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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
вЂў Blocking is a technique for dealing with
controllable nuisance variables
вЂў A 2k factorial design with n replicates.
вЂў This is the same scenario discussed previously
(Chapter 5, Section 5-6)
вЂў If there are n replicates of the design, then each
replicate is a block
вЂў Each replicate is run in one of the blocks (time
periods, batches of raw material, etc.)
вЂў Runs within the block are randomized
2
вЂў Example 7.1
Consider the
example from
Section 6-2; k = 2
factors, n = 3
replicates
This is the вЂњusualвЂќ
method for
calculating a block
sum of squares
3
SS B locks пЂЅ
пѓҐ
i пЂЅ1
2
Bi
4
2
пЂ­
y ...
12
пЂЅ 6.50
3
вЂў The ANOVA table of Example 7.1
4
7.3 Confounding in the 2k Factorial
Design
вЂў 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.
вЂў Cause information about certain treatment effects
to be indistinguishable from (confounded with)
blocks.
вЂў Consider the construction and analysis of the 2k
5
7.4 Confounding the 2k Factorial
blocks.
вЂ“ Block 1: (1) and ab
вЂ“ Block 2: a and b
вЂ“ AB is confounded with blocks!
вЂ“ See Page 275
вЂ“ 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
вЂў Defining contrast:
L пЂЅ пЃЎ 1 x1 пЂ« пЃЎ 2 x 2 пЂ« пЃЊ пЂ« пЃЎ k x k
вЂ“ xi is the level of the ith factor appearing in a
particular treatment combination
вЂ“ пЃЎi is the exponent appearing on the ith factor in
the effect to be confounded
вЂ“ Treatment combinations that produce the same
value of L (mod 2) will be placed in the same
block.
вЂ“ See Page 277
вЂў Group:
вЂ“ Principal block: Contain the treatment (1)
8
9
вЂў Estimation of error:
10
вЂў Example 7.2
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12
13
7.6 Confounding the 2k Factorial
вЂў 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
вЂў The generalized interaction:
вЂ“ ADE, BCE and ABCD are all confounded with
blocks.
15
7.7 Confounding the 2k Factorial
вЂў Choose p independent effects to be confounded.
вЂў Exact 2p -p -1 other effects will be confounded
with blocks.
16
17
```
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