LAB wednesday o finish ch 7 n-way factorial -like 2-way more complex -how work with 1 obs per cell-orthogonal trick & normal plot - Taguchi approach - rules 9-16 about C.I. will discuss via example on 3-way for beads - also more on interaction plots o only one (could do with 2) obs - too expensive? - have size (a=3), finger (b=3) and color (c=2) * note: old reps given last time, say were with black beads - ANOVA table Source SS df S a-1=2 F b-1=2 C c-1=1 S*F (a-1)(b-1)=4 S*C (a-1)(c-1)=2 F*C (b-1)(c-1)=2 S*F*C (a-1)(b-1)(c-1)=4 Error ------------------------------ total n-1=17 - do orthogonal trend contrast trick A2(equal ss) - see main effects set up and interaction by multiplication LINES; s 1 t 64 1 o 1 100 -1 1 -1 1 1 s 1 a 72 2 o 1 140 -1 1 0 -2 1 s 1 b 76 3 o 1 93 -1 1 1 1 1 m 2 t 70 1 o 1 85 0 -2 -1 1 1 m 2 a 71 2 o 1 45 0 -2 0 -2 1 m 2 b 79 3 o 1 105 0 -2 1 1 1 l 3 t 70 1 o 1 40 1 1 -1 1 1 l 3 a 75 2 o 1 37 1 1 0 -2 1 l 3 b 77 3 o 1 35 1 1 1 1 1 s 1 t 63 1 b 2 119 -1 1 -1 1 -1 s 1 a 72 2 b 2 127 -1 1 0 -2 -1 s 1 b 79 3 b 2 70 -1 1 1 1 -1 m 2 t 69 1 b 2 49 0 -2 -1 1 -1 m 2 a 71 2 b 2 133 0 -2 0 -2 -1 m 2 b 80 3 b 2 86 0 -2 1 1 -1 l 3 t 65 1 b 2 44 1 1 -1 1 -1 l 3 a 75 2 b 2 35 1 1 0 -2 -1 l 3 b 77 3 b 2 53 1 1 1 1 -1 ; - before collapse a few interactions for error, now... - theory: sum of (c(i)*Y(~.)) is N(sum of (c(i)*m), ?) ........ so sum(c(i)*Y(~.) divided by 2 sqrt(sum c(i) /r) is 2 N(0,sigma ) under no effects at all - 1. find normalized contrasts (see first handout) 2. plot them on normal prob paper a) order low to high b) find (rank-.375)/(n+.25) c) find Pr(zsignificant effect 4) assumes normality, independence and equal variance note: slope is sigma Example: quad on size (est1,?) 1 VLA1 -2.35744 -1.79619 2 VLA2 -0.67692 -1.02715 3 VLB1 -0.03936 0.14583 4 VLB2 0.04326 0.61852 5 VLC1 -0.27789 -0.14583 6 VLA1B1 0.09743 1.02715 7 VLA1B2 0.71942 1.79619 8 VLA1C1 -0.34835 -0.29484 9 VLA2B1 -0.53016 -0.80592 10 VLA2B2 0.00682 0.29484 11 VLA2C1 0.02625 0.45074 12 VLB1C1 -0.23935 0.00000 13 VLB2C1 0.16955 1.31531 14 VLA1B1C1 -0.48516 -0.61852 15 VLA2B1C1 0.07394 0.80592 16 VLA1B2C1 -0.36729 -0.45074 17 VLA2B2C1 -1.04130 -1.31531 Plot of EST1*NSCORE. Legend: A = 1 obs, B = 2 obs, etc. 1 + | A | | A 0 + A AA A A A | AA A A | A A | A -1 + A | | | -2 + | A | | -3 + | -+-----------+-----------+-----------+-----------+- -2 -1 0 1 2 RANK FOR VARIABLE o Taguchi approach - controversial since abit ad hoc - objective: choose a best mean performer where variability is small, particularly for effect which maybe noise - mostly visual - Example S, F, C; F is noise plot size*color against finger choice? all 3*2=6 lines over t,a,b o 3-way - get 18 more people to string s,m,l; t,a,b; black and orange(see 2nd handout) 1 s 1 t 64 1 o 1 100 orignl -0.04976 11 sto 2 s 1 a 72 2 o 1 140 orignl 0.38872 12 sao 3 s 1 b 76 3 o 1 93 orignl -0.12276 13 sbo 4 m 2 t 70 1 o 1 85 orignl -0.21019 21 mto 5 m 2 a 71 2 o 1 45 orignl -0.94228 22 mao 6 m 2 b 79 3 o 1 105 orignl 0.00140 23 mbo 7 l 3 t 70 1 o 1 40 orignl -1.19402 31 lto 8 l 3 a 75 2 o 1 37 orignl -1.46099 32 lao 9 l 3 b 77 3 o 1 35 orignl -1.81535 33 lbo 10 s 1 t 63 1 b 2 119 oldrep 0.14540 11 stb 11 s 1 a 72 2 b 2 127 oldrep 0.23190 12 sab 12 s 1 b 79 3 b 2 70 oldrep -0.39772 13 sbb 13 m 2 t 69 1 b 2 49 oldrep -0.80764 21 mtb 14 m 2 a 71 2 b 2 133 oldrep 0.30099 22 mab 15 m 2 b 80 3 b 2 86 oldrep -0.19893 23 mbb 16 l 3 t 65 1 b 2 44 oldrep -0.98286 31 ltb 17 l 3 a 75 2 b 2 35 oldrep -1.81535 32 lab 18 l 3 b 77 3 b 2 53 oldrep -0.70206 33 lbb 19 s 1 t 64 1 o 1 106 newrep 0.01159 11 sto 20 s 1 a 72 2 o 1 125 newrep 0.20978 12 sao 21 s 1 b 76 3 o 1 92 newrep -0.13340 13 sbo 22 m 2 t 70 1 o 1 94 newrep -0.11218 21 mto 23 m 2 a 71 2 o 1 51 newrep -0.75209 22 mao 24 m 2 b 79 3 o 1 133 newrep 0.30099 23 mbo 25 l 3 t 70 1 o 1 43 newrep -1.02729 31 lto 26 l 3 a 75 2 o 1 43 newrep -1.02729 32 lao 27 l 3 b 77 3 o 1 37 newrep -1.46099 33 lbo 28 s 1 t 63 1 b 2 105 newrep 0.00140 11 stb 29 s 1 a 72 2 b 2 115 newrep 0.10370 12 sab 30 s 1 b 79 3 b 2 86 newrep -0.19893 13 sbb 31 m 2 t 69 1 b 2 63 newrep -0.50473 21 mtb 32 m 2 a 71 2 b 2 58 newrep -0.59464 22 mab 33 m 2 b 80 3 b 2 95 newrep -0.10166 23 mbb 34 l 3 t 65 1 b 2 39 newrep -1.26666 31 ltb 35 l 3 a 75 2 b 2 37 newrep -1.46099 32 lab 36 l 3 b 77 3 b 2 34 newrep -2.24181 33 lbb - 3-way interaction? look at plots oer color. Note: // ism one way SSABC=0 ------------------------------- COLCAT=1 ------------------------------- Plot of AV_HTIME*SIZECAT. Symbol is value of FINGCAT. 1 + | | | |2 | 3 0 +1 |3 1 | AV_HTIME | | | 2 -1 + | 1 | | | 3 | -2 + -+----------------------------+----------------------------+- 1 2 3 SIZECAT NOTE: 1 obs hidden. ------------------------------- COLCAT=2 ------------------------------- Plot of AV_HTIME*SIZECAT. Symbol is value of FINGCAT. 1 + | | | | |2 0 +1 | 2 |3 AV_HTIME | | 1 | -1 + | 1 | | 3 | 2 | -2 + -+----------------------------+----------------------------+- 1 2 3 SIZECAT NOTE: 1 obs hidden. - test H0: (abc)(ijk)-(abc)(i.k) -(abc)(.jk)-(abc)(ij.)+ (ab)(ij.)+(ac)(i.k)+ (bc)(.jk)-(abc)(...)=0 for all i,j,k how? think rules and Y-bars - check SAS printout General Linear Models Procedure Class Level Information Class Levels Values SIZECAT 3 1 2 3 FINGCAT 3 1 2 3 COLCAT 2 1 2 Number of observations in data set = 36 General Linear Models Procedure Dependent Variable: HTIME Sum of Mean Source DF Squares Square F Value Pr > F Model 17 15.11 0.88 7.86 0.0001 Error 18 2.034960 0.113053 Corrected Total 35 17.150313 R-Square C.V. Root MSE HTIME Mean 0.881346 -60.86701 0.3362 -0.5524 Dependent Variable: HTIME Source DF Type I SS Mean Square F Value Pr > F SIZECAT 2 12.67 6.33 56.06 0.0001 FINGCAT 2 0.05 0.02 0.23 0.7949 COLCAT 1 0.03 0.03 0.29 0.5941 SIZECAT*FINGCAT 4 1.32 0.33 2.92 0.0502 SIZECAT*COLCAT 2 0.00 0.00 0.02 0.9829 FINGCAT*COLCAT 2 0.06 0.03 0.29 0.7507 SIZECA*FINGCA*COLCAT 4 0.96 0.24 2.13 0.1195 Source DF Type III SS Mean Square F Value Pr > F SIZECAT 2 12.67 6.33 56.06 0.0001 FINGCAT 2 0.05 0.02 0.23 0.7949 COLCAT 1 0.03 0.03 0.29 0.5941 SIZECAT*FINGCAT 4 1.32 0.33 2.92 0.0502 SIZECAT*COLCAT 2 0.00 0.00 0.02 0.9829 FINGCAT*COLCAT 2 0.06 0.03 0.29 0.7507 SIZECA*FINGCA*CAT 4 0.96 0.24 2.13 0.1195 Level of ------------HTIME------------ SIZECAT N Mean SD 1 12 0.01582649 0.21553326 2 12 -0.30174538 0.41663482 3 12 -1.37130438 0.43213225 Level of ------------HTIME------------ FINGCAT N Mean SD 1 12 -0.49974423 0.52588144 2 12 -0.56821165 0.79216469 3 12 -0.58926740 0.80638888 Level of ------------HTIME------------ COLCAT N Mean SD 1 18 -0.52200532 0.68717079 2 18 -0.58281019 0.73121879 - note: most interactions resoundingly negligible what of S*F? and S*C? p-value=.05 p-value=.98 what is plot difference? - need means st sa sb mt ma mb lt la lb o -.0 .4 -.1 -.2 -.9 .0 -1.2 -1.5 -1.8 o .0 .2 -.1 -.1 -.7 .3 -1.0 -1.0 -1.5 b .1 .2 -.4 -.8 .3 -.2 1.0 -1.8 - .7 b .0 .1 -.2 -.5 -.6 -.1 -1.2 -1.3 -2.2 look at plot S*F average over color(down columns: x-axis: sml, symbol=f ) look at plot S*C average over finger (top 2x3, next 2x3 etc.: x-axis:sml, symbol=c) - just accept all interactions are negligible then test main effects - "use small beads" - scheffe follow-up what if beads shipped mixed: 30% small & 70% large versus medium