Chapter 5 Introduction to Factorial Designs

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```					Chapter 5 Introduction to 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

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Regression Model &
The Associated
Response Surface

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The Effect of
Interaction on the
Response Surface
Suppose that we add an
interaction term to the
model:

Interaction is actually
a 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 the
significance 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 effects
A+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
• 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: 32 factorial 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? 9
• The data for the Battery Design:

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• Completely randomized design: a levels of factor
A, b levels of factor B, n replicates

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• Statistical (effects) model:

m is an overall mean, ti is the effect of the ith level
of the row factor A, bj is the effect of the jth
column of column factor B and (t b)ij is the
interaction between ti and bj .
• Testing hypotheses:

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• 5.3.2 Statistical Analysis of the Fixed Effects
Model

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• Mean squares

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• The ANOVA table:

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Example 5.1
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

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• 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 Page 174
– Compare all ab cells means to determine which
one differ significantly

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5.3.3 Model Adequacy Checking
• Residual analysis:

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5.3.4 Estimating the Model Parameters
• The model is

• The normal equations:

• Constraints:
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• Estimations:

• The fitted value:

• Choice of sample size: Use OC curves to choose
the proper sample size.
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• Consider a two-factor model without interaction:
– Table 5.8
– The fitted values:

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• 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 (tb)ij = rtibj (Page 183)
– Example 5.2

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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.

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• 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 186)
– Example 5.3
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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 life
– Hierarchy principle

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– Involve both quantitative and qualitative factors
– This can be accounted for in the analysis to produce
regression models for the quantitative factors at each
level (or combination of levels) of the qualitative
factors
A = Material type
B = Linear effect of Temperature
B2 = Quadratic effect of
Temperature
AB = Material type – TempLinear
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)

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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

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