CHEN4860 2-Factorial Example
All Slides in this presentation are copyrighted by
StatEase, Inc. and used by permission
Edme section 1
1
Two-Level Full Factorials
 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.
Edme section 1
2
Agenda Transition
 DOE – Process and design
construction
Introduce the process for designing
factorial experiments and motivate
their use.
Screening
Known
Factors
Vital few
Characterization
Factor effects
and interactions
 Step-by-step analysis (popcorn)
 Popcorn analysis via computer
Edme section 1
Curvature?
yes
Optimization
 Advantage over one-factor-at-a-time
(OFAT)
Trivial
many
Screening
no
 Multiple response optimization
Unknown
Factors
Response
Surface
Methods
Verification
Confirm?
Celebrate!
no
Backup
yes
3
Design of Experiments
Controllable Factors “x”
Process
DOE (Design of Experiments) is:
“A systematic series of tests,
in which purposeful changes
are made to input factors,
Responses “y”
so that you may identify
causes for significant changes
in the output responses.”
Noise Factors “z”
Edme section 1
4
Iterative Experimentation
Conjecture
Analysis
Design
Experiment
Expend no more than 25% of budget on the 1st cycle.
Edme section 1
5
DOE Process (1 of 2)
1. Identify opportunity and define objective.
2. State objective in terms of measurable responses.
a. Define the change (Dy) that is important to detect
for each response.
b. Estimate experimental error (s) for each
response.
c. Use the signal to noise ratio (Dy/s) to estimate
power.
3. Select the input factors to study. (Remember that
the factor levels chosen determine the size of Dy.)
Edme section 1
6
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
7
Design of Experiments
Controllable Factors “x”
Let’s brainstorm.
What process might you
experiment on for best payback?
Process
Responses “y”
How will you measure the
response?
Noise Factors “z”
What factors can you control?
Write it down.
Edme section 1
8
Central Limit Theorem
Compare Averages NOT Individuals
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.”
Edme section 1
9
Central Limit Theorem
Compare Averages NOT Individuals

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.
2
s 2y 
s
n
s (sigma) symbolizes true standard deviation

The mean of the distribution of averages is the same as the mean of
distribution of individuals.
 y   yi
 (mu) symbolizes true population mean
The CLT applies regardless of the distribution of the individuals.
Edme section 1
10
Central Limit Theorem
Illustration using Dice
Individuals are uniform;
averages tending toward
normal!
2
1
3
4
5
6
Averages of Two Dice
_
Example: "snakeyes" [1/1] is the
only way to get an
average of one.
5
4
3
2
1
3
1
2
4
3
1
2
3
1 2 2 2 3 24
1 1 1 2 1 3 1 4 1 5
1
Edme section 1
2
3
_
_
_ 62
_
_ 53
_
_ 44
_
_ 35
_
_ 26
4
6
5
3
4
6
4
4 5 5 5 6
5
36 4 6 5
6 6
6
5
6
11
Central Limit Theorem
Uniform Distribution
 As the sample size (n)
becomes large, the
distribution of averages
becomes approximately
normal.
n=1
 The variance of the
n=2
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 n=5
individuals.
Edme section 1
s
Individuals
s
Y
Averages
Averages

sY
s
2
s
5
12
Motivation for Factorial Design
 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.
Edme section 1
13
Two-Level Full Factorial Design
Run all high/low combinations of 2 (or more) factors
Use statistics to identify the critical factors
22 Full Factorial
What could be simpler?
Edme section 1
14
Design Construction
23 Full Factorial
7
3
8
4
5
6
Std
A
B
C
AB
AC
BC
ABC
1
–
–
–
+
+
+
–
y1
2
+
–
–
–
–
+
+
y2
3
–
+
–
–
+
–
+
y3
4
+
+
–
+
–
–
–
y4
5
–
–
+
+
–
–
+
y5
6
+
–
+
–
+
–
–
y6
7
–
+
+
–
–
+
–
y7
8
+
+
+
+
+
+
+
y8
2
1
B
C
A
Edme section 1
15
Agenda Transition
 DOE – Process and design
construction
 Step-by-step analysis (popcorn)
Screening
Known
Factors
 Multiple response optimization
 Advantage over one-factor-at-a-time
(OFAT)
Edme section 1
Trivial
many
Screening
Vital few
Characterization
Factor effects
and interactions
Learn benefits and basics of
two-level factorial design by working
through a simple example.
 Popcorn analysis via computer
Unknown
Factors
no
Curvature?
yes
Optimization
Response
Surface
Methods
Verification
Confirm?
Celebrate!
no
Backup
yes
16
Two Level Factorial Design
As 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
17
Two Level Factorial Design
As Easy As Popping Corn!
Run
Ord
A
Brand
expense
B
Time
minutes
C
Power
percent
R1
Taste
rating*
R2
UPKs
oz.
Std
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
* Average scores multiplied by 10 to make the calculations easier.
Edme section 1
18
Two Level Factorial Design
As Easy As Popping Corn!
Run
Ord
A
Brand
expense
B
Time
minutes
C
Power
percent
R1
Taste
rating
R2
UPKs
oz.
Std
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
Factors shown in coded values
Edme section 1
19
R1 - Popcorn Taste
Average A-Effect
There are four comparisons of factor A (Brand), where
levels of factors B and C (time and power) are the
same:
42
Time
71
32
80
77
74
75
Brand
Edme section 1
Po
w
er
81
75 – 74 = + 1
80 – 71 = + 9
77 – 81 = – 4
32 – 42 = – 10
1  9  4  10
Dy A 
 1
4
20
R1 - Popcorn Taste
Average A-Effect
42
Time
71
32
80
77
75
Brand
P
74
ow
er
81
y

Effect  Dy  
n

y



n
75  80  77  32 74  71  81  42
Dy A 

 1
4
4
Edme section 1
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R1 - Popcorn Taste
Analysis Matrix in Standard Order
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
 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.
Edme section 1
22
Popcorn Taste
Compute the effect of C and BC
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
Dy
-1
-20.5
0.5
-6
y

Dy 
n

y


Edme section 1
n

Dy C 
DyBC 
-3.5
81  77  42  32 74  75  71  80


4
4


4




4


23
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
24
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
25
Half Normal Probability Paper
Sorting the vital few from the trivial many.
92.86
Significant effects (the vital
few) are outliers. They are
too big to be explained by
noise.
78.57
Pi
64.29
They’re "keepers"!
50.00
35.71
21.43
7.14
0
|Effect|
Negligible effects (the trivial many) will be N(0, s), so they fall
near zero on straight line. These are used to estimate error.
Edme section 1
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Half Normal Probability Paper
Sorting the vital few from the trivial many.
BC
92.86
Significant effects:
78.57
B
The model terms!
Pi
C
64.29
50.00
35.71
21.43
7.14
0
|Effect|
Negligible effects: The error estimate!
Edme section 1
27
Half Normal Probability Paper (taste)
Sorting the vital few from the trivial many.
i
Pi
|Dy|
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
78.57
|–20.5|
B
5
6
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.
7
Pi 
Edme section 1
100
i

m
1
2

i  1,2,...,m
m7
28
Half Normal Probability Paper (taste)
Sorting the vital few from the trivial many.
92.86
78.57
Pi
64.29
50.00
35.71
21.43
7.14
0
5
10
15
20
25
|Effect|
Edme section 1
29
Analysis of Variance (taste)
Sorting the vital few from the trivial many.
i
Pi
|Dy|
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|
6
78.57
|–20.5|
7
92.86
|–21.5|
Compute Sum of Squares for C and BC:
Edme section 1
C
840.5
B
BC
SS 

N
Dy 2
4

N8
30
Analysis of Variance (taste)
Sorting the vital few from the trivial many.
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”.
Source
Sum of
Squares
df
Mean
Square
Model
2343.0
3
781.0
Residual
99.0
4
24.8
Cor Total
2442.0
7
Edme section 1
F
Value
31.5
p-value
Prob > F
0.001<p<0.005
31
Edme section 1
32
Edme section 1
33
Edme section 1
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Analysis of Variance (taste)
Sorting the vital few from the trivial many.
F-value = 31.5
0.001 < p-value < 0.005
df = (3, 4)
10%
0
Edme section 1
4.19
5%
6.59
1%
0.5%
16.69 24.26
0.1%
31.5
56.18
35
Analysis of Variance (taste)
Sorting the vital few from the trivial many
Null Hypothesis:
There are no effects, that is: H0: DA= DB=…= DABC= 0
F-value:
If the null hypothesis is true (all effects are zero) then
the calculated
F-value is  1.
As the model effects (DB, DC and DBC) 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
36
Popcorn Taste
BC Interaction
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
B
C
–
–
+
–
–
+
+
+
Edme section 1
Taste
74
75
74.5
37
Popcorn Taste
BC Interaction
Taste
80
70
60
50
B
C
Taste
–
–
74
75
74.5
+
–
71
80
75.5
–
+
81
77
79.0
+
+
42
32
37.0
40
30
B- 4 min
Edme section 1
B+ 6 min
38
Agenda Transition
Screening
Known
Factors
 DOE – Process and design
construction
 Step-by-step analysis (popcorn)
 Popcorn analysis via computer
Learn to extract more information
from the data.
 Multiple response optimization
Unknown
Factors
Vital few
Characterization
Factor effects
and interactions
no
Curvature?
yes
Optimization
Response
Surface
Methods
Verification
Confirm?
 Advantage over one-factor-at-a-time
(OFAT)
Edme section 1
Trivial
many
Screening
Celebrate!
no
Backup
yes
39
Popcorn via Computer!
Use Design-Expert to build and analyze the
popcorn DOE:
Std
ord
A: Brand
expense
B: Time
minutes
C: Power
percent
R1: Taste
rating
R2: UPKs
oz.
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
40
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.
Edme section 1
Leave delta and
sigma blank to
skip power
calculations.
Power will be
introduced in
section 2! 41
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
42
Popcorn Analysis – Taste
Effects Button - View, Effects List
  y  y 
Dy  


n
n

 

n4
SS 
 
N 2
D
4
N8
Edme section 1
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
43
Popcorn Analysis – Taste
Effects - View, Half Normal Plot of Effects
Design-Expert® Software
Taste
99
Half-Normal % Probability
Shapiro-Wilk test
W-value = 0.973
p-value = 0.861
A: Brand
B: Time
C: Power
Positive Effects
Negative Effects
Half-Normal Plot
95
BC
90
80
B
70
C
50
30
20
10
0
0.00
5.38
10.75
16.13
21.50
|Standardized Effect|
Edme section 1
44
Popcorn Analysis – Taste
Effects - View, Pareto Chart of “t” Effects
Pareto Chart
BC
B
6.11
C
Bonferroni Limit 5.06751
t-Value of |Effect|
4.58
t  0.05



3.06
t-Value Limit 2.77645
t 0.05

2
2

df  4 


k 7
df  4

 5.06751
 2.77645
1.53
0.00
1
2
3
4
5
6
7
Rank
Edme section 1
45
Popcorn Analysis – Taste
ANOVA button
Analysis of variance table [Partial sum of squares]
Sum of
Mean
F
Source
Squares
df
Square
Value
Model
2343.00
3
781.00
31.56
B-Time
840.50
1
840.50
33.96
C-Power
578.00
1
578.00
23.35
BC
924.50
1
924.50
37.35
Residual
99.00
4
24.75
Cor Total
2442.00
7
Edme section 1
Prob > F
0.0030
0.0043
0.0084
0.0036
46
Popcorn Analysis – Taste
ANOVA (summary statistics)
Std. Dev.
Mean
C.V. %
PRESS
Edme section 1
4.97
66.50
7.48
396.00
R-Squared
Adj R-Squared
Pred R-Squared
Adeq Precision
0.9595
0.9291
0.8378
11.939
47
Popcorn Analysis – Taste
ANOVA Coefficient Estimates
Coefficient Estimate: One-half of the
factorial effect (in coded units)
95% CI
Low
61.62
-15.13
-13.38
-15.63
95% CI
High
71.38
-5.37
-3.62
-5.87
VIF
1.00
1.00
1.00
Effect
Standard
DF
Error
1
1.76
1
1.76
1
1.76
1
1.76
Response
Factor
Intercept
B-Time
C-Power
BC
Coefficient
Estimate
66.50
-10.25
-8.50
-10.75
Coefficient  Dy / Dx  Dy / 2
-1
0
+1
Factor Level (Coded)
Edme section 1
48
Popcorn Analysis – Taste
Predictive Equation (Coded)
Final Equation in Terms of Coded Factors:
Taste =
Std
B
C
Pred y
+66.50
1
−
−
74.50
-10.25*B
2
−
−
74.50
-8.50*C
3
+
−
75.50
-10.75*B*C
4
+
−
75.50
5
−
+
79.00
6
−
+
79.00
7
+
+
37.00
8
+
+
37.00
Edme section 1
49
Popcorn Analysis – Taste
Predictive Equation (Actual)
Final Equation in Terms of Actual Factors:
Taste =
Std
B
C
Pred y
1
4 min
75%
74.50
2
4 min
75%
74.50
+3.62*Power
3
6 min
75%
75.50
-0.86*Time*Power
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
-199.00
+65.00*Time
Edme section 1
50
Popcorn Analysis – Taste
Predictive Equations
Coded Factors:
Taste =
+66.50
-10.25*B
-8.50*C
-10.75*B*C
Actual Factors:
Taste =
-199.00
+65.00*Time
+3.62*Power
-0.86*Time*Power
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.
Edme section 1
51
Factorial Design
Residual Analysis
Data
(Observed Values)
Signal
Noise
Analysis
Filter signal
Model
(Predicted Values)
Signal
Residuals
(Observed – Predicted)
Noise
Independent N(0,s2)
Edme section 1
52
Popcorn Analysis – Taste
Diagnostic Case Statistics
Diagnostics → Influence → Report
Diagnostics Case Statistics
Std
Order
1
2
3
4
5
6
7
8
Actual
Value
74.00
75.00
71.00
80.00
81.00
77.00
42.00
32.00
Predicted
Value Residual
74.50
-0.50
74.50
0.50
75.50
-4.50
75.50
4.50
79.00
2.00
79.00
-2.00
37.00
5.00
37.00
-5.00
Leverage
0.500
0.500
0.500
0.500
0.500
0.500
0.500
0.500
Internally
Studentized
Residual
-0.142
0.142
-1.279
1.279
0.569
-0.569
1.421
-1.421
Externally
Studentized
Residual
-0.123
0.123
-1.441
1.441
0.514
-0.514
1.750
-1.750
Influence on
Fitted Value
DFFITS
-0.123
0.123
-1.441
1.441
0.514
-0.514
1.750
-1.750
Cook's
Distance
0.005
0.005
0.409
0.409
0.081
0.081
0.505
0.505
Run
Order
8
1
2
4
3
5
7
6
See “Diagnostics Report – Formulas & Definitions”
in your “Handbook for Experimenters”.
Edme section 1
53
Factorial Design
ANOVA Assumptions
Additive treatment effects
• Model F-test
Factorial: An interaction model will adequately
represent response behavior.
• Lack-of-Fit
• Box-Cox plot
Independence of errors
Knowing the residual from one experiment gives
no information about the residual from the next.
S Residuals
versus
Run Order
Studentized residuals N(0,s2):
• Normally distributed
Normal Plot of
S Residuals
• Mean of zero
• Constant variance, s2=1
S Residuals
versus
Predicted
Check assumptions by plotting studentized residuals!
Edme section 1
54
Popcorn Analysis – Taste
Diagnostics - ANOVA Assumptions
oftware
Design-Expert® Software
aste
Normal Plot ofT Residuals
ue of
Residuals vs. Predicted
Color points by value of
T aste:
81.0
99
3.00
Internally Studentized Residuals
32.0
Normal % Probability
95
90
80
70
50
30
20
10
5
1.50
0.00
-1.50
1
-3.00
-0.71
-1.42
0.00
0.71
Internally Studentized Residuals
Edme section 1
1.42
37.00
47.50
58.00
68.50
79.00
Predicted
55
Popcorn Analysis – Taste
Diagnostics - ANOVA Assumptions
oftware
Design-Expert® Software
T aste
Residuals vs.
Run
lue of
Predicted vs. Actual
Color points by value of
T aste:
81.0
3.00
81.00
1.50
68.75
Predicted
Internally Studentized Residuals
32.0
0.00
56.50
-1.50
44.25
-3.00
32.00
1
2
3
4
5
Run Number
Edme section 1
6
7
8
32.00
44.25
56.50
68.75
81.00
Actual
56
Popcorn Analysis – Taste
Diagnostics - ANOVA Assumptions
Design-Expert® Software
T aste
Box-Cox Plot for Power Transforms
Lambda
Current = 1
Best = 1.77
Low C.I. = -0.24
High C.I. = 4.79
8.36
Details in
section 3
Recommend transform:
None
(Lambda = 1)
Ln(ResidualSS)
7.38
6.41
5.43
4.46
-3
-2
-1
0
1
2
3
Lambda
Edme section 1
57
Popcorn Analysis – Taste
Influence
Software
Design-Expert®
Software
Residuals
Externally Studentized
DFFITS vs. Run
Taste
value of
2.00
Color points by value of
Taste:
81.0
2.63
1.00
32.0
DFFITS
Externally Studentized Residuals
5.26
0.00
0.00
-2.63
-1.00
-5.26
-2.00
1
2
3
4
5
Run Number
Edme section 1
6
7
8
1
2
3
4
5
6
7
8
Run Number
58
Popcorn Analysis – Taste
Influence
DFBETAS for B vs. Run
2.00
2.00
1.00
1.00
DFBETAS for B
DFBETAS for Intercept
DFBETAS for Intercept vs. Run
0.00
0.00
-1.00
-1.00
-2.00
-2.00
1
2
3
4
5
6
7
8
1
2.00
1.00
1.00
0.00
-1.00
-2.00
-2.00
Edme section 1
3
4
5
Run Number
6
5
6
7
8
0.00
-1.00
2
4
Run Number
2.00
1
3
DFBETAS for BC vs. Run
DFBETAS for BC
DFBETAS for C
Run Number
DFBETAS
for C vs. Run
2
7
8
1
2
3
4
5
Run Number
6
7
8
59
Popcorn Analysis – Taste
Influence
Design-Expert® Software
T aste
Cook's Distance
Color points by value of
T aste:
81.0
1.00
32.0
Cook's Distance
0.75
0.50
0.25
0.00
1
2
3
4
5
6
7
8
Run Number
Edme section 1
60
Popcorn Analysis – Taste
Influence
Tool
Description
WIIFM*
Internally Studentized
Res.
Residual divided by the estimated
standard deviation of that residual
Normality,
constant s2
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
Influence
without a run; the influence a run has
on the predictions
DF Betas (difference
in betas)
Change in each model coefficient
(beta) with and without a run
Edme section 1
Influence
61
Popcorn Analysis – Taste
Model Graphs - Factor “B” Effect Plot
Design-Expert® Software
One Factor
T aste
Warning! Factor inv olv ed in an interaction.
82.0
X1 = B: T ime
Actual Factors
A: Brand = Cheap
C: Power = 87.50
Taste
69.5
57.0
44.5
Don’t make one factor
plot of factors involved
in an interaction!
32.0
4.00
4.50
5.00
5.50
6.00
B: Time
Edme section 1
62
Popcorn Analysis – Taste
Model Graphs – View, Interaction Plot (BC)
Design-Expert® Software
Interaction
T aste
C: Power
Design Points
86.0
C- 75.000
C+ 100.000
X1 = B: T ime
X2 = C: Power
72.0
Taste
Actual Factor
A: Brand = Cheap
58.0
44.0
30.0
4.00
4.50
5.00
5.50
6.00
B: Time
Edme section 1
63
Popcorn Analysis – Taste
Model Graphs – View, Contour Plot
and 3D Surface (BC)
Software
Taste
100.00
80.0
40.0
ts
45.0
50.0
69.3
55.0
93.75
C: Power
75.0
Taste
60.0
ap
65.0
87.50
58.5
47.8
70.0
37.0
81.25
75.0
75.00
4.00
4.50
5.00
B: Time
5.50
6.00
C:
100.00
93.75
Power 87.50
81.25
75.00 4.00
4.50
5.00
6.00
5.50
B: Time
Edme section 1
64
Popcorn Analysis – Taste
BC Interaction Plot Comparison
Interaction
C: Power
86.0
80.0
C+
C-
69.3
Taste
Taste
72.0
58.0
58.5
47.8
44.0
37.0
30.0
4.00
4.50
5.00
B: Time
5.50
6.00
C:
100.00
93.75
Power 87.50
81.25
75.00 4.00
4.50
5.00
5.50
6.00
B: Time
Edme section 1
65
Popcorn Analysis – UPKs
Your 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
66
Popcorn Revisited!
Choose factor levels to try to simultaneously satisfy all
requirements. Balance desired levels of each response
against overall performance.
Interaction
Interaction
C: Power
C: Power
90.0
3.6
C80.0
C+
2.7
C-
UPKs
Taste
70.0
60.0
1.9
50.0
1.0
C+
40.0
30.0
0.1
4.00
4.50
5.00
B: Time
Edme section 1
5.50
6.00
4.00
4.50
5.00
5.50
6.00
B: Time
67
Agenda Transition
 DOE – Process and design
construction
Screening
Known
Factors
Unknown
Factors
 Step-by-step analysis (popcorn)
Vital few
Characterization
 Popcorn analysis via computer
 Multiple response optimization
Learn to use numerical search
tools to find factor settings to
optimize tradeoffs among
multiple responses.
Trivial
many
Screening
Factor effects
and interactions
no
Curvature?
yes
Optimization
Response
Surface
Methods
Verification
Confirm?
Celebrate!
no
Backup
yes
 Advantage over one-factor-at-a-time
(OFAT)
Edme section 1
68
Popcorn Optimization
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”.
Edme section 1
69
Popcorn Optimization
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.
Edme section 1
70
Popcorn Optimization
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:
Design-Expert® Software
Interaction
Desirability
C: Power
Design Points
1.000
C- 75.000
C+ 100.000
X1 = B: Time
X2 = C: Power
0.750
Desirability
Actual Factor
A: Brand = Costly
Prediction 0.64
0.500
0.250
0.000
4.00
4.50
5.00
5.50
6.00
B: Time
Edme section 1
71
Popcorn Optimization
5. Choose “Contour” and “3D Surface” from the “View”
menu :
Design-Expert® Software
Desirability
X1 = B: T ime
X2 = C: Power
Expert® Software
Predi cti 0.64
lity
gn Points
Actual Factor
Desirability Contour
100.00
A: Brand = Cheap
0.7
0.500
0.6
T ime
Power
0.5
0.400
actor
d = Cheap
93.75
0.300
Desirability
0.4
0.100
C: Power
0.200
87.50
0.3
0.2
0.1
0
0.200
0.100
81.25
100.00
0.300
93.75
6.00
87.50
C: Power
75.00
4.00
4.50
5.00
5.50
5.50
5.00
81.25
4.50
6.00
75.00
4.00
B: Time
B: T ime
Edme section 1
72
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
73
Agenda Transition
 DOE – Process and design
construction
Screening
Known
Factors
Unknown
Factors
Trivial
many
Screening
 Step-by-step analysis (popcorn)
Vital few
Characterization
Factor effects
and interactions
 Popcorn analysis via computer
no
 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
Curvature?
yes
Optimization
Response
Surface
Methods
Verification
Confirm?
Celebrate!
no
Backup
yes
74
Traditional Approach to DOE
One 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
75
Relative Efficiency
Factorial versus OFAT
B
B
A
A
Relative efficiency = 6/4 = 1.5
B
B
C
C
A
A
Relative efficiency = 16/8= 2.0
Edme section 1
76
2k Factorial Design
Advantages
 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.
Edme section 1
 Form base for more complex designs.
Second order response surface design.
77