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The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs

The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs. Developed by Don Edwards, John Grego and James Lynch Center for Reliability and Quality Sciences Department of Statistics University of South Carolina 803-777-7800. Part I. Full Factorial Designs.

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The Essentials of 2-Level Design of Experiments Part I: The Essentials of Full Factorial Designs

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  1. The Essentials of 2-Level Design of ExperimentsPart I: The Essentials of Full Factorial Designs Developed by Don Edwards, John Grego and James LynchCenter for Reliability and Quality SciencesDepartment of StatisticsUniversity of South Carolina803-777-7800

  2. Part I. Full Factorial Designs • 24 Designs • Introduction • Analysis Tools • Example • Violin Exercise • 2k Designs

  3. 24 DesignsU-Do-It - Violin Exercise

  4. 24 DesignsU-Do-It - Violin Exercise How to Play the Violin in 176 Easy Steps1,2 • A very scientifically-inclined violinist was interested in determining what factors affect the loudness of her instrument. She believed these might include: • A: Pressure (Lo,Hi) • B: Bow placement (near,far) • C: Bow Angle (Lo,Hi) • D: Bow Speed (Lo,Hi) • The precise definition of factor levels is not shown, but they were very rigidly defined and controlled in the experiment. • Eleven replicates of the full 24 were performed, in completely randomized order. Analyze the data! 1176=11x162Data courtesy of Carla Padgett

  5. 24 DesignsU-Do-It - Violin Exercise Report Form • Responses are Averages of 11 Independent Replicates • All 176 trials were randomly ordered • Analyze and Interpret the Data

  6. 24 DesignsU-Do-It Solution - Violin Exercise Signs Table

  7. U-Do-It ExerciseU-Do-It Solution - Violin Exercise Cube Plot • Factors • A: Pressure (Lo,Hi) • B: Bow Placement (near,far) • C: Bow Angle (Lo,Hi) • D: Bow Speed (Lo,Hi)

  8. 24 DesignsU-Do-It Solution - Violin Exercise Effects Normal Probability Plot • Factors • A: Pressure (Lo,Hi) • B: Bow Placement (near,far) • C: Bow Angle (Lo,Hi) • D: Bow Speed (Lo,Hi)

  9. 24 DesignsU-Do-It Solution - Violin Exercise Interpretation • The interaction between A and B is so weak that it is probably ignorable and will not be included initially. This simplifies the analysis since, when there are no interactions, the observed changes in the response will be the sum of the individual changes in the main effects, i.e, the main effects are additive.

  10. 24 DesignsU-Do-It Solution - Violin Exercise Interpretation • When the AB interaction is ignored, we expect • A loudness increase of 3.3 decibels when increasing bow speed from Lo to Hi. • A loudness increase of about 5 decibels when changing the bow placement from “near” to “far”. • A loudness increase of 4.8 decibels when changing pressure from Lo to Hi. • The loudness seems unaffected by the angle factor; this “non-effect” is in itself interesting and useful.

  11. U-Do-It ExerciseU-Do-It Solution Violin Exercise: Including the AB Interaction • We now include the AB interaction for comparison purposes. Since the interaction is so weak, it does not appreciably change the analysis

  12. U-Do-It ExerciseU-Do-It Solution - Violin Exercise AB Interaction Table

  13. U-Do-It ExerciseU-Do-It Solution - Violin Exercise AB Interaction Table/Plot

  14. 24 DesignsU-Do-It Solution - Violin Exercise Interpretation • If We Include the AB Interaction, We Expect • Loudness to increase 3.3 when bowing speed, D, increases from Lo to Hi. • Since the lines in the AB interaction are nearly parallel, the effect of the interaction is weak. This is reflected in our estimates of the EMR.

  15. 24 DesignsU-Do-It Solution - Violin Exercise EMR • Let us calculate the EMR if we want the response to be the quietest. • If We Don’t Include the AB Interaction, • Set A, B and D at their Lo setting, -1. • EMR = 76.1 - (4.8+4.9+3.34)/2 = 69.56 • If We Include the AB Interaction, • 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 both -1, AB is +1. • EMR = 76.1 - (4.8+4.9+3.34)/2 +(-1.3)/2 =68.93

  16. 2k DesignsIntroduction • Suppose the effects of k factors, each having two levels, are to be investigated. • How many runs (recipes) will there be with no replication? • 2k runs • How may effects are you estimating? • There will be 2k-1 columns in the Signs Table • Each column will be estimating an Effect • k main effects, A, B, C,... • k(k-1)/2 two-way interactions, AB, AC, AD,... • k(k-1)(k-2)/3! three-way interactions • ... • k (k-1)-way interactions • one k-way interaction

  17. 2k DesignsAnalysis Tools • Signs Table to Estimate Effects • 2k-1 columns of signs; first k estimate the k main effects and remaining 2k-k -1 estimate interactions • 2k - 1 Effects Normal Probability Plots to Determine Statistically Significant Effects • Interaction Tables/Plots to Analyze Two-Way Interactions • EMR Computed as Before

  18. 2k DesignsConcluding Comments • Know How to Design, Analyze and Interpret Full Factorial Two-Level Designs • This means that • The design is orthogonal • The run order is totally randomized

  19. 2k DesignsOrthogonality • (Hard to Explain) If a Design is Orthogonal, Each Factor’s “Effect” can be Estimated Without Interference From the Others...

  20. 2k DesignsOrthogonality - Checking Orthogonality 1. Use the -1 and 1 Design Matrix. 2. Pick Any Pair of Columns 3. Create a New Column by Multiplying These Two, Row by Row. 4. Sum the New Column; If the Sum is Zero, the Two Columns/Factors Are Orthogonal. 5. If Every Pair of Columns is Orthogonal, the Design is Orthogonal.

  21. 2k DesignsRandomization • It is Highly Recommended That the Trials be Carried Out in a Randomized Order!!!

  22. 2k DesignsRandomization devices • Slips of Paper in a Bowl • Multi-Sided Die • Coin Flips • Table of Random Digits • Pseudo-Random Numbers on a Computer

  23. 2k DesignsRandomization - Why randomize order? • It’s MAE’s fault... M=A + E (M easured response)= (A ctual effect of factor combination) + (E verything else-”random error”)

  24. 2k DesignsRandomization - Beware the convenient sample! • Randomize Run Order to Protect Against the Unknown Factors Which are not Either • varied as experimental factors, or • fixed as background effects. • Try Hard to Determine What These Unknown Factors Are!

  25. 2k DesignsRandomization - Instructions for Operators • Having Randomized the Run Order, Present the Operator With Easy-to-Follow Instructions. • Tell Him/Her Not to "Help" by Rearranging the Order for Convenience!

  26. 2k DesignsRandomization - Partial Randomization • In Certain Situations It May Not Be Possible to Totally Randomize All the Runs • 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 other factor levels • This Leads to Blocks of Runs Within Which The Factor Settings Can Be Totally Randomized • The Analysis of Blocked Designs Will Be Discussed in a Later Module • Remember An Important Goal of a DOE is to Get Good Data • Randomization Protects Us From Background Sources of Variation Of Which We May Not Be Aware • Blocking Allows Us to Include Known But Hard to Control Sources So That We Estimate Their Effect. We Can Then Remove Their Effect and Analyze the other Factor Effects

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