design of experiment using minitab book_1

DuPont Quality Management and Technology SOE/MTB - 1.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000

Section 1

INTRODUCTION


WHAT IS DESIGN OF EXPERIMENTS (DOE)?

DOE is a strategic process, with supporting methods and tools, for guiding the

- planning- execution- analysis of results- application of results

of experimental or developmental programs.


BENEFITS OF DOE

Highest leverage quality tool available --- use to design quality in up front

Reduces product/process development cycle time Most efficient strategy for gaining process

understanding Develops true cause-and-effect relationships Provides solution to current problem plus

information for solving future problems An objective, fact-based system for decision

making, complete with quantitative measures of uncertainty


WHERE CAN DOE BE APPLIED?

DOE is useful in every stage of product life cycle– Product and Process Development– Process Scale-Up, Operations Startup, Customer

Verification– Process Control, Product and Process

Improvement


APPLICATIONS IN PRODUCT ANDPROCESS DEVELOPMENT

Define and translate customer needs Design robust products Design robust processes Reduce time to commercialization Develop test methods Define operating procedures Enhance business integration of R&D with

Operations


APPLICATIONS IN SCALE-UP, STARTUPAND CUSTOMER VERIFICATION

Reduce time to process qualification Identify key process variables Determine product specifications Design product field tests Develop standard operating conditions and

standard operating procedures


APPLICATIONS IN PROCESS CONTROL

Develop process models Calibrate process control knobs Adjust to changing customer needs Troubleshoot process problems Develop sampling protocols


APPLICATIONS IN PRODUCT AND PROCESS IMPROVEMENT

Reduce product variability Improve first-pass first-quality yield Increase capacity Reduce transition time Make process more robust Improve test methods


WHAT YOU SHOULD GET OUT OF SOE

An appreciation of the basic underlying concepts of DOE

A general strategy for approaching experimentation

A set of efficient, widely-applicable tools for designing and analyzing experiments

Hands-on experience at using these tools Some sense of when and where to seek

expert help

DuPont Quality Management and Technology SOE/ECHIP - 2.1© 2000 E. I. du Pont de Nemours and Company Revised 3/10/2000

Section 2

WORKSHOP 1


WORKSHOP 1 PROBLEM

A new polymer is being readied for a plant process. One major problem remains: the COLOR (yellowness) has often been unacceptable in experimental production to date. The COLOR value should be made as low as possible. Prior work has indicated that COLOR may be affected by the following variables.

EXP. RANGEABBREV. VARIABLE NAME LOW HIGH UNITS

C Catalyst Concentration 1.0 1.8 %T Reactor Temperature 130 190 deg CA Additive Amount 1 5 kg

Also from prior work, it is known that MODULUS can be predicted by the following equation over the experimental range of interest:

MODULUS = -69.5 + 100*C + 0.15*T - 5.0*A


Your problem is:1. To demonstrate an approximate set of conditions

to obtain low COLOR together with low MODULUS, and

2. To support your conclusion with a description of the effects of the factors on the response, COLOR.

Your boss’s best guess of a good place to start is:CATALYST = 1.25 %TEMPERATURE = 137 deg CADDITIVE = 3 kg

WORKSHOP 1 PROBLEM


WORKSHOP 1 TEAM REPORTS

TeamRec. Settings

Cata Temp Addi Color Mod.# ofRuns

123456789

10

ResultsCata Temp Addi

Effects on ColorMethod/Comments


WORKSHOP 1 TEAM REPORTS

TeamRec. Settings

Cata Temp Addi Color Mod.# ofRuns

11121314151617181920

ResultsCata Temp Addi

Effects on ColorMethod/Comments


WORKSHOP 1:Concepts Introduced

Experimental Variability Is a Fact of Life Properties Can Be Represented As

Functions of Control Variables Geometry of Experimental Region Contour Plots Interaction Multiple Responses


Section 3

FOUNDATIONS OF THE STRATEGY


UNDERLYING PRINCIPLES

World is multivariate Experimental error is a fact of life Experimentation is a process Multi-stage approach Statistical strategy The 6 B’s of DOE


WORLD IS MULTIVARIATE

Almost always more than one factor of interest that can be varied

e.g. pressure, temperature, pH, flow rate, hold time, screw speed

Interactions often present --- factor effects not additive i.e. synergistic or antagonistic effects

Usually, several responses (outcomes) of intereste.g. viscosity, yield, assay, color, hardness, modulus, dyeability

Tradeoffs between responses often necessary


VARIABLES

RESPONSESFACTORS LURKINGVARIABLES

variables that aredeliberately controlled

in the experiment

outcome variablesthat are measured

during the experiment

variables that areunidentified oruncontrolled

synonymsDEPENDENT VARIABLES

PROPERTIESCHARACTERISTICS

OUTCOMES

INDEPENDENT VARIABLESPREDICTORS

KNOBSPROCESS VARIABLES

TREATMENTSDOSES

NOISEUNCONTROLLED VARIABLES


EXPERIMENTAL ERROR

Experimental error is the “noise” in the system --- the catchall term used to explain why results are not identical from replicate to replicate

2 types of experimental error:– systematic or bias error– random error


RANDOM ERROR

0 0

•Unpatterned variability

•Unpredictable

•Multiple unassignable causes

•Normal error distribution

•Standard deviation represents “typical” error


SYSTEMATIC OR BIAS ERROR

•Patterned variation

•May be predictable

•Due to single assignable cause–e.g. shift, raw material lot, day, tool wear, ambient temperature


DEALING WITH EXPERIMENTAL ERROR

EXPERIMENTAL ERROR

RANDOM ERROR SYSTEMATIC ERROR

CAUSE Unassignable Assignable

NATURE Unpatterned Patterned

REMEDY Replication BlockingRandomization


REPLICATION

true relationship

fitted relationship

random error

random error

true relationship

fitted relationship

average

average

•Through averaging of replicates the impact of random error is reduced

•Two forms of replication–Hidden replication --- feature of all good designs

–Pure replication (as above)

FACTOR FACTOR

RES

PON

SE

RES

PON

SE


BLOCKING

Used in presence of identifiable source of potential bias (“blocking factor”)

– e.g. day, raw material lot Split the experiment up into blocks

representing different levels of the blocking factor

Keep the blocks balanced with respect to the experimental factors. This prevents confounding, or confusing, any block effect with that of an experimental factor.


RANDOMIZATION

“Insurance” protection against potential unidentified sources of bias

Randomize the order of experimental runs If experiment is blocked, randomize within

blocks May require constrained randomization if

some experimental factors hard to change


EXPERIMENTATION IS A PROCESS

Gather Information

Define Objectives

Design Experiment

Run Experiment

Analyze Experiment

Interpret Results

Perform Confirmation Runs

Go tonext stage of

experimentation?

Apply Results

Update Information

yes

no


MULTI-STAGE APPROACH

SCREENING INTERACTION RESPONSEDESIGNS DESIGNS SURFACE

DESIGNS

Evolution of the Experimental Environment

NUMBER OF 6 or more 3 - 8 2 - 6FACTORS

OBJECTIVE Identify key factors Understand factor Prediction modelinteractions Optimization

COMMON Plackett-Burman Full Factorial Box-BehnkenDESIGNS Fractional Factorial Fractional Factorial Central Composite

(resolution 3 or 4) (resolution 5) Face Center Cube


STATISTICAL STRATEGYVS. ONE-FACTOR-AT-A-TIME

Y

X1

Y

X1

Levelsof X2

Levelsof X2

One-Factor-At-A-Time Statistical


COMPARING THE STRATEGIES

1-FACTOR-AT-A-TIME STATISTICAL

DESIGN

FITTED MODEL

EXPERIMENTAL ERROR

INTERACTIONS

Vary only 1 factor at time,in multiple small increments,keeping all others fixed

Curves fitted through datapoints, separately for each factor

Ignored

Not considered; not estimable

Vary all factors jointly in balanced bite-sized space-filling design

Simple empirical models based on low-order polynomials

Recognized and estimated

Dealt with as appropriate


6 B’S OF DOE

Bite Size– Just enough runs to meet objectives, achieve desired sensitivity, and

estimate experimental error Boldness

– Vary experimental factors over wide range– Measure all relevant responses

Balance– Use balanced designs to maximize efficiency and minimize confounding

Bias Error– Take countermeasures such as randomization and blocking

Blunders– Avoid through careful planning and execution

Batting Average– Improve your odds of success through statistical designs and empirical

modeling


Section 4

FACTORIAL GEOMETRY


EVOLUTION OF THE ENVIRONMENT:Intermediate Stage

SCREENING RESPONSEDESIGNS SURFACE

DESIGNS






INTERACTIONDESIGNS


TWO-LEVEL FACTORIAL DESIGNS

2k Distinct Runs Easy to Plan and Analyze Usable for Either Continuous or Discrete

Factors with Two Levels Uniformly Spread Through Factor Space Permit Estimation of Both Main Factor

Effects and Interaction Effects


23 FACTORIAL DESIGN

X3

X2

X1

(HI, HI, HI)

(HI, LO, HI)

(LO, HI, HI)

(LO, LO , HI)

(HI, HI, LO)

(HI, LO, LO)

(LO, HI, LO)

(X1, X2, X3) =(LO, LO, LO)


3 FACTOR 2 LEVEL FULL-FACTORIALExperiment Data

CATALYST TEMPERATURE ADDITIVE COLOR MODULUS1.81.01.81.01.81.81.01.0

190190130190130190130130

51551151

7248374836746261

11454

10534

1251342545


23 FACTORIAL DESIGNwith Workshop 1 COLOR Data

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36


WHAT IS AN EFFECT ?

An effect is the difference in the averagesof two groups of observations.

Let YLOW = average of values on LOW plane.Let YHIGH = average of values on HIGH plane.

EFFECT = YHIGH - YLOW


SAMPLE CALCULATION: EFFECT OF X1

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

72 + 37 + 74 + 36 48 + 62 + 48 + 614 4

= 24 + (-25) + 26 + (-25)4

= 0

24

-25

26

-25



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

48 + 72 + 48 + 74 62 + 37 + 61 + 364 4

= -14 + 35 + (-13) + 384

= 11.5

-14

-13

35

38



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

48 + 72 + 62 + 37 48 + 74 + 61 + 364 4

= 0 + (-2) + 1 + 14 = 0

0 -2

1 1


ONE FACTOR AT A TIME

X3

X2

X1

No Hidden ReplicationNot Space Filling


INTERACTION OF X1 AND X2

Y

X1

Low X2

High X2

Y

X1

Low X2

High X2

Y

X1

Low X2

High X2

Y

X1

Low X2

High X2

INTERACTION

NO INTERACTION


INTERACTION GEOMETRYX1 Effect at High and Low X2

X1

X2

X1*X2 Interaction Effect

= [(YD - YC) - (YB - YA)] / 2

(YA + YD) (YB + YC)=

2 2

YA

YB

YC

YD


INTERACTION GEOMETRYX2 Effect at High and Low X1

X1

X2

X1*X2 Interaction Effect

= [(YD - YB) - (YC - YA)] / 2

(YA + YD) (YB + YC)=

2 2

YA

YB

YC

YD


SAMPLE CALCULATION: X1*X2 INTERACTION EFFECT

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

48

37

74

61 36

72 + 74 + 62 + 61 48 + 37 + 48 + 364 4

= 25



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

37 + 72 + 48 + 61 48 + 62 + 74 + 364 4

= -0.5



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

48 + 72 + 61 + 36 62 + 37 + 48 + 744 4

= -1


SAMPLE CALCULATION: X1*X2*X3 INTERACTION EFFECT

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

72 + 62 + 48 + 36 48 + 37 + 74 + 614 4

= -0.5


HALF OF THREE-FACTOR INTERACTION SHOWS BALANCE IN ALL FACTOR PAIRS

X2X1

X3

X3

X2

X1


USES OF HIGHER-ORDERINTERACTION GEOMETRY

Blocking– Basis for Splitting Experiment into

Smaller Blocks– Factors Are Balanced Within Blocks

Screening– Cut Experiment in Half by Using Only

One of Blocks– Factors Are Balanced


FACTORIAL DESIGNS: Summary

The “Cube” Approach Each Dimension Is a Factor Coding: Low = “-” and High = “+” Effects Are Comparisons of “Planes” Hidden Replication Efficiency: All Data Used to Calculate

Each Effect High-Order Interactions

Du Pont Quality Management and Technology© 2000 E. I. du Pont de Nemours and Company

SOE/MTB - 5.1Revised 5/09/2000

Section 5

FACTORIAL EXAMPLE:Design

Presenter

Presentation Notes

We will now generate a full-factorial design using Minitab using the same problem as we used in Workshop 1.



Follow the example to design a 2-level full-factorial experiment forthe workshop 1 problem.

Factors RangeCATALYST CONCENTRATION 1.0 to 1.8 REACTOR TEMPERATURE 130 to 190AMOUNT OF ADDITIVE 1 to 5

ResponsesCOLOR MODULUS

Design = Full-Factorial with center pointsModel = Linear + 2 factor interaction terms

FACTORIAL DESIGN EXAMPLEDesign Phase

Link to Workshop 1 Problem

Presenter

Presentation Notes

In this example, we will Generate a 2-level full-factorial design for the above three factors. Run some overall center points to test for overall curvature (all the factors are continuous) Replicate the entire design -- have 2 runs at each of the corner points. Thus we will have 2*(23) =16 runs among the corner points and 4 runs of the overall center point for a total of 20 runs. Randomize the entire design (as you should always do as much as possible) Later we will enter response data and analyze it.



X3

X2

X1

THREE-FACTOR CUBE PLOT

Presenter

Presentation Notes

This cube plot shows the points at which we will be collecting data. There will be 2 runs at each of the corners and 4 runs of the overall center point.



TEST FOR CURVATURE

+0-

YDifference IsCurvature

..............

X

Presenter

Presentation Notes

What will we do with the data we collect from the overall center point ? After the response data is collected, we will fit a model that does NOT include curvature. We will then be able to test whether or not the model we have fit adequately fits the center region of the cube. To do this, we essentially compare what the model predicts for the center with the results of the center point runs.



Presenter

Presentation Notes

Start up Minitab Note that this session is set up with 3 windows: The session window shows commands that have been run and non-graphics output The worksheet window is a spreadsheet for the data The project manager window summarized major steps and results we have completed.



Presenter

Presentation Notes

We want to generate a design so from the Stat pull-down window select DOE Factorial Create Factorial Design...



Presenter

Presentation Notes

Select 2-level factorial (default generators) We want the Number of factors to be 3 Click on Designs… to further specify the desired design We want a Full Factorial (8 runs) with 4 runs of the center point and 2 replicates of the corner points (2 runs of each point) and just 1 block (the default) This will give us 2 * 23 + 4 = 20 runs total Click on the OK button



Presenter

Presentation Notes

Click the Factors… button to specify the factor names and levels for the design Enter the factor names and low and high levels Click the OK button when finished



Presenter

Presentation Notes

Click on the OK button to produce the design



Presenter

Presentation Notes

The design has been generated and is stored in columns C1-C7 of the worksheet. Expand the worksheet window to get a better view of the design



Presenter

Presentation Notes

Review the worksheet for operability and reorder as necessary. Print the worksheet and run the design in the order specified by the worksheet. This can be done here by using the Print Worksheet… item of the File pull-down menu. Reduce the worksheet size again to view the other windows.



Presenter

Presentation Notes

Don’t forget to save the project From the File pull-down menu, select Save Project As… and specify a location and name for the file.



Presenter

Presentation Notes

Run the design on the process and return here when you are ready to enter response data into the worksheet. For purposes of this workshop Enter names for the response columns (COLOR and MODULUS) - use columns other than those used by the design process (C1-C7) in this case. Editor>Enable Commands Type %WK1REV in the session window. This will run a simulator that will enter response data into the worksheet response columns. To view the response data, maximize the worksheet window again.



Presenter

Presentation Notes

Reduce the worksheet window size when done to see the other windows again.



Presenter

Presentation Notes

Save the project again so that the response data is saved with the design data. This can be done with the CTRL/S keys, clicking on the diskette button, or through the File pull-down menu. We will return to this data later for analysis.


Section 6

ANALYSIS OF TWO-LEVEL FACTORIAL DESIGNS


SECTION 6 OVERVIEW

How do we know the effects are real ?

Computer analysis of the effects

How well have we explained the behaviorof Y?


HOW DO WE KNOW THE EFFECTS ARE “REAL” ?If each corner is an average of replicated runs, we can

study the repeatability of the effects.

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

Rep. 1

Rep. 262

48 72

37

48 74

61 36


EXAMINE THE REPLICATES FOR REPEATABILITY (Case 1)

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

4650

7371

4947

7573

3834

6260

6163

3638



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

5292

3363

44104

5418

2498

19105

5717

6630


THE IMPACT OF EXPERIMENTAL ERROR

Effect

AverageRelationship

Low (-) High (+)Factor

Response Experimental Error


HOW “REAL” ARE THE EFFECTS ?

Must estimate the size of the experimental error.

STANDARD DEVIATION: A measure of variability

(Y1- Y)2 + (Y2 - Y)2 + . . . + (Yn - Y)2

n - 1Std. Dev. = s =



X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

4650

7371

4947

7573

3834

6260

6163

3638

s=2.8

s=1.4

s=2.8s=1.4

s=1.4

s=1.4

s=1.4s=1.4


POOLED STANDARD DEVIATION

Based on a weighted average of individual squared standard deviations

Assumes homogeneous error --- size ofexperimental error is uniform throughoutdesign region

A more reliable estimate of the overall standard deviation

(n1 - 1) s12 + (n2 - 1) s2

2 + . . . + (nk - 1) sk2

(n1 - 1) + (n2 - 1) + . . . + (nk - 1)spooled =


COMPUTE THE POOLED STANDARD DEVIATIONFOR THE 23 EXAMPLE: (Case 1)

spooled =

3.5 = 1.87

1*2 + 1*2 + 1*8 + 1*2 + 1*8 + 1*2 + 1*2 + 1*21 + 1 + 1 + 1 + 1 + 1 + 1 + 1

= 28

8

=


INTERPRETING THE STANDARD DEVIATION

-4 -3 -2 -1 0 1 2 3 4 SD units68%95%

99.7%


DISTRIBUTION OF AVERAGES

The distribution of the means of samples of size n is:–More nearly normal

–Narrower σY = σY n


THE STANDARD ERROR OF A FACTOR EFFECT

Represents the PRECISION in an estimated factor effect

Can be used to “test” if a factor effect is “statistically significant”

STD. ERROR = S-Pooled * SQRT(1/n1 + 1/n2)where

n1 = number of observations forming YLOW

n2 = number of observations forming YHIGH


EXAMPLE CALCULATION OF STANDARD ERROR

Consider the C*T interaction effect The estimated effect = 25 Standard error = 1.87 * SQRT(1/8 + 1/8) = 0.9

(1.87 is the pooled standard deviation calculatedearlier with 8 degrees of freedom)

Tabled t-value for 8 df, 95% confidence level

= 2.31 (t-table on next page)


t-DISTRIBUTION VALUES FOR TWO-SIDED CONFIDENCE INTERVALS

99%

63.709.925.844.604.03

3.713.503.363.253.17

3.113.053.012.982.95

2.922.902.882.862.85

2.832.822.812.802.79

2.752.702.662.622.58

95%

12.704.303.182.782.57

2.452.362.312.262.23

2.202.182.162.142.13

2.122.112.102.092.09

2.082.072.072.062.06

2.042.022.001.981.96

90%

6.312.922.352.132.01

1.941.891.861.831.81

1.801.781.771.761.75

1.751.741.731.731.72

1.721.721.711.711.71

1.701.681.671.661.64

DF

12345

6789

10

1112131415

1617181920

2122232425

304060

120∞


SIGNIFICANCE OF AN EFFECTBY SIGNIFICANCE TEST

Consider the C*T interaction effect Observed t-ratio = (estimated effect)/(std error)

= 25.0 / 0.9 = 28 Compare observed t-ratio to tabled t-value:

Here |t-ratio| > tabled t-value (28 > 2.31)So the C*T interaction is statistically significantat the 0.95 (or 95%) confidence level

Note that the C*T interaction is also significant at the0.99 confidence level (28 > 3.36).

How high can we take the confidence level and still besignificant? This is related to the P-value.


P-VALUE

The P-value of an effect is the chance of having observed an effect that large purely due to random experimental error

The larger the effect, the smaller the P-value P-value is commonly displayed by most statistical

packages, usually as a decimal (i.e. between 0 and 1) P-value = 1 - (maximum confidence level at which

the effect is significant) EXAMPLE: P-value of an effect = 0.02

So the effect is significant at 98% confidence level For the C*T interaction effect, the P-value is 0.0000 !


SIGNIFICANCE OF AN EFFECTBY CONFIDENCE INTERVAL

Consider the C*T interaction effect Confidence interval is:

Estimated effect +/- (t-value * standard error) For this example 95% confidence interval

is25 +/- (2.31 * 0.9) = 25 +/- 2.1 or 22.9 to 27.1

Whenever ZERO is not included in the confidence interval, the effect is significant at the specified confidence level

Graphical display of uncertainty


SECTION 6 OVERVIEW

How do we know the effects are “real” ?– Estimate the experimental error (s-pooled)

and the standard error of the effect– Compare the estimated effect with its standard

error– t-test or– confidence interval

Computer analysis of the effects How well have we explained the

behavior of Y ?


COMPUTER ANALYSIS OF THEESTIMATED EFFECTS

Two common approaches: EFFECTS ANALYSIS

– Displays each effect and t-test or confidence interval

– Follows approach just covered earlier REGRESSION ANALYSIS

– Expresses each effect as a slope of a line or coefficient of a model term

– Statistically equivalent to an effects analysis


STANDARD FACTOR CODINGRegression coefficient (slope) = 1/2 effect

Effect

AverageRelationshipSlope = ∆ / 2

Low (-1) 0 High (+1)

Factor

Response


MEANING OF COEFFICIENTS:Temperature Effect

190180170160150140130

75

65

55

45

35

CO

LO

R

Temperature

130 140 150 160 170 180 190

35

45

5

65

75Slope of this line =Regression coefficient = 5.75(using centered/scaled factor settings)

For this example:Effect = 11.5 = expected

increase in COLOR fromTemperature=130 to 190(-1 to +1)

Coefficient = 5.75 = expectedincrease in COLOR fromTemperature=160 (middle)to 190 (0 to +1). Usesorthogonally scaled(-1 to +1) factor settings.

(-1) (+1)(0)


3 FACTOR EXAMPLE (Case 1)Minitab Effects Table for COLOR Response

Fractional Factorial Fit: COLOR versus Catalyst, Temperature, Additive

Estimated Effects and Coefficients for COLOR (coded units)

Term Effect Coef SE Coef T P

Constant 54.7500 0.4488 122.00 0.000

Catalyst 0.0000 0.0000 0.4488 0.00 1.000

Temperat 11.5000 5.7500 0.4488 12.81 0.000

Additive 0.0000 0.0000 0.4488 0.00 1.000

Catalyst*Temperat 25.0000 12.5000 0.4488 27.85 0.000

Catalyst*Additive -0.5000 -0.2500 0.4488 -0.56 0.591

Temperat*Additive -1.0000 -0.5000 0.4488 -1.11 0.294


ITEMS IN THE MINITABEFFECTS & COEFFICIENTS TABLE

Term: Name for model term.

Effect: Expected change in the response over the entire range of the term (high “plane” - low “plane” averages).

Coef: Regression coefficients for the model terms. Expected change in the response per unit change in the term. Note carefully the centering and scaling used for these coefficients for interpretation.

SE Coef: Standard error of the coefficient -- uncertainty around the coefficients due to experimental error.

T: The ratio of the coefficients divided by their standard errors.

P: Probability of observing a coefficient of that magnitude when the true coefficient is zero. Low values (<.05) imply significance.

– If significant, we are pretty sure that at least the sign (direction) of the relationship is correct


3 Factor Example (Case 1)Graphical Representation of Factor Effects

Catalyst Temperature Additive

1.0 1.8 130 190 1 5

50.0

52.5

55.0

57.5

60.0

CO

LOR

Main Effects Plot (data means) for COLOR


3 Factor Example (Case 1)Graphical Representation of the Interactions

1 1.8 130 190 1 5

40

55

70

40

55

70

40

55

70Catalyst

Temperature

Additive

1

1.8

130

190

1

5

Interaction Plot (data means) for COLOR


3 FACTOR EXAMPLE (Case 1)Minitab Cube Plot for COLOR Response

36

37

74

72

61

62

48

48

1.0 1.8

Catalyst

Temperature

Additive

130

190

1

5

Cube Plot (data means) for COLOR


3 FACTOR EXAMPLE (Case 1)Pareto Plot of Effects on COLOR


3 FACTOR EXAMPLE (Case 1)Normal Probability Plot for Effects on COLOR

0 10 20

-1

0

1

Standardized Effect

Nor

mal

Sco

re

AB

B

Normal Probability Plot of the Standardized Effects(response is COLOR, Alpha = .10)

A: CatalystB: TemperatC: Additive


SECTION 6 OVERVIEW

How do we know the effects are “real” ?

Computer analysis of the effects– Effects analysis– Regression analysis

How well have we explained the behaviorof Y ?


TWO ESTIMATES OF EXPERIMENTAL ERROR

Pure Error Estimate (PE)– Based on variability among replicate runs under

fixed X-settings– Same as the pooled standard deviation

Lack of Fit Estimate (LOF)– Based on how well the estimated effects explain

the variation in the observed data.

When both the PE and LOF estimates are available, they can be used to test if additional effects shouldbe accounted for.


WHEN ARE P.E. AND L.O.F. ESTIMATES AVAILABLE ?

Not Available Available

NotAvailable

Available

Lack of Fit

Pure

Err

or

df: 2, 0, 0 df: 2, 0, 1

df: 2, 2, 0 df: 2, 3, 1

Key:df: #, #, #

Fit

PureError

Lack ofFit


THE ANALYSIS OF VARIANCE

Statistical technique for evaluating PURE ERROR and LACK-OF-FITexperimental error

Characterizes how well the estimatedeffects explain the data


ONE FACTOR EXAMPLEExperiment Worksheet

Row FACTOR RESPONSE1 -1 252 -1 353 0 604 0 805 1 456 1 55


ONE FACTOR EXAMPLEData Graph with Fitted Line

10-1

80

70

60

50

40

30

Factor X

Resp

onse

Y = 50 + 10XY


ONE FACTOR EXAMPLECoefficients and Analysis of Variance Results

The regression equation isRESPONSE = 50.0 + 10.0 FACTOR

Predictor Coef SE Coef T PConstant 50.000 7.906 6.32 0.003FACTOR 10.000 9.682 1.03 0.360

S = 19.36 . . .

Analysis of Variance

Source DF SS MS F PRegression 1 400.0 400.0 1.07 0.360Residual Error 4 1500.0 375.0Lack of Fit 1 1200.0 1200.0 12.00 0.041Pure Error 3 300.0 100.0

Total 5 1900.0


FURTHER EXPLANATION ON SELECTEDCOEFFICIENT TABLE ITEMS

Regression F-test: Tests whether any of the model terms explain the behavior of Y (overall test). If the model has some significant terms, the p-value should be small (<.05).

Lack of Fit F-test: Tests whether model can be improved with existing data. Might be significant due to missing terms, outliers, a need for a transformation, lack of measurement precision, etc. An appropriate model will have a non-significant lack of fit F-test (high p-value).

Square Root of MS Pure Error: Pooled standard deviation from replicates. Estimates SD of experimental error at any fixed set of conditions.


REGRESSION AND LACK OF FIT SIGNIFICANCEExample 1

Regression p-value = .0000Lack of fit p-value = .5514

FACTOR.X252015

14

12

10

8

6

4

2

0

FACTOR.X

RESP

.Y1




252015

5.5

5.0

4.5

FACTOR.X

RESP

.Y2




252015

9.5

8.5

7.5

6.5

5.5

4.5

3.5

2.5

FACTOR.X

RESP

.Y3




252015

12.6

11.6

10.6

9.6

8.6

7.6

6.6

5.6

4.6

3.6

FACTOR.X

RESP

.Y4


BACK TO OUR 3 FACTOR EXAMPLE . . .Case 1

X3 (Additive)

X2 (Temperature)

X1 (Catalyst)

62

48 72

37

48 74

61 36

4650

7371

4947

7573

3834

6260

6163

3638


3 FACTOR EXAMPLE (Case 1)Coefficients Table for COLOR Response

Fractional Factorial Fit: COLOR versus Catalyst, Temperature, Additive

Estimated Effects and Coefficients for COLOR (coded units)

Term Effect Coef SE Coef T P

Constant 54.7500 0.4488 122.00 0.000

Catalyst 0.0000 0.0000 0.4488 0.00 1.000

Temperat 11.5000 5.7500 0.4488 12.81 0.000

Additive 0.0000 0.0000 0.4488 0.00 1.000

Catalyst*Temperat 25.0000 12.5000 0.4488 27.85 0.000

Catalyst*Additive -0.5000 -0.2500 0.4488 -0.56 0.591

Temperat*Additive -1.0000 -0.5000 0.4488 -1.11 0.294


3 FACTOR EXAMPLE (Case 1)Analysis of Variance for COLOR Response

Analysis of Variance for COLOR (coded units)

Source DF Seq SS Adj SS Adj MS F P

Main Effects 3 529.00 529.00 176.333 54.72 0.000

2-Way Interactions 3 2505.00 2505.00 835.000 259.14 0.000

Residual Error 9 29.00 29.00 3.222

Lack of Fit 1 1.00 1.00 1.000 0.29 0.608

Pure Error 8 28.00 28.00 3.500

Total 15 3063.00


SECTION 6 OVERVIEW

How do we know the effects are “real” ?

Computer analysis of the effects

How well have we explained thebehavior of Y ?

– Analysis of variance results–Regression F-test–Lack of fit test



Section 7

FACTORIAL EXAMPLE:Analysis

Presenter

Presentation Notes

We now resume the workshop 1 problem (revisited) by analyzing the response data for the design we generated previously in Minitab.



Recall the workshop 1 problem (summarized below) for which we generated a design earlier in ECHIP. We already have response data entered so we’renow ready to analyze the data.

Factors RangeCATALYST CONCENTRATION REACTOR TEMPERATURE AMOUNT OF ADDITIVE

Design = Full-Factorial with center pointsModel = Linear + two-factor interaction terms

1.0 to 1.8130 to 190

1 to 5

FACTORIAL DESIGN EXAMPLEAnalysis Phase

ResponsesCOLORMODULUS



Additive

Temperature

Catalyst

THREE-FACTOR CUBE PLOT OFEXAMPLE WORKSHOP 1 DATA

6059

5857

4849

5051

3435

3638

7374

7271

38 4142 43

45

25

54

34

125

105

134

114

80

KEY: Numbers inside circles = COLOR valuesNumbers outside circles = MODULUS values



Workshop 1 RevisitedTeam

COLOR -- Coefficients (circle if significant)Cata Temp Addi C*T C*A T*A

Signif.of LOFtest

ReplicateStandardDeviation

123456789

10

Comments



Presenter

Presentation Notes

Retrieve your workshop 1 revisited data into Minitab using either the Open Project … item or the previous projects list in the File pull-down menu. Recall that we called the project WKSHOP1REV.



Presenter

Presentation Notes

To analyze factorial data, select from the Stat pull-down menu DOE Factorial Analyze Factorial Design



Presenter

Presentation Notes

Select the response columns to analyze by highlighting the columns in the left window and clicking the Select button.



Presenter

Presentation Notes

Click on the Terms… button to review or edit the list of effects to be estimated.



Presenter

Presentation Notes

With a full factorial design, we can estimate main effects, 2-factor interactions, as well as higher order interactions. We will estimate the main effects and all interactions but remove the term for the center point from the model. Uncheck the Include center points in the model box This way, the variability from the replicates and from curvature will be combined when assessing the significance of the effects. We do not have the proper data to assess curvature of each of the individual factors with this design. Click OK



Presenter

Presentation Notes

Click OK to fit the model



Fractional Factorial Fit: COLOR versus CATALYST, TEMPERATURE, ADDITIVEEstimated Effects and Coefficients for COLOR (coded units)

Term Effect Coef SE Coef T PConstant 51.450 1.535 33.52 0.000CATALYST 0.125 0.063 1.716 0.04 0.972TEMPERAT 13.875 6.938 1.716 4.04 0.002ADDITIVE 0.125 0.062 1.716 0.04 0.972CATALYST*TEMPERAT 22.875 11.438 1.716 6.66 0.000 CATALYST*ADDITIVE 0.125 0.063 1.716 0.04 0.972TEMPERAT*ADDITIVE -0.125 -0.063 1.716 -0.04 0.972CATALYST*TEMPERAT*ADDITIVE -2.125 -1.063 1.716 -0.62 0.547

Analysis of Variance for COLOR (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 3 770.19 770.19 256.729 5.45 0.0132-Way Interactions 3 2093.19 2093.19 697.729 14.81 0.0003-Way Interactions 1 18.06 18.06 18.063 0.38 0.547Residual Error 12 565.51 565.51 47.126

Curvature 1 546.01 546.01 546.012 308.01 0.000Pure Error 11 19.50 19.50 1.773

Total 19 3446.95

Unusual Observations for COLOR

Obs COLOR Fit SE Fit Residual St Resid4 38.0000 51.4500 1.5350 -13.4500 -2.01R

R denotes an observation with a large standardized residual . . .

Presenter

Presentation Notes

Scroll up a bit in the session window to view the COLOR response analysis. Note that the TEMPERATURE effect and CATALYST*TEMPERATURE interaction are both significant. Although we haven’t specifically asked for overall curvature to be added to the model, Minitab separates the variability due to curvature from the pure error (replicate) variability in the analysis of variance table. It is highly significant indicating that we need to get further data to determine where the curvature is coming from. Observation 4 is marked as a potential outlier. Although its standardized residual is just outside [-2,2], it is not of concern.



Fractional Factorial Fit: MODULUS versus CATALYST, TEMPERATURE, ADDITIVE

Estimated Effects and Coefficients for MODULUS (coded units)

Term Effect Coef SE Coef T PConstant 79.60 0.05774 1378.71 0.000CATALYST 80.00 40.00 0.06455 619.68 0.000TEMPERAT 9.00 4.50 0.06455 69.71 0.000ADDITIVE -20.00 -10.00 0.06455 -154.92 0.000CATALYST*TEMPERAT -0.00 -0.00 0.06455 -0.00 1.000CATALYST*ADDITIVE 0.00 0.00 0.06455 0.00 1.000TEMPERAT*ADDITIVE 0.00 0.00 0.06455 0.00 1.000CATALYST*TEMPERAT*ADDITIVE 0.00 0.00 0.06455 0.00 1.000

Analysis of Variance for MODULUS (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 3 27524.0 27524.0 9174.67 1E+05 0.0002-Way Interactions 3 0.0 0.0 0.00 * *3-Way Interactions 1 0.0 0.0 0.00 * *Residual Error 12 0.8 0.8 0.07

Curvature 1 0.8 0.8 0.80 Pure Error 11 0.0 0.0 0.00

Total 19 27524.8. . .

Presenter

Presentation Notes

The analysis of the MODULUS response is not very interesting here since we have added no variability into its simulation.



Presenter

Presentation Notes

We’ll produce a plot of the factor effects for each response. From the Stat pull-down menu, select DOE Factorial Factorial Plots ...



Presenter

Presentation Notes

Check the boxes for the Main Effects graph and the Interaction graph. Click on the Setup... button for the Main Effects graph



Presenter

Presentation Notes

We will ask for a graph showing the main effects all factors for both responses. Select both response columns and click on Select to enter them into the Responses entry box Use the >> button to select all of the factors. Click on the OK button We are now back in the previous window. Click on the Setup… button for the interaction plot.



Presenter

Presentation Notes

We will look at a plot of the significant interaction that we saw in the analysis of the COLOR response. That was the Catalyst*Temperature interaction. No interactions were of interest with the MODULUS response, so select COLOR as the response for the interaction plot Select the 2 factors involved in a significant interaction. In this case we want CATALYST, and TEMPERATURE. Click on the OK button Click on the OK button in the previous window also to produce the graphs.



1.0 1.8

190190130130

70

60

50

40

TEMPERATURE

CATALYST

Mea

nInteraction Plot (data means) for COLOR

Centerpoint

Presenter

Presentation Notes

The first graph we see is the interaction plot between Catalyst and Temperature on the COLOR response. The lines here show the Temperature effect on Color at low and high Catalyst levels. The lines have markedly different slopes illustrating this significant interaction. Additionally, a point is plotted at the average response for the center point runs. That point is quite far from where a model using main effects and interactions would predict it to be (the center of the lines) also indicating curvature which we need to investigate further with additional data. Close the graph window when done.



ADDITIVETEMPERATURECATALYST

511901301.81 .0

120

100

80

60

40

MO

DU

LUS

Main Effects Plot (data means) for MODULUSCenterpoint

Presenter

Presentation Notes

Here we see the main effects for the MODULUS response. Note that the center point runs fall right on the lines indicating no unexplained curvature. Recall that the MODULUS response had no random variability in it. Close the graph window to view the next graph.



ADDITIVETEMPERATURECATALYST

511901301.81 .0

60

55

50

45

40

CO

LOR

Main Effects Plot (data means) for COLORCenterpoint

Presenter

Presentation Notes

This main effects plot for the COLOR response shows the large TEMPERATURE main effect and the fact that the center point runs are not fit well with the current list of terms (model). Close this last graph window



Presenter

Presentation Notes

Save the project again by clicking on the diskette button. We are done with this data for now.


Section 8

GOOD EXPERIMENTALPRACTICE


GOOD EXPERIMENTAL PRACTICE

1. Assess the environment2. Consider the factors3. Consider the responses4. Choose an appropriate design5. Consider strategies for bias error6. Create the experimental plan7. Review the plan for operability8. Avoid blunders9. Plan for the analysis

10. Report the recommendations


1. ASSESS THE ENVIRONMENT

Gather basic information Determine current state of understanding Define experimental objectives Define physical environment and constraints Consider experimental error


2. CONSIDER THE FACTORS

How many and which factors to vary? Which factors to keep fixed, and at what

level? Are factors continuous or discrete? How many levels of each factor? How bold in choice of levels?


3. CONSIDER THE RESPONSES

Consider all responses of potential interest Are responses continuous or discrete? What is effect size of interest? Is measurement error absolute or relative? Anticipate measurement issues


4. CHOOSE AN APPROPRIATE DESIGN

Identify underlying model Choose appropriate design class Consider desired sensitivity (resolution) in

choosing size of design Consider extra runs Practical constraints?


NUMBER OF RUNS VS. SENSITIVITY

SOE Rule of Thumb for balanced 2-level factorial designs:

n = 7 or 8∆ / σ( )2

∆ = smallest size effect worth detectingσ = standard deviation of experimental error

∆ / σ is “signal-to-noise ratio”

∆ / σ 0.5 1.0 1.5 2.0

n 196-256 49-64 22-28 12-16


5. STRATEGIES FOR BIAS ERROR

Consider blocking to protect against bias due to identifiable variables

– e.g. time, material batch, operator Use randomization to protect against bias

due to unidentified variables Consider possible constraints on

randomization, do restricted randomization if necessary

Assess randomization, adjust if necessary


6. CREATE THE EXPERIMENTAL PLAN

Create experimental worksheet, in randomized run order

Express factor levels in actual physical units

Include blank columns for responses Write detailed protocol with explicit

instructions for every step


7. REVIEW PLAN FOR OPERABILITY

Include all relevant parties in review– Planner– Executor– Lab analyst– Local experts– Statistician

Examine all runs for operability, adjust if necessary

Consider constraints on experiment, feasibility of schedule


8. AVOID BLUNDERS

Anticipate problems in advance Emphasize importance of adhering to

experimental protocol Avoid shortcuts Understand importance of each run Avoid stopping short Record any deviations from plan


9. PLAN FOR THE ANALYSIS

Consider analysis issues in advance– Models– Software– Method of analysis– Organization of data

Plot the data


10. REPORT RECOMMENDATIONS

Relate recommendations to the objective Recognize value of negligible effects Avoid extrapolation Report raw data for credibility Provide direction for future action


Section 9

SCREENING DESIGNS


EVOLUTION OF THE ENVIRONMENT:Early Stage

INTERACTION RESPONSEDESIGNS SURFACE

DESIGNS






SCREENINGDESIGNS


CHARACTERISTICS OF SCREENING DESIGNS

Number of runs n only a few more than number of factors k

Factors considered at two levels each Use a fraction of the full 2k factorial design Designs balanced (orthogonal) Main factor effects clear of each other Interactions generally not estimable Two-way interactions may be fully or

partially confounded with main effects


CLASSES OF SCREENING DESIGNS

Plackett-Burman (P-B) designs

Fractional Factorial (FF) designs


PLACKETT-BURMAN DESIGNS

Available in sizes n which are multiples of 4 Can handle up to n-1 factors, although

recommended maximum is n-5 Tables available for commonly used sizes of

n=12, 20, 24, 28 For n equal to power of 2 (e.g. 8, 16, 32,...),

same as Fractional Factorial designs Two-way interactions partially confounded

with each main effect (for n not a power of 2)


12 RUN PLACKETT-BURMAN DESIGNTrial X1 X2 X3 X4 X5 X6 X7 X8 X9 X10 X11

1 + + - + + + - - - + -2 + - + + + - - - + - +3 - + + + - - - + - + +4 + + + - - - + - + + -5 + + - - - + - + + - +6 + - - - + - + + - + +7 - - - + - + + - + + +8 - - + - + + - + + + -9 - + - + + - + + + - -10 + - + + - + + + - - -11 - + + - + + + - - - +12 - - - - - - - - - - -

last row all minuses

each rowa cyclicpermutationof previousrow


BALANCE OF 12 RUN P-B DESIGN

Illustrate using factors X1 and X2Rows rearranged according to level of X1

Trial X1 X21 + +2 + -4 + +5 + +6 + -

10 + -

3 - +7 - -8 - -9 - +

11 - +12 - -

equal numberof + and -

equal numberof + and -

SAME BALANCE TRUEFOR ANY PAIR OF FACTORS(COLUMNS ARE “ORTHOGONAL”)


PARTIAL CONFOUNDING IN 12 RUN P-B

Illustration showing partial confounding between X1 and X2*X3 interaction.Rows rearranged according to level of X1.

Trial X1 X2 X3 X2*X31 + + - -2 + - + -4 + + + +5 + + - -6 + - - +10 + - + -

3 - + + +7 - - - +8 - - + -9 - + - -

11 - + + +12 - - - +

2 +’s4 -’s

4 +’s2 -’s

not completely balanced

If there is an X2*X3 interaction effect, it will slightly bias the estimate of the X1 effect(as well as all other main effects except X2 and X3).


USING PLACKETT-BURMAN DESIGNS

If using tabled design, assign factors to any columns of the design

If blocking, use additional column(s) to define blocks

Save at least 4 columns for estimating experimental error

Assess sensitivity of design --- if inadequate:– Use larger design– Add reflected design


FRACTIONAL FACTORIAL DESIGNS

Available design sizes in powers of 2 For k factors, 2k-p FF design is a 1/2p fraction

of a full 2k factorial design 16 and 32 run designs most useful for

screening Confounding between any two effects is

either total or absent Degree of confounding determined by

Resolution of design


CONSTRUCTING FF DESIGNS

TO CONSTRUCT 2k-p DESIGN, WHERE k = NUMBER OF FACTORS:

1. Construct a full factorial design in k-p of the factors.

2. Define each of the p additional factors as equal to, or the negativeof, a judiciously selected interaction among some of the first k-pfactors. These are called the defining relations of the fractionalfactorial design.

For given values of k and p, tables are available of good choicesfor the defining relations which will result in the highest possibleresolution design.


EXAMPLE 1: CONSTRUCTING A 24-1 DESIGN

4 FACTORS --- FOR SIMPLICITY, DENOTE FACTORS BY 1, 2, 3, 41. Write down full factorial 23 design in factors 1, 2, 32. Define factor 4 equal to 123 interaction1 2 3 4=123 Defining Relation 4=123 induces- - - - additional relations:+ - - + 1=234 2=134 3=124- + - + 12=34 13=24 14=23+ + - - 1234=I (I =column of all +)- - + + NOTE THAT EFFECTS ARE+ - + - CONFOUNDED IN PAIRS:- + + - main effects with 3-way+ + + + 2-way with 2-way

4-way with meanTHIS IS A RESOLUTION 4 DESIGN


EXAMPLE 2: CONSTRUCTING A 25-2 DESIGN

5 FACTORS --- 1, 2, 3, 4, 51. Write down full factorial 23 design in factors 1, 2, 32. Defining relations: 4 =12 , 5=13 1 2 3 4=12 5=13 Defining Relations induce- - - + + additional relations:+ - - - - 1=24=35=12345 2=14=345=1235- + - - + 3=15=245=1234 4=12=235=1345+ + - + - 5=13=234=1245 23=45=134=125- - + + - 25=34=123=145 I=124=135=2345+ - + - + NOTE THAT EFFECTS ARE NOW - + + - - CONFOUNDED IN GROUPS OF 4+ + + + + Each main effect is now confounded

with a 2-way interactionTHIS IS A RESOLUTION 3 DESIGN


RESOLUTION OF A DESIGN

Let R denote resolution of design.Two effects of order a and b are unconfounded if a+b < R.However, if a+b ≥ R, then the effects may be confounded.

In particular, a main effect is unconfounded with any effects oforder less than R-1 but may be confounded with an effect of orderequal to R-1.

RESOLUTION PROPERTIES OF DESIGN

3 Main effects clear of each other but confounded with some2-way interactions

4 Main effects clear of each other and of 2-way interactionsbut 2-way interactions confounded with each other

5 or more Main effects and 2-way interactions all clear of each other


AVAILABLE DESIGN SIZES

# OF FRACTIONAL FACTORIAL PLACKETT-BURMAN FACTORS R=3 R=4 R=5 MIN ≥ 5 DF ERROR

5 8 16 16 8 12

6 8 16 32 8 12

7 8 16 64 8 16

8 16 16 64 12 16

9 16 32 128 12 16

10 16 32 128 12 16

11 16 32 128 12 20

12 16 32 256 16 20

Note: All designs listed assuming no replicates and no center points

DuPont Quality Management & Technology© 2000 E. I. du Pont de Nemours and Company

SOE/MTB 10.1Revised 9/26/2000

Section 10

SCREENING EXAMPLE

Presenter

Presentation Notes

We will now demonstrate a Plackett-Burman design in Minitab.



Screening Design Example

This example is a 2-level screening experiment for a product called GLOOP.Factors Abbreviation SettingsTENSION CONTROL TENS manual, automatic

MACHINE MACH #1, #2THROUGHPUT TPUT 10 to

20 gal/min MIXING MIX single, double TEMPERATURE TEMP 200°to 250° F MOISTURE MOIST 20% to 80%Response Abbreviation Expected RangePRODUCT HARDNESS Hardness 10 to 200 (Gauge)

Design = Plackett-Burman Model = Main Effects onlySpecial Notes: In the past day-to-day differences with this process have been observed. Since we can only make 8 product items per day we would also like to block this design on day. Thus, this study really involves 7 factors including the blocking variable (DAY). A preliminary standard deviation estimate of 13 for hardness has been obtained. Detecting changes in hardness of at least 30 units is of interest.

Presenter

Presentation Notes

We have 6 factors plus an additional factor for blocking (DAY) making 7 factors in total.



What Design Size Is Needed?

Using our sample size formula we need at least:

The smallest Plackett-Burman design for n ≥ 10 is the 12 run design

The smallest Fractional-Factorial design for n ≥10 is the 16 run design

n 730/13

9.2 10

n 10

2

≥

= ⇒

≥

Presenter

Presentation Notes

In order to detect an effect of 30 Hardness units with noise on the order of 13, we need a design with at least 10 runs. Another general rule to follow in design size is to have at least 5 more runs than the number of factors. This allows for a reasonable estimate for the standard deviation on which you will be assessing the significance of the effects. For 7 factors, we want at least 7 + 5 = 12 runs



Presenter

Presentation Notes

Start Minitab From the Stat pull-down menu, select DOE Factorial Create Factorial Design



Presenter

Presentation Notes

A full factorial design in 7 factors would include 27= 128 runs. As this is typically impractical, we will choose a screening design. As fractional-factorial designs are more complicated, we’ll Select a Plackett-Burman design. We have 7 factors Click on Display Available Designs… to view the various options for Plackett-Burman designs



Presenter

Presentation Notes

This is just an informational display -- no information is entered here. The text below the table indicates that Plackett-Burman designs are available for 2-7 factors between 8 and 48 runs. Note that Plackett-Burman designs where the runs are powers of 2 (4, 8, 16, 32, 64,…) are really fractional-factorial designs and should be handled as fractional-factorial designs. From the portable power calculation, we determined that we needed at least 10 runs to detect the effects we are interested in. To also get a reasonable estimate of the standard deviation, we need at least 12 runs. A 12 run Plackett-Burman should satisfy our requirements. Click OK



Presenter

Presentation Notes

Click on the Designs… button to specify the size of the Plackett-Burman design to generate.



Presenter

Presentation Notes

Select the 12 run design. Plackett-Burman designs will typically not include center points as frequently discrete factors are included in screening designs. We want 1 run of each of the design points. Click on the OK button



Presenter

Presentation Notes

Now we need to specify the levels of the factors. Click on Factors… Enter the factor names or abbreviations with their low and high levels. Note that character values can be entered for levels of factors for factorial and screening designs. Click on the OK button when done.



Presenter

Presentation Notes

We are now ready to generate the design. It will automatically be randomized. Click on the OK button.



Presenter

Presentation Notes

A 12 run Plackett-Burman design has been generated and is stored in columns C1-C11. Expand the worksheet window to better view the design and do any necessary editing.



Presenter

Presentation Notes

Note that Minitab does not currently have blocking for Plackett-Burman designs. We entered the block variable DAY as a regular variable, but when the design is randomized, that variable is randomized also. We need to sort the design by the DAY column.



Presenter

Presentation Notes

Select the Sort … item from the Manip pull-down menu.



Presenter

Presentation Notes

We want to sort all the columns (C1-C11) by the DAY (C11) column and put them back into the same columns. Select C1-C11 for columns to be sorted Specify that we want the sorted results to be put back into C1-C11. We want to sort by column C11 (DAY) Click OK



Presenter

Presentation Notes

For any design, you should review the worksheet for operability. This includes ensuring that the factor settings don’t remain constant for too long. (We want to get a realistic estimate of experimental error which includes variability due to changing the settings of the factors). In this random ordering, the Tension factor isn’t changed more than once on day 2 and Temperature is held at 200 for 4 runs in a row. We will reorder some runs to increase the number of changes of those variables while still keeping the DAY variable sorted. To reorder the rows, you want to modify the RunOrder (C2) column and then sort the worksheet by the revised C2 column.



Presenter

Presentation Notes

We’ve now entered different RunOrder values into C2 so that when the data is sorted by that column we should have more changes to TENS and TEMP while keeping the values sorted by DAY.



Presenter

Presentation Notes

As before, we select Sort … from the Manip pull-down menu. Specify the columns to be sorted, where to put the sorted columns (we’re putting them back into the original columns -- this is best for Minitab to know where your design is stored) We sort by C2 and click OK



Presenter

Presentation Notes

The resulting sorted data looks better. All variables are changed reasonably often except DAY which is our blocking variable. In practice, you may want to print this worksheet for use in running the design. We will go back to the session window by reducing the size of this window.



Presenter

Presentation Notes

Save the Minitab project using the Save Project As … item in the File pull-down menu. Specify a location and file name Now it’s time to run the experiment on the process according to the worksheet design.



Presenter

Presentation Notes

We are now returning to the worksheet assuming that the design has been run on the process and we have response data to enter. Enter the response name(s) and data. Return to the session window when done entering response data.



Presenter

Presentation Notes

We are now ready to analyze the response data we just entered. From the Stat pull-down menu, select DOE Factorial Analyze Factorial Design



Presenter

Presentation Notes

Minitab knows which columns contain the design, but we need to tell it the column numbers or names for the responses. Select the column containing the response. Click the Graphs… button We will skip the other buttons as we do not need to change anything in them here. The model to be fit will contain linear effect terms as are appropriate for Plackett-Burman designs.



Presenter

Presentation Notes

Check the box for the Pareto Plot. We will discuss the other plots here later. Click OK to close this window Click OK to perform the analysis



Presenter

Presentation Notes

This plot displays the standardized effects in rank order (by magnitude). Those effects that cross the red line are statistically significant with at least 90% confidence. Look at the output in the session window for additional analysis results.



Fractional Factorial Fit: Hardness versus TENS, MACH, ...

Estimated Effects and Coefficients for Hardness (coded units)

Term Effect Coef SE Coef T PConstant 70.00 3.592 19.49 0.000TENS 1.67 0.83 3.592 0.23 0.828MACH 19.33 9.67 3.592 2.69 0.055TPUT 26.33 13.17 3.592 3.67 0.021MIX -0.67 -0.33 3.592 -0.09 0.931TEMP 61.67 30.83 3.592 8.58 0.001MOIST 7.00 3.50 3.592 0.97 0.385DAY -20.00 -10.00 3.592 -2.78 0.050

Analysis of Variance for Hardness (coded units)

Source DF Seq SS Adj SS Adj MS F PMain Effects 7 15966.7 15966.7 2281.0 14.73 0.010Residual Error 4 619.3 619.3 154.8Total 11 16586.0. . .

Presenter

Presentation Notes

This is the the part of the analysis output for the Hardness response that appears in the session window that we are most interested in. If we use 90% confidence (or a p-value of .10), we would say that MACH, TPUT, TEMP, and DAY all have significant effects on Hardness. Our estimate of experimental error (standard deviation) from this data would be sqrt(154.8)=12.4



Presenter

Presentation Notes

We can produce a plot of the factor effects by: Select from the Stat pull-down menu DOE Factorial Factorial Plots …



Presenter

Presentation Notes

We do not have data to support much investigation of interactions and have too many factors for a cube plot so we will Select the Main effects graph of the Data Means Click on the Setup … button for the graph to specify which effects



Presenter

Presentation Notes

We want to plot the effects for all 6 factors versus the response Hardness. Click on the OK button when done Back in the previous menu, click on the OK button again to produce the graph.



Presenter

Presentation Notes

This plot shows dots at the average data value for the low and high levels of each of the factors and connects those dots with a line. The difference between the average value at the high level of a factor and the average value at the low level of a factor would be the effect. So this is a graphical representation of the effects. The larger the effect (in magnitude), the more significant the relationship between the factor and the response for the region studied.



Presenter

Presentation Notes

We are now done with the analysis of this data. Click on the diskette picture to Save the project using the current name.



Thought Questions

What would happen if you analyzed this dataignoring the day effect?

Can we get any information on interactionsamong the significant effects with this

data?


Section 11

WORKSHOP 2:Glyxel Screening


Use a screening design to determine which of the following twelve factors are the most important in their effects on the two observed responses.

RESPONSES Bend Resistance (BR) -- valid range approximately 50-80 psigoal = target to 68 (acceptable range is 65-71)

Area Shrinkage (AS) -- valid range approximately 0-3 %goal = minimize (a maximum of 1% is desirable)

FACTORS LOW LIMIT HIGH LIMIT UNITSThroughput (TP) 200 800 kg/hr

Additive A Concentration (AC) 4 8 %Additive A Impurity (AI) 0.7 3.2 % impurity

Catalyst Amount (CA) 0.1 0.3 %Reactor Pressure (RP) 100 150 psiDryer Temperature (DT) 120 150 °C

Extruder Temperature (ET) 180 190 °CQuench Water Temperature (QT) 10 15 °CQuench Water Flow Rate (QF) 10 20 l/minPress Temperature (PT) 140 160 °CStorage Temperature (ST) 10 25 °CBlock (BL) - see description 2 blocks needed week

Workshop 2 - Problem Description


Workshop 2 - Background Information - I

Your R&D team has been assigned to develop a new product for a critical aerospace use. There is also a potential market in the military aeronautic and automotive industries.The product, called Glyxel, will be produced and sold in sheet form. Two critical property goals must be met.

Bend Resistance (BR) - (acceptable range 65 - 71 psi, target of 68) Gives a proper balance between end-use strength and customer processing needs.

Area Shrinkage (AS) - (1.0% maximum, lower is better) Low shrinkage is required to maintain dimensional stability through customers’ processing.

Below are brief variable descriptions and key learnings from early R&D work. First, to meet the expected market demand, property goals should be met with the highest possible throughput (TP). It is expected that at least 500 kg/hr will be needed for acceptable profitability.Preliminary R&D work has indicated two process variables likely to aid in meeting bending resistance and area shrinkage goals are reactor pressure (RP) and extruder temperature (ET).Additive A concentration (AC) is suspected to be important for obtaining low area shrinkage. Impurity levels of this additive (AI) are also believed to affect area shrinkage. Impurity levels are dictated by the lot number. From our supplier we receive an an accurate estimate of impurity level through a Certificate Of Analysis for each lot. However, since we have 20 lots to choose from that span the range of impurity levels listed, you may experiment with impurity level and treat it as a continuous variable.Two other variables, quench water temperature (QT) and quench water flow rate (QF) have been suggested as potentially affecting either bending resistance or area shrinkage.


It is expected that, in operation, quench water would be supplied from a well with an average yearly temperature range from 10 to 15 deg C. However, you have a temperature controlled water supply (+/- 1 deg) available for experimental use. Hopefully, your team can better understand how typical quench water temperature variation will affect the process. Another related variable that may enhance quenching is quench water flow rate (QF).

The catalyst has been thought to enhance molecular structure development. Since our catalyst is expensive, a lower level of CA is desirable if property goals can be satisfied.

Materials exiting the reactor are fed into a dryer to remove moisture prior to extrusion. Dryer temperature (DT) may influence final product properties as well.

Once the product is formed into sheets, it is stored for up to two weeks in a warehouse until shipped. High storage temperature (ST) may be related to product property deterioration.

After our customer receives our product in sheet form, it is fed into presses for final shaping. The press design temperature (PT) is 150 deg C. However, due to press-to-press differences and temperature control, the actual temperature can vary from 140 to 160 deg C. Your R&D team has acquired a similar press for the duration of the experimental program.

Lastly, due to the complicated nature of this process and time required to make process and recipe changes, it is only possible to experiment with about 12 run combinations in a week. Because a knowledgeable technician has suggested product properties vary from week to week, we would like to test this claim and, more importantly, remove this potential source of variation by blocking (BL) the experiment. Since it is not possible to complete our 12-factor screening design in one week, you will need to run it over two weeks or blocks.

Workshop 2 - Background Information - II


Workshop 2 - Screening Design Assignment

Choose a Plackett-Burman screening design to identify the most important factors affecting BR and AS. The smallest Plackett-Burman design which could handle 12 factors is the 16 run design which is actually a fractional-factorial design. As this would only have 3 degrees of freedom to estimate experimental error and should be treated as a resolution 3 fractional-factorial design (which are more complex to deal with), the next design might be preferable. This will be a 20-run design (with zero replicates). Make sure you define factors and responses by the 2-letter abbreviations (factors =>TP, AC, AI, ... ,BL responses => BR, AS).

Preliminary standard deviation estimates for BR and AS are 1.5 psi and 0.1 %, respectively. Assuming we are interested in detecting effects (least important difference) of at least 3.0 psi for BR and 0.2 % for AS, will our proposed screening design will have adequate sensitivity?

Type BR and AS as the names of two empty columns in the worksheet. Generate the response data using the simulator: %GLYXEL Examine the results in detail. Be prepared to answer the following questions and report your teams

results (see team report spreadsheets). Which factors have significant effects on product properties? What was the experimental error from

your results? Perhaps the most important question is: What factors have you selected to be included in further process optimization work? You may identify up to 5 factors to carry forward to the next design stage. Fill in your team results on the appropriate row of the team report spreadsheets. Note, we will pick up here in the next workshop where we will follow up with a response surface design to optimize this process.


Workshop 2: Glyxel Screening - Team Reports

Effects for BEND RESISTANCE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL

1

2

3

4

5

6

7

8

9

10

Exp. Error(Resid. SD)


Effects for BEND RESISTANCE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL

11

12

13

14

15

16

17

18

19

20




Effects for AREA SHRINKAGE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL

1

2

3

4

5

6

7

8

9

10




Effects for AREA SHRINKAGE - circle if significantTeam TP AC AI CA RP DT ET QT QF PT ST BL

11

12

13

14

15

16

17

18

19

20




Circle Factors To Be Studied in Workshop 3Team

1

2

3

4

5

6

7

8

9

10

COMMENTS

TP AC AI CA RP DT ET QT QF PT ST BL










(note that a maximum of 5 factors may be circled)



Circle Factors To Be Studied in Workshop 3Team

11

12

13

14

15

16

17

18

19

20

COMMENTS











(note that a maximum of 5 factors may be circled)



Section 12

RESPONSE SURFACE DESIGNS


SCREENING INTERACTIONDESIGNS DESIGNS






EVOLUTION OF THE ENVIRONMENT:Advanced Stage

RESPONSESURFACEDESIGNS


USES OF RESPONSE SURFACE MODELS

Quantitative Understanding

Prediction

Optimization

Conditions for Stability

Calibration

Process Control Adjustments


FACE-CENTERED CUBE DESIGN for 3 factors

X3

X2

X1

Block 1(First Half-Fraction)

Block 2(Second Half Fraction)

Block 3(Face Points)

Center Points


BOX-BEHNKEN DESIGNfor 3 factors

X3

X2

X1

Edge Centers

Center Point


SETTING UP A RESPONSESURFACE EXPERIMENT

Assess the Environment Consider the Factors Consider the Responses Choose an Appropriate Design Consider Strategies for Bias Error Rewrite the Experimental Schedule in

Physical Units Review the Experiment for Operability Avoid Blunders Plan for the Analysis Report the Recommendations


RESPONSE SURFACE DESIGNS: Summary

Quadratic Polynomial Models

Danger of Extrapolation

Shape of Experimental Region

– Cubical: Face-Centered Cube

– Spherical: Central Composite, Box-Behnken

Space-Filling, Balanced, and Robust

Link to Model Diagnostics

Link to RS Example

Link to RS Workshop


UNDERLYING ASSUMPTIONS

Fitted model form is “approximately correct”– More important for Response Surface models

Deviations from model have no systematic component (bias error)

Experimental error (random error) is approximately normally distributed

SD of experimental error is homogeneous throughout experimental space


DIAGNOSTICS AND REMEDIES

ASSUMPTION DIAGNOSTIC TOOL POSSIBLE REMEDY

Correct model form Lack-of-fit test Model augmentation

Transformation

No bias error Residual plots Bias modeling

Outlier handling

Normal error Residual histogram Transformation& normal plot

Fundamental knowledge

Homogeneous error Residual plots Transformation

Fundamental knowledge Weighted regression


RESIDUAL PLOTS IN MINITAB

Minitab can calculate 3 kinds of residuals: Regular, Standardized, or Studentized [“Deleted”]. Standardized and “Deleted” residuals use a standard deviation scale.

Available plots of residuals include:HistogramNormal Probability Plot vs. Time vs. Fits (Predicted Values) vs. Variables


Minitab: Transformations

• Transformations in Minitab are calculated using the Calculator… item in the Calcpull-down menu.

• Store results of transformations in separatecolumns so that original values are retained.


R-Squared & Adjusted R-Squared

Both estimate percent or proportion of observed variability explained by model

R-squared adjusted is generally a more honest appraisal as it adjusts for the number of terms in the model.

Minitab will not display R-Squared values for the analyses of 2-level designs but will display them for response surface designs.



Section 14

RESPONSE SURFACE EXAMPLE



A new compound is being developed for a coating process. Threefactors are under consideration:

Factor RangeADDITIVE AMOUNT 0 to 70 gramsREACTION TIME 20 to 60 minutesREACTION TEMPERATURE 100 to 180 degrees C

The yield of the compound is measured. The compound is added to a fixed amount of formula and the coating process completed. Adhesion is then measured. Specifications on the responses are:

YIELD ≥ 91%ADHESION ≥ 45 grams

Find settings of the factors for which these conditions can be achieved. Resources are available for a maximum of 24 runs.

RESPONSE SURFACE EXAMPLEProblem Statement



EXAMPLE DATA PLOTTED ON CUBES

Tem

pera

ture

Additive

Tem

pera

ture

Additive

YIELD ADHESION

68 3

82,87,87,82,85,85

37,41,40,40,42,42

40 37

50 4090 39

81 1065 48

51 40

68 2475 4480 38

92 41

75 31

77 44

75 31



Response Surface Regression: YIELD versus Additive, Time, Temperature

The analysis was done using coded units.

Estimated Regression Coefficients for YIELD

Term Coef SE Coef T PConstant 84.55 0.6964 121.403 0.000Additive 4.90 0.6406 7.649 0.000Time 6.40 0.6406 9.991 0.000Temperat -0.80 0.6406 -1.249 0.240Additive*Additive -12.86 1.2216 -10.530 0.000Time*Time 1.64 1.2216 1.340 0.210Temperat*Temperat -8.36 1.2216 -6.847 0.000Additive*Time 0.75 0.7162 1.047 0.320Additive*Temperat 13.50 0.7162 18.849 0.000Time*Temperat -0.25 0.7162 -0.349 0.734

S = 2.026 R-Sq = 98.9% R-Sq(adj) = 97.9%

Analysis of Variance for YIELD

Source DF Seq SS Adj SS Adj MS F PRegression 9 3746.71 3746.71 416.302 101.45 0.000Linear 3 656.10 656.10 218.700 53.29 0.000Square 3 1627.61 1627.61 542.538 132.21 0.000Interaction 3 1463.00 1463.00 487.667 118.84 0.000

Residual Error 10 41.04 41.04 4.104 Lack-of-Fit 5 15.70 15.70 3.141 0.62 0.694Pure Error 5 25.33 25.33 5.067

Total 19 3787.75

...



Response Surface Regression: ADHESION versus Additive, Time, TemperatureThe analysis was done using coded units.Estimated Regression Coefficients for ADHESION

Term Coef SE Coef T PConstant 40.24 0.5847 68.816 0.000Additive 8.80 0.5378 16.362 0.000Time 2.90 0.5378 5.392 0.000Temperat 5.90 0.5378 10.970 0.000Additive*Additive -6.09 1.0256 -5.939 0.000Time*Time -0.59 1.0256 -0.576 0.577Temperat*Temperat -2.59 1.0256 -2.526 0.030Additive*Time 0.75 0.6013 1.247 0.241Additive*Temperat -10.25 0.6013 -17.046 0.000Time*Temperat -0.50 0.6013 -0.831 0.425

S = 1.701 R-Sq = 98.8% R-Sq(adj) = 97.7%

Analysis of Variance for ADHESION

Source DF Seq SS Adj SS Adj MS F PRegression 9 2399.87 2399.87 266.653 92.18 0.000Linear 3 1206.60 1206.60 402.200 139.04 0.000Square 3 346.27 346.27 115.424 39.90 0.000Interaction 3 847.00 847.00 282.333 97.60 0.000

Residual Error 10 28.93 28.93 2.893 Lack-of-Fit 5 11.59 11.59 2.319 0.67 0.665Pure Error 5 17.33 17.33 3.467

Total 19 2428.80

Unusual Observations for ADHESIONObservation ADHESION Fit SE Fit Residual St Resid

3 37.000 40.236 0.585 -3.236 -2.03RR denotes an observation with a large standardized residual. ...



Do any of the initial design points satisfy the specified performance criteria for YIELD and ADHESION ?

Which model terms are statistically significant ? What confidence level did you use to judge significance ?

How well does the model fit the data ? How did you assess this ?

At a given set of experimental conditions (X settings), what is the experimental error in YIELD and ADHESION ?

THOUGHT QUESTIONSPart 1



Click on OK and then OK againin the next frame to get to:



706050403020100

180

170

160

150

140

130

120

110

100

Additive

Tem

pera

ture

Hold values: Time: 60.0 Hold values: Time: 60.0 Hold values: Time: 60.0 Hold values: Time: 60.0

Overlaid contours for desired values of Yield, Adhesion

YIELD

ADHESION

91100

45100

Lower BoundUpper Bound

White area: feasible region



Link to RS Workshop


Section 15

WORKSHOP 3:Glyxel Response Surface


This workshop is a continuation of the Glyxel problem described in workshop 2, however, now we will use a response surface design to optimize the process. Here we will study in more detail the key factors identified through your screening design.

The original list of factors examined in workshop 2 are given below. From your previous assignment you reduced this list down to the critical few variables (up to 5) to investigate in this response surface design.

Factors (original list) Low Limit High Limit UnitsThroughput (TP) 200 800 kg/hr

Additive A Concentration (AC)4 8% Additive A

Impurity (AI) 0.7 3.2 % impurityCatalyst Amount (CA) 0.1 0.3 %

Reactor Pressure (RP)100 150 psiDryer Temperature (DT) 120 150 °C

Extruder Temperature (ET) 180 190 °C

Quench Water Temperature (QT) 1015 °C Quench

Water Flow Rate (QF) 10 20 l/minPress Temperature (PT) 140 160 °C

St T t (ST)

Workshop 3 - Problem Description


Workshop 3 - Background Information

If needed, refer back to the original problem description (Section 11 - workshop 2 background information slides) to refresh your memory on the design variable descriptions.

Recall the two critical property goals.Bend Resistance (BR) - (acceptable range 65 - 71 psi, target of 68) Gives a proper

balance between end-use strength and customer processing needs.Area Shrinkage (AS) - (1.0% maximum, lower is better) Low shrinkage is required to

maintain dimensional stability through customers’ processing.

Additionally, recall that based on acceptable profitability and market demand expectations we need to satisfy property goals with the highest possible throughput. It was previously stated that we needed to achieve at least 500 kg/hr if possible with higher levels being more desirable.

Several of the factors originally described fall into the category of environmental variables. Such variables may include uncontrolled, noise, ambient, raw materials, or customer use variables. Although environmental variables are not typically controlled in operation, we may choose to explicitly control them within the context of a designed experiment to understand their potential impact on product properties.

Which factors explored in the Glyxel problem in workshop 2 are environmental variables? It is also possible that some factors retained for the optimization study in this workshop are environmental variables. What are they? The importance of Identifying environmental factor(s) and understanding their nature and how they might be treated during optimization will become clear in the assignment discussion and thought questions.


Workshop 3 - RS Design Assignment (Part 1)

Select a response surface design. Choose either a CENTRAL COMPOSITE or BOX-BEHNKEN design. Design sizes will vary according to the number of factors studied, design type, and the number of replicates.

Verify that your intended design will be large enough to detect the size effects we are interested in. Recall that you want to detect a 3 psi change in BR and a 0.2% change in AS. You now have 2 standard deviation estimates for BR and AS, the preliminary estimates (given in workshop 2) and the estimates obtained from the screening design. Which estimate would you use in your design size or sensitivity calculations here and why?

Type BR and AS as column names for two unused columns in the worksheet. Generate the response data with the simulator by typing: %GLYXELRS

in the session window. This simulator will ask for settings for your fixed factors. Enter * for factors that are in the design.

Proceed to analyze.

Examine the results in detail to understand which effects are important and how well the model fits. Fill in your teams results on the appropriate row in the BR and AS regression model results spreadsheets and answer all thought questions (slide 7) prior to generating any contour plots.


Workshop 3 - RS Design Assignment (Part 2)

Generate a few contour plots to get a feel for the behavior of the response surfaces. Keep in mind that knowing the key model effects can help point you to a more promising portion of the design space. Try to identify design regions where property goals can be satisfied.

Initially you may want to assume that any environmental factor(s) will vary across the full design range. Under this assumption you might begin by generating contour plots that leave environmental factor(s) as off-axis variables set to their midpoints and then explore the limits of other design factors to see if our property goals can be met. Next, you may want to investigate the range of environmental factor(s) either as on-axis or off-axis variable(s). Finally, you may choose to relax this assumption and explore optimizing with regard to all factors (including environmental ones) , to see if this makes a difference in meeting process goals albeit recognizing greater control of such factors(s) may be required.

Identify recommended factor settings (where predictions satisfy stated goals) to test model predictions with confirmatory runs. Obtaining independent data for validation is a critical step for building confidence in the predictive capabilities of our models in the region of interest. Wait for instructions on how to collect and process the confirmatory runs.

Finally, answer all thought questions on slide 12 and fill in your results on the team report spreadsheets with recommended settings, predicted response levels, and results obtained from confirmatory runs. Also, be prepared to discuss (or assign a spokesperson from your team to discuss) your teams approach, results, and recommendations.


Perform initial design size or sensitivity calculations Design the experiment Run the experiment on the process and enter the data Analyze the experimental data

– fit the model to the responses– determine the important effects and check model adequacy (examine the

various tabular summaries and residuals; any evidence of LOF?)– model look OK?

Generate plots describing the model– 2D and 3D contour plots– do any predictions from the 2D contour plots satisfy process goals?– optimize the individual responses and then simultaneously optimize to meet

the combined process goals Verify predictions with check point runs

Flowchart for Experimental Design and Analysis


Thought Questions for Workshop 3(Before contour plotting)

Do any of your initial design points satisfy the property and process goals?

Which model terms are statistically significant? What confidence level did you use to judge significance?

How well does the model fit the data? Be prepared to justify your answer.

Is there evidence of lack of fit? At a given set of experimental conditions (X settings),

what is the experimental error in BR and AS (in terms of a standard deviation)?

Based on your results, which factors or set of factors would you choose as on-axis factors in contour plots?


Workshop 3 - Response Surface Model Results (BR)

Team

1

2

3

4

5

6

7

8

9

10

ResidsOK?

DesignUsed

Significant Effects (list factor abbreviations)Main Effects Inter. Effects Quad. Effects

LOF?R2Adj

ResdSD

RepSD


Team

11

12

13

14

15

16

17

18

19

20

ResidsOK?

DesignUsed


LOF?R2Adj

ResdSD

RepSD

Workshop 3 - Response Surface Model Results (BR)


Team

1

2

3

4

5

6

7

8

9

10

ResidsOK?

DesignUsed


LOF?R2Adj

ResdSD

RepSD

Workshop 3 - Response Surface Model Results (AS)


Team

11

12

13

14

15

16

17

18

19

20

ResidsOK?

DesignUsed


LOF?R2Adj

ResdSD

RepSD

Workshop 3 - Response Surface Model Results (AS)


Thought Questions for Workshop 3(After contour plotting)

Which contour plot(s) were the most informative? What is the most promising region of the design space for

satisfying the property goals? Are throughput levels OK? What confirmatory (check point) runs did you make?

Were your predictions supported? Requirements met? What conclusions can you make, if any, on the impact the

environmental factor(s) may have on meeting your process goals?

What recommendations would you make, if any, to improve the process by exercising greater control of the environmental factor(s)?

What plot(s) and / or tables would you include in a report of your results?


Workshop 3 (Glyxel Response Surface) - Team Reports

Recommended Settings (use MP for MidPoint of excluded factors)

Team TP AC AI CA RP DT ET QT QF PT ST

1

2

3

4

5

6

7

8

9

10

PredictedBR AS

ObtainedBR AS


Workshop 3 (Glyxel Response Surface) - Team Reports

11

12

13

14

15

16

17

18

19

20

Recommended Settings (use MP for MidPoint of excluded factors)

Team TP AC AI CA RP DT ET QT QF PT STPredictedBR AS

ObtainedBR AS


Section 16

OTHER EXPERIMENTALENVIRONMENTS


SOME OTHER TYPES OFEXPERIMENTAL SITUATIONS

Categorical factors with more than 2 levels Constrained regions Mixture problems Incomplete block designs Split plot designs Nested designs Supersaturated designs

FOR THESE TYPES OF SITUATIONSCONSULT AN EXPERT!


CATEGORICAL FACTORSWITH MULTIPLE LEVELS

Catalysts

Electronic components

Suppliers

Operators

Machines

Brands or types of formulation ingredients


MIXTURE EXAMPLES

Drugs

Gasoline Blends

Metal Alloys

Rocket Propellants

Aerosol Propellants

Herbicides

Paints

Dyes

Textile Fiber Blends

Concrete

Cake Mixes

Composite Materials


A CLASSIC MIXTURE

5 Parts Gin 1 Part Vermouth


MIXTURE CONSTRAINT

Mixture Components Cannot Be Varied Independently

Factorial and Response Surface Designs Cannot Be Used

Σj=1

q

Xj = 10 ≤ Xj ≤ 1

Σso Xq = 1 - Xjj=1

q-1


FACTOR SPACE IN TWO VARIABLES

FACTOR SPACE IN THREE VARIABLESX1

X2

Independent

X1

X2

Mixture

- + 0 1

1

0

X1

X2

X3

X1

X2

X3

Independent Mixture

X1+X2+X3=1

+

-


FACTOR SPACE FOR A FOUR-COMPONENT MIXTURE

0,1,0,0

0,0,1,0 1,0,0,0

0,0,0,1


QUADRATIC RESPONSE-SURFACEMODEL FOR TWO FACTORS

Y = a0 + a1X1 + a2X2 + a12X1X2 + a11X12 + a22X2

2

The Mixture ConstraintX1 + X2 = 1X1

2 = X1*X1 = X1(1 - X2) = X1 - X1*X2

X22 = X2*X2 = X2(1 - X1) = X2 - X1*X2

Quadratic Mixture Model (Scheffé)Y = b1X1 + b2X2 + b12X1X2


TWO-COMPONENT MIXTUREScheffé Linear & Quadratic Model

d = 0.25*b12

Y

b1

b2

X1 1.0 0.5 0.0

X2 0.0 0.5 1.0


THREE-COMPONENT MIXTURE DESIGN

X1 = 1

X3 = 1X2 = 1

Pure Component

Binary Blend

Ternary Blend

Check Points

X1 = 0


ROCKET PROPELLANT CONTOUR PLOT

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

*

* *

* *

*

*

* *

*

800900

X2

X3

700

1000

1000

800

Maximum Near(0.2, 0.3, 0.5)


MINIMUM COMPONENT LEVELS

Concrete

Cake

Steel


MINIMUM COMPONENT LEVELS

0.0 0.50.0

0.5

0.0

0.5

X2 = 1

X1 = 1

X3 = 1

X1 = 0.18

X3 = 0.11

X2 = 0.14Requirements:

X1 ≥ 0.18X2 ≥ 0.14X3 ≥ 0.11


0.0 0.50.0

0.5

0.0

0.5

X3 = 1

X1 = 0.51

X3 = 0.58

X2 = 0.52

X3 = 0.11

X2 = 0.14

X1 = 0.18

X2 = 1

X1 = 1

MINIMUM AND MAXIMUM COMPONENT LEVELS

Requirements:0.18 ≤ X1 ≤ 0.510.14 ≤ X2 ≤ 0.520.11 ≤ X3 ≤ 0.58


FLARE EXPERIMENT: DESIGN

X2 = NaNO3

X3 = SrNO3 X1 = Mg

X4 = Binder

Face Centers

Vertices

Center


AEROSOL PROPELLANT STUDY

X2= 1

X1 = 1

X3 = 1

MixtureHighly

Flammable


EXAMPLE OF CONSTRAINED REGION:Petroleum Fractionation Process

Amountof TolueneIn Solvent

Solvent/Solute

Regionof Interest

Equipment Fouling:No Phase Separation

UnfavorableEconomics


STEPS FOR DESIGNS IN IRREGULAR REGIONS

Define Region

Incorporate the Principles of Good Design

Identify Candidate Runs– Include extreme points

Select Runs– By inspection, if geometry simple– Using computer-aided algorithmic design

(e.g. D-Optimal Design)


OTHER EXPERIMENTAL ENVIRONMENTS:Summary

Discrete Factors

Mixture Designs

Constrained Factor Spaces


Section 18

SUMMARY


EXPERIMENTAL DESIGN VERSUSONE-FACTOR-AT-A-TIME EXPERIMENTS

Drawbacks of one-factor-at-a-time experiments– Not space-filling

– Ignores interactions

– Ignores experimental error

– Inefficient due to lack of hidden replication

– Limited system understanding

– Potential bias error due to lack of randomization


EXPERIMENTAL DESIGN VERSUSANALYSIS OF HISTORICAL DATA

Drawbacks of historical data analysis– Correlations between factors– Unrecorded control actions may create misleading

effects, confusion of cause and effect– Typical lack of boldness in factor settings– Data collection problems

– Missing data– Bad observations

– Large bias errors due to lack of randomization– At best describes “What is” instead of “What is

possible”


WHEN SHOULD I USE DOE ?

Discovery Research and Scouting Product/Process Design and Development Process Scale-up, Startup, and

Qualification Process Control and Calibration Product/Process Improvement


PRINCIPLES OF GOOD EXPERIMENTAL STRATEGY

Diagnosis of the Environment (objectives, prior knowledge, number & nature of factors)

Balanced Statistical Designs Measure All Relevant Responses Bite-Sized Experiments Boldness Randomization and Blocking Estimate Experimental Error Avoid Blunders Plan Ahead for Statistical Analysis


EXPERIMENTAL EVOLUTION

Screening DesignsMany factorsDistinguish “critical few”

from “trivial many”Linear model: Y = b0 + b1X1 + b2X2

+ b3X3 + b4X4 + . . .Plackett-Burman and

Fractional FactorialUsually only a few moreruns than factors

Interaction DesignsFewer factorsIdentify/exploit interactionsModel contains linear terms

and at least some interactions:Y = b0 + b1X1 + b2X2

+ b3X3 + b12X1*X2+ b13X1*X3 + b23X2*X3

Full and Fractional Factorial

Response Surface DesignsSmall number of factors (3-6)Used for prediction, optimization,modelling, . . .

Quadratic model:Y = b0 + b1X1 + b2X2

+ b3X3 + b12X1*X2+ b13X1*X3 + b23X2*X3+ b11X1

2 + b22X22 + b33X3

2

Face-centered cube, Box-Behnken,others


DOE APPLICATION PROCESSStrategy of Experimentation

Gather Information

Define Experimental Objectives

Design the Experiment

Run Experiment

Analyze Experiment

Interpret Results

Perform Confirmation Runs

Go to Next Stage of Experimentation?

Apply Results

IdentifyBusinessNeeds

Assess, Document & Communicate

Business Results

UpdateInformation


ANALYZING EXPERIMENTS: Flowchart

Rank Effects ExploitInteractions

Find “ Optimum”

ScreeningDesigns

InteractionDesigns

Response Sur faceDesigns

Main Effects Main Effectsand Interactions

QuadraticModel

ENTER DATA

DISPLAY DATA

PLOT/VERIFY DATA

FIT MODEL

ASSESS SIGNIFICANCE

VALIDATE ANALYSIS

PLOT RESULTS


WHEN DO I NEED A D-OPTIMAL DESIGN ?

You may need a d-optimal design for the following situations:

– Discrete/Qualitative factors at more than 2 levels– Constrained regions (including mixtures)– A special model (mixed number of levels of factors or models

with some terms excluded due to your assuming that they have negligible effects on the responses of interest)

– To augment an existing design to be able to estimate a larger model (in some cases) -- assuming no change in process other than possibly a level shift (use blocking) between when the existing and new data are collected

Do not overuse d-optimal designs. Use standard designs whenever they are appropriate.



Section 19

MIXTURES IN MINITAB



TYPES OF MIXTURE DESIGNSAVAILABLE IN MINITAB

Simplex centroid– use when components have no upper bounds (lower bounds are OK)– includes:

• 2q-1 runs for q components• all pure component (100% of component) runs, all binary (½, ½) blends,

all ternary ( , , ) blends, … Simplex lattice

– use when components have no upper bounds (lower bounds are OK)– includes:

• degree 1 design: pure component runs• degree 2 design: pure component runs, binary blends• degree 3 design: pure component runs, binary blends, ternary runs, all

( , ) blends Extreme vertices

– use when components have upper bounds

1 31 31 3

1 3 2 3



CREATING MIXTURE DESIGNSIN MINITAB

Start a new Minitab project From the Stat pull-down menu, select

DOEMixtureCreate Mixture Design

Select the number of components Select the type of mixture design Click on Designs… to select the points to be

included



EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:

Choose type of design

Effects

X1X2X3

Ranges

.18 to .51

.14 to .52

.11 to .58



Extreme Vertices Design Options:Vertices and Center Point only



Extreme Vertices Design Options:Vertices, Center Point, Axial Points



Extreme Vertices Design Options:Vertices, Center Point, Binary Blends




Defining factors in Minitab

Effects

X1X2X3

Ranges

.18 to .51

.14 to .52

.11 to .58




Minitab worksheet from extreme vertices design with degree=2 and center point



VISUALIZING THE MIXTURE DESIGN IN MINITAB

After producing the design: Select from the Stat pull-down menu

DOEMixtureSimplex Design Plot

Click OK to usethe default graphsettings



ANALYSIS OF MIXTURE DATAIN MINITAB

Select from the Stat pull-down menuDOEMixtureAnalyze Mixture Design

Select the response column to analyze The default model is quadratic -- click OK




Minitab analysis resultsRegression for Mixtures: Response versus X1, X2, X3Estimated Regression Coefficients for Response (component proportions)

Term Coef SE Coef T P VIFX1 44.14 24.42 * * 243.89X2 4.60 20.66 * * 168.54X3 -31.81 13.44 * * 82.35X1*X2 194.74 77.83 2.50 0.041 280.33X1*X3 209.88 62.86 3.34 0.012 160.19X2*X3 165.30 49.35 3.35 0.012 77.67

S = 2.0298 PRESS = 85.656R-Sq = 98.26% R-Sq(pred) = 94.85% R-Sq(adj) = 97.02%

Analysis of Variance for Response (component proportions)

Source DF Seq SS Adj SS Adj MS F PRegression 5 1632.89 1632.8886 326.5777 79.26 0.000

Linear 2 1561.86 79.9464 39.9732 9.70 0.010Quadratic 3 71.03 71.0328 23.6776 5.75 0.027

Residual Error 7 28.84 28.8406 4.1201Total 12 1661.73

Unusual Observations for Response

Observation Response Fit SE Fit Residual St Resid1 46.000 49.697 1.321 -3.697 -2.40R

R denotes an observation with a large standardized residual



EXAMPLE WITH LOWER AND UPPER BOUNDS ON COMPONENTS:Minitab Response Trace Plot

Minitab’s response trace plot is similar to an effects plot for mixture components.

From the Stat pull-down menu, selectDOEMixtureResponse Trace Plot

Click OK for default settings



Produce a contour plot of the model fit by selecting from the Stat pull-down menu:DOEMixtureContour/Surface (Wireframe) plots

Click on Contour plotSetupOK (for default settings)

3-D plots and optimizations are available also


Minitab contour plot

Note that this plot was produced by requesting specific contour levels.



THOUGHT QUESTION

What if sample sizes test tells you only need 8 design points -- and each experiment will be very expensive.

What would you do?



MIXTURE EXAMPLE 2

Create a Mixture Design with the FollowingConstraints

– Variable 1 must be less than 0.60.– Variable 2 must be less than 0.70.– Variable 3 must be GREATER than 0.20.

You will need an extreme vertices design due to the maximums given for variables 1 and 2.



MIXTURE EXAMPLE 2:2 Design Alternatives

Minitab’s defaultdesign with axial

points

Minitab’s degree=2design without axial

points



MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS

Four factors with their ranges.Poly 0.10 to 0.30Comp1 0.00 to 0.15Comp2 0.00 to 0.15Filler 0.55 to 0.85

We have the following additional constraintsB + C <= 0.15B + C >= 0.05.



MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Define Variables



MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Define Constraints



MIXTURE EXAMPLE WITH LINEAR CONSTRAINTS: Extreme Vertices

with Axial PointsRun Poly Comp1 Comp2 Filler

1 0.15 0.050 0.025 0.775

2 0.25 0.050 0.025 0.675

3 0.25 0.100 0.025 0.625

4 0.15 0.025 0.050 0.775

5 0.10 0.000 0.050 0.850

6 0.25 0.025 0.100 0.625

7 0.20 0.050 0.050 0.700

8 0.10 0.050 0.000 0.850

9 0.30 0.000 0.050 0.650

10 0.10 0.000 0.150 0.750

11 0.30 0.050 0.000 0.650

12 0.15 0.025 0.100 0.725

13 0.15 0.100 0.025 0.725

14 0.10 0.150 0.000 0.750

15 0.30 0.000 0.150 0.550

16 0.25 0.025 0.050 0.675

17 0.30 0.150 0.000 0.550


Section 20

GLOSSARY OF TERMS


GLOSSARY OF TERMS

algorithmic designsDesigns that are computer-generated for very specific situations, often using an algorithmtied to some particular optimality criterion, such as D-optimality.

analysis of variance (ANOVA)Procedure for analyzing data which involves partitioning the total variation into portionsexplainable by a model and unexplainable. Through appropriate partitioning, the statisticalsignificance of particular model terms or groups of terms can be tested using F-ratios.

axial pointsPoints in a central composite design having the property that all but one of the factors areset at their middle level. Syn: star points.

balanceDesirable characteristic of an experimental design wherein design points are allocated ina manner that is balanced with respect to the center of the factor space.

boldnessThe recommended practice in experimentation of investigating factors over wide ranges.

bias errorVariability in the response data that is of a systematic, patterned nature, often due to asingle assignable cause. Syn: systematic error.

blockingRunning the experiment in specially chosen subgroups, called blocks, within which theexperimental conditions or material is expected to be more homogeneous than betweenblocks.

Box-Behnken designsA particular class of response surface designs which are spherical in shape.

categorical factorA factor that can assume only a finite number of possible values or levels. The levels maybe numerical or non-numerical labels, and are usually treated as unordered. Syn:discrete factor.

centeringThe procedure of re-expressing the scale of a continuous factor by subtracting a centralvalue, such as the midpoint of the factor’s experimental range, from the factor values.Fitting polynomial models in the centered factors results in more easily-interpretedcoefficients, and reduces correlations among the coefficients.

central composite designA class of response surface designs which consist of corner points, axial points, and theoverall centroid.

central composite in cube designThe same as a face-centered cube. These are a special class of central compositedesigns in which the axial points are located in the centers of the faces of the cube.

central composite in sphere designA special class of central composite designs in which the axial points are located outsidethe faces of the cube, on the surface of the circumscribed sphere.

confidence intervalAn interval within which an unknown population parameter is estimated to lie. Theconfidence level (e.g. 95%) associated with the interval represents the long-runpercentage of times that the interval will actually include the population parameter.


GLOSSARY OF TERMSconfounding

A property of a design wherein the estimates of certain effects are correlated with eachother. When two effects are totally confounded they are inseparable from each other --they are both estimated with the same contrast. Syn: aliasing.

continuous factorA factor which can take on any value over some numerical range.

contour plotA two-dimensional plot of the relationship between two continuous factors and aresponse, in which the plot axes represent the two factors, and points of constantresponse are connected by curves, called contour lines.

correlationA measure of the degree to which values of one variable change in concert with values ofanother variable.

D-optimal designA design that minimizes the volume of the region of uncertainty of the unknown modelparameters. Usually generated algorithmically.

degrees of freedom (d.f.)The number of independent pieces of information used to fit a model (model d.f.) orestimate experimental error (residual or replicate d.f.)

designThe set of specific factor combinations to be run in the experiment. Usually specified in adesign table with factors assigned to columns and runs to rows which indicate the factorcombinations to be run.

design of experiments (DOE)A strategic process, with supporting methods and tools, for guiding the planning,execution, analysis, and application of results of experimental or developmentalprograms.

discrete factorsee categorical factor

duplicateA repeated run that does not repeat all elements of the 'run' process. Examples: makingone piece of product and measuring it more than once, or setting conditions once andmaking multiple pieces of product and measuring each. Note distinction from replicate.

effectA difference of averages: high level average - low level average. The expected change inthe response as you go from the low to high level of the factor/interaction/etc.

efficiencyA comparison of the current design vs. a theoretical optimum. May be based on any ofseveral commonly-used criteria (e.g. D-efficiency or G-efficiency). Usually expressed as apercentage between 0 and 100; 100% efficiency may not always be achievable.

environmental factorFactor that may affect product functionality but is not controlled during normal productionor use. Syn: noise factor.

experimental errorLack of repeatability (variability) in the experimental outcomes.

extrapolationMaking predictions outside the range covered by the current data.

F-testA ratio of variances or mean squares used to compare means, or to compare variances,or to test the significance of terms or groups of terms in models.


GLOSSARY OF TERMSface centered cube design

See central composite in cube design.factor

A variable which is deliberately manipulated in an experiment. Syn: independentvariable, knob, predictor, input variable, controlled variable.

factorial designExperimental design that is generated by using all possible combinations of each of thelevels of the factors; most commonly used in cases where the factors each have twolevels.

foldover designA design obtained by adding the reflection of a design to the original design (therebydoubling the number of runs); the reflection is obtained by reversing all ‘+’ and ‘-’signs in the coded design table. A foldover Plackett-Burman design will isolate the maineffects from two-way interactions. Syn: reflected design.

fractional factorial designSubset of a full-factorial design that is formed by totally confounding factor effects withcertain high-order interactions.

Gaussian distributionSee normal distribution.

hidden replicationThe apparent repeating of factor combinations when a balanced design is collapsed overvariables not involved in the effect of interest.

high-order interactionsInteraction terms that involve several (typically 3 or more) variables simultaneously.Frequently used as a basis for blocking or determining fractional factorial designs.

historical dataData taken during the normal operation of a process where factors are not varied in adeliberate, planned manner.

influenceA numerical measure of the importance of an observation in determining the fitted model.Various commonly-used measures are available, which may depend on either the locationof the point in the design space, or the response value at that point, or both.

inoperable regionA subset of the design space in which response data can not be collected/used.

interactionA condition involving two or more factors in which the effect of one factor depends on thelevels of the other(s).

interaction modelA model which consists of main effects and 2-factor interactions (higher-order interactionsgenerally not included).

interpolationMaking predictions within the range covered by the current data.

lack-of-fit testAn analysis that compares residual variability versus replicate variability to assesswhether the model could somehow be significantly improved using the current data.

linear modelA model which includes main effects only.

meanThe average.


GLOSSARY OF TERMSmixture experiment

An experimental environment where the factors of interest are the proportions of variousingredients in a formulation.

modelA mathematical representation of the relationship between the factors and the response.

normal distributionA symmetric, bell-shaped function that represents the expected frequency of data values,often used to model the distribution of random error. It is specified by a mean andstandard deviation. Syn: Gaussian distribution.

one-factor-at-a-time experimentationAn experimental strategy which involves holding all factors constant except one, which isvaried across a range. This process is then repeated in turn for each factor.

orthogonal designA design in which the columns of the design matrix are uncorrelated with each other.

orthogonal coding or scalingExpressing the range of a factor on a -1 to +1 scale.

outlierAn observation that appears to be far removed from the range of variation of the otherobservations in the data set. May suggest a possible error or anomaly.

p-value of an estimated effectThe probability of observing an effect that large purely by chance i.e. when the true effectis actually zero. The smaller the p-value, the stronger the evidence of a real effect.

parameterAn unknown constant associated with the population.

Plackett-Burman designA class of two-level screening designs that exist in multiples of 4 runs. Factor effects arenot completely independent of 2-factor interactions. Designs where the number of runs isa power of 2 should be treated as fractional factorials.

pooled standard deviationA combined estimate of experimental error variability based on replicating more than oneset of experimental conditions. Assumes that the true variance is about the same for allfactor combinations in the experiment.

populationThe hypothetical set of all possible data values of a variable.

practical significanceThe degree to which an estimated effect or parameter represents something meaningfulor useful in the context of the experimental environment.

pure error standard deviationSee replicate standard deviation.

quadratic modelA continuous factor model which consists of main effects, 2-factor interactions, andcurvature terms in each factor.

qualityThe totality of features and characteristics of a product or service that bear on its ability tosatisfy stated or implied needs.

R-squaredThe proportion of the total variation of the response explained by fitting the model. Alsoknown as the coefficient of determination. Increases with the addition of model terms,regardless of their significance.


GLOSSARY OF TERMSR-squared adjusted

An adjusted form of R-squared that takes into account the number of terms in the modelvis-a-vis the total number of observations. Provides a more equitable measure forcomparing models of different sizes than the unadjusted R-squared. Can be negative formodels that have no value.

random errorVariability in the response data which exhibits no systematic pattern. It can not beattributable to any single cause.

randomizationThe deliberate scrambling of the run order of the design so that any bias present isunlikely to be confounded with any factor effects, but will appear as random variation.

reflected designSee foldover design.

regression (least squares)A method of fitting a model to a set of data by minimizing the sum of squares of thedeviations from the model.

regression F-testA test that assesses the overall significance of the model. Compares the varianceexplained by the model to the variance unexplained by the model.

replicateA repeated run that includes repetition of all components of the 'run' process (i.e.changingfactor settings, making product, measuring product, etc.) Note distinction from duplicate.

replicate standard deviationA pooled standard deviation computed using all replicated sets of runs. Provides anestimate of the standard deviation of experimental error. Syn: pure error standarddeviation.

residualThe difference between an observed response value and the response predicted from themodel.

residual standard deviationA standard deviation based on the set of all residual values. Under the assumption thatthe fitted model form is valid, provides an estimate of the standard deviation ofexperimental error.

resolution (of fractional-factorial design)A number which indicates the level of confounding in a fractional-factorial design. Highernumbers imply less confounding. A design of resolution R is one in which no p-factoreffect is confounded with any other effect containing fewer than R-p factors.

responseA variable which is observed/measured whose value may depend upon the settings of thedesign factors. Syn: dependent variable, property, characteristic.

response surface experimentA stage of experimentation where the experimental data is to be used for optimization,calibration, prediction, etc. Quadratic models are typically used as the fitted models.

robustInsensitive to changes in environmental conditions.

screening experimentA stage of experimentation where the experimenter needs relatively crude informationabout which of a relatively large number of factors are important. Models containing maineffects only are typically used here.


GLOSSARY OF TERMS

signal-to-noise ratioDelta/s where delta is the minimum change in the response that is desired to be detectedand s is the standard deviation of experimental error.

standard deviationA measure of the variability in a population or set of data. The square root of thevariance.

standard errorThe variability associated with an estimated effect or coefficient.

standardized residualThe residual divided by the estimated standard deviation of the residuals where allobservations contribute to the standard deviation.

statisticA numerical characteristic of a sample; for example, mean and standard deviation.

statistical significanceA conclusion made from statistical analysis of data that a difference or effect is real.

studentized residualThe residual divided by the estimated standard deviation of the residual where the currentobservation is omitted from the standard deviation calculation.

transformationA function applied to a variable (typically a response) to help improve the fit of the model,or to make the statistical analysis more valid.

variableA measurement or observation for which any of various possible data values can occur.

varianceA measure of the variability in a population or set of data. The square of the standarddeviation.


SOE/MTB 21.1Revised: 5/08/2000

Section 21

REFERENCE



CONTENTS

Catalogue of Designs Blank Cube Diagrams Defining Relations for Fractional-Factorial Designs Miscellaneous Formulas Selected DOE Bibliography DOE Related Accession Reports Consultant List (hand-out)



DESIGN CATALOGUE: Two-level Designs

DesignNumber ofDistinct Points

Confoundingof Model Terms

Full Factorial 2k none

Fractional Factorial 2k-m either total or none - see pageslater in this section

Plackett-Burman multiples of 4 main effects partially that are not powers of 2 confounded w/interactions

Plackett-Burman multiples of 8 main effects clear of 2-factorplus Reflection that are not powers of 2 interactions

k = number of factors in designm = degree of fractionation



DESIGN CATALOGUE:Response Surface Designs

DesignFace-Centered Cube 3 Cubical

Box-Behnken 3 Spherical

Spherical Central- 5 SphericalComposites

FactorLevels

Shape of Design Space



23 FACTORIAL DESIGN



TWO 23 FACTORIAL DESIGNS



33 FACTORIAL DESIGN



TWO 33 FACTORIAL DESIGNS



THREE-FACTOR FACE-CENTERED CUBE



TWO THREE-FACTOR FACE-CENTERED CUBES



THREE-FACTOR BOX-BEHNKEN DESIGN



TWO THREE-FACTOR BOX-BEHNKEN DESIGNS



FRACTIONAL-FACTORIAL DESIGNSAVAILABLE WITH VARIOUS RESOLUTIONS

Number of Resolution Resolution Resolution FullFactors III IV V or more Factorial

3

4

5

6

7

8

9

10

11

12

4

-

8

8

8

-

16

16

16

16

-

8

-

16

16

16

32

32

32

32

-

-

16

32

64

64

128

128

128

256

Number of runs required for:

8

16

32

64

128

256

512

1024

2048

4096



Standard Error of a Factor Effect

√ n1 n2where n1 and n2 are the number of observations in the low and high “halves” of the factor effect.

Confidence Intervalestimate +/- t * standard errort is a tabled Student-t quantile whose value depends

on the degrees of freedom & confidence usedWhen zero is not included in the confidence interval, the effect is

statistically significant

Experiment SizeFor 2-level designs

7 or 8 2 where ∆ is effect size desired to

∆ / s detect and s = estimate of std dev.For 3-level designs

about 50% more than for 2 levels

Sample MeanY1 + Y2 + . . . Yn Yi

n nSample Variance

(Y1-Y)2 + (Y2-Y)2 + . . . + (Yn-Y)2

n - 1Sample Pooled Variance

(n1-1)s12+(n2-1)s2

2+ . . .+(nk-1)sk2

(n1-1)+(n2-1)+ . . .+(nk-1)where s1

2 , s22 , . . . sk

2 are the individual variancesand n1 , n2 , . . . nk are the number of replicate measurements at each combination

Sample Standard Deviationss = √ s2 sp = √ sp

2

Estimated EffectYhigh - Ylow

Y =

)(n =

MISCELLANEOUS HANDY FORMULAS

1 1+Σi=1

n

=

s2 =

sp2 =

sFE = sp



Explained Variation (%) in Regression (R-Squared)

Explained Variation in Regression (R-Squared adjusted)

MISCELLANEOUS HANDY FORMULAS(continued)

Σi=1

n

(Yi - Y)2_

(Ypredicted - Y)2Σi=1

n

Σi=1

n

(Yi - Y)2

(Yi - Ypred)2Σi=1

n

_

_

R2(adj) = 100 1 -

where k = number of model terms (including constant)

R2(adj)= 100 (1 - (varianceresidual/ variancetotal))

)(

100

/(n-k)

/(n-1)

R2 =



SELECTED DOE BIBLIOGRAPHY Design and Analysis of Experiments, Douglas C. Montgomery, 3rd ed.,

1990, John Wiley and Sons, Inc. Empirical Model Building and Response Surfaces, George E. P. Box

and Norman R. Draper, 1987, John Wiley and Sons, Inc. Experiments with Mixtures, John Cornell, 2nd ed., 1990, John Wiley and

Sons, Inc. Response Surface Methodology, Raymond H. Myers, 1976, Virginia

Polytechnic Institute and State University. Statistical Design & Analysis of Experiments With Applications to

Engineering and Science, Robert L. Mason, Richard F. Gunst, James L. Hess, 1989, John Wiley & Sons, Inc.

Statistics for Experimenters, George E. P. Box, William G. Hunter and J. Stuart Hunter, 1978, John Wiley and Sons, Inc.

Strategy of Experimentation, Course Text (1988), DuPont Quality Management & Technology Center, Wilmington, DE.



AVAILABLE DOE-RELATED MATERIALS Design of Experiments: A Competitive Advantage

– Introduction in question and answer format– Accession Report #17934

Design of Experiments Application Guide– Outline of DOE as a strategic process. How and when to apply the

tools and methods effectively.– Accession Report #17960

Design of Experiments Quick Reference Guide– Quick reference guide for trained users of DOE as a strategic

process. Includes tools and methods, examples and catalogs.– Accession Report #17961

Design of Experiments Overview (presentation)

DuPont Quality Management and Technology SOE/MTB© 2000 E. I. du Pont de Nemours and Company Revised 5/09/2000

TABLE OF CONTENTS

SECTION TITLE

1 Introduction2 Workshop 13 Foundations of the Strategy4 Factorial Geometry5 Factorial Example: Design6 Analysis of Two-Level Factorial Designs7 Factorial Example: Analysis8 Good Experimental Practice9 Screening Designs10 Screening Example11 Workshop 2 - Glyxel Screening12 Response Surface Designs13 Model Diagnostics14 Response Surface Example15 Workshop 3 - Glyxel Response Surface16 Other Experimental Environments17 Algorithmic Design18 Summary19 Mixtures in Minitab20 Glossary of Terms21 Reference

Agenda Additions:

Questions & AnswersYour Data Session

design of experiment using minitab book_1

Documents