chapter 6design & analysis of experiments 8e 2012 montgomery 1 design of engineering experiments...
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Design & Analysis of Experiments 8E 2012 Montgomery
1Chapter 6
Design of Engineering ExperimentsPart 5 – The 2k Factorial Design
• Text reference, Chapter 6• Special case of the general factorial design; k
factors, all at two levels• The two levels are usually called low and high (they
could be either quantitative or qualitative)• Very widely used in industrial experimentation• Form a basic “building block” for other very useful
experimental designs (DNA)• Special (short-cut) methods for analysis• We will make use of Design-Expert
Design & Analysis of Experiments 8E 2012 Montgomery
2Chapter 6
Design & Analysis of Experiments 8E 2012 Montgomery
3Chapter 6
The Simplest Case: The 22
“-” and “+” denote the low and high levels of a factor, respectively
• Low and high are arbitrary terms
• Geometrically, the four runs form the corners of a square
• Factors can be quantitative or qualitative, although their treatment in the final model will be different
Design & Analysis of Experiments 8E 2012 Montgomery
4Chapter 6
Chemical Process Example
A = reactant concentration, B = catalyst amount, y = recovery
Design & Analysis of Experiments 8E 2012 Montgomery
5Chapter 6
Analysis Procedure for a Factorial Design
• Estimate factor effects• Formulate model– With replication, use full model– With an unreplicated design, use normal probability
plots• Statistical testing (ANOVA)• Refine the model• Analyze residuals (graphical)• Interpret results
Design & Analysis of Experiments 8E 2012 Montgomery
6Chapter 6
Estimation of Factor Effects
12
12
12
(1)
2 2[ (1)]
(1)
2 2[ (1)]
(1)
2 2[ (1) ]
A A
n
B B
n
n
A y y
ab a b
n nab a b
B y y
ab b a
n nab b a
ab a bAB
n nab a b
See textbook, pg. 235-236 for manual calculations
The effect estimates are: A = 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
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Estimation of Factor EffectsForm Tentative Model
Term Effect SumSqr % ContributionModel InterceptModel A 8.33333 208.333 64.4995Model B -5 75 23.2198Model AB 1.66667 8.33333 2.57998Error Lack Of Fit 0 0Error P Error 31.3333 9.70072
Lenth's ME 6.15809 Lenth's SME 7.95671
Design & Analysis of Experiments 8E 2012 Montgomery
8Chapter 6
Statistical Testing - ANOVA
The F-test for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?
Design & Analysis of Experiments 8E 2012 Montgomery
9Chapter 6
Design-Expert output, full model
Design & Analysis of Experiments 8E 2012 Montgomery
10Chapter 6
Design-Expert output, edited or reduced model
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11Chapter 6
Residuals and Diagnostic Checking
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12Chapter 6
The Response Surface
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13Chapter 6
The 23 Factorial Design
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Effects in The 23 Factorial Design
etc, etc, ...
A A
B B
C C
A y y
B y y
C y y
Analysis done via computer
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15Chapter 6
An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
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16Chapter 6
Table of – and + Signs for the 23 Factorial Design (pg. 218)
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17Chapter 6
Properties of the Table
• Except for column I, every column has an equal number of + and – signs
• The sum of the product of signs in any two columns is zero• Multiplying any column by I leaves that column unchanged (identity
element)• The product of any two columns yields a column in the table:
• Orthogonal design• Orthogonality is an important property shared by all factorial designs
2
A B AB
AB BC AB C AC
Design & Analysis of Experiments 8E 2012 Montgomery
18Chapter 6
Estimation of Factor Effects
Design & Analysis of Experiments 8E 2012 Montgomery
19Chapter 6
ANOVA Summary – Full Model
Design & Analysis of Experiments 8E 2012 Montgomery
20Chapter 6
Model Coefficients – Full Model
Design & Analysis of Experiments 8E 2012 Montgomery
21Chapter 6
Refine Model – Remove Nonsignificant Factors
Design & Analysis of Experiments 8E 2012 Montgomery
22Chapter 6
Model Coefficients – Reduced Model
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23Chapter 6
Model Summary Statistics for Reduced Model
• R2 and adjusted R2
• R2 for prediction (based on PRESS)
52
5
25
5.106 100.9608
5.314 10
/ 20857.75 /121 1 0.9509
/ 5.314 10 /15
Model
T
E EAdj
T T
SSR
SS
SS dfR
SS df
2Pred 5
37080.441 1 0.9302
5.314 10T
PRESSR
SS
Design & Analysis of Experiments 8E 2012 Montgomery
24Chapter 6
Model Summary Statistics
• Standard error of model coefficients (full model)
• Confidence interval on model coefficients
2 2252.56ˆ ˆ( ) ( ) 11.872 2 2(8)
Ek k
MSse V
n n
/ 2, / 2,ˆ ˆ ˆ ˆ( ) ( )
E Edf dft se t se
Design & Analysis of Experiments 8E 2012 Montgomery
25Chapter 6
The Regression Model
Design & Analysis of Experiments 8E 2012 Montgomery
26Chapter 6
Model Interpretation
Cube plots are often useful visual displays of experimental results
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27Chapter 6
Cube Plot of Ranges
What do the large ranges
when gap and power are at the high level tell
you?
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28Chapter 6
Design & Analysis of Experiments 8E 2012 Montgomery
29Chapter 6
The General 2k Factorial Design• Section 6-4, pg. 253, Table 6-9, pg. 25• There will be k main effects, and
two-factor interactions2
three-factor interactions3
1 factor interaction
k
k
k
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30Chapter 6
6.5 Unreplicated 2k Factorial Designs
• These are 2k factorial designs with one observation at each corner of the “cube”
• An unreplicated 2k factorial design is also sometimes called a “single replicate” of the 2k
• These designs are very widely used• Risks…if there is only one observation at each
corner, is there a chance of unusual response observations spoiling the results?
• Modeling “noise”?
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31Chapter 6
Spacing of Factor Levels in the Unreplicated 2k Factorial Designs
If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data
More aggressive spacing is usually best
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32Chapter 6
Unreplicated 2k Factorial Designs
• Lack of replication causes potential problems in statistical testing– Replication admits an estimate of “pure error” (a better
phrase is an internal estimate of error)– With no replication, fitting the full model results in zero
degrees of freedom for error• Potential solutions to this problem– Pooling high-order interactions to estimate error– Normal probability plotting of effects (Daniels, 1959)– Other methods…see text
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33Chapter 6
Example of an Unreplicated 2k Design
• A 24 factorial was used to investigate the effects of four factors on the filtration rate of a resin
• The factors are A = temperature, B = pressure, C = mole ratio, D= stirring rate
• Experiment was performed in a pilot plant
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34Chapter 6
The Resin Plant Experiment
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35Chapter 6
The Resin Plant Experiment
Design & Analysis of Experiments 8E 2012 Montgomery
36Chapter 6
Design & Analysis of Experiments 8E 2012 Montgomery
37Chapter 6
Estimates of the Effects
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38Chapter 6
The Half-Normal Probability Plot of Effects
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39Chapter 6
Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D
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40Chapter 6
The Regression Model
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Model Residuals are Satisfactory
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42Chapter 6
Model Interpretation – Main Effects and Interactions
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43Chapter 6
Model Interpretation – Response Surface Plots
With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates
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Design & Analysis of Experiments 8E 2012 Montgomery
45Chapter 6
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46Chapter 6
Outliers: suppose that cd = 375 (instead of 75)
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47Chapter 6
Dealing with Outliers
• Replace with an estimate• Make the highest-order interaction zero• In this case, estimate cd such that ABCD =
0• Analyze only the data you have• Now the design isn’t orthogonal• Consequences?
Design & Analysis of Experiments 8E 2012 Montgomery
48Chapter 6
Design & Analysis of Experiments 8E 2012 Montgomery
49Chapter 6
The Drilling Experiment Example 6.3
A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
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Normal Probability Plot of Effects –The Drilling Experiment
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51Chapter 6
Residual Plots
DESIGN-EXPERT Plotadv._rate
Predicted
Re
sid
ua
ls
Residuals vs. Predicted
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69 4.70 7.70 10.71 13.71
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52Chapter 6
• The residual plots indicate that there are problems with the equality of variance assumption
• The usual approach to this problem is to employ a transformation on the response
• Power family transformations are widely used
• Transformations are typically performed to – Stabilize variance– Induce at least approximate normality– Simplify the model
Residual Plots
*y y
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53Chapter 6
Selecting a Transformation
• Empirical selection of lambda• Prior (theoretical) knowledge or experience can
often suggest the form of a transformation• Analytical selection of lambda…the Box-Cox
(1964) method (simultaneously estimates the model parameters and the transformation parameter lambda)
• Box-Cox method implemented in Design-Expert
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(15.1)
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The Box-Cox MethodDESIGN-EXPERT Plotadv._rate
LambdaCurrent = 1Best = -0.23Low C.I. = -0.79High C.I. = 0.32
Recommend transform:Log (Lambda = 0)
Lambda
Ln
(Re
sid
ua
lSS
)
Box-Cox Plot for Power Transforms
1.05
2.50
3.95
5.40
6.85
-3 -2 -1 0 1 2 3
A log transformation is recommended
The procedure provides a confidence interval on the transformation parameter lambda
If unity is included in the confidence interval, no transformation would be needed
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56Chapter 6
Effect Estimates Following the Log Transformation
Three main effects are large
No indication of large interaction effects
What happened to the interactions?
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57Chapter 6
ANOVA Following the Log Transformation
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58Chapter 6
Following the Log Transformation
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59Chapter 6
The Log Advance Rate Model
• Is the log model “better”?• We would generally prefer a simpler model
in a transformed scale to a more complicated model in the original metric
• What happened to the interactions?• Sometimes transformations provide insight
into the underlying mechanism
Design & Analysis of Experiments 8E 2012 Montgomery
60Chapter 6
Other Examples of Unreplicated 2k Designs
• The sidewall panel experiment (Example 6.4, pg. 274)– Two factors affect the mean number of defects– A third factor affects variability– Residual plots were useful in identifying the dispersion
effect
• The oxidation furnace experiment (Example 6.5, pg. 245)– Replicates versus repeat (or duplicate) observations?– Modeling within-run variability
Design & Analysis of Experiments 8E 2012 Montgomery
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• Example 6.6, Credit Card Marketing, page 278– Using DOX in marketing and marketing
research, a growing application– Analysis is with the JMP screening platform
Chapter 6
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62Chapter 6
Other Analysis Methods for Unreplicated 2k Designs
• Lenth’s method (see text, pg. 262)– Analytical method for testing effects, uses an estimate
of error formed by pooling small contrasts– Some adjustment to the critical values in the original
method can be helpful– Probably most useful as a supplement to the normal
probability plot
• Conditional inference charts (pg. 264)
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Overview of Lenth’s method
For an individual contrast, compare to the margin of error
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65Chapter 6
Adjusted multipliers for Lenth’s method
Suggested because the original method makes too many type I errors, especially for small designs (few contrasts)
Simulation was used to find these adjusted multipliers
Lenth’s method is a nice supplement to the normal probability plot of effects
JMP has an excellent implementation of Lenth’s method in the screening platform
Design & Analysis of Experiments 8E 2012 Montgomery
66Chapter 6
Design & Analysis of Experiments 8E 2012 Montgomery
67Chapter 6
The 2k design and design optimality
The model parameter estimates in a 2k design (and the effect estimates) are least squares estimates. For example, for a 22 design the model is
0 1 1 2 2 12 1 2
0 1 2 12 1
0 1 2 12 2
0 1 2 12 3
0 1 2 12 4
(1) ( 1) ( 1) ( 1)( 1)
(1) ( 1) (1)( 1)
( 1) (1) ( 1)(1)
(1) (1) (1)(1)
(1) 1 1 1 1
1 1 1 1, ,
1 1 1
y x x x x
a
b
ab
a
b
ab
y = Xβ + ε y X
0 1
1 2
2 3
12 4
, ,1
1 1 1 1
β ε
The four observations from a 22 design
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The least squares estimate of β is
1
0
14
2
12
ˆ
4 0 0 0 (1)
0 4 0 0 (1)
0 0 4 0 (1)
0 0 0 4 (1)
(1)
4ˆ (1) (ˆ (1)1ˆ (1)4
(1)ˆ
a b ab
a ab b
b ab a
a b ab
a b ab
a b ab a ab ba ab b
b ab a
a b ab
-1β = (X X) X y
I
1)
4(1)
4(1)
4
b ab a
a b ab
The matrix is diagonal – consequences of an orthogonal design
X X
The regression coefficient estimates are exactly half of the ‘usual” effect estimates
The “usual” contrasts
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69Chapter 6
The matrix has interesting and useful properties:X X
2 1
2
ˆ( ) (diagonal element of ( ) )
4
V
X XMinimum possible value for a four-run
design
|( ) | 256 X XMaximum possible value for a four-run
design
Notice that these results depend on both the design that you have chosen and the model
What about predicting the response?
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70Chapter 6
21 2
1 2 1 2
22 2 2 2
1 2 1 2 1 2
1 2
21 2
1 2
2
1 2
ˆ[ ( , )]
[1, , , ]
ˆ[ ( , )] (1 )4
The maximum prediction variance occurs when 1, 1
ˆ[ ( , )]
The prediction variance when 0 is
ˆ[ ( , )]
V y x x
x x x x
V y x x x x x x
x x
V y x x
x x
V y x x
-1x (X X) x
x
4What about prediction variance over the design space?average
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71Chapter 6
Average prediction variance1 1
21 2 1 2
1 1
1 12 2 2 2 2
1 2 1 2 1 2
1 1
2
1ˆ[ ( , ) = area of design space = 2 4
1 1(1 )
4 4
4
9
I V y x x dx dx AA
x x x x dx dx
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Design-Expert® Software
Min StdErr Mean: 0.500Max StdErr Mean: 1.000Cuboidalradius = 1Points = 10000
FDS Graph
Fraction of Design Space
Std
Err
Me
an
0.00 0.25 0.50 0.75 1.00
0.000
0.250
0.500
0.750
1.000
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73Chapter 6
For the 22 and in general the 2k
• The design produces regression model coefficients that have the smallest variances (D-optimal design)
• The design results in minimizing the maximum variance of the predicted response over the design space (G-optimal design)
• The design results in minimizing the average variance of the predicted response over the design space (I-optimal design)
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74Chapter 6
Optimal Designs
• These results give us some assurance that these designs are “good” designs in some general ways
• Factorial designs typically share some (most) of these properties
• There are excellent computer routines for finding optimal designs (JMP is outstanding)
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75Chapter 6
Addition of Center Points to a 2k Designs
• Based on the idea of replicating some of the runs in a factorial design
• Runs at the center provide an estimate of error and allow the experimenter to distinguish between two possible models:
01 1
20
1 1 1
First-order model (interaction)
Second-order model
k k k
i i ij i ji i j i
k k k k
i i ij i j ii ii i j i i
y x x x
y x x x x
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Design & Analysis of Experiments 8E 2012 Montgomery
77Chapter 6
no "curvature"F Cy y
The hypotheses are:
01
11
: 0
: 0
k
iii
k
iii
H
H
2
Pure Quad
( )F C F C
F C
n n y ySS
n n
This sum of squares has a single degree of freedom
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Example 6.7, Pg. 286
4Cn
Usually between 3 and 6 center points will work well
Design-Expert provides the analysis, including the F-test for pure quadratic curvature
Refer to the original experiment shown in Table 6.10. Suppose that four center points are added to this experiment, and at the points x1=x2 =x3=x4=0 the four observed filtration rates were 73, 75, 66, and 69. The average of these four center points is 70.75, and the average of the 16 factorial runs is 70.06. Since are very similar, we suspect that there is no strong curvature present.
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Design & Analysis of Experiments 8E 2012 Montgomery
80Chapter 6
ANOVA for Example 6.7
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81Chapter 6
If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design
for fitting a second-order response surface model
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82Chapter 6
Practical Use of Center Points (pg. 289)
• Use current operating conditions as the center point
• Check for “abnormal” conditions during the time the experiment was conducted
• Check for time trends• Use center points as the first few runs when there
is little or no information available about the magnitude of error
• Center points and qualitative factors?
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83Chapter 6
Center Points and Qualitative Factors