# Design of Engineering Experiments Part 5 � The 2k Factorial Design - PowerPoint

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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

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