Stat 5303 (Oehlert): Fractional Factorials 1

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Lionel Miles
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1 Stat 5303 (Oehlert): Fractional Factorials 1 Cmd> gen<-matrix(vector(1,1,1,0,0,1,0,0,1,1),5) MacAnova has several commands for working with fractioned designs. For two-series fractions, we need to be able to specify the generators. The generator matrix has a column for each factor and a row for each generator. The elements can be±1 or 0. A nonzero element means that the factor is in the generator, and the product of the nonzero elements in a generator determines the sign of the generator. Cmd> gen Here we have ABC and ADE as generators. (1,1) (2,1) Cmd> doff2(basis:gen,ialiases:t) doff2() can take a generator matrix and find the aliases (of I). Statistics :: Design of Experiments :: Fractioning is a dialog-based front end to doff2(). (1) "I" (2) "ABC" (3) "ADE" (4) "BCDE" Cmd> doff2(basis:gen,allaliases:t) You can instead request that doff2() print the complete alias structure for the design. This can be a very long output. Aliases: (1) "I = ABC = ADE = BCDE" (2) "A = BC = DE = ABCDE" (3) "B = AC = ABDE = CDE" (4) "AB = C = BDE = ACDE" (5) "D = ABCD = AE = BCE" (6) "AD = BCD = E = ABCE" (7) "BD = ACD = ABE = CE" (8) "ABD = CD = BE = ACE" Cmd> doff2(k:7,p:4,ialiases:t,aber:t) You can also have doff2() choose a design. Here we ask for a design. We also ask for all the aliases of I and for the aberration. Aliases of I: (1) "I" (2) "ABD" (3) "ACE" (4) "BCDE" (5) "BCF" (6) "ACDF" (7) "ABEF" (8) "DEF" (9) "ABCG" (10) "CDG" (11) "BEG" (12) "ADEG" (13) "AFG" (14) "BDFG" (15) "CEFG"

2 Stat 5303 (Oehlert): Fractional Factorials 2 (16) "ABCDEFG" Aberration: (1) (6) 0 1 Cmd> gen<-matrix(vector( 0,1,1,1,1,0,0,0, 1,0,1,1,0,1,0,0,\ 1,1,1,0,0,0,1,0, 1,1,0,1,0,0,0,1),8) This is the generator for a Cmd> gen The generators are BCDE, ACDF, ABCG, and ABDH. (1,1) (2,1) (3,1) (4,1) Cmd> doff2(basis:gen,ialiases:t) The aliases of I. Aliases of I: (1) "I" (2) "BCDE" (3) "ACDF" (4) "ABEF" (5) "ABCG" (6) "ADEG" (7) "BDFG" (8) "CEFG" (9) "ABDH" (10) "ACEH" (11) "BCFH" (12) "DEFH" (13) "CDGH" (14) "BEGH" (15) "AFGH" (16) "ABCDEFGH" Cmd> allaliases2(gen) All the aliases. Sorry about the squint print. Aliases: (1) "I = BCDE = ACDF = ABEF = ABCG = ADEG = BDFG = CEFG = ABDH = ACEH = BCFH = DEFH = CDGH = BEGH = AFGH = ABCDEFGH" (2) "A = ABCDE = CDF = BEF = BCG = DEG = ABDFG = ACEFG = BDH = CEH = ABCFH = ADEFH = ACDGH = ABEGH = FGH = BCDEFGH" (3) "B = CDE = ABCDF = AEF = ACG = ABDEG = DFG = BCEFG = ADH = ABCEH = CFH = BDEFH = BCDGH = EGH = ABFGH = ACDEFGH" (4) "AB = ACDE = BCDF = EF = CG = BDEG = ADFG = ABCEFG = DH = BCEH = ACFH = ABDEFH = ABCDGH = AEGH = BFGH = CDEFGH" (5) "C = BDE = ADF = ABCEF = ABG = ACDEG = BCDFG = EFG = ABCDH = AEH = BFH = CDEFH = DGH = BCEGH = ACFGH = ABDEFGH" (6) "AC = ABDE = DF = BCEF = BG = CDEG = ABCDFG = AEFG = BCDH = EH = ABFH = ACDEFH = ADGH = ABCEGH = CFGH = BDEFGH" (7) "BC = DE = ABDF = ACEF = AG = ABCDEG = CDFG = BEFG = ACDH = ABEH = FH = BCDEFH = BDGH = CEGH = ABCFGH = ADEFGH" (8) "ABC = ADE = BDF = CEF = G = BCDEG = ACDFG = ABEFG = CDH = BEH = AFH = ABCDEFH = ABDGH = ACEGH = BCFGH = DEFGH" (9) "D = BCE = ACF = ABDEF = ABCDG = AEG = BFG = CDEFG = ABH = ACDEH = BCDFH = EFH = CGH = BDEGH = ADFGH = ABCEFGH" (10) "AD = ABCE = CF = BDEF = BCDG = EG = ABFG = ACDEFG = BH = CDEH = ABCDFH = AEFH = ACGH = ABDEGH = DFGH = BCEFGH" (11) "BD = CE = ABCF = ADEF = ACDG = ABEG = FG = BCDEFG = AH = ABCDEH = CDFH = BEFH = BCGH = DEGH = ABDFGH = ACEFGH" (12) "ABD = ACE = BCF = DEF = CDG = BEG = AFG = ABCDEFG = H = BCDEH = ACDFH = ABEFH = ABCGH = ADEGH = BDFGH = CEFGH" (13) "CD = BE = AF = ABCDEF = ABDG = ACEG = BCFG = DEFG = ABCH = ADEH = BDFH = CEFH = GH = BCDEGH = ACDFGH = ABEFGH" (14) "ACD = ABE = F = BCDEF = BDG = CEG = ABCFG = ADEFG = BCH = DEH = ABDFH = ACEFH = AGH = ABCDEGH = CDFGH = BEFGH" (15) "BCD = E = ABF = ACDEF = ADG = ABCEG = CFG = BDEFG = ACH = ABDEH = DFH = BCEFH = BGH = CDEGH = ABCDFGH = AEFGH" (16) "ABCD = AE = BF = CDEF = DG = BCEG = ACFG = ABDEFG = CH = BDEH = ADFH = ABCEFH = ABGH = ACDEGH = BCDFGH = EFGH"

3 Stat 5303 (Oehlert): Fractional Factorials 3 Cmd> doff2(basis:gen,showfrac:t) doff2() can also determine which factor/level combinations are used in the fraction for a given set of generators. Treatments: (1) "(1)" (2) "afgh" (3) "begh" (4) "abef" (5) "cefg" (6) "aceh" (7) "bcfh" (8) "abcg" (9) "defh" (10) "adeg" (11) "bdfg" (12) "abdh" (13) "cdgh" (14) "acdf" (15) "bcde" (16) "abcdefgh" Cmd> gen2<-gen;gen2[4,8]<- -1 Cmd> gen2 Here we ve changed the generators so that the last one is ABDH instead of ABDH. (1,1) (2,1) (3,1) (4,1) Cmd> doff2(basis:gen2,ialiases:t) The same sets of letters appear as before, but now there are some negatives. (1) "I" (2) "BCDE" (3) "ACDF" (4) "ABEF" (5) "ABCG" (6) "ADEG" (7) "BDFG" (8) "CEFG" (9) "-ABDH" (10) "-ACEH" (11) "-BCFH" (12) "-DEFH" (13) "-CDGH" (14) "-BEGH" (15) "-AFGH" (16) "-ABCDEFGH"

4 Stat 5303 (Oehlert): Fractional Factorials 4 Cmd> doff2(basis:gen2,allaliases:t) The same sets of letters appear as before, but now there are some negatives. Aliases: (1) "I = BCDE = ACDF = ABEF = ABCG = ADEG = BDFG = CEFG = -ABDH = -ACEH = -BCFH = -DEFH = -CDGH = -BEGH = -AFGH = -ABCDEFGH" (2) "A = ABCDE = CDF = BEF = BCG = DEG = ABDFG = ACEFG = -BDH = -CEH = -ABCFH = -ADEFH = -ACDGH = -ABEGH = -FGH = -BCDEFGH" (3) "B = CDE = ABCDF = AEF = ACG = ABDEG = DFG = BCEFG = -ADH = -ABCEH = -CFH = -BDEFH = -BCDGH = -EGH = -ABFGH = -ACDEFGH" (4) "AB = ACDE = BCDF = EF = CG = BDEG = ADFG = ABCEFG = -DH = -BCEH = -ACFH = -ABDEFH = -ABCDGH = -AEGH = -BFGH = -CDEFGH" (5) "C = BDE = ADF = ABCEF = ABG = ACDEG = BCDFG = EFG = -ABCDH = -AEH = -BFH = -CDEFH = -DGH = -BCEGH = -ACFGH = -ABDEFGH" (6) "AC = ABDE = DF = BCEF = BG = CDEG = ABCDFG = AEFG = -BCDH = -EH = -ABFH = -ACDEFH = -ADGH = -ABCEGH = -CFGH = -BDEFGH" (7) "BC = DE = ABDF = ACEF = AG = ABCDEG = CDFG = BEFG = -ACDH = -ABEH = -FH = -BCDEFH = -BDGH = -CEGH = -ABCFGH = -ADEFGH" (8) "ABC = ADE = BDF = CEF = G = BCDEG = ACDFG = ABEFG = -CDH = -BEH = -AFH = -ABCDEFH = -ABDGH = -ACEGH = -BCFGH = -DEFGH" (9) "D = BCE = ACF = ABDEF = ABCDG = AEG = BFG = CDEFG = -ABH = -ACDEH = -BCDFH = -EFH = -CGH = -BDEGH = -ADFGH = -ABCEFGH" (10) "AD = ABCE = CF = BDEF = BCDG = EG = ABFG = ACDEFG = -BH = -CDEH = -ABCDFH = -AEFH = -ACGH = -ABDEGH = -DFGH = -BCEFGH" (11) "BD = CE = ABCF = ADEF = ACDG = ABEG = FG = BCDEFG = -AH = -ABCDEH = -CDFH = -BEFH = -BCGH = -DEGH = -ABDFGH = -ACEFGH" (12) "ABD = ACE = BCF = DEF = CDG = BEG = AFG = ABCDEFG = -H = -BCDEH = -ACDFH = -ABEFH = -ABCGH = -ADEGH = -BDFGH = -CEFGH" (13) "CD = BE = AF = ABCDEF = ABDG = ACEG = BCFG = DEFG = -ABCH = -ADEH = -BDFH = -CEFH = -GH = -BCDEGH = -ACDFGH = -ABEFGH" (14) "ACD = ABE = F = BCDEF = BDG = CEG = ABCFG = ADEFG = -BCH = -DEH = -ABDFH = -ACEFH = -AGH = -ABCDEGH = -CDFGH = -BEFGH" (15) "BCD = E = ABF = ACDEF = ADG = ABCEG = CFG = BDEFG = -ACH = -ABDEH = -DFH = -BCEFH = -BGH = -CDEGH = -ABCDFGH = -AEFGH" (16) "ABCD = AE = BF = CDEF = DG = BCEG = ACFG = ABDEFG = -CH = -BDEH = -ADFH = -ABCEFH = -ABGH = -ACDEGH = -BCDFGH = -EFGH" Cmd> doff2(basis:gen2,showfrac:t) The treatments in the design change completely however. Where ever there was an h before, it is now gone, and where ever h was missing before, it is now present. (1) "h" (2) "afg" (3) "beg" (4) "abefh" (5) "cefgh" (6) "ace" (7) "bcf" (8) "abcgh" (9) "def" (10) "adegh" (11) "bdfgh" (12) "abd" (13) "cdg" (14) "acdfh" (15) "bcdeh" (16) "abcdefg" Cmd> y<-vector(40,20,17,12,31,19,22,20,36,25,34,11,37,21,29,19) Data from a from Garcia-Diaz and Phillips. Cmd> a<-factor(rep(rep(run(2),rep(1,2)),8)) I am entering things in standard order for factors A through D. This command is a bit more baroque than is strictly required (you could get the same thing with factor(rep(run(2),8))), but it fits the general pattern that I use to build up the 1 s and 2 s. Cmd> b<-factor(rep(rep(run(2),rep(2,2)),4)) Cmd> c<-factor(rep(rep(run(2),rep(4,2)),2)) Cmd> d<-factor(rep(rep(run(2),rep(8,2)),1))

5 Stat 5303 (Oehlert): Fractional Factorials 5 Cmd> e<-factor(2,1,1,2,1,2,2,1,1,2,2,1,2,1,1,2) Here we enter the levels of E directly. The generator is I=ABCDE. Cmd> apm<-2*(a-1.5) In this example there are only 16 observations so typing them in isn t too bad. For larger designs, it might be helpful to have a method to compute the aliased factor(s) without typing them. MacAnova stores two-series factors as 1 s and 2 s. By subtracting 1.5 and multiplying the difference by 2 we get -1 s and 1 s. Products of the±terms can give us the ± form of the aliased factor. Cmd> bpm<-2*(b-1.5) Cmd> cpm<-2*(c-1.5) Cmd> dpm<-2*(d-1.5) Cmd> hconcat(a,apm,b,bpm)[run(5),] (1,1) (2,1) (3,1) (4,1) (5,1) Cmd> epm<-apm*bpm*cpm*dpm E (in±form) is the product of A, B, C, and D (in±form). Cmd> epm (1) (9) Cmd> e (1) (9) Cmd> e3<-factor(epm/2+1.5) Divide the±form by 2 and add 1.5 to get back the 1 s and 2 s that MacAnova likes for factors.

6 Stat 5303 (Oehlert): Fractional Factorials 6 Cmd> e3 Same as before. (1) (9) Cmd> anova("y=a*b*c*d") Here is the usual ANOVA using A, B, C and D (and their interactions). All 15 degrees of freedom among the 16 responses are fit. Model used is y=a*b*c*d DF SS MS CONSTANT a b a.b c a.c b.c a.b.c d a.d b.d a.b.d c.d a.c.d b.c.d a.b.c.d ERROR1 0 0 undefined Cmd> anova("y=b*c*d*e") Here we use B, C, D, and E (and their interactions). These sums of squares are for the same 15 contrasts, they ve just been relabeled. For example, the in the ANOVA above (labeled ABCD) is in this ANOVA but labeled E. Of course, E is aliased with ABCD. Model used is y=b*c*d*e DF SS MS CONSTANT b c b.c d b.d c.d b.c.d e b.e c.e b.c.e d.e b.d.e c.d.e b.c.d.e ERROR1 0 0 undefined

7 Stat 5303 (Oehlert): Fractional Factorials 7 Cmd> anova("y=(a+b+c+d+e)ˆ2") In an effort to get some degrees of freedom for error, we might try fitting just main effects and two-factor interactions, but for this design that is again all 15 degrees of freedom. Model used is y=(a+b+c+d+e)ˆ2 WARNING: summaries are sequential DF SS MS CONSTANT a b c d e a.b a.c a.d a.e b.c b.d b.e c.d c.e d.e ERROR1 0 0 undefined Cmd> anova("y=a+b+c+d+e",pvals:t) Just look at main effects. It looks like A and B are significant. Keep in mind, however, that this error isn t really pure error, it is a pooling of interaction terms. Model used is y=a+b+c+d+e DF SS MS P-value CONSTANT e-09 a b c d e ERROR Cmd> anova("y=a*b",pvals:t) The AB interaction doesn t look big. Model used is y=a*b DF SS MS P-value CONSTANT e-10 a b a.b ERROR

8 Stat 5303 (Oehlert): Fractional Factorials 8 Cmd> yatesplot(y) I prefer to look at a rankit plot or halfnormal plot of the effects (computed via yates()). The data were entered in standard order for A, B, C, and D, so the effects will be the effects for those factors in standard order. Keep in mind that there is aliasing, so an ABCD interaction is also the E main effect! Absolute effects Half Normal plot of absolute effects B ABD D BC ABCD AC AB AD BCD ABCBD A C CDACD Half normal scores Cmd> y<-vector(142,106,88,109,113,162,200,79,101,108,146,72,200,83,115,118) These data are again a from Davies, but run in two blocks of size 8 with ABD= CE confounded with blocks. Factors a, b, c, d are the same as before, due to standard order. Cmd> e<-factor(1,2,2,1,2,1,1,2,2,1,1,2,1,2,2,1) The generator here is E= ABCDE. Cmd> bl<-factor(1,2,2,1,1,2,2,1,2,1,1,2,2,1,1,2) ABD= CE as block.

9 Stat 5303 (Oehlert): Fractional Factorials 9 Cmd> yatesplot(y) Do the analysis via halfnormal plotting of effects. Data in standard order for A, B, C, and D. Effects ABCD (aliased with E), A, C, and ABD (confounded with blocks) look like possible outliers. Absolute effects Half Normal plot of absolute effects ABCD A C ABD BCD AC B AD AB D CDABC BC BDACD Half normal scores Cmd> anova("y=bl+a*c*e",fstats:t) This design has resolution V, so we can project it down onto factors A, C and E. Unfortunately, CE (= ABD) is confounded with blocks. Only the main effects look significant, but AE is starting to get close. Model used is y=bl+a*c*e WARNING: summaries are sequential DF SS MS F P-value CONSTANT e e < 1e-08 bl a c a.c e e-05 a.e c.e 0 0 undefined undefined undefined a.c.e ERROR

10 Stat 5303 (Oehlert): Fractional Factorials 10 Cmd> gen<-vector(1,1,2) We can also find aliases for three series designs. The generator matrix has one column for every factor, and one row for every generator. The elements are 0, 1, or 2. Cmd> gen This is the A 1 B 1 C 2 effect. (1,1) Cmd> aliases3(gen) Aliases of I. Note that the generator gets repeated. (1) "I" (2) "Aˆ1 Bˆ1 Cˆ2 " (3) "Aˆ1 Bˆ1 Cˆ2 " Cmd> aliases3(gen,effect:vector(1,0,0)) Aliases of A. (1) "Aˆ1 " (2) "Aˆ1 Bˆ2 Cˆ1 " (3) "Bˆ1 Cˆ2 " Cmd> aliases3(gen,effect:vector(0,1,0)) Aliases of B. (1) "Bˆ1 " (2) "Aˆ1 Bˆ2 Cˆ2 " (3) "Aˆ1 Cˆ2 " Cmd> aliases3(gen,effect:vector(0,0,1)) Aliases of C. (1) "Cˆ1 " (2) "Aˆ1 Bˆ1 " (3) "Aˆ1 Bˆ1 Cˆ1 " Cmd> aliases3(gen,effect:vector(1,0,1)) Aliases of A 1 C 1. There are nine units in a 3 3 1, so 8 degrees of freedom between them, grouped into four sets of two. We now have all the aliases. (1) "Aˆ1 Cˆ1 " (2) "Aˆ1 Bˆ2 " (3) "Bˆ1 Cˆ1 "

11 Stat 5303 (Oehlert): Fractional Factorials 11 Cmd> aliases3(gen,effect:vector(1,0,0)) Here are the A aliases again. (1) "Aˆ1 " (2) "Aˆ1 Bˆ2 Cˆ1 " (3) "Bˆ1 Cˆ2 " Cmd> aliases3(gen,effect:vector(0,1,2)) These are the aliases of B 1 C 2 (one of A s aliases). (1) "Bˆ1 Cˆ2 " (2) "Aˆ1 Bˆ2 Cˆ1 " (3) "Aˆ1 " Cmd> aliases3(gen,effect:vector(0,2,1)) aliases3() doesn t mind if you use a leading exponent of 2, it figures things out. (1) "Bˆ1 Cˆ2 " (2) "Aˆ1 Bˆ2 Cˆ1 " (3) "Aˆ1 "

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