Class Meeting 25 by HC121109064610

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```									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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