Download Lecture 9.2 - University of South Carolina

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 Exercise
Rope Pull Study* - 23 with Replication

Purpose of the Design
– Test Hose to Determine the Effect of
Several Factors on an Important Quality
Hosiery Characteristic, Rope Pull
– Response y = Upper Boot Rope Pull (in
inches)

Factors:
– A: Vacuum level
(Lo, Hi)
– B: Needle Type
(EX, GB)
– C: Upper Boot Speed (1000,1200)

Two Replicates of the Full 23 Were
Performed
*Empirical basis for this data was motivated by a much
larger study performed by the developers at Sara Lee
Hosiery
Replication
Why?

Average values have less variability as the
number of things you average increases
– Estimated effects will be reliably closer to true effects
– More of the mid-sized and small effects will be
distinguishable from error
Data from replicated experiments can be used
to estimate the amount of variability in the
process (This allows more formal test for “real”
effects—ANOVA).
 Data from replicated experiments can be used to
determine not only which factors affect the
mean of the process, but which factors affect
the variability of the process.

Replication
Analysis of a Replicated 23


Replication means repeating the entire set of 8 runs, but (for the
analysis as described below), the entire collection of runs should be
done in random order (be it 16, or 24, or 48, etc. runs); if you want
to do them in complete sets of 8, you should analyze the results in
blocks—explained later).
For our analysis, you can reduce the data to averages over each of
the 8 treatment combinations; use these averages as your “y’s” in
the rest of the analysis.
– Discussion of shortcomings of this approach to follow

Effects plot, interaction plots, and EMR calculations are done as
before using these estimated effects.
Replication Example to Follow!
U-Do-It Exercise
Rope Pull Study - Experimental Report Form
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
std. order
Run #
5
2
1
7
3
3
6
8
4
5
6
1
7
2
4
8
Vacuum
level
LO
HI
LO
LO
LO
LO
HI
HI
HI
LO
HI
LO
LO
HI
HI
HI
Needle
T ype
EX
EX
EX
GB
GB
GB
EX
GB
GB
EX
EX
EX
GB
EX
GB
GB
U.B.
Speed
1200
1000
1000
1200
1000
1000
1200
1200
1000
1200
1200
1000
1200
1000
1000
1200
Boot
rope pull
(in.)
94.8
109.8
100.3
92.1
102.3
99.2
95.4
94.7
110.1
92.7
97.6
100.4
92.7
111.9
108.3
96.2
U-Do-It Exercise
Rope Pull Study - The Analysis

To do: Analyze the data. This should include...
– Fill in the table on the next slide.
– Analyze the averages in Minitab:
 Create a 3-factor 2-level design, enter the averages as a
response variable; compute factor effects and construct a
normal probability plot of the effects.
 If appropriate, graph interaction plots.
 Compute EMR using only the significant terms
U-Do-It Exercise
Rope Pull Study - The Analysis
Run
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
std. order
Run #
5
2
1
7
3
3
6
8
4
5
6
1
7
2
4
8
Vacuum
level
LO
HI
LO
LO
LO
LO
HI
HI
HI
LO
HI
LO
LO
HI
HI
HI
Needle
T ype
EX
EX
EX
GB
GB
GB
EX
GB
GB
EX
EX
EX
GB
EX
GB
GB
U.B.
Speed
1200
1000
1000
1200
1000
1000
1200
1200
1000
1200
1200
1000
1200
1000
1000
1200
Boot
rope pull
(in.)
94.8
109.8
100.3
92.1
102.3
99.2
95.4
94.7
110.1
92.7
97.6
100.4
92.7
111.9
108.3
96.2
Stardard
Order Run
#
1
2
3
4
5
6
7
8
First y
Second y Average
U-Do-It Exercise Solution
Rope Pull Study
Stardard
Order Run
#
1
2
3
4
5
6
7
8

First y
100.3
109.8
102.3
110.1
94.8
95.4
92.1
94.7
Second y Average
100.4
111.9
99.2
108.3
92.7
97.6
92.7
96.2
100.35
110.85
100.75
109.2
93.75
96.5
92.4
95.45
The signs table, cube plot, effects normal probability plot and
AC interaction table and graph are given on the next few pages.
–
–
The cube plot leads us to expect a negative main effect for A (Vacuum level),
and a positive main effect for C (upper boot speed). Note that the changes in
the response for changes in A are much larger at Lo C than at Hi C, which
suggests an AC interaction. Estimated effects from the response table and
the normal probability plot of effects support this observation.
An AC interaction table and plot are therefore called for, and have been
constructed.
U-Do-It Exercise Solution
Rope Pull Study - Completed Cube Plot and Signs
Table
Main Effects
95.45
92.40
109.20
100.75
+
Actual
Run
Vacuum
A
100.35
110.85
100.75
109.20
93.75
96.50
92.40
95.45
Sum
799.25
Divisor
8
Effect
99.9
-1
1
-1
1
-1
1
-1
1
24.75
4
6.1875
Needle
T ype
B
-1
-1
1
1
-1
-1
1
1
-3.65
4
-0.9125
Interaction Effects
Upper
Boot Speed AB
AC
BC
ABC
C
-1
1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
-1
1
-1
1
-1
-1
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
-1
-1
1
-1
1
1
1
1
1
-43.05
-1.750 -13.15 -1.150 2.35
4
4
4
4
4
-10.7625 -0.4375 -3.2875 -0.2875 0.5875
96.50
93.75
B
+
C
_
110.85
100.35
_
A

Factors:
– A: Vacuum Level
(Lo, Hi)
– B: Needle Type
(EX, GB)
– C: Upper Boot Speed (1000,1200)
_
+

Response:
– Rope Pull (in inches)
U-Do-It Exercise Solution
Rope Pull Study -Completed Seven Effects Paper

Factors:
– A: Vacuum Level
(Lo, Hi)
– B: Needle Type
(EX, GB)
– C: Upper Boot Speed (1000,1200)
7 Effects Plo t
7
A
6
5
4
3
2
AC
C
1
-9
-6
-3
Effects
0
3
6
Ordered Effects -10.7625, -3.2875, -0.9125, -0.4375, -0.2875, 0.5875, 6.1875
U-Do-It Exercise Solution
Rope Pull Study - Completed AC Interaction Table
C: Upper Boot Speed
2
1
A:
Vacu um
Level
100.35
100.75
201.1
1
A C = 100.55
1 1
2
110.85
109.20
220.05
A C = 110.025
2 1
93.75
92.40
186.15
A 1C2 = 93.075
96.50
95.45
191.95
A 2C2= 95.975
U-Do-It Exercise Solution
Rope Pull Study - AC Interaction Plot

Factors
A: Vacuum Level
(Lo, Hi)
C: Upper Boot Speed (1000,1200)
U-Do-It Exercise Solution
Rope Pull Study - Interpretation of the Experiment


There is non-ignorable interaction between A = Vacuum level and C =
Upper boot speed, so we should not interpret main effects for these
factors individually. For example, a Hi Vacuum level greatly increases the
effect of a change from 1000 to 1200 RPM in Upper boot speed. Judging
from the interaction plot,
– At Lo Vacuum level, we expect a decrease of about 7” in rope pull
when changing Upper boot speed from 1000 to 1200 RPM.
– At Hi Vacuum level, we expect a decrease of about 14’’ in rope pull
when changing Upper boot speed from 1000 to 1200 RPM.
– At 1000 RPM Upper boot speed, we expect an increase of about 10”
in rope pull when changing Vacuum level from Lo to Hi.
– At 1200 RPM Upper boot speed, we expect an increase of about 3:
in rope pull when changing Vacuum level from Lo to Hi.
The above interpretations hold for both needle types (Factor B). There is
no detectable difference in mean rope pull between the two needle types.
Replication
Why? It Allows You To See Things More Clearly!
Seven Effects Plo t - Fi rst Re pl i cate

1.5

0.5
Below Are the Normal Probability
Plots for the First and Second
Replication
Notice How Hard it is to see that
the AC Interaction is Significant
-0.5
Seven Effects Plo t - Seco nd Re pl i ca te
-1.5
-10
-5
0
Effects
5
1.5
0.5
-0.5
-1.5
-10
-5
0
Effects
5
Replication
Why? It Allows You To See Things More Clearly!
Seven Effects Plo t - Seco nd Re pl i ca te


Below Are the Normal Probability
Plots for the Individual Replicates
and the one based on the Averages
Notice How Replication Makes it
Easier to see that the AC Interaction
is Significant
1.5
A
0.5
-0.5
-10
0
5
Seven Effects Pl ot - Ave ra ge s
1.5
A
A
0.5
0.5
AC
-0.5
-1.5
-5
Effects
Seven Effects Plo t - Fi rst Re pl i cate
1.5
C
-1.5
C
-10
AC
-0.5
-1.5
-5
0
Effects
5
C
-10
-5
0
Effects
5
Replication
Why? It Allows You To Use ANOVA!


The Small p-Values in the ANOVA Table Indicate that there are
Significant Main Effects and that the Interaction is Significant
The zero p-values in the Factor Effects Table Indicate that both A
and B are Significant
Factor
y-bar
A
B
C
AB
AC
BC
ABC
Effect
99.906
6.188
-0.913
-10.763
-0.438
-3.288
-0.288
0.587
P-value
0.000
0.000
0.213
0.000
0.535
0.000
0.681
0.409
Analysis of Variance Table
Source
Main Effects
2-W ay Interactions
3-W ay Interactions
DF
3
3
1
SS
619.797
44.327
1.381
MS
206.599
14.776
1.381
Residual Error
T otal
8
15
14.565
680.070
1.821
F
113.48
8.12
0.76
P-value
0.000
0.008
0.409