# Statistical Analysis

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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
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

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
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

-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
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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