ii.4 sixteen run fractional factorial designs introduction resolution reviewed design and...
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II.4 Sixteen Run II.4 Sixteen Run Fractional Factorial Fractional Factorial
DesignsDesigns IntroductionIntroduction Resolution ReviewedResolution Reviewed Design and AnalysisDesign and Analysis Example: Five Factors Affecting Example: Five Factors Affecting
Centerpost Gasket Clipping TimeCenterpost Gasket Clipping Time Example / Exercise: Seven Factors Example / Exercise: Seven Factors
Affecting a Polymerization ProcessAffecting a Polymerization Process DiscussionDiscussion
II.4 Sixteen Run II.4 Sixteen Run Fractional Factorial Fractional Factorial
Designs: IntroductionDesigns: Introduction With 16 runs, up to 15 Factors may be With 16 runs, up to 15 Factors may be
analyzed at Resolution III. analyzed at Resolution III. – Resolution IV is possible with 8 or fewer factors.Resolution IV is possible with 8 or fewer factors.– Resolution V is possible with 5 or fewer factors.Resolution V is possible with 5 or fewer factors.
These designs are very useful for These designs are very useful for ““screeningscreening”” situations: determine situations: determine whichwhich factors have factors have strong main effectsstrong main effects
20% rule20% rule
II.4 Sixteen Run II.4 Sixteen Run Designs: Designs:
Resolution ReviewedResolution Reviewed Q: What is a Resolution III design? Q: What is a Resolution III design?
– A: a design in which main effects are not A: a design in which main effects are not confounded with other main effects, but at least confounded with other main effects, but at least one main effect is confounded with a 2-way one main effect is confounded with a 2-way interactioninteraction
Resolution III designs are the riskiest fractional Resolution III designs are the riskiest fractional factorial designs…but the most useful for screening factorial designs…but the most useful for screening – ““damn the interactions….full speed ahead!damn the interactions….full speed ahead!””
II.4 Sixteen Run II.4 Sixteen Run Designs:Designs:
Resolution ReviewedResolution Reviewed Q: What is a Resolution IV design?Q: What is a Resolution IV design?– A: a design in which main effects are not A: a design in which main effects are not
confounded with other main effects or 2-way confounded with other main effects or 2-way interactions, but either (a) at least one main effect is interactions, but either (a) at least one main effect is confounded with a 3-way interaction, or (b) at least confounded with a 3-way interaction, or (b) at least one 2-way interaction is confounded with another 2-one 2-way interaction is confounded with another 2-way interaction.way interaction.
Hence, in a Resolution IV design, if 3-way and higher Hence, in a Resolution IV design, if 3-way and higher interactions are negligible, all main effects are interactions are negligible, all main effects are estimable with no confounding.estimable with no confounding.
II.4 Sixteen Run II.4 Sixteen Run Designs: Designs:
Resolution Reviewed Resolution Reviewed Q: What is a Resolution V design?Q: What is a Resolution V design?– A: a design in which main effects are not A: a design in which main effects are not
confounded with other main effects or 2- or 3-confounded with other main effects or 2- or 3-way interactions, and 2-way interactions are not way interactions, and 2-way interactions are not confounded with other 2-way interactions. confounded with other 2-way interactions. There is either (a) at least one main effect There is either (a) at least one main effect confounded with a 4-way interaction, or (b) at confounded with a 4-way interaction, or (b) at least one 2-way interaction confounded with a 3-least one 2-way interaction confounded with a 3-way interaction.way interaction.
II.4 Sixteen Run II.4 Sixteen Run Designs: Designs:
Resolution Reviewed Resolution Reviewed Hence, in a Resolution V design, if 3-way Hence, in a Resolution V design, if 3-way
and higher interactions are negligible, all and higher interactions are negligible, all main effects and 2-way interactions are main effects and 2-way interactions are estimable with no confounding.estimable with no confounding.
16 Run Signs Table16 Run Signs Table
ActualOrder y A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD
-1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 11 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1
-1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -11 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1
-1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -11 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1
-1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 11 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1
-1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -11 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1
-1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 11 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1
-1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 11 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1
-1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -11 1 1 1 1 1 1 1 1 1 1 1 1 1 1
SumDivisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect
II.4 Sixteen Run DesignsII.4 Sixteen Run DesignsExample: Five Factors Example: Five Factors
Affecting Centerpost Gasket Clipping Time*Affecting Centerpost Gasket Clipping Time* y = clip time (secs) for 16 parts from the sprue (injector y = clip time (secs) for 16 parts from the sprue (injector
for liquid molding process) for liquid molding process) Factors and levels Factors and levels -- ++
– A: TableA: Table NoNo YesYes– B: ShakeB: Shake NoNo YesYes– C: PositionC: Position SittingSitting StandingStanding– D: CutterD: Cutter SmallSmall LargeLarge– E: GripE: Grip UnfoldUnfold FoldFold
*Contributed by Rodney Phillips (B.S. 1994), at that time *Contributed by Rodney Phillips (B.S. 1994), at that time working for Whirlpool. This was a STAT 506 (Intro. To DOE) working for Whirlpool. This was a STAT 506 (Intro. To DOE) projectproject..
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping TimeAffecting Centerpost Gasket Clipping Time
Design the Experiment: associate factors with carefully Design the Experiment: associate factors with carefully chosen columns in the 16-run signs matrix to generate a chosen columns in the 16-run signs matrix to generate a design matrixdesign matrix– Always associate A, B, C, D with the first four columnsAlways associate A, B, C, D with the first four columns– With five factors, E = ABCD is universally With five factors, E = ABCD is universally
recommended (or E= -ABCD)recommended (or E= -ABCD)
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping TimeAffecting Centerpost Gasket Clipping Time
I=ABCDEI=ABCDE
A=BCDEA=BCDE
B=ACDEB=ACDE
C=ABDEC=ABDE
D=ABCED=ABCE
E=ABCDE=ABCD
AB=CDEAB=CDE
AC=BDEAC=BDE
AD=BCEAD=BCE
AE=BCDAE=BCD
BC=ADEBC=ADE
BD=ACEBD=ACE
BE=ACDBE=ACD
CD=ABECD=ABE
CE=ABDCE=ABD
DE=ABCDE=ABC
Full Alias Structure for the design E=ABCDFull Alias Structure for the design E=ABCD
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping TimeAffecting Centerpost Gasket Clipping Time
Completed Operator Report FormCompleted Operator Report FormStd. Actual A = B= C= D= E= y = Clip
Order Order Table Shake Position Cutter Grip Time (s)12 1 Yes Yes Sitting Large Unfold 46.3016 2 Yes Yes Standing Large Fold 27.353 3 No Yes Sitting Small Unfold 54.892 4 Yes No Sitting Small Unfold 40.054 5 Yes Yes Sitting Small Fold 28.82
13 6 No No Standing Large Fold 45.995 7 No No Standing Small Unfold 57.696 8 Yes No Standing Small Fold 29.499 9 No No Sitting Large Unfold 44.191 10 No No Sitting Small Fold 31.55
10 11 Yes No Sitting Large Fold 28.4711 12 No Yes Sitting Large Fold 29.168 13 Yes Yes Standing Small Unfold 36.017 14 No Yes Standing Small Fold 39.51
14 15 Yes No Standing Large Unfold 36.6015 16 No Yes Standing Large Unfold 52.41
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping TimeAffecting Centerpost Gasket Clipping Time
Completed Signs Table with Estimated EffectsCompleted Signs Table with Estimated Effects
ActualOrder
y =cliptime
A B C D AB AC AD BC BD CD ABC=DE
ABD=CE
ACD=BE
BCD=AE
ABCD=E
10 31.55 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 14 40.05 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -13 54.89 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -15 28.82 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 17 57.69 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -18 29.49 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1
14 39.51 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 113 36.01 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -19 44.19 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1
11 28.47 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 112 29.16 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 11 46.30 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -16 45.99 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1
15 36.60 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -116 52.41 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -12 27.35 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Sum 628.5 -82.4 0.40 21.6 -7.52 7.36 -50.0 16.24 -29.4 -0.48 6.88 10.72 26.96 -21.8 18.08 -107.8Divisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 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
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping TimeAffecting Centerpost Gasket Clipping Time
Normal Plot of Estimated EffectsNormal Plot of Estimated Effects
EffectsEffects
OrdereOrdered d
Effects:Effects:-13.48 -13.48 -10.28 -10.28 -6.25 -6.25 -3.68 -3.68 -2.72 -2.72
-0.94 -0.94 -0.06 -0.06
0.05 0.05 0.86 0.86 0.92 0.92 1.341.34
2.03 2.03 2.262.26
2.702.70 3.37 3.37
-20 -10 0 10
E=ABCD
A=BCDE
AC=BDE
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTime
The Normal Plot indicates three The Normal Plot indicates three effects distinguishable from error. effects distinguishable from error. These areThese are– E = ABCD (estimating E+ABCD)E = ABCD (estimating E+ABCD)– A = BCDE (estimating A+BCDE)A = BCDE (estimating A+BCDE)– AC = BDE (estimating AC+BDE), AC = BDE (estimating AC+BDE),
marginal.marginal.
Preliminary InterpretationPreliminary Interpretation
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTime
Since it is unusual for four-way Since it is unusual for four-way interactions to be active, the first interactions to be active, the first two are attributed to E and Atwo are attributed to E and A
Since A is active, the AC+BDE effect Since A is active, the AC+BDE effect is attributed to ACis attributed to AC– We should calculate an AC We should calculate an AC
interaction table and plotinteraction table and plot
Preliminary InterpretationPreliminary Interpretation
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTimeAC Interaction Table and PlotAC Interaction Table and Plot
C
1 2
31.55 57.6954.89 39.5144.19 45.9929.16 52.4139.95 48.90
1
40.05 29.4928.82 36.0128.47 36.6046.30 27.3535.91 32.26
A
2
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTimeAC Interaction Table and PlotAC Interaction Table and Plot
-1 = no
1 = yes
-1=sitting 1=standing
32
37
42
47
C=Position
A=TableInteraction Plot for y = clip time (s)
-1 = no
1 = yes
-1=sitting 1=standing
32
37
42
47
C=Position
A=TableInteraction Plot for y = clip time (s)
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTime
E = -13.5. Hence, the clip time is E = -13.5. Hence, the clip time is reduced an average of about 13.5 reduced an average of about 13.5 seconds when the worker uses the seconds when the worker uses the low level of E (the folded grip, as low level of E (the folded grip, as opposed to the unfolded grip). This opposed to the unfolded grip). This seems to hold regardless of the seems to hold regardless of the levels of other factors (E does not levels of other factors (E does not seem to interact with anything).seem to interact with anything).
InterpretationInterpretation
Example: Five Factors Example: Five Factors Affecting Centerpost Gasket Clipping Affecting Centerpost Gasket Clipping
TimeTime
The effects of A (table) and C The effects of A (table) and C (position) seem to interact. The (position) seem to interact. The presence of a table reduces presence of a table reduces average clip time, but the average clip time, but the reduction is larger (16.6 seconds) reduction is larger (16.6 seconds) when the worker is standing than when the worker is standing than when he/she is sitting (4.0 when he/she is sitting (4.0 seconds)seconds)
InterpretationInterpretation
II.4 Sixteen Run DesignsII.4 Sixteen Run DesignsExample / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a
Polymerization ProcessPolymerization Process
y = blender motor maximum amp loady = blender motor maximum amp load Factors and levels Factors and levels -- ++
– A: Mixing SpeedA: Mixing Speed LoLo HiHi– B: Batch SizeB: Batch Size SmallSmall LargeLarge– C: Final temp.C: Final temp. LoLo HiHi– D: Intermed. Temp.D: Intermed. Temp. LoLo HiHi– E: Addition sequenceE: Addition sequence 11 22– F: Temp. of modiferF: Temp. of modifer LoLo HiHi– G: Add. Time of modifierG: Add. Time of modifier LoLo HiHi
Contributed by Solomon Bekele (Cryovac). This was part of a STAT Contributed by Solomon Bekele (Cryovac). This was part of a STAT 706 (graduate DOE) project.706 (graduate DOE) project.
Example / Exercise: Seven Factors Example / Exercise: Seven Factors Affecting a Polymerization ProcessAffecting a Polymerization Process
Design the Experiment: associate additional Design the Experiment: associate additional factors with columns of the 16-run signs matrix factors with columns of the 16-run signs matrix
For 6, 7, or 8 factors, we assign the For 6, 7, or 8 factors, we assign the additional factors to the 3-way interaction additional factors to the 3-way interaction columnscolumns
For this 7-factor experiment, the following For this 7-factor experiment, the following assignment was usedassignment was used
E = ABC, F = BCD, G = ACDE = ABC, F = BCD, G = ACD
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Runs tableRuns tableStd Order A B C D E=AB
CG=ACD F=BCD
1 -1 -1 -1 -1 -1 -1 -1
2 1 -1 -1 -1 1 1 -1
3 -1 1 -1 -1 1 -1 1
4 1 1 -1 -1 -1 1 1
5 -1 -1 1 -1 1 1 1
6 1 -1 1 -1 -1 -1 1
7 -1 1 1 -1 -1 1 -1
8 1 1 1 -1 1 -1 -1
9 -1 -1 -1 1 -1 1 1
10 1 -1 -1 1 1 -1 1
11 -1 1 -1 1 1 1 -1
12 1 1 -1 1 -1 -1 -1
13 -1 -1 1 1 1 -1 -1
14 1 -1 1 1 -1 1 -1
15 -1 1 1 1 -1 -1 1
16 1 1 1 1 1 1 1
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Determine the designDetermine the design’’s alias structures alias structure– There will again be 16 rows in the full alias There will again be 16 rows in the full alias
table, but now 2table, but now 277 = 128 effects (including = 128 effects (including I)! Each row of the full table will have 8 I)! Each row of the full table will have 8 confounded effects! Here is how to start: confounded effects! Here is how to start: find the full defining relation:find the full defining relation:
– Since E = ABC, we have I = ABCE.Since E = ABC, we have I = ABCE.– But also F = BCD, so I = BCDFBut also F = BCD, so I = BCDF– Likewise G = ACD, so I = ACDGLikewise G = ACD, so I = ACDG– Likewise I = I x I = (ABCE)(BCDF) = ADEF !Likewise I = I x I = (ABCE)(BCDF) = ADEF !
Example / Exercise: Seven Factors Example / Exercise: Seven Factors Affecting a Polymerization ProcessAffecting a Polymerization Process
Continue in this fashion until you findContinue in this fashion until you find I = ABCE = BCDF = ACDG = ADEF = BDEG = ABFG = I = ABCE = BCDF = ACDG = ADEF = BDEG = ABFG =
CEFGCEFG
We have verified that this design is We have verified that this design is of Resolution IV (why?)of Resolution IV (why?)
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Determine the alias table: multiply the defining relation Determine the alias table: multiply the defining relation (rearranged alphabetically here)(rearranged alphabetically here)
I = ABCE = ABFG = ACDG = ADEF = BCDF = BDEG = CEFG I = ABCE = ABFG = ACDG = ADEF = BCDF = BDEG = CEFG by A for the second row:by A for the second row:
A = BCE = BFG = CDG = DEF = ABCDF = ABDEG = ACEFGA = BCE = BFG = CDG = DEF = ABCDF = ABDEG = ACEFG by B for the third row:by B for the third row:
B = ACE = AFG = ABCDG = ABDEF = CDF = DEG = BCEFGB = ACE = AFG = ABCDG = ABDEF = CDF = DEG = BCEFG and so on; after all seven main effects are done, start and so on; after all seven main effects are done, start
with two way interactions:with two way interactions:
AB = CE = FG = BCDG = BDEF = ACDF = ADEG = ABCEFGAB = CE = FG = BCDG = BDEF = ACDF = ADEG = ABCEFG
and so on...(what a pain!)…until you have 16 rows.and so on...(what a pain!)…until you have 16 rows.
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFGI + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG
A + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFGA + BCE + BFG + CDG + DEF + ABCDF + ABDEG + ACEFG
B + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFGB + ACE + AFG + CDF + DEG + ABCDG + ABDEF + BCEFG
C + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEGC + ABE + ADG + BDF + EFG + ABCFG + ACDEF + BCDEG
D + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFGD + ACG + AEF + BCF + BEG + ABCDE + ABDFG + CDEFG
E + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEFE + ABC + ADF + BDG + CFG + ABEFG + ACDEG + BCDEF
F + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFGF + ABG + ADE + BCD + CEG + ABCEF + ACDFG + BDEFG
G + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFGG + ABF + ACD + BDE + CEF + ABCEG + ADEFG + BCDFG
AB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFGAB + CE + FG + ACDF + ADEG + BCDG + BDEF + ABCEFG
AC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEGAC + BE + DG + ABDF + AEFG + BCFG + CDEF + ABCDEG
AD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFGAD + CG + EF + ABCF + ABEG + BCDE + BDFG + ACDEFG
AE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEFAE + BC + DF + ABDG + ACFG + BEFG + CDEG + ABCDEF
AF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFGAF + BG + DE + ABCD + ACEG + BCEF + CDFG + ABDEFG
AG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFGAG + BF + CD + ABDE + ACEF + BCEG + DEFG + ABCDFG
BD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFGBD + CF + EG + ABCG + ABEF + ACDE + ADFG + BCDEFG
ABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFGABD + ACF + AEG + BCG + BEF + CDE + DFG + ABCDEFG
Full Alias Structure for the 2Full Alias Structure for the 2IVIV7-37-3 design design
E = ABC, F = BCD, G = ACDE = ABC, F = BCD, G = ACD
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
I + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFGI + ABCE + ABFG + ACDG + ADEF + BCDF + BDEG + CEFG
Reduced Alias Structure (up to 2-way interactions) Reduced Alias Structure (up to 2-way interactions) for the 2for the 2IVIV
7-37-3 design E = ABC, F = BCD, G = ACD design E = ABC, F = BCD, G = ACD
AABBCCDDEEFFGG(***) ((***) ( three-way and higher ints.) three-way and higher ints.)
AB + CE + FGAB + CE + FGAC + BE + DGAC + BE + DGAD + CG + EFAD + CG + EFAE + BC + DFAE + BC + DFAF + BG + DEAF + BG + DEAG + BF + CDAG + BF + CDBD + CF + EGBD + CF + EG
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Std Order
Y (amps) A B C D E=ABC
G=ACD
F=BCD
1 130 -1 -1 -1 -1 -1 -1 -1
2 232 1 -1 -1 -1 1 1 -1
3 135 -1 1 -1 -1 1 -1 1
4 235 1 1 -1 -1 -1 1 1
5 128 -1 -1 1 -1 1 1 1
6 184 1 -1 1 -1 -1 -1 1
7 133 -1 1 1 -1 -1 1 -1
8 249 1 1 1 -1 1 -1 -1
9 130 -1 -1 -1 1 -1 1 1
10 225 1 -1 -1 1 1 -1 1
11 143 -1 1 -1 1 1 1 -1
12 270 1 1 -1 1 -1 -1 -1
13 132 -1 -1 1 1 1 -1 -1
14 198 1 -1 1 1 -1 1 -1
15 138 -1 1 1 1 -1 -1 1
16 249 1 1 1 1 1 1 1
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Completed Signs Table with Estimated EffectsCompleted Signs Table with Estimated Effects
ActualOrder
y =maxamps
A B C D AB=CE=FG
AC=BE=DG
AD=CG=EF
BC=AE=DF
BD=CF=EG
CD=AG=BF
ABC=E
ABD ACD=G
BCD=F
ABCD=AF=BG=DE
Un- 130 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1known 232 1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1
135 -1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1235 1 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 1128 -1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1184 1 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1133 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 1249 1 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1130 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1225 1 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 1143 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1270 1 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -1132 -1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1198 1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1138 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1249 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Sum 2911 773 193 -89 59 135 -75 25 61 37 -13 75 19 -15 -63 -49Divisor 16 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8Effect 181.9 96.6 24.1 -11.1 7.4 16.9 -9.4 3.1 7.6 4.6 -1.6 9.4 2.4 -1.9 -7.9 -6.1
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Analyze the Experiment: as an exercise,Analyze the Experiment: as an exercise,– construct and interpret a Normal construct and interpret a Normal
probability plot of the estimated effects; probability plot of the estimated effects; – if any 2-way interactions are if any 2-way interactions are
distinguishable from error, construct distinguishable from error, construct interaction tables and plots for these;interaction tables and plots for these;
– provide interpretationsprovide interpretations
EffectsEffects0 20 60 80 100-20
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
Solution: Normal Plot of Estimated EffectsSolution: Normal Plot of Estimated Effects
OrdereOrdered d
Effects:Effects:-11.1-11.1
-9.4-9.4-7.9-7.9-6.1-6.1-1.9-1.9-1.6-1.62.42.43.13.14.64.67.47.47.67.69.49.4
16.916.924.124.196.696.6
AB
Example / Exercise: Seven Factors Affecting a Example / Exercise: Seven Factors Affecting a Polymerization ProcessPolymerization Process
The effect of mixing speed is A = 96.6 amps. The effect of mixing speed is A = 96.6 amps. Hence, when we change the mixing speed from Hence, when we change the mixing speed from its low setting to its high setting, we expect the its low setting to its high setting, we expect the motormotor’’s max amp load to increase by about 97 s max amp load to increase by about 97 amps.amps.
The effect of batch size is B = 24.1 amps. The effect of batch size is B = 24.1 amps. Hence, when we change the batch size from Hence, when we change the batch size from small to large, we expect the motorsmall to large, we expect the motor’’s max amp s max amp load to increase by about 24 amps.load to increase by about 24 amps.
None of the other factors seems to affect the None of the other factors seems to affect the motormotor’’s max amp load.s max amp load.
Suggested InterpretationSuggested Interpretation
II.4 DiscussionII.4 Discussion As in 8-run designs, we can always As in 8-run designs, we can always ““fold overfold over””
a 16 run fractional factorial design. There are a 16 run fractional factorial design. There are several variations on this technique; in several variations on this technique; in particular, for any 16-run Resolution III design, particular, for any 16-run Resolution III design, it is always possible to add 16 runs in such a it is always possible to add 16 runs in such a way that the pooled design is Resolution IV.way that the pooled design is Resolution IV.
There are a great many other fractional There are a great many other fractional factorial designs; in particular, the Plackett-factorial designs; in particular, the Plackett-Burman designs have runs any multiple of Burman designs have runs any multiple of four (4,8,12,16,20, etc.) up to 100, and in n four (4,8,12,16,20, etc.) up to 100, and in n runs can analyze (n-1) Factors at Resolution III.runs can analyze (n-1) Factors at Resolution III.
II.4 ReferencesII.4 References Daniel, Cuthbert (1976). Daniel, Cuthbert (1976). Applications of Applications of
Statistics to Industrial ExperimentationStatistics to Industrial Experimentation. New . New York: John Wiley & Sons, Inc.York: John Wiley & Sons, Inc.
Box, G.E.P. and Draper, N.R. (1987). Box, G.E.P. and Draper, N.R. (1987). Empirical Empirical Model-Building and Response SurfacesModel-Building and Response Surfaces. New . New York: John Wiley & Sons, Inc.York: John Wiley & Sons, Inc.
Box, G.E.P., Hunter, W. G., and Hunter, J.S. Box, G.E.P., Hunter, W. G., and Hunter, J.S. (1978). (1978). Statistics for ExperimentersStatistics for Experimenters. New York: . New York: John Wiley & Sons, Inc.John Wiley & Sons, Inc.
Lochner, R.H. and Matar, J.E. (1990). Lochner, R.H. and Matar, J.E. (1990). Designing Designing for Qualityfor Quality. Milwaukee: ASQC Quality Press.. Milwaukee: ASQC Quality Press.