FRACTIONAL FACTORIAL (

Transcription

1 FRACTIONAL FACTORIAL ( # : ; ) STUDIES Page 1 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 Cœa measure of product cold crack resistance "& # œ $#ß(')!!!!!! "Solution": Collect data for only some (a fraction) of all possible s of levels of the factors.

2 ( ) Factor B Qualitative Points That Ought to be "Obvious" +:riori: Page 2 ì necessary information loss (relative to the full factorial) ì some ambiguity inevitable because of the loss ì careful planning and wise analysis needed to hold this to a minimum # " Example (hypothetical) #... a half fraction of a # # factorial ( + ) b ab (1) a ( ) Factor A ( + )

3 ( ) $ " $ Example (hypothetical) a #... Suppose that # factorial effects and means are as below: Page 3. á œ"!,!# œ$, "# œ", ## œ#,!"## œ#,!### œ!, "### œ!,!"# ### œ! µ bc = 8 µ abc = 18 ( + ) Factor B µ b = 4 µ (1) = 6 ( ) µ c = 10 Factor A µ ab = 14 µ a = 8 ( + ) ( ) µ ac = 12 Factor C ( + ) Suppose further that one gets data adequate to essentially reveal the mean responses for s a, b, c and abc (the % corners circled above) but has no data on the other s.! # œ "right face average" "grand average" A "half-fraction version" of this might be! * # œ "available right face average" "available grand average" œ"$ "! œ$ # * œ #!!!!! Here!!!!! Something for nothing?

4 A similar calculation for the C main effect however gives: Page 4 # * # œ "available back face average" "available grand average" œ%???? ## * Á ## The general story behind this situation is that for this * * # # ## # # ##! œ! "# and # œ #!" # $ " fractional factorial Confounding/aliasing... ambiguity

5 Issues to be Addressed in Order to Use # : ; Fractional Factorials: Page 5 " ì how to rationally choose #; out of # s for study ì how to determine the corresponding aliasing/confoundingpattern ì how to do data analysis : First 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 s of the "first" : " factors. Then "multiply" these together for a given of the "first" factors to arrive at a corresponding level to use for the "last" factor. Example (# % " ) With % two-level factors A, B, C and D one proceeds as per A B C Product (used for D) Combination (1) ad bd ab cd ac bc abcd

6 Example Snee in 1985 ASQC Technical Supplement Page 6 Ð Ñ Ð Ñ A 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 Cœ color index e a b abe c ace bce abc C Þ'$ #Þ&" #Þ') "Þ'' #Þ!' "Þ## #Þ!* "Þ*$ d ade bde abd cde acd bcd abcde C 'Þ(* 'Þ%( $Þ%& &Þ') &Þ## *Þ$) %Þ$! %Þ!& These are data from half of all $# s of # levels of each of the & factors (half of all possible labels of s based on the & letters a,b,c,d and e are given above, namely those involving an odd number of letters). Snee followed the standard recommendation for choosing the half fraction

7 Page 7 Determining the "alias structure" of the half fraction (the implied pattern of ambiguities): Use a method of formal multiplication, beginning from a so-called "generator" that represents the way in which the half fraction was chosen. The generator is of the form name of "last" factor Ç The rules of multiplication are that letter IÇ the same letter letter same letterç I product of names of "first" factors Example (the # $ " numerical example used above) The generator here is C We can multiply through by C to obtain the so called "defining relation" I Ç Ç AB ABC This first says that the ABC $ factor interaction!"# ### is aliased with the grand mean. That is, only can be estimated, not!"# ### alone.. á!"#### Multiplying through the defining relation by any set of lettersofinterest produces a statement of what effect(s) are "aliasedwith" thecorresponding effect. For example, we see that A Ç (read "the A main effect is aliased withthe BC 2 factor interaction). Similarly BC C Ç AB

8 as was illustrated earlier. In fact, the whole alias structure is Page 8 I Ç ABC A Ç BC B Ç AC C Ç AB $ # 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 generator the defining relation is I D Ç Ç ABC ABCD From this, e.g., we see that the AB # factor interaction is aliased with the CD 2-factor interaction. Example Snee's and hence defining relation # & " study had generator E Ç ABCD I Ç ABCDE From this one sees, e.g., that the AB #-factor interaction is aliased with the CDE $ -factor interaction.

9 Data Analysis for Standard Half Fractions: Page 9 Initially temporarily ignore the "last" factor and treating the data as a full 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 of appropriate sums of # : effects. Where there is some replication (not all sample sizes are 1) confidence intervals can be made for the (sums of) effects. where # : " effect ^ 1 " " " " > = pooled â 2: " Ë Ð"Ñ a b ab # pooled = œ #!aa8 "= b b!a8 " b and the appropriate degrees of freedom for > are "a8 " b œ8 # : " Lacking any replication, normal plotting of the output of the Yates algorithm (ignoring the "last" factor) can be used in judging statistical significance.

10 Example (another hypothetical # $ " ) Page 10 Suppose 8,, a œ" Ca œ& 8b œ# Cb œ$, = b # œ"þ& 8c œ", Cc œ#þ& and, # 8 œ$ C œ&þ&, = œ"þ). abc abc abc Yates applied to: #Þ& & $ &Þ& # = œ! Ð# "Ñ"Þ&! Ð$ "Ñ"Þ) pooled! Ð# "Ñ! Ð$ "Ñ Intervals: effect ^ " " " " " > $ = pooled Ê # $ " " # " $

11 & " Page 11 Example Snee's # had no replication Þ Ignoring factor E temporarily, Yates can be applied to the "' responses exactly as listed earlier. The result is estimates as belowþ C estimate ("' divisor) e Þ'$ #Þ)(& a #Þ&" Þ)#$ b #Þ') "Þ#&$ abe "Þ'' Þ!&& c #Þ!' Þ$)% ace "Þ## Þ!'% bce #Þ!* Þ!%" abc "Þ*$ Þ!!" d 'Þ(* #Þ(*$ ade 'Þ%( Þ!*& bde $Þ%& Þ!%& abd &Þ') Þ#)) cde &Þ## Þ$"% acd *Þ$) Þ")' bcd %Þ$! Þ$!' abcde %Þ!& Þ)(" Normal plot the ( last "&) estimates...

12 Tentative engineering conclusion of Snee study: For uniform color index, attention must be paid to controlling/reducing variation in "st, Factor D, Reactant Purity #nd, Factor B, Catalyst/Reactant Ratio $rd, Factor E, ph of Reactant %th, Factor A, Solvent/Reactant Ratio Page 12

13 : : ; Page 13 Smaller (than half) Fractions of # Studies (# Fractional Factorials) ;œ" ;œ# ;œ$ etc. half fractions quarter fractions eighth fractions... Issues (still) À " : ì how to rationally choose #; of # possible sof levels of :#- level factors ì how to determine the corresponding aliasing/confoundingpattern ì how to do data analysis Answers: the natural generalizations of the half fraction answers just discussed

14 " # Choice of standard fractions of factorials: ; # : Page 14 Write out signs for specifying levels for all possible s of the "first" : ; factors. Pick ; different groups of the first : ; factors. Use the products of the signs corresponding to members of the groups to specify levels for the "last" ; factors. Example Best and Hanson1986 ASA Meeting Presentation development of a catalyst for producing ethyleneamines by theamination of monoethanolamine ÞÞÞ:œ& factors A Ni/Re Ratio #Î" vs #!Î" B Precipitant (NH%)CO # $ vs none C Calcining Temp $!! vs &!! D Reduction Temp $!! vs &!! E Support Used alpha-alumina vs silica alumina Cœ% water produced " ;œ# i.e. a % fraction contemplated... i.e. # œ) out of the # œ$# possible A, B, C, D, E s The (somewhat arbitrary) choice was made to use ABC signproducts to choose levels of D, and BC sign products to chooselevels of E. (Other choices are possible and lead to different aliasing patterns that might for some other studies be preferred bythe engineer in charge.) & # &

15 Page 15 A B C ABC Product (for D) BC Product (for E) Combination + e + ade bd ab cd ac bce abcde The last column specifies those 8 s actually used in the study. The data obtained were as below. e ade bd ab cd ac bce abcde C C = )Þ(!ß""Þ'!ß*Þ!! *Þ('( #Þ&%$ #'Þ)! #'Þ)!! #%Þ)) #%Þ))! $$Þ"& $$Þ"&! #)Þ*!ß$!Þ*) #*Þ*%! #Þ"'$ $!Þ#! $!Þ#!! )Þ!!ß)Þ'* )Þ$%& Þ#$) #*Þ$! #*Þ$!! #

16 Determining the "alias structure" of the " # ; fraction: Page 16 Use the method of formal multiplication, beginning from ; generators that " represent the way in which the # fraction was chosen. To find the ; defining relation (the list of all products 'equivalent to' I) first convert the generators to statements of products equivalent to I, and then multiply these in pairs, then in triples, then in sets of four, etc. The letter I will ; : ; have # " equivalent products... i.e. effects are aliased in # different groups of # ; each. Example Hanson and Best again further, multiplying these two we get i.e. D Ç ABC so I Ç ABCD E Ç BC so I Ç BCE I I Ç (ABCD) (BCE) I Ç ADE So the defining relation for the catalyst study is I Ç ABCD Ç BCE Ç ADE and therefore effects are aliased in ) groups of %. For example, multiplying through the defining relation by A gives A Ç BCD Ç ABCE Ç DE and we see that, for example, the A main effect is aliased with the DE # interaction. factor

17 Data analysis for standard # : ; studies: Page 17 Initially ignore the "last" ; factors, and treating the data as a full factorial in the "first" : ; factors, judge the statistical significance and practical importance of estimates produced by the Yates algorithm. Then interpret these in light of the alias structure as estimates of appropriate sums of # : effects. With some replication, confidence intervals can be made for the (sums of) effects and used in the process of judging statistical significance. where (as always) effect ^ 1 " " " " > = pooled â 2: ; Ë Ð"Ñ a b ab # pooled = œ #!aa8 "= b b!a8 " b and the appropriate degrees of freedom for > are "a8 " b œ8 # : ; Lacking any replication, one can normal plot estimates, looking for ones clearly of larger order of magnitude than the rest (and therefore larger than background noise as well).

18 Example Hanson and Best catalyst study Page 18 The ) sample means, C, listed before were in Yates standard order forfactors A, B and C (the "first" : ;œ$) ignoring D and E (the"last" ;œ#). So the Yates algorithm can be applied to them inthe order listed. C estimate sum estimated e *Þ('( #%Þ!%) grand mean aliases ade #'Þ)!! &Þ)"& A main effect aliases bd #%Þ))! Þ"#* B main effect aliases ab $$Þ"&! "Þ%*# AB interaction aliases cd #*Þ*%! Þ$** C main effect aliases ac $!Þ#!! Þ&"" AC interaction aliases bce )Þ$%& &Þ%*& BC interaction aliases abcde #*Þ$!! $Þ')# ABC interaction aliases statistical significance/detectability of these? # ( $ " )(#Þ&%$) (# ")(#Þ"'$) (# ")( Þ#$) ) = pooled œ œ"þ)(# ( $ " ) (# ") (# ") So = œ È pooled "Þ)(# œ"þ$'),and this can be used as ameasure of background noise and as a basic ingredient ofconfidence intervals for the sums of effects.

19 Page 19 = pooled has % associated degrees of freedom. So if ß e.g. ß*& %confidence intervals for the sums of effects are desired, the" / part" of the confidence interval formula becomes " " " " " " " " " #Þ(('("Þ$')) # $ Ê $ " " " # " # " i.e. "Þ"*& We therefore might judge any estimate larger in absolute valuethan "Þ"*& to represent a sum of effects clearly large enoughto see above the background experimental variation. Note the "detectable" sums are (in order of magnitude): Tentative interpretations? sum estimate!# "#$ ###!"#% #### $% ## "###!$ ## % #!"#$% #####!"#### $ #!% ## "#$%#### &Þ)"& &Þ%*& $Þ')#!" #$!#% "$% "Þ%*# ## ## ### ### A main effect?? E main effect?? D main effect?? or AE interaction???????? (happily much smaller than the other sums) (And there are other equally plausible interpretations of the 3 large sums!) In fact a follow-up study confirmed the importance of the D main effect.

20 Page 20 If the A (Ni/Re ratio) main effect, the E (Support Type) main effect and the D (Reduction Temp) main effect are indeed the most important determiners of C, and large C is desirable, the signs of the estimates indicate the need for "high A" (#!Î" Ni/Re ratio), "low E" (alpha-alumina support) and "high D" (&!! reduction temp). Notice!!!! The larger ;, the larger the inevitable ambiguity of interpretation of the fractional factorial results and the more likely the need for follow-up study. Small fraction are really most useful as screening studies, to pick a few likely candidates out of many potentially important factors for subsequent more detailed study. End with an extreme example of large ;, i.e. a small fraction. Example Hendrix Chemtech study mentioned at the beginning Cœcold crack resistance of a product :œ"& factors A, B, C, D, E, F, G, H, J, K, L, M, N, O, P (factor names and levels given earlier) % : ;œ%, i.e., only # œ"' s were run!!!!! " " #"& % #!%) This was a œ fraction!!!! The 11 generators used were: E Ç ABCD F Ç BCD GÇ ACD H Ç ABC J Ç ABD K Ç CD L Ç BD M Ç AD N ÇBC O ÇAC P Ç AB These led to the "' s and (ultimately) the databelow:

21 Page 21 eklmnop aghjkln bfhjkmo abefgkp cfghlmp acefjlo bcegjmn abchnop dfgjnop adefhmn bdeghlo abdjlmp cdehjkp acdgkmo bcdfkln abcdefghjklmnop C "%Þ) "'Þ$ #$Þ& #$Þ* "*Þ' ")Þ' ##Þ$ ##Þ# "(Þ) ")Þ* #$Þ" #"Þ) "'Þ' "'Þ( #$Þ& #%Þ* Pretty clearly it isn't sensible to write out the whole definingrelation here... "" effects are going to be aliased in "' groupsof # œ#!%) effects. But for a most tentative interpretation, let's see what wemight glean if the physical system is so simple that only main effects dominate.(physically reasonable???? Ask theengineer, not the statistician!) The "' observations are listed in Yates order for factors A,B,C and D (ignoring the rest). We therefore begin by running them through the Yates algorithm, with the results below.

22 C estimate ("' divisor) sum estimated eklmnop "%Þ) #!Þ#) grand mean â aghjkln "'Þ$ Þ"$ A â bfhjkmo #$Þ& #Þ)( B â abefgkp #$Þ* Þ!) AB P â cfghlmp "*Þ' Þ#( C â acefjlo ")Þ' Þ!) AC O â bcegjmn ##Þ$ Þ"* BC N â abchnop ##Þ# Þ$' ABC H â dfgjnop "(Þ) Þ"$ D â adefhmn ")Þ* Þ!$ AD M â bdeghlo #$ Þ" Þ!% BD L â abdjlmp #"Þ) Þ!' ABD J â cdehjkp "'Þ' Þ#' CD K â acdgkmo "'Þ( Þ#* ACD G â bcdfkln #$Þ& "Þ!' BCD F â abcdefghjklmnop #%Þ* Þ"" ABCD E â Page 22 There is no replication in this data set... so we're driven tonormal plotting in order to judge statistical significance of theseestimates.

23 A normal plot of the ( last 15) estimates is: Page 23 Tentative interpretation: The most important factors appearto be B (Solvent) and F (# of Chill Rolls) and for large cold crack resistance "high B" (refined solvent)and "high F" (# chill rolls) appear best. (Note that the analysis does point out what is in retrospect quite obvious, namely that it is those s in the data set with "high B" and "high F" that have the largest C's.)