Design and Analysis of Engineering Experiments in Practice
Chapter 1
Based on Design & Analysis of
Experiments 7E 2009 Montgomery
1
Design and Analysis of
Engineering Experiments
Ali Ahmad, PhD
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
2
Design of Engineering Experiments
Blocking and Confounding Systems
for Two-level factorials
•
Blocking
is a technique for dealing with
controllable
nuisance
variables
•
Two cases are considered
–
Replicated designs
–
Unreplicated designs
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
3
Blocking a Replicated Design
•
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
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
4
Blocking a Replicated Design
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
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
5
ANOVA for the Blocked Design
Page 267
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
6
Confounding in Blocks
•
Now consider the
unreplicated
case
•
Clearly the previous discussion does not
apply, since there is only one replicate
•
To illustrate, consider the situation of
Example 6.2, the resin plant experiment
•
This is a 2
4
,
n
= 1 replicate
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
7
Experiment from Example 6.2
Suppose only 8 runs can be made from one batch of raw material
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
8
The Table of + & - Signs, Example 6-4
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
9
ABCD is Confounded with Blocks
(Page 279)
Observations in block 1 are reduced by 20
units…this is the simulated “block effect”
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
10
Effect Estimates
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
11
The ANOVA
The
ABCD
interaction (or the block effect) is not considered as part of
the error term
The reset of the analysis is unchanged from the original analysis
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
12
Another Illustration of the Importance of Blocking
Now the
first eight
runs (in run
order) have
filtration
rate reduced
by 20 units
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
13
The interpretation is
harder; not as easy to
identify the large
effects
One important
interaction is not
identified (
AD
)
Failing to block when
we should have causes
problems in
interpretation the
result of an
experiment and can
mask the presence of
real factor effects
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
14
Confounding in Blocks
•
More than two blocks (page 282)
–
The two-level factorial can be confounded in 2,
4, 8, … (2
p
,
p >
1) blocks
–
For
four
blocks, select
two
effects to confound,
automatically confounding a
third
effect
–
See example, page 282
–
Choice of confounding schemes non-trivial; see
Table 7.9, page 285
•
Partial confounding (page 285)
Chapter 7
Design & Analysis of Experiments
7E 2009 Montgomery
15
General Advice About Blocking
•
When in doubt, block
•
Block out the nuisance variables you know about,
randomize as much as possible and rely on randomization
to help balance out unknown nuisance effects
•
Measure the nuisance factors you know about but can’t
control (ANCOVA)
•
It may be a good idea to conduct the experiment in blocks
even if there isn't an obvious nuisance factor, just to
protect against the loss of data or situations where the
complete experiment can’t be finished
This is a basic course blah, blah, blah…
Explore the fundamentals of engineering experiments, including blocking and confounding systems for two-level factorials. Learn about replicated and unreplicated designs, the importance of blocking in a replicated design, ANOVA for blocked designs, and considerations for confounding in blocks. Dive into examples and calculations to enhance your understanding of experimental design concepts.
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Presentation Transcript
Design and Analysis of Engineering Experiments Ali Ahmad, PhD Chapter 1 Based on Design & Analysis of Experiments 7E 2009 Montgomery 1
Design of Engineering Experiments Blocking and Confounding Systems for Two-level factorials Blocking is a technique for dealing with controllable nuisance variables Two cases are considered Replicated designs Unreplicated designs Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 2
Blocking a Replicated Design 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 Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 3
Blocking a Replicated Design 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 i 2 ... 3 B y = SS Blocks 1 4 6.50 12 = i = Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 4
ANOVA for the Blocked Design Page 267 Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 5
Confounding in Blocks Now consider the unreplicated case Clearly the previous discussion does not apply, since there is only one replicate To illustrate, consider the situation of Example 6.2, the resin plant experiment This is a 24, n = 1 replicate Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 6
Experiment from Example 6.2 Suppose only 8 runs can be made from one batch of raw material Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 7
The Table of + & - Signs, Example 6-4 Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 8
ABCD is Confounded with Blocks (Page 279) Observations in block 1 are reduced by 20 units this is the simulated block effect Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 9
Effect Estimates Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 10
The ANOVA The ABCD interaction (or the block effect) is not considered as part of the error term The reset of the analysis is unchanged from the original analysis Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 11
Another Illustration of the Importance of Blocking Now the first eight runs (in run order) have filtration rate reduced by 20 units Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 12
The interpretation is harder; not as easy to identify the large effects One important interaction is not identified (AD) Failing to block when we should have causes problems in interpretation the result of an experiment and can mask the presence of real factor effects Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 13
Confounding in Blocks More than two blocks (page 282) The two-level factorial can be confounded in 2, 4, 8, (2p, p > 1) blocks For four blocks, select two effects to confound, automatically confounding a third effect See example, page 282 Choice of confounding schemes non-trivial; see Table 7.9, page 285 Partial confounding (page 285) Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 14
General Advice About Blocking When in doubt, block Block out the nuisance variables you know about, randomize as much as possible and rely on randomization to help balance out unknown nuisance effects Measure the nuisance factors you know about but can t control (ANCOVA) It may be a good idea to conduct the experiment in blocks even if there isn't an obvious nuisance factor, just to protect against the loss of data or situations where the complete experiment can t be finished Chapter 7 Design & Analysis of Experiments 7E 2009 Montgomery 15
