









With 16 runs, up to 15 Factors may be analyzed at Resolution III.
– Resolution IV is possible with 8 or fewer factors.
– Resolution V is possible with 5 or fewer factors.
These designs are very useful for “ screening ” situations: determine main effects
factors have strong
20% rule



Q: What is a Resolution III design?
– A: a design in which main effects are not confounded with other main effects, but at least one main effect is confounded with a 2-way interaction
Resolution III designs are the riskiest fractional factorial designs…but the most useful for screening
– “ damn the interactions….full speed ahead!
”



Q: What is a Resolution IV design?
– A: a design in which main effects are not confounded with other main effects or 2-way interactions, but either (a) at least one main effect is confounded with a 3-way interaction, or
(b) at least one 2-way interaction is confounded with another 2-way interaction.
Hence, in a Resolution IV design, if 3-way and higher interactions are negligible, all main effects are estimable with no confounding.


Q: What is a Resolution V design?
– A: a design in which main effects are not confounded with other main effects or 2- or 3-way interactions, and
2-way interactions are not confounded with other 2way interactions. There is either (a) at least one main effect confounded with a 4-way interaction, or (b) at least one 2-way interaction confounded with a 3-way interaction.


Hence, in a Resolution V design, if 3-way and higher interactions are negligible, all main effects and 2-way interactions are estimable with no confounding.

Actual
Order y
Sum
Divisor
Effect
16
A
-1
1
-1
1
-1
1
-1
1
1
-1
1
-1
1
-1
-1
1
8
C
1
1
1
1
-1
-1
-1
-1
-1
1
1
-1
-1
-1
1
1
8
B
-1
-1
1
1
-1
-1
1
1
1
-1
-1
-1
-1
1
1
1
8
AB
1
-1
-1
1
1
-1
-1
1
1
1
-1
1
-1
-1
-1
1
8
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
8
AD
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
8
AC
-1
1
-1
1
1
-1
1
-1
-1
-1
1
1
-1
1
-1
1
8
BC
-1
-1
1
1
-1
-1
1
1
-1
-1
-1
1
1
-1
1
1
8
BD
-1
-1
1
1
-1
-1
1
1
1
-1
-1
-1
-1
1
1
1
8
CD ABC ABD ACD BCD ABCD
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
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
-1
1
1
-1
1
-1
-1
1
-1
1
-1
-1
1
1
-1
1
8 8 8 8 8 8



Example: Five Factors
Affecting Centerpost Gasket Clipping Time* y = clip time (secs) for 16 parts from the sprue (injector for liquid molding process)
Factors and levels
– A: Table
– B: Shake
– C: Position
– D: Cutter
– E: Grip
-
No
No
Sitting
Small
Unfold
+
Yes
Yes
Standing
Large
Fold
*Contributed by Rodney Phillips (B.S. 1994), at that time working for Whirlpool. This was a STAT 506 (Intro. To
DOE) project .

Example: Five Factors
Affecting Centerpost Gasket Clipping Time

Design the Experiment: associate factors with carefully chosen columns in the 16run signs matrix to generate a design matrix
– Always associate A, B, C, D with the first four columns
– With five factors, E = ABCD is universally recommended (or E= -
ABCD)

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Full Alias Structure for the design E=ABCD
I=ABCDE
A=BCDE
B=ACDE
C=ABDE
D=ABCE
E=ABCD
AB=CDE
AC=BDE
AD=BCE
AE=BCD
BC=ADE
BD=ACE
BE=ACD
CD=ABE
CE=ABD
DE=ABC

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Std.
Order
12
16
3
2
4
13
5
1
10
6
9
11
8
7
14
15
Completed Operator Report Form
Actual
Order
1
2
3
4
10
11
8
9
5
6
7
12
13
14
15
16
A =
Table
Yes
Yes
No
Yes
Yes
No
No
Yes
No
No
Yes
No
Yes
No
Yes
No
B= C=
Shake Position
Yes Sitting
Yes Standing
Yes
No
Sitting
Sitting
Yes Sitting
No Standing
No Standing
No Standing
No Sitting
No
No
Sitting
Sitting
Yes Sitting
Yes Standing
Yes Standing
No Standing
Yes Standing
D=
Cutter
E= y = Clip
Grip Time (s)
Large Unfold
Large Fold
Small Unfold
Small Unfold
46.30
27.35
54.89
40.05
Small Fold
Large Fold
Small Unfold
Small Fold
Large Unfold
Small
Large
Fold
Fold
Large Fold
Small Unfold
Small Fold
Large Unfold
Large Unfold
28.82
45.99
57.69
29.49
44.19
31.55
28.47
29.16
36.01
39.51
36.60
52.41

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Completed Signs Table with Estimated Effects
Actual
Order y = clip time
A B C D AB AC AD BC BD CD ABC
=DE
ABD
=CE
ACD
=BE
BCD
=AE
ABCD
=E
6
15
16
2
9
11
12
1
7
8
14
13
10
4
3
5
31.55
40.05
54.89
28.82
57.69
29.49
39.51
36.01
44.19
28.47
29.16
46.30
45.99
36.60
52.41
27.35
-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
-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
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
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
-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
1
-1
1
-1
-1
1
-1
1
Sum
Divisor
628.5
16
-82.4
8
0.40
8
21.6
8
-7.52
8
7.36
8
-50.0
8
16.24
8
-29.4
8
-0.48
8
6.88
8
10.72
8
26.96
8
-21.8
8
18.08
8
-107.8
8
Effect 39.28
-10.3
0.05
2.70
-0.94
0.92
-6.25
2.03
-3.68
-0.06
0.86
1.34
3.37
-2.72
2.26
-13.48
-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

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Normal Plot of Estimated Effects
Ordered
Effects:
-13.48
-10.28
-6.25
-3.68
-2.72
-0.94
-0.06
0.05
0.86
0.92
1.34
2.03
2.26
2.70
3.37
AC=BDE
A=BCDE
E=ABCD
-20 -10
Effects
0 10

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Preliminary Interpretation

The Normal Plot indicates three effects distinguishable from error. These are
– E = ABCD (estimating E+ABCD)
– A = BCDE (estimating A+BCDE)
– AC = BDE (estimating AC+BDE), marginal.

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Preliminary Interpretation


Since it is unusual for four-way interactions to be active, the first two are attributed to E and A
Since A is active, the AC+BDE effect is attributed to AC
– We should calculate an AC interaction table and plot

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
AC Interaction Table and Plot
C
1 2
1
31.55
54.89
44.19
29.16
39.95
57.69
39.51
45.99
52.41
48.90
A
2
40.05
28.82
28.47
46.30
35.91
29.49
36.01
36.60
27.35
32.26

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
AC Interaction Table and Plot
Interaction Plot for y = clip time (s)
A=Table
-1 = no
1 = yes
47
42
37
32
-1=sitting
C=Position
1=standing

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Interpretation

E = -13.5. Hence, the clip time is reduced an average of about 13.5 seconds when the worker uses the low level of E (the folded grip, as opposed to the unfolded grip). This seems to hold regardless of the levels of other factors (E does not seem to interact with anything).

Example: Five Factors
Affecting Centerpost Gasket Clipping Time
Interpretation


Example / Exercise: Seven Factors Affecting a
Polymerization Process

 y = blender motor maximum amp load
Factors and levels -
– A: Mixing Speed
– B: Batch Size
Lo
Small
– C: Final temp.
– D: Intermed. Temp.
– E: Addition sequence
– F: Temp. of modifer
Lo
Lo
1
Lo
– G: Add. Time of modifier Lo
Hi
Hi
2
Hi
Hi
+
Hi
Large
Contributed by Solomon Bekele (Cryovac). This was part of a STAT
706 (graduate DOE) project.

Example / Exercise: Seven Factors Affecting a Polymerization Process
Design the Experiment: associate additional factors with columns of the 16-run signs matrix


For 6, 7, or 8 factors, we assign the additional factors to the 3-way interaction columns
For this 7-factor experiment, the following assignment was used
E = ABC, F = BCD, G = ACD

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Runs table
Std Order A B C D E=ABC G=ACD F=BCD
9
10
7
8
11
12
13
14
15
16
1
2
3
4
5
6
-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 -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
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


Example / Exercise: Seven Factors Affecting a
Polymerization Process
Determine the design ’ s alias structure
– There will again be 16 rows in the full alias table, but now 2 7 = 128 effects (including I)!
Each row of the full table will have 8 confounded effects! Here is how to start: find the full defining relation:
– Since E = ABC, we have I = ABCE.
– But also F = BCD, so I = BCDF
– Likewise G = ACD, so I = ACDG
– Likewise I = I x I = (ABCE)(BCDF) = ADEF !

Example / Exercise: Seven Factors Affecting a Polymerization Process


Continue in this fashion until you find
I = ABCE = BCDF = ACDG = ADEF = BDEG = ABFG = CEFG

We have verified that this design is of
Resolution IV (why?)

Example / Exercise: Seven Factors Affecting a
Polymerization Process

Determine the alias table: multiply the defining relation
(rearranged alphabetically here)

I = ABCE = ABFG = ACDG = ADEF = BCDF = BDEG = CEFG by A for the second row:
A = BCE = BFG = CDG = DEF = ABCDF = ABDEG = ACEFG
 by B for the third row:
B = ACE = AFG = ABCDG = ABDEF = CDF = DEG = BCEFG
 and so on; after all seven main effects are done, start with two way interactions:
AB = CE = FG = BCDG = BDEF = ACDF = ADEG = ABCEFG and so on...(what a pain!)…until you have 16 rows.

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Full Alias Structure for the 2
IV
7-3 design
E = ABC, F = BCD, G = ACD
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG +
CEFG
A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG
B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG
D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF
F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG
AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG
AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF
AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG
BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Reduced Alias Structure (up to 2-way interactions) for the 2
IV
7-3 design E = ABC, F = BCD, G = ACD
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG
C
D
E
A
B
AB + CE + FG
AC + BE + DG
AD + CG + EF
AE + BC + DF
AF + BG + DE
F AG + BF + CD
G BD + CF + EG
(***) (  three-way and higher ints.)

Example / Exercise: Seven Factors Affecting a
Polymerization Process
A B C D
11
12
13
14
15
16
Std Order Y (amps)
1
2
3
130
232
135
6
7
4
5
8
9
10
249
130
225
235
128
184
133
143
270
132
198
138
249
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
-1
1
-1
-1
-1
1
-1
1
-1
1
-1
1
-1
1
1
-1
1
1
-1
-1
1
-1
1
-1
E=ABC G=ACD F=BCD
-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
-1
1
-1
-1
-1
-1
1
1

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Completed Signs Table with Estimated Effects
Actual
Order y = max amps
Unknown
Sum
Divisor
Effect
225
143
270
132
198
138
249
130
232
135
235
128
184
133
249
130
2911
16
181.9
A B 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
773 193
8 8
-89
8
96.6
24.1
-11.1
-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
8
AB
=CE
=FG
AC
=BE
=DG
-1
-1
1
1
-1
-1
1
-1
-1
1
1
1
-1
-1
1
1
59 135
8
-1
1
-1
-1
1
-1
1
1
-1
1
1
1
-1
1
-1
-1
-75
8
7.4
16.9
-9.4
AD
=CG
=EF
25
8
3.1
1
-1
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
BC
=AE
=DF
61
8
7.6
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
37
8
-13
8
4.6
-1.6
-1
-1
-1
1
1
1
1
-1
-1
-1
-1
1
1
1
1
-1
BD
=CF
=EG
CD
=AG
=BF
ABC
=E
75
8
9.4
1
1
-1
1
-1
-1
1
-1
-1
1
-1
-1
1
1
-1
1
ABD ACD
=G
BCD
=F
-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
19
8
-15
8
-63
8
2.4
-1.9
-7.9
1
-1
-1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
1
1
ABCD
=AF
=BG
=DE
1
1
-1
1
-1
-1
1
1
1
-1
-1
1
-1
-1
1
-1
-49
8
-6.1

Example / Exercise: Seven Factors Affecting a
Polymerization Process

Analyze the Experiment: as an exercise,
– construct and interpret a Normal probability plot of the estimated effects;
– if any 2-way interactions are distinguishable from error, construct interaction tables and plots for these;
– provide interpretations

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Solution: Normal Plot of Estimated Effects
Ordered
Effects:
-11.1
-9.4
-7.9
-6.1
-1.9
-1.6
7.6
9.4
16.9
24.1
96.6
2.4
3.1
4.6
7.4
-20
B
A
0 20
Effects
60 80 100

Example / Exercise: Seven Factors Affecting a
Polymerization Process
Suggested Interpretation



The effect of mixing speed is A = 96.6 amps.
Hence, when we change the mixing speed from its low setting to its high setting, we expect the motor ’ s max amp load to increase by about 97 amps.
The effect of batch size is B = 24.1 amps.
Hence, when we change the batch size from small to large, we expect the motor ’ s max amp load to increase by about 24 amps.
None of the other factors seems to affect the motor ’ s max amp load.



As in 8-run designs, we can always “ fold over ” a
16 run fractional factorial design. There are several variations on this technique; in particular, for any 16-run Resolution III design, it is always possible to add 16 runs in such a way that the pooled design is Resolution IV.
There are a great many other fractional factorial designs; in particular, the Plackett-Burman designs have runs any multiple of four
(4,8,12,16,20, etc.) up to 100, and in n runs can analyze (n-1) Factors at Resolution III.

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Daniel, Cuthbert (1976). Applications of Statistics to Industrial Experimentation . New York: John
Wiley & Sons, Inc.
Box, G.E.P. and Draper, N.R. (1987). Empirical
Model-Building and Response Surfaces . New
York: John Wiley & Sons, Inc.
Box, G.E.P., Hunter, W. G., and Hunter, J.S.
(1978). Statistics for Experimenters
John Wiley & Sons, Inc.
. New York:
Lochner, R.H. and Matar, J.E. (1990). Designing for Quality . Milwaukee: ASQC Quality Press.