design of engineering experiments chapter 6 – the 2k...
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Montgomery_Chap6 1
Design of Engineering ExperimentsChapter 6 – 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
Montgomery_Chap6 2
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
Montgomery_Chap6 3
Chemical Process Example
A = reactant concentration, B = catalyst amount, y = recovery
Montgomery_Chap6 4
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
Montgomery_Chap6 5
Estimation of Factor Effects
1
1
1
(1)2 2[ (1)]
(1)2 2[ (1)]
(1)2 2
[ (1) ]
A A
n
B B
n
n
A y y
ab a bn n
ab a bB y y
ab b an n
ab b aab a bAB
n nab a b
+ −
+ −
= −
+ += −
= + − −
= −
+ += −
= + − −
+ += −
= + − −
See textbook, pg. 221 For manual calculations
The effect estimates are: A = 8.33, B = -5.00, AB = 1.67
Practical interpretation?
Design-Expert analysis
Montgomery_Chap6 6
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 6.15809Lenth's SME 7.95671
Montgomery_Chap6 7
Statistical Testing - ANOVAResponse:Conversion
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 291.67 3 97.22 24.82 0.0002A 208.33 1 208.33 53.19 < 0.0001B 75.00 1 75.00 19.15 0.0024AB 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11
Std. Dev. 1.98 R-Squared 0.9030Mean 27.50 Adj R-Squared 0.8666C.V. 7.20 Pred R-Squared 0.7817
PRESS 70.50 Adeq Precision 11.669
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?
Montgomery_Chap6 8
Statistical Testing - ANOVA
Coefficient Standard 95% CI 95% CI
Factor Estimate DF Error Low High VIFIntercept 27.50 1 0.57 26.18 28.82A-Concent 4.17 1 0.57 2.85 5.48 1.00B-Catalyst -2.50 1 0.57 -3.82 -1.18 1.00AB 0.83 1 0.57 -0.48 2.15 1.00
General formulas for the standard errors of the model coefficients and the confidence intervals are available. They will be given later.
Montgomery_Chap6 9
Refine ModelResponse:Conversion
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 283.33 2 141.67 32.14 < 0.0001A 208.33 1 208.33 47.27 < 0.0001B 75.00 1 75.00 17.02 0.0026Residual 39.67 9 4.41Lack of Fit 8.33 1 8.33 2.13 0.1828Pure Error 31.33 8 3.92Cor Total 323.00 11
Std. Dev. 2.10 R-Squared 0.8772Mean 27.50 Adj R-Squared 0.8499C.V. 7.63 Pred R-Squared 0.7817
PRESS 70.52 Adeq Precision 12.702
There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component
Montgomery_Chap6 10
Regression Model for the ProcessCoefficient Standard 95% CI 95% CI
Factor Estimate DF Error Low High VIFIntercept 27.5 1 0.60604 26.12904 28.87096A-Concent 4.166667 1 0.60604 2.79571 5.537623 1B-Catalyst -2.5 1 0.60604 -3.87096 -1.12904 1
Final Equation in Terms of Coded Factors:
Conversion =27.5
4.166667 * A-2.5 * B
Final Equation in Terms of Actual Factors:
Conversion =18.333330.833333 * Concentration
-5 * Catalyst
Montgomery_Chap6 11
Residuals and Diagnostic CheckingDESIGN-EXPERT Plo tConversion
Res idual
Nor
mal
% p
roba
bilit
yNormal plot of residuals
-2.83333 -1.58333 -0.333333 0.916667 2.16667
1
5
10
20
30
50
7080
90
95
99
DESIGN-EXPERT P lo tConversion
22
Predicted
Res
idua
ls
Residuals vs. Predicted
-2.83333
-1.58333
-0.333333
0.916667
2.16667
20.83 24.17 27.50 30.83 34.17
Montgomery_Chap6 12
The Response Surfacet Conversion
A: Concentration
B: C
atal
yst
15.00 17.50 20.00 22.50 25.00
1.00
1.25
1.50
1.75
2.00
23.0556
25.2778
27.529.7222
31.9444
3 3
3 3 n
20.8333
24.1667
27.5
30.8333
4.1667
C
on
vers
ion
15.00
17.50
20.00
22.50
25.00
1 .00
1.25
1.50
1.75
2.00
A : Concen tra tion B : Ca ta lyst
3
Montgomery_Chap6 13
The 23 Factorial Design
Montgomery_Chap6 14
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
Montgomery_Chap6 15
An Example of a 23 Factorial Design
A = carbonation, B = pressure, C = speed, y = fill deviation
Montgomery_Chap6 16
Table of – and + Signs for the 23 Factorial Design (pg. 231)
Factorial Effect
TreatmentCombination
I A B AB C AC BC ABC
(1) = -4 + - - + - + + -
a = 1 + + - - - - + +
b = -1 + - + - - + - +
ab = 5 + + + + - - - -
c = -1 + - - + + - - +
ac = 3 + + - - + + - -
bc = 2 + - + - + - + -
abc = 11 + + + + + + + +
Contrast 24 18 6 14 2 4 4
Effect 3.00 2.25 0.75 1.75 0.25 0.50 0.50
Montgomery_Chap6 17
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 ABAB BC AB C AC× =
× = =
Montgomery_Chap6 18
Estimation of Factor Effects
Term Effect SumSqr % ContributionModel InterceptError A 3 36 46.1538Error B 2.25 20.25 25.9615Error C 1.75 12.25 15.7051Error AB 0.75 2.25 2.88462Error AC 0.25 0.25 0.320513Error BC 0.5 1 1.28205Error ABC 0.5 1 1.28205Error LOF 0Error P Error 5 6.41026
Lenth's ME 1.25382Lenth's SME 1.88156
Montgomery_Chap6 19
ANOVA Summary – Full ModelResponse:Fill-deviation
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 73.00 7 10.43 16.69 0.0003A 36.00 1 36.00 57.60 < 0.0001B 20.25 1 20.25 32.40 0.0005C 12.25 1 12.25 19.60 0.0022AB 2.25 1 2.25 3.60 0.0943AC 0.25 1 0.25 0.40 0.5447BC 1.00 1 1.00 1.60 0.2415ABC 1.00 1 1.00 1.60 0.2415Pure Error 5.00 8 0.63Cor Total 78.00 15
Std. Dev. 0.79 R-Squared 0.9359Mean 1.00 Adj R-Squared 0.8798C.V. 79.06 Pred R-Squared 0.7436
PRESS 20.00 Adeq Precision 13.416
Montgomery_Chap6 20
Model Coefficients – Full Model
CoefficientStandard 95% CI 95% CI
Factor Estimate DF Error Low High VIF
Intercept 1.00 1 0.20 0.54 1.46
A-Carbonation 1.50 1 0.20 1.04 1.96 1.00B-Pressure 1.13 1 0.20 0.67 1.58 1.00C-Speed 0.88 1 0.20 0.42 1.33 1.00AB 0.38 1 0.20 -0.081 0.83 1.00AC 0.13 1 0.20 -0.33 0.58 1.00BC 0.25 1 0.20 -0.21 0.71 1.00ABC 0.25 1 0.20 -0.21 0.71 1.00
Montgomery_Chap6 21
Refine Model – Remove Nonsignificant FactorsResponse: Fill-deviation
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 70.75 4 17.69 26.84 < 0.0001A 36.00 1 36.00 54.62 < 0.0001B 20.25 1 20.25 30.72 0.0002C 12.25 1 12.25 18.59 0.0012AB 2.25 1 2.25 3.41 0.0917Residual 7.25 11 0.66LOF 2.25 3 0.75 1.20 0.3700Pure E 5.00 8 0.63C Total 78.00 15
Std. Dev. 0.81 R-Squared 0.9071Mean 1.00 Adj R-Squared 0.8733C.V. 81.18 Pred R-Squared 0.8033
PRESS 15.34 Adeq Precision 15.424
Montgomery_Chap6 22
Model Coefficients – Reduced Model
Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low HighIntercept 1.00 1 0.20 0.55 1.45A-Carbonation 1.50 1 0.20 1.05 1.95B-Pressure 1.13 1 0.20 0.68 1.57C-Speed 0.88 1 0.20 0.43 1.32AB 0.38 1 0.20 -0.072 0.82
Montgomery_Chap6 23
Model Summary Statistics (pg. 239)• R2 and adjusted R2
• R2 for prediction (based on PRESS)
2
2
73.00 0.935978.00
/ 5.00 / 81 1 0.8798/ 78.00 /15
Model
T
E EAdj
T T
SSRSS
SS dfRSS df
= = =
= − = − =
2 20.00PRESSR = − = − =Pred 1 1 0.743678.00TSS
Montgomery_Chap6 24
Model Summary Statistics (pg. 239)
• Standard error of model coefficients
• Confidence interval on model coefficients
2 0.625ˆ ˆ( ) ( ) 0.202 2 2(8)
Ek k
MSse Vn nσ
β β= = = = =
/ 2, / 2,ˆ ˆ ˆ ˆ( ) ( )
E Edf dft se t seα αβ β β β β− ≤ ≤ +
Montgomery_Chap6 25
The Regression ModelFinal Equation in Terms of Coded Factors:
Fill-deviation =+1.00+1.50 * A+1.13 * B+0.88 * C+0.38 * A * B
Final Equation in Terms of Actual Factors:
Fill-deviation =+9.62500-2.62500 * Carbonation-1.20000 * Pressure+0.035000 * Speed+0.15000 * Carbonation * Pressure
Montgomery_Chap6 26
Residual Plots are SatisfactoryDESIGN-EXPERT PlotFi l l -devia tion
Studentized Res iduals
Nor
mal
% p
roba
bilit
y
Normal plot of residuals
-1.67 -0.84 0.00 0.84 1.67
1
5
10
20
30
50
7080
90
95
99
Montgomery_Chap6 27
Model InterpretationDESIGN-EXPERT Plot
Fi l l -deviation
X = A: CarbonationY = B: Pressure
B- 25.000B+ 30.000
Actual FactorC: Speed = 225.00
B: Pres s ureInteraction Graph
Fill-
devi
atio
n
A: Carbonation
10.00 10.50 11.00 11.50 12.00
-3
-0.75
1.5
3.75
6
Moderate interaction between carbonation level and pressure
Montgomery_Chap6 28
Model InterpretationDESIGN-EXPERT P lo t
Fi l l -deviationX = A: CarbonationY = B: PressureZ = C: Speed
Cube GraphFill-deviation
A: Carbonation
B: P
ress
ure
C : Speed
A- A+B-
B+
C-
C+
-2.13
-0.37
-0.63
1.13
0.12
1.88
3.13
4.88 Cube plots are often useful visual displays of experimental results
Montgomery_Chap6 29
Contour & Response Surface Plots –Speed at the High Level
DESIGN-EXPERT P lo t
Fi l l -deviationX = A: CarbonationY = B: Pressure
Design Poin ts
Actua l FactorC: Speed = 250.00
Fill-deviation
A: Carbonation
B: P
ress
ure
10.00 10.50 11.00 11.50 12.00
25.00
26.25
27.50
28.75
30.00
0.5
1.375
2.25
3.125
2 2
2 2
-0.375
0.9375
2.25
3.5625
4.875
Fill
-dev
iatio
n
10.00
10.50
11.00
11.50
12.00
2 5.00
26.25
27.50
28.75
30.00
A: Carbonation B: Pressure
Montgomery_Chap6 30
The General 2kFactorial Design• Section 6-4, pg. 242, Table 6-9, pg. 243• There will be k main effects, and
two-factor interactions2
three-factor interactions3
1 factor interaction
k
k
k
−M
Montgomery_Chap6 31
Unreplicated 2kFactorial Designs
• These are 2k factorial designs with oneobservation 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”?
Montgomery_Chap6 32
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
Montgomery_Chap6 33
Unreplicated 2kFactorial 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, pp. 246
Montgomery_Chap6 34
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
Montgomery_Chap6 35
The Resin Plant Experiment
Montgomery_Chap6 36
The Resin Plant Experiment
Montgomery_Chap6 37
Estimates of the EffectsTerm Effect SumSqr % Contribution
Model InterceptError A 21.625 1870.56 32.6397Error B 3.125 39.0625 0.681608Error C 9.875 390.062 6.80626Error D 14.625 855.563 14.9288Error AB 0.125 0.0625 0.00109057Error AC -18.125 1314.06 22.9293Error AD 16.625 1105.56 19.2911Error BC 2.375 22.5625 0.393696Error BD -0.375 0.5625 0.00981515Error CD -1.125 5.0625 0.0883363Error ABC 1.875 14.0625 0.245379Error ABD 4.125 68.0625 1.18763Error ACD -1.625 10.5625 0.184307Error BCD -2.625 27.5625 0.480942Error ABCD 1.375 7.5625 0.131959
Lenth's ME 6.74778Lenth's SME 13.699
Montgomery_Chap6 38
The Normal Probability Plot of EffectsDESIGN-EXPERT PlotFi l tra tion Rate
A: T em peratureB: PressureC: ConcentrationD: Sti rring Rate
Normal plot
Nor
mal
% p
roba
bilit
y
Effect
-18.12 -8.19 1.75 11.69 21.62
1
5
10
20
30
50
70
80
90
95
99
A
CD
AC
AD
Montgomery_Chap6 39
The Half-Normal Probability Plot DESIGN-EXPERT PlotFi l tra tion Rate
A: T em peratureB: PressureC: ConcentrationD: Sti rring Rate
Half Normal plot
Hal
f Nor
mal
% p
roba
bilit
y
|Effect|
0.00 5.41 10.81 16.22 21.63
0
20
40
60
70
80
85
90
95
97
99
A
CD
AC
AD
Montgomery_Chap6 40
ANOVA Summary for the ModelResponse:Filtration Rate
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob >FModel 5535.81 5 1107.16 56.74 < 0.0001A 1870.56 1 1870.56 95.86 < 0.0001C 390.06 1 390.06 19.99 0.0012D 855.56 1 855.56 43.85 < 0.0001AC 1314.06 1 1314.06 67.34 < 0.0001AD 1105.56 1 1105.56 56.66 < 0.0001Residual 195.12 10 19.51Cor Total 5730.94 15
Std. Dev. 4.42 R-Squared 0.9660Mean 70.06 Adj R-Squared 0.9489C.V. 6.30 Pred R-Squared 0.9128
PRESS 499.52 Adeq Precision 20.841
Montgomery_Chap6 41
The Regression Model
Final Equation in Terms of Coded Factors:
Filtration Rate =+70.06250+10.81250 * Temperature+4.93750 * Concentration+7.31250 * Stirring Rate-9.06250 * Temperature * Concentration+8.31250 * Temperature * Stirring Rate
Montgomery_Chap6 42
Model Residuals are SatisfactoryDESIGN-EXPERT PlotFi l tration Rate
Studentized Res iduals
Nor
mal
% p
roba
bilit
y
Normal plot of residuals
-1.83 -0.96 -0.09 0.78 1.65
1
5
10
20
30
50
7080
90
95
99
Montgomery_Chap6 43
Model Interpretation – InteractionsDESIGN-EXPERT P lo t
Fi l tra tion Rate
X = A: T em peratureY = C: Concentra tion
C- -1 .000C+ 1.000
Actua l FactorsB: Pressure = 0 .00D: Sti rring Rate = 0 .00
C: ConcentrationInteraction Graph
Filtr
atio
n R
ate
A: Tem perature
-1.00 -0.50 0.00 0.50 1.00
41.7702
57.3277
72.8851
88.4426
104
DESIGN-EXPERT Plo t
Fi l tra tion Rate
X = A: T em pera tureY = D: S ti rring Rate
D- -1 .000D+ 1.000
Actua l FactorsB: Pressure = 0 .00C: Concentration = 0.00
D: Stirring RateInteraction Graph
Filtr
atio
n R
ate
A: Tem perature
-1.00 -0.50 0.00 0.50 1.00
43
58.25
73.5
88.75
104
Montgomery_Chap6 44
Model Interpretation – Cube PlotIf one factor is dropped, the unreplicated 24
design will projectinto two replicates of a 23
Design projection is an extremely useful property, carrying over into fractional factorials
DESIGN-EXPERT Plo t
Fi l tra tion RateX = A: T em peratureY = C: ConcentrationZ = D: Sti rring Rate
Actua l FactorB: Pressure = 0.00
Cube GraphFiltration Rate
A: Tem perature
C: C
once
ntra
tion
D : Stirring Rate
A- A+C-
C+
D-
D+
46.25
44.25
74.25
72.25
69.38
100.63
61.13
92.38
Montgomery_Chap6 45
Model Interpretation – Response Surface Plots
DESIGN-EXPERT P lo t
Fi l tra tion RateX = A: T em peratureY = D: S ti rring Rate
Actua l FactorsB: Pressure = 0 .00C: Concentra tion = -1.00
44.25
58.3438
72.4375
86.5313
100.625
Filt
ratio
n R
ate
-1.0 0
-0.50
0.00
0.50
1.00
-1 .00
-0.50
0.00
0.50
1.00
A : T em perature D: Sti rring Rate
DESIGN-EXPERT Plo t
Fi l tra tion RateX = A: T em peratureY = D: Sti rring Rate
Actua l FactorsB: Pressure = 0 .00C: Concentra tion = -1 .00
Filtration Rate
A: Tem perature
D: S
tirrin
g R
ate
-1.00 -0.50 0.00 0.50 1.00
-1.00
-0.50
0.00
0.50
1.00
64.62571
77.375
83.75
90.125
56.93551.9395
With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates
Montgomery_Chap6 46
The Drilling Experiment Example 6-3, pg. 257
A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
Montgomery_Chap6 47
Effect Estimates - The Drilling Experiment
Term Effect SumSqr % ContributionModel InterceptError A 0.9175 3.36722 1.28072Error B 6.4375 165.766 63.0489Error C 3.2925 43.3622 16.4928Error D 2.29 20.9764 7.97837Error AB 0.59 1.3924 0.529599Error AC 0.155 0.0961 0.0365516Error AD 0.8375 2.80563 1.06712Error BC 1.51 9.1204 3.46894Error BD 1.5925 10.1442 3.85835Error CD 0.4475 0.801025 0.30467Error ABC 0.1625 0.105625 0.0401744Error ABD 0.76 2.3104 0.87876Error ACD 0.585 1.3689 0.520661Error BCD 0.175 0.1225 0.0465928Error ABCD 0.5425 1.17722 0.447757
Lenth's ME 2.27496Lenth's SME 4.61851
Montgomery_Chap6 48
Half-Normal Probability Plot of EffectsDESIGN-EXPERT Plotadv._rate
A: loadB: flowC: speedD: m ud
Half Normal plot
Hal
f Nor
mal
% p
roba
bilit
y
|Effect|
0.00 1.61 3.22 4.83 6.44
0
20
40
60
70
80
85
90
95
97
99
B
C
D
BCBD
Montgomery_Chap6 49
Residual Plotsot
Res idual
Nor
mal
% p
roba
bilit
y
Normal plot of residuals
-1.96375 -0.82625 0.31125 1.44875 2.58625
1
5
10
20
30
50
7080
90
95
99
t
Predicted
Res
idua
ls
Residuals vs. Predicted
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69 4.70 7.70 10.71 13.71
DE SIG N- EXP ERT Pl o ta d v._ r a te
P redi cted
Re
sid
uals
Residuals vs. P redicted
-1.96375
-0.82625
0.31125
1.44875
2.58625
1.69 4.70 7.70 10.71 13.71
Montgomery_Chap6 50
Residual Plots• 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 normality– Simplify the model
*y yλ=
Montgomery_Chap6 51
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
Montgomery_Chap6 52
The Box-Cox MethodDESIGN-EXPERT Plotadv._rate
Lam bdaCurrent = 1Best = -0 .23Low C.I. = -0 .79High C.I. = 0 .32
Recom m end transform :Log (Lam bda = 0)
Lam bda
Ln(R
esid
ualS
S)
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
Montgomery_Chap6 53
Effect Estimates Following the Log Transformation
DESIGN-EXPERT PlotLn(adv._rate)
A: loadB: flowC: speedD: m ud
Half Normal plotH
alf N
orm
al %
pro
babi
lity
|Effect|
0.00 0.29 0.58 0.87 1.16
0
20
40
60
70
80
85
90
95
97
99
B
C
D
Three main effects are large
No indication of large interaction effects
What happened to the interactions?
Montgomery_Chap6 54
ANOVA Following the Log TransformationResponse: adv._rate Transform: Natural logConstant: 0.000
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]Sum of Mean F
Source Squares DF Square Value Prob > FModel 7.11 3 2.37 164.82 < 0.0001B 5.35 1 5.35 371.49 < 0.0001C 1.34 1 1.34 93.05 < 0.0001D 0.43 1 0.43 29.92 0.0001Residual 0.17 12 0.014Cor Total 7.29 15
Std. Dev. 0.12 R-Squared 0.9763Mean 1.60 Adj R-Squared 0.9704C.V. 7.51 Pred R-Squared 0.9579
PRESS 0.31 Adeq Precision 34.391
Montgomery_Chap6 55
Following the Log Transformation
Final Equation in Terms of Coded Factors:
Ln(adv._rate) =+1.60+0.58 * B+0.29 * C+0.16 * D
Montgomery_Chap6 56
Following the Log TransformationDESIGN-EXPERT Plo tLn(adv._rate)
Res idual
Nor
mal
% p
roba
bilit
yNormal plot of residuals
-0.166184 -0.0760939 0.0139965 0.104087 0.194177
1
5
10
20
30
50
7080
90
95
99
DESILn(adv
GN-EXPERT Plo t._rate)
PredictedR
esid
uals
Residuals vs. Predicted
-0.166184
-0.0760939
0.0139965
0.104087
0.194177
0.57 1.08 1.60 2.11 2.63
Montgomery_Chap6 57
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
Montgomery_Chap6 58
Other Examples of Unreplicated 2k Designs
• The sidewall panel experiment (Example 6-4, pg. 260)– 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. 265)– Replicates versus repeat (or duplicate) observations?– Modeling within-run variability
Montgomery_Chap6 59
Other Analysis Methods for Unreplicated 2k Designs
• Lenth’s method (see text, pg. 254)– 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. 255 & 256)
Montgomery_Chap6 60
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
β β β ε
β β β β ε
= = >
= = > =
= + + +
= + + + +
∑ ∑∑
∑ ∑∑ ∑
Montgomery_Chap6 61
no "curvature"F Cy y= ⇒
The hypotheses are:
01
11
: 0
: 0
k
iiik
iii
H
H
β
β
=
=
=
≠
∑
∑2
Pure Quad( )F C F C
F C
n n y ySSn n
−=
+
This sum of squares has a single degree of freedom
Montgomery_Chap6 62
Example 6-6, Pg. 273
5Cn =
Usually between 3 and 6 center points will work well
Design-Expert provides the analysis, including the F-test for pure quadratic curvature
Montgomery_Chap6 63
ANOVA for Example 6-6Response: yield
ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum of Mean FSource Squares DF Square Value Prob > FModel 2.83 3 0.94 21.92 0.0060A 2.40 1 2.40 55.87 0.0017B 0.42 1 0.42 9.83 0.0350AB 2.500E-003 1 2.500E-003 0.058 0.8213Curvature 2.722E-003 1 2.722E-003 0.063 0.8137Pure Error 0.17 4 0.043Cor Total 3.00 8
Std. Dev. 0.21 R-Squared 0.9427Mean 40.44 Adj R-Squared 0.8996
C.V. 0.51 Pred R-Squared N/A
PRESS N/A Adeq Precision 14.234
Montgomery_Chap6 64
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
Montgomery_Chap6 65
Practical Use of Center Points (pg. 275)
• 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?
Montgomery_Chap6 66
Center Points and Qualitative Factors