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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
1
Design of Engineering Experiments Two-Level Factorial Designs
• 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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chemical Process Example
A = reactant concentration, B = catalyst amount, y = recovery
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
5
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. 209-210 For manual calculations
The effect estimates are: A = 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Design-Expert output, full model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Design-Expert output, edited or reduced model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Residuals and Diagnostic Checking
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Response Surface
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The 23 Factorial Design
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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An Example of a 23 Factorial Design
A = gap, B = Flow, C = Power, y = Etch Rate
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Table of – and + Signs for the 23 Factorial Design (pg. 218)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Estimation of Factor Effects
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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ANOVA Summary – Full Model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Model Coefficients – Full Model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Refine Model – Remove Nonsignificant Factors
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Model Coefficients – Reduced Model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
23
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Regression Model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Model Interpretation
Cube plots are often useful visual displays of experimental results
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Cube Plot of Ranges
What do the large ranges
when gap and power are at the high level tell
you?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The General 2k Factorial Design
• Section 6-4, pg. 227, Table 6-9, pg. 228
• There will be k main effects, and
two-factor interactions2
three-factor interactions3
1 factor interaction
k
k
k
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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”?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
30
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
33
The Resin Plant Experiment
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Resin Plant Experiment
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Estimates of the Effects
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Half-Normal Probability Plot of Effects
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Regression Model
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Model Residuals are Satisfactory
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Model Interpretation – Main Effects and Interactions
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Outliers: suppose that cd = 375 (instead of 75)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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The Drilling Experiment Example 6.3
A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Normal Probability Plot of Effects –The Drilling Experiment
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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• 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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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(15.1)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Effect Estimates Following the Log Transformation
Three main effects are large
No indication of large interaction effects
What happened to the interactions?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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ANOVA Following the Log Transformation
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Following the Log Transformation
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Other Examples of Unreplicated 2k Designs
• The sidewall panel experiment (Example 6.4, pg. 245)– 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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Other Analysis Methods for Unreplicated 2k Designs
• Lenth’s method (see text, pg. 235)– 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. 236)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Overview of Lenth’s method
For an individual contrast, compare to the margin of error
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
64
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
65
The matrix has interesting and useful properties:X X
2 1
2
ˆ( ) (diagonal element of ( ) )
4
V
X X
Minimum 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?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
66
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
67
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
68
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
73
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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
74
Example 6.6, Pg. 248
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.
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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ANOVA for Example 6.6 (A Portion of Table 6.22)
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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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
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
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Practical Use of Center Points (pg. 260)
• 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?
Chapter 6 Design & Analysis of Experiments 7E 2009 Montgomery
79
Center Points and Qualitative Factors