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Design of Engineering Experiments
– 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
• Special (short-cut) methods for analysis
1
Design of Engineering Experiments
– The 2k Factorial Design
• Assumptions
• The factors are fixed
• The designs are completely randomized
• Usual normality assumptions are satisfied
• It provides the smallest number of runs can be
studied in a complete factorial design – used as factor
screening experiments
• Linear response in the specified range is assumed
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
3
Chemical Process Example
A = reactant concentration, B = catalyst amount,
y = recovery
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
5
Estimation of Factor Effects
A y A y A
ab a b (1) See textbook, pg. 221 For
manual calculations
2n 2n
1
[ab a b (1)] The effect estimates are: A
2n = 8.33, B = -5.00, AB = 1.67
B yB yB
Practical interpretation
ab b a (1)
2n 2n
1
[ab b a (1)]
2n
ab (1) a b
AB
2n 2n
1
[ab (1) a b]
2n 6
Estimation of Factor Effects
Effects (1) a b ab
A -1 +1 -1 +1
B -1 -1 +1 +1
AB +1 -1 -1 +1
• “(1), a, b, ab” – standard order
• Used to determine the proper sign for each treatment combination
Treatment Factorial Effect
combination I A B AB
(1) + - - +
a + + - -
b + - + -
ab + + + + 7
Statistical Testing - ANOVA
Response: Conversion
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of Mean F
Source Squares DF Square Value Prob > F
Model 291.67 3 97.22 24.82 0.0002
A 208.33 1 208.33 53.19 < 0.0001
B 75.00 1 75.00 19.15 0.0024
AB 8.33 1 8.33 2.13 0.1828
Pure Error 31.33 8 3.92
Cor Total 323.00 11
Std. Dev. 1.98 R-Squared 0.9030
Mean 27.50 Adj R-Squared 0.8666
C.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?
8
Estimation of Factor Effects
Form Tentative Model
Term Effect SumSqr % Contribution
Model Intercept
Model A 8.33333 208.333 64.4995
Model B -5 75 23.2198
Model AB 1.66667 8.33333 2.57998
Error Lack Of Fit 0 0
Error P Error 31.3333
9
Regression Model
y = bo + b1x1 + b2x2 + b12x1x2 + e
or let x3 = x1x2, b3 = b12
y = bo + b1 x1 + b2 x2 + b3 x3 + e
A linear regression model.
Coded variables are related to natural variables by
Conc. (Conclow Conchigh ) / 2
x1
(Conclow Conchigh ) / 2
Catalyst (Catalystlow Catalysthigh ) / 2
x2
(Catalystlow Catalysthigh ) / 2
Therefore,
x1 : [1,1] Conc. : [Conclow , Conchigh ]
x2 : [1,1] Catalyst : [Catalystlow , Catalysthigh ]
10
Statistical Testing - ANOVA
Coefficient Standard 95% CI 95% CI
Factor Estimate DF Error Low High VIF
Intercept 27.50 1 0.57 26.18 28.82
A-Concent 4.17 1 0.57 2.85 5.48 1.00
B-Catalyst -2.50 1 0.57 -3.82 -1.18 1.00
AB 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.
11
Refined/reduced Model
y = bo + b1 x1 + b2 x2 + e
Response: Conversion
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of Mean F
Source Squares DF Square Value Prob > F
Model 283.33 2 141.67 32.14 < 0.0001
A 208.33 1 208.33 47.27 < 0.0001
B 75.00 1 75.00 17.02 0.0026
Residual 39.67 9 4.41
Lack of Fit 8.33 1 8.33 2.13 0.1828
Pure Error 31.33 8 3.92
Cor Total 323.00 11
Std. Dev. 2.10 R-Squared 0.8772
Mean 27.50 Adj R-Squared 0.8499
C.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
12
Regression Model for the Process
Coefficient Standard 95% CI 95% CI
Factor Estimate DF Error Low High VIF
Intercept 27.5 1 0.60604 26.12904 28.87096
4.166667
A-Concentration 1 0.60604 2.79571 5.537623 1
B-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 =
18.33333
0.833333 * Concentration
-5 * Catalyst
13
Residuals and Diagnostic Checking
DESIGN-EXPERT Plot Normal plot of residuals DESIGN-EXPERT Plot Residuals vs. Predicted
Conversion Conversion
2.16667
99
95
0.916667
90
Norm al % probability
80
70
Res iduals
50 -0.333333
30
20
10
-1.58333
5
2
1
-2.83333
20.83 24.17 27.50 30.83 34.17
-2.83333 -1.58333 -0.333333 0.916667 2.16667
Predicted
Res idual
14
PERT Plot The Response Surface
DESIGN-EXPERT Plot 3 Conversion 3
centration 2.00
lyst Conversion
X = A: Concentration
34.1667 Y = B: Catalyst
23.0556
30.8333 Design Points
1.75
27.5
24.1667
Conversion
25.2778
B: Catalys t
20.8333 27.5
1.50
29.7222
1.25
2.00 31.9444
25.00
1.75
22.50
1.50
20.00 3 3
1.00
B: Catalyst 1.25 17.50
A: Concentration 15.00 17.50 20.00 22.50 25.00
1.00 15.00
A: Concentration
Direction of potential improvement for a process (method of
steepest ascent) 15
