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.3 The Essentials of 2-Cubed Designs

Methodology
– Cube Plots
– Estimating Main Effects
– Estimating Interactions (Interaction
Tables and Graphs)
Part I.3 The Essentials of 2-Cubed Designs





Statistical Significance:
When is an Effect “Real”?
An Example With Interactions
A U-Do-It Case Study
Replication
Rope Pull Exercise
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”.

Statistical Significance
When is an Effect “Real”?
(As Opposed to Being “Due to Error”)
Introduction

We will Discuss Tools to Help in This
Decision
– Normal Probability Plots of Estimated
Effects
– Replication
– ANOVA
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?
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: -1 is Less Than 2, 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.
Statistical Significance
When is an Effect “Real”?
Normal Probability Plots - Seven Effects Paper
7 Effects Plot
7
6
5
4
3
2
1
Effects
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”.

Statistical Significance
When is an Effect “Real”?
Normal Probability Plots - Example 2
7 Effects Plot
A
7
6
5
4
3
2
1
-1
0
1
2
Effects
Ordered Effects: -1, -.5, 0, .5, .5, 1.5, 5
3
4
5
Methodology
Example 3 - PC Response Time
Objective
Reduce Company’s PC Response Time
 Factors

– A: Cache (Two Levels Lo, Hi)
– B: Machine (Lo - 200MHz, 64 MB RAM,
Hi - 400MHz, 1GB RAM
– C: Line (Lo - 56K modem, Hi - LAN)
Methodology
Example 3 - PC Response Time
Response: PC Response Time
 Factors

–
–
–
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo - 200MHz, 64 MB RAM,
Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem, Hi - LAN)
A
Lo
Hi
Lo
Hi
Lo
Hi
Lo
Hi
Factors
B
Lo
Lo
Hi
Hi
Lo
Lo
Hi
Hi
C
Lo
Lo
Lo
Lo
Hi
Hi
Hi
Hi
Response
y
51
29.5
39.8
13.5
25.5
25.8
7
6.8
Methodology
Example 3 - PC Response Time
Cube Plot
Response: PC Response Time
 Factors

–
–
–
–
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo - 200MHz, 64 MB
RAM, Hi - 400MHz, 1GB RAM)
C: Line (Lo - 56K modem,
Hi - LAN)
7
6.8
39.8
13.5
+
25.8
25.5
B
+
C
_
29.5
51
_
A
+
_
Methodology
Example 3 - PC Response Time
Estimating the Effects - Signs Tables
Main Effects
Actual
Run
5
2
1
4
3
6
8
7
Sum
Divisor
Effect
Time
y
51
29.5
39.8
13.5
25.5
25.8
7
6.8
198.9
8
24.9
Cache
A
-1
1
-1
1
-1
1
-1
1
-47.7
4
-11.9
Machine
B
-1
-1
1
1
-1
-1
1
1
-64.70
4
-16.2
Interaction Effects
Line
C
-1
-1
-1
-1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
-68.7
-5.3
47.9
-10.3
4
-17.2
4
-1.3
4
12
4
-2.6
ABC
-1
1
1
-1
1
-1
-1
1
4.3
4
1
Methodology
Example 3 - PC Response Time
Effects Normal Probability Plot
1.5
Norm al
Scores
AC
0.5
A
- 0.5
B
C
- 1.5
- 20
- 10
0
Effects
Ord ered Effects: -17. 2, -16 .2, -11. 9, -2.6 , -1. 3, 1, 12
10
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
1
A-1B-1C-1
A-1B1C-1
A-1B-1C1
A-1B1C1
A-1B-1
A:
1
A1B-1
A-1B1
A1B-1C-1
A1B1C-1
A1B-1C1
A1B1C1
A1B1
Methodology
Example 3 - PC Response Time
AC Interaction Table
Time
y
51
29.5
39.8
13.5
25.5
25.8
7
6.8
Cache
A
-1
1
-1
1
-1
1
-1
1
Line
C
-1
-1
-1
-1
1
1
1
1
C: Line
-1
-1
1
51.0
25.5
39.8
7.0
90.8
32.5
A-1C-1 = 45.4
A: Cache
1
A-1C1 = 16.25
29.5
25.8
13.5
6.8
43.0
32.6
A1C-1 = 21.5
A-1C1 = 16.3
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).

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.

Methodology
Response: PC Response Time
 Factors

Example 3
AC Interaction Table and Graph
–
–
–
–
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo - 200MHz, 64 MB
RAM, Hi - 400MHz, 1GB RAM)
C: Line
(Lo - 56K modem, Hi - LAN)
C: Line
-1
-1
1
51.0
25.5
39.8
7.0
90.8
32.5
A-1C-1 = 45.4
A: Cache
1
A-1C1 = 16.25
29.5
25.8
13.5
6.8
43.0
32.6
A1C-1 = 21.5
A-1C1 = 16.3
Methodology
Example 3 - AC Interaction Graph
Response: PC Response Time
 Factors

–
–
–
A: Cache (Two Levels Lo,Hi)
B: Machine (Lo - 200MHz, 64 MB
RAM, Hi - 400MHz, 1GB RAM)
C: Line
(Lo - 56K modem, Hi - LAN)
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 - 200MHz, 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.

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 = 24.9 + [(+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 = -8.55 = A2C2 – y; so EMR=24.98.55=16.35
Main Effects

For Calculating EMR Include:
– Significant Main Effects
– Significant Interactions,
and All Their Lower
Order Interactions and
Main Effects
Actual
Run
5
2
1
4
3
6
8
7
Sum
Divisor
Effect
Time
y
51
29.5
39.8
13.5
25.5
25.8
7
6.8
198.9
8
24.9
Cache
A
-1
1
-1
1
-1
1
-1
1
-47.7
4
-11.9
Machine
B
-1
-1
1
1
-1
-1
1
1
-64.70
4
-16.2
Interaction Effects
Line
C
-1
-1
-1
-1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
-68.7
-5.3
4
-17.2
4
-1.3
AC
1
-1
1
-1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
47.9
-10.3
4
12
4
-2.6
ABC
-1
1
1
-1
1
-1
-1
1
4.3
4
1
Methodology
U-Do-It: Example 3 - Estimate the Response
A = +1, B = +1, C = +1 and A = +1, B = +1, C = -1
Main Effects
Actual
Run
5
2
1
4
3
6
8
7
Sum
Divisor
Effect
Time
y
51
29.5
39.8
13.5
25.5
25.8
7
6.8
198.9
8
24.9
Cache
A
-1
1
-1
1
-1
1
-1
1
-47.7
4
-11.9
Machine
B
-1
-1
1
1
-1
-1
1
1
-64.70
4
-16.2
Interaction Effects
Line
C
-1
-1
-1
-1
1
1
1
1
AB
1
-1
-1
1
1
-1
-1
1
AC
1
-1
1
-1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
1
-68.7
-5.3
47.9
-10.3
4
-17.2
4
-1.3
4
12
4
-2.6
ABC
-1
1
1
-1
1
-1
-1
1
4.3
4
1