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Edme section 1 1
CHEN4860 2-Factorial Example
All Slides in this presentation are copyrighted by StatEase, Inc. and used by permission
Edme section 1 2
DOE – Process and design construction
Step-by-step analysis (popcorn)
Popcorn analysis via computer
Multiple response optimization
Advantage over one-factor-at-a-time (OFAT)
1. Mark Anderson and Pat Whitcomb (2000), DOE Simplified, Productivity Inc., chapter 3.
2. Douglas Montgomery (2006), Design and Analysis of Experiments, 6th edition, John Wiley, sections 6.1 – 6.3.
Two-Level Full Factorials
Edme section 1 3
Agenda Transition
DOE – Process and design constructionIntroduce the process for designing factorial experiments and motivate their use.
Step-by-step analysis (popcorn)
Popcorn analysis via computer
Multiple response optimization
Advantage over one-factor-at-a-time (OFAT)
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
Edme section 1 4
Process
Noise Factors “z”
Controllable Factors “x”
Responses “y”
DOE (Design of Experiments) is:
“A systematic series of tests,
in which purposeful changes
are made to input factors,
so that you may identify
causes for significant changes
in the output responses.”
Design of Experiments
Edme section 1 5
Expend no more than 25% of budget on the 1st cycle.
Conjecture
Design
Experiment
Analysis
Iterative Experimentation
Edme section 1 6
DOE Process (1 of 2)
1. Identify opportunity and define objective.
2. State objective in terms of measurable responses.
a. Define the change (y) that is important to detect for each response.
b. Estimate experimental error () for each response.
c. Use the signal to noise ratio (y/) to estimate power.
3. Select the input factors to study. (Remember that the factor levels chosen determine the size of y.)
Edme section 1 7
DOE Process (2 of 2)
4. Select a design and:
Evaluate aliases (details in section 4).
Evaluate power (details in section 2).
Examine the design layout to ensure all the factor combinations are safe to run and are likely to result in meaningful information (no disasters).
We will begin using and flesh out the DOE Process in the next section.
Edme section 1 8
Process
Noise Factors “z”
Controllable Factors “x”
Responses “y”
Let’s brainstorm.
What process might you experiment on for best payback?
How will you measure the response?
What factors can you control?
Write it down.
Design of Experiments
Edme section 1 9
Jacob Bernoulli (1654-1705)
The ‘Father of Uncertainty’
“Even the most stupid of men, by some instinct of nature, by himself and without any instruction, is convinced that the more observations have been made, the less danger there is of wandering from one’s goal.”
Central Limit TheoremCompare Averages NOT Individuals
Edme section 1 10
As the sample size (n) becomes large, the distribution of averages becomes approximately normal.
The variance of the averages is smaller than the variance of the individuals by a factor of n.
(sigma) symbolizes true standard deviation
The mean of the distribution of averages is the same as the mean of distribution of individuals.
(mu) symbolizes true population mean
22y n
iy y
The CLT applies regardless of the distribution of the individuals.
Central Limit TheoremCompare Averages NOT Individuals
Edme section 1 11
Individuals are uniform; averages tending toward normal!
Example:"snakeyes" [1/1] is the only way to get an average of one.
Central Limit TheoremIllustration using Dice
1 1 1
13
3
2 2
1 5
2 4
33
42
51
2 6
3 5
44
53
62
4 6
5 5
64
6 61 2
2 1
65
65
3 6
4 5
54
63
1 4
2 3
32
41
__
__
__
__
__
__
1 2 4 5 63
Averages of Two Dice
1 2 4 5 63
Edme section 1 12
Averages
Averages
Individuals
Y
2
Y
5
n=1
n=2
n=5
As the sample size (n) becomes large, the distribution of averages becomes approximately normal.
The variance of the averages is smaller than the variance of the individuals by a factor of n.
The mean of the distribution of averages is the same as the mean of distribution of individuals.
Central Limit TheoremUniform Distribution
Edme section 1 13
Want to estimate factor effects well; this implies estimating effects from averages.Refer to the slides on the Central Limit Theorem.
Want to obtain the most information in the fewest number of runs.
Want to estimate each factor effect independent of the existence of other factor effects.
Want to keep it simple.
Motivation for Factorial Design
Edme section 1 14
Run all high/low combinations of 2 (or more) factors
Use statistics to identify the critical factors
22 Full Factorial
What could be simpler?
Two-Level Full Factorial Design
Edme section 1 15
Std A B C AB AC BC ABC
1 – – – + + + – y1
2 + – – – – + + y2
3 – + – – + – + y3
4 + + – + – – – y4
5 – – + + – – + y5
6 + – + – + – – y6
7 – + + – – + – y7
8 + + + + + + + y8
1 2
5 6
3 4
87
B
A
C
Design Construction23 Full Factorial
Edme section 1 16
Agenda Transition
DOE – Process and design construction
Step-by-step analysis (popcorn)Learn benefits and basics of two-level factorial design by working through a simple example.
Popcorn analysis via computer
Multiple response optimization
Advantage over one-factor-at-a-time (OFAT)
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
Edme section 1 17
Two Level Factorial DesignAs Easy As Popping Corn!
Kitchen scientists* conducted a 23 factorial experiment on microwave popcorn. The factors are:
A. Brand of popcorn
B. Time in microwave
C. Power setting
A panel of neighborhood kids rated taste from one to ten and weighed the un-popped kernels (UPKs).
* For full report, see Mark and Hank Andersons' Applying DOE to Microwave Popcorn, PI Quality 7/93, p30.
Edme section 1 18
Two Level Factorial DesignAs Easy As Popping Corn!
* Average scores multiplied by 10 to make the calculations easier.
A B C R1 R2
Run Brand Time Power Taste UPKs StdOrd expense minutes percent rating* oz. Ord
1 Costly 4 75 75 3.5 2
2 Cheap 6 75 71 1.6 3
3 Cheap 4 100 81 0.7 5
4 Costly 6 75 80 1.2 4
5 Costly 4 100 77 0.7 6
6 Costly 6 100 32 0.3 8
7 Cheap 6 100 42 0.5 7
8 Cheap 4 75 74 3.1 1
Edme section 1 19
Two Level Factorial DesignAs Easy As Popping Corn!
Factors shown in coded values
A B C R1 R2
Run Brand Time Power Taste UPKs StdOrd expense minutes percent rating oz. Ord
1 + – – 75 3.5 2
2 – + – 71 1.6 3
3 – – + 81 0.7 5
4 + + – 80 1.2 4
5 + – + 77 0.7 6
6 + + + 32 0.3 8
7 – + + 42 0.5 7
8 – – – 74 3.1 1
Edme section 1 20
R1 - Popcorn TasteAverage A-Effect
75 – 74 = + 1
80 – 71 = + 9
77 – 81 = – 4
32 – 42 = – 10
42 32
7781
74 75
7 1 80
Brand
Tim
e
A
1 9 4 10y 1
4
There are four comparisons of factor A (Brand), where levels of factors B and C (time and power) are the same:
Edme section 1 21
y yEffect y
n n
A
75 80 77 32 74 71 81 42y 1
4 4
42 32
7781
74 75
71 80
Brand
Tim
e
R1 - Popcorn TasteAverage A-Effect
Edme section 1 22
R1 - Popcorn TasteAnalysis Matrix in Standard Order
I for the intercept, i.e., average response.
A, B and C for main effects (ME's).These columns define the runs.
Remainder for factor interactions (FI's)Three 2FI's and One 3FI.
Std.Order I A B C AB AC BC ABC
Taste rating
1 + – – – + + + – 74
2 + + – – – – + + 75
3 + – + – – + – + 71
4 + + + – + – – – 80
5 + – – + + – – + 81
6 + + – + – + – – 77
7 + – + + – – + – 42
8 + + + + + + + + 32
Edme section 1 23
Popcorn TasteCompute the effect of C and BC
y yy
n n
C
BC
81 77 42 32 74 75 71 80y
4 4
y4 4
Std. Taste
Order A B C AB AC BC ABC rating
1 – – – + + + – 74
2 + – – – – + + 75
3 – + – – + – + 71
4 + + – + – – – 80
5 – – + + – – + 81
6 + – + – + – – 77
7 – + + – – + – 42
8 + + + + + + + 32
y -1 -20.5 0.5 -6 -3.5
Edme section 1 24
Sparsity of Effects Principle
Do you expect all effects to be significant?
Two types of effects: • Vital Few:
About 20 % of ME's and 2 FI's will be significant.
• Trivial Many:The remainder result from random variation.
Edme section 1 25
Estimating Noise
How are the “trivial many” effects distributed?
Hint: Since the effects are based on averages you can apply the Central Limit Theorem.
Hint: Since the trivial effects estimate noise they should be centered on zero.
How are the “vital few” effects distributed?
No idea! Except that they are too large to be part of the error distribution.
Edme section 1 26
Half Normal Probability PaperSorting the vital few from the trivial many.
7.14
21.43
35.71
50.00
64.29
78.57
92.86
Pi
|E ffect|0
Significant effects (the vital few) are outliers. They are too big to be explained by noise.
They’re "keepers"!
Negligible effects (the trivial many) will be N(0, ), so they fall near zero on straight line. These are used to estimate error.
Edme section 1 27
Half Normal Probability PaperSorting the vital few from the trivial many.
7.14
21.43
35.71
50.00
64.29
78.57
92.86
Pi
0|Effect|
BC
B
C
Significant effects:
The model terms!
Negligible effects: The error estimate!
Edme section 1 28
i Pi |y| ID
1 7.14 0.5 AB
2 21.43 |–1.0| A
3 35.71 |–3.5| ABC
4 50.00 |–6.0| AC
5
6 78.57 |–20.5| B
7
1. Sort absolute value of effects into ascending order, “i”. Enter C & BC effects.
2. Compute Pis for effects. Enter Pis for i = 5 & 7.
3. Label the effects. Enter labels for C & BC effects.
i12
100 P i i 1,2,...,m m 7
m
Half Normal Probability Paper (taste)Sorting the vital few from the trivial many.
Edme section 1 29
Half Normal Probability Paper (taste) Sorting the vital few from the trivial many.
7.14
21.43
35.71
50.00
64.29
78.57
92.86
Pi
0 5 10 15 20 25
|Effect|
Edme section 1 30
Analysis of Variance (taste)Sorting the vital few from the trivial many.
Compute Sum of Squares for C and BC: 2NSS y N 8
4
i Pi |y| SS ID
1 7.14 0.5 0.5 AB
2 21.43 |–1.0| 2.0 A
3 35.71 |–3.5| 24.5 ABC
4 50.00 |–6.0| 72.0 AC
5 64.29 |–17.0| C
6 78.57 |–20.5| 840.5 B
7 92.86 |–21.5| BC
Edme section 1 31
Analysis of Variance (taste)Sorting the vital few from the trivial many.
Source
Sum of Squares
df
Mean Square
F Value
p-value Prob > F
Model 2343.0 3 781.0 31.5 0.001<p<0.005
Residual 99.0 4 24.8
Cor Total 2442.0 7
1. Add SS for significant effects: B, C & BC.
Call these vital few the “Model”.
2. Add SS for negligible effects: A, AB, AC & ABC.
Call these trivial many the “Residual”.
Edme section 1 32
Edme section 1 33
Edme section 1 34
Edme section 1 35
Analysis of Variance (taste)Sorting the vital few from the trivial many.
6.59 56.18
5%0.1%
d f = (3, 4)
4.19
10%
16.69
1%
31.50 24.26
0.5%
F-value = 31.5
0.001 < p-value < 0.005
Edme section 1 36
Analysis of Variance (taste)Sorting the vital few from the trivial many
Null Hypothesis:There are no effects, that is: H0: A= B=…= ABC= 0
F-value:
If the null hypothesis is true (all effects are zero) then the calculated F-value is 1.
As the model effects (B, C and BC) become large the calculated F-value becomes >> 1.
p-value:
The probability of obtaining the observed F-value or higher when the null hypothesis is true.
Edme section 1 37
Popcorn TasteBC Interaction
Std. Taste Order I A B C AB AC BC ABC rating
1 + – – – + + + – 74
2 + + – – – – + + 75
3 + – + – – + – + 71
4 + + + – + – – – 80
5 + – – + + – – + 81
6 + + – + – + – – 77
7 + – + + – – + – 42
8 + + + + + + + + 32
B C Taste
– – 74 75 74.5
+ –
– +
+ +
Edme section 1 38
Popcorn TasteBC Interaction
B C Taste
– – 74 75 74.5
+ – 71 80 75.5
– + 81 77 79.0
+ + 42 32 37.0
B 4 m in B+ 6 m in
80
70
60
50
40
30
Taste
Edme section 1 39
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
Agenda Transition
DOE – Process and design construction
Step-by-step analysis (popcorn)
Popcorn analysis via computerLearn to extract more information from the data.
Multiple response optimization
Advantage over one-factor-at-a-time (OFAT)
Edme section 1 40
Popcorn via Computer!
Use Design-Expert to build and analyze the popcorn DOE:
Stdord
A: Brandexpense
B: Timeminutes
C: Powerpercent
R1: Tasterating
R2: UPKsoz.
1 Cheap 4.0 75.0 74.0 3.1
2 Costly 4.0 75.0 75.0 3.5
3 Cheap 6.0 75.0 71.0 1.6
4 Costly 6.0 75.0 80.0 1.2
5 Cheap 4.0 100.0 81.0 0.7
6 Costly 4.0 100.0 77.0 0.7
7 Cheap 6.0 100.0 42.0 0.5
8 Costly 6.0 100.0 32.0 0.3
Edme section 1 41
Popcorn Analysis via Computer!Instructor led (page 1 of 2)
1. File, New Design.
2. Build a design for 3 factors, 8 runs.
3. Enter factors:
4. Enter responses. Leave delta and sigma blank to
skip power calculations.
Power will be introduced in
section 2!
Edme section 1 42
Popcorn Analysis via Computer!Instructor led (page 2 of 2)
5. Right-click on Std column header and choose “Sort by Standard Order”.
6. Type in response data (from previous page) for Taste and UPKs.
7. Analyze Taste. Taste will be instructor led; you will analyze the
UPKs on your own.
8. Save this design.
Edme section 1 43
Popcorn Analysis – Taste Effects Button - View, Effects List
Term Stdized Effect SumSqr % Contribution
Require Intercept
Error A-Brand -1 2 0.0819001
Error B-Time -20.5 840.5 34.4185
Error C-Power -17 578 23.6691
Error AB 0.5 0.5 0.020475
Error AC -6 72 2.9484
Error BC -21.5 924.5 37.8583
Error ABC -3.5 24.5 1.00328
Lenth's ME 33.8783
Lenth's SME 81.0775
2
y yy
n n
n 4
NSS
4
N 8
Edme section 1 44
Popcorn Analysis – TasteEffects - View, Half Normal Plot of Effects
Design-Expert® SoftwareTaste
Shapiro-Wilk testW-value = 0.973p-value = 0.861A: BrandB: TimeC: Power
Positive Effects Negative Effects
Half-Normal Plot
Ha
lf-N
orm
al %
Pro
ba
bili
ty
|Standardized Effect|
0.00 5.38 10.75 16.13 21.50
0102030
50
70
80
90
95
99
B
C
BC
Edme section 1 45
Popcorn Analysis – TasteEffects - View, Pareto Chart of “t” Effects
Pareto Chartt-
Va
lue
of
|Eff
ect
|
Rank
0.00
1.53
3.06
4.58
6.11
Bonferroni Limit 5.06751
t-Value Limit 2.77645
1 2 3 4 5 6 7
BCB
C
0.05 df 42t 2.77645
0.052 df 4k 7
t 5.06751
Edme section 1 46
Popcorn Analysis – Taste ANOVA button
Analysis of variance table [Partial sum of squares]Sum of Mean F
Source Squares df Square Value Prob > FModel 2343.00 3 781.00 31.56 0.0030
B-Time 840.50 1 840.50 33.96 0.0043C-Power 578.00 1 578.00 23.35 0.0084
BC 924.50 1 924.50 37.35 0.0036Residual 99.00 4 24.75Cor Total 2442.00 7
Edme section 1 47
Popcorn Analysis – Taste ANOVA (summary statistics)
Std. Dev. 4.97 R-Squared 0.9595
Mean 66.50 Adj R-Squared 0.9291
C.V. % 7.48 Pred R-Squared 0.8378
PRESS 396.00 Adeq Precision 11.939
Edme section 1 48
Popcorn Analysis – Taste ANOVA Coefficient Estimates
Coefficient Standard 95% CI 95% CIFactor Estimate DF Error Low High VIFIntercept 66.50 1 1.76 61.62 71.38B-Time -10.25 1 1.76 -15.13 -5.37 1.00C-Power -8.50 1 1.76 -13.38 -3.62 1.00BC -10.75 1 1.76 -15.63 -5.87 1.00
-1 0 +1R
espo
nse
Eff
ect
F actor Level (Cod ed )
Coefficient Estimate: One-half of the factorial effect (in coded units)
Coefficient y / x y / 2
Edme section 1 49
Final Equation in Terms of Coded Factors:
Taste =
+66.50
-10.25*B
-8.50*C
-10.75*B*C
Std B C Pred y
1 − − 74.50
2 − − 74.50
3 + − 75.50
4 + − 75.50
5 − + 79.00
6 − + 79.00
7 + + 37.00
8 + + 37.00
Popcorn Analysis – Taste Predictive Equation (Coded)
Edme section 1 50
Final Equation in Terms of Actual Factors:
Taste =
-199.00
+65.00*Time
+3.62*Power
-0.86*Time*Power
Popcorn Analysis – Taste Predictive Equation (Actual)
Std B C Pred y
1 4 min 75% 74.50
2 4 min 75% 74.50
3 6 min 75% 75.50
4 6 min 75% 75.50
5 4 min 100% 79.00
6 4 min 100% 79.00
7 6 min 100% 37.00
8 6 min 100% 37.00
Edme section 1 51
Popcorn Analysis – Taste Predictive Equations
For process understanding, use coded values:
1. Regression coefficients tell us how the response changes relative to the intercept. The intercept in coded values is in the center of our design.
2. Units of measure are normalized (removed) by coding. Coefficients measure half the change from –1 to +1 for all factors.
Actual Factors: Taste =
-199.00+65.00*Time
+3.62*Power-0.86*Time*Power
Coded Factors: Taste =
+66.50-10.25*B
-8.50*C-10.75*B*C
Edme section 1 52
Analysis
Filter signal
Data(Observed Values)
Signal
Noise
Model(Predicted Values)
Signal
Residuals
(Observed – Predicted)
Noise
Independent N(0,)
Factorial DesignResidual Analysis
Edme section 1 53
Popcorn Analysis – Taste Diagnostic Case Statistics
Diagnostics → Influence → ReportDiagnostics Case Statistics
Internally Externally Influence on
Std Actual Predicted Studentized Studentized Fitted Value Cook's Run
Order Value Value Residual Leverage Residual Residual DFFITS Distance Order
1 74.00 74.50 -0.50 0.500 -0.142 -0.123 -0.123 0.005 8
2 75.00 74.50 0.50 0.500 0.142 0.123 0.123 0.005 1
3 71.00 75.50 -4.50 0.500 -1.279 -1.441 -1.441 0.409 2
4 80.00 75.50 4.50 0.500 1.279 1.441 1.441 0.409 4
5 81.00 79.00 2.00 0.500 0.569 0.514 0.514 0.081 3
6 77.00 79.00 -2.00 0.500 -0.569 -0.514 -0.514 0.081 5
7 42.00 37.00 5.00 0.500 1.421 1.750 1.750 0.505 7
8 32.00 37.00 -5.00 0.500 -1.421 -1.750 -1.750 0.505 6
See “Diagnostics Report – Formulas & Definitions” in your “Handbook for Experimenters”.
Edme section 1 54
Factorial DesignANOVA Assumptions
Additive treatment effects
Factorial: An interaction model will adequately represent response behavior.
Independence of errors
Knowing the residual from one experiment givesno information about the residual from the next.
Studentized residuals N(0,2):
• Normally distributed
• Mean of zero
• Constant variance, 2=1
Check assumptions by plotting studentized residuals!
• Model F-test
• Lack-of-Fit
• Box-Cox plot
S Residualsversus
Run Order
Normal Plot ofS Residuals
S Residualsversus
Predicted
Edme section 1 55
Popcorn Analysis – Taste Diagnostics - ANOVA Assumptions
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Internally Studentized Residuals
No
rma
l % P
rob
ab
ility
Normal Plot of Residuals
-1.42 -0.71 0.00 0.71 1.42
1
5
10
20
30
50
70
80
90
95
99
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Predicted
Inte
rna
lly S
tud
en
tize
d R
esi
du
als
Residuals vs. Predicted
-3.00
-1.50
0.00
1.50
3.00
37.00 47.50 58.00 68.50 79.00
Edme section 1 56
Popcorn Analysis – Taste Diagnostics - ANOVA Assumptions
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Run Number
Inte
rna
lly S
tud
en
tize
d R
esi
du
als
Residuals vs. Run
-3.00
-1.50
0.00
1.50
3.00
1 2 3 4 5 6 7 8
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Actual
Pre
dic
ted
Predicted vs. Actual
32.00
44.25
56.50
68.75
81.00
32.00 44.25 56.50 68.75 81.00
Edme section 1 57
Design-Expert® SoftwareTaste
LambdaCurrent = 1Best = 1.77Low C.I. = -0.24High C.I. = 4.79
Recommend transform:None (Lambda = 1)
Lambda
Ln
(Re
sid
ua
lSS
)
Box-Cox Plot for Power Transforms
4.46
5.43
6.41
7.38
8.36
-3 -2 -1 0 1 2 3
Popcorn Analysis – Taste Diagnostics - ANOVA Assumptions
Details in section 3
Edme section 1 58
Popcorn Analysis – Taste Influence
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Run Number
Ext
ern
ally
Stu
de
ntiz
ed
Re
sid
ua
ls
Externally Studentized Residuals
-5.26
-2.63
0.00
2.63
5.26
1 2 3 4 5 6 7 8
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Run Number
DF
FIT
S
DFFITS vs. Run
-2.00
-1.00
0.00
1.00
2.00
1 2 3 4 5 6 7 8
Edme section 1 59
Popcorn Analysis – Taste Influence
DF
BE
TA
S f
or
Inte
rce
pt
DFBETAS for Intercept vs. Run
-2.00
-1.00
0.00
1.00
2.00
1 2 3 4 5 6 7 8
DF
BE
TA
S f
or
B
DFBETAS for B vs. Run
-2.00
-1.00
0.00
1.00
2.00
1 2 3 4 5 6 7 8
Run Number
DF
BE
TA
S f
or
C
DFBETAS for C vs. Run
-2.00
-1.00
0.00
1.00
2.00
1 2 3 4 5 6 7 8
DF
BE
TA
S f
or
BC
DFBETAS for BC vs. Run
-2.00
-1.00
0.00
1.00
2.00
1 2 3 4 5 6 7 8
Edme section 1 60
Popcorn Analysis – Taste Influence
Design-Expert® SoftwareTaste
Color points by value ofTaste:
81.0
32.0
Run Number
Co
ok'
s D
ista
nce
Cook's Distance
0.00
0.25
0.50
0.75
1.00
1 2 3 4 5 6 7 8
Edme section 1 61
Popcorn Analysis – Taste Influence
Tool Description WIIFM*
Internally Studentized Res.
Residual divided by the estimated standard deviation of that residual
Normality, constant 2
Externally Studentized Res.
Residual divided by the estimated std dev of that residual, without the ith case
Outlier detection
Cook’s Distance Change in joint confidence ellipsoid (regression) with and without a run
Influence
DF Fits (difference in fits)
Change in predictions with and without a run; the influence a run has on the predictions
Influence
DF Betas (difference in betas)
Change in each model coefficient (beta) with and without a run
Influence
Edme section 1 62
Popcorn Analysis – Taste Model Graphs - Factor “B” Effect Plot
Don’t make one factor plot of factors involved in an interaction!
Design-Expert® Software
Taste
X1 = B: Time
Actual FactorsA: Brand = CheapC: Power = 87.50
4.00 4.50 5.00 5.50 6.00
32.0
44.5
57.0
69.5
82.0
B: Time
Ta
ste
One FactorWarning! Factor inv olv ed in an interaction.
Edme section 1 63
Popcorn Analysis – Taste Model Graphs – View, Interaction Plot (BC)
Design-Expert® Software
Taste
Design Points
C- 75.000C+ 100.000
X1 = B: TimeX2 = C: Power
Actual FactorA: Brand = Cheap
C: Power
4.00 4.50 5.00 5.50 6.00
Interaction
B: Time
Ta
ste
30.0
44.0
58.0
72.0
86.0
Edme section 1 64
Popcorn Analysis – Taste Model Graphs – View, Contour Plot
and 3D Surface (BC)
Design-Expert® Software
TasteDesign Points81
32
X1 = B: TimeX2 = C: Power
Actual FactorA: Brand = Cheap
4.00 4.50 5.00 5.50 6.00
75.00
81.25
87.50
93.75
100.00Taste
B: Time
C:
Po
we
r
40.0
45.0
50.0
55.0
60.0
65.0
70.0
75.0
75.0
4.00 4.50 5.00 5.50 6.0075.00
81.25
87.50
93.75
100.00
37.0
47.8
58.5
69.3
80.0
T
aste
B: Time
C: Power
Edme section 1 65
Popcorn Analysis – Taste BC Interaction Plot Comparison
4.00 4.50 5.00 5.50 6.0075.00
81.25
87.50
93.75
100.00
37.0
47.8
58.5
69.3
80.0
T
aste
B: Time
C: Power
C: Power
4.00 4.50 5.00 5.50 6.00
Interaction
B: Time
Ta
ste
30.0
44.0
58.0
72.0
86.0
C-
C+
Edme section 1 66
Popcorn Analysis – UPKsYour Turn!
1. Analyze UPKs:Use the “Factorial Analysis Guide” in your “Handbook for Experimenters” – page 2-1.
2. Pick the time and power settings that maximize popcorn taste while minimizing UPKs.
Edme section 1 67
Choose factor levels to try to simultaneously satisfy all requirements. Balance desired levels of each response against overall performance.
Popcorn Revisited!
C: Power
4.00 4.50 5.00 5.50 6.00
Interaction
B: Time
Ta
ste
30.0
40.0
50.0
60.0
70.0
80.0
90.0
C-
C+
C: Power
4.00 4.50 5.00 5.50 6.00
Interaction
B: Time
UP
Ks
0.1
1.0
1.9
2.7
3.6
C-
C+
Edme section 1 68
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
Agenda Transition
DOE – Process and design construction
Step-by-step analysis (popcorn)
Popcorn analysis via computer
Multiple response optimizationLearn to use numerical search tools to find factor settings to optimize tradeoffs among multiple responses.
Advantage over one-factor-at-a-time (OFAT)
Edme section 1 69
1. Go to the Numerical Optimization node and set the goal for Taste to “maximize” with a lower limit of “60” and an upper limit of “90”.
2. Set the goal for UPKs to “minimize” with a lower limit of “0” and an upper limit of “2”.
Popcorn Optimization
Edme section 1 70
3. Click on the “Solutions” button:
Solutions
# Brand* Time Power Taste UPKs Desirability
1 Costly 4.00 100.00 79.0 0.70 0.642 Selected
2 Cheap 4.00 100.00 79.0 0.70 0.642
3 Cheap 6.00 75.00 75.5 1.40 0.394
4 Costly 6.00 75.00 75.5 1.40 0.394
*Has no effect on optimization results.
Take a look at the “Ramps” view for a nice summary.
Popcorn Optimization
Edme section 1 71
4. Click on the “Graphs” button and by right clicking on the factors tool pallet choose “B:Time” as the X1-axis and “C:Power” as the X2-axis:
Popcorn Optimization
Design-Expert® Software
Desirability
Design Points
C- 75.000C+ 100.000
X1 = B: TimeX2 = C: Power
Actual FactorA: Brand = Costly
C: Power
4.00 4.50 5.00 5.50 6.00
Interaction
B: Time
De
sira
bili
ty
0.000
0.250
0.500
0.750
1.000
Prediction 0.64
Edme section 1 72
5. Choose “Contour” and “3D Surface” from the “View” menu :
Popcorn Optimization
Design-Expert® Software
DesirabilityDesign Points
X1 = B: TimeX2 = C: Power
Actual FactorA: Brand = Cheap
4.00 4.50 5.00 5.50 6.00
75.00
81.25
87.50
93.75
100.00Desirability Contour
B: Time
C: Power
0.100
0.100
0.200
0.200
0.300
0.300
0.400
0.500
Predicti 0.64
Design-Expert® Software
Desirability
X1 = B: TimeX2 = C: Power
Actual FactorA: Brand = Cheap
4.00 4.50
5.00 5.50
6.00
75.00
81.25
87.50
93.75
100.00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Desira
bility
B: Time
C: Power
Edme section 1 73
Popcorn Optimization
To learn more about optimization:
Read Derringer’s article from Quality Progress:
www.statease.com/pubs/derringer.pdf
Attend the “RSM” workshop on response surface methodology!
Edme section 1 74
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
yes
Factor effectsand interactions
ResponseSurfaceMethods
Curvature?
Confirm?
KnownFactors
UnknownFactors
Screening
Backup
Celebrate!
no
no
yes
Trivialmany
Vital few
Screening
Characterization
Optimization
Verification
Agenda Transition
DOE – Process and design construction
Step-by-step analysis (popcorn)
Popcorn analysis via computer
Multiple response optimization
Advantage over one-factor-at-a-time (OFAT)Summarize the benefits factorial design has over one-factor-at-a-time experimentation.
Edme section 1 75
Traditional Approach to DOEOne Factor at a Time (OFAT)
“There aren't any interactions."
“I'll investigate that factor next.”
“It's too early to use statistical methods.”
“A statistical experiment would be too large.”
“My data are too variable to use statistics.”
“We'll worry about the statistics after we've run the experiment.”
“Lets just vary one thing at a time so we don't get confused.”
Edme section 1 76
Relative EfficiencyFactorial versus OFAT
A
B
CA
B
C
A
B
A
B
Relative efficiency = 6/4 = 1.5
Relative efficiency = 16/8= 2.0
Edme section 1 77
2k Factorial DesignAdvantages
What could be simpler? Minimal runs required.
Can run fractions if 5 or more factors. Have hidden replication. Wider inductive basis than OFAT experiments. Show interactions.
Key to Success - Extremely important! Easy to analyze.
Do by hand if you want. Interpretation is not too difficult.
Graphs make it easy. Can be applied sequentially. Form base for more complex designs.
Second order response surface design.