Testing for outliers in nonlinear longitudinal data models based on M-estimation

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1 ISS , England, UK Journal of Informaton and Computng Scence Vol 1, o, 017, pp estng for outlers n nonlnear longtudnal data models based on M-estmaton Huhu Sun 1 1 School of Mathematcs and Statstcs, Yancheng eachers Unversty, Yancheng, 400, Chna, E-mal: sunhuhu1@163com (Receved September 1, 016, accepted January 04, 017) Abstract In ths paper we propose and analyze nonlnear mxed-effects models for longtudnal data, obtanng robust maxmum lelhood estmates for the parameters by ntroducng Huber s functon n the log-lelhood functon Furthermore, the test for outlers n the model based on robust estmaton s nvestgated through generalzed Coo s dstance he obtaned results are llustrated by plasma concentratons data presented n Davdan and Gltman, whch was analyzed under the non-robust stuaton Keywords: M-estmaton; nonlnear mxed models; longtudnal data; testng for outlers; generalzed Coo s dstance 1 Introducton onlnear mxed models can model mechanstc relatonshps between ndependent and dependent varables and can estmate more physcally nterpretable parameters (Pnhero and Bates 000), whch are mportant to the analyss of longtudnal data, mult-level data and repeated survey data and wdely used n the feld of economcs, bo-pharmaceutcals, agrculture Recently, some dfferent nonlnear mxed effects models and nference procedures have been proposed Russo et al (009) proposed nonlnear ellptcal models for longtudnal data and presented dagnostc results based on resdual dstances and local nfluence (Coo, 1986 and 1987) We and Zhong (001) focused on nfluence analyss n nonlnear models wth random effects In standard analyss of data well-modeled by a nonlnear mxed model, an outlyng observaton can greatly dstort parameter estmates and subsequent standard errors Consequently, nferences about parameters are msleadng hen, a robust procedure s needed for accurate results n the presence of outlers M-estmaton s the most wdely used robust estmaton method, whch was frstly ntroduced n Huber s artcle (1981) on regresson Mancn et al (005) and Muler and Yoha (008) proposed a robust M- estmator that assgns a much lower weght to outlers than tradtonal maxmum lelhood estmators does Pnhero et al (001) and Staudenmayer et al (009) studed robust estmaton technques n whch both random effects and errors have multvarate Student-t dstrbutons Whle, less alternatves have been studed for outler accommodaton n the context of nonlnear mxed-effects models Yeap and Davdan (001), who proposed a two-stage robust estmaton n nonlnear mxed-effects models when outlers are presented, s one of the few references that address ths case Meza et al (01) presented an extenson of a Gaussan nonlnear mxed-effects model usng heavy-taled multvarate dstrbutons for both random effects and resdual errors James et al (015) proposed an outler robust method based on lnearzaton to estmate fxed effects parameters and varance components n nonlnear mxed model However, lttle attenton has been pad to the nfluence dagnostc for nonlnear mxed models n the current lterature In ths artcle, we ntroduce a robust method by utlzng a robust verson of the log-lelhood for the nonlnear mxed model and nvestgate the test for outlers n the model based on robust estmaton by generalzed Coo s dstance, extendng and expandng the studes of Gll (000) and James et al (015) Our results show that the generalzed Coo s dstance based on robust estmaton can successfully detect the masng effects that appear n the data set he rest of the artcle s organzed as follows Secton ntroduces the nonlnear mxed effects model dscussed n ths paper and uses Fsher scorng method to get M-estmaton of parameters And the asymptotc propertes s also establshed In Secton 3, we nvestgate the test for outlers n nonlnear model based on robust estmates In Secton 4, as an llustraton, we apply the proposed method to analyze an observatonal data set Fnally, some conclusons are gven and possble future wor s dscussed n Secton 5 Publshed by World Academc Press, World Academc Unon

2 108 Model and robust estmaton Huhu Sun: estng for outlers n nonlnear longtudnal data models based on M-estmaton Assume that response measurements are collected on subjects and the -th subject beng observed on n tme ponts, thus M n 1 s the total number of measurements In the matrx notaton, the model for measurements from subject s y f ( X, ) C e, 1,,,, (1) where y ( y1,, yn ) s a vector of length n contanng observable response varable from subject ; s a nown second order dfferentable nonlnear functon of the regresson vector, whch s a f (, ) vector of q unnown but fxed parameters wth nown desgn matrx the n r desgn matrx for the random effects of subject, s a X, and X ( x 1,, x ) ; n C r 1 vector of random effects assumed to be sampled from a multvarate normal dstrbuton wth mean 0 and covarance matrx 1 n e ( e,, e ) s an n 1 unobservable random error and e ~ (0, ) It s also assumed that and e are ndependent from each other hen, cov( y ) C C Let denote the vector of unnown parameters n, and the log-lelhood for the nonlnear mxed model s l(, y) cons tan t M log log, () 1 1 whch s obtaned from the margnal model from (1) (Russo et al, 009), where 1 1/ ( y f ( X, )) ote that the last term of () s a half sum of squares and grows qucly Followng the M-estmaton theory expressed n Huber (1981), the log-lelhood functon l s robustfed by replacng t wth a functon that ncreases much slower In ths paper, Huber functon s chosen to bound the nfluence of outlyng observatons on the estmaton, whch s defned by 1 f c ( ), 1 c c f c where c s the Huber tunng constant and usually c [07, ], here c=1345 (Rpley, 004) he frst dervatve of ( ) wth respect to, s gven by f c ( ) ( ) / csgn( ) f c herefore, the robustfed verson of () s gven by n 1 1 (, y) cons tan t 1M log 1log ( j ), (3) 1 1 j1 where 1 E( ( )) Pr( c) s the consstency correcton factor and s needed to mae the estmatng equatons have zero expectaton hen, we can obtan robust maxmum lelhood estmaton (RMLE) through Fsher scorng method (Gll, 000 and James et al, 015) based on (3) ote that the robust maxmum lelhood estmaton becomes the classcal maxmum lelhood estmaton as c approaches nfnty n Huber functon ow we study the asymptotc propertes of RMLE Let (,, ), by Domowtz and Whte (198), we consder that the RMLE of s obtaned by maxmzng an objectve functon n the form 1 G( y, ) g( y, ), 1 and the estmatng equaton for s as follows s JIC emal for contrbuton: edtor@jcorgu

3 Journal of Informaton and Computng Scence, Vol 1(017) o, pp ' 1 G( y, ) g( y, ) 0, 1 where g( y, ) g( y, ) / Consder that s obtaned by maxmzng the objectve functon 0 G ( ) E[ G ( y, )] ' So we have E[ G ( y, )] 0 ow usng the mean value theorem, we get 0 ' ' " 0 0 G ( y, ) G ( y, ) G ( y, )( ) (4) ' where (1 ) 0,0 1 Because G ( y, ) 0, (4) gves ( ) [ G ( y, )] ( G ( y, )) " 1 ' 0 0 Property 1 shows that the M-estmaton for has the same propertes as MLE Property 1 Under some assumptons (Domowtz and Whte, 198), accordng to Sanjoy (004), the followng consstency and asymptotc normalty property can be got ( a) a s 0 ( b) ( ) ( 0J, ( )) where J ( 0 ) M Q M, M G ( 0 ), G ( ) E[ G ( y, )], Q var[ G ( y, 0 )] Whte (198) gves the detaled proof 3 Case deleton model and generalzed Coo s dstance, Domowtz and he data deleton model s one of the most basc statstcal dagnoss model he basc method commonly used to study outlers or strong nfluence ponts s to compare the dfference of the estmator between the data pont deleted before and after For nonlnear mxed effects model of longtudnal data, there are two types of data deleton: one s to delete an observaton value of an ndvdual, that s ( y, x ), and the other s to delete the whole set of observatons of an ndvdual, e ( y, x ) For convenence, ths paper only dscusses the second case When the whole data set of the -th ndvdual s deleted, model (1) becomes y f ( X, ) C e, j (31) j j j j j he robustfed log-lelhood functon s gven by n 1 1 [ ] (,, ) constant 1( M n ) log 1 log ( j ), (3) 1, 1, j1 where the subscrpt [] denotes the deleton of observatons of the -th ndvdual Let estmate of after deletng the -th ndvdual he dfference between [] [] j j be the robust and can be compared to measure the nfluence of the -th ndvdual on If the dfference s large, the -th ndvdual may be a strong nfluence ndvdual Accordng to Coo and Wesberg (198), defne the generalzed Coo s dstance to measure - [] Defnton 31 he generalzed Coo s dstance based on the Fsher nformaton matrx of for model (1) and ts case deleton model (31) s gven by GC ( [ ] ) E ( [ ] ) If the value of generalzed Coo s dstance correspondng to the -th ndvdual values, the -th ndvdual s consdered to be a strong nfluental ndvdual (33) GC s larger than other JIC emal for subscrpton: publshng@wauorgu

4 110 4 An llustratve example Huhu Sun: estng for outlers n nonlnear longtudnal data models based on M-estmaton Plasma drug penetraton data was collected by the followng experment: Frstly, sx volunteers were njected the same dose of a drug through vens hen, the drug concentraton n ther plasma was measured at eleven tmes wthn eght hours Davdan and Gltman (1995) appled the followng double exponental model to descrbe the data: 1 3 y e exp( e x ) e exp( e 4 x ) e, 1,,,6; j 1,,,11, where y j j j j j denotes the drug concentraton of the -th volunteer at the j-th tme; th volunteer measure the j-th tme drug concentraton; ; (0, ) e j s the random error that ndependent wth x j s the tme nterval of the - s the drug effect of the -th volunteer and and we suppose e (0, ) Desgn matrx X (05,050,075,100,15, 00,300, 400,500,600,800) Frst, we get the RMLE of parameters by the method we proposed n the paper he results are shown n able 1 Parameters able 1 RMLE of parameters RMLE MSE ow calculate the generalzed Coo s dstance for each ndvdual usng formula 33 Consder the scatter plot of the generalzed Coo s dstance as shown n Fgure 1 he generalzed Coo s dstance s calculated based on RMLE n Fgure 1(a) and based on MLE n Fgure 1(b) As can be seen from the two fgures, the generalzed Coo s dstance for the thrd ndvdual s the largest, followed by the forth ndvdual So t can be consdered that the thrd ndvdual s a strong nfluental ndvdual In addton, we can see from Fgure 1(a) and 1(b) that the generalzed Coo s dstance for the forth ndvdual s larger n Fgure 1(a) herefore, the generalzed Coo s dstance presented n the paper based on M-estmaton may dentfy potental outlers, whch could overcome the masng phenomenon j Fgure 1 generalzed Coo s dstance JIC emal for contrbuton: edtor@jcorgu

5 Journal of Informaton and Computng Scence, Vol 1(017) o, pp Concluson onlnear mxed effects models are very useful statstcal tools n analyzng nonlnear data wth random effects However, when there are outlers n the data, current nonrobust estmaton methods may produce msleadng results, so a robust method s needed In ths artcle, the log-lelhood functon of the nonlnear mxed effects model s altered by ncorporatng an approprate loss functon,, of the resduals Fsher scorng method s appled to get M-estmaton of the parameters for the means model and for the varance components hen, generalzed Coo s dstance based on M-estmaton s presented n the paper Here Huber functon s used to bound the nfluence of outlyng observatons An alternatve functon s the one suggested by Andrews et al(197), called the Bsquare functon, s 3 c 1 ( ) 1 f c 6 c ( ), c f c 6 where c s the tunng constant he frst dervatve of wth respect to, denoted ( ) ( ) /, s gven by 1 f c ( ) c 0 f c We can also use perturbaton dagnostcs based on M-estmaton to overcome the masng phenomenon when dentfyng potental outlers, such as mean shft perturbaton, case weghts perturbaton, perturbaton of covarate n random effects and so on he ey pont of all the above perturbatons s to fnd 6 Acnowledgement ( ) he project supported by atonal Statstcal Scence Research Project of Chna (015LZ7) 7 References Andrews D F, Bcel P J, Hampel F R, Huber P J, Rogers W H and uey J W Robust Estmates of Locaton: Survey and Advances Prnceton Unversty Press: ew Jersey (197) Banerjee M, Frees E W Influence dagnostcs for longtudnal models Journal of the Amercan Statstcal Assocaton 439: (1997) Coo, D Assessment of local nfluence Journal of the Royal Statstcal Socety - Seres B 48(): (1986) Coo, D Influence assessment Journal of Appled Statstcs 14(): (1987) Dggle, P J, Lang, K Y and Zeger, S Analyss of longtudnal data ew Yor: Oxford Unversty Press (00) Davdan M, Gltman D M onlnear Models for Repeated Measurement Data London: Chaman and Hall (1995) Domowtz, I, Whte, H Msspecfed models wth dependent observatons Journal of econometrcs 0: 35-58(198) Fung WK,Kwan CW A note on local nfluence based on normal curvature J R Statst Soc B 59: (1997) Gll, P S A robust mxed lnear model analyss for longtudnal data Statstcs n Medcne 19: (000) Huber, PJ Robust Statstcs ew Yor: Wley (1981) James DW,Jeffrey B B and Abdel-Salam G A Outler robust nonlnear mxed model estmaton Statstcs n Medcne 34: (015) Lesaffre E, Verbee G Local nfluence n lnear mxed models Bometrcs 54: (1998) F JIC emal for subscrpton: publshng@wauorgu

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