introduction factorial designs
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Factorial Designs
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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
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Definition of a factor effect: The change in the mean response when
the factor is changed from low to high
40 52 20 30
212 2
30 52 20 4011
2 2
52 20 30 401
2 2
A A
B B
A y y
B y y
AB
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50 12 20 401
2 2
40 12 20 509
2 2
12 20 40 5029
2 2
A A
B B
A y y
B y y
AB
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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
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
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The Effect of
Interaction on the
Response SurfaceSuppose that we add an
interaction term to the
model:
1 2
1 2
35.5 10.5 5.5
8
y x x
x x
Interactionis actuallya form of curvature
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When an interaction is large, the corresponding
main effects have little practical meaning.
A significant interaction will often mask thesignificance of main effects.
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5.2 The Advantage of Factorials
One-factor-at-a-time desgin
Compute the main effects of factors
A: A+B-- A-B-
B: A-B-- A-B+
Total number of experiments: 6
Interaction effectsA+B-, A-B+> A-B-=> A+B+is
better???
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5.3 The Two-Factor Factorial Design
5.3.1 An Example
alevels for factor A, blevels for factor B and n
replicates Design a battery: the plate materials (3 levels) v.s.
temperatures (3 levels), and n = 4: 32factorial design
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?
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The data for the Battery Design:
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Completely randomized design: alevels of factor
A, blevels of factor B, nreplicates
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Statistical (effects) model:
is an overall mean, iis the effect of theith level
of the row factor A, jis the effect of thejthcolumn of column factor B and ()ijis the
interaction between iand j.
Testing hypotheses:
1,2,...,
( ) 1, 2,...,
1,2,...,
ijk i j ij ijk
i a
y j b
k n
0)(oneleastat:v.s.,0)(:
0oneleastat:v.s.0:
0oneleastat:v.s.0:
10
110
110
ijij
jb
ia
HjiH
HH
HH
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5.3.2 Statistical Analysis of the Fixed Effects
Model
a
i
b
j
n
k
ijk
ij
ij
n
k
ijkij
ja
i
j
n
k
ijkj
ib
j
i
n
k
ijki
abn
yyyy
n
yyyy
an
yyyy
bnyyyy
1 1
......
1
...
.
.
1
.
..
1
..
1
..
..
1
..
1
..
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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 ja 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 E SS SS SS SS SS
df
abn a b a b ab n
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Mean squares
2
1 1
2
2
1
2
2
1
2
2
))1(
()(
)1)(1(
)(
))1)(1(
()(
1
))1/(()(
1))1/(()(
nab
SSEMSE
ba
n
ba
SSEMSE
b
an
bSSEMSE
a
bn
aSSEMSE
EE
a
i
b
j
ij
ABAB
b
j
j
BB
a
i
i
AA
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The ANOVA table:
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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
Example 5.1
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DESIGN-EXPERT Plot
Life
X = B: Temperature
Y = A: Material
A1 A1
A2 A2
A3 A3
A: Material
Interaction Graph
Life
B: Temperature
15 70 125
20
62
104
146
188
2
2
22
2
2
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Multiple Comparisons:
Use the methods in Chapter 3.
Since the interaction is significant, fix the factorB at a specific level and apply Turkeys test to
the means of factor A at this level.
See Page 174
Compare all abcells means to determine which
one differ significantly
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5.3.3 Model Adequacy Checking
Residual analysis: ijijkijkijkijk yyyye
DESIGN-EXPERT PlotLife
Residual
Normal%p
robab
ility
Normal plot of residuals
-60.75 -34.25 -7.75 18.75 45.25
1
5
10
20
30
50
70
80
90
95
99
DESIGN-EXPERT Plot
Life
Predicted
Residuals
Residuals vs. Predicted
-60.75
-34.25
-7.75
18.75
45.25
49.50 76.06 102.62 129.19 155.75
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DESIGN-EXPERT Plot
Life
Material
Residuals
Residuals vs. Material
-60.75
-34.25
-7.75
18.75
45.25
1 2 3
DESIGN-EXPERT Pl ot
Life
Temperature
Res
iduals
Residuals vs. Temperature
-60.75
-34.25
-7.75
18.75
45.25
1 2 3
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5.3.4 Estimating the Model Parameters
The model is
The normal equations:
Constraints:
ijkijjiijky )(
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
0,0,01111
b
j
ij
a
i
ij
b
j
j
a
i
i
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Estimations:
The fitted value:
Choice of sample size: Use OC curves to choose
the proper sample size.
yyyy
yyyy
y
jiijij
jj
ii
ijijjiijk yy
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Consider a two-factor model without interaction:
Table 5.8
The fitted values: yyyy jiijk
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One observation per cell:
The error variance is not estimable because the
two-factor interaction and the error can not beseparated.
Assume no interaction. (Table 5.9)
Tukey (1949): assume ()ij= r
i
j(Page 183)
Example 5.2
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Degree of freedom:
Main effect: # of levels1
Interaction: the product of the # of degrees offreedom 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 186)
Example 5.3
ijklijkjkik
ijkjiijkly
)()()(
)(
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Example 5.3: Three factors: the percent
carbonation (A), the operating pressure (B); the
line speed (C)
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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 lifeHierarchy principle
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Involve both quantitativeand qualitativefactors
This can be accounted for in the analysis to produce
regression modelsfor the quantitative factors at each
level (or combination of levels) of the qualitativefactors
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A= Material type
B= Linear effect of Temperature
B2= Quadratic effect of
Temperature
AB= Material typeTempLinear
AB2= Material type - TempQuad
B3= Cubic effect of
Temperature (Aliased)
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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 5.20)
ijkkijjiijky )(
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Example 5.6:
Two factors: ground clutter and filter type
Nuisance factor: operator
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Two randomization restrictions: Latin square
design
An example in Page 200.
Model:
Tables 5.23 and 5.24
ijklljkkjiijkly )(