1

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

2

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

3

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

4

Chemical Process Example

A = reactant concentration, B = catalyst amount, y = recovery

5

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

6

Estimation of Factor Effects

See textbook, pg. 206 For manual calculations

The effect estimates are: A = 8.33, B = -5.00, AB = 1.67

Practical interpretation

])1([21

22)1(

)]1([21

2)1(

2

)]1([21

2)1(

2

baabn

nba

nabAB

ababn

na

nbab

yyB

baabn

nb

naab

yyA

BB

AA

7

Estimation of Factor EffectsEffects (1) a b ab

A -1 +1 -1 +1

B -1 -1 +1 +1AB +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 + + + +

8

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?

9

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

10

Regression Modely = o + 1x1 + 2x2 + 12x1x2 +

or let x3 = x1x2, 3 = 12

y = o + 1x1 + 2x2 + 3x3 +

A linear regression model.Coded variables are related to natural variables by

2/)(2/)(.

1highlow

highlow

ConcConcConcConcConc

x

2/)(2/)(

2highlow

highlow

CatalystCatalystCatalystCatalystCatalyst

x

Therefore,

],[:]1,1[:

],[:.]1,1[:

2

1

highlow

highlow

CatalystCatalystCatalystx

ConcConcConcx

11

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.

12

Refined/reduced Modely = o + 1x1 + 2x2 +

Response: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

13

Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low High VIFIntercept 27.5 1 0.60604 26.12904 28.87096A-Concentration4.166667 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

Regression Model for the Process

14

Residuals and Diagnostic CheckingDESIGN-EXPERT PlotConversion

Residual

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

70

80

90

95

99

DESIGN-EXPERT PlotConversion

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

15

The Response SurfaceDESIGN-EXPERT Plot

ConversionX = A: ConcentrationY = B: Catalyst

Design Points

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

25.277827.5

29.7222

31.9444

3 3

3 3

DESIGN-EXPERT Plot

ConversionX = A: ConcentrationY = B: Catalyst

20.8333

24.1667

27.5

30.8333

34.1667

Con

vers

ion

15.00

17.50

20.00

22.50

25.00

1.00

1.25

1.50

1.75

2.00

A: Concentration B: Catalyst

Direction of potential improvement for a process (method of steepest ascent)