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					Statistical Analysis


      Professor Lynne Stokes
   Department of Statistical Science

             Lecture 14
     Sequential Experimentation,
     Screening Designs, Fold-Over
               Designs
 Don’t Risk all Experimental Resources
 on a Single Comprehensive Experiment



•Usually many inert factors, few dominant ones
•Unexpected effects may be found early
•Experiment could be terminated early with
  substantial cost savings
•Comprehensive evaluation of a few dominant
  factors is usually more informative than
  little information on many factors
              Conduct a Screening Experiment
               to Identify Dominant Factors




 Augment the Screening
 Experiment to Identify            Conduct a RV Experiment
   Strong Two-Factor               with the Dominant Factors
      Interactions



              Comprehensive Experiment with
             a Few Factors and Multiple Levels
                             or
                 Design to Quantitatively
             Characterize the Response Surface

Figure 7.7 A simple strategy for a sequence of experiments.
Sequential Experimentation

   Large experiments
   Design so that key fractions can be run in
    sequence
       Key fractions : Resolution III, IV, or V
   Analyze each sequence of data as it is
    completed
   Based on the results of the analysis
       Continue experiment
       Terminate
       Redesign with dominant/new factors
Acid Plant Corrosion Study

                                     Coded Level
        Factor                      -1          +1
 Raw Material Feed Rate          3,000pph     6,000pph
 Gas Temperature                 100oC        220oC
 Scrubber Water                  5%           20%
 Reactor Bed Acid                20%          30%
 Exit Temperture                 300oC        360oC
 Reactant Distribution Point     East         West

            Plant must cease commercial
    production during experimentation -- test runs
                 must be minimized
                                                MGH Table 7.1
Screening Experiments


  Highly effective for isolating vital few strong effects
  should be used ONLY under the proper circumstances


     Very few test runs
     Ability to assess main effects only
     Generally leads to a comprehensive
      evaluation of a few dominant factors
     Potential for bias
Plackett-Burman Screening
Designs
   Any number of factors, each having 2 levels
   Interactions nonexistent or negligible Relative
    to main effects
   Number of test runs is a multiple of 4
   At least 6 more test runs than factors should
    be used
Construction
   Determine the number of factors (k) to be included
    in the design
   Determine the experiments size : at least k + 6
       6df for error
   Select the design generator from Table 7A2
   Generate the rows of the design
       Design generator is the first row
       Move all levels in the previous row one position to the left;
        move the first level of the previous row to the last position
       Continue the previous step until n - 1 rows are filled
       The last row has all levels equal to -1
Construction (con’t)
   Randomize
       Randomly Assign Factors to Columns; Delete Unassigned
        Columns
       Randomly Permute the Rows
Acid Plant Corrosion Study

                                     Coded Level
       Factor                      -1          +1
Raw Material Feed Rate          3,000pph     6,000pph
Gas Temperature                 100oC        220oC
Scrubber Water                  5%           20%
Reactor Bed Acid                20%          30%
Exit Temperture                 300oC        360oC
Reactant Distribution Point     East         West


   Seeking identification of dominant main effects
 Plackett-Burman Design :
 Corrosion Study
    k = 6 Factors
    n = 12 (Minimum Recommended)
                                                  Design Generator

Run No.   A    B    C    D    E     F   G    H        I    J     K
   1      1    1    -1    1   1     1   -1   -1      -1    1     -1
   2      1    -1   1     1   1    -1   -1   -1      1     -1    1
   3      -1   1    1     1   -1   -1   -1   1       -1    1     1
   4      1    1    1    -1   -1   -1   1    -1      1     1     -1
   5      1    1    -1   -1   -1    1   -1   1       1     -1    1
   6      1    -1   -1   -1   1    -1   1    1       -1    1     1
   7      -1   -1   -1    1   -1    1   1    -1      1     1     1
   8      -1   -1   1    -1   1     1   -1   1       1     1     -1
   9      -1   1    -1    1   1    -1   1    1       1     -1    -1
  10      1    -1   1     1   -1    1   1    1       -1    -1    -1
  11      -1   1    1    -1   1     1   1    -1      -1    -1    1
  12      -1   -1   -1   -1   -1   -1   -1   -1      -1    -1    -1
 Plackett-Burman Design :
 Corrosion Study
    k = 6 Factors
    n = 12 (Minimum Recommended)

Run No.   A    B    C    D    E     F   G    H     I   J    K
   1      1    1    -1    1   1     1   -1   -1   -1   1    -1
   2      1    -1   1     1   1    -1   -1   -1   1    -1   1
   3      -1   1    1     1   -1   -1   -1   1    -1   1    1
   4      1    1    1    -1   -1   -1   1    -1   1    1    -1
   5      1    1    -1   -1   -1    1   -1   1    1    -1   1
   6      1    -1   -1   -1   1    -1   1    1    -1   1    1
   7      -1   -1   -1    1   -1    1   1    -1   1    1    1
   8      -1   -1   1    -1   1     1   -1   1    1    1    -1
   9      -1   1    -1    1   1    -1   1    1    1    -1   -1
  10      1    -1   1     1   -1    1   1    1    -1   -1   -1
  11      -1   1    1    -1   1     1   1    -1   -1   -1   1
  12      -1   -1   -1   -1   -1   -1   -1   -1   -1   -1   -1
 Plackett-Burman Design :
 Corrosion Study
    k = 6 Factors
    n = 12 (Minimum Recommended)

Run No.   A    B    C    D    E     F   G    H     I   J    K
   1      1    1    -1    1   1     1   -1   -1   -1   1    -1
   2      1    -1   1     1   1    -1   -1   -1   1    -1   1
   3      -1   1    1     1   -1   -1   -1   1    -1   1    1
   4      1    1    1    -1   -1   -1   1    -1   1    1    -1
   5      1    1    -1   -1   -1    1   -1   1    1    -1   1
   6      1    -1   -1   -1   1    -1   1    1    -1   1    1
   7      -1   -1   -1    1   -1    1   1    -1   1    1    1
   8      -1   -1   1    -1   1     1   -1   1    1    1    -1
   9      -1   1    -1    1   1    -1   1    1    1    -1   -1
  10      1    -1   1     1   -1    1   1    1    -1   -1   -1
  11      -1   1    1    -1   1     1   1    -1   -1   -1   1
  12      -1   -1   -1   -1   -1   -1   -1   -1   -1   -1   -1
 Plackett-Burman Design :
 Corrosion Study
    k = 6 Factors
    n = 12 (Minimum Recommended)

Run No.   A    B    C    D    E     F   G    H     I   J    K
   1      1    1    -1    1   1     1   -1   -1   -1   1    -1
   2      1    -1   1     1   1    -1   -1   -1   1    -1   1
   3      -1   1    1     1   -1   -1   -1   1    -1   1    1
   4      1    1    1    -1   -1   -1   1    -1   1    1    -1
   5      1    1    -1   -1   -1    1   -1   1    1    -1   1
   6      1    -1   -1   -1   1    -1   1    1    -1   1    1
   7      -1   -1   -1    1   -1    1   1    -1   1    1    1
   8      -1   -1   1    -1   1     1   -1   1    1    1    -1
   9      -1   1    -1    1   1    -1   1    1    1    -1   -1
  10      1    -1   1     1   -1    1   1    1    -1   -1   -1
  11      -1   1    1    -1   1     1   1    -1   -1   -1   1
  12      -1   -1   -1   -1   -1   -1   -1   -1   -1   -1   -1
 Plackett-Burman Design :
 Corrosion Study

               Scrub.        Distn.        Bed     Exit    Gas              Feed
Run No.   A    Water    C    Point    E    Acid   Temp.   Temp.    I   J    Rate
   1       1     1      -1     1      1     1       -1      -1    -1   1     -1
   2       1     -1     1      1      1     -1      -1      -1    1    -1     1
   3      -1     1      1      1      -1    -1      -1      1     -1   1      1
   4       1     1      1      -1     -1    -1      1       -1    1    1     -1
   5       1     1      -1     -1     -1    1       -1      1     1    -1     1
   6       1     -1     -1     -1     1     -1      1       1     -1   1      1
   7      -1     -1     -1     1      -1    1       1       -1    1    1      1
   8      -1     -1     1      -1     1     1       -1      1     1    1     -1
   9      -1     1      -1     1      1     -1      1       1     1    -1    -1
  10       1     -1     1      1      -1    1       1       1     -1   -1    -1
  11      -1     1      1      -1     1     1       1       -1    -1   -1     1
  12      -1     -1     -1     -1     -1    -1      -1      -1    -1   -1    -1


                        Randomly assign factors to columns
Plackett-Burman Design :
Corrosion Study

 Original   Scrub.    Distn.    Bed       Exit       Gas    Feed
 Run No.    Water     Point     Acid     Temp.      Temp.   Rate
    11         1        -1       1         1          -1      1
    3          1        1        -1        -1         1       1
    8         -1        -1       1         -1         1      -1
    1          1        1        1         -1         -1     -1
    4          1        -1       -1        1          -1     -1
    12        -1        -1       -1        -1         -1     -1
    6         -1        -1       -1        1          1       1
    10        -1        1        1         1          1      -1
    7         -1        1        1         1          -1      1
    5          1        -1       1         -1         1       1
    2         -1        1        -1        -1         -1      1
    9          1        1        -1        1          1      -1


                     Eliminate unassigned columns
                        randomly permute rows
Plackett-Burman Design :
Corrosion Study

 Original   Scrub.   Distn.   Bed     Exit    Gas        Feed
 Run No.    Water    Point    Acid   Temp.   Temp.       Rate
    11        20     East      30     360     100        6000
    3         20     West      20     300     220        6000
    8         5      East      30     300     220        3000
    1         20     West      30     300     100        3000
    4         20     East      20     360     100        3000
    12        5      East      20     300     100        3000
    6         5      East      20     360     220        6000
    10        5      West      30     360     220        3000
    7         5      West      30     360     100        6000
    5         20     East      30     300     220        6000
    2         5      West      20     300     100        6000
    9         20     West      20     360     220        3000




                                        Resolution III
Human Performance Testing

      Response
           Eye Focus Time (ms)

         Predictors
               (A) Acuity or Sharpness of Vision
               (B) Distance from Eye to Target
2 Levels Each  (C) Target Shape
               (D) Illumination Level
               (E) Target Size
               (F) Target Density
               (G) Subject
               Only a few effects anticipated,
                      no interactions
Design Considerations
   Complete factorial : 27 + repeats = 128 +
    repeats
   Very few effects expected, no interactions

                     Solution
              Fractional Factorial RIII
Human Performance Testing

                  Design:   2 7 4
                              III
                                             n=8



   Defining Equation
         I = ABD = ACE = BCF = ABCG

   Added Factors
        D = AB , E = AC , F = BC , G = ABC

   Implicit Equations
         24 - 4 - 1 = 11
Human Performance Testing

Complete Defining Relation
   I = ABD = ACE = BCF = ABCG = BCDE = ACDF
     = CDG = ABEF = BEG = AFG = DEF = ADEG
     = CEFG = BDFG = ABCDEFG        Implicit Contrasts
Human Performance Testing

Complete Defining Relation
   I = ABD = ACE = BCF = ABCG = BCDE = ACDF
     = CDG = ABEF = BEG = AFG = DEF = ADEG
     = CEFG = BDFG = ABCDEFG        Implicit Contrasts


             Main-Effect Aliases
               A = BD = CE = FG
               B = AD = CF = EG
               C = AE = BF = DG        Assuming No 3fi
               D = AB = CG = EF
               E = AC = BG = DF
               F = BC = AG = DE
               G = CD = BE = AF
Human Performance Testing

Complete Defining Relation
   I = ABD = ACE = BCF = ABCG = BCDE = ACDF
     = CDG = ABEF = BEG = AFG = DEF = ADEG
     = CEFG = BDFG = ABCDEFG        Implicit Contrasts


             Main-Effect Aliases
               A + BD + CE + FG
               B + AD + CF + EG    Alternative Interpretation
               C + AE + BF + DG           of Aliasing:
               D + AB + CG + EF     Each Measured Effect is
               E + AC + BG + DF     the Sum of Four Effects
               F + BC + AG + DE
               G + CD + BE + AF
  Human Performance Testing

Run      A     B     C     D=AB E=AC F=BC G=ABC Time
  1     -1    -1    -1      +1     +1     +1     -1       85.5
  2     +1    -1    -1      -1     -1     +1     +1       75.1
  3     -1    +1    -1      -1     +1     -1     +1       93.2
  4     +1    +1    -1      +1     -1     -1     -1      145.4
  5     -1    -1    +1      +1     -1     -1     +1       83.7
  6     +1    -1    +1      -1     +1     -1     -1       77.6
  7     -1    +1    +1      -1     -1     +1     -1       95.0
  8     +1    +1    +1      +1     +1     +1     +1      141.8
Effects 20.63 38.38 -.28   28.88   -.28   -.63   -2.43
Human Performance Testing

 Conclusions
       A, B, and D are the primary factors that affect
       eye focus times
Human Performance Testing

 Conclusions
 A, B, and D are the primary factors that affect
 eye focus times

              Key Main-Effect Aliases
                     A = BD
                     B = AD
                     D = AB


           Could the primary effects be only
          two factors and their interaction ?
Human Performance Testing

                              7 4
          Fold-Over Design: 2 III



        Reverse the signs on all levels
         of all factors in the design

   Defining Equation
         I = -ABD = -ACE = -BCF = -ABCG

   Added Factors
        -D = AB , -E = AC , -F = BC , -G = ABC
  Human Performance Testing:
  Fold-Over Design
Run      A      B     C      D=-AB E=-AC F=-BC G=-ABC Time
  1      +1     +1    +1      -1     -1    -1     +1     91.3
  2      -1     +1    +1      +1     +1    -1     -1     136.7
  3      +1     -1    +1      +1     -1    +1     -1     82.4
  4      -1     -1    +1      -1     +1    +1     +1     73.4
  5      +1     +1    -1      -1     +1    +1     -1     94.1
  6      -1     +1    -1      +1     -1    +1     +1     143.8
  7      +1     -1    -1      +1     +1    -1     +1     87.3
  8      -1     -1    -1      -1     -1    -1     -1     71.9
Effects -17.68 37.73 -3.33   29.88   .53   1.63   2.68
Human Performance Testing:
Fold-Over Design
Combined Effects:
     Original Design  A + BD + CE + FG
     Fold-Over Design A - BD - CE - FG
     Average          A
     Difference/2         BD + CE + FG


                           Conclusion:
              Reversing ALL signs in a second fraction
     unaliases ALL main effects from two-factor interactions
      (still assumes higher-order interactions are negligible)
 Human Performance Testing:
 Fold-Over Design

                                                    n = 16
             Contrasts Average    Difference / 2
                   A   A= 1.48    BD+CE+FG= 19.15
                   B   B= 38.05   AD+CF+EG= .33
                   C   C= -1.80   AE+BF+DG= 1.53
Original Design:
    D = AB
                   D   D= 29.38   AB+CG+EF= -.50
                   E   E= .13     AC+BG+DF= -.40
                   F   F= .50     BC+AG+DE= -1.53
                   G   G= .13     CD+BE+AF= -2.55

                         Conclusions ?
Fold-Over Designs

   Reverse the signs on one or more factors
   Run a second fraction with the sign reversals
   Use the confounding pattern of the original
    and the fold-over design to determine the
    alias structure
       Averages
       Half-Differences
Human Performance Testing

Complete Defining Relation Reversing the Signs on B
   I = -ABD = ACE = -BCF = -ABCG = -BCDE = ACDF
     = CDG = -ABEF = -BEG = AFG = DEF = ADEG
     = CEFG = -BDFG = -ABCDEFG

            Main-Effect Aliases
               A - BD + CE + FG
              -B + AD + CF + EG
               C + AE - BF + DG
               D - AB + CG + EF
               E + AC - BG + DF
               F - BC + AG + DE
               G + CD - BE + AF
Human Performance Testing:
Fold-Over Design
Combined Effects:
     Original Design  A + BD + CE + FG
                                                             Similar
     Fold-Over Design A - BD + CE + FG                       With All
     Average          A      + CE + FG                      Main Effects
     Difference/2         BD                                 Except B


      Original Design   B + AD + CF + EG
      Fold-Over Design -B + AD + CF + EG
      Average               AD + CE + FG
      Difference/2      B
                             Conclusion:
      Reversing the signs on ONE factor in a second fraction
    unaliases its main effect and ALL its two-factor interactions

				
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