II.4 Sixteen Run Fractional Factorial Designs Introduction Resolution Reviewed Design and...

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.


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.


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)


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


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


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


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


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


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


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)


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


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


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?)


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.


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


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


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


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


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


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


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.

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