding is a design technique for arranging
a complete factorial experiment in blocks, where block size is smaller than the number of treatment combinations in one replicate.
• Cause information about certain treatment effects to be indistinguishable from (confounded with) blocks.
• Consider the construction and analysis of the 2k factorial design in 2p incomplete blocks with p < k
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7.4 Confounding the 2k Factorial Design in Two Blocks• For example: Consider a 22 factorial design in 2
blocks.– Block 1: (1) and ab– Block 2: a and b– AB is confounded with blocks! – See Page 275– How to construct such designs??
7BlockbaabAB
ababB
baabA
2/])1([
2/)]1([
2/)]1([
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• Defining contrast:
– xi is the level of the ith factor appearing in a
particular treatment combination
i is the exponent appearing on the ith factor in
the effect to be confounded – Treatment combinations that produce the same
value of L (mod 2) will be placed in the same block.
– See Page 277• Group:
– Principal block: Contain the treatment (1)
kk xxxL 2211
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• Estimation of error:
• Example 7.2
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7.6 Confounding the 2k Factorial Design in Four Blocks• Two defining contrasts: Consider 25 design.
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532
541
2:
1:
xxxLBCE
xxxLADE
• The generalized interaction:
(ADE)(BCE) = ABCD– ADE, BCE and ABCD are all confounded with
blocks.
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7.7 Confounding the 2k Factorial Design in 2p Blocks• Choose p independent effects to be confounded.• Exact 2p -p -1 other effects will be confounded
with blocks.
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