Transcript Tabular and Graphical Methodology for 23 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.3 The Essentials of 2-Cubed Designs

Methodology
– Cube Plots
– Estimating Main Effects
– Estimating Interactions (Interaction
Tables and Graphs)
 Statistical Significance:




When is an Effect “Real”?
An Example With Interactions
A U-Do-It Case Study
Replication
Rope Pull Exercise
U-Do-It Case Study
Ball Bearing Example*

Purpose of the Design
– Test (Under Accelerated Conditions) New Bearing Prototypes for Use
in a Specific Application for Which the Current Design’s Performance
Was “Unsatisfactory”.
– Response of Interest: y - Bearing Life (h).

Design Factors:
– Factor
A: Cage Design
B: Outer Ring Osculation
C: Inner Ring Heat Treatment.

Levels (Lo,Hi)
Current, New
Current, New
Current, New
The 8 Standard Runs of the 23 Design Were Randomly
Ordered, and Each Prototype Bearing Tested.
*Empirical Basis for this data was motivated by C. Hellstrand’s article “The necessity of modern
quality improvement and some experiences with implications in the manufacture of ball bearings
(1989, Philos. Trans. Royal Society London, A 327, 529-537)
U-Do-It Case Study
Ball Bearing Example - A Typical Ball Bearing
U-Do-It Case Study
Ball Bearing Example - Operator Report Form
Experimental
Run Order
1
2
3
4
5
6
7
8
Standard
Order Trial
Number
4
7
1
8
3
2
5
6
Cage
Design
A
new
current
current
new
current
new
current
new
Outer Ring
Osculation
B
new
new
current
new
new
current
current
current
Inner Ring Heat
Treatment
C
current
new
current
new
current
current
new
new
Bearing
Life (h)
Y
21
99
18
112
25
16
24
17
U-Do-It Case Study
Ball Bearing Example - Exercise Instructions

In Class
– Put the results of the experiment in
standard order and enter the data into a
cube plot (in Minitab—see handout)
– Estimate the factor effects (in Minitab)
– Construct and interpret a normal
probability plot of the factor effects (in
Minitab)
U-Do-It Case Study
Ball Bearing Example - Exercise Instructions

In Class
– Construct BC interaction graph in Minitab; use
table and graph to interpret BC interaction
– Determine the factor settings that maximize
bearing life and estimate the Mean Response
(EMR) at these settings. How close is your
answer to the observed mean response at your
optimal settings?
– If you would like to do hand calculations, blank
signs tables, cube plots, etc. are provided over
the next several slides
U-Do-It Case Study
Ball Bearing Example - Signs Table
Main Effects
Actual
Run
Sum
Divisor
Effect
8
Interaction Effects
A
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
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
ABC
-1
1
1
-1
1
-1
-1
1
4
4
4
4
4
4
4
U-Do-It Case Study
Ball Bearing Example - Cube Plot
+
B
+
C
_
_
_
A
+
U-Do-It Case Study
Ball Bearing Example - Seven Effects Paper
7 Effects Plot
7
6
5
4
3
2
1
Effects
U-Do-It Case Study
Ball Bearing Example - Interaction Table
1
1
2
2
U-Do-It Case Study Solution
Ball Bearing Example - Cube Plot
Bearing Lifetimes (h) Shown


Factor
A: Cage Design
B: Outer Ring Osculation
C: Inner Ring Heat Treatment
Levels: Lo = Current, Hi = New
99
112
21
25
+
17
24
B
+
C
_
18
16
_
A
+
_
U-Do-It Case Study Solution
Ball Bearing Example - Signs Table
Main Effects
Actual
Run
Lifetime Cage
Y
Design
A
3
18
-1
6
16
1
5
25
-1
1
21
1
7
24
-1
8
17
1
2
99
-1
4
112
1
Sum
332
0
Divisor
8
4
Effect
41.5
0
Outer
Ring
B
-1
-1
1
1
-1
-1
1
1
182
4
45.5
Inner
Ring
C
-1
-1
-1
-1
1
1
1
1
172
4
43
Interaction Effects
AB
AC
BC
ABC
1
-1
-1
1
1
-1
-1
1
18
4
4.5
1
-1
1
-1
-1
1
-1
1
12
4
3
1
1
-1
-1
-1
-1
1
1
158
4
39.5
-1
1
1
-1
1
-1
-1
1
22
4
5.5
U-Do-It Case Study Solution
Ball Bearing Example - Probability Plot
7 Effects Plot
7
B
6
C
5
BC
4
3
2
1
0
8
16
24
32
40
48 Effects
Ordered Effects 0, 3, 4.5, 5.5, 39.5, 43, 45.5
U-Do-It Case Study Solution
Ball Bearing Example - Completed BC Interaction
Table
B: Outer Ring Osculation
2
1
C:
Inner
Ring
Heat
Trt.
18
16
34
1
B 1C1 = 17
2
24
17
41
B1C2 = 20.5
25
21
46
B2C1 = 23
99
112
211
B 2C2 = 105.5
U-Do-It Case Study Solution
Ball Bearing Example - BC Interaction Plot
o
o

Factors
B: Outer Ring Osculation
C: Inner Ring Heat Treatment
Levels: Lo = Current, Hi = New
Interpretation
– Choose the Hi
Level for both
B and C to
Maximize the
Bearing Life
U-Do-It Case Study Solution
Ball Bearing Example - Expected Mean Response


Since the BC Interaction is
Significant, the Main Effects B
and C and the BC Interaction
are Included
Factor A is NOT Included
Since it was Not Significant
o
For B = +1, C = +1,
EMR = 41.5 + [(+1)(45.5)+(+1)(43)+(+1)(39.5)]/2
= 105.5 (vs (99+112)/2 = 105.5 Observed MR)
Main Effects
Actual
Run
Lifetime Cage
Y
Design
A
3
18
-1
6
16
1
5
25
-1
1
21
1
7
24
-1
8
17
1
2
99
-1
4
112
1
Sum
332
0
Divisor
8
4
Effect
41.5
0
Outer
Ring
B
-1
-1
1
1
-1
-1
1
1
182
4
45.5
Inner
Ring
C
-1
-1
-1
-1
1
1
1
1
172
4
43
Interaction Effects
AB
AC
BC
ABC
1
-1
-1
1
1
-1
-1
1
18
4
4.5
1
-1
1
-1
-1
1
-1
1
12
4
3
1
1
-1
-1
-1
-1
1
1
158
4
39.5
-1
1
1
-1
1
-1
-1
1
22
4
5.5
U-Do-It Case Study Solution
Ball Bearing Example - Interpretation of the
Experiment



Unexpected Interaction Discovered (Would Not Have Been
Discovered Using “One-at-a-Time” Experimentation).
Results May Carry Over to Other Bearing Designs.
Contrary to Existing Beliefs, the Two Cage Designs had
Very Similar Lifetimes. This was Very Important Since
Bearings Were Much Cheaper to Produce Under One of the
Two Cage Designs.
New Design’s Performance (In the Specific Application
Under Investigation) Far Superior to That of the Current
Bearing Being Used.