A NOTE ON GLMM AND GEE IN LONGITUDINAL DATA ANALYSIS

Transcription

1 Jural Karya Asl Loreka Ahl Matematk Vol. No. ( page 5-5. Jural Karya Asl Loreka Ahl Matematk A NOE ON GLMM AND GEE IN LONGIUDINAL DAA ANALYSIS Noor Akma Ibrahm, Sulad Isttute for Mathematcal Research ad Dept. of Mathematcs Faculty of Scece Uverst Putra Malaysa UPM Serdag, Selagor Darul Ehsa, Malaysa 44 Departmet of Statstcs, Badug Islamc Uversty Jl. amasar No. Badug 46 - Idoesa akma@putra.upm.edu.my, sulad@gmal.com Abstract : hs paper dscusses two of the most used methods aalyzg logtudal data, the geeralzed lear mxed model (GLMM ad geeralzed estmatg equato (GEE. As kow, logtudal data are correlated ad the varaces are heterogeeous. he geeralzed lear model (GLM has some lmtato to hadle such data. wo methods that are the most popular are GLMM ad GEE. GLMM s also called subject specfc effect model, ad GEE s a margal model. For cotuous data, the two methods are smlar, especally the terpretato of the regresso coeffcets, eve though the estmates of the regresso coeffcets are ot exactly the same. Whlst for other types of data, GLMM ad GEE deftely have dfferet terpretato. Keywords: logtudal study, correlated data, geeralzed lear mxed model, geeralzed estmatg equato.. Itroducto Researchers some areas of applcato, especally ecoomcs, epdemology or clcal trals, use desg where the respose s measured repeatedly through tme or occaso. I ths study subjects are followed over tme or several occasos to collect the respose varables. Data whch are collected from ths research are called logtudal data ad the study s kow as logtudal study. hs study would be useful to explore the patter of the respose varables over tmes. he scetfc questo of terest s ot oly how the respose dffers across treatmets but also how the respose chages over tme the presece of covarates. he commo aspects of modelg logtudal data are Characterzg the mea of the resposes the model a way that best captures how the mea of the respose chages wth tme whch mght deped o other factors. akg to accout the mportat sources of varato by characterzg the ature of radom devato. here are several advatageous of logtudal study. hs study may be more formatve tha cross-sectoal study, because measuremets o the same subject at dfferet tmes may be more alke tha measuremets o dfferet subject. I ths study, assessmet o wth-subject chage of respose ca be observed. hs s the oly type of study whch t s possble to obta formato about dvdual patter of chage. I ths study, subject ca serve as ther ow cotrols, whch the respose ca be measured uder cotrol or treatmet codtos oe subject. I ths case the betwee subjects varablty ca be excluded from the expermetal error. he cosequece s that a more effcet estmator ca be obtaed tha from cross sectoal study (Dobso, ; Davs, ; Ftzmaurce et al, 4. he characterstc of data logtudal study s that there usually exst postve correlatos amog wth subject measuremets. Subject wth hgh respose wll be above average ad s expected to have hgh respose aother occaso. herefore, there s depedece amog measuremets from a subject ad ths correlato must be accommodated the aalyss. hus model that accommodates the correlato are eeded order to make vald statstcal fereces. Aother usual problem s that the varaces are ot costat over the durato of the study. Ufortuately, the usual lear model assumes that all observatos are depedet ad have costat varace. hus Jural Karya Asl Loreka Ahl Matematk Publshed by Pustaka Ama Press Sd. Bhd.

2 Jural KALAM Vol., No., Page 5-5 the usual lear model ca ot be used to aalyze logtudal data. here are three sources of varablty logtudal data (Ftzmaurce et al, 4: Betwee subjects varablty. Betwee subjects varablty reflects the atural varato dvduals propesty to respose. For specfc treatmet or factor, subject usually has cosstet respose. Hece subject wth hgher (lower respose wll cosstetly have hgher (lower respose the the others over tme. hs property s a source of usual postve correlato logtudal data. Wth subject varablty. hs varablty ca be thought as realzato of some bologcal processes or teracto betwee some bologcal processes ad tme. Measuremet error. hs varablty ca happe all type of study, ot oly logtudal study. he source s the mperfect or cosstecy measuremet, due to the devce or huma error. May types of data ca be aalyzed usg class of geeralzed lear model (GLM where data are assumed to follow expoetal famly dstrbuto. GLM captures the aalyss of cotuous, cout, ad categorcal data. GLM uses maxmum lkelhood estmatg the model parameters wth the assumpto that observatos are depedet. hus there are two problems to be addressed logtudal data aalyss, ( method that allows depedet observatos ad accommodates the correlato ad ( method that has the ablty to estmate the correlato. Method that has ablty to estmate correlato s mportat, sce the correlato s also the terest questo of a researcher. Multvarate aalyss of varace ca be used to aalyze correlated cotuous data, whch ca hadle the problem of depedece ad heterogeeous of varace. But ths method s ot geeral to hadle logtudal data due to the characterstcs of the data. Logtudal data are characterzed by ( varato amog expermetal uts wth respect to the umber ad tmg of observatos; ( mssg data; ad ( tme-depedet covarates. Such features make the classcal multvarate procedure dffcult to apply (Davs,. hs paper dscusses two mportat methods aalyzg logtudal data, geeralzed lear mxed model (GLMM ad geeralzed estmatg equato (GEE. We gve short revews of these methods ad dscuss ther advatageous. he outle of ths paper s as follows. Secto cosders estmato of GLMM for cotuous data ad geeral expoetal famly data. Secto dscusses the populato average model kow as geeralzed estmatg equato (GEE. I Secto 4 we dscuss some types of correlato structures logtudal data followed by the dscusso of the applcato of GEE ad GLMM Secto 5.. Mxed Models or Radom Coeffcet Models I ths secto we brefly dscuss the radom coeffcet model. We separate the model for cotuous data uder ormal dstrbuto ad the model for geeral data uder expoetal famly dstrbuto. he frst model s called lear mxed model (LMM ad the secod oe s called geeralzed lear mxed model (GLMM. Reaso for ths separato s that the multvarate ormal dstrbuto s already kow thus drect estmato usg maxmum lkelhood ca be doe. Meawhle, the multvarate dstrbuto for other expoetal famly has ot bee well establshed yet. he dea of the mxed model s that some of the regresso parameters vary from oe dvdual to aother (Ftzmauce et al, 4. I mxed effect models the mea respose acts as a combato of the regresso parameters that characterzed the populato ad the subject specfc effect parameter that uquely characterzed each subject (dvdual. It explas the sources of varablty the populato, that s the populato parameters expla how the meas respose of populato chage over tme, ad the subject specfc effects expla how the dvdual chage over tme uquely.. Lear Mxed Effect Models Suppose there are m depedet subjects ad each measured oce. Let y = (y, y,, y m be a m vector of depedet observatos. he classcal lear model s 6

3 Ibrahm ad Sulad y = X + ( where X s a m p of covarate matrx, s a p vector of ukow parameters, ad s a m vector of depedet errors. he vector s assumed depedet wth mea ad costat varace. hus E(y = X. he usual lear model above caot be appled logtudal data sce the wth subject observatos are ot depedet aymore ad the varace mght ot be homogeeous. Now suppose there are m subjects ad each subject s measured tmes, for =,,..., m. Wthout loss of geeralty, suppose each subject s measured tmes. Lear mxed effect model was troduced by Lard & Ware (98 usg maxmum lkelhood. I ths model, the wth subject correlato s modeled by troducg radom effects model (. he lear mxed model s defed as y = X + Z v + ( where =,,, m, y = (y, y,, y s vector of resposes of the -th subject, X s a p desg matrx for the fxed effect of the -th subject, Z s a k desg matrx for the radom effect of the -th subject, s p vector of the fxed effect coeffcets, v s k vector of the radom effect, ad s vector of error. I (, there are two radom compoets, v ad. he assumptos of these radom vectors are: v depedet detcally dstrbuted (d multvarate ormal (, ; depedet detcally dstrbuted (d multvarate ormal (, I; ad ad v are depedet, the we have E(y = X ad var(y = ZZ + I. hus y MN(X, ZZ + I. he estmato of,, ad are the cetral ssue, that eed hgh level computato. Sce the multvarate ormal dstrbuto has already bee establshed, the the estmato s based o ths dstrbuto. here are two methods to estmate these parameters, maxmum lkelhood (ML ad restrcted maxmum lkelhood (REML ad by usg several algorthm that has bee developed,.e Newto-Raphso, Fsher Scorg, EM, ad Geeralzed EM algorthm (Demdeko, 4; Davs,. A specal case of model ( s called the radom tercept model that allows subject to have dvdual fluece as a represetato of the devato from the populato mea. he radom tercept model s y j = + x j k x kj + v + j ( where s the tercept of the populato;,..., k are the regresso coeffcets for the covarates, v s the tercept for the -th subject, ad j s the error the -th subject ad j-th tme measuremet. Model ( mples that every subject has ts ow tercept, v, that represets the devace from the populato tercept. We may treat the populato of v s as a radom effect ad has a dstrbuto, whch usually s assumed ormally dstrbuto wth mea ad varace v. hs model assumes that j ad v are depedet. From ths, we have E(y j = x j; var(y j = + v; cov(y j,y k = v for j k; cov(y j,y hk = for h ; ad corr(y j,y k = v /( + v =. hus the result s a costat correlato amog repeated measuremets wth a subject. hs model s commoly called compoud symmetry (exchageable correlato structure. hs model usually s ot preferable logtudal data aalyss because the correlatos logtudal data are usually ot costat over tme,.e the correlato decreases f the lag tme creases. We ca exted the radom tercept model to radom tercept ad slope model. I ths model we assume that dvdual (subject has ts ow slope that vary from subject to subject ad has certa dstrbuto. Radom tercept ad slope model has the same form as (. Suppose v = (v, v ad assumed bvarate ormal dstrbuto N(,, where 7

4 Jural KALAM Vol., No., Page 5-5 v vv vv v ad depedet of, Var(v = v, Var(v = v, ad Cov(v, v = vv. We have E(y = X ad var(y = ZZ + I. hus y MN(X, Z Z + I. he depedecy s explaed the form of matrx Cov(v =.. Geeralzed Lear Mxed Models Geeralzed lear mxed model (GLMM ca be see as geeralzato of geeralzed lear model to o depedet data, or exteso of LMM model to the more geeral expoetal famly dstrbuto. he dea s smlar to the LMM model the prevous secto, where some regresso parameters have geeral value for all populato members, ad some of other parameters are dfferet for each subject. hus each subject has specfc parameters that descrbe the patter or profle of specfc subject over tme. As dscussed before, there s o problem for cotuous data that are usually assumed to follow multvarate ormal dstrbuto ad the parameter estmato s based o ths dstrbuto. he lkelhood fucto for the -th subject L(y, y,, y eeds multvarate dstrbuto of y = (y, y,, y. Except ormal dstrbuto, other multvarate expoetal famly dstrbutos such as bary or couted data have ot bee establshed yet. hus the parameters estmato eeds other approach. he dea that ca be adopted s that gve v, the y v, y v,, y v are depedetly dstrbuted. Suppose the margal dstrbuto of the respose follows expoetal famly dstrbuto j yj b( j f ( y, exp j c( y, a( j, wth E(y j = j = b ( j ad var(y j = b ( j a(. he relatoshp betwee the respose ad the fxed ad radom covarates s through a lk fucto g( v x z v. j j j Gve v the y v, y v,, y v are depedetly dstrbuted. hus the codtoal lkelhood fucto for the -th subject s j j f ( yj,, v. L( y v f ( y, It s assumed that v, v,, v k d N k (, hece the margal lkelhood for the -th subject s h( y k R ( L( Y v f ( v dv k / / k R j expl j f ( y j j,, v.exp.5v v dv. Usg caocal lk fucto, g(=, ad assumg a( =, the log-lkelhood for all subject s Nk N N l(, l( l l k exp xj zj v yj b xj zj v v v du R ( (.5. j 8

5 Ibrahm ad Sulad he problem estmatg ad s that there s tegrato the log-lkelhood fucto, whch drect dfferetato caot be doe drectly. here are some methods to approxmate the tegrato part, such as Laplace approxmato, fxed sample lkelhood approxmato, ad adaptve Gauss- Hermte quadrature (see Demdeko, 4 for detal.. Iterpretato of the Regresso Parameters of GLMM Eve though the lear mxed effect ca be see as a specal model of GLMM but the terpretato of the regresso parameters of LMM ad GLMM are deftely dfferet. I lear mxed effect we ca terpret the regresso parameters as populato average (also called as margal effect ad as subject-specfc effect (codtoal mea respose. he codtoal mea or the subject specfc mea of the respose for the -th subject, gve that the subject specfc effect s v s E(y v = X + Z v, that shows the profle for the -th subject gve X ad Z covarates. he margal or populato averaged mea of y s gve by E(y = = E{E(y v } = X + Z E(v = X. he populato average shows the average of the respose of all subjects the populato for specfc covarate X. hus we have two models, subject specfc model ad populato average model, that descrbe the relatoshp betwee the covarate ad the respose for populato ad for dvdual over tme respectvely. he terpretato of LMM parameters s ot as flexble as GLMM parameters sce GLMM geerally uses o-lear (o-detty lk fucto as LMM model. he geeral form of the lk fucto s but f g(. s olear/o-detty the g{e(y j x j, v } = x j + z jv g{ E(y j x j } x j. Because of ths reaso, GLMM s called a subject-specfc model, where the regresso parameter ca be terpreted oly for the specfc subject (dvdual ad caot be terpreted for populato mea. he smple example below wll gve a easer uderstadg of the terpretato problem GLMM. Suppose we have logstc regresso wth radom tercept. For the -th subject, gve v ad x j = x chage to x j = x+, whle other covarates are fxed. We have P( y v, x x P( y v, x x j j j j log v x xj p x jp; ad log v ( x xj p xjp. P( yj v, xj x P yj v xj x (, he regresso coeffcet k ca be terpreted as the chage of the log odds f the covarate x jk s chaged oe ut measuremet, for the (specfc -th dvdual. It s specfc for the -th dvdual sce t depeds o the gve value v, the -th subject specfc effect. he odds rato s OR v = Exp(. Sce the terpretato of fxed effect, k, depeds o the dvdual radom effect, v, the ths model s also called as subject-specfc model. hus GLMM s most useful f the ma objectve of the research s to make ferece o dvdual rather tha populato. 9

6 Jural KALAM Vol., No., Page 5-5 he terpretato of fxed effect regresso parameter s dffcult f the covarate s the form of tme-varat or subject covarate, such as sex or blood type. Sce the value of ths covarate s fxed over tme, or may be dfferet oly for dfferet subject, the t s mpossble to gve the terpretato. he example below wll gve a clearer uderstadg. log odds( A v, sex Sex v a a v ; ad a log odds( B v, sex Sex v b b v b If we assume that subject specfc effect of dvdual A ad B are the same (v a = v b the regresso coeffcet of Sex,, ca be terpreted as the dfferet log odds of Sex = ad Sex =. If we dd ot assume that v a = v b, the s dffcult to be terpreted (See Ftzmaurce, 4 chap. for detal.. Margal Models As dscussed the prevous secto GLMM has lmtato o the terpretato of the regresso parameter, whch leads to GLMM beg a subject-specfc models. Sce the focus of some studes mght be the populato rather tha dvdual thus GLMM s o loger approprate. Margal model s also kow as populato average model. he method to aalyze ths model s the geeralzed estmatg equato (GEE tally proposed by Lag & Zeger (986. hey troduced the populato average model for the geeral expoetal famly dstrbuto for logtudal data. hs approach ca be see as a exteso of quas-lkelhood proposed by Wedderbur 974, by troducg workg correlato matrx quas-lkelhood estmatg equato. hs workg correlato acts as assocato measuremet. Geeralzed estmatg equato results cosstet estmator eve though there s mspecfcato o the workg correlato. he estmate s stll cosstet as log as the mea s specfed correctly. hs method receved a lot of attetos ad commets, focusg the effcecy ad the estmato of the workg correlato. Some efforts to mprove the orgal GEE have bee proposed. Pretce (988 proposed the secod set of estmatg equato for the assocato parameter. hs secod estmatg equato for the assocato parameters assumes that the regresso coeffcet ad the assocato parameter are orthogoal. he momet estmator of Lag & Zeger for assocato parameters are specal case of Pretce s. We refer ths as GEE. I GEE, the assocato parameters are treated as usace parameter. hs meas that the ma terest of the study s the regresso parameter (regresso coeffcets. But t s ofte that a study, the assocato parameter s also the focus of terest. For ths reaso ad to crease the effcecy, Zhao & Pretce (99 ad Pretce & Zhao (99 proposed the ew GEE usg quadratc expoetal model that allows jot estmato of mea ad covarace parameter oe estmatg equato. It s also a lkelhood-based approach kow as pseudo maxmum lkelhood (PML, because the use of quadratc expoetal famly dstrbuto. hs method volves rather dffcult computato, sce t eeds the computato of the thrd ad fourth momet. hs approach oly cosders two-way assocato ( jk ad set the thrd ad hgher order of assocato to zero ( jkl =,, jklm... =,. Correct specfcato of the mea ad the assocato should be fulflled, stead of oly correct specfcato of mea as Lag & Zeger. We refer ths as GEE. Hall & Sever (998 used dea from exteded quas-lkelhood to derve a set of estmatg equato for regresso ad assocato parameters, ad called t EGEE (exteded GEE. he advatage of EGEE s that the assocato parameter s also the focus of terest, t does ot eed the thrd ad hgher order of momet, ad the cosstecy of regresso parameter holds eve f the assocato parameter s msspecfed.. GEE Let y = (y, y,, y be vector of respose for the -th subject wth correspodg covarate X = (x, x,, x wth x j = (x j, x j,, x jp. Assume that the margal desty of y j follows expoetal famly dstrbuto wth caocal parameter j. E(y j = μ j s related to the

7 Ibrahm ad Sulad covarate through a lk fucto such that g(μ j = j = x j. Defed μ = (μ, μ,, μ ad = (,,, p be the p x vector of regresso coeffcets. Lag & Zeger (986 proposed geeralzed estmatg equato (GEE by extedg quas-lkelhood estmatg equato by troducg workg correlato to the estmatg equato. Let R( be matrx whch fulflls the requremet of beg correlato matrx ad be a s vector whch fully characterzed R(. R( s called workg correlato matrx. Defed V = A / R(A /, where A = dag{var(y j }. hus V wll be cov(y f R( s the true correlato of y. he geeralzed estmatg equato s defed by K D V ( y where D = μ / = (μ / ( / ( /. he teratve procedure to obta s by usg modfed Fsher scorg for ad the momet estmator for ad. hus ths method treats ad as usace parameter. he teratve procedure s K K ~ ~ s s D ( ˆ s V ( ˆ s D ( ˆ s D ( ˆ s V ˆ ( s ( y. Lag & Zeger (986 pot out that the cosstecy of ad depeds o the correct specfcato of mea ot the choce of correlato matrx. hus for ay gve R(, ˆ ad ˆ var( ˆ var( ˆ are stll cosstet. GEE uses pearso resdual to estmate. Choosg R( closer to the true correlato wll crease the effcecy. hs GEE has ts advatageous ad also drawbacks. he advatages are, frst t oly eeds correct specfcato of the mea. he parameter estmate s stll cosstet eve though there s msspecfcato of the workg correlato. Secod, we may obta a cosstet ad robust varace estmate eve f the workg correlato s correct. But the effect of the msspecfcato of assocato wll affect the effcecy. Hgh effcecy wll be obtaed f the assocato s correctly specfed. Aother drawback s, ths approach the ma terest s the regresso parameter treatg the assocato as a usace parameter. I some study, t s ofte that both parameters, regresso ad assocato parameter are the focused of terest. GEE produces estmates of correlato parameter wth low effcecy relatve to the maxmum lkelhood estmato. 4. GEE Zhao & Pretce (99 ad Pretce & Zhao (99 proposed a ew GEE usg quadratc expoetal model that allows jot estmato of the mea ad the covarace parameter oe estmatg equato. It s a lkelhood-based approach ad kow as pseudo maxmum lkelhood (PML, because t employs quadratc expoetal famly dstrbuto. hs method eeds rather dffcult computato, sce t eeds the computato of the thrd ad fourth order of momet. hs approach oly cosders two-way assocato ( jk ad set the thrd ad hgher order of assocato to zero ( jkl =,, jklm... =,. Correct specfcato of the mea ad the assocato should be fulflled, rather tha correct specfcato of mea oly as GEE of Lag & Zeger (986. Cosder a m-sample of depedet observatos y = (y, y,, y, =,,, m. Estmatg equato for regresso parameter ad covarace matrx of the respose s geerated uder quadratc expoetal model: Pr(y,, exp { y w c ( y }

8 Jural KALAM Vol., No., Page 5-5 where w = (y, y y,, y, y y,, c (. s the shape fucto, ad be the ormalzato costat wth ad as the caocal parameters. Score estmatg equato for the regresso parameter,, ad the assocato parameter,, s gve by K D V f, / var( y cov( y,s y where D ; V ; f / / cov(s, y var( s s, = E(s, ad s s a vector that represets Corr(y or Cov(y (for detals, see Zhao & Pretce (99 ad Pretce & Zhao (99.. Exteded Geeralzed Estmatg Equato (EGEE GEE treat the correlato as usace parameter ad treat the regresso ad correlato coeffcet orthogoal, eve though f they are ot. I GEE, the regresso ad correlato coeffcets are estmated smultaeously. hs meas that these two parameters are the focus of terest. Although the correlato estmate of GEE s more effcet tha GEE, GEE eeds the correct specfcato of the mea ad the covarace structure order to obta cosstet estmate of regresso parameter. It also eeds the thrd ad fourth order of momet. Eve though the msspecfcato of the thrd ad fourth orders of momet does ot affect the cosstecy of regresso parameter, but these affect the effcecy (Hall & Sever, 998. Hall & Sever (998 proposed a estmatg equato, called the exteded geeralzed estmatg equato (EGEE, based o exteded quas-lkelhood to estmate regresso ad correlato parameters smultaeously. he advatages of ths method are, ( the regresso ad correlato parameters are estmated smultaeously, thus the correlato s also the ma terest of study ( t does ot eed the thrd ad forth order of momet ( t does ot eed the correct specfcato of covarace, oly eeds correct specfcato of mea. Let y j be the respose of scalar ad x j be p vector of covarates for the -th subject, =,,, m ad the j-th measuremet, j =,,,. Let vector of respose for the -th subject, y = (y, y,, y, vector of mea E(y = μ = (μ, μ,, μ ad p matrx of covarates x = (x, x,, x. Suppose the relato betwee resposes ad covarates through lk fucto g(μ s descrbed by the vector of parameters = (,,, p ad the wth subject assocatos (correlatos s descrbed by s vector of = (,,, s. Suppose the relato betwee respose ad covarate through lk fucto, the form g(μ j = x j. Assumed cov( y / / V (, A R( A, where A ad R( as defed 4.. Hall & Sever (998 proposed estmatg equato for = (, usg exteded quas-lkelhood fucto for, wth estmatg equato gve by K U ( U (, where U ( = D V - (y - ; U ( = (U (, U (,, U ( s, ad U ( r = -(y- V - (V / r V - (y- + r(v - (V / r..4 Iterpretato of the Regresso Coeffcets GEE s also kow as a margal model where the regresso coeffcets are terpreted as populato parameter smlar to the geeralzed lear model (GLM. Hece, ths method s

9 Ibrahm ad Sulad approprate f the focus of study s populato rather tha dvdual or subject specfc effect. As a example, the case of logstc regresso wth oe covarate. Gve X j = x ad X j = x +, the P( yj X j log P( yj X j x x x P( y P( y' X x x ' j' ' j' ; ad log x j' X Hece exp( s terpreted as the odds rato f the covarate X creases by oe ut measuremet. I ths case, the log odds do ot deped o the subject or dvdual. It s the populato chage f we chage the covarate. 4. Some Correlato Structures Logtudal Data he depedecy of the wth-subject measuremets s exhbted by the correlato. I some studes, ths correlato s also the ma terest of the researcher. he problem of correlated data s ot oly how to estmate the correlato, but also the problem of ferece. Suppose there are m subjects ad each subject s measured tmes, for =,,..., m. Wthout loss of geeralty, suppose each subject s measured tmes. he covarace ad correlato structure of Y = (Y, Y,, Y are ' j' Cov(Y ; Corr( Y. hs ustructured covarace matrx has varace parameter ad (-/ covarace parameters. We may use ths structure, but t s too may parameters to be estmated. here are some correlato structures that ca be used to reduce the umber of parameters to be estmated (Ftzmurce et al,. We dscuss some of them below. 4. Compoud Symmetry or Exchageable I ths structure, t s assumed that Corr(Y j, Y k = for all j k ad j, k =,,...,. Usually ths structure s approprate for logtudal data, whch correlato s expected to decay wth creasg dfferet tme measuremet. hs s usually approprate for repeated measuremet area study. If we assume that varace s costat for all tme measuremet, say Var(Y j = for j =,,,, the we oly eed to estmate two parameters, ad. If we assume that varace s heterogeeous, Var(Y j = for j =,,,, the we eed + parameters to be estmated. he correlato ad geeral covarace structures are gve below. Corr ( Y ; Cov( Y. 4. oepltz hs structure assumes that correlatos of two measuremets are equal f those measuremets have the same of dfferet tme. Precsely, f d = j-k, the Corr(Y j, Y k = d for all j k ad j, k =,,...,. hs structure s approprate for logtudal data wth the same terval tme of measuremet. If we assume that varace s costat across measuremets, the the umber of parameter that should be estmated s parameters, oe for varace ( ad - parameters for

10 Jural KALAM Vol., No., Page correlato. We may assume that varace s ot costat,.e. Var(Y j = for j =,,,. hus the umber of parameter that should be estmated s - parameters: parameters for the varace ad - parameters for the correlato. he correlato ad geeral covarace structures for ths structure are ( ; ( Y Cov Y Corr. 4. Autoregressve Autoregressve correlato structure has the form Cov(Y j, Y k = j-k. If assumed that the varace s costat, Var(Y j = for j =,,,, the t oly eeds two parameters to be estmated,.e. the varace ad the correlato. hus t s very parsmoous estmatg parameter. We may assume that the varace s heterogeeous, where Var(Y j = for j =,,,. I ths case we eed + parameters to be estmated. he correlato ad geeral covarace structures are ( ; ( Y Cov Y Corr, 4.4 Baded Baded structure assumes that the correlatos are zero after some specfc terval, say k. hus Corr(Y j, Y j = for j-j < k, ad Corr(Y j, Y j = for j-j k. We ca combe ths structure wth other correlato structure. For example, the combato of baded- wth autoregressve structure has correlato structure ( Y Corr 5. Dscusso o the choce of margal model ad subject specfc model Margal model ad subject specfc model (GLMM are the most commo methods used the aalyss of logtudal data. But some questos may arse from the user about whch method should be used aalyzg logtudal or repeated measuremet data. So far, there s o agreemet amog statstcas about ths matter. Davs (, p95 stated that margal models may be the most approprate may study rather tha subject specfc models, sce most study the parameter of populato s the ma of terest. Whlst Ldsey ad Lambert (998 advocated that margal model should be used wth care. Demdeko (4, p4 also preferred usg subject specfc model sce t mples a vald statstcal model, whle the drawback assocated wth dffcultes of estmato ca be overcome usg some approxmato method. Some authors have poted out that the method that should be used depeds o the objectves of the research (Hard & Hlbe, p 8; Ftzmaurce et al, 4 chap. If the focus of study s the effect of the covarates dvdually (subject-by-subject the the approprate method s radom effect or subject specfc models. Whlst f the ma terest of study s ferece o populato, the margal model s the most approprate method.

11 Ibrahm ad Sulad he authors agree wth the opo that the choce of method depeds o the stuato. For cotuous data, we prefer to use lear mxed effect (radom effect model. hs s because the estmato s based o vald dstrbuto,.e. multvarate ormal dstrbuto. Besdes, from lear mxed model we ca get the margal model ad subject specfc model drectly. But for other types of data, such as cout ad categorcal data, the use of margal or subject specfc model depeds o the objectve of the research. If we wat to aalyze the average of chage of the respose the populato, the the margal model s approprate. Whle f the objectve s aalyzg the effect of the covarate to the chage of respose dvdually, the subject specfc (GLMM s approprate. As a example, take a study to aalyze the effect of smokg to the exstece of lug cacer. If the questo s whether there wll be ay rsk dfferece gettg lug cacer betwee smoker ad osmoker, the the margal model s the approprate method. But f the questo s what s the effect f a dvdual chage the habt from smokg to o-smokg (or vce versa, the subject specfc model (GLMM should be used. Ackowledgemet he authors are very grateful to the Edtor ad Referees for costructve commets ad suggestos. hs paper s supported by Scece Fud grat Vote from Mstry of Scece, echology ad Iovato Malaysa. Refereces [] Davs, Charles S. (. Statstcal Methods for the Aalyss of Repeated Measuremets. Sprger-Verlag. New York. USA. [] Demdeko, Eugee. (4. Mxed Models. heory ad Applcatos. Jo Wley & Sos, Ic. New Jersey. USA. [] Dobso, Aette J. (. A Itroducto to Geeralzed Lear Models. d Edto. Chapma ad Hall. New York. USA. [4] Ftzmaurce, G. M., Na M. Lard, ad James H. Ware. (4. Appled Logtudal Aalyss. Jo Wley & Sos, Ic. New Jersey. USA. [5] Hard, James W. ad Joseph M Hlbe. (. Geeralzed Estmatg Equatos. Chapma & Hall/CRC. Washgto, DC. USA. [6] Hall, Dael B ad homas A. Sever. (998. Exteded Geeralzed Estmatg Equato for Clustered Data. Joural of the Amerca Statstcal Assocato 9: [7] Lard, Na M. ad James H. Ware. (98. Radom Effects Models for Logtudal Data. Bometrcs 8: [8] Lag, K. Y. ad S. L. Zeger. (986. Logtudal Data Aalyss usg Geeralzed Lear Models. Bometrka 7: -. [9] Ldsay, J.K. ad P. Lambert. (998. O the Approprateess of Margal Models for Repeated Measuremets Clcal rals. Statstcs Medce 7: [] Pretce, Ross L. (988. Correlated Bary Regresso wth Covarate Specfc to Each Bary Observato. Bometrcs 44:-48. [] Pretce, Ross L. ad Lue Pg Zhao. (99. Estmatg Equatos for Parameters Meas ad Covaraces of Multvarate Dscrete ad Cotuous Resposes. Bometrcs 47: [] Zeger, Scott L. ad Kug-Yee Lag. (986. Logtudal Data Aalyss for Dscrete ad Cotuous Outcomes. Bometrcs 4: -. [] Zhao, Lue Pg ad Ross L. Pretce. (99. Correlated bary regresso usg a quadratc expoetal model. Bometrka 77: