Chapter 6Design & Analysis of Experiments 7E 2009 Montgomery 1 Design of Engineering Experiments Two-Level Factorial Designs Text reference, Chapter 6

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


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

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


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 6.15809 Lenth's SME 7.95671


7

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?


8

Design-Expert output, full model


9

Design-Expert output, edited or reduced model


10

Residuals and Diagnostic Checking


11

The Response Surface


12

The 23 Factorial Design


13

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


14

An Example of a 23 Factorial Design

A = gap, B = Flow, C = Power, y = Etch Rate


15

Table of – and + Signs for the 23 Factorial Design (pg. 218)


16

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


17

Estimation of Factor Effects


18

ANOVA Summary – Full Model


19

Model Coefficients – Full Model


20

Refine Model – Remove Nonsignificant Factors


21

Model Coefficients – Reduced Model


22

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


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


24

The Regression Model


25

Model Interpretation

Cube plots are often useful visual displays of experimental results


26

Cube Plot of Ranges

What do the large ranges

when gap and power are at the high level tell

you?


27


28

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


29

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”?


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


31

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


32

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


33

The Resin Plant Experiment


34

The Resin Plant Experiment


35


36

Estimates of the Effects


37

The Half-Normal Probability Plot of Effects


38

Design Projection: ANOVA Summary for the Model as a 23 in Factors A, C, and D


39

The Regression Model


40

Model Residuals are Satisfactory


41

Model Interpretation – Main Effects and Interactions


42

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


43

Outliers: suppose that cd = 375 (instead of 75)


44

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?


45


46

The Drilling Experiment Example 6.3

A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill


47

Normal Probability Plot of Effects –The Drilling Experiment


48

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


49

• 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


50

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


51

(15.1)


52

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


53

Effect Estimates Following the Log Transformation

Three main effects are large

No indication of large interaction effects

What happened to the interactions?


54

ANOVA Following the Log Transformation


55

Following the Log Transformation


56

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


57

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


58

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)


59

Overview of Lenth’s method

For an individual contrast, compare to the margin of error


60


61

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


62


63

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


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


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?


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


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


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


69

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)


70

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)


71

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


72


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


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.


75


76

ANOVA for Example 6.6 (A Portion of Table 6.22)


77

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


78

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?


79

Center Points and Qualitative Factors

Documents

Chapter 6Design & Analysis of Experiments 7E 2009 Montgomery 1 Design of Engineering Experiments Two-Level Factorial Designs Text reference, Chapter 6