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II.4 Sixteen Run Fractional Factorial Designs Introduction Resolution Reviewed Design and Analysis Example: Five Factors Affecting Centerpost Gasket Clipping Time Example / Exercise: Seven Factors Affecting a Polymerization Process Discussion

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II.4 Sixteen Run Fractional Factorial Designs: Introduction With 16 runs, up to 15 Factors may be analyzed at Resolution III. 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 which factors have strong main effects These designs are very useful for “screening” situations: determine which factors have strong main effects 20% rule 20% rule

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II.4 Sixteen Run Designs: Resolution Reviewed Q: What is a Resolution III design? 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 Resolution III designs are the riskiest fractional factorial designs…but the most useful for screening –“damn the interactions….full speed ahead!”

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II.4 Sixteen Run Designs: Resolution Reviewed Q: What is a Resolution IV design? 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. Hence, in a Resolution IV design, if 3-way and higher interactions are negligible, all main effects are estimable with no confounding.

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II.4 Sixteen Run Designs: 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 confounded with other main effects or 2- or 3-way interactions, and 2-way interactions are not confounded with other 2- way 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.

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II.4 Sixteen Run Designs: Resolution Reviewed 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. 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.

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16 Run Signs Table

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II.4 Sixteen Run Designs Example: Five Factors Affecting Centerpost Gasket Clipping Time* y = clip time (secs) for 16 parts from the sprue (injector for liquid molding process) y = clip time (secs) for 16 parts from the sprue (injector for liquid molding process) Factors and levels -+ Factors and levels -+ –A: TableNoYes –B: ShakeNoYes –C: PositionSittingStanding –D: CutterSmallLarge –E: GripUnfoldFold *Contributed by Rodney Phillips (B.S. 1994), at that time working for Whirlpool. This was a STAT 506 (Intro. To DOE) project.

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Example: Five Factors Affecting Centerpost Gasket Clipping Time Design the Experiment: associate factors with carefully chosen columns in the 16- run signs matrix to generate a design matrix Design the Experiment: associate factors with carefully chosen columns in the 16- run 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)

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Example: Five Factors Affecting Centerpost Gasket Clipping Time I=ABCDEA=BCDEB=ACDEC=ABDED=ABCEE=ABCDAB=CDEAC=BDEAD=BCEAE=BCDBC=ADEBD=ACEBE=ACDCD=ABECE=ABDDE=ABC Full Alias Structure for the design E=ABCD

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Example: Five Factors Affecting Centerpost Gasket Clipping Time Completed Operator Report Form

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Example: Five Factors Affecting Centerpost Gasket Clipping Time Completed Signs Table with Estimated Effects

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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 2.032.26 2.70 2.70 3.37 3.37 -20 -10010 E=ABCD A=BCDE AC=BDE

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Example: Five Factors Affecting Centerpost Gasket Clipping Time The Normal Plot indicates three effects distinguishable from error. These are 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. Preliminary Interpretation

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

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Example: Five Factors Affecting Centerpost Gasket Clipping Time AC Interaction Table and Plot

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Example: Five Factors Affecting Centerpost Gasket Clipping Time AC Interaction Table and Plot

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Example: Five Factors Affecting Centerpost Gasket Clipping Time 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). 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). Interpretation

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Example: Five Factors Affecting Centerpost Gasket Clipping Time The effects of A (table) and C (position) seem to interact. The presence of a table reduces average clip time, but the reduction is larger (16.6 seconds) when the worker is standing than when he/she is sitting (4.0 seconds) The effects of A (table) and C (position) seem to interact. The presence of a table reduces average clip time, but the reduction is larger (16.6 seconds) when the worker is standing than when he/she is sitting (4.0 seconds) Interpretation

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II.4 Sixteen Run Designs Example / Exercise: Seven Factors Affecting a Polymerization Process y = blender motor maximum amp load y = blender motor maximum amp load Factors and levels -+ Factors and levels -+ –A: Mixing SpeedLoHi –B: Batch SizeSmallLarge –C: Final temp.LoHi –D: Intermed. Temp.LoHi –E: Addition sequence12 –F: Temp. of modiferLoHi –G: Add. Time of modifierLoHi Contributed by Solomon Bekele (Cryovac). This was part of a STAT 706 (graduate DOE) project.

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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

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Example / Exercise: Seven Factors Affecting a Polymerization Process Runs table Std OrderABCDE=ABCG=ACDF=BCD 1 21 11 3 1 1 1 411 11 5 1 111 61 1 1 7 11 1 8111 1 9 1 11 101 11 1 111 111 12111 13 111 14111 1 15111 1 161111111

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Example / Exercise: Seven Factors Affecting a Polymerization Process Determine the design’s alias structure 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 !

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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?)

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Example / Exercise: Seven Factors Affecting a Polymerization Process Determine the alias table: multiply the defining relation (rearranged alphabetically here) 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: by A for the second row: A = 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 = BCEFG and so on; after all seven main effects are done, start with two way interactions: 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.

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Example / Exercise: Seven Factors Affecting a Polymerization Process 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 Full Alias Structure for the 2 IV 7-3 design E = ABC, F = BCD, G = ACD

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

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Example / Exercise: Seven Factors Affecting a Polymerization Process Std OrderY (amps)ABCDE=ABCG=ACDF=BCD 1130 22321 11 31351 1 1 423511 11 5128 1 111 618411 1 713311 1 82491111 9130 1 11 102251 11 1 111431 111 12270111 13132 111 14198111 1 15138111 1 162491111111

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Example / Exercise: Seven Factors Affecting a Polymerization Process Completed Signs Table with Estimated Effects

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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

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0206080100-20 Example / Exercise: Seven Factors Affecting a Polymerization Process Solution: Normal Plot of Estimated Effects Ordered Effects: -11.1-9.4-7.9-6.1-1.9-1.62.43.14.67.47.69.416.924.196.6 A B

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Example / Exercise: Seven Factors Affecting a Polymerization Process 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 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. 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. None of the other factors seems to affect the motor’s max amp load. Suggested Interpretation

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II.4 Discussion 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. 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. 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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II.4 References Daniel, Cuthbert (1976). Applications of Statistics to Industrial Experimentation. New York: John Wiley & Sons, Inc. 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. 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. New York: John Wiley & Sons, Inc. Box, G.E.P., Hunter, W. G., and Hunter, J.S. (1978). Statistics for Experimenters. New York: John Wiley & Sons, Inc. Lochner, R.H. and Matar, J.E. (1990). Designing for Quality. Milwaukee: ASQC Quality Press. Lochner, R.H. and Matar, J.E. (1990). Designing for Quality. Milwaukee: ASQC Quality Press.

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DOX 6E Montgomery1 Design of Engineering Experiments Part 7 – The 2 k-p Fractional Factorial Design Text reference, Chapter 8 Motivation for fractional.

DOX 6E Montgomery1 Design of Engineering Experiments Part 7 – The 2 k-p Fractional Factorial Design Text reference, Chapter 8 Motivation for fractional.

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