Chapter 5 Introduction to Factorial Designs

1

5.1 Basic Definitions and Principles

• Study the effects of two or more factors.

• Factorial designs • Crossed: factors are arranged in a factorial design • Main effect: the change in response produced by a change in the level of the factor 2

Definition of a factor effect: The change in the mean response when the factor is changed from low to high

A



y A

 

y A

 

B



y B

 

y B

 

AB

 2  2 2 2  2    1 2  21  11 3

A



y A

 

y A

 

B



y B

 

y B

 

AB

  2 2 2  2    29 2 2  1   9 4

Regression Model & The Associated Response Surface

y

   0  

x

1 1 

x x

12 1 2     2

x

2

y

 The least squares fit is 

x

1  5.5

x

2   0.5

x x

1 2

x

1  5.5

x

2 5

The Effect of Interaction on the Response Surface

Suppose that we add an interaction term to the model:

y

  8

x x

1 2 

x

1  5.5

x

2

Interaction

is actually a form of

curvature

6

• When an interaction is large, the corresponding main effects have little practical meaning.

• A significant interaction will often mask the significance of main effects. 7

5.3 The Two-Factor Factorial Design

• 5.3.1 An Example

a

levels for factor A, replicates

b

levels for factor B and

n

• Design a battery: the plate materials (3 levels) v.s. temperatures (3 levels), and

n = 4

• Two questions: – What effects do material type and temperature have on the life of the battery?

– Is there a choice of material that would give uniformly long life regardless of temperature?

8

• The data for the Battery Design: 9

• Completely randomized design:

a

levels of factor A,

b

levels of factor B,

n

replicates 10

• Statistical (effects) model:

y ijk j

 (  )

ij

 

ijk

  

k i j

  1, 2,..., 1, 2,..., 1, 2,...,

a b n

• Testing hypotheses:

H

0 :  1    

a

 0 v.s.

H

1 : at least one 

i

 0

H

0 :  1    

b

 0 v.s.

H

1 : at least one 

j

 0

H

0 : (  )

ij

 0 

i

,

j

v.s.

H

1 : at least one (  )

ij

 0 11

• 5.3.2 Statistical Analysis of the Fixed Effects Model

i a

  1

j b

 1

k n

 1 (

y ijk



y

...

) 2 

bn i a

  1 (

y i

..



y

...

) 2 

an j b

  1 (

y



n i a b

  1

j

 1 (

y ij

.



y i

..



y



y

...

) 2 

y

...

) 2 

i a

  1

j b

 1

k n

 1 (

y ijk



y ij

.

) 2

SS T



SS A



SS B



SS AB



SS E df

breakdown:

abn a

1

b

1 (

a

1)(

b

 1)

12

• Mean squares

E

(

MS A

) 

E

(

SS A

/(

a

 1 ))   2

E

(

MS B

) 

E

(

SS B

/(

b

 1 ))   2 

a bn

  1

i a

  1

i

2 

an j b

  1

b

 1 

j

2

E

(

MS AB

) 

E

( (

a SS AB

 1 )(

b

 1 ) )   2 

n i b a

  1  1

j

(  )

ij

2 (

a

 1 )(

b

 1 )

E

(

MS E

) 

E

(

ab SS

(

n E

 1 ) )   2 13

• The ANOVA table: • See Page 180 • Example 5.1

14

Response: Life ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Source

Model

A B AB

Sum of Squares

59416.22

10683.72

39118.72

9613.78

DF

8

2 2 4

Pure E 18230.75 27 C Total 77646.97 35

Mean Square

7427.03 11.00 < 0.0001

5341.86

19559.36

2403.44

675.21

F Value

7.91

28.97

3.56

Prob > F

0.0020

< 0.0001

0.0186

Std. Dev. 25.98

Mean 105.53

C.V.

24.62

PRESS 32410.22

R-Squared Adj R-Squared Pred R-Squared Adeq Precision 0.7652

0.6956

0.5826

8.178

15

DESIGN-EXPERT Plot Life X = B: Temperature Y = A: Material A1 A1 A2 A2 A3 A3 188 146 104 Interaction Graph A: Material 62 20 15 70 B: Tem perature 125 16

• Multiple Comparisons: – Use the methods in Chapter 3.

– Since the interaction is significant, fix the factor B at a specific level and apply Turkey’s test to the means of factor A at this level.

– See Pages 182, 183 – Compare all

ab

cells means to determine which one differ significantly 17

5.3.3 Model Adequacy Checking • Residual analysis:

e ijk



y ijk



y

ˆ

ijk



y ijk



y ij

 DESIGN-EXPERT Plot Life Normal plot of residuals DESIGN-EXPERT Plot Life 45.25

Residuals vs. Predicted 99 95 90 80 70 50 30 20 10 5 1 18.75

-7.75

-34.25

-60.75

-60.75

-34.25

-7.75

18.75

45.25

Res idual 49.50

76.06

102.62

129.19

155.75

Predicted 18

DESIGN-EXPERT Plot Life 45.25

Residuals vs. Run 18.75

-7.75

-34.25

-60.75

1 6 11 16 21 26 31 36 Run Num ber 19

DESIGN-EXPERT Plot Life 45.25

Residuals vs. Material DESIGN-EXPERT Plot Life 45.25

Residuals vs. Temperature 18.75

18.75

-7.75

-7.75

-34.25

-34.25

-60.75

1 2 Material 3 -60.75

1 2 Tem perature 3 20

5.3.4 Estimating the Model Parameters • The model is

y ijk

   

i

 

j

 (  )

ij

 

ijk

• The normal equations:  :

abn

 

bn i a

  1 

i



an j b

  1 

j



n i

 1

j b a

  1 (  )

ij



y

  

i

:

bn

 

bn



i



n j b

  1 

j



n j b

  1 (  )

ij



y i

  

j

(  : )

ij an

 :

n

  

n i a

  1 

n



i i

 

an



n



j



j



n i a

  1

n

(  )

ij

(   )

ij y ij

 • Constraints:

i a

  1 

i

 0 ,

j b

  1 

j

 0 ,

i a

  1  

ij



y



j

 

j b

  1  

ij

 0 21

• Estimations:  ˆ 

y

  ˆ

i



y i

   ˆ 

j

 

ij y



j

   

y ij



y

 

y

  

y i

 

y



j

 

y

  • The fitted value:

y

ˆ

ijk

  ˆ   ˆ

i

  ˆ

j

  

ij



y ij

 • Choice of sample size: Use OC curves to choose the proper sample size.

22

• Consider a two-factor model without interaction: – Table 5.8

– The fitted values:

y

ˆ

ijk



y i

 

y



j

 

y

  – Figure 5.15

• One observation per cell: – The error variance is not estimable because the two-factor interaction and the error can not be separated. – Assume no interaction. (Table 5.9) – Tukey (1949): assume

(

– Example 5.2



) ij = r



i



j

(Page 192) 23

5.4 The General Factorial Design

• More than two factors:

a

levels of factor B,

c

levels of factor A,

b

levels of factor C, …, and

n

replicates.

• Total

abc … n

observations.

• For a fixed effects model, test statistics for each main effect and interaction may be constructed by dividing the corresponding mean square for effect or interaction by the mean square error.

24

• Degree of freedom: – Main effect: # of levels – 1 – Interaction: the product of the # of degrees of freedom associated with the individual components of the interaction.

• The three factor analysis of variance model: –

y ijkl

   

i

 

j

 

k

 (  )

ij

 (  )  (  )  (  )

ik jk

– The ANOVA table (see Table 5.12)

ijk

 

ijkl

– Computing formulas for the sums of squares (see Page 196) – Example 5.3

25

5.5 Fitting Response Curves and Surfaces

• An equation relates the response (

y

) to the factor (

x

).

• Useful for interpolation.

• Linear regression methods • Example 5.4

– Study how temperatures affects the battery life – Hierarchy principle • Example 5.5 26

5.6 Blocking in a Factorial Design

• A nuisance factor: blocking • A single replicate of a complete factorial experiment is run within each block.

• Model:

y ijk

   

i

 

j

 (  )

ij

 

k

 

ijk

– No interaction between blocks and treatments • ANOVA table (Table 5.18) • Example 5.6

27

• Two randomization restrictions: Latin square design • An example in Page 209 • Model:

y ijkl

   

i

 

j

 

k

 (  )

jk

 

k

 

ijk

• Table 5.22

28