Columbia University. Columbia University Biostatistics Technical Report Series. A Note on the Censoring Problem in Empirical Case-Outcome Studies

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1 olmba versy olmba versy Bosascs echcal epor eres Year 2006 aper A Noe o he esorg roblem Emprcal ase-ocome des Mchael O. kelse Brce Lev a W. McKeage We-Ya sa olmba versy, Brce.Lev@olmba.ed hs workg paper s hosed by he Berkeley Elecroc ress bepress ad may o be commercally reprodced who he permsso of he copyrgh holder. hp://bosas.bepress.com/colmbabosa/ar opyrgh c 2006 by he ahors.

2 A Noe o he esorg roblem Emprcal ase-ocome des Mchael O. kelse, Brce Lev, a W. McKeage, ad We-Ya sa Absrac sdes of he legal sysem vesgaors may collec formao abo cases wh a sdy wdow ad comple sascal formao abo her ocomes. Becase here s freqely a log delay bewee he sar me for cases ad her resolo, a sgfca mber of cases may be pedg a he close of he sdy wdow. f here s a correlao bewee he ocome varable ad beg cesored, exclso of cesored cases may bas he aalyss he sese ha he repored ocomes wll be sysemacally dffere from wha wold be repored f all he cesored cases were followed o compleo ad clded he daa. A prme example, whch we wll se o llsrae or approach, s he ladmark sdy of reversals deah pealy cases he sae cors ha was ahored by a eam led by rofessor James. Lebma of olmba Law chool. wo eqvale ways of esmag ocome raes accog for he cesored cases s he sbjec of hs arcle.

3 A NOE ON HE ENONG OBLEM N EMAL AE-OOME DE Mchael O. kelse, * Brce Lev, ** a W. McKeage, *** ad We-Ya sa **** * Member of he New York Bar; Adjc acly, olmba Law chool. Emal: <mofkelse@homal.com ** rofessor ad har, Deparme of Bosascs, Malma chool of blc Healh, olmba versy. Address correspodece o rofessor Brce Lev, Deparme of Bosascs, Malma chool of blc Healh, 722 Wes 68 h ree, oom 626a, New York, NY Emal: <brce.lev@colmba.ed. *** rofessor, Deparme of Bosascs, Malma chool of blc Healh, olmba versy. Emal: <m23@colmba.ed **** rofessor, Deparme of Bosascs, Malma chool of blc Healh, olmba versy. Emal: <w5@colmba.ed We hak James. Lebma ad Yves hree for kdly provdg her daa ad asssace. Hosed by he Berkeley Elecroc ress

4 A NOE ON HE ENONG OBLEM N EMAL AE-OOME DE sdes of he legal sysem vesgaors may collec formao abo cases wh a sdy wdow ad comple sascal formao abo her ocomes. Becase here s freqely a log delay bewee he sar me for cases ad her resolo, a sgfca mber of cases may be pedg a he close of he sdy wdow. We refer o cases ha are compleed wh he sdy perod as observed wh respec o ocome. Ad coformy wh bosascal pracce we refer o hose ha are sll pedg as cesored. f here s a correlao bewee he ocome varable ad beg cesored, exclso of cesored cases may bas he aalyss he sese ha he repored ocomes wll be sysemacally dffere from wha wold be repored f all he cesored cases were followed o compleo ad clded he daa. wo eqvale ways of esmag ocome raes accog for he cesored cases s he sbjec of hs arcle. A prme example, whch we wll se o llsrae or approach, s he ladmark sdy of reversals deah pealy cases he sae cors ha was ahored by a eam led by rofessor James. Lebma of olmba Law chool. he Lebma sdy prpored o clde all capal seeces wh cera excepos haded dow by he sae cors he ed aes from 973 o 995. here were 5,760 sch cases ad he rae of reversals he compleed cases was a asoshg 68%. B may cases were sll pedg whe he sdy wdow closed 995, ad so were cesored, wh he ocomes remag kow. he vesgaors debaed wha o do abo hose cases ad fally decded o smply exclde hem from he aalyss. 2 hs led a scholarly crc o compla ha he 68% reversal fgre cold be based ad was o secre. 3 he oly safe corse, he arged, was o assme ha all he pedg cases wold be affrmed, whch eve he reversal rae wold drop o 40%. he ahors repled ha he assmpo of o reversals pedg cases 2 3 Lebma, J.., e al A Broke ysem: Error aes apal ases, New York: olmba versy chool of Law, avalable a <hp://www2.law.colmba.ed/ srcoal_servces/lebma/. Lebma, J.., e al A Broke ysem ar : Why here s o Mch Error apal ases, ad Wha a Be Doe Abo. New York: olmba versy chool of Law, avalable a <hp://www2.law.colmba.ed/ brokesysem2/. he mpes for hs arcle came from a colm o he Lebma sdy ad s afermah kelse, M.O. ad Lev, B he machery of deah. hace 8: Gelma, A., e al A broke sysem: he persse paers of reversals of deah seeces he ed aes. Joral of Emprcal Legal des : Hoffma, J.L Volece ad he rh. daa Law Joral 76: hp://bosas.bepress.com/colmbabosa/ar

5 was oladsh, b proposed o mehod for dealg wh he cesored cases, oher ha exclso. he ahors also rejeced aoher way of cog compleers-oly cases rged by oppoes of he deah pealy ha wold have led o a 88% rae of reversal, o w, cog oly cases ha were reversed a he frs wo sages of he revew process ad were eher affrmed or reversed a he fal sage of he process. Aoher example s a major sdy of mae lawss by rofessor Margo chlager, he a Harvard Law chool, whch colleced daa o he effec of he rso Lgao eform Ac LA o prsoer lgao. 4 aempg o measre he effec o lgao ocomes afer he LA, whch was passed 996, rofessor chlager ecoered he problem ha, whe she dd her research, ocome daa were oly avalable hrogh fscal 200. hs mea ha for each year of case flgs from 998 o, some sgfca poro of he mae docke remas o be resolved. Ad, becase dsmssals parclar occr qckly, he resolved cases are more lkely ha hose wh recorded ocomes o be plaffs vcores or selemes. 5 he addressed he problem by leavg o lae-resolved cases from he earler years order o mach smlar cases avodable omssos from laer years. 6 Her solo has he mer of rasparecy ad smplcy, b dscards daa ad also assmes, as she pos o, ha he relave me o dsposo for plaffs sccessfl cases vs. ohers has o bee chaged by he sae or oher rece facors. he aalyss of cesored daa has a rch lerare he felds of bosascs, dsral lfeesg, ad ecoomercs, where he mehods of srvval aalyss, compeg rsks, proporoal moraly, ad correco for seleco bas are germae. bosascal ad lfe-esg segs, eres has hsorcally ceered somewha more o esmag he dsrbo of he mes o eves, e.g., lfemes or draos, or he effecs of compeg rsks, ha has o he ocome varable, whch wold be cosdered axlary srvval sdes, alhogh of prmary eres legal sdes e.g., he reversal or affrmace of he deah seece he Lebma sdy. 7 4 chlager, M mae Lgao. Harvard Law evew 6: d. a d. 7 he ocome varable oher segs wold be called he compeg rsk ype, case of deah, or mark varable. ee, e.g., Hag, Y. ad Los,.A Noparamerc esmao of he jo dsrbo of srvval me ad mark varables. Bomerka 85: Glber,., McKeage,.W., ad, Y ess for comparg mark-specfc hazards ad cmlave cdece fcos. Lfeme Daa Aalyss 0: Hosed by he Berkeley Elecroc ress

6 he ecoomercs lerare, James J. Heckma has sded a geeral class of models ha allows for varos ypes of cesorg, rcao, ad based sample seleco mechasms. 8 he ecoomerc approach, however, ypcally adops assmpos ha are approprae or a leas sppored wh regard o daa legal sdes. or example, he ecoomerc approach, as appled o he Lebma daa, wold reqre he assmpo ha case draos have a ormal dsrbo he bell-shaped crve, whe he dsrbo of draos s decdedly o-ormal; he reaso for a case beg of drao log eogh o be cesored s kow ad ca be modeled mahemacally, whereas he reasos are kow ad cao be modeled; ad he bas reslg from cesorg ca be modeled mahemacally, whe ha cao be doe who makg arbrary ad sppored assmpos. ally, all of he ecoomerc mehods are dagly complex ad wold be a black box o a legal adece. We am o crcmve hose dffcles by offerg a mehod ha ca be grasped vely wh js a lle explaao, s easy o calclae, prodces cosse resls.e., s esseally based large samples, ad reqres a mmm of assmpos. Noparamerc srvval aalyss mees hese crera ad herefore has appeal he legal area, where we wllgly sacrfce a modcm of sascal effcecy exchage for greaer valdy. he mehod of esmao we dscss here s oparamerc becase makes o assmpo abo he mahemacal form of he dsrbo of case draos or ocomes as a fco of draos. A echcal descrpo of or esmaor s gve he Appedx o hs paper. Here we gve a formal descrpo o make he ve appeal of or mehod accessble o he omahemacal reader. here are wo geeral paradgms bosascs o deal wh possble bas creaed by he cesorg of cases daa. he frs s o se he observed cases o mpe vales for he cesored cases. he mehod we develop here o se ha approach we call he self-cosse esmaor, for reasos ha wll appear. he oher approach s o wegh he observed cases so ha he esmae for he ocome of eres s based or oly slghly based. We call hs he verse probably weghg W esmaor, aga for reasos ha wll appear. We descrbe here or self-cosse esmaor ad he gve a bref descrpo of he W esmaor. rs o ha he wo mehods are eqvale for or problem. 8 ee, e.g., Heckma, J.J ample seleco bas as a specfcao error. Ecoomerka 47:53-6. or a dscsso of seleco bas he epdemologc lerare, see, e.g., Herá, Herádez-Díaz, ad obs A srcral approach o seleco bas. Epdemology 5: hp://bosas.bepress.com/colmbabosa/ar

7 o see how he self-cosse esmaor works, mage ha we array all he cesored cases by seecg dae he sar dae, begg wh he earles case. f all we kow s he sar year, he cesored cases wold be groped by year he dscree-me case. ppose ha he frs grop of cesored cases has a sar dae of 980 ad a 5 year followp o he ed of he sdy he followp me, whch was 995. hs he me from sar dae o decso drao s more ha 5 years, he exac fgre beg kow becase of cesorg. o esmae he probably of reversal afer 5 years for cases he grop, we look a he reversal rae for cases wh sar daes pror o 980 ad draos of more ha 5 years. All of hese cases are observed becase 980 s he earles sar dae for a cesored case. Alhogh we are forced o look o cases wh sar-ps pror o 980, we may do so who rodcg bas becase we assme boh he probably of reversal ad he drao of he case are sascally depede of sar dae. 9 ppose here are 0 sch cases ad 5 of hem were reversed. We wold he esmae he probably of reversal for cesored cases wh more ha 5 years drao as f here were 20 sch cesored cases 980, he expeced mber of reversals he grop wold corbe wold be 0.50 x effec, hs approach assmes ha, wh respec o ocome, he cesored cases were seleced a radom from all cases wh draos of more ha 5 years. 0 We ow se hs resl o proceed o he secod oldes grop of cesored cases. Le s say ha here are 2 sch cases wh a sar dae of 985, whch gves hem a followp me of e years. Aga we look a he reversal raes for all cases wh sar daes pror o 985 ad draos of more ha e years. ppose here are 40 sch cases. Of he cases hs grop, 20 are observed cldg he 0 cosdered above ad 20 are he earler grop of cesored cases. ppose ha here are 3 reversals amog he 20 observed cases. he he reversal rae for he secod cesored grop, sg he mped rae for he frs grop, s 3 0/ he expeced mber of reversals hs grop s 2 x We proceed hs fasho, workg forward sar me whch s backward oward zero he followp me of he cesored cases o calclae esmaed reversal probables ad expeced mbers of reversals for each grop of cesored cases, 9 We dscss how o proceed whe hs assmpo does o hold a p. XX fra. 0 f he seecg dae for each case were avalable, separae reversal probables ad he expeced mber of reversals for a sgle case, he fraco of he case correspodg o he probably of reversal cold be comped for each. 5 Hosed by he Berkeley Elecroc ress

8 each calclao sg he resls of he prevos oe. We ca he esmae he probably of reversal for all cases.e. cases wh greaer ha zero drao as he mber of reversals for he observed cases pls he mber of mped reversals for he cesored cases, dvded by he oal mber of cases, observed ad cesored. hs s a esmae of he reversal rae ha wold be observed f all he cesored cases had bee followed o compleo ad clded he daa wh he observed cases. We refer o hs as a self-cosse esmaor becase he mehod sed o mpe reversals for he cesored cases a each sep ses he resl of he same mehod a prevos seps. he resl has he desrable propery of beg a maxmm lkelhood esmae of he overall probably of reversal, whch meas ha hese esmaes maxmze he lkelhood of he observed daa o drao of cases ad her ocomes. ch esmaes are also desrable becase, amog all oparamerc procedres.e., procedres ha make o assmpo abo he dsrbo of draos or ocomes as a fco of draos, hey make he mos effce se of he formao coaed he daa. he oher mehod for dealg wh cesored daa s, as we have sad, o wegh he ocomes of he observed cases sch a way as o redce or elmae he bas esmaes based o hem. geeral, whe some observaos a daa se are mssg, he wegh o assg a observed case s he recprocal of probably of s beg observed,.e., he probably wold o be mssg. he logc here s hs: sppose a gve case ocome has a 20% probably of beg observed. hs for hs case, ad every case lke hs oe ha s observed, here wll be 4 oher cases o average ha are mssg. herefore he observed case shold sad for self ad hese 4 ohers, so shold be weghed by a facor of /0.25. Wha are he weghs for or problem? A case wh drao s observed f ad oly f s cesorg me he me from s sar dae o he close of he sdy wdow s greaer ha. f we regard he cesorg me as a radom varable, he he probably ha a case wh drao wold o be cesored,.e. wold be observed, s he probably ha s cesorg me wold be greaer ha. f hs probably s small, mos cases wh drao wold be cesored. hs, he relavely few observed cases wh draos ms sad for he larger mber of cesored cases ad shold be weghed more heavly he esmaes. hs s accomplshed by weghg he hs s a mahemacal arfce sce s he sar dae of he case, o he edg dae of he sdy ha ca be cosdered radom. 6 hp://bosas.bepress.com/colmbabosa/ar

9 observed case wh drao by he verse recprocal of he probably ha a cesorg me wold exceed. 2 oversely, f he probably ha a case s cesorg me wold be greaer ha s large, mos cases wh drao wold be observed; ad sce he observed case wold sad for fewer cesored cases wold receve less wegh he daa. osder he mercal example dscssed above, ad, for smplcy, sppose ha hese were he oly daa avalable. he frs sep s o oba he Kapla-Meer esmae of he probably ha a cesorg me exceeds 0 years. Becase here were 52 cases a rsk of beg cesored a he sar, of whch 2 were cesored a 0 years ad 40 were o, he sample proporo of cases o cesored s 40/52, or 0/3. he recprocal of hs,.3, s he wegh ha wll apply o he e cases wh observed draos bewee 0 ad 5 years. he ex cesorg me occrs a 5 years, a whch me 20 cases were cesored ad 0 o, here beg oly 30 cases a rsk of beg cesored 52 cases ms he 2 cesored a 0 years ms he 0 cases observed pror o 5 years. hs he Kapla-Meer esmae of he probably ha a cesorg me exceeds 5 years s 0/3 x 0/30 0/39. he recprocal of hs, 3.9, s he wegh ha wll apply o he e cases wh observed draos greaer ha 5 years. ally we calclae he weghed average of he observed reversal ocomes : {.3 x x 5} / {.3 x x 0} 29.9/ he W esmae hs yelds a 57.5% reversal rae. he self-cosse esmae hs llsrao s exacly he same: he meraor we have 6.9 mped reversals for he 2 cases cesored a 0 years; pls 8 reversals observed wh draos bewee 0 ad 5 years; pls 0.0 mped reversals for he 20 cases cesored a 5 years; pls 5 reversals observed wh draos greaer ha 5 years, for a oal of 29.9 observed or mped reversals. hs dvded by 52 cases oal yelds he esmae based solely o he observed cases hs example s 85/000.65, so gorg he cesored cases makes a dfferece. 2 Esmag he probably ha a cesorg me wold exceed s a sadard problem srvval aalyss. he Kapla-Meer esmaor s commoly sed for hs prpose. Alhogh he sal Kapla-Meer applcao s o esmae he srvval fco of draos whle accog for cesored cases, for he prpose of fdg he weghs he W mehod, he roles of draos ad cesored followp mes are reversed. hs he Kapla-Meer esmae of he probably ha a cesorg me wold exceed s he prodc of he sample proporos of cases o cesored amog cases a rsk of beg cesored a each elapsed me a or pror o a whch a case s cesored. ee, e.g., Mchael O. kelse & Brce d Lev 200, p.38. ascs for Lawyers, 2 Ed. New York: prger-verlag. 7 Hosed by he Berkeley Elecroc ress

10 A geeral formla for applyg he W mehod s gve he Appedx. As we oed earler s mahemacally eqvale o he self-cosse esmaor. ce he wo mehods are eqvale we wll descrbe or applcaos below erms of he somewha more ve oo of mpao of mssg ocomes based o he self-cossecy prcple. he self-cosse mehod of calclao does have oe addoal praccal advaage provdes ermedae esmaes of reversal probables for cases wh draos greaer ha ay gve legh, whch ca be ploed o reveal he effec of drao o he probably of reversal. We appled he self-cosse esmaor o smlaed daa ad o daa from he Lebma sdy. o smlae he daa we assmed ha sar daes were formly dsrbed over he erval of he sdy, ha he drao of cases was eher expoeally or formly dsrbed, ha he probably of reversal decled eher lear or o-lear fasho wh followp me, ad ha he degree of cesorg was eher moderae or heavy. he resls showed ha he adjsed or complee-case esmaor whch merely delees cesored cases was clearly based, from bewee 4 ad 6 perceage pos depedg o he degree of cesorg. Whe reversal probables were comped sg he self-cosse esmaor, bas became eglgble. Deals are gve he Appedx. was more challegg o apply he esmaor o he Lebma sdy daa. he ahors were helpfl spplyg s wh daa for 5,356 cases, whch was 404 cases shy of he pblshed mber of 5,760. hey cold o expla hs shorfall, whch mgh be de o he fac of cesorg. a mber of cases, codg coveos were ambgos o wheher a case was pedg or had ermaed wh a affrmace ad for sll oher cases he codg was formave ad arbrary decsos o ocome had o be made. he boom le was hs: for he daa we had, he compleecase esmae of he reversal rae was 67.8% whch s qe close o he pblshed esmae of 68% ad he self-cosse esmae was 62.2%. here were 2,27 cesored cases 40% of all cases he aalyss. hs s heavy cesorg ha shold o be gored. he ermedae esmaes reveal a edecy for cases wh loger draos o have a lower probably of reversal: he reversal probably reached a mmm of abo 49% for cases wh draos more ha 7 years. ee gre. Evdely here was some bas from cesorg, b o so mch as o affec he basc fdgs of he sdy. 8 hp://bosas.bepress.com/colmbabosa/ar

11 hese dffereces are o de o radom varao. he complee-case esmae has a sadard error of 0.8%, ad he sadard error of he self-cosse esmae of.% s slghly larger, reflecg he ceray of he mped daa for cesored cases. A 95% cofdece erval for he re reversal rae based o he self-cosse esmaor s 60.%, 64.4%, whch excldes he vale of 68% gve by gorg cesored cases. he Appedx descrbes how o oba he sadard error of he self-cosse esmae. gre Esmaed probably of reversal for cases wh draos greaer ha years verss for he Lebma daa eversal Drao 0.65: : : O O : O 0.60: : : O : : 0.55: : O : : O O : 0.50: : O O O O O : : : 0.45: : : : : : : : : : : : : years Now le s look a some of or assmpos. Or ocome varable of reversal s bary akg oly wo vales b or mehod s o lmed o bary varables. ca be sed o esmae he average of ay coos ocome as well, e.g., damages awarded a case. Nex, he reader may have oced ha we assmed ha here were observed cases ha had earler sar mes ad loger draos ha he followp me of he earles cesored case. hs does seem reasoable, b may occr ha here are o sch cases, so ha we wold eed aoher way o sar he calclao. here are varos ways of dealg wh hs problem, b large daa ses he effec of he sarg esmae wll wear o as he daa accmlae wh laer 9 Hosed by he Berkeley Elecroc ress

12 grops of cases so ha by he ed of he calclao he sarg esmae makes very lle dfferece. he mporace of he sarg esmae ca be esed by compg he reversal probably for all cases by sg 0 ad as alerave sarg pos. hs was o reqred he Lebma sdy as he cesored cases wh he earles sar me bega 985 as bes as ca be deermed he daa, whereas he loges observed drao was 9 years, so he calclao cold beg wh a esmaed reversal rae of 52.3% based o 28 cases wh draos more ha 0 years. f here are o observed cases wh draos a leas as log as he followp me for he earles cesored cases, we sgges he followg: ake he reversal rae for observed cases wh he maxmm drao regardless of sar dae ad se ha rae for all grops of cesored cases ha have loger followp mes. hs bases he esmae for he cesored cases o he observed daa closes o hem drao eares eghbor mpao 3. he for all grops of cesored cases wh shorer followp mes proceed as we have descrbed above. shold be oed ha for he desg of sdes wh fxed observao wdows sch as we are cosderg, he wdow shold be se sffcely wde o allow observao of complee draos for mos of he early cases. learly, f he wdow s so arrow ha a o-eglgble fraco of eve he earles cases are cesored, may be mpossble o avod bas he overall esmaes. f oly a arrow wdow may be sed, he resls wll be based for he overall reversal rae oly f he reversal probably s cosa for all cases wh draos greaer ha he wdh of he wdow. O he oher had, a very wde wdow may rodce perod or cohor effecs, whch we dscss below. vesgaors shold be aware of sch offseg cosderaos desgg her sdes. he oher mpora assmpo s me homogeey,.e., ha he drao of he case ad he probably of reversal are sascally depede of sar dae. Whe hs s qesoable here are dffere ways o proceed depedg o he op of he sdy. f he op s a oe-mber smmary of a average effec across he wdow he wold seem preferable o compe he selfcosse esmaor he same way eve he face of some deparres from me homogeey. 3 hs s a form of ho-deck mpao, a recogzed echqe sed by he.. ess Brea. ee, e.g., Lle,.J.A. ad b, D.B ascal Aalyss wh Mssg Daa, 2 d Ed., New York: Joh Wley & os. ah v. Evas, , he.. preme or pheld he cess brea's eares-eghbor mpao of demographc characerscs o orespodg hoseholds he deceal cess. 0 hp://bosas.bepress.com/colmbabosa/ar

13 Boh hecomplee-case ad self-cosse esmaors wold be averages, b akg acco of cesored cases by meas of he self-cosse esmaor shold geerally lead o a less based descrpo ha he complee-case esmaor, whch gores he pedg cases. Of corse, whe codos are srogly flx or aalyc assmpos are clearly volaed, he ay sgle-mber smmary may be less sefl as a descrpo ha ca be appled osde he wdow of he daa. Whe here are mpora shfs ocome raes or he sdy s prpose s o exame a perod effec or a cohor effec wh he sdy wdow, vesgaors wll eed o dvde he wdow o sbervals ad apply he self-cosse esmaor separaely o each sberval. A perod effec wold exs f a chage jdcal pracce e.g., a resrco o pos-covco habeas corps revew occrs a a po me ad affecs pedg cases. A cohor effec wold exs f a chage he basele characerscs of cases e.g., he qaly of defese cosel mproves de o greaer fdg affecs he case mx he sample over me wh respec o hese characerscs. A perod or cohor effec may lead o a chage boh he drao of cases ad her ocomes, or o oe who he oher. 4 he boom le s he reversal rae. or esg wheher here s a perod effec o he reversal rae, a raspare mehod s o esmae ad compare he reversal raes for cases wh sar mes each of wo perods of eqal legh. he earler perod s chose o ed a some seleced me before or a he chage po ad he laer perod s chose o sar a or afer he chage po. or cohor effecs ha resrco s o ecessary. eher case, draos from he earler perod ms be admsravely cesored a he maxmm drao observable from cases he laer perod. ha s, we eoally gore all followp ad ocomes of cases from he earler perod ha exceed he admsrave cesorg po eve hogh hey may be had. each perod we mpe vales o he cases admsravely cesored. hs way we ca lze he formao from pedg cases each perod as well as he compleed cases. We oba a vald comparso of reversal raes from cases sarg he earler perod o hose begg he laer perod, alhogh as oed above, he ocome raes esmaed ad compared by hs mehod are hose resrced o he cases wh he shorer draos observable 4 Whe comparg wo perods for a sascally sgfca dfferece draos, a ox model may be employed wh a me-depede covarae. Whe comparg wo cohors a log-rak es may be sed. ee, e.g., Lawless, J.. 982, pp. 354, 392ff. ascal Models & Mehods for Lfeme Daa. New York: Joh Wley & os. Hosed by he Berkeley Elecroc ress

14 he laer perod. Becase he esmaes wold reflec he effec of mpg ocomes for cesored cases based o he observed cases wh draos o greaer ha he shorer wdow wdh, oe wold o expec he esmaes beg compared o agree wh he ocome raes esmaed from a loger observao wdow for eher perod; sch agreeme wold occr oly f ocome probables dd o deped o drao beyod he wdow wdh. he case of rofessor chlager s rso Lgao eform Ac dy, or proposal o esmae a perod effec wold allow s o clde all of he avalable ad formave daa sbjec o admsrave cesorg he aalyss of he pre-ac perod who deleo of he lae-resolved cases for earler years. A more comprehesve mehod for esmag cohor effecs he ocome rae whch corporaes several fxed or me-depede covaraes awas frher developme. * * * 2 hp://bosas.bepress.com/colmbabosa/ar

15 AENDX O A NOE ON HE ENONG OBLEM N EMAL AE-OOME DE M.O. kelse, B. Lev,.M. McKeage, ad W.-Y. sa or ma example we cosder capal cases ha have a sarg dae,, a dae of decso,, ad a bary ocome, Y ε {0,}, where Y dcaes a reversal of he deah seece. We refer o he wag me o decso, 0, as a drao. Assme we ca oly observe cases wh sarg daes a gve wdow caledar me, say he half-ope erval 0,, ad, frhermore, ha we are able o observe he ocome oly f he dae of decso s also he wdow. By sbracg 0 from all daes caledar me, we ca ake he observao wdow o be 0, for 0 who loss of geeraly. We he refer o he sarg dae as a sarg me ad he decso dae as a decso me. Heceforh we shall assme 0 < for all cases who explcly dcag so he oao. f, we say ha he me o decso ad he drao are cesored.e., we do o observe excep for or kowledge ha s greaer ha ad ha he ocome s cesored as well.e., Y s observed. f, we observe he par of varables,y. We se he followg oao for he avalable daa, wh cases dexed by,,. or ay cesored case, we observe he rple,,y, wh, ad we wre for hese cases. or ay cesored case wh, we observe oly he par,, where s he observed follow-p me oly a poro of he re drao, ad we wre 0 for hese cases. We also se o deoe he followp me for cesored cases wh for compleely observed draos. Or goal s o esmae Y, he overall reversal rae, rrespecve of drao. Gve he cosras o he observaos, may o be obvos how o esmae, or ha s esmable a all. he problem s ha, geeral, ad Y may be assocaed for example, cases ha ake loger o decde may be less lkely o resl a reversal. f so, ad f he observao wdow s so arrow ha excldes some draos, wll be mpossble o esmae basedly who frher assmpos. Heceforh we assme A ha s sffcely large so ha s eglgble. We frher assme a me-homogeey codo, o w, A2 ha he jo dsrbo of he drao ad ocome s sascally depede of he sarg me:,y. of 5 Hosed by he Berkeley Elecroc ress

16 Below we dscss a esmaor whch follows from wo sadard approaches o mssg daa: mpao va a self-cossecy prcple ad verse probably weghg W. We show ha he self-cosse esmaor of he reversal rae s eqal o boh he oparamerc maxmm lkelhood esmaor NMLE ad he W esmaor, ad provde several ways o calclae. We emphasze he self-cossecy approach here for s ease of erpreao ad adapably o deals of or problem. elf-cosse esmaor A key qay of eres wha follows s he rae of reversals amog resdal wag mes, Y. Noe ha 0. f were cosa, a reasoable esmaor of wold be he complee-case esmaor, ~, whch ses daa oly from he cesored cases: ~ Y Y. hs esmaor, however, wold geerally be based f were o cosa, say, for example, mooocally decreasg. ha case, for ay gve sar dae, becase cesored draos wold all be loger ha, he cesored ocomes wold have reversal probables less ha, whle cesored ocomes wold have larger reversal probables. ce hs s re for each, he complee-case esmaor wold overesmae. he lagage of mssg daa, he cesored ocomes are o mssg a radom, ad so a esmaor based oly o complee cases s based. We cosder aoher esmaor of whch largely correcs he bas he complee-case esmaor by addg a esmaed fraco of reversals from he cesored cases. ollowg ermology rodced by Efro 967, he esmaor,, s called self-cosse for reasos ha wll soo be made clear. o movae he esmaor, cosder ha for ay fxed vale of ad ay vale of sch ha <, we ca wre he eve as he o of dsjo eves 2 of 5 hp://bosas.bepress.com/colmbabosa/ar

17 ad, becase he eve ad max,. hs we have,,, Y Y Y Y } { } { E Y Y E 2 { } { } E Y E. hs sggess ha we ca esmae 0 by replacg he kow vales of ad he eqao above by esmaed vales ad back-solvg, as follows. or ay 0 <, defe he se of dces {: < }. 3 Noe ha f ε ad s cesored, he becase. Noe also ha ' for ' <. he we defe he self-cosse esmaor of he fco o be { } { } { } { } Y Y. 4a words, he meraor of he self-cosse esmae adds o he ally of reversals from cesored ocomes wh sar mes pror o ad draos greaer ha he esmaed mber of addoal reversals ha wold be expeced from cesored ocomes wh cesored draos greaer ha. Becase a cesored followp me of he form wh ε sasfes, as oed above, a expresso eqvale o 4 s { } { } { } { } Y Y # : : :. 4b 3 of 5 Hosed by he Berkeley Elecroc ress

18 he esmaor s called self-cosse becase s defo a me ses vales of self o mpe ocomes for cesored cases wh laer followp mes of he form. alclao We solve for vales of he self-cosse esmaor a 0 ad a cesored followp mes by sarg wh he larges cesored followp me ad workg backwards. o avod deermaces we assme for he mome ha he loges observed drao s greaer ha he loges cesored follow-p me. symbols, assmpo A3 s ha * max{ : } max{ : 0} m{ : 0}, where we reserve he oao j for dsc ordered sar mes < 2 < amog cesored observaos oly. A3 mples ha he sar me correspodg o *, say *, s less ha becase * * <, whch mples ha he dex correspodg o * s coaed, so ha s o-empy, ad ha a leas oe sasfes for. However, here are o vales of wh cesored followp mes, becase f here were, he for sch we wold have <, coradcg he mmaly of amog cesored cases. hs Y Y. 5 By dco, f we have vales of j for all dsc cesored followp mes j,, k, he for we reqre for cesored followp mes wh ε k so ha k < k k, b he s amog,, k, so s avalable, havg bee prevosly calclaed. We coe hs way o calclae for each of he dsc cesored follow-p mes. ally, og 0 {all,,}, we compe 4 of 5 hp://bosas.bepress.com/colmbabosa/ar

19 0 { Y } { } { Y }. 6 Noe ha f were assmed o be cosa, say 0, he we wold se all. he eqao 6 becomes 0 for 0 { Y 0 } { } ad solvg for 0 yelds he complee-case esmaor ~. We ca ow relax assmpo A3. ppose ha he loges observed drao * sasfes k < * k for some k ; assmpo A3 s he case k. Now here are o vales of wh ad for k. follows from 4 ha he vales of for,, k wold be all eqal b deermae. o resolve he deermacy, we se he se * {: ad * } of cases possbly oly oe wh loges observed draos o esmae vales of for * wh he complee-case esmaor a me *,.e., we se k * Y L 7 * 5 of 5 Hosed by he Berkeley Elecroc ress

20 ad coe back-solvg for, ec., sg 4a or 4b. Noe ha fe samples here may be a small bas rodced by seg he frs k esmaes eqal o he complee-case esmaor 7. However hs bas vashes large samples as we ex demosrae. k ossecy earragg erms eqao 2, we see ha he self-cosse esmaor process sasfes he sysem of esmag eqaos 0} : { 0 } { } { Y. 8 We clam ha hs s a based sysem of esmag eqaos, becase for ay ε 0,, he codoal expeced vale of he esmag fco o he lef had sde, gve for ε, eqals zero: Y Y E } {, Y Y E } {,, Y Y Y Y by depedece 0 } { Y, aga sg he fac ha s he o of dsjo eves ad. herefore he codoal expeced vale of he esmag fco s also eqal o 0. follows ha s a cosse esmaor of for ay fxed coaed he sppor of as becomes large. 6 of 5 hp://bosas.bepress.com/colmbabosa/ar

21 Noparamerc maxmm lkelhood esmae We also oe ha he self-cosse esmaor s he oparamerc maxmm lkelhood esmae NMLE of. o see hs, defe he sb-srvval fcos, Y ad 0, Y0, for 0, so ha he srvval fco 0. Express erms of he sb-srvval fcos as follows:, Y Y. 0 he NMLE s of he sb-srvval fcos sasfy he par of self-cossecy eqaos ad Y 9a 0 0 Y, 9b where max,. Addg 9a ad 9b gves he self-cossecy eqaos for he NMLE of he srvval fco,. 0 B hs s he well-kow self-cossecy eqao for he Kapla-Meer esmaor, so we ca sbse ha esmaor for Ŝ back-solvg 9a for a each cesored followp me, whch allows calclao of 0 a he fal sep. f he larges observed drao s less ha or 7 of 5 Hosed by he Berkeley Elecroc ress

22 eqal o he larges cesored followp me, we proceed as before, seg eqal o he complee-case esmaor a me * for j,,k, where k < * k. / j j o see ha agrees wh he solo from 4, we oe ha f ad, he, so ha. hs all he o-zero erms of are already 0 0 coaed, ha s,. mlarly,. Also, f 0 ad, he, so, whereas f 0 ad Y Y /, he. hs we have { } / so ha we ca represe as follows: { } /. Noe ha he deomaor of may happe ha f s a cesored followp me. We merely represe hs way, we do o calclae hs way s calclaed as he sal Kapla-Meer esmaor see below for alerave expressos. mlarly, Ŝ / Y, so ha /. 2 8 of 5 hp://bosas.bepress.com/colmbabosa/ar

23 Dvdg by 2, he deomaors cacel ad we have { } Y. We ow arge by dco. or cesored followp mes j for j,,k wh k < * k we have se j j j eqal o he compleecase esmaor a *. Assme he ha we have show j j j p o some dex j j k. he for dex j, wh j, we have ha { } { } { } j j j Y Y becase as oed above, j ad s herefore of he form h for some dex h j, wh by he dcve hypohess. he same argme he shows ha Hag ad Los 998, page 789 derve a explc expresso for he NMLE of whe he larges followp me s o cesored. hey show hs esmaor s cosse, asympocally ormally dsrbed, ad ha he boosrap apples. her resl s < j Y j j j, 3 where are he ordered vales of all followp mes, ad where ad Y are he correspodg vales of he cocoma varables,.e., f j for some j, he j ad Y Y j. he above expresso es wh observed eve mes or wh cesorg mes are ordered arbrarly, whle ed eve ad cesorg mes are spl wh he eve occrrg frs. 9 of 5 Hosed by he Berkeley Elecroc ress

24 hs a explc expresso for s gve by 3 dvded by, ad a explc expresso for whe he las observao s cesored s: 0 j j Y j< j. 4 Whe he loges followp me s a cesored observao, he self-cosse verso 6 wh modfcao 7 shold be sed, or, aleravely, expresso 5 wh modfcao 6 below. verse probably weghg Herá, Herádez-Díaz, ad obs 2004 dscss he W approach for addressg varos forms of seleco bas epdemologc sdes. hs approach, sead of mpg vales for he mssg ocomes for cesored cases, each observed ocome s gve a wegh o represe self as well as oher ocomes mssg hrogh cesorg, ad he a weghed average of he ocomes provdes he esmae of. We recogze 4 as js sch a W esmaor, wh weghs gve by w j j j j <. A small smplfcao occrs by og he followg. rs, a alerave expresso for he Kapla-Meer esmaor Ŝ s obaed from 3 by omg Y alogeher, yeldg a form slghly dffere from he famlar eqvale expresso. rom he alerave expresso we see ha w 0. ecod, for ay dex wh, he correspodg wegh w eqals j j d he more famlar eqvale expressos are where here are d j j observed draos ed a ordered dsc eve mes wh cases pedg decso js pror o me, cog cesored followp mes ed a as par of he deomaor. 0 of 5 hp://bosas.bepress.com/colmbabosa/ar

25 w j j< j j j j< j j< j j. j B we recogze he deomaor o he rgh-had sde of he above expresso as he famlar form of he Kapla-Meer esmaor of he srvval fco for he cesored followp mes, evalaed js o he lef of he observed drao a me becase of he src eqaly j< he prodc raher ha he reqred j see foooe o prevos page. Le KM deoe hs esmaor s rgh-coos form, obaed by reag he cesored followp mes as f hey were observed eve mes ad reag he observed followp mes as f hey were cesorg eves. As sal, f a cesorg me exceeds he loges observed eve me hs reversed sese,.e., f he loges acally observed drao exceeds he loges acal cesored followp me, he KM s ake o rema a he level aaed a s las jmp po. hs for observed cases we ca wre he weghs smply as w. he oao dcaes ha he KM evalao of KM akes place js o he lef of s argme,.e., KM lm 0 KM ε ε. hs breaks es bewee observed draos ad cesored followp mes for dscree me daa. ally, omg he cosa /, 4 yelds he smplfed W form of he esmaor cases where he larges observed drao exceeds he larges cesored followp me: : : wy w where w /KM. 5 f he larges observed drao * sasfes k < * k for some k, as he oao leadg p o 7, he we make a slgh modfcao o he weghs order for 5 o agree wh he self-cosse esmaor 6 wh modfcao 7. As before, deoe by * he se of observed cases ed for he larges observed drao, * {: ad * }, ad le r deoe he mber of elemes *. or each dex *, each case wegh w eqals /KM * w, say. he modfcao cosss of replacg each of hese r weghs by of 5 Hosed by he Berkeley Elecroc ress

26 * k k * w w KM. 6 r r he weghed average of he observed case ocomes wh he modfed weghs he agrees wh gve by 6 ad 7. adard errors A sadard error for may be derved by large sample mehods, b he formla s complcaed. 2 Aleravely, we may se he followg smple boosrap echqe. We resample he daa vecors {,, Y, } a radom ad wh replaceme o oba a boosrap sample of daa vecors. he self-cosse esmaor for hs boosrap sample s calclaed ad recorded, ad he he process s repeaed B mes. he boosrap sadard error of he esmaed reversal rae s obaed as he sadard devao of he self-cosse esmaes obaed from he B boosrap samples. he sadard error for he esmaed reversal rae he Lebma sdy was calclaed hs way, resamplg he daa vecors B500 mes. Examples We prese for examples. he frs llsraes he case of moderae cesorg, whle he oher llsrae he case of heavy cesorg. he hrd ad forh examples llsrae how he probably of Y depeds o. each example we se a sample of,000 cases ad,000 replcaos per example o esmae he bas ad roo mea sqared error of boh he complee-case ad self-cosse esmaors. each example we ake 23 years correspodg o he perod 973 hrogh 995, ad we ake o be formly dsrbed o he erval 0,23. he frs wo examples we assme 2 he case ha here are o es.e., here are dsc eve mes, we have a smple expresso for a esmae of he varace of. eferrg o he oao of foooe, 2 Var. { Y Y } 2 of 5 hp://bosas.bepress.com/colmbabosa/ar

27 ha depede of, follows a expoeal dsrbo wh parameer β whch we rcae o he erval 0,23 so as o assre all re draos wh early sar mes have some probably of observao. or he frs example, we ake β0.2 so ha he mea re drao who cesorg wold be E β {expβ } years, ad he proporo of cesored observaos wold be β {expβ } dervaos of formlas omed. hs represes moderae cesorg. We frher assme a lear decle he probably of reversal as a fco of drao: p Y /, whch decles from cera reversal for shor draos o cera affrmao for log draos. hs Y E/ { exp β} β or he secod example, we ake β0.0 sch ha he mea re drao s.06 years, he cesorg fraco s heavy cesorg, ad he re or he hrd ad forh examples, we coe o assme f 0, ad ow ake formly dsrbed o 0, as well. hs we have ha he mea re drao who cesorg wold be E /2 ad heavy cesorg of ½. or he hrd example, we ake p / as above, whch case Y E/ ½, whereas for he forh example we ake p γ /, wh γ ad ¼, whch decles o zero rapdly oly for close o. ha case we have Y γ/ 0.8. he resls of he smlao sdy are coaed able below. We ca see each example ha he complee-case esmae s serosly based pward. he frs example wh moderae cesorg he bas s a b der 5 perceage pos; he remag examples wh heavy cesorg, he bas s abo 6 or 7 perceage pos examples 2 ad 3 ad 9 perceage pos example 4. he self-cosse esmaor has a larger sadard devao ha he complee-case esmaor by abo 24% example, 34% example 2, 27% example 3, ad 58% example 4, b s esseally who bas. he roo mea sqared error ME of he self-cosse esmaor s abo 66% smaller ha he ME of he complee-case esmaor he example, abo 83% smaller example 2, 84% smaller example 3, ad 6% smaller example 4. he ME of he self-cosse esmaor cosss prmarly of s sadard devao compoe, whle he ME of he complee-case esmaor cosss prmarly of s bas compoe. 3 of 5 Hosed by he Berkeley Elecroc ress

28 he small bas see example 4 ca be arbed o he cases he smlao whch he self-cosse esmaor reqred L k for k o be esmaed from he complee-case esmaor a *. hese cases he esmaed vales for,, k based oly o observed draos less ha k wold be pwardly based, becase he re p fco falls precposly oward 0 for close o. able here. efereces Efro, B he wo sample problem wh cesored daa. roc. 5 h Berkeley ymp. Mah. as. Ad robab., Vol. 4, Berkeley: v. of alfora ress. Herá, M.A., Herádez-Díaz,., ad obs, J.M A srcral approach o seleco bas. Epdemology 5: Hag, Y. ad Los,.A Noparamerc esmao of he jo dsrbo of srvval me ad mark varables. Bomerka 85: of 5 hp://bosas.bepress.com/colmbabosa/ar

29 able mlao resls. Example : Moderae cesorg, form, rcaed expoeal, lear Y /. Esmaor Average Bas adard devao oo Mea adard error of he esmaor sqared error of smlao ~ ~ Example 2: Heavy cesorg, form, rcaed expoeal, lear Y /. Esmaor Average Bas adard devao oo Mea adard error of he esmaor sqared error of smlao ~ ~ Example 3: Heavy cesorg, form ad, lear Y /. Esmaor Average Bas adard devao oo Mea adard error of he esmaor sqared error of smlao ~ ~ Example 4: Heavy cesorg, form ad, Y / /4. Esmaor Average Bas adard devao oo Mea adard error of he esmaor sqared error of smlao ~ ~ of 5 Hosed by he Berkeley Elecroc ress