Much ado about nothing: the mixed models controversy revisited

Transcription

1 Much do out nothing: the mixed model controvery reviited Vivin etriz Lencin eprtmento de Metodologí e Invetigción, FM Univeridd Ncionl de Tucumán Julio d Mott Singer eprtmento de Ettític, IME Univeridde de São Pulo Edwrd J. Stnek III eprtment of iottitic nd Epidemiology, SPH Univerity of Mchuett t Amhert. - Introduction Coniderle ttention h een given to the o clled mixed model controvery, which i pprently relted to the exitence of two different wy for teting the min effect of the rndom fctor in the nlyi of n r-replicte fctoril experiment, with fixed fctor (A) nd rndom fctor (). The prolem tem from two common model ued in the nlyi nd h een recently ddreed y Schwrz (993), Vo (999) nd dicued y Hinkelmnn (000), Wolfinger nd Stroup (000). Adopting the uul ANOVA nottion, one of the model, termed the uncontrined prmeter (UP) model y Vo (999), my e expreed,...,, =,...,, k =,..., r with ( 0, ) Y = µ + α + + (α) + E () ik i i ik αi = 0 nd ~ ( 0, ), ( ) ~ ( 0, ) N α i N α nd E ik ~ N repreenting independent rndom vrile. Under model (), i the term ocited with (the rndom) fctor. Mny uthor (like Serle (97), Milliken nd Johnon

2 (994) nd the SAS (990) oftwre, for exmple) tke H : 0 the hypothei of inexitence of fctor min effect, concluding tht 0 = MS MSA i the pproprite tet ttitic. The competing model, termed the contrined prmeter (CP) model y Vo (999) correpond to Y = η + τ + + (τ) + E () ik i i ik,...,, =,...,, k =,..., r with = 0 τ i, ~ N( 0, ), ( τ) i N{ 0, ( ) τ } nd ~ N( 0, ) E ik repreenting independent rndom vrile; dditionlly, ( τ ) = 0, i =,...,, which implie Cov( ( τ) i, ( τ) i ) = τ for i i. Under model (), i the term ocited with fctor, nd the hypothei H : 0 i uully interpreted tht of 0 = inexitence of fctor min effect; the tndrd tet reect H 0 if MS MSE i ufficiently lrge. Thi i the pproch conidered in Neter et l. (996), Montgomery (99) nd other. The two tet my generte conflicting reult for teting wht i thought the fctor min effect. The ource of the "controvery" i ocited to the choice of the pproprite model. Tking µ = η, αi = τi, = τ nd =, Hocking (973) how tht the α τ UP nd CP model re equivlent, nd tht oth re pecil ce of more generl model propoed y Scheffé (959). Therefore, the hypothei H = under the UP model i : 0 0 equivlent to H : / 0 0 τ = under the CP model; lterntively, the hypothei H = under the CP model i equivlent to : 0 0 H : / 0 0 α + = under the UP model. Thu, it eem tht the mixed model controvery i not relted to the choice etween two contending model ut to the definition of the rndom fctor min effect nd it expreion in

3 term of the component of ech model. Hocking (973) ddree thi iue, ut he ugget tht the definition of the min effect hould e hndled y the reercher. Vo (999), on the other hnd, ttempt to reolve the controvery y emedding oth model in finite popultion etting (which he term uperpopultion) in which ll poile level of the rndom fctor () re included nd define the min effect of in fixed effect model. He then relte either the rndom vrile in the CP model, or the rndom vrile + ( ) α ) ( α, with ) (α, nd ( ) α denoting ( ), α nd i = = ( α ) repectively, in the UP model to the min effect. Then he how vi i conditioning rgument tht under either model E ( MS) meure error vriility plu "min effect of ", concluding tht the pproprite tet ttitic for the inexitence of fctor min effect i MS MSE in either model. Wolfinger nd Stroup (000) comment tht plcing oth the UP nd the CP model in the finite popultion etup doe not reolve the controvery ecue thi i where the CP model h it theoreticl i. They recommend ue of the UP model coupled with likelihood-ed inference pproch nd dicu the hypothee in term of the covrince tructure. They do not try to interpret the min effect of the rndom fctor, nor the vrince component in term of potentilly oervle quntitie. We dd two criticim to the pproch tken y Vo (999): firt, hi rgument tht E ( MS) meure error vriility plu "min effect" doe not eem to follow clerly, ince he ocite with the min effect without explicitly howing how. Furthermore, he doe not extend hi reult to the ce where the numer of level of i infinite, i.e. to the ce where oth model my e et ccommodted. The poor or indequte definition of the min effect of the rndom fctor eem to e the mor cue of the miundertnding nd the min oective of thi pper i to pecify the

4 reltion etween the min effect of the rndom fctor nd the prmeter in oth model. Here we ring into perpective the relevnce of teting the min effect of in the preence of interction. Thi my e reonle lterntive when the interction i non-eentil indicted in Neter et l. (996), for exmple. Keeping in mind tht min effect my e eily undertood chnge on expected vlue, in Section we refer to Scheffé (959) nd define min effect nd interction under oth the UP nd CP model howing how they re relted to the correponding prmeter. In Section 3, we indicte how the (finite) uperpopultion pproch conidered in Vo (999) my e extended to the infinite ce. Finlly, in Section 4 we dicu computtionl pect nd extenion to the unlnced ce.. efinition of min effect nd interction in mixed model Hocking (973) how tht the UP nd CP model re pecil ce of the following model preented y Scheffé (959),...,, =,...,, k =,..., r, with ( 0, ) Y = µ + α + M + E, (3) ik i i ik αi = 0, M N i (0, p + q ), i i' Cov( M, M ) = q nd E ik ~ N denoting independent rndom vrile which in turn re independent of the M i. We mut ditinguih two ource of vrition in Y ik, nmely, the vriility due to the rndom election of the level of fctor nd the nturl vriility of the repone vrile (error vriility). From model (3) it follow tht Vr( M,, M ) = pi+ qj where I denote n identity mtrix nd J denote mtrix with ll element equl to. Under the UP prmeteriztion, we hve M ( α ) = + with N( 0, ) nd ( ) i ~ N( 0, α) i i α o tht p α = nd

5 q =. On the other hnd, under the CP prmeteriztion, Mi = + ( τ ) i with ~ N( 0, ), ( τ) i N{ 0,( ) τ }, Cov = o tht (( τ ),( i τ ) i' ) τ / p = τ nd q = τ /. Given the ove etup, we my define min effect nd interction in term of expected vlue, ut in the ce of fixed effect model. Let Ω denote the et of ll level of fctor nd Θ denote the et of ll unit nd oerve tht Y ik, M i nd E ik re rel vlued rndom vrile defined on the pce Ω Θ. Alo, note tht for fixed ω Ω, Yik ( ω,) i, Mi( ω,) i nd Eik ( ω,) i re rndom vrile defined on Θ. Furthermore, oerve tht M i depend only on the level of fctor, i.e. Mi( ωθ, ) = Mi ( ωθ, ) for ll θ, θ Θ. Under thi context, the min effect of level i of fctor A my e defined ( ii ) ( ii ) E Y (,) E Y (,) = ( µ + α ) µ = α Ω Θ iii Ω Θ iii i i o tht the hypothei of no fctor A min effect i α i = 0,,...,. Similrly, for every ω Ω, the min effect of level ω of fctor my e defined ( ω ) i i ( i i ) EΘ Y (,) i E E Y (,) Ω Θ ii = ( ω). Given tht the rndom vector ( M,, M ) re independent nd identiclly ditriuted, it follow tht o re the Y i i nd conequently, E E ( Y (,)) E E ( Y (,)) ii = iii ii. Thu, from Ω Θ i i Ω Θ (3), we hve ( ω) = EΘ ( Mi( ω,) ) = EΘ( Mi ( ω,) ) i i, which i function only of ω Ω. Therefore, teting tht there re no fctor min effect correpond to teting tht (the rndom vrile) E ( M ω ) i (,) i 0 lmot urely, which, in turn, i equivlent to Θ = ( ( ( ))) VrΩ EΘ Mi ii, = 0. From () nd (), it i ey to prove tht thi correpond to

6 + α / = 0 under the UP prmeteriztion or to = 0 under the CP prmeteriztion, o tht in oth ce, the pproprite tet i ed on MS / MSE. 3. The Finite Popultion Model Here we follow Vo (999) pproch ed on finite popultion of ( ) level of the rndom fctor. Conidering ll fctor level, we hve fixed effect model like Yi ω k µ i ω εi ω k = + (4) where, for i =,...,, ω =,..., nd k,..., r = we let ( ) E Y ω i k = µ i ω nd iω k ε e independent ( 0, ) N rndom vrile. Inexitence of fctor min effect correpond to µ µ = 0, ω =,..., iω ii. For the prolem under invetigtion, we ctully oerve the repone only on imple rndom mple without replcement (r) of the level of fctor. The correponding model my e expreed Yik = M i + Eik (5) i =,...,, =,..., ( < ), k,..., r = where nd M = U µ, E = U ε i ω iω ik ω iωk ω= ω= U ω = if the th elected level of i level ω nd 0 U ω = otherwie. Thi implie tht Eiω k~ N( 0, ) i =,...,, ω =,...,, k,..., r = re mutully independent like the εi ωk in the fixed effect model (4). Thi model will e clled the uperpopultion model in Vo (999). Since the level of fctor re mpled ccording to r cheme, it i ey to how U tifie E ( U ) =, V ( ) = I J tht = ( U U U ) Cov ( U ) U, = I J. ( ) U nd

7 efining the min effect for level ω of fctor in Section, we hve ( ω ) = µ µ o () = M () µ, iω ii i i i rndom vrile nd conequently, under model (5), Vr( M ) = 0 lo chrcterize the inexitence of fctor min effect. Letting ( i) = ì i I J ì i nd A ( i, i ) = ì i I J ì i repectively denote the finite popultion vrince of {,..., i } µ µ nd the finite popultion covrince etween ì = ( µ µ µ ) nd = ( µ µ µ ) ( ) i i i i i' i' i' i ' V M Vr M i ì, it follow tht ( ) ( ) i = i = + () i A ( ii, ) ( ) i = i i Conequently, Cov (, µ ) 0 i = i i = µ chrcterize the inexitence of min effect of the rndom fctor under (5). We my compre the role plyed y the different prmeter in the different model vi n exmintion of the correponding covrince tructure, i.e. V ( Y), ( Y) UP CP V nd ( Y ) lim V ( Y) V = with = lim nd lim A( i, i = ) A ( i, i. The relevnt expreion ) ( i) ( i) re diplyed in Tle. Tle : Centrl econd order moment under UP, CP nd the limit verion of the Finite Popultion Model with ( ). Cov Moment UP Model CP Model ( ) Y ik V + α + ( Y Y ), k k ik, ik α + τ + + τ Finite Popultion Model with ( i) + + (i)

8 Cov Cov Noting tht V ( Y Y ), i i ik, ik τ ( Y Y ), ik i k A( i, i ), ( M ) = lim V ( M ) = lim + lim ( ) ( i) ( ) i = i i A ( i, i ), nd following Wilk nd Kempthorne (955), Kempthorne (957), rownlee (960) nd Hocking (973), who interpret the mixed model under finite popultion perpective, we my conclude from Tle, tht V ( M ) = { + + ( ) } = + under the UP model, nd V α ( M ) = + + ( ) under the CP model. α τ τ = Given the forementioned equivlence etween the prmeter of the different model, we my conclude tht teting for the inexitence of the rndom fctor effect correpond to teting tht ( M ) = 0 V under the limiting form of the finite popultion model, + = 0 α under the UP model or = 0 under the CP model. 4. Teting vrince component under the UP or the CP model Although it eem cler tht oth the UP nd the CP model my e employed equivlently to tet for the inexitence of the rndom fctor min effect, mny uthor like Hinkelmnn (000) nd Wolfinger nd Stroup (000) dicourge the ue of the CP model on

9 ccount of the lck of flexiility of the reulting LUP, the ville oftwre for the UP model well the difficulty in deling with incomplete or miing dt. It i true tht Proc Mixed in SAS i ueful tool for the nlyi of unlnced mixed model, ut we hve to keep in mind tht it doe not provide exct tet for null vrince component. In fct, the ville tet re only roughly pproximte nd do not ccount for the fct tht under the null hypothei, the prmeter lie in the oundry of the prmetric pce. Approprite lrge mple tet mut e contructed ccording to the uggetion of Strm nd Lee (994). Öfverten (993) propoe n exct tet for vrince component in unlnced mixed liner model tht coincide, in lnced ce, to the uul exct F tet. Thi tet cn e ued either under the UP or the CP model, ut it i not ville in ttiticl oftwre yet. Uing dt et preented in McLen et l. (99), we compre the different tet ville in SAS with tht propoed y Öfverten (993). For the unlnced dt the only exct tet for vrince component i tht preented y the ltter uthor. The reult re diplyed in Tle nd 3. Tle : Tet for vrince component for the lnced dt exmple in McLen et l. (993) Vrince Component ANOVA proc ANOVA proc GLM F =.95 (df =, 6) p = F = 3.85 (df =, ) p = 0.06 F = 3.0 (df =, 6) p = 0.88 Exct Tet Ofverten (993) F =.95 (df=, 6). p = Wld Tet proc MIXE Z = 0.9 P= F = 3.85 (df =, ) p = 0.06 Z = 0.7 P= F = 3.0 (df =, 6). p = 0.88 Z = 0.67 P= 0.56 Tle 3: Tet for vrince component for the unlnced dt exmple in McLen et l. (993)

10 Vrince Component Approximte Tet proc GLM F = 9.77 (df =, 5) p = F:= 5.55 (df =, ) p = 0.57 F =.76 (df =, 5). P= Exct Tet Öfverten (993) F = 9.46 (df =, 5) p = Wld Tet proc MIXE Z = 0.89 p = F = 9. (df =, ) p = Z = 0.8 p = 0.00 F =.76 (df =, 5). p = Z = 0.9 p =: icuion The pprent controvery for the ANOVA for lnced fctor mixed model tem from the imilrity etween the two competing model (UP nd CP) nd the uul fixed effect model. The uthor dvocting one or the other model eem to e their concluion on the term ocited to the rndom fctor ( in the UP model nd tht teting whether the correponding prmeter the inexitence of fctor min effect. in the CP model), nd ume or re null i equivlent to teting for Vo (999) ttempt to reolve the controvery y emedding the prolem in etting where the numer of level of the rndom fctor i finite ut eem not to ddre the crux of the prolem which, we elieve, lie in n pproprite definition of the rndom fctor min effect. Chrcterizing inexitence of fctor min effect y erting tht lmot ll fctor men level re equl, we my recll the work of Scheffé (959), who define the effect of the rndom fctor under more generl model, nd pecifie the definition in the context of the more generl model. Thi eem to reolve the controvery.

11 An lterntive pproch to the prolem i to e the nlyi on the choice of ome pecil tructure for the covrince mtrix for the dt nd pecifying the relevnt hypothee directly in term of the diperion prmeter. Under uch n pproch, no ttention i plced on the mening of the min effect of the rndom fctor. We my tet if ome vrince component i zero, ut relting thi to the inexitence of min effect of ome fctor i not o trightforwrd. Note tht y teting = 0 under the UP model, we re teting whether the covrince etween the expected repone under different level of fctor for two different level in fctor A i null, or ccording to Wolfinger nd Stroup (000), teting tht = 0 correpond to teting whether oervtion hring the me level of re equicorrelted. Intuitively, if there re min effect of fctor, then the covrince will e poitive. In fct, = 0 i relted to the min effect of fctor, ut it doe not ummrize them completely when 0. Note tht thi i the itution of interet, ince if = 0, there i no controvery t ll. α Finlly, we note tht the choice of the UP model dvocted y mny uthor i recommended in view of it computtionl dvntge well tht of eing more eily employed in the nlyi of unlnced ce. McLen et l. (99) hve hown tht when the purpoe i the evlution of etimle function nd the ocited tndrd error, mixed model procedure (MMP) my e ued without ny ditinction etween lnced nd unlnced ce. When the interet i to evlute whether ome vrince component i null under unlnced ce, we cn ue the exct tet propoed y Öfverten (993), either under the UP model or the CP model; lterntively, lrge mple likelihood rtio tet my e conider, ut then the pproprite null limiting ditriution i mixture of chi-qured ditriution indicted y Self nd Ling (987) nd Strm nd Lee (994). α

12 6. - Reference rownlee, K.A. (960). Sttiticl Theory nd Methodology in Science nd Engineering. New York: Wiley. Hinkelmnn, K. (000). Reolving the Mixed Model Controvery, Comment. The Americn Sttiticin, 54, 8. Hocking, R.R. (973). A icuion of the Two-Wy Mixed Model. The Americn Sttiticin, 7, Kempthorne, O. (957). An Introduction to Genetic Sttitic. New York: Wiley. Longford, N.T. (993). Rndom Coefficient Model. Oxford: Clrendon Pre. McLen, R.A., Snder W.L. nd Stroup, W.W. (99). A Unified Approch to Mixed Liner Model. The Americn Sttiticin, 45, Montgomery,.C. (99). eign nd Anlyi of Experiment. New York: Wiley. Neter, J., Wermn, W., Kutner, M.H. nd Nchtheim, C.J. (996). Applied Liner Sttiticl Model (4 rd ed.). Homewood, Ill: Irwin. Öfverten, J. (993). Exct Tet for Vrince Component in Unlnced Mixed Liner Model. iometric, 49, SAS Intitute. Inc. (990). SAS Uer Guide: Sttitic (Verion 6). Cry. NC: Author. Scheffé, H. (959). The Anlyi of Vrince. New York: Wiley. Schwrz, C.J. (993). The Mixed-Model ANOVA: The Truth, the Computer Pckge, the ook-prt I: lnced t. The Americn Sttiticin, 47, Serle, S.R. (97). Liner Model. New York: Wiley. Self, S.G. nd Ling, K-Y. (987). Aymptotic Propertie of Mximum Likelihood Etimtor nd Likelihood Rtio Tet Under Nontndrd Condition. Journl of the Americn Sttiticil Aocition, 8, Strm,.O. nd Lee, J.W. (994). Vrince component tetting in the longitudinl mixed effect model. iometric, 50, Vo,.T. (999). Reolving the Mixed Model Controvery. The Americn Sttiticin, 5, Wilk, M.. nd Kempthorne, O. (955). Fixed, Mixed nd Rndom Model. Journl of the Americn Sttiticil Aocition, 50, Wolfinger, R. nd Stroup, W.W. (000). Reolving the Mixed Model Controvery, Comment. The Americn Sttiticin, 54, 8.