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Chapter 5 Introduction to Factorial Designs

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Chapter 5 Introduction to Factorial Designs Powered By Docstoc
					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




                                                                  3
4
Regression Model &
The Associated
Response Surface




                     5
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



                           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.




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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???
                                        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: 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




                                                     11
• 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:




                                                        12
• 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:
                                        24
• Estimations:




• The fitted value:



• Choice of sample size: Use OC curves to choose
  the proper sample size.
                                                   25
• 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


                                                  27
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
                                                       29
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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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posted:4/1/2014
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