Class Meeting 25 by HC121109064610

VIEWS: 5 PAGES: 17

									CHEN 3010
Applied Data Analysis
Class Meeting #24
Tuesday, Nov. 25th


     Today's Topics

    Experimental Design
          Background
          Factorial design
          Details on 22 design

  Reading for Thursday: pages 333-349
  Homework #12 posted, due Tuesday, Dec. 7th OR
        Wednesday Dec. 8th by 9 a.m.
                                                  1
Experimental Design
        (a.k.a. Statistical Design of Experiments, SDOE, DOE )
   inputs                   Process                  outputs
   (a.k.a.                     or                    (a.k.a.
   factors)                 Product                  responses)
some inputs can be                         some outputs represent
set/controlled while                       primary product quality
others cannot – latter                     measures, others relate
called environmental                       to secondary constraints
Possible scenarios
  preliminary product/process design
      10’s of factors – need to screen to find important ones
      only crude linear models req’d
  final product/process design
       handful of factors, all important
       nonlinear models usually req’d
  on-line optimization and control                               2
       “on the fly ” adjustments during production
Need for Miminal Experimentation
  running experimental tests costly and time consuming
  practical need to extract as much information as posssible
         with a minimal amount of experimentation
  naïve approach to experimentation yields far too many
         experimental runs
  proper approach to experimental design an absolute
         necessity in R&D, design and production
What is done in experimental design?
  determine the factors that are going to be varied and the
  levels to be set for each factor
  determine the experimental plan: the schedule for
  experimental runs and what are the factor levels for each run
  set up the scheme for processing the experimental results
                                                               3
Experimental Designs Are Sequential
                                  final
             screening
                                 design
types
        fractional factorial   full factorial
of
                               central composite
designs


 Where to start?
    simple factorial designs
    ANOVA to determine significant factors
    linear regression to determine simple models



                                                   4
Naïve Designs
   3 factors
   10 levels chosen for each factor
   1000 experimental runs req’d     (usually infeasible)
Factorial Designs
   3 factors
   3 levels chosen for each factor
   27 experimental runs req’d

Categorization of Simple Factorial Designs

                                                          k
    l        no. of levels for each factor            l
    k         no. of factors

        no. of runs req’d for full factorial design           5
The 22 Factorial Design
      Bhigh                                     2 factors
                 b                ab
                                                2 levels each
                                                4 experiments
    Factor
                                            the various experimental
       B
                                            arrangements are called
                                            “treatments”
                (1)                a
         Blow                                   (1)    Alow Blow
             Alow      Factor A    Ahigh        a      Ahigh Blow
                                                b      Alow Bhigh
Note: factors may be quantitative               ab     Ahigh Bhigh
or qualitative ( “categorical” )
                                         alternately
  Here, the experimental                         (1)    -1    -1
  levels for factors A and B                     a     +1     -1
  are normalized or “coded”                      b      -1    +1
  to the range –1 to +1                          ab    +1     +1   6
Computing the “Effects”                       Bhigh
(effects are change estimates)                        b                  ab

Main Effects (average of 2 changes)
         a   1    ab  b           Factor
     A           
                                             B
                    2
       b   1    ab  a 
                                                      (1)                 a
                                              Blow
     B         
                                                 Alow                     Ahigh
                   2                                          Factor A

 With replicated experiments (experimental set run n times)
                                                                   this part is
         a   1    ab  b  1
     A                             ab  a  b   1         called
                   2n              2n                    
                                                                   “contrastA”
        b   1    ab  a  1
     B                                                          “contrastB”
                                     ab  b  a   1 
                  2n              2n                    
                                                                           7
Interaction Effect
  Does the change with respect to A depend on the level of B?
             1
       AB      ab   1  a  b 
            2n                    
  Note: the change below will equal zero if there is no interaction
    ab  b   a  1 
Response                         Response
                     at Blow                           at Blow



      at Bhigh                          at Bhigh
      Alow       A       Ahigh          Alow       A       Ahigh

    no apparent interaction            yes, apparent interaction8
Layout Table for Experimental Design
                             Effect
   Treatment        A          B         AB

       (1)          -1         -1        +1                 Note: this
       a           +1          -1         -1                column turns
                                                            out to be the
       b            -1        +1          -1                product of the
                                                            A and B cols
       ab          +1         +1         +1
                                                        and the dot-product
                                                        with the treatment
                                                        column gives the
           dot-product gives contrast                   interaction contrast
     1   1  a   1  b   1  ab   1
              ab  b  a   1                                        9
Example: Problem 7-3, page 332
Development of ultrafiltration membrane for separating
separating proteins & peptide drugs from fermentation broth
                                  Time (hours)
                             1                    3
                    69.6         71.5      80.0       81.6
               2
    Additive        70.0         69.0      83.0       84.3
     (wt%)          91.0         93.0      92.3       93.4
               5
                    93.2         87.2      88.5       95.6
                                 entries: separation values in %
 Coding the factor levels:
                Additive  2                       Time  1
       x1  2               1           x2  2           1
                   52                               31
            2             7
       x1  Additive                     x2  Time  2
            3             3                                        10
Arranging the design & results in standard table form:
                      Effects                          Replicate Results
Treatment   A            B          AB        1          2           3      4
   (1)      -1           -1          1       69.6       71.5       70.0    69.0
    a        1           -1         -1       91.0       93.2       93.0    87.2
    b       -1            1         -1       80.0       81.6        8.0    84.3
   ab        1            1          1       92.3       93.4       88.5    95.6

Summary statistics:
                 Treatment      Total    Average    StdDev
                    (1)         280.1     70.0       1.07
                     a          364.4     91.1       2.78
                     b          325.9     81.5       2.03
                    ab          369.8     92.5       2.97

Computing the A effect:
    1                        1
A
   2n                     2  4 369.8  364.4  325.9  280.1  16.0
       ab  a  b  1  
      
                                                                            11
Computing the effects:
                               Effects
    Treatment        A            B         AB
       (1)           -1           -1         1
        a             1           -1        -1
        b            -1            1        -1
       ab             1            1         1
     Contrast         128.2          51.2     -40.4
      Effect            16.0          6.4      -5.1
                                                                  variance of
Significance of the effect values?                         e2    individual
    [need estimate of uncertainty]                                responses
                                   e2
        Var  Effect                                n  2k 2   degrees of
                               n  2k 2                          freedom
Compute variance estimate for                          Treatment Variance
each treatment level – estimate                           (1)      1.14
of “local error” – then pool estimates                     a       7.75
                                                           b       4.12
                       5.45                               ab       8.82
      s.e.effect         22
                               1.17
                      42                                 Pooled   5.45         12
Statistical significance of the effects
                                            Effects
                                 A             B        AB
                    Effect           16.0         6.4     -5.1
                        t0           13.7         5.5     -4.3

                     alpha            5%
       t(alpha/2,2^l*(n-1))          2.18

All significant at 95%-confidence limit
ANOVA will be generally used to determine significance
of effects
Linear regression model       y  0  1 x1   2 x2  12 x1 x2
  ˆ
  0  y..  83.8              ˆ  Beffect  3.20
                              2
                                     2
 ˆ  Aeffect  8.01
 1                           ˆ  ABeffect  2.53
                               12
       2                               2                            13
Confirm with Excel’s Data Analysis Regression
   A       B         AB    Response
   -1      -1         1      69.6
    1      -1        -1      91.0
   -1       1        -1      80.0
    1       1         1      92.3
   -1      -1         1      71.5
    1      -1        -1      93.2             Y-output range
   -1       1        -1      81.6
    1       1         1      93.4
   -1      -1         1      70.0
    1
   -1
           -1
            1
                     -1
                     -1
                             93.0
                             80.0
                                             X-input range
    1       1         1      88.5
   -1      -1         1      69.0
    1      -1        -1      87.2
   -1       1        -1      84.3
    1       1         1      95.6



                               Standard                        Lower     Upper
                Coefficients     Error      t Stat   P-value    95%       95%
    Intercept           83.8         0.58    143.47 0.000%      82.49      85.03
           A            8.01         0.58      13.72 0.000%       6.74      9.28
           B             3.2         0.58       5.48 0.014%       1.93      4.47
          AB           -2.53         0.58      -4.32 0.099%      -3.80     -1.25
                                                                               14
                                      confirmed!
“Un”-coding the Model
    y  83.8  8.01x1  3.2x2  2.53x1 x2
           2           7
       x1  Additive                   x2  Time  2
           3           3
   y   46.9  8.71 Additive  9.10 Time  1.69 Additive Time

                               (done with Mathcad symbolics)




                                                               15
Analysis of Residuals
                                                                                                             Normal Scores Plot

                 2.5
                                                                                                                         Residuals vs Predicted Response

                   2
                                              4
                                                                                                                                                                       Residuals vs A
                 1.5
                                              3                                                    4

                   1                                                                                                                                                                                  Residuals vs B

                                              2                                                    3
                                                                                                                                         4
                 0.5
 Normal Scores




                                                                                                                                                     Predicted vs Measured Response
                                                                                                   2
                   0                          1
                                                                             95                                                          3
                             Residual Value




                 -0.5                                                                              1
                                              0
                                                                                  Residual Value




                                                                                                                                         2
                                                                             90
                  -1                                                                               0
                                              -1
                                                                                                                                         1
                                                                                                                        Residual Value




                 -1.5
                                                                                                   -1
                                              -2                             85
                                                                                                                                         0
                                                        Predicted Response




                  -2
                                                                                                   -2
                                              -3                                                                                         -1
                 -2.5
                                                                             80
                        -4                    -3                             -2                             -1                            0             1                  2                      3                4
                                                                                                   -3
                                              -4                                                                    Data Values
                                                                                                                       -2
                                                   65                                              70                75                                     80                        85                      90              95
                                                                             75                    -4
                                                                                                                                                  Predicted Response
                                                                                                     -1.5                        -1                      -0.5                     0                     0.5             1                1.5
                                                                                                                                   -3
                                                                                                                                                                                  A


                                                                             70                                                          -4
                                                                                                                                           -1.5                  -1                        -0.5                    0               0.5          1        1.5
                                                                                                                                                                                                                   B

                                                                             65
                                                                               65.0                              70.0                                75.0                      80.0                    85.0            90.0              95.0
                                                                                                                                                                      Measured Response                                                             16
Two-way ANOVA                             Excel’s Data Analysis
                                          2-way ANOVA with replication
  ANOVA
  Source of Variation SS       df         MS     F      P-value F crit
  Additive               979.7        1    979.7   185.6 0.000%       4.7
  Time                   183.6        1    183.6    34.8 0.007%       4.7
  Interaction            117.7        1    117.7    22.3 0.049%       4.7
  Within                  63.3       12      5.3

  Total                 1344.4       15



                    same conclusion: all 3 effects significant

  Where from here?
     No. of factors > 2             More detailed coverage of
     No. of levels > 2              ANOVA approach (standard)

                                                                         17

								
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