A Semi-parametric Approach for Analyzing Longitudinal Measurements with Non-ignorable Missingness Using Regression Spline

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1 Avalable at Appl. Appl. Math. ISSN: Vol., Issue (June 5), pp Applatons and Appled Mathemats: An Internatonal Journal (AAM) A Sem-parametr Approah for Analyzng Longtudnal Measurements wth Non-gnorable Mssngness Usng Regresson Splne Taban Baghfalak * Department of Statsts Tarbat Modares Unversty Tehran, Iran t.baghfalak@modares.a.r * Correspondene author Saede Sefd and Mojtaba Ganjal Department of Statsts Shahd Behesht Unversty Tehran, Iran Reeved: Aprl 4, 4; Aepted: February, 5 Abstrat In longtudnal studes wth mssngness, shared parameter models (SPM) provde approprate framework for the jont modelng of the measurements and mssngness proess. These models use a set of random effets to aount for the nterdependene between two proesses. Sometmes the longtudnal responses may not be ftted well by usng a lnear model and some non-parametr methods have to be used. Also, parametr assumptons are typally made for the random effets dstrbuton, and volaton of those may affet the parameter estmates and standard errors. To overome these problems, we propose a sem-parametr model for the jont modellng of longtudnal markers and a mssng not at random mehansm. In ths model, beause of the flexblty n nonparametr regresson models, the relatonshp between the response varables and the ovarates has been modeled by sem-parametr mxed effet model. Also, we do not assume any parametr assumpton for the random effets dstrbuton and we allow t to be unspefed. The parameter estmatons are made usng a vertex exhange method. In order to evaluate the performane of the proposed model, we ompare SPM usng regresson splne (Splne-SPM) and sem-parametr SPM (SpSPM) models. We also ondut a smulaton study wth dfferent parametr assumptons for the random effets dstrbuton. A real example from a reent HIV study s analyzed for llustraton of the proposed approah. Keywords: Jont modelng; Longtudnal data; Mssng mehansm; Nonparametr model; Regresson splne; Random effets; Vertex exhange method. MSC No.: 6J, 6H 95

2 96 Taban Baghfalak et a.. Introduton In longtudnal studes, ndvduals are followed over a duraton of tme and for eah ndvdual, data are olleted at multple tme ponts. These repeated measurements may share a ommon haraterst and may be orrelated, although measurements on dfferent ndvduals ould be assumed to be ndependent. Consderaton of orrelatons wthn measurements of the same ndvdual expresses the key haraterst of longtudnal data. Mssngness s a problem of longtudnal data. In some ases, a subjet may be mssng n one or several measurement oasons. Rubn (976) provded a framework for the nomplete data by ntrodung the mportant lassfaton of mssng data mehansms, whh onsst of mssng ompletely at random (MCAR), mssng at random (MAR) and mssng not at random (MNAR). A mehansms s alled MCAR f the mssng mehansm s ndependent of the unobserved and also observed data, MAR f, ondtonal on the observed data, the mssng mehansm s ndependent of the mssng measurements; otherwse the mssng proess s termed MNAR. As an example of MNAR, a patent dedes not to show up at some of the sheduled vsts beause of her/hs very bad urrent health ondtons. The mssngness depends on unobserved responses. In suh ases, analyzng the longtudnal measurements for dsease evaluatons usng, e.g., a mxed effets model, where gnorng the mssngness proess, leads to based nferenes. For the jont modelng of these two proesses, Shared Parameter Model (SPM) (Wu and Carroll, 988; Follmann and Wu, 995) an be used. In ths approah, the two models are lnked through some ommon unknown varables. Shared parameter models suppose that a set of random effets ndue the nterdependene. In partular, onsder the vetor Y as a omplete longtudnal o m response and based on the mssngness proess, R dvde t nto two parts of Y and Y whh are the observed and mssng omponents, respetvely. Under the SPM framework, the jont densty of the measurement proess Y and the mssngness proess R may be ompleted as o m o m f ( Y, Y, R ) = f ( Y, Y Y, b) f ( R b, R) f ( b b ) db, () where f (.) denotes a probablty densty funton, b s a vetor of random effets, and = ( Y, R, b ). In, Y s the parameter vetor of the model for Y gven b, R s the vetor of parameters of R gven b and b s the vetor of parameters of the dstrbuton of b. Ths fatorzaton shows that gven the random effet b, the vetor of response varable (Y ) and mssngness proess (R) are ndependent. Aordng to De Gruttola and Tu (994) and Lttle (995), SPMs are approprate when mssngness s due to an underlyng proess by whh the longtudnal responses are measured wth error. The sze of ths measurement error determnes the strength of the dependene of the mssngness on the latent varable b. The onstruton of an SPM mssngness mehansm leads to a mssng not at random (Rubn, 976) proess, where the mssng data mehansm () s gven by o m o m f ( R Y, Y, ) = f ( R b, R) f ( b Y, Y, b ) db, ()

3 AAM: Intern. J., Vol., Issue (June 5) 97 whh shows that the probablty of nonresponse depends on f b Y o, Y m, ). Therefore, the ( b random effets are the man omponent n the modelng of the mssng data. However, msspefaton for dstrbuton of random effets an severely affet our nferene. Fndng sutable parametr dstrbuton assumpton for the random effets s however dffult. Beause, the potental dependene of the random effets on unobserved ovarates ndues heterogenety that annot be aptured by ommon parametr assumptons (Tsonaka et al., 9). Several authors have proposed jont models that are not dependent on strong parametr assumptons for the random effets, and are also robust to some dstrbutonal assumptons. In partular, n the ontext of jont modelng of longtudnal measurements and survval data, Song et al. () have gven a shared latent omponent whh s the produt of a polynomal term and the standard normal densty. In mssng data analyss, Ln et al. () and Beunkens et al. (8) assume that the random effets have a fnte mxture of normal dstrbuton. Also they offer some nsght n the shape of the random effets dstrbuton, whh helps n determnng a potental subpopulaton struture n the data, and produes enhaned subjet-spef predtons. In ths paper, we propose to leave the random effets dstrbuton ompletely unspefed. The estmaton of ths model s based on a sem-parametr method that assumes the random effets dstrbuton to be dsrete wth unknown support szes. To effetvely maxmze the loglkelhood wth respet to the random effets dstrbuton, we apply the Vertex Exhange Method (VEM) (Bohnng, 985). For longtudnal data, parametr mxed-effets models, suh as lnear and nonlnear mxed-effets models are a natural tool. Lnear mxed-effets (LME) models are used when the relatonshp between a longtudnal response varable and ts ovarates an be expressed va a lnear model. Nonlnear mxed-effets (NLME) models are used when the relatonshp between a longtudnal response varable and ts ovarates annot be expressed va a lnear model. A parametr regresson model requres an assumpton that the form of the underlyng regresson funton s known exept for the values of a fnte number of parameters. A dsadvantage of parametr modelng s that a parametr model may be too restrtve n some applatons. The use of an napproprate parametr model leads to msleadng results. For suh a longtudnal data set, we do not assume a parametr model for the relatonshp between the response varable and the tme as a ovarate. Instead, we just assume that the ndvdual and the populaton mean funtons are smooth funtons of tme t, and let the data themselves determned the form of the underlyng funton. There are many nonparametr regresson and smoothng method. The most popular methods, thenlne kernel smoothng, loal polynomal fttng, regresson splne, smoothng splne and penalzed splnes (Zhang et al., 998; Wu and Zhang, ). Tsonaka et al. (9) use LME model for measurements proess, alled the SpSP model (sem-parametr shared parameter model). But, the proess that the data are generated from (suh as our data) may not be lnear, thus for analyzng ths knd of data set, the LME model s napplable. Therefore, for measurements proess the modelng we use s the regresson splne for nonparametr fxedeffets omponent of the sem-parametr model. We alled t the Splne-SpSP model (semparametr shared random effets model usng regresson splne). Also, the VEM s used for jont modelng of the mssngness proess and longtudnal measurements.

4 98 Taban Baghfalak et a. The paper s organzed as follows: the nonparametr regresson for longtudnal data s onsdered n Seton. Seton 3 presents the proposed modelng framework. Also, ths Seton summarzes some theoretal results and gves the detals for the estmaton proedure. The performane of the proposed method s evaluated va some smulaton studes n Seton 4. The proposed approah s appled for analyzng a real data set n Seton 5. The fnal seton nludes some onludng remarks.. Sem-parametr mxed-effets model The parametr models are usually restrtve and less robust aganst modfaton of model assumpton, but they are advantageous and effent when models are orretly spefed. In ontrast, nonparametr models are more robust aganst the model assumpton than a parametr model, but they are usually more omplex and less effent. Sem-parametr models are performs well and retan ne features of both parametr and nonparametr models. In semparametr models the parametr omponents are often used to model mportant fators that affet the responses parametrally and the nonparametr omponents are often used for nusane fators whh are usually less mportant (Wu and Zhang, 4)... Models spefaton A longtudnal data set an be expressed n a ommon form as ( j j t, y ), =,,..., n, j =,,..., n, (3) where t j denotes desgnsated tme ponts, y j the observed response at tme t j, n the number of observatons for the th subjet and n s the number of subjets. In the sem-parametr mxedeffets model (SpME), the mean response funton at tme t j depends on tme t j nonparametrally va a smooth funton (t), and lnearly on some other observable ovarates = (,..., ) j j p j, where p s the number of ovarates observed at tme t j. The random effet omponents at tme t j may depend on tme t j nonparametrally va a smooth proess (.) and lnearly on some other ovarates, namely h j= ( h j,..., hq j ), where q p. The resultng model may be wrtten as y j = ( t ) h b ( t ), j =,..., n, =,..., n, (4) j j j j j where and (.) are smooth funtons of tme, the ovarate vetor b = ( b,..., b ) q onssts of the oeffents of h j, (t) s smooth proess of tme, and j s the error at tme t j that s not explaned by ether the fxed-effets omponent t ) or the random effets omponent h b j ( t j j ). Other speal SpME models are obtaned when one or two SpME omponents are dropped from the general SpME model (4). When only the nonparametr random-effets ( j

5 AAM: Intern. J., Vol., Issue (June 5) 99 omponent s dropped, the SpME model (4) redues to the followng SpME model y j = ( t ) h b, j =,..., n ; =,..., n. (5) j j j j Ruppert et al. (3) dealt wth a smple verson (wth q = ) of ths type of SpME model usng penalzed splnes. For the longtudnal responses Y, the SpME model an be wrtten as Y = C H b, (6) where Y = ( y, y,..., yn ), = ( ( t ), ( t),..., ( tn )), = (,,..., n, H = ( h, h,..., hn ) C ) and N(, ). The error terms are assumed ndependent of b and = I n. We an approxmately express (t) as a regresson splne. In regresson splne smoothng, loal neghborhoods are spefed by a group of loatons, say,,,..., K, K n the range of nterest, suh that, an nterval [ a, b] an be onsdered as: a = K K b < <... < < =. (7) These loatons are known as knots; and r, r =,,..., K are alled nteror knots or smply knots. A regresson splne an be onstruted usng the followng so alled k th degree trunated power bass wth K knots,,..., K k k k p( t) = (, t,..., t,( t ),...,( t K ) ), (8) k k where k s hosen or 3 and a = ( a) denotes power k of the postve part of a, a = max(, a) and p = K k denotes the number of the bass funtons nvolve whh are alled smoothng parameters. We an express ( t ) ( t p ), where = (,..., s the assoated oeffents vetor. For loatng the knots, we an use equally spaed sample quantles as knots. Let t, ( l ) l =,,..., M be the order statsts of the pooled desgn tme ponts, where p ) n M = n. = Then the K knots are defned as

6 Taban Baghfalak et a. = t, )]) r =,...,, (9) r ([ rm/( K K where [a] denotes the nteger part of a. For smoothng parameter seleton, a good seletor usually tres to selet a good smoothng parameter p to trade-off the goodness of ft of the smoother and ts model omplexty. Generalzed ross-valdaton (GCV) s a smoothng parameter seletor whh s defned as follows n T GCV ( p) = ( y ˆ ) ( ˆ y y y )/( p/ M ). () = Note that the numerator n the GCV sore, s the SSE (sum of squared errors), representng the goodness of ft, and denomnator s assoated wth the model omplexty, where p s the model omplexty n regresson splne... Spefaton of mssngness model Consder a general pattern of mssng data and let R be the assoated matrx of the mssngness ndator related to the Y matrx and R j = f Y j s observed and otherwse R j =. For the mssngness proess R, probablty of response, p Pr( R = b ), s modeled usng a mxed effets logst regresson model as follows: where vetor, logt ( pj j j j = j w j s the j th row of the fxed effets desgn matrx z j the j th row of Z, and = dag ( ) ) = w z b, () W, the regresson oeffent. As above, ovarates n Z are not nluded n W. The measurements and mssngness proesses are lnked through the random effets term and ther assoaton s quantfed by the parameter vetor. 3. Random effets estmate In ths paper we make no parametr assumptons for the random effets dstrbuton and leave t ompletely unspefed. We assume that b G, wth G M, where M s the set of all dstrbuton funtons on the parameter spae M of b (Tsonaka et al., 9). Thus margnal densty for Y and R s gven by: f ( Y, R G, ) = f ( Y Y, b ) f ( R b, R) dg( b ). () m In general, G an be a dsrete or a ontnuous dstrbuton. However, Lard (978) and Lndsay (983) have shown that the nonparametr maxmum lkelhood estmate (NPMLE) of

7 AAM: Intern. J., Vol., Issue (June 5) the unknown G s dsrete wth fnte support and thus M redues to nludes all dsrete dstrbutons. So, () would be f Y, R G, ) = f ( Y, ) f ( R, ), (3) ( Y R where = ( Y, R) nludes the parameter vetor for the Y and for all R proesses, = (,,...) s the support ponts and = (,,...) s the orrespondng weghts of G. We all the model defned by equaton (3) Splne sem-parametr shared parameter model (Splne- SpSP). Ths s due to havng parametr assumptons for the nvolved submodels, but we have the random effets dstrbuton unspefed. 3.. Estmaton Proedure A two-step proedure has been developed that s terated untl onvergene. In the frst step, G s estmated for fxed at ts urrent estmate ˆ and n a seond step s updated by maxmzng the profle lkelhood l( G ˆ ), where Ĝ denote the estmated G of the frst step. The latter step an be easly mplemented usng an optmzaton method of R software. Estmate of G an be obtaned usng a VEM algorthm. The VEM s a dretonal dervatve-based algorthm that teratvely maxmzes the log-lkelhood l ( G ) n the set M of all dsrete dstrbutons over a prespefed grd,,..., ) wth C large. ( C The man dea of VEM s to searh n eah teraton for the dreton that maxmzes the loglkelhood nrease = l( G ) l( G ) (where G and G denote the urrent and updated estmates of G, respetvely), and exhange weghts between the grd ponts that ontrbute the least and the most to. These ponts are dentfed based on the propertes of the dretonal dervatve of the log-lkelhood from one dstrbuton G to another G. When G s degenerate at, =,... C, then G = G. In partular, the dretonal dervatve D( G, G ) of l (G) at, G n the dreton of G s defned as D( G l(( s) G sg ) l( G ), G ) = lm. (4) s s ( For eah grd pont, wth =,..., C, we evaluate the dretonal dervatve, for fxed the ase of the proposed Splne-SpSP model takes the form ˆ t ), n for proof, let D( G n f ( Y, R G, ˆ), G ) = n, (5) f ( Y, R G, ˆ) n = l( G) = log f ( Y, R G, ˆ) = log f ( Y, R G, ˆ), (6) = n =

8 Taban Baghfalak et a. We use (4) and (6) for D( G, G ), so that D G G s f Y R G ˆ sf Y R G ˆ n (, ) = lm log [( ) (,, ) (,, )] s s = n log (,, ˆ f Y R G ) = ˆ ˆ = lm log. (7) s s n ( s) f ( Y, R G, ) sf ( Y, R G, ) ˆ = f ( Y, R G, ), Usng the L hoptal rule, equaton (7) lead to (5). Also, we have C (,, ˆ) = ˆ (,, ˆ f Y R ). = f Y R G (8) So equaton (8), for the eah teraton, an be wrtten as D( Gˆ t n ˆt f ( Y, R, ), G ) = n. (9) C = ( t) ˆ f ( Y, R, ˆt ) = As a frst step, we spefy the grd. b s a q dmensonal vetor. Thus a grd for b defned q n [, ] = [, ] [, ] wth of order 4 or 5 would n most ases be suffent. For eah, wth =,..., C, we get a q varate vetor, where omponents must be hosen n U [4,4], suh that ( )/ k, wth k =.. Then, (9) s omputed for all s. Note that ntal value for the parameters ( Y, R ) of the Y and R proesses, an be obtaned by fttng the approprate gnorable mxed effets models,.e., a lnear mxed model and a mxed effets logst regresson, respetvely. Intal values for the orrespondng weghts of the support ponts are / C. After spefyng all dretonal dervatves, and as follows ˆ t = arg mn (, ), = arg max ( ˆ t D G G D G, G ), () and ther weghts updated aordng to ( t) * ( t) ( t) * ( t) ( t) ˆ = ( s ) ˆ, ˆ = s ˆ ˆ, () where ( s * [,]) denote the step length defned as

9 AAM: Intern. J., Vol., Issue (June 5) 3 s = arg max[ l{ Gˆ ˆt ( s) } l{ Gˆ ˆ }]. * t t t s () The estmaton of * s s mplemented usng a lne searh method. ˆ * Note that f s = then = and thus s exluded from the grd and the grd sze redue to C. ˆ ( t ) After estmatng G n the frst step, n the seond step by usng for example optm" funton n the R software, the vetor s estmated. These two steps are repeated teratvely untl onvergene. The algorthm onverges when the followng ondtons are satsfed ˆ ( ) max D( G t, G ) < whh guarantees that ˆ ( t) ˆ( t) ( ) ( ˆ ( t) ˆ( t) l G l G ) <. Note that the submodels (6) and () requre the mean of the random effets to be zero,.e., C E( b ) =. = To ensure dentfablty, we fx through the optmzaton proedure that the models nterepts follow 4. Smulaton study C = Sˆ, = Sˆ. (3) new b new b = = A smulaton study s mplemented to nvestgate the performane of the proposed method. The performane of our model s evaluated wth the use of varous dstrbutonal assumptons for the random effets omponent. We ompare the Splne-SpSP model wth two other models. The frst model s Splne-SPM, where the longtudnal proess s modeled wth the splne and the seond model s the SpSPM, where the longtudnal proess s modeled by a lnear model. We show robustness of the Splne-SpSP model wth respet to dstrbuton assumptons of the random effets and nonlnearty of the model. The longtudnal proess Y s smulated from the followng sem-parametr model: C Y j = tj tj 3( tj ) 4( tj ) 5 b j, (4) where the subsrpts measurements, where =,..., n denotes the subjet, and j =,..., N denotes the repeated N = max n, t j s the tme varable that takes values n [,3], s the bnary ovarate and b s the random effets omponent. The parameter vetor s taken as =.5, =, =.4, =.8, = and =.5. For the error omponent, we 3 4 assume j N(, Y ) wth Y =.5. Two sample szes n = and n = 5 wth N = 5 equally spaed vst tmes s assumed. 5

10 4 Taban Baghfalak et a. A model that set s for R proess s the non-monotone mssngness model. The bnary ndator R s smulated from a mxed effets logst regresson j logtp ( Rj = b ) = t b, j where =., = and =.5. Usng ths logst model, we generate a matrx ontanng zero and one. Ths matrx s alled the mssngness matrx. The Y j that orresponds to the zero elements of the matrx go to be mssng values n the data set. The assumed values for the regresson parameters are hosen suh that they lead to approxmately % of the mssng. The shared random nterepts b lnked the Y and R proesses, also we assume =.5. For random effet b three senaros are onsdered: a dstrbuton N (,), a mxture of two normal omponents,.5n(.35,.6 ).5N(.35,. ), a dsrete dstrbuton wth support at.5,.5,. 75,.75 and orrespondng weghts.3,.8,.8 and.3. For eah of these three senaros and 5 samples are smulated. Eah sample was ftted under Splne-SpSP, Splne-SP and SpSP models. The SpSP model whh s used for analyzng the generated data set s: Y j = j 5 j t b. (5) Comparsons between estmates are based on the root mean squared error (RMSE) and relatve bases (RB) whh are defned as: RMSE ( ) = N * N * ˆ ˆ ( ), ( ) = ( ). * RB * N N = = The results of ths smulaton are presented n Tables and. These smulaton studes show that the Splne-SpSP model s robust to the volaton of dstrbutonal assumptons of the random effets. When the random effets dstrbuton s normal, parameter estmates of Splne-SpSP and Splne-SP models are smlar. But when the random effets dstrbuton departs from normalty assumpton, dfferene of the two models are unfolded and the Splne-SpSP model gves parameter estmates that are loser to real values than parameter estmates n Splne-SP and SpSP models. Moreover, the RMSE and RB of Splne-SpSP model s lower than Splne-SP and SpSP models. Also t an be seen n Fgure A. that our approah offers an nformatve nsght on the assumed shape of the random effets dstrbuton. 5. Applaton We apply the Splne-SpSP, Splne-SP and SpSP models to the analyss of the HIV- RNA data (Sun and Wu, 5 and Hammer et al., ) from an AIDS lnal tral study for omparng a sngle protease nhbtor (PI) versus a double-pi antretrovral regmens n treatng HIV-nfeted patents. In ths study, all subjets start the antretrovral treatment at tme and HIV- RNA

11 AAM: Intern. J., Vol., Issue (June 5) 5 levels n plasma (vral load) was measured repeatedly over tme. The sheduled vsts for the measurements were at weeks,, 4, 8, 6 and 4. A total of 48 patents were entered n the lsted study, wth 66 total vsts. Indvdual profles for patent are shown n Fgure. From ths plot, t s dffult to attan any useful nformaton. It an be seen that the ndvdual RNA level are outrght nosy n any tme t. We usually expet that the RNA levels would nrease f treatment was effetve. But from ths plot, t s not easy to see any patterns among the ndvdual patents RNA levels. We wll use nonparametr regresson for the relatonshp between the response varable and tme n the model. Fgure. Profle for patents The response varable Y s the hange of the HIV- RNA level usng a log sale at tme t whh showed the advane of a dsease. As regards the relatonshp between the response varable Y and tme we see that t annot be expressed va a lnear model. Therefore, we use the regresson splne for onsderng t n the model. In ths study, the four treatment groups are used for patents. We evaluate treatment groups and tme n the response varable., and 3 are ndator varables (dummy varables) suh that f treatment s used = o.w. and f treatment s used = o.w. f treatment 3 s used 3 = o.w. The sem-parametr model for measurements proess an be wrtten as

12 6 Taban Baghfalak et a. y j = tj tj 3( tj ) 4( tj ) b, j where =,...,48, j =,...,n and n =,...,6. We use the trunated power based on (8) wth k =, and adopted the equally spaed sample quantles as knots" method to spefy the knots. Naturally, ths model s jonted to the non-gnorable mssngness model note that the perentage of mssngness s around %. The probablty of response s modeled usng a mxed effets logst regresson as follows logt ( Pr( rj = b )) = tj 3 43 b. (6) The Y and R proesses are lnked through the shared random effet b, and ther assoaton s measured by the parameter. If =, the Y and R proesses are ndependent. The estmated parameters and ther standard devatons (omputed by the bootstrap method) are presented n Table 3. These two models are ompared by Akake nformaton rteron (Akake 973) and Bayesan nformaton rteron (Shwartz 978). These are defned as AIC = Loglk df, BIC = Loglk log ( n) df, where Loglk s the logarthm of the lkelhood funton and df s the model omplexty whh s the number of bass funton p together wth p ovarates observed at tme t (Wu and Zhang, 4). It an be seen n Table 3 that AIC and BIC of the Splne-SpSP model s smaller than those of the Splne-SP and SpSP models. The model produes relable parameter estmates under any dstrbutonal assumpton for the random effets. Also aordng to Table 3, the Splne-SpSP model shows that treatment and treatment are not sgnfant. But, tme s an effent varable; suh that the more tme, the less vral load measurements. Also, s a sgnfant parameter,.e. mssngness s found to be non-gnorable. The ftted log (RNA) for some randomly hosen subjets are presented n Fgure A.. To summarze, these results suggest that the Splne-SpSP model provde prese predton for the dataset thanthe two other models. 6. Conluson In ths paper, we have foused on the use of a sem-parametr model n longtudnal data. At frst we explan shared parameter models as an appealng framework for the jont modelng of the measurements and mssngness proesses, partulary n the nonmonotone mssngness ase. We take a sem-parametr model for the measurment proess and logst regresson as a model for mssngness mehansm. Wth the usage of a NPMLE method also alled a vertex exhange method, we estmate the random effet dstrbuton. We use the Splne-SpSP model n some sets of smulated data and onsdered the varous dstrbutonal assumptons for the random effets. Our study uses the Splne-SpSP model framework applyng the nonmonoton non-gnorable mssngness. Our smulaton studes show that the proposed model s robust to the varous dstrbutonal assumptons onsdered for the random effets. We also observed that the proposed model produes estmates wth RMSE and S.E. whh are lower than those obtaned by the

13 AAM: Intern. J., Vol., Issue (June 5) 7 Splne-SP and SpSP models. REFERENCES Akake, H. (973). Informaton theory as an extenson of the maxmum lkelhood prnple. Pages 67-8 n B. N. Petrov, and F. Csak, (Eds.) Seond Internatonal Symposum on Informaton Theory. Akadema Kado, Budapest. Beunkens, C., Molenberghs, G., Verbeke, G., and Mallnkrodt, C. (8). A latent-lass mxture model for nomplete longtudnal Gaussan data. Bometrs, 64, Bohnng, D. (985). Numeral estmaton of a probablty measure. Journal of Statstal Plannng and Inferene,, De Gruttola, V., and Tu, X. M. (994). Modellng progresson of CD-4 lymphoyte ount and ts relatonshp to survval tme. Bometrs, 5, 3-4. Follmann, D., and Wu, M. (995). An approxmate generalzed lnear model wth random effets for nformatve mssng data, Bometrs, 55, Hammer SM, Vada F, Bennett KK et al. (). Dual vs sngle protease nhbtor therapy followng antretrovral treatment falure: a randomzed tral. JAMA, 88, Lard, N. (978). Nonparametr maxmum lkelhood estmaton of a mxng dstrbuton. Journal of the Ameran Statstal Assoaton, 73, Ln, H., Turnbull, B. W., MCulloh, C. E., Turnbull, B. W., Slate, E. H., and Clark, L. (). A latent lass mxed model for analysng bomarker trajetores wth rregularly sheduled observatons. Statsts n Medene, 9(), Lndsay, B. G. (983), The geometry of mxture lkelhoods: A general theory. The Annals of Statsts,, Lttle, R. (995). Modelng the drop-out mehansm n repeated measures studes. Journal of the Ameran Statstal Assoaton, 9, Rubn, D. B. (976). Inferene and mssng data (wth dsusson). Bometrka, 63, Ruppert, D., Wand, M. P., and Carrol, R. j. (3). Sem-parametr Regresson, the press of london Cambrdge Unversty. Shwartz, G. (978). Estmatng the dmenson of a model. Annals of statsts, 6, Song, X., Davdan, M., and Tsats, A. A. (). A sem-parametr lkelhood approah to jont modellng of longtudnal and tme to event data. Bometrs, 58, Sun, Y. and Wu, H. (5). Semparametr Tme-Varyng Coeffents Regresson Model for Longtudnal Data, San. J. Statst., 3, -47. Tsonaka, R., Verbeke, G., and Lesaffre, E. (9). A Sem- Parameter shared parameter model to handle nonmonotone non-gnorable mssngness. Bometers, 65, Wu, M. C., and Carroll, R. (988). Estmaton and omparson of hanges n the presene of nformatve rght ensorng by modelng the ensorng proess. Bometrs, 44, Wu, H., and Zhang, J. T. (). Loal polynomal mxed-effets models for longtudnal data analyss. Journal of the Ameran Statstal Assoaton, 97, Wu, H., and Zhang, J. T. (4). Nonparametr regresson methods for longtudnal data analyss. Wley Seres n Probablty and Statsts, New York. Zhang, D., Ln, X., Rez, J. and Sowers, M. (998). Semparametr stohast mxed models for longtudnal data. Journal of the Ameran Statstal Assoaton, 93, 7-79.

14 8 Taban Baghfalak et a. Table. Results of the smulaton study: Evaluaton of the Splne-SpSP model and omparson wth the Splne-SPM and SpSP models. Mean (Est.), standard error (SE) and Root Mean Square Error (RMSE) for sample sze 5 Splne-SpSP model Splne-SP model SpSP model Par Real Est. S.E. RMSE RB Est. S.E RMSE RB Est. S.E. RMSE RB b b b Normal dstrbuton Mxture of two normal dstrbutons Dsrete dstrbuton

15 AAM: Intern. J., Vol., Issue (June 5) 9 Table. Results of the smulaton study: evaluaton of the Splne-SpSP model and omparson wth the Splne- SPM and SpSP models. Mean (Est.), standard error (SE) and Root Mean Square Error (RMSE) for sample sze Splne-SpSP model Splne-SP model SpSP model Par Real Est. S.E. RMSE RB Est. S.E RMSE RB Est. S.E. RMSE RB b b Normal dstrbuton Mxture of two normal dstrbutons

16 Taban Baghfalak et a. Table. Contnues b Dsrete dstrbuton Table 3. The random nterepts analyss of the AIDS lnal tral study. The estmates (Est.) and standard devaton (S.D.) are presented for the proposed Splne-SpSP, Splne-SP and the ommon SP models Splne-SPSP model Splne-SP model SPSP model parameters Est. S.D. Est. S.D. Est. S.D b AIC BIC

17 AAM: Intern. J., Vol., Issue (June 5) APPENDIX A Fgure A.. True dstrbuton (a) normal, (b) mxture of two normal dstrbutons and () dsrete: barharts are of NPMLE of the random effets dstrbuton for randomly seleted ftted data set Fgure A.. Indvdual vral load trajetory estmates for sx randomly hosen subjets after fttng the three models. The flled rles are the observed values for ndvduals