Multiple Fto ANOVA Notes Pge wo Fto Anlsis Anlsis of Vine fo Multiple Ftos Conside two ftos (tetments) A nd B with A done t levels nd B done t levels. Within given tetment omintion of A nd B levels, leled i nd j espetivel, epeted mesuements e mde of esponse vile. Sine the nume of mesuements fo eh i, j level is the sme, this kind of expeimentl design is lled lned. he k th soe in the i, j tetment omintion is designted s ijk,, i is the the tetment fto index fo fto A ; i j is the the tetment fto index fo fto B ; j k lels the soe the within the i, j level ; k. he null hpothesis ssets tht no tetment popultion diffes fom n othe. hus, the soes in the tetment levels should hve the sme men nd vine. Fo theoetil onveniene, we will ssume the soes e nomll distiuted with ommon vine within eh tetment level, ut the tests whih follow e fil oust fo deptues fom nomlit. Null Hpothesis: H 0 : μij, μ0 fo i, j, whee μ0 is the ommon popultion men of the tetment popultions. hus, ll of the vition seen in the mesued soes is due to smple vitions, i.e., ndomness. he totl nume of soes is n. he vege of ll soes is the gnd men nd is given ijk,, i j k. () he smple men of the A B tetment omintion i, j is given ijk,, k AB ij,. () he smple men of tetment A t level i is ijk,, j k A i. (3) Simill, the smple men of the B tetment t level j is ijk,, i k B j. (4) Now, the definitions of these mens impl tht the following sums of devitions vnish. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge ( A i ) 0 (5) i ( B j ) 0 (6) j ( AB ij, A i) 0 (7) j ( AB ij, B j) 0 (8) i ( AB ij, A i B ) 0 (9) i ( AB ij, A i B ) 0 (0) j ( ijk,, AB ij, ) 0 () k An effets due to tetments should mnifest themselves in the devitions of the tetment smple mens fom the gnd men. he effets n e listed s follows: he Effet due to A t level i: A i he Effet due to B t level j: B j he Intetion Effet of A t level i nd B t level j: ( ) ( ) ( ) + AB ij, A i B j AB ij, A i B j he Residul o Eo Effet: ijk,, AB ij, he AB Intetion Effet mesues how A nd B when oth pesent n use n effet diffeent fom thei septe effets. he Effet due to Eo mesues the ssumed ndom vition of the esponse vile within the tetment omintion i, j. Now, fom eqution (5), onl - of the devitions A i Simill, the devitions B j n e independentl speified. hve - degees of feedom. he eqution fo the AB intetion effet, AB ij, A i B, defines the omponents of n mtix. Fom equtions (9) nd (0) oth the ow nd olumn sums of this mtix e onstined. As onsequene, in the fist - ows thee e onl - independent enties. he enties in the lst ow e then ompletel detemined the olumn sum equiement. hus, the AB intetion hs ( ) ( ) degees of feedom. Finll, fom eqution (), fo eh of the A nd B fto omintions thee e - independent devitions fom the smple men AB ij,. he following tle summizes the degees of feedom,ν, ssoited with eh effet. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 3 Effet Degees of Feedom Gnd Men ν Fto A ν A Fto B ν B Intetion of Ftos AB ν AB ( )( ) Residul o Eo ν ( ) Note tht the sum of the degees of feedom is the totl nume of soes n. Fomll, we n wite the following. ( ) ( ) ( ) ( ) + + + + + () ijk,, A i B j AB ij, A i B j ijk,, AB ij, If the null hpothesis is tue nd ll of the tetment omintions give the sme men esult, then should e fi estimte of eve soe nd the effets just listed ll mesue ndom vition. Eqution () n e thought of n othogonl veto deomposition of the esponse vile. Eh of the mesuements n e thought of s omponent of n dimensionl olumn veto.,, A B AB, A B +,, AB,,, A B AB, A B +,, AB,,, A B AB, A B +,, AB,,, A B AB, A B +,, AB,,, A B AB, A B +,, AB, +,, + + A B AB, A B + +,, AB, (3) ij,, B A i j AB ij, A i B ij,, AB ij, ij,, A i B j AB ij, A i B ij,, AB ij,,, A B AB, A B +,, AB, Fom the devition onstints of equtions (5) though () the dot podut of n two diffeent olumn vetos on the ight hnd side of eqution (3) must e zeo, i.e., the olumn vetos on the ight side of eqution (3) e mutull othogonl. he olumn veto whih gives the totl devition fom the gnd men, ijk,,, n e deomposed s the sum VA + VB + VAB + VE, whee the mutull othogonl olumn vetos e given the following equtions. E Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 4 V A A B AB, A B + A B AB, A B + A B AB, A B + A B AB, A B + A B AB, A B + ; V B ; V A B AB AB, A B + ; V E A i B j AB ij, A i B ij A i B j AB ij, A i B ij A B AB, A B +,, AB,,, AB,,, AB,,, AB,,, AB,,, AB,,, AB ij,,, AB ij,,, AB, Beuse of the othogonlit ondition on the fou V vetos, the totl vition in n e deomposed into ontiutions ssoited with effets A, B, AB nd Eo. ( i, j, k ) ( VA + VB + VAB + VE ) i( VA + VB + VAB + VE ) i j k (4) A + B + AB + E ( ) A VAi VA A i (5) i ( ) B VBi VB B j (6) j ( ) AB VABi VAB AB ij, A i B (7) i j E V ( ) Ei VE i, j, k AB i, j (8) i j k his deomposition llows us to test the null hpothesis. If the null hpothesis is tue then the men sque of the Effets A, B nd AB ll estimte the sme ommon popultion vine of eh A B fto omintion. he men sques e omputed s vition divided the ssoited degees of feedom. ( ) A i he Men Sque due to Fto A: A i A (9) Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 5 ( ) B j he Men Sque due to Fto B: B j B he Men Sque due to Intetion of A with B: ( AB ij, A i B ) AB i j AB () he Eo o Residul Men Sque: ( ) (0) ( ijk,, AB ij, ) E i j k ( ) () o test the null hpothesis tht ll effets mesue the sme ommon ndom vition ssoited with the Men Sque due to Eo, we ompute the oseved Fishe F soe. Effet Men Sque Fos (3) Fo given level of signifine,α, this vlue is omped ginst itil soe lulted fom n F distiution used to ompe two vines otined fom smpling vines fom two nomll distiuted popultions. he numeto degees of feedom is ppopite to the effet in question nd the denominto degees of feedom is (-). his infomtion is esil summized in wo-fto ANOVA tle. Soue Sum of Sques Degees of Feedom Fto A A - Fto B B - Intetion AB AB ( - )( - ) Men Sque Eo E ( - ) E otl n - - s n F os A A A B B B AB AB AB ( )( ) ( ) Eh omputed F os is omped ginst F α ( νeffet, ν E ) fo stted level of signifine,α. If Fos F α ( νeffet, ν E ) ndom eo (vition). If F F α ( ν ν ) H nd onlude the effet in <, we fil to ejet the null hpothesis tht the effet is just n lis fo os > effet, E, we ejet 0 question is el. Note: It is possile tht the AB Intetion Effet n test s signifint even when one o oth of the septe A Fto nd B Fto Effets e insignifint. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 6 wo Fto Anlsis with Replition A polem tht often ous, ptiull in indust o mnuftuing, is tht of Replition. his efes to seemingl unontolled vition in mesuements mde on wht e nominll smples set up unde identil tetments. his n sometimes e ttiuted to unontolled ftos tht v t the time of unit onstution suh s ometi pessue, humidit, humn sseml vitions, et. Replition n e thought of s inluding time s fto in podut pefomne. As esult, if two tetment ftos A nd B e led eing onsideed, Replition n e teted s thid fto to e inluded in thee fto nlsis (see the next setion). A simple, if less omplete, ppoh is to ignoe Replition intetion effets with the othe tetment ftos, ut still intodue min effet due to Replition. Conside two ftos (tetments) A nd B with A done t levels nd B done t levels. Within given tetment omintion of A nd B levels leled i, j, thee wee eplitions done in ndom ode t diffeent times of epeted mesuements of esponse vile. One gin, sine the nume of mesuements fo eh i, j eplition level is the sme, the expeimentl design is lned. he k th soe in the i, j tetment omintion nd g th eplition is designted s ijgk i is the the tetment fto index fo fto A ; i j is the the tetment fto index fo fto B ; j g is the eplition index ; g k lels the soe the within the i, j, g level ; k. he null hpothesis ssets tht no tetment popultion diffes fom n othe. hus, the soes in the tetment levels/eplitions should ll hve the sme men nd vine. Agin we will ssume the soes e nomll distiuted with ommon vine within eh tetment/ eplition level. Null Hpothesis: H 0 : μijg,, μ0 fo i, j, g, whee μ0 is the ommon popultion men of the tetment popultions. All of the vition seen in the mesued soes is due to ndomness. he totl nume of soes is n. he gnd men is given ijgk i j g k. (4) he smple men of the A B tetment omintion i, j nd eplition g is given ijgk k AB ijg,,. (5) he smple men of the A B tetment omintion i, j ove ll eplitions is given ijgk g k AB ij,. (6) Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 7 he smple men of tetment A t level i is ijgk j g k A i. (7) Simill, the smple men of the B tetment t level j is ijgk i g k B j. (8) One gin these definitions impl devition onstints. ( A i ) 0 (9) i ( B j ) 0 (30) j ( AB ij, A i) 0 (3) j ( AB ij, B j) 0 (3) i ( AB ij, A i B ) 0 (33) i ( AB ij, A i B ) 0 (34) j ( AB ijg,, AB ij, ) 0 (35) g ( ijgk AB ijg,, ) 0 (36) k he effets of the etments/replitions e listed s follows: he Effet due to A t level i: A i he Effet due to B t level j: B j he Intetion Effet of A t level i nd B t level j: ( ) ( ) ( ) AB ij, A i B j AB ij, A i B he Replition Effet: AB ijg,, AB ij, he Residul o Eo Effet: ijgk AB ijg,, Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 8 he degees of feedom ssoited with Effets A, B, nd AB e the sme s in the lst setion. Fom eqution (35), fo eh of the omintions of A nd B ftos, thee e - independent devitions, AB ijg,, AB ij,. Fom eqution (36), fo eh of the A nd B Fto/Replition omintions thee e - independent devitions fom the smple men AB ijg,,. he following tle summizes the degees of feedom,ν, ssoited with eh effet. Effet Degees of Feedom Gnd Men ν Fto A ν A Fto B ν B Intetion of Ftos AB ν AB ( )( ) Replition G ν G ( ) ν Residul o Eo ( ) Note tht the sum of the degees of feedom is still the totl nume of soes n. E Fomll, we n wite the following. ijgk + ( A i ) + ( B j ) + ( AB ij, A i B ) + ( AB ijg,, AB ij, ) ( ijgk AB ijg,, ) + (37) If the null hpothesis is tue nd ll of the tetment omintions give the sme men esult, then the effets just listed ll mesue the sme ndom deptue fom the gnd men. he onstint equtions (9) though (36) veif tht eqution (37) desies n othogonl veto deomposition of the esponse vile. Eh of the mesuements n e thought of s omponent of n dimensionl olumn veto. he olumn veto whih gives the totl devition fom the gnd men, ijgk, n e deomposed s the sum of the five mutull othogonl olumn vetos, VA + VB + VAB + VG + VE. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 9 A B AB, A B + A B AB, A B + A B AB, A B + A B AB, A B + A B AB, A B + V A ; V B ; V A B AB AB, A B + A i B j AB ij, A i B A i B j AB ij, A i B A B AB, A B + AB,, AB,,,, AB,, AB,, AB,,,, AB,, AB,, AB,,,, AB,, AB,, AB,,,, AB,, AB,, AB,,,, AB,, V G AB,, AB, ; VE,,, AB,, AB ijg,, AB ij, ijg AB ijg,, AB ijg,, AB ij, ijg AB ijg,, AB,, AB, AB,, Beuse of the othogonlit ondition on the five V vetos, the totl vition in n e deomposed into ontiutions ssoited with effets A, B, AB, G(Replition) nd Eo. ( i, j, g, k ) ( VA+ VB+ VAB+ VG+ VE) i( VA+ VB+ VAB+ VG+ VE) i j g k (38) A + B + AB + G + E ( ) A VAi VA A i (39) i Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 0 ( ) B VBi VB B j (40) j ( ) AB VABi VAB AB ij, A i B (4) i j ( ) G VGi VG AB ijg,, AB ij, (4) i j g E V ( ) Ei VE ijgk AB ijg,, (43) i j g k If the null hpothesis is tue then the men sque of the Effets A, B, AB nd G ll estimte the sme ommon popultion vine of eh A B fto omintion. he men sques e omputed s vition divided the ssoited degees of feedom. he Men Sque due to Fto A: ( ) A i (44) A i A he Men Sque due to Fto B: ( ) B j (45) B j B he Men Sque due to Intetion of A with B: ( AB ij, A i B ) AB i j AB ( ) he Men Sque due to Replition: ( AB ijg,, AB ij, ) G i j g G ( ) ( ) he Eo o Residul Men Sque: ( ijgk AB ijg,, ) E i j g k ( ) (46) (47) (48) Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge o test the null hpothesis tht ll effets mesue the sme ommon ndom vition ssoited with the Men Sque due to Eo, we ompute the oseved Fishe F soe. Effet Men Sque Fos (49) Fo given level of signifine,α, this vlue is omped ginst itil soe lulted fom n F distiution used to ompe two vines otined fom smpling vines fom two nomll distiuted popultions. he numeto degees of feedom is ppopite to the effet in question nd the denominto degees of feedom is (-). his infomtion is summized in wo-fto ANOVA tle. Soue Sum of Sques Degees of Feedom Fto A A - Fto B B - Intetion AB AB ( - )( - ) Replition G (-) Men Sque F os A A A AB AB B B B G G ( ) Eo E ( - ) E otl n - - s n ( ) AB G Eh omputed F os is omped ginst F α ( νeffet, ν E ) fo stted level of signifine, α. If Fos < F α ( νeffet, ν E ), we fil to ejet the null hpothesis tht the effet is just n lis fo ndom eo (vition). If F F α ( ν ν ) H nd onlude the effet in os > effet, E, we ejet 0 question is el. Note: It is possile tht eithe the AB Intetion Effet o the Replition Effet n test s signifint even when one o oth of the septe A Fto nd B Fto Effets e insignifint. o filitte the tul lultions, we define the following intemedite viles. ijg,, ijgk (50) Q ( ) ijg,, ijgk (5) k k ij, ijg,, (5) Qij, Qijg,, (53) g g C ( ) A i i, j, g (54) A i i, j (55) j g j Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge B j ij, (56) ij, (57) i i j ( ) A i i, j (58) QA i Qi, j (59) j j W ijg,, ijg,, (60) Now, ( ) ij, AB ij, (6) A i A i (6) B j B j (63) (64) n A i A ( A i) A i ( A ) i + + i i i i A i ( B j ) B j B ( B j) B j ( B j) + + j j j j ij, A i B j AB + i j A i A B j B j, i ij ij, + i j A i ij, B j A i B j ij, + i j j i j A i ( A i ) ( ij, ) ij, + B j + B j i j j j ( ij, ) ( A i) ( B j) B j + i j i j ( ij, ) ( A i) ( B j) + i j i j W A i ( A i) ( B j) + i i j Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 3 ( ij) ( ij) ijg ij ij,, G ( ijg,, ) ijg,, + i j g i j g g, WA i ( ijg,, ) CA i i j g i ( ijg) ( ijg) Qijg ( ijg),, ijg ijg,,,, E ijgk ( ijgk ) ijgk + i j g k i j g k k,,,, C ( ijgk ),, A i QA i i j g k i j g i Finll, s hek, the otl Vition must e the sum of the Effet Vitions. i, j, g, k ( i, j, g, k) QA i i j g k i j g k i A + B + AB + G + E ( A i) ( B j) + + WA i A ( i) ( B j) + i j i i j + CA i WA i + QA i CA i QA i i i i i i So sheme to lulte wo-fto with Replition nlsis of vine is to l out the dt in spedsheet, with djent ows in given olumn epesenting the diffeent A B Fto nd Replition omintions (i, j, g). Fo eh suh gouping of soes, lulte the sum of soes, ijg,, ijgk, nd the sum of sques of soes, Q ( ) ijg,, ijgk. hen lulte the k k ( ) ijg,, Q ijg,, ijg,, / smple men, AB ijg,, nd the smple vine, sab ijg,,. he ltte is lulted fo inspetion puposes. Bsed on the vlues of the smple vines is it esonle to ssume tht ll of the tetment nd eplition omintions hve the sme popultion vine? Next lulte the sums given equtions (5) though (59). Fom these esults onstut the wo-fto ANOVA tle nd test whethe n of the Effets e signifint. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 4 Soue Sum of Sques Men Sque F os A A ( A i ) A A A i B B ( B j ) B B B j AB AB W A i ( A i) ( B j) + AB AB AB ( )( ) i i j G Eo otl WA i G CA i i CA i E QA i i A + B + AB + G + E QA i i G G ( ) E ( ) s G hee Fto Anlsis Now onside thee ftos (tetments) A, B nd C with A done t levels nd B done t levels, C done t levels. Within given tetment omintion of A, B nd C levels leled i, j, m thee wee epeted mesuements of esponse vile. One gin, sine the nume of mesuements fo eh i, j, m level is the sme, the expeimentl design is lned. he k th soe in the i, j, m tetment omintion is designted s ijmk i is the tetment fto index fo fto A ; i j is the tetment fto index fo fto B ; j m is the tetment fto index fo fto C; m k lels the soe the within the i, j, m level ; k. he null hpothesis ssets tht no tetment popultion diffes fom n othe. hus, the soes in the tetment levels should ll hve the sme men nd vine. Agin we will ssume the soes e nomll distiuted with ommon vine within eh i, j, m tetment omintion. Null Hpothesis: H 0 : μijm,, μ0 fo i, j, m, whee μ0 is the ommon popultion men of the tetment popultions. All of the vition seen in the mesued soes is due to ndomness. he totl nume of soes is n. he gnd men is given Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 5 ijmk i j m k. (65) he smple men of the A B C tetment omintion i, j, m is given ijmk k ABC ijm,,. (66) he thee smple mens of the two-w omintions A nd B t levels i nd j, A nd C t levels i nd m, nd B nd C t levels j nd m e given the following. ijmk (67) m k AB ij, ijmk (68) j k AC im, ijmk (69) i k BC jm, Simil lultions give the smple mens of tetment A t level i, tetment B t level j, nd tetment C t level m. ijmk (70) j m k A i B j ijmk (7) i m k ijmk (7) i j k C m Agin these definitions impl onstints fo the sums of devitions. ( A i ) 0 (73) ( B j ) i ( C m ) 0 (75) ( AB ij, A i) m ( AB ij, B j) 0 i ( AC im, C m) 0 i ( BC jm, C m) 0 j 0 (74) j 0 (76) j 0 (78) m BC jm, B j 0 (80) m ABC i, j, m BC j, m (8) i (77) ( AC im, A i) (79) ( ) (8) ( ) 0 ( ABC i, j, m AC i, m ) 0 (83) ( ABC i, j, m ABi, j ) 0 j (84) m ( ABC ijm,, A i) 0 (85) ( ABC ijm,, B j) j m 0 (86) i m Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 6 ( ABC ijm,, C m) 0 (87) ( i, j, m, k ABC i, j, m ) 0 i j he effets in the hee Fto nlsis of e listed s follows: (88) k he Effet due to Fto A t level i: he Effet due to Fto B t level j: he Effet due to Fto C t level m: A i B j C m he wo-fto Intetion Effet of A t level i nd B t level j: ( ) ( ) ( ) AB ij, A i B j AB ij, A i B he wo-fto Intetion Effet of A t level i nd C t level m: ( ) ( ) ( ) + AC im, A i C m AC im, A i C m he wo-fto Intetion Effet of B t level j nd C t level m: ( ) ( ) ( ) BC jm, B j C m BC jm, B j C m+ he hee-fto Intetion Effet of A t level i, B t level j, nd C t level m: ABC ijm,, AB ij, A i B AC im, A i C m+ BC jm, B j C m+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + + A i B j C m ABC i, j, m AB i, j AC i, m BC j, m A i B j C m he Residul o Eo Effet: ijmk AB ijm,, Now sine the sums of devitions is zeo, i.e., fom equtions (73) though (87), the sum ove n ouing index of the Fto effets, the wo-fto intetions nd the hee-fto intetion vnishes. Fo exmple, the sum ove j of the hee-fto Intetion is zeo equtions (83), (76), (8) nd (74). ( ABC ijm,, AB ij, AC im, BC jm, + A i+ B C m ) j ( ABC ijm,, AC im, ) + ( A i AB ij, ) + ( C m BC jm, ) + ( B j ) j j j j 0+ 0+ 0+ 0 0 hus, fo the hee-fto intetion onl the fist (-) of the j omponents e independent. Simil onstints ppl to the i nd m omponents, so the degees of feedom of the hee- Fto intetion is (-)(-)(-). he degees of feedom ssoited with Effets A, B, C nd the wo-fto intetions e wht we would expet fom the wo Fto Anlsis. Fom eqution (88), fo eh of the omintions of A, B nd C ftos, thee e - independent devitions, ijmk ABC ijm,,. Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 7 he following tle summizes the degees of feedom,ν, ssoited with eh effet. Effet Degees of Feedom Gnd Men ν Fto A ν A Fto B ν B Fto C ν C wo-fto Intetion of Ftos AB ν AB ( )( ) wo-fto Intetion of Ftos AC ν AC ( )( ) wo-fto Intetion of Ftos BC ν BC ( )( ) hee-fto Intetion of Ftos ABC ν ABC ( )( )( ) Residul o Eo ν ( ) Note tht the sum of the degees of feedom is still the totl nume of soes n. If the null hpothesis is tue nd ll of the tetment omintions give the sme men esult, nd the eight effets just listed ll mesue the sme ndom deptue fom the gnd men. Fomll, we n wite the following. E ijgk + ( A i ) + ( B j ) + ( C m ) ( AB ij, A i B j ) ( AC ij, A i C m ) ( BC im, B j C m ) ( ABC ijm,, AB ij, AC im, BC jm, A i B j C m ) ( ijmk ABC ijm,, ) + + + + + + + + + + + (89) he zeo sum onstint of eh effet insues tht eqution (89) desies n othogonl veto deomposition of the esponse vile. he olumn veto whih gives the totl devition fom the gnd men, ijmk, n e deomposed s the sum of the eight mutull othogonl olumn vetos, VA + VB + VC + VAB + VAC + VBC + VABC + VE, with eh veto ssoited with n effet. he othogonlit ondition on the eight V vetos mens tht the totl vition in n e deomposed into ontiutions ssoited with the eight effets A, B, C, AB, AC, BC, ABC nd Eo. ( i, j, m, k ) i j m k ( V V V V V V V V ) i( V V V V V V V V ) + + + + + + + + + + + + + + A B C AB AC BC ABC E A B C AB AC BC ABC + + + + + + + E A B C AB AC BC ABC A A A A i i ( ) V i V E (90) (9) Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 8 B Bi B ( B j ) (9) j C Ci C ( C m ) (93) m ( ) AB ABi AB AB ij, A i B (94) i j AC AC i AC ( AC im, A i C m+ ) (95) i m ( ) BC BCi BC BC jm, B j C m+ (96) j m ( ) ABC ABC i ABC ABC ijm,, AB ij, AC im, BC jm, + A i+ B C m (97) i j m ( ) Ei E i, j, m, k AB i, j, m (98) i j m k V V V V V V V V V V V V E V V If the null hpothesis is tue then the men sque of the Effets A, B, C, AB, AC, BC nd ABC ll estimte the sme ommon popultion vine of eh A B fto omintion. he men sques e omputed s vition divided the ssoited degees of feedom. he Men Sque due to Fto A: he Men Sque due to Fto B: he Men Sque due to Fto C: A B C ( ) A i A i ( ) B j B j ( ) C m C m (99) (00) (0) he Men Sque due to thewo-fto Intetion of A with B: ( AB ij, A i B ) (0) AB i j AB Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 9 he Men Sque due to thewo-fto Intetion of A with C: ( AC im, A i C m+ ) (03) AC i m AC he Men Sque due to thewo-fto Intetion of B with C: ( BC jm, B j C m+ ) (04) BC j m BC he Men Sque due to thehee-fto Intetion of A,B nd C: ABC ABC ( )( )( ) ( ABC ijm,, AB ij, AC im, BC jm, + A i+ B C m ) i j m ( ) ( )( )( ) he Eo o Residul Men Sque: ( ijmk ABC ijm,, ) E i j m k ( ) (05) (06) o test the null hpothesis tht ll effets mesue the sme ommon ndom vition ssoited with the Men Sque due to Eo, we ompute the oseved Fishe F soe. Effet Men Sque Fos (07) Fo given level of signifine,α, this vlue is omped ginst itil soe lulted fom n F distiution used to ompe two vines otined fom smpling vines fom two nomll distiuted popultions. he numeto degees of feedom is ppopite to the effet in question nd the denominto degees of feedom is (-). his infomtion is summized in hee-fto ANOVA tle. Soue Sum of Sque s Degees of Feedom Fto A A - Fto B B - Fto C C - Men Sque F os A A A C C B B B C Al Lehnen Mdison Ae ehnil College 5/3/00
Multiple Fto ANOVA Notes Pge 0 Intetion AB AB ( - )( - ) Intetion AC AC ( - )( - ) Intetion BC BC ( - )( - ) AB AC BC AB ( )( ) AC ( )( ) BC ( )( ) ABC Intetion ABC ABC ( - )( - ) (-) ( )( )( ) ABC AB AC BC ABC Eo E ( - ) E otl n - - s n ( ) Eh omputed F os is omped ginst F α ( νeffet, ν E ) fo stted level of signifine, α. If Fos < F α ( νeffet, ν E ), we fil to ejet the null hpothesis tht the effet is just n lis fo ndom eo (vition). If F F α ( ν ν ) H nd onlude the effet in question is el. os > effet, E, we ejet 0 As this pesenttion demonsttes when thee e mn ftos the nlsis is quite omplited due to ll of the intetion effets. In ddition, the nume of expeimentl uns ineses s the podut of the nume of levels of eh fto. o ese time, expense nd nlsis often onl two levels e onsideed fo eh fto. Of ouse this esults in loss of detil, howeve, when thee e mn ftos pilot expeiment done t two levels pe fto n e used to eliminte insignifint ftos. Suh n expeiment un with n ftos is lled n Ftoil Expeimentl Design. he use of onl two levels onsidel edues the omplexit of the nlsis. Still n is ig nume fo lge n. Fo exmple, ten ftos, whih in n industil poess is not unesonle, would equie 04 tetment goups ignoing eplitions! his is usull still too expensive. Fotuntel, most highe ode fto intetions tun out to e insignifint. Using the Design Mtix of the n Ftoil Design, one n lis ftos with speifi highe ode intetions to edue the nume of tetment goups. his poedue is lled Ftionl Ftoil Expeimentl Design. he speifi detils of n Ftoil Expeimentl Designs nd Ftionl Ftoil Expeimentl Designs e eond the sope of these notes ut e povided in the text. Al Lehnen Mdison Ae ehnil College 5/3/00