Edme section 11 DOE – Process and design construction Step-by-step analysis (popcorn) Popcorn analysis via computer Multiple response optimization.

Edme section 1 1

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 2

Agenda Transition

DOE – Process and design constructionIntroduce the process for designing factorial experiments and motivate their use.





yes

Factor effectsand interactions

ResponseSurfaceMethods

Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

Edme section 1 3

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 4

Expend no more than 25% of budget on the 1st cycle.

Conjecture

Design

Experiment

Analysis

Iterative Experimentation

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 (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 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

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 8

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 9

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 10

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 11

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 12

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 13

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 14

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 15

Agenda Transition


Step-by-step analysis (popcorn)Learn benefits and basics of two-level factorial design by working through a simple example.




yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

Edme section 1 16

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 17


* 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 18


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 19

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 20

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 21

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 22

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 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 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 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

0|Effect|

BC

B

C

Significant effects:

The model terms!

Negligible effects: The error estimate!

Edme section 1 27

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 28

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 29

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 30


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 31


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 32

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 33

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 34

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 35

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

Agenda Transition



Popcorn analysis via computerLearn to extract more information from the data.



Edme section 1 36

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 37

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 38

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 39

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 40

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 41

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 42

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 43

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 44

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 45

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 46

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 47

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 48

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 49

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 50

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 51

Popcorn Analysis – Taste Diagnostics - ANOVA Assumptions


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



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 52




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



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 53


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


Details in section 3

Edme section 1 54

Popcorn Analysis – Taste Influence



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



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 55


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 56




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 57


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 58

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 59

Popcorn Analysis – Taste Model Graphs – View, Interaction Plot (BC)


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 60

Popcorn Analysis – Taste Model Graphs – View, Contour Plot

and 3D Surface (BC)


TasteDesign Points81

32



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 61

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 62

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 63

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 64

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

Agenda Transition




Multiple response optimizationLearn to use numerical search tools to find factor settings to optimize tradeoffs among multiple responses.


Edme section 1 65

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 66

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 67

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:



Desirability

Design Points

C- 75.000C+ 100.000


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 68

5. Choose “Contour” and “3D Surface” from the “View” menu :



DesirabilityDesign Points



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


Desirability



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 69


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 70

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

yes



Curvature?

Confirm?

KnownFactors

UnknownFactors

Screening

Backup

Celebrate!

no

no

yes

Trivialmany

Vital few

Screening

Characterization

Optimization

Verification

Agenda Transition





Advantage over one-factor-at-a-time (OFAT)Summarize the benefits factorial design has over one-factor-at-a-time experimentation.

Edme section 1 71

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 72

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 73

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.

Edme section 11 DOE – Process and design construction Step-by-step analysis (popcorn) Popcorn analysis via computer Multiple response optimization.

Documents

edme section

analysis of experiments

design layout

factor of

distribution of averages

time ofat slide

agenda transition doe

input factors