Linear Minimum Variance Unbiased Estimation of Individual and Population slopes in the presence of Informative Right Censoring

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1 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 ISSN Lear Mu Varace Uased Esao of Idvdual ad Populao slopes he presece of Iforave Rgh Cesorg VswaahaN * RavaaR ** * Depare of Sascs Presdecy College Chea 65 Ida ** Depare of Sascs Presdecy College Chea 65 Ida Asrac- Esao of dvdual ad populao slopes usg Lear Mu Varace Uased Esao y Wu ad Baley (988) s appled for coplee logudal sudes I hese sudes he rgh cesorg process s cosdered forave whch he uer of oservaos ade for each sujec s assued o vary depedg o he rae of chage or slope of he dvdual respose varale Ths process of ssg o a rado poses proles he esao of he slope paraeers The proposed ehod provdes eer esaes uder dffere paraerc odels for he cesorg dsruo The ehod s perforace s suded hrough a sulao sudy uder Sadard Noral ad Ufor cesorg dsruos Idex Ters- Cuulave dsruo fuco Eprcal Bayes esao Sadard Noral dsruo Pro fuco Iforave Rgh cesorg Sulao C I INTRODUCTION haracerzao of rae of chage he sudy varale over e s of prary eres logudal sudes where repeaed easurees of he sudy varales are ake o he sujecs The rae of chage of hese varales over e s oe of he prary eress of he researchers as helps us rack he dvduals who are a rsk or hose who eed edae aeo The classcal ehods of esao fal uder he codo of ssg o a rado or whe he cesorg dsruo s depede o he rae of chage of he sudy varale of he dvdual sujec Logudal sudes prolog for ay years ad eal dropous due o deahs grao ad relaed facors There are several proposed ehodologes o accoodae coplee daa y Orchard ad Woodury (98) Kleau (973) ad Lard ad Ware (98) These ehods assue paers ha accoodae dropous ha are depede of he sudy varale I hs paper he paer of ssg daa cosdered s relaed o he sudy varale The er 'forave rgh cesorg' he coex of slope esao refers o he suao where he cesorg proaly of each dvdual relaes o hs or her uderlyg rae of chage (Wu ad Baley 988)They showed ha uder a lear rado effec odel he weghed leas square esae of slope ca e severely ased f he rgh cesorg process s forave alhough s os effce f he cesorg process s o-forave I a effor o reduce he as roduced y he presece of forave rgh cesorg Wu ad Carroll (988) derved a Pro Pseudo-Maxu Lkelhood Esaor (PPMLE) of he populao rae of chage uder a lear rado effec odel where he rgh cesorg process follows a pro fuco oh of al value ad of he slope of he dvdual sujecs Wu ad Baley (989) adaped a codoal lear odel approach ad proposed a Lear Mu Varace Uased (LMVUB) ad Lear Mu Mea Square Error (LMMSE) esaors of he populao rae of chage whch hey showed were copeve wh he PPMLE her sulao sudy The prese sudy focuses he effecveess of he LMVUB esaor uder dffere cesorg dsruos whose paraeer depeds o he rae of chage of he dvdual sudy varale LMVUB esaor s copared wh he geeral Eprcal Bayes esaor U-weghed ad Weghed esaors To easure s effecveess he sudy cosders oh he dvdual ad populao slopes Sulao resuls are cosdered ad fereces are ade wh suggesos for he fuure research II THE ESTIMATION PROCEDURE The LMVUB esaor s derved fro he geeral case of he Eprcal Bayes esao procedure Ths odel assues ha here are sujecs ad ha for each sujec ( = ) he followg codos hold: () () ~ N( V ) ~ N( * A) where he slope ad up V Var ( s he Ordary Leas Square (OLS) esae of ) s he axu e he h sujec s followed j j where s a kow cosa ad A= Var ( ) s also assued o e kow Le 3 deoes he vecor of e pos for he h sujec I s assued ha he las easuree e s he wwwjsrporg

2 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 ISSN cesorg e The argal dsruo of s gve y ~ N( A V ) ad where are he paraeers of he cesorg dsruo gve The poseror dsruo of s ~ N( V B ) where B B V B wh V A ˆ B ˆ e B ˆ arx represeao s gve y B B X X ' DX X ' DB ˆ e gve y B ; = 3 Is The correspodg esaor of he populao slope s ˆ pop e ˆ e resuls he LMVUB esao Le ad X ' 33 Here he secod row represes he dvdual cesorg e of he sujecs V A V A Le D V A 3 V A ad B B B B 3 The paraeer s esaed usg he Weghed Leas Square (WLS) ehod ad s gve y ˆ X ' DX X ' DB The Eprcal Bayes esae for he dvdual slope s gve y B Var The assupo of a codoal lear odel s gve y wh LMVUB( ) ˆ ˆ The E E ad Bˆ If reduces o he Uweghed esaor ad s uw ˆ populao slope s gve y The weghed ˆ w B B ˆ w dvdual slope esaor s wh he correspodg populao slope esaor ˆ w B B III SIMULATION STUDY The sulao sudy s carred ou o copare he perforace of he four esaors: Eprcal Bayes LMVUB Weghed ad Uweghed Esaors The sudy copares oh dvdual ad populao slopes Sulaos are carred ou for a seg slar o ha already proposed ad dscussed he leraure y Wu ad Baly (988) R sofware s used o perfor he sulao process Paraeers used he sulao are slar o hose of Wu ad Baly who oaed he esaes fro a feasly sudy for a a-phoolyc replacee herapy ral coduced y he Workshop o he Naural Hsory of Pz Ephysea The sudy varale hs ral s rae of decle he FFV easurees We geeraed daa se each coag oservaos accordg o he followg specfcaos: Measuree error sadard devao =55; slope = 9; he expeced slope = - 9 I s sadard devao assued ha he sudy durao exeds o 3 years wh 4 easurees for each year Furher s assued ha rgh wwwjsrporg

3 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 3 ISSN cesorg occurs oly a he ddle of each year afer he secod easuree The proaly of rgh cesorg or cesorg j dsruo durg a specfed e erval gve he slope o e j where s he cuulave proaly dsruo fuco of N ( ) The cesorg j paraeer for dffere e ervals are deoed y j = 3 The sulaos are carred ou uder 3 dffere codos: () = = - 4 = - 7 ad 3 = - 5; () = - 3 = - 34 = - 4 ad 3 = - 38; ad (3) = - 38 = - 36 = - 6 ad 3 = - 6; he aove scearo he frs se of paraeers correspod o o-forave rgh cesorg ad he secod ad hrd codos correspod o Iforave rgh cesorg The sulao process s carred ou wh he followg sequece () ~ Noral (- 9 9 ); PrT( ) () j j ; ad (3) V Var ( ) ~ Noral V j j 55 where The crera used o evaluae he perforace are as ad wo ypes of Mea Square Error MSE (a) ad MSE () defed respecvely as: Bas (a) = MSE (a) = MSE () = R r ˆ R r R ˆ r R r R ˆ r r R r where R s he oal uer of replcaos ad s he uer of oservaos each daa se MSE (a) easures he closeess of he esaor o he populao slope ad MSE () easures he closeess of he esaor o he dvdual slopes 3 Rado Geerao of Cesorg Dsruo: The cesorg dsruo correspodg o he rado varale s geeraed usg he Pr( T ) assupo j j Thus he cesorg dsruo for each sujec vares as vares fro sujec o sujec As deoe he cuulave dsruo fuco of sadard oral varae he followg procedure s used o geerae values Four oservaos are ade hree years aoug o axu ( ) = Also s assued ha rgh cesorg occurs oly a he ddle of each year afer he secod easuree Ths ples akes values 6 For each of he hree years he oral cuulave j proales wh varyg j= 3 are calculaed v The proaly for = s ake as he cuulave proaly calculaed for j= The proaly for = s ake as he dfferece ewee uy ad he cuulave proaly calculaed for j= 3 The proaly for = 6 s ake as he dfferece ewee he cuulave proales for j = ad j = The proaly for = s ake as he dfferece ewee he cuulave proales for j=3 ad j= v I case of Ufor cuulave dsruo fuco he upper ad lower ls of he dsruo are fxed y Lower l = Mu ( )* + Mu ( 3 ) Upper l = Maxu ( )* + Maxu ( 3 ) The aove procedure represes oe of he ways of pleeg Pr( ) T j j o geerae rado values for Sulao resuls correspodg o sadard oral cesorg dsruo are provded Tale 3 Ths suarzes resuls regardg as ad ea square error The resul reveals rrespecve of Iforave or No- Iforave rgh cesorg process he LMVUB perfors eer copared o he oher hree esaors By oservg he MSE correspodg o populao ad dvdual slopes he LMVUB esaor has wwwjsrporg

4 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 4 ISSN Tale 3 Coparsos of as ad ea square error: Uweghed Weghed Eprcal Bayes ad LMVUB for Sadard Noral dsruo Esaor Bas MSE(a) MSE()* 3 = Uweghed Weghed Eprcal Bayes LMVUB = - 3 Uweghed Weghed Eprcal Bayes LMVUB = - 38 Uweghed Weghed Eprcal Bayes LMVUB he u MSE followed y Weghed Eprcal Bayes ad Uweghed esaors ha order Ths red s slar oh he populao ad dvdual slopes Also o e oed s he agude of reduco he MSE values I case of Noforave rgh cesorg he MSE correspodg o he secod es weghed esaor for populao slope s early fve es as ha of he LMVUB esaor I case of he dvdual slopes he MSE of Weghed esaor s 44 es as ha of he LMVUB esaor I he presece of forave rgh cesorg wh - 3 he LMVUB perfors eer copared o he oher hree esaors The MSE has a slar paer hs case u wh a reduced arg of he ulplcave facors Cosderg he populao slope he MSE of he secod es weghed esaor s early 6 es as ha of he correspodg LMVUB esaor For he dvdual slopes he MSE of he weghed esaor s early 3 es as ha of he correspodg LMVUB esaor Wh he scearo of forave rgh cesorg wh - 38 he MSE of he populao slope for he secod es weghed esaor s early 68 es as ha of he correspodg LMVUB esaor For he correspodg dvdual slopes he MSE of he weghed esaor s early 48 es as ha of he correspodg LMVUB esaor Coparg he as all he hree scearos s see ha he Eprcal Bayes esaor perfors eer ha LMVUB esaor cases wh ad - 3 I case of - 38 he as of he LMVUB esaor s he u I os of he esaors s see ha over esao of he paraeers akes place Sulao resuls correspodg o he Ufor cesorg dsruo are provded Tale 3 I hs case aga he doace of LMVUB esaor ca e see all u he case for populao slope wh By oservg he MSE correspodg o populao ad dvdual slopes he LMVUB esaor has he u followed y Weghed Eprcal Bayes ad Uweghed esaors ha order Ths red s slar oh he populao slope ad wwwjsrporg

5 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 5 ISSN Tale 3 Coparsos of as ad ea square error: Uweghed Weghed Eprcal Bayes ad LMVUB for Ufor ( ) dsruo Esaor Bas MSE(a) MSE()* 3 Uweghed Weghed Eprcal Bayes LMVUB Uweghed Weghed Eprcal Bayes LMVUB Uweghed Weghed Eprcal Bayes LMVUB as well he dvdual slopes Also hs paer s slar o ha of he oe see usg he Sadard Noral cuulave dsruo I case of No-forave rgh cesorg wh he populao slope he weghed esaor has he leas MSE The oher hree has argally hgher values I case of he dvdual slopes he MSE of Weghed esaor s 59 es as ha of he LMVUB esaor ha has he u MSE I he presece of forave rgh cesorg wh - 3 he LMVUB perfors eer copared o he oher hree esaors Cosderg he populao slope he MSE of he secod es weghed esaor for he populao slope s early 94 es as ha of he correspodg LMVUB esaor For he dvdual slopes he MSE of he secod es weghed esaor s early 666 es as ha of he correspodg LMVUB esaor The scearo of forave rgh cesorg wh - 38 he MSE of he populao slope for he secod es weghed esaor s early 58 es as ha of he correspodg LMVUB esaor For he correspodg dvdual slopes he MSE of he weghed esaor s early 5 es as ha of he correspodg LMVUB esaor Coparg he as all he hree scearos s see ha he Weghed esaor perfors eer he o-forave suao Uweghed esaor perforg eer case of forave rgh cesorg wh - 3 ad he LMVUB perforg eer he case of - 38 I he case of he Ufor cesorg dsruo for os of he esaors over esao of he paraeer values akes place IV DISCUSSION Coparso wh respec o he MSE uder sulao sudy reveals he eer perforace of LMVUB esaor as a alerave o ha of he Uweghed Weghed ad Eprcal Bayes esaors oh he forave ad No-forave Rgh cesorg scearos LMVUB esaor perfors eer esag oh dvdual ad populao slopes Ths advaage of LMVUB ay e he resul of he codoal lear odellg of he oserved slopes wh he cesorg e ha provdes a eer judgee of he rae of chage he respose varale As Lle (988) poed ou he coex of codoal lear odels he a dea of he Eprcal Bayes esaor adaped here s o shrk o owards a coo ea u owards a regresso le where he ea s a lear fuco of he cesorg e The coparso wh respec o he Sadard Noral cuulave ad Ufor dsruos dcaes he effecveess of LMVUB he chagg paer of he dropous I case of he Ufor dsruo he dropous are spread a ore eve aer copared o ha of he Noral dsruo ha has a oooc dropou paer LMVUB s capale of accoodag oh he paers ad has u MSE oh wwwjsrporg

6 Ieraoal Joural of Scefc ad Research Pulcaos Volue 4 Issue Ocoer 4 6 ISSN he scearos Wu ad Baley (989) show ha he codoal expecao of he slope gve he cesorg e s a oooc fuco of he cesorg e uder he lear rado effecs odel wh pro cesorg dsruo Furher heorecal work s ecessary he areas of esg he presece of forave rgh cesorg Ths requres esag he sadard errors for he populao ad dvdual slopes The case whe he varace paraeers are ukow eeds o e addressed separaely Sujecs wh oe oservao eed o e accoodaed ay e usg oher ha slope cosruc o fully ulze he avalale forao REFERENCES [] Efro B ad Morrs C (975) Daa aalyss usg Se s esaor ad s geeralzaos Joural of he Aerca Sascal Assocao [] Fear T (975) A Bayesa approach o growh curves Boerka [3] Hu S L ad Berger J (983) Eprcal Bayes esao of raes logudal sudes Joural of he Aerca Sascal Assocao [4] Kleau D G (973) A geeralzao of he growh curve odel whch allows ssg daa Joural of Mulvarae Aalyss [5] Lard N M ad Ware J H (98) Rado-effecs odels for logudal daa Boercs [6] Morrs C N (983) Paraerc eprcal Bayes ferece: Theory ad applcaos Joural of he Aerca Sascal Assocao [7] Orchard T ad Woodury M A (97) A ssg forao prcple: Theory ad applcaos Proceedgs of he 6h Berkeley Syposu o Mehods Sascs ad Proaly [8] Searle S R (97) Lear Models Wley New York [9] Wu M C ad Baley K (988) Aalyzg chages he presece of forave rgh cesorg caused y Deah ad whdrawal Sascs Medce [] Wu M C ad Baley K (989) Esao ad coparso of chages he presece of forave rgh Cesorg: Codoal lear odel Boercs [] Wu M C ad Carroll R J (988) Esao ad coparso of chages he presece of forave rgh Cesorg y odelg he cesorg process Boercs AUTHORS Frs Auhor VswaahaN MSc Asssa Professor Depare of Sascs Presdecy College Chea ealvsusas@galco Secod Auhor Dr R Ravaa MSc MPhl PhD Assocae Professor & Head Depare of Sascs Presdecy College Chea eal-ravaaasas@galco wwwjsrporg