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1
Chapter 5 Introduction to Factorial
Designs
2
5.1 Basic Definitions and Principles
• Study the effects of two or more factors.
• Factorial designs
• Crossed: factors ...
3
Definition of a factor effect: The change in the mean response when
the factor is changed from low to high
40 52 20 30
2...
4
50 12 20 40
1
2 2
40 12 20 50
9
2 2
12 20 40 50
29
2 2
A A
B B
A y y
B y y
AB
+ −
+ −
+ +
= − = − =
+ +
= − = − = −
+ +
...
5
Regression Model &
The Associated
Response Surface
0 1 1 2 2
12 1 2
1 2
1 2
1 2
The least squares fit is
ˆ 35.5 10.5 5.5...
6
The Effect of
Interaction on the
Response Surface
Suppose that we add an
interaction term to the
model:
1 2
1 2
ˆ 35.5 1...
7
• When an interaction is large, the corresponding
main effects have little practical meaning.
• A significant interactio...
8
5.3 The Two-Factor Factorial Design
5.3.1 An Example
• a levels for factor A, b levels for factor B and n
replicates
• D...
9
• The data for the Battery Design:
10
• Completely randomized design: a levels of factor
A, b levels of factor B, n replicates
11
• Statistical (effects) model:
• Testing hypotheses:
1,2,...,
( ) 1,2,...,
1,2,...,
ijk i j ij ijk
i a
y j b
k n
µ τ β ...
12
• 5.3.2 Statistical Analysis of the Fixed Effects
Model
2 2 2
... .. ... . . ...
1 1 1 1 1
2 2
. .. . . ... .
1 1 1 1 1...
13
• Mean squares
2
1 1
2
2
1
2
2
1
2
2
)
)1(
()(
)1)(1(
)(
)
)1)(1(
()(
1
))1/(()(
1
))1/(()(
σ
τβ
σ
β
σ
τ
σ
=
−
=
−−
+=
...
14
• The ANOVA table:
• See Page 180
• Example 5.1
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Response: Life
ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]
Sum of Mean F
Sou...
16
A: M a te ria l
Interaction G raph
Life
B : Te m p e ra tu re
15 70 125
20
62
104
146
188
2
2
22
2
2
17
• Multiple Comparisons:
– Use the methods in Chapter 3.
– Since the interaction is significant, fix the
factor B at a s...
18
5.3.3 Model Adequacy Checking
• Residual analysis: ⋅−=−= ijijkijkijkijk yyyye ˆ
R e s id u a l
Normal%probability
Norm ...
19
R u n N u m b e r
Residuals
Residuals vs. Run
-60.75
-34.25
-7.75
18.75
45.25
1 6 11 16 21 26 31 36
20
M a te ria l
Residuals
Residuals vs. M aterial
-60.75
-34.25
-7.75
18.75
45.25
1 2 3
T e m p e ra tu re
Residuals
Resid...
21
5.3.4 Estimating the Model Parameters
• The model is
• The normal equations:
• Constraints:
ijkijjiijky ετββτµ ++++= )(...
22
• Estimations:
• The fitted value:
• Choice of sample size: Use OC curves to choose
the proper sample size.
( ) ⋅⋅⋅⋅⋅⋅⋅...
23
• Consider a two-factor model without interaction:
– Table 5.8
– The fitted values:
– Figure 5.15
• One observation per...
24
5.4 The General Factorial Design
• More than two factors: a levels of factor A, b
levels of factor B, c levels of facto...
25
• Degree of freedom:
– Main effect: # of levels – 1
– Interaction: the product of the # of degrees of
freedom associate...
26
5.5 Fitting Response Curves and
Surfaces
• An equation relates the response (y) to the factor
(x).
• Useful for interpo...
27
5.6 Blocking in a Factorial Design
• A nuisance factor: blocking
• A single replicate of a complete factorial
experimen...
28
• Two randomization restrictions: Latin square
design
• An example in Page 209
• Model:
• Table 5.22
ijkkjkkjiijkly εδτ...
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Introduction to factorial designs

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Introduction to factorial designs

  1. 1. 1 Chapter 5 Introduction to Factorial Designs
  2. 2. 2 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
  3. 3. 3 Definition of a factor effect: The change in the mean response when the factor is changed from low to high 40 52 20 30 21 2 2 30 52 20 40 11 2 2 52 20 30 40 1 2 2 A A B B A y y B y y AB + − + − + + = − = − = + + = − = − = + + = − = −
  4. 4. 4 50 12 20 40 1 2 2 40 12 20 50 9 2 2 12 20 40 50 29 2 2 A A B B A y y B y y AB + − + − + + = − = − = + + = − = − = − + + = − = −
  5. 5. 5 Regression Model & The Associated Response Surface 0 1 1 2 2 12 1 2 1 2 1 2 1 2 The least squares fit is ˆ 35.5 10.5 5.5 0.5 35.5 10.5 5.5 y x x x x y x x x x x x β β β β ε = + + + + = + + + ≅ + +
  6. 6. 6 The Effect of Interaction on the Response Surface Suppose that we add an interaction term to the model: 1 2 1 2 ˆ 35.5 10.5 5.5 8 y x x x x = + + + Interaction is actually a form of curvature
  7. 7. 7 • When an interaction is large, the corresponding main effects have little practical meaning. • A significant interaction will often mask the significance of main effects.
  8. 8. 8 5.3 The Two-Factor Factorial Design 5.3.1 An Example • a levels for factor A, b levels for factor B and n replicates • 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?
  9. 9. 9 • The data for the Battery Design:
  10. 10. 10 • Completely randomized design: a levels of factor A, b levels of factor B, n replicates
  11. 11. 11 • Statistical (effects) model: • Testing hypotheses: 1,2,..., ( ) 1,2,..., 1,2,..., ijk i j ij ijk i a y j b k n µ τ β τβ ε =  = + + + + =  = 0)(oneleastat:v.s.,0)(: 0oneleastat:v.s.0: 0oneleastat:v.s.0: 10 110 110 ≠∀= ≠=== ≠=== ijij jb ia HjiH HH HH τβτβ βββ τττ  
  12. 12. 12 • 5.3.2 Statistical Analysis of the Fixed Effects Model 2 2 2 ... .. ... . . ... 1 1 1 1 1 2 2 . .. . . ... . 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) a b n a b ijk i j i j k i j a b a b n ij i j ijk ij i j i j k y y bn y y an y y n y y y y y y = = = = = = = = = = − = − + − + − − + + − ∑∑∑ ∑ ∑ ∑∑ ∑∑∑ breakdown: 1 1 1 ( 1)( 1) ( 1) T A B AB ESS SS SS SS SS df abn a b a b ab n = + + + − = − + − + − − + −
  13. 13. 13 • Mean squares 2 1 1 2 2 1 2 2 1 2 2 ) )1( ()( )1)(1( )( ) )1)(1( ()( 1 ))1/(()( 1 ))1/(()( σ τβ σ β σ τ σ = − = −− += −− = − +=−= − +=−= ∑∑ ∑ ∑ = = = = nab SS EMSE ba n ba SS EMSE b an bSSEMSE a bn aSSEMSE E E a i b j ij AB AB b j j BB a i i AA
  14. 14. 14 • The ANOVA table: • See Page 180 • Example 5.1
  15. 15. 15 Response: Life 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 59416.22 8 7427.03 11.00 < 0.0001 A 10683.72 2 5341.86 7.91 0.0020 B 39118.72 2 19559.36 28.97 < 0.0001 AB 9613.78 4 2403.44 3.56 0.0186 Pure E 18230.75 27 675.21 C Total 77646.97 35 Std. Dev. 25.98 R-Squared 0.7652 Mean 105.53 Adj R-Squared 0.6956 C.V. 24.62 Pred R-Squared 0.5826 PRESS 32410.22 Adeq Precision 8.178
  16. 16. 16 A: M a te ria l Interaction G raph Life B : Te m p e ra tu re 15 70 125 20 62 104 146 188 2 2 22 2 2
  17. 17. 17 • 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
  18. 18. 18 5.3.3 Model Adequacy Checking • Residual analysis: ⋅−=−= ijijkijkijkijk yyyye ˆ R e s id u a l Normal%probability Norm al plot of residuals -60.75 -34.25 -7.75 18.75 45.25 1 5 10 20 30 50 70 80 90 95 99 P re d icte d Residuals Residuals vs. P redicted -60.75 -34.25 -7.75 18.75 45.25 49.50 76.06 102.62 129.19 155.75
  19. 19. 19 R u n N u m b e r Residuals Residuals vs. Run -60.75 -34.25 -7.75 18.75 45.25 1 6 11 16 21 26 31 36
  20. 20. 20 M a te ria l Residuals Residuals vs. M aterial -60.75 -34.25 -7.75 18.75 45.25 1 2 3 T e m p e ra tu re Residuals Residuals vs. Tem perature -60.75 -34.25 -7.75 18.75 45.25 1 2 3
  21. 21. 21 5.3.4 Estimating the Model Parameters • The model is • The normal equations: • Constraints: ijkijjiijky ετββτµ ++++= )( ⋅ ⋅⋅ == ⋅⋅⋅ == ⋅⋅⋅ = === =+++ =+++ =+++ =+++ ∑∑ ∑∑ ∑∑∑∑ ijijjiij j a i ijj a i ij i b j ij b j jii a i b j ij b j j a i i ynnnn ynannan ynnbnbn ynanbnabn )(:)( )(: )(: )(: 11 11 1 111 τββτµτβ τββτµβ τββτµτ τββτµµ ( ) ( ) 0,0,0 1111 ==== ∑∑∑∑ ==== b j ij a i ij b j j a i i τβτββτ
  22. 22. 22 • Estimations: • The fitted value: • Choice of sample size: Use OC curves to choose the proper sample size. ( ) ⋅⋅⋅⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅⋅⋅ ⋅⋅⋅ +−−= −= −= = yyyy yy yy y jiijij jj ii τβ β τ µ ˆ ˆ ˆ ( ) ⋅=+++= ijijjiijk yy τββτµ ˆˆˆˆ
  23. 23. 23 • Consider a two-factor model without interaction: – Table 5.8 – The fitted values: – 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 (τβ)ij = rτi βj (Page 192) – Example 5.2 ⋅⋅⋅⋅⋅⋅⋅ −+= yyyy jiijk ˆ
  24. 24. 24 5.4 The General Factorial Design • More than two factors: a levels of factor A, b levels of factor B, c 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.
  25. 25. 25 • 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: – – The ANOVA table (see Table 5.12) – Computing formulas for the sums of squares (see Page 196) – Example 5.3 ijklijkjkik ijkjiijkly ετβγβγτγ τβγβτµ ++++ ++++= )()()( )(
  26. 26. 26 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
  27. 27. 27 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: – No interaction between blocks and treatments • ANOVA table (Table 5.18) • Example 5.6 ijkkijjiijky εδτββτµ +++++= )(
  28. 28. 28 • Two randomization restrictions: Latin square design • An example in Page 209 • Model: • Table 5.22 ijkkjkkjiijkly εδτββταµ ++++++= )(

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