Transcript Statistical Data Analysis: Primer

Stat 470-15
•
Today: Start Chapter 4
•
Assignment 3: 3.14 a, b (do normal qq-plots only),c, 3.16, 3.17
•
Additional questions: 3.14 b (also use the IER version of Lenth’s
method and compare to the qq-plot conclusions), 3.19
Fractional Factorial Designs at 2-Levels
• Use a 2k-p fractional factorial design to explore k factors in 2k-p trials
• In general, can construct a 2k-p fractional factorial design from the full
factorial design with 2k-p trials
• Set the levels of the first (k-p) factors similar to the full factorial design
with 2k-p trials
• Next, use the interaction columns between the first (k-p) factors to set
levels of the remaining factors
Example
• Suppose have 7 factors, each at 2-levels, but only enough resources to
run 16 trials
• Can use a 16-run full factorial to design the experiment
• Use the 16 unique treatments for 4 factors to set the levels of the first 4
factors (A-D)
• Use interaction columns from the first 4 factors to set the levels of the
remaining 3 factors
Example
A
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
B
-1.00
-1.00
-1.00
-1.00
1.00
1.00
1.00
1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
1.00
1.00
C
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
D
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
AB
1.00
1.00
1.00
1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
1.00
1.00
AC
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
AD
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
BC
1.00
1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
1.00
1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
BD
-1.00
1.00
-1.00
1.00
1.00
-1.00
1.00
-1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
1.00
-1.00
CD
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
ABC
-1.00
-1.00
1.00
1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
-1.00
-1.00
1.00
1.00
ABD
1.00
-1.00
1.00
-1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
1.00
-1.00
ACD
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
BCD
1.00
-1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
1.00
-1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
ABCD
-1.00
1.00
1.00
-1.00
1.00
-1.00
-1.00
1.00
1.00
-1.00
-1.00
1.00
-1.00
1.00
1.00
-1.00
Example
• Would like to have as few short words as possible
• Why?
• How can we compare designs?
• Resolution
• Minimum aberration (MA):
Example
• Suppose have 7 factors, each at 2-levels, but only enough resources to
run 32 trials
• Can use a 27-2 fractional factorial design
• Which one is better?
– D1: I=ABCDF=ABCEG=DEFG
– D2: I=ABCF=ADEG=BCDEG
Example
• Suppose have 8 factors (A-H), each at 2-levels, but only enough
resources to run 32 trials
• Can use a 28-3 fractional factorial design
• Table 4A gives the minimum aberration (MA) designs for 8, 16, 32 and
64 runs
• From Table 4A.3, MA design gives:
– 6=123
– 7=124
– 8=1345
Example
• Table 4A.3, MA design is:
– 6=123
– 7=124
– 8=1345
• Design for our factors:
• Word length pattern:
Example
• Speedometer cables can be noisy because of shrinkage in the plastic
casing material
• An experiment was conducted to find out what caused shrinkage
• Engineers started with 6 different factors:
–
–
–
–
–
–
A
B
C
D
E
F
braiding tension
wire diameter
liner tension
liner temperature
coating material
melt temperature
Example
• Response is percentage shrinkage per specimen
• There were two levels of each factor
• A 26-2 fractional factorial
• The purpose of such an experiment is to determine which factors
impact the response
Example
• Constructing the design
– Write down the 16 run full factorial
– Use interaction columns to set levels of the other 2 factors
• Which interaction columns do we use?
• Table 4A.2 gives 16 run MA designs
– E=ABC; F=ABD
Example
A
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
C
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
D
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
AB
1
1
1
1
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
AC
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
AD
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
1
-1
BC
1
1
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
BD
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
1
-1
CD
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
ABC
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
ABD
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
ACD
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
-1
1
1
-1
BCD
1
-1
-1
1
-1
1
1
-1
1
-1
-1
1
-1
1
1
-1
ABCD
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
-1
1
1
-1
Example
• Results
A
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
B
-1
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
C
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
D
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
E
-1
-1
1
1
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
F
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
Y
0.611
0.701
0.359
0.439
0.331
0.271
0.499
0.439
0.194
0.189
0.191
0.251
0.239
0.344
0.131
0.211
Example
• Which effects can we estimate?
• Defining Contrast Sub-Group: I=ABCE=ABDF=CDEF
• Word-Length Pattern:
• Resolution:
Example
• Effect Estimates and QQ-Plot:
Estimate
-0.238
-0.059
-0.045
-0.036
-0.144
-0.053
0.084
0.000
-0.024
0.069
0.020
-0.004
0.013
0.010
-0.006
E
s
t
i
m
a
e
d
f
c
t
s
-0.25 -0.2 -0.15 -0.1 -0.5 0. 0.5
Effect
A
B
C
D
E
F
AB=CE=DF
AC=BE
AD=BF
AE=BC
AF=BD
CD=EF
CF=DE
ACF=BEF=BCD=ADE
ACD=BDE=BCF=AEF
- 1
0
1
Q u a n tile s
Comments
• Use defining contrast subgroup to determine which effects to estimate
• Can use qq-plot or Lenth’s method to evaluate the significance of the
effects
• Fractional factorial designs allow you to explore many factors in
relatively few trials
• Trade-off run-size for information about interactions