Page 1FRACTIONAL FACTORIAL () STUDIES#:•;Motivation: For factors, even gets big fast:#:for :œ"!#œ"!#%:Example Hendrix 1979 Chemtech A Coating Roll Temp 115° vs 125° B Solvent Recycled vs Refined C Polymer X-12 Preheat No vs Yes D Web Type LX-14 vs LB-17 E Coating Roll Tension 30 vs 40 F Number of Chill Rolls 1 vs 2 G Drying Roll Temp 75° vs 80° H Humidity of Air Feed 75% vs 90% J Feed Air to Dryer Preheat Yes vs No K Dibutylfutile in Formula 12% vs 15% L Surfactant in Formula .5% vs 1% M Dispersant in Formula .1% vs .2% N Wetting Agent in Formula 1.5% vs 2.5% O Time Lapse 10min vs 30min P Mixer Agitation Speed 100rpm vs 250rpm a measure of product cold crack resistanceCœ !!!!!!#œ$#ß(')"&"Solution": Collect data for only some (a fraction) of all possible combinations of levels of the factors.Page 2Qualitative Points That Ought to be "Obvious" +:riori: necessary information loss (relative to the full factorial)ì some ambiguity inevitable because of the lossì careful planning and wise analysis needed to hold this to a minimumìExample (hypothetical) ... a half fraction of a factorial##‚##•"(1) a b ab Factor A Factor B ( ) − ( ) + ( ) + ( ) −Page 3Example (hypothetical) a ... Suppose that factorial effects and##$•"$combination means are as below:.!"#!"!#"#á#########œ"!œ$œ"œ#œ#œ!œ!, , , , , , ,!"####œ!a 8 µ = b 4 µ = Factor A Factor B ( ) − ( ) + ( ) + ( ) − ab 14 µ = (1) 6 µ = bc 8 µ = abc 18 µ = ac 12 µ = c 10 µ = Factor C ( ) + ( ) − Suppose further that one gets data adequate to essentially reveal the meanresponses for combinations a, b, c and abc (the corners circled above) but%has no data on the other combinations. "right face average""grand average"!#œ•A "half-fraction version" of this might be "available right face average""available grand average"!#*œ• œ"$•"! œ$!!!!! Here !!!!!##*œSomething for nothing?Page 4A similar calculation for the C main effect however gives: "available back face average""available grand average"##*œ• œ%???? ####*ÁThe general story behind this situation is that for this fractional factorial#$•" and !!"###!"########**œ€œ€Confounding/aliasing ... ambiguityPage 5Issues to be Addressed in Order to Use Fractional Factorials#:•;:ì#how to rationally choose out of combinations for study"#:;ì how to determine the corresponding aliasing/confoundingpatternì how to do data analysisFirst consider these in the context of half fractions ... then for general .;Choice of standard half fractions of factorials#::Write out signs for specifying levels for all possible combinations of the"first" factors. Then "multiply" these together for a given:•"combination of the "first" factors to arrive at a corresponding level to usefor the "last" factor.Example ()#%•"With two-level factors A, B, C and D one proceeds as per%ABCProduct (used for D)Combination(1)adbdabcdacbcabcd••••€••€•€•€€€••••€€€•€••€€•€€€€Page 6Example Snee in 1985 ASQC Technical Supplement ЕÑЀÑA Solvent/Reactant low vs high B Catalyst/Reactant .025 vs .035 C Temperature 150° vs 160° D Reactant Purity 92% vs 96% E pH of Reactant 8.0 vs 8.7 color indexCœcombinationcombinationedaadebbdeabeabdccdeaceacdbcebcdabcabcdeCC•Þ'$'Þ(*#Þ&"'Þ%(•#Þ')$Þ%&•"Þ''&Þ')#Þ!'&Þ##"Þ##*Þ$)•#Þ!*%Þ$!"Þ*$%Þ!&These are data from half of all combinations of levels of each of the $##&factors (half of all possible labels of combinations based on the letters&a,b,c,d and e are given above, namely those involving an odd number ofletters).Snee followed the standard recommendation for choosing the half fractionPage 7Determining the "alias structure" of the half fraction (the implied patternof ambiguities):Use a method of formal multiplication, beginning from a so-called"generator" that represents the way in which the half fraction waschosen. The generator is of the formname of "last" factorproduct of names of "first" factorsÇThe rules of multiplication are that • letterIthe same letter‚Ç • lettersame letterI‚ÇExample (the numerical example used above)#$•"The generator here isCABÇWe can multiply through by C to obtain the so called "defining relation"IABCÇThis first says that the ABC factor interaction is aliased with the$ !"####grand mean. That is, only.!"# á###€can be estimated, not alone.!"####Multiplying through the defining relation by any set of lettersofinterestproduces a statement of what effect(s) are "aliasedwith" thecorrespondingeffect. For example, we see thatABCÇ(read "the A main effect is aliased withthe BC 2 factor interaction).SimilarlyCABÇPage 8as was illustrated earlier.In fact, the whole alias structure is IABCÇ ABCÇ BACÇ CABÇ#%$ effects are aliased in pairs.The technicalmeaning of aliasing is that only sums of effects can beestimated,not individual effects.Example (the again)#%•"With the generatorDABCÇthe defining relation isIABCDÇFrom this, e.g., we see that the AB factor interaction is aliased with the CD#2-factor interaction.Example Snee's study had generator#&•"EABCDÇand hence defining relationIABCDEÇFrom this one sees, e.g., that the AB -factor interaction is aliased with the#CDE -factor interaction.$Page 9Data Analysis for Standard Half Fractions:Initially temporarily ignore the "last" factor and treating the data as afull factorial in the "first" factors, judge the statistical significance:•"and practical importance of estimates derived from the Yates algorithm.Then interpret these estimates in light of the alias structure as estimates ofappropriate sums of effects.#:Where there is some replication (not all sample sizes are 1) confidence#:•"intervals can be made for the (sums of) effects.effect ^12„>†=€€€€â""""8888pooledabab:•"Ð"ÑËwhere=œ8•"=8•"pooledcombinationcombinationcombination##!abab!aband the appropriate degrees of freedom for are>"ab8•"œ8•#combination:•"Lacking any replication, normal plotting of the output of the Yates algorithm(ignoring the "last" factor) can be used in judging statistical significance.Page 10Example (another hypothetical )#$•"Suppose , , , , –8œ"Cœ&8œ#Cœ$=œ"Þ&8œ"Cœ#Þ&aabccbb#and , , .–8œ$Cœ&Þ&=œ"Þ)abcabcabc#Yates applied to: #Þ& & $