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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5.3 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?

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• 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
• 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
• 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 Source Squares
DF Square Value Prob gt F Model
59416.22 8 7427.03 11.00 lt 0.0001 A
10683.72 2 5341.86 7.91 0.0020 B
39118.72 2 19559.36 28.97 lt
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 Turkeys
  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

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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
• 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 (??)ij r?i?j (Page 192)
• 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 196)
• Example 5.3

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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
• Example 5.5

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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 5.18)
• Example 5.6

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• Two randomization restrictions Latin square
  design
• An example in Page 209
• Model
• Table 5.22