S3-. The Defto of Fctor Effects Chpter 3 Supplemetl Text Mterl As oted Sectos 3- d 3-3, there re two wys to wrte the model for sglefctor expermet, the mes model d the effects model. We wll geerlly use the effects model =,,, yj = µ + τ + εjt j =,,, where, for smplcty, we re workg wth the blced cse (ll fctor levels or tretmets re replcted the sme umber of tmes). Recll tht wrtg ths model, the th fctor level me µ s broke up to two compoets, tht s µ µ + τ, where µ τ s the th tretmet effect d µ s overll me. We usully defe µ = = d ths mples tht τ = = 0. Ths s ctully rbtrry defto, d there re other wys to defe the overll me. For exmple, we could defe R S µ = wµ where w = = = Ths would result the tretmet effect defed such tht w τ = 0 = Here the overll me s weghted verge of the dvdul tretmet mes. Whe there re uequl umber of observtos ech tretmet, the weghts w could be tke s the frctos of the tretmet smple szes /. S3-. Expected Me Squres I Secto 3-3. we derved the expected vlue of the me squre for error the sglefctor lyss of vrce. We gve the result for the expected vlue of the me squre for tretmets, but the dervto ws omtted. The dervto s strghtforwrd. Cosder ow for blced desg d the model s F HG E( MS E SS Tretmets) = SS Tretmets Tretmets = y y =... I K J =
y = µ + τ + ε j j I ddto, we wll fd the followg useful: ow R S T =,,, j =,,, E( ε ) = E( ε ) = E( ε ) = 0, E( ε ) = σ, E( ε ) = σ, E( ε ) = j... j... ESS ( ) = E( y ) E( y ) Tretmets =.. Cosder the frst term o the rght hd sde of the bove expresso: E( y ) = E ( + + µ τ ε ).. = = Squrg the expresso pretheses d tkg expectto results E( y. ) [ ( ) = µ + τ + σ ] = = = µ + τ + σ becuse the three cross-product terms re ll zero. ow cosder the secod term o the rght hd sde of ( ): sce ESS Tretmets = F I HG K J = + + E y E.. ( µ τ ε.. ) = E ( µ + ε.. ) τ = 0. Upo squrg the term pretheses d tkg expectto, we obt = = F I HG K J = + E y.. [( µ ) σ ] = µ + σ sce the expected vlue of the cross-product s zero. Therefore, ESS ( ) = E( y ) E( y ) Tretmets = = µ + τ + σ ( µ + σ ) = σ ( ) +... = τ = σ
Cosequetly the expected vlue of the me squre for tretmets s Ths s the result gve the textbook. F HG E( MS E SS Tretmets) = Tretmets τ = σ ( ) + = τ = σ + I K J S3-3. Cofdece Itervl for σ I developg the lyss of vrce (AOVA) procedure we hve observed tht the error vrce σ s estmted by the error me squre; tht s, σ = σ SSE We ow gve cofdece tervl for. Sce we hve ssumed tht the observtos re ormlly dstrbuted, the dstrbuto of s where. Therefore, d P α/, α/, F SS σ SS E σ E I α/, α/, HG K J = re the lower d upper α/ percetge pots of the dstrbuto wth - degrees of freedom, respectvely. ow f we rerrge the expresso sde the probblty sttemet we obt P F HG SS SS E E σ α/, α/, I α α KJ = Therefore, 00(-α) percet cofdece tervl o the error vrce σ s SS E σ SS α/, α/, Ths cofdece tervl expresso s lso gve Chpter o expermets wth rdom effects. E
Sometmes expermeter s terested upper boud o the error vrce; tht s, how lrge could σ resobly be? Ths c be useful whe there s formto bout σ from pror expermet d the expermeter s performg clcultos to determe smple szes for ew expermet. A upper 00(-α) percet cofdece lmt o σ s gve by σ SS E α, If 00(-α) percet cofdece tervl o the stdrd devto σ s desred sted, the σ SS E α /, S3-4. Smulteous Cofdece Itervls o Tretmet Mes I secto 3-3.3 we dscuss fdg cofdece tervls o tretmet me d o dffereces betwee pr of mes. We lso show how to fd smulteous cofdece tervls o set of tretmet mes or set of dffereces betwee prs of mes usg the Boferro pproch. Essetlly, f there re set of r cofdece sttemets to be costructed the Boferro method smply replces α/ by α/(r). ths produces set of r cofdece tervls for whch the overll cofdece level s t lest 00(-α) percet. To see why ths works, cosder the cse where r = ; tht s, we hve two 00(-α) percet cofdece tervls. Let E deote the evet tht the frst cofdece tervl s ot correct (t does ot cover the true me) d E deote the eve tht the secod cofdece tervl s correct. ow PE ( ) = PE ( ) = α The probblty tht ether or both tervls s correct s PE ( E) = PE ( ) + PE ( ) PE ( E) From the probblty of complmetry evets we c fd the probblty tht both tervls re correct s ow we kow tht PE ( E) Boferro equlty PE ( E) = PE ( E) 0 = PE ( ) PE ( ) + PE ( E ), so from the lst equto bove we obt the PE ( E) PE ( ) PE ( ) I the cotext of our exmple, the left-hd sde of ths equlty s the probblty tht both of the two cofdece tervl sttemets s correct d PE ( ) = PE ( ) = α, so
PE ( E) α α α Therefore, f we wt the probblty tht both of the cofdece tervls re correct to be t lest -α we c ssure ths by costructg 00(-α/) percet dvdul cofdece tervl. If there re r cofdece tervls of terest, we c use mthemtcl ducto to show tht PE ( E E) PE ( ) r r = rα As oted the text, the Boferro method works resobly well whe the umber of smulteous cofdece tervls tht you desre to costruct, r, s ot too lrge. As r becomes lrger, the legths of the dvdul cofdece tervls crese. The legths of the dvdul cofdece tervls c become so lrge tht the tervls re ot very formtve. Also, t s ot ecessry tht ll dvdul cofdece sttemets hve the sme level of cofdece. Oe mght select 98 percet for oe sttemet d 9 percet for the other, resultg two cofdece tervls for whch the smulteous cofdece level s t lest 90 percet. S3-5. Regresso Models for Qutttve Fctor Regresso models re dscussed detl Chpter 0, but they pper reltvely ofte throughout the book becuse t s coveet to express the reltoshp betwee the respose d qutttve desg vrbles terms of equto. Whe there s oly sge qutttve desg fctor, ler regresso model reltg the respose to the fctor s y = β + β x+ ε 0 where x represets the vlues of the desg fctor. I sgle-fctor expermet there re observtos, d ech observto c be expressed terms of ths model s follows: y = β + β x + ε, =,,, 0 The method of lest squres s used to estmte the ukow prmeters (the β s) ths model. Ths volves estmtg the prmeters so tht the sum of the squres of the errors s mmzed. The lest squres fucto s L= ε = ( y β0 βx ) = = To fd the lest squres estmtors we tke the prtl dervtves of L wth respect to the β s d equte to zero:
L = ( y β0 βx) = 0 β 0 = L = ( y β0 βx) x = 0 β After smplfcto, we obt the lest squres orml equtos = β + β x = 0 = = x β0 + βx = xy = = = where β 0 d β re the lest squres estmtors of the model prmeters. So, to ft ths prtculr model to the expermetl dt by lest squres, ll we hve to do s solve the orml equtos. Sce there re oly two equtos two ukows, ths s frly esy. I the textbook we ft two regresso models for the respose vrble etch rte (y) s fucto of the RF power (x); the ler regresso model show bove, d qudrtc model y = β + β x+ β x + ε x x 3 β + β + β x = x y 0 The lest squres orml equtos for the qudrtc model re x β + β + β x = 0 = = = 0 = = = = x 3 β β x 4 + + β x = x 0 = = = = y Obvously s the order of the model creses d there re more ukow prmeters to estmte, the orml equtos become more complcted. I Chpter 0 we use mtrx methods to develop the geerl soluto. Most sttstcs softwre pckges hve very good regresso model fttg cpblty. y y S3-6. More About Estmble Fuctos I Secto 3-9. we use the lest squres pproch to estmtg the prmeters the sgle-fctor model. Assumg blced expermetl desg, we fe the lest squres orml equtos s Equto 3-48, repeted below:
µ + τ + τ + + τ = y µ + τ = y µ + τ = y µ + τ = = j= j j= j j= where = s the totl umber of observtos. As oted the textbook, f we dd the lst of these orml equtos we obt the frst oe. Tht s, the orml equtos re ot lerly depedet d so they do ot hve uque soluto. We sy tht the effects model s overprmeterzed model. Oe wy to resolve ths s to dd other lerly depedet equto to the orml equtos. The most commo wy to do ths s to use the equto j= y j j τ = 0. Ths s cosstet wth defg the fctor effects s devtos from the overll me µ. If we mpose ths costrt, the soluto to the orml equtos s µ = y τ = y y, =,,, Tht s, the overll me s estmted by the verge of ll smple observto, whle ech dvdul fctor effect s estmted by the dfferece betwee the smple verge for tht fctor level d the verge of ll observtos. Aother possble choce of costrt s to set the overll me equl to costt, sy µ = 0. Ths results the soluto µ = 0 τ = y, =,,, Stll thrd choce s τ = 0. Ths s the pproch used the SAS softwre, for exmple. Ths choce of costrt produces the soluto µ = y τ = y y, =,,, τ = 0 There re fte umber of possble costrts tht could be used to solve the orml equtos. Fortutely, s observed the book, t relly does t mtter. For ech of the three solutos bove (deed for y soluto to the orml equtos) we hve µ = µ + τ = y, =,,, =
Tht s, the lest squres estmtor of the me of the th fctor level wll lwys be the smple verge of the observtos t tht fctor level. So eve f we cot obt uque estmtes for the prmeters the effects model we c obt uque estmtors of fucto of these prmeters tht we re terested. Ths s the de of estmble fuctos. Ay fucto of the model prmeters tht c be uquely estmted regrdless of the costrt selected to solve the orml equtos s estmble fucto. Wht fuctos re estmble? It c be show tht the expected vlue of y observto s estmble. ow E( y j )= µ + τ so s show bove, the me of the th tretmet s estmble. Ay fucto tht s ler combto of the left-hd sde of the orml equtos s lso estmble. For exmple, subtrct the thrd orml equto from the secod, yeldg τ τ. Cosequetly, the dfferece y two tretmet effect s estmble. I geerl, y cotrst the tretmet effects c τ where c = = = 0 s estmble. otce tht the dvdul model prmeters µ, τ,, τ re ot estmble, s there s o ler combto of the orml equtos tht wll produce these prmeters seprtely. However, ths s geerlly ot problem, for s observed prevously, the estmble fuctos correspod to fuctos of the model prmeters tht re of terest to expermeters. For excellet d very redble dscusso of estmble fuctos, see Myers, R. H. d Mlto, J. S. (99), A Frst Course the Theory of the Ler Model, PWS-Ket, Bosto. MA. S3-7. The Reltoshp Betwee Regresso d AOVA Secto 3-9 explored some of the coectos betwee lyss of vrce (AOVA) models d regresso models. We showed how lest squres methods could be used to estmte the model prmeters d how the AOVA c be developed by regressobsed procedure clled the geerl regresso sgfcce test c be used to develop the AOVA test sttstc. Every AOVA model c be wrtte explctly s equvlet ler regresso model. We ow show how ths s doe for the sgle-fctor expermet wth = 3 tretmets. The sgle-fctor blced AOVA model s The equvlet regresso model s y = µ + τ + ε j j R S T y = β + β x + β x + ε j 0 j j j = 3,, j =,,, R S T = 3,, j =,,,
where the vrbles x j d x j re defed s follows: x x j j R f observto j s from tretmet = S T R 0 otherwse f observto j s from tretmet = S T 0 otherwse The reltoshps betwee the prmeters the regresso model d the prmeters the AOVA model re esly determed. For exmple, f the observtos come from tretmet, the x j = d x j = 0 d the regresso model s y j 0 = β + β () + β ( 0) + ε = β + β + ε 0 j Sce the AOVA model these observtos re defed by y j = µ + τ + ε j, ths mples tht β 0 + β = µ = µ + τ Smlrly, f the observtos re from tretmet, the y j 0 = β + β ( 0) + β ( ) + ε = β + β + ε 0 j d the reltoshp betwee the prmeters s β 0 + β = µ = µ + τ Flly, cosder observtos from tretmet 3, for whch the regresso model s d we hve y3 j 0 = β + β ( 0) + β ( 0) + ε 3 = β + ε 0 3j β 0 = µ 3 = µ + τ 3 Thus the regresso model formulto of the oe-wy AOVA model, the regresso coeffcets descrbe comprsos of the frst two tretmet mes wth the thrd tretmet me; tht s β = µ 0 3 β = µ µ β = µ µ I geerl, f there re tretmets, the regresso model wll hve regressor vrbles, sy y = β + β x + β x + + β x + ε j 0 j j j 3 3 j j j R S T =,,, j =,,,
where x j = R S T f observto js from tretmet 0 otherwse Sce these regressor vrbles oly tke o the vlues 0 d, they re ofte clled dctor vrbles. The reltoshp betwee the prmeters the AOVA model d the regresso model s β = µ 0 β = µ µ, =,,, Therefore the tercept s lwys the me of the th tretmet d the regresso coeffcet β estmtes the dfferece betwee the me of the th tretmet d the th tretmet. ow cosder testg hypotheses. Suppose tht we wt to test tht ll tretmet mes re equl (the usul ull hypothess). If ths ull hypothess s true, the the prmeters the regresso model become β µ = 0 β = 0, =,,, Usg the geerl regresso sgfcce test procedure, we could develop test for ths hypothess. It would be detcl to the F-sttstc test the oe-wy AOVA. Most regresso softwre pckges utomtclly test the hypothess tht ll model regresso coeffcets (except the tercept) re zero. We wll llustrte ths usg Mtb d the dt from the plsm etchg expermet Exmple 3-. Recll ths exmple tht the egeer s terested determg the effect of RF power o etch rte, d he hs ru completely rdomzed expermet wth four levels of RF power d fve replctes. For coveece, we repet the dt from Tble 3- here: RF Power Observed etch rte (W) 3 4 5 60 575 54 530 539 570 80 565 593 590 579 60 00 600 65 60 637 69 0 75 700 75 685 70 The dt ws coverted to the x j 0/ dctor vrbles s descrbed bove. Sce there re 4 tretmets, there re oly 3 of the x s. The coded dt tht s used s put to Mtb s show below:
x x x3 Etch rte 0 0 575 0 0 54 0 0 530 0 0 539 0 0 570 0 0 565 0 0 593 0 0 590 0 0 579 0 0 60 0 0 600 0 0 65 0 0 60 0 0 637 0 0 69 0 0 0 75 0 0 0 700 0 0 0 75 0 0 0 685 The Regresso Module Mtb ws ru usg the bove spredsheet where x through x3 were used s the predctors d the vrble Etch rte ws the respose. The output s show below. Regresso Alyss: Etch rte versus x, x, x3 The regresso equto s Etch rte = 707-56 x - 0 x - 8.6 x3 Predctor Coef SE Coef T P Costt 707.000 8.69 86.54 0.000 x -55.80.55-3.49 0.000 x -9.60.55-0.35 0.000 x3-8.60.55-7.06 0.000 S = 8.675 R-Sq = 9.6% R-Sq(dj) = 9.% Alyss of Vrce Source DF SS MS F P Regresso 3 6687 90 66.80 0.000 Resdul Error 6 5339 334
otce tht the AOVA tble ths regresso output s detcl (prt from roudg) to the AOVA dsply Tble 3-4. Therefore, testg the hypothess tht the regresso coeffcets β = β = β 3 = β 4 = 0 ths regresso model s equvlet to testg the ull hypothess of equl tretmet mes the orgl AOVA model formulto. Also ote tht the estmte of the tercept or the costt term the bove tble s the me of the 4 th tretmet. Furthermore, ech regresso coeffcet s just the dfferece betwee oe of the tretmet mes d the 4 th tretmet me.