1. 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

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

3. 24 Designs U-Do-It - Violin Exercise

4. 24 Designs U-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! =11x16 2Data courtesy of Carla Padgett
A recent 706 project duplicated this result

5. 24 Designs U-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 Designs U-Do-It Solution - Violin Exercise Signs Table

7. U-Do-It Exercise U-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 Designs U-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)
Show this slide to explain why we collapse on C, and not D

9. 24 Designs U-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. When the AB interaction is ignored, we expect
24 Designs U-Do-It Solution - Violin Exercise Interpretation (Main effects only)
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 Exercise U-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
1) Prepare the AB interaction plot. 2) EMR for violin to be quiet (using only B, A, D, AB) 3) Demonstrate EMR by computer

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

13. U-Do-It Exercise U-Do-It Solution - Violin Exercise AB Interaction Table/Plot
Interaction preserves main effect, but not synergistic. Surprisingly subtle. Show both interaction plots.

14. If We Include the AB Interaction, We Expect
24 Designs U-Do-It Solution - Violin Exercise Interpretation (Interaction Model)
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 Designs U-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 = ( )/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 = ( )/2 +(-1.3)/2 =68.93

16. 2k Designs Introduction
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 Designs Analysis 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 Designs Concluding 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 Designs Orthogonality
(Hard to Explain) If a Design is Orthogonal, Each Factor’s “Effect” can be Estimated Without Interference From the Others...
Effects are same whether other terms are included in the model or not. Use Terms in Minitab to show this. Natural here, because we’re used to it, but think of multiple linear regression context.

20. 2k Designs Orthogonality - 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.
3. Note that this column will always be one of our other effects. 4. Demonstrate.

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

22. 2k Designs Randomization devices
Slips of Paper in a Bowl
Multi-Sided Die
Coin Flips
Table of Random Digits
Pseudo-Random Numbers on a Computer
Show use of Sample command in Minitab.

23. 2k Designs Randomization - 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”)
We don’t want A to be A+E

24. 2k Designs Randomization - 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!
Second bullet—If we don’t identify unknown factors, randomization may work, but still introduce lots of noise.

25. 2k Designs Randomization - 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 Designs Randomization - 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