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.3 The Essentials of 2-Cubed Designs
Methodology
Cube Plots
Estimating Main Effects
Estimating Interactions (Interaction Tables and Graphs)

3. Part I.3 The Essentials of 2-Cubed Designs Lecture Topics
Statistical Significance: When is an Effect “Real”?
An Example With Interactions
A U-Do-It Case Study
Replication
Rope Pull Exercise

4. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? (As Opposed to Being “Due to Error”) Introduction
The Effects (Main and Interactions) We Compute are Really Estimates of the “True Effects” (Remember MAE).
All the True Effects are Probably Nonzero, but Some are Very Small - It is More Correct to Ask If an Effect is “Distinguishable from Error” or “Indistinguishable from Error”.

5. We will Discuss Tools to Help in This Decision
Statistical Significance When is an Effect “Real”? (As Opposed to Being “Due to Error”) Tools
We will Discuss Tools to Help in This Decision
Normal Probability Plots of Estimated Effects
Replication
ANOVA

6. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? Normal Probability Plots of Estimated Effects
What if all the true effects were zero, so that estimated effects represented only random error?
Show normal scores figures from website—show where ORDERED effect “live”. Plots show “well-behaved” normal sample.

7. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? Normal Probability Plots - Background
In 1959, Cuthbert Daniel Found a Way to Plot the Estimated Effects so that Effects Due to Random Error Fall (Roughly) on a Straight Line in the Plot
To Construct a Normal Probability Plot of the Effects
1. Order the Estimated Effects from Smallest to Largest (Minus Signs Count: -2 is Less Than 1, For Example).
2. Plot the Points (Ei,Pi), i = 1,..,m on Normal Probability Paper, Where m = Number of Effects, Ei is the ith Smallest Effect (Put the E’s on the Horizontal Axis), and Pi = 100(i-0.5)/m.
3. Normal Probability Paper is on the next Slide for m = 7.
To Use This Paper
Scale the x-axis (Horizontal Axis) to Cover the Range of the Effects
Plot the Smallest Value on Line 1, the Next Smallest on Line 2, etc.
Plot ordered effects against expected positions for ordered effects from a normal distribution. Vertical scale may be presented as percentiles for some normal probability paper (e.g., Minitab). Vertical scale depends only on m=7.

8. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? Normal Probability Plots - Seven Effects Paper
Need new vertical scale each time m changes.

9. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? Normal Probability Plots - Interpretation
If There are Enough Effects Plotted, and Some are Due to Random Error, These Will Lie Approximately on a Straight Line Centered at 0. Sketch in the Line.
Identify Any Effects That Fall off the Line to the Upper Right and Lower Left. These Effects are Probably Not Due to Noise; They are “Distinguishable from Error”.

10. Statistical Significance When is an Effect “Real”
Statistical Significance When is an Effect “Real”? Normal Probability Plots - Example 2
Will study Lenth’s test later. Humans are bad at sorting.

11. Methodology Example 3 - PC Response Time Design
Objective Reduce Company’s PC Response Time
Factors
A: Cache (Two Levels Lo, Hi)
B: Machine (Lo MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM
C: Line (Lo - 56K modem, Hi - LAN)
NCR example. Cache—small local storage. LAN—Local area network. One of the machine settings is really old

12. Methodology Example 3 - PC Response Time
Response: PC Response Time
Factors
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)

13. Methodology Example 3 - PC Response Time Cube Plot
Response: PC Response Time
Factors
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo MHz, 64 MB
RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)
Looks like AC is important. BC too, but the analysis says otherwise.

14. Methodology Example 3 - PC Response Time Estimating the Effects - Signs Tables
BC effect is rather small!

15. Methodology Example 3 - PC Response Time Effects Normal Probability Plot
Choosing the correct line is hard—shallow slope with everything as noise, or steep slope based on only three negligible factors? Remove Minitab’s line, and construct line by hand.

16. Methodology Interaction Tables and Graphs
Tools for Aiding Interpretation of SIGNIFICANT Two-Way Interactions
At the Right is a Blank AB Interaction Table
In the Table, 1 Corresponds to the Lo Level and 2 to the Hi Level B: -1 1 A:
A-1B-1C-1
A-1B1C-1
A-1B-1C1
A-1B1C1
A-1B-1
A-1B1
A1B-1C-1
A1B1C-1
A1B-1C1
A1B1C1
A1B-1
A1B1
AB here, but the significant AC interaction follows.

17. Methodology Example 3 - PC Response Time AC Interaction Table
C: Line -1 1
A: Cache
51.0
25.5
39.8
7.0
90.8
32.5
A-1C-1 = 45.4
A-1C1 = 16.25
29.5
25.8
13.5
6.8
43.0
32.6
A1C-1 = 21.5
A-1C1 = 16.3

18. Methodology Interaction Tables and Graphs Interaction Plots - Construction
1. For a Given Pair of Factors (Say A and B) Find the Average Response at Each of Their Four Level Combinations.
2. Plot These with Response on the Vertical Axis, Using One of the Factor’s Levels (Say B) on the Horizontal Axis. Connect and Label the Averages with the Same Level of the Other Factor (A).

19. 1. If the Lines are Roughly Parallel, There is No Strong Interaction.
Methodology Interaction Tables and Graphs Interaction Plots - Interpretation
1. If the Lines are Roughly Parallel, There is No Strong Interaction.
2. If There is Interaction, the Plot Shows Clearly the Effect of a Factor at Each of the Levels of the Other Factor.
3. Maximizing and Minimizing Combinations of the Factors are Easily Identified on the Plot and in the Table.

20. Methodology Example 3 AC Interaction Table and Graph
Response: PC Response Time
Factors
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo MHz, 64 MB
RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)
C: Line -1 1
A: Cache
51.0
25.5
39.8
7.0
90.8
32.5
A-1C-1 = 45.4
A-1C1 = 16.25
29.5
25.8
13.5
6.8
43.0
32.6
A1C-1 = 21.5
A-1C1 = 16.3
U-Do-It—reverse role of A and C. It will be similar in appearance, but it often pays to look at both plots. Minitab can do both. A low, C low is bad.

21. Methodology Example 3 - AC Interaction Graph
Response: PC Response Time
Factors
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)

22. Methodology Example 3 - AC Interaction Interpretation Noise Factors versus Control Factors
Response: PC Response Time
Factors
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo MHz, 64 MB RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)
To minimize the response, choose B Hi and C Hi. When C is Hi, the effect of A is negligible.
Refer back to the cube plot—the (B Hi, C Hi) edge had the two lowest readings. Our analysis shows that this was not due to a BC interaction, but to a significant B main effect and the particular form of the significant AC interaction.

23. Methodology Example 3 - Estimating the Mean Response: A = +1, B = -1, C = +1
Estimated Mean Response (EMR) = y + [(Sign of A)(Effect of A)+(Sign of B)(Effect of B) +(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2
For A = +1, B = -1, C = +1, EMR = [(+1)(-11.9)+(-1)(-16.2)+(+1)(-17.2)+(1)(12)]/2 = 24.4
Notice that for A = +1 and C = +1, [(Sign of A)(Effect of A)+(Sign of C)(Effect of C)+(Sign of AC)(Effect of AC)]/2 = [(+1)(-11.9)+(+1)(-17.2)+(1)(12)]/2 = -17.1/2 = = A2C2 – y; so EMR= =16.35
For Calculating EMR Include:
Significant Main Effects
Significant Interactions, and All Their Lower Order Interactions and Main Effects
The third bullet shows how to compute EMR for collapsed data.

24. Methodology U-Do-It: Example 3 - Estimate the Response A = +1, B = +1, C = +1 and A = +1, B = +1, C = -1
First example: Second example: 13.45