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Chapter 7 Blocking and Confounding in the 2k Factorial Design

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Chapter 7 Blocking and Confounding
in the 2k Factorial Design
1
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
2
• 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
3
SS B locks пЂЅ

i пЂЅ1
2
Bi
4
2
пЂ­
y ...
12
пЂЅ 6.50
3
• The ANOVA table of Example 7.1
4
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
5
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??
6
A пЂЅ [ ab пЂ« a пЂ­ b пЂ­ (1)] / 2
B пЂЅ [ ab пЂ« b пЂ­ a пЂ­ (1)] / 2
AB пЂЅ [ ab пЂ« (1) пЂ­ a пЂ­ b ] / 2 пЂЅ Block
7
• Defining contrast:
L пЂЅ пЃЎ 1 x1 пЂ« пЃЎ 2 x 2 пЂ« пЃЊ пЂ« пЃЎ k x k
– 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)
8
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• Estimation of error:
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• Example 7.2
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12
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7.6 Confounding the 2k Factorial
Design in Four Blocks
• Two defining contrasts: Consider 25 design.
ADE : L1 пЂЅ x1 пЂ« x 4 пЂ« x 5
BCE : L 2 пЂЅ x 2 пЂ« x 3 пЂ« x 5
14
• The generalized interaction:
(ADE)(BCE) = ABCD
– ADE, BCE and ABCD are all confounded with
blocks.
15
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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