The Essentials of 2-Level Design of The Essentials of 2-Level Design of ExperimentsExperiments

Part I: The Essentials of Full Factorial DesignsPart I: The Essentials of Full Factorial Designs

Developed by Don Edwards, John Grego and James Developed by Don Edwards, John Grego and James LynchLynch

Center for Reliability and Quality SciencesCenter for Reliability and Quality SciencesDepartment of StatisticsDepartment of Statistics

University of South CarolinaUniversity of South Carolina803-777-7800803-777-7800

Part I. Full Factorial DesignsPart I. Full Factorial Designs

2244 Designs Designs– IntroductionIntroduction– Analysis ToolsAnalysis Tools– ExampleExample– Violin ExerciseViolin Exercise

22kk Designs Designs

2244 Designs DesignsU-Do-It - Violin ExerciseU-Do-It - Violin Exercise

2244 Designs DesignsU-Do-It - Violin Exercise U-Do-It - Violin Exercise

How to Play the Violin in 176 Easy StepsHow to Play the Violin in 176 Easy Steps1,21,2

A very scientifically-inclined violinist was interested in A very scientifically-inclined violinist was interested in determining what factors affect the loudness of her determining what factors affect the loudness of her instrument. She believed these might include:instrument. She believed these might include:– A: Pressure (Lo,Hi)A: Pressure (Lo,Hi)– B: Bow placement (near,far)B: Bow placement (near,far)– C: Bow Angle (Lo,Hi)C: Bow Angle (Lo,Hi)– D: Bow Speed (Lo,Hi)D: Bow Speed (Lo,Hi)

The precise definition of factor levels is not shown, but they The precise definition of factor levels is not shown, but they were very rigidly defined and controlled in the experiment.were very rigidly defined and controlled in the experiment.

Eleven replicates of the full 2Eleven replicates of the full 244 were performed, in completely were performed, in completely randomized order. Analyze the data! randomized order. Analyze the data! 1 1176=11x16176=11x16 22Data courtesy of Carla PadgettData courtesy of Carla Padgett

2244 Designs DesignsU-Do-It - Violin Exercise Report FormU-Do-It - Violin Exercise Report Form

Responses are AveragesResponses are Averages of 11 Independent of 11 Independent Replicates Replicates– All 176 trials were All 176 trials were

randomly orderedrandomly ordered Analyze and Interpret Analyze and Interpret

the Datathe Data

APressure

BBow

Placement

CBow Angle

DBow Speed

YLoudness(Decibels)

-1 -1 -1 -1 69.31 -1 -1 -1 75.3

-1 1 -1 -1 75.91 1 -1 -1 79.3

-1 -1 1 -1 67.41 -1 1 -1 74.9

-1 1 1 -1 74.41 1 1 -1 78.8

-1 -1 -1 1 73.41 -1 -1 1 77.5

-1 1 -1 1 78.81 1 -1 1 81.6

-1 -1 1 1 72.31 -1 1 1 78.8

-1 1 1 1 78.11 1 1 1 81.5

2244 Designs DesignsU-Do-It Solution - Violin Exercise Signs TableU-Do-It Solution - Violin Exercise Signs Table

Main Effects Interaction EffectsActualOrder y A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD

69.3 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 175.3 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -175.9 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -179.3 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 167.4 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -174.9 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 174.4 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 178.8 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -173.4 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -177.5 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 178.8 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 181.6 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -172.3 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 178.8 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -178.1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -181.5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Sum 1217.3 38.1 39.5 -4.9 26.7 -10.1 5.5 -4.5 -.7 -3.5 3.7 -2.3 1.3 .5 -1.3 -1.3Divisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 76.1 4.8 4.9 -.6 3.34 -1.3 .7 -.6 -.1 -.4 .5 -.3 .2 0 -.2 -.2

U-Do-It ExerciseU-Do-It Exercise U-Do-It Solution - Violin Exercise Cube PlotU-Do-It Solution - Violin Exercise Cube Plot

FactorsFactors– A: Pressure (Lo,Hi)A: Pressure (Lo,Hi)– B: Bow PlacementB: Bow Placement

(near,far) (near,far)– C: Bow Angle (Lo,Hi)C: Bow Angle (Lo,Hi)– D: Bow Speed (Lo,Hi)D: Bow Speed (Lo,Hi)

D

B

A

+

+

+

_

_

_

74.9 75.3

78.8 77.5

72.3 73.4

67.4 69.3

74.4 75.9

78.8 79.3

81.5 81.6

78.1 78.8

Note:

Hi C Lo C

y A B C D69.3 -1 -1 -1 -175.3 1 -1 -1 -175.9 -1 1 -1 -179.3 1 1 -1 -167.4 -1 -1 1 -174.9 1 -1 1 -174.4 -1 1 1 -178.8 1 1 1 -173.4 -1 -1 -1 177.5 1 -1 -1 178.8 -1 1 -1 181.6 1 1 -1 172.3 -1 -1 1 178.8 1 -1 1 178.1 -1 1 1 181.5 1 1 1 1

2244 Designs DesignsU-Do-It Solution - Violin Exercise Effects Normal U-Do-It Solution - Violin Exercise Effects Normal

Probability PlotProbability Plot

543210-1EffectsOrdered Effects: -1.3 -0.6 -0.6 -0.4 -0.3

-0.2 -0.2 -0.1 0.0 0.2 0.5 0.7 3.34 4.8 4.9

BA

D

AB

FactorsFactors– A: Pressure (Lo,Hi)A: Pressure (Lo,Hi)– B: Bow PlacementB: Bow Placement

(near,far) (near,far)– C: Bow Angle (Lo,Hi)C: Bow Angle (Lo,Hi)– D: Bow Speed (Lo,Hi)D: Bow Speed (Lo,Hi)

2244 Designs DesignsU-Do-It Solution - Violin Exercise U-Do-It Solution - Violin Exercise

InterpretationInterpretation The interaction between A and B is so The interaction between A and B is so

weak that it is probably ignorable and weak that it is probably ignorable and will not be included initially. This will not be included initially. This simplifies the analysis since, when there simplifies the analysis since, when there are no interactions, the observed are no interactions, the observed changes in the response will be the sum changes in the response will be the sum of the individual changes in the main of the individual changes in the main effects, i.e, the main effects are effects, i.e, the main effects are additive.additive.

2244 Designs DesignsU-Do-It Solution - Violin Exercise U-Do-It Solution - Violin Exercise

InterpretationInterpretation When the AB interaction is ignored, we When the AB interaction is ignored, we

expect expect – A loudness increase of 3.3 decibels when A loudness increase of 3.3 decibels when

increasing bow speed from Lo to Hi.increasing bow speed from Lo to Hi.– A loudness increase of about 5 decibels when A loudness increase of about 5 decibels when

changing the bow placement from “near” to changing the bow placement from “near” to “far”.“far”.

– A loudness increase of 4.8 decibels when A loudness increase of 4.8 decibels when changing pressure from Lo to Hi.changing pressure from Lo to Hi.

– The loudness seems unaffected by the angle The loudness seems unaffected by the angle factor; this “non-effect” is in itself interesting factor; this “non-effect” is in itself interesting and useful.and useful.

U-Do-It ExerciseU-Do-It Exercise U-Do-It Solution U-Do-It Solution

Violin Exercise: Including the AB InteractionViolin Exercise: Including the AB Interaction We now include the AB interaction for We now include the AB interaction for

comparison purposes. Since the comparison purposes. Since the interaction is so weak, it does not interaction is so weak, it does not appreciably change the analysisappreciably change the analysis

U-Do-It ExerciseU-Do-It Exercise U-Do-It Solution - Violin Exercise AB Interaction TableU-Do-It Solution - Violin Exercise AB Interaction Table

1

1 2

2

B: Bow Placement

A: Pressure

A B = 282.4/4 = 70.60

69.3 67.4 73.4 72.3 282.4

1 1

75.9 74.4 78.8 78.1 307.2 A B =

307.2/4 = 76.801 2

79.3 78.8 81.6 81.5 321.2

75.3 74.9 77.5 78.8 306.5A B =

306.5/4 = 76.6252 1 A B =

321.2/4 = 80.302 2

y A B69.3 -1 -175.3 1 -175.9 -1 179.3 1 167.4 -1 -174.9 1 -174.4 -1 178.8 1 173.4 -1 -177.5 1 -178.8 -1 181.6 1 172.3 -1 -178.8 1 -178.1 -1 181.5 1 1

U-Do-It ExerciseU-Do-It Exercise U-Do-It Solution - Violin Exercise AB Interaction U-Do-It Solution - Violin Exercise AB Interaction

Table/PlotTable/Plot

-1 1-1 1

1 1-1-1

80

76

72

B

AAB Interaction Plot

- -

y = 76.1_

1

1 2

2

B: Bow Placement

A: Pressure

A B = 282.4/4 = 70.60

69.3 67.4 73.4 72.3 282.4

1 1

75.9 74.4 78.8 78.1 307.2 A B =

307.2/4 = 76.801 2

79.3 78.8 81.6 81.5 321.2

75.3 74.9 77.5 78.8 306.5A B =

306.5/4 = 76.6252 1 A B =

321.2/4 = 80.302 2

2244 Designs DesignsU-Do-It Solution - Violin Exercise U-Do-It Solution - Violin Exercise

InterpretationInterpretation If We Include the AB Interaction, If We Include the AB Interaction,

We ExpectWe Expect– Loudness to increase 3.3 when Loudness to increase 3.3 when

bowing speed, D, increases from Lo bowing speed, D, increases from Lo to Hi. to Hi.

– Since the lines in the AB interaction Since the lines in the AB interaction are nearly parallel, the effect of the are nearly parallel, the effect of the interaction is weak. This is reflected interaction is weak. This is reflected in our estimates of the EMR.in our estimates of the EMR.

2244 Designs DesignsU-Do-It Solution - Violin Exercise EMRU-Do-It Solution - Violin Exercise EMR

Let us calculate the EMR if we want the response to be Let us calculate the EMR if we want the response to be the quietest.the quietest.

If We Don’t Include the AB Interaction, If We Don’t Include the AB Interaction, – Set A, B and D at their Lo setting, -1.Set A, B and D at their Lo setting, -1.– EMR = 76.1 - (4.8+4.9+3.34)/2 = 69.56EMR = 76.1 - (4.8+4.9+3.34)/2 = 69.56

If We Include the AB Interaction,If We Include the AB Interaction,– Set D at its Lo setting, -1. The AB Interaction Table and Plot show Set D at its Lo setting, -1. The AB Interaction Table and Plot show

that A and B still should be set Lo, -1. Note that when A and B are that A and B still should be set Lo, -1. Note that when A and B are both -1, AB is +1.both -1, AB is +1.

– EMR = 76.1 - (4.8+4.9+3.34)/2 +(-1.3)/2 =68.93EMR = 76.1 - (4.8+4.9+3.34)/2 +(-1.3)/2 =68.93

y A B C D AB69.3 -1 -1 -1 -1 175.3 1 -1 -1 -1 -175.9 -1 1 -1 -1 -179.3 1 1 -1 -1 167.4 -1 -1 1 -1 174.9 1 -1 1 -1 -174.4 -1 1 1 -1 -178.8 1 1 1 -1 173.4 -1 -1 -1 1 177.5 1 -1 -1 1 -178.8 -1 1 -1 1 -181.6 1 1 -1 1 172.3 -1 -1 1 1 178.8 1 -1 1 1 -178.1 -1 1 1 1 -181.5 1 1 1 1 11217.3 38.1 39.5 -4.9 26.7 -10.116 8 8 8 8 8

76.1 4.8 4.9 -.6 3.34 -1.3

y A B C D AB-1 -1 -1 -1 1-1 -1 1 -1 1

1217.3 38.1 39.5 -4.9 26.7 -10.116 8 8 8 8 8

76.1 4.8 4.9 -.6 3.34 -1.3

y A B C D AB-1 -1 -1 -1 1-1 -1 1 -1 1

1217.3 38.1 39.5 -4.9 26.7 -10.116 8 8 8 8 8

76.1 4.8 4.9 -.6 3.34 -1.3

22kk Designs DesignsIntroductionIntroduction

Suppose the effects of k factors, each having two levels, Suppose the effects of k factors, each having two levels, are to be investigated.are to be investigated.

How many runs (recipes) will there be with no replication?How many runs (recipes) will there be with no replication?– 22kk runs runs

How may effects are you estimating?How may effects are you estimating?– There will be 2There will be 2kk-1 columns in the Signs Table-1 columns in the Signs Table– Each column will be estimating an EffectEach column will be estimating an Effect

k main effects, A, B, C,... k main effects, A, B, C,... k(k-1)/2 two-way interactions, AB, AC, AD,... k(k-1)/2 two-way interactions, AB, AC, AD,... k(k-1)(k-2)/3! three-way interactionsk(k-1)(k-2)/3! three-way interactions ..

..

.. k (k-1)-way interactionsk (k-1)-way interactions one k-way interactionone k-way interaction

22kk Designs DesignsAnalysis ToolsAnalysis Tools

Signs Table to Estimate EffectsSigns Table to Estimate Effects– 22kk-1 columns of signs; first k estimate the k main effects and remaining -1 columns of signs; first k estimate the k main effects and remaining

22kk-k -1 estimate interactions-k -1 estimate interactions

22kk - 1 Effects Normal Probability Plots - 1 Effects Normal Probability Plots to Determine Statistically Significant to Determine Statistically Significant EffectsEffects

Interaction Tables/Plots to Analyze Interaction Tables/Plots to Analyze Two-Way InteractionsTwo-Way Interactions

EMR Computed as BeforeEMR Computed as Before

22kk Designs DesignsConcluding CommentsConcluding Comments

Know How to Design, Analyze and Know How to Design, Analyze and Interpret Full Factorial Two-Level Interpret Full Factorial Two-Level DesignsDesigns

This means thatThis means that– The design is orthogonalThe design is orthogonal– The run order is totally randomizedThe run order is totally randomized

22kk Designs DesignsOrthogonalityOrthogonality

(Hard to Explain) If a Design is (Hard to Explain) If a Design is OrthogonalOrthogonal, Each Factor’s “Effect” , Each Factor’s “Effect” can be Estimated Without can be Estimated Without Interference From the Others...Interference From the Others...

22kk Designs DesignsOrthogonality - Checking OrthogonalityOrthogonality - Checking Orthogonality

1. Use the -1 and 1 Design Matrix.1. Use the -1 and 1 Design Matrix.2. Pick Any Pair of Columns2. Pick Any Pair of Columns3. Create a New Column by Multiplying 3. Create a New Column by Multiplying These Two, Row by Row.These Two, Row by Row.4. Sum the New Column; If the Sum is Zero, 4. Sum the New Column; If the Sum is Zero, the Two Columns/Factors Are Orthogonal.the Two Columns/Factors Are Orthogonal.5. If Every Pair of Columns is Orthogonal, the 5. If Every Pair of Columns is Orthogonal, the Design is Orthogonal.Design is Orthogonal.

22kk Designs Designs RandomizationRandomization

It is It is HighlyHighly Recommended That the Recommended That the Trials be Carried Out in a Randomized Trials be Carried Out in a Randomized Order!!!Order!!!

22kk Designs Designs Randomization devicesRandomization devices

Slips of Paper in a BowlSlips of Paper in a Bowl Multi-Sided DieMulti-Sided Die Coin FlipsCoin Flips Table of Random DigitsTable of Random Digits Pseudo-Random Numbers on a Pseudo-Random Numbers on a

ComputerComputer

22kk Designs Designs Randomization - Why randomize order? Randomization - Why randomize order?

It’s MAE’s fault...It’s MAE’s fault...M=A + EM=A + E

(M easured response)=(M easured response)=(A ctual effect of factor (A ctual effect of factor

combination)combination)+ (E verything else-”random + (E verything else-”random

error”)error”)

22kk Designs Designs Randomization - Beware the convenient sample!Randomization - Beware the convenient sample!

Randomize Run Order to Protect Randomize Run Order to Protect Against the Against the Unknown Factors Unknown Factors Which Which are not Either are not Either – varied as experimental factors, orvaried as experimental factors, or– fixed as fixed as background effectsbackground effects..

Try Hard to Determine What These Try Hard to Determine What These Unknown Factors Are!Unknown Factors Are!

22kk Designs Designs Randomization - Instructions for OperatorsRandomization - Instructions for Operators

Having Randomized the Run Order, Having Randomized the Run Order, Present the Operator With Easy-to-Present the Operator With Easy-to-Follow Instructions. Follow Instructions.

Tell Him/Her Not to "Help" by Tell Him/Her Not to "Help" by Rearranging the Order for Rearranging the Order for Convenience!Convenience!

22kk Designs Designs Randomization - Partial RandomizationRandomization - Partial Randomization

In Certain Situations It May Not Be Possible to Totally In Certain Situations It May Not Be Possible to Totally Randomize All the RunsRandomize All the Runs– e.g., it may be too costly to completely randomize the temperatures e.g., it may be too costly to completely randomize the temperatures

of a series of ovens while one may be able to totally randomize the of a series of ovens while one may be able to totally randomize the other factor levelsother factor levels

This Leads to Blocks of Runs Within Which The Factor This Leads to Blocks of Runs Within Which The Factor Settings Can Be Totally RandomizedSettings Can Be Totally Randomized– The Analysis of Blocked Designs Will Be Discussed in a Later ModuleThe Analysis of Blocked Designs Will Be Discussed in a Later Module

Remember An Important Goal of a DOE is to Get Good Remember An Important Goal of a DOE is to Get Good DataData– Randomization Protects Us From Background Sources of Variation Of Randomization Protects Us From Background Sources of Variation Of

Which We May Not Be AwareWhich We May Not Be Aware– Blocking Allows Us to Include Known But Hard to Control Sources So Blocking Allows Us to Include Known But Hard to Control Sources So

That We Estimate Their Effect. We Can Then Remove Their Effect That We Estimate Their Effect. We Can Then Remove Their Effect and Analyze the other Factor Effectsand Analyze the other Factor Effects