Testing Quasi-independence for Truncation Data

Transcription

1 MPA Munch Personl epec Archve Testng Qus-ndependence for Truncton Dt Tkesh Emur nd eng ng 3 Jul 9 Onlne t MPA Pper No 5858 posted 7 September 4 4:3 UTC

2 Testng Qus-ndependence for Truncton Dt Tkesh Emur Eml: Ntonl Cho-Tung Unverst Hsn-Chu Twn nd eng ng Eml: Abstrct Qus-ndependence s common ssumpton for nlzng truncted dt To verf ths condton we propose clss of weghted log-rnk tpe sttstcs tht ncludes estng tests proposed b Ts 99 nd Mrtn nd Betensk 5 s specl cses To choose n pproprte weght functon tht m led to more power test we derve score test when the dependence structure under the lterntve hpothess s modeled v the odds rto functon proposed b Cheb vest nd Abdous 6 Asmptotc propertes of the proposed tests re estblshed bsed on the functonl delt method whch cn hndle more generl stutons thn results bsed on rnk-sttstcs or U-sttstcs Etenson of the proposed methodolog under two dfferent censorng settngs s lso dscussed Smultons re performed to emne fnte-smple performnces of the proposed method nd ts compettors Two dtsets re nlzed for llustrtve purposes Ke words nd phrses: Condtonl lkelhood; Kendll s tu; Mntel-Henszel test; Power; ght-censorng; Survvl dt; Two-b-two tble The reserch ws fnncll supported b the reserch grnt 95-8-M-9-5 funded b Ntonl Scence Councl of Twn - -

3 INTODUCTION Truncted dt re commonl seen n studes of bomedcne epdemolog stronom nd econometrcs Such dt occur when the vrbles of nterest cn be observed f ther vlues stsf certn crter In ths rtcle we dscuss the stuton tht pr of lfetme vrbles cn be ncluded n the smple onl f The vrble s sd to be left-truncted b nd s rght-truncted b Sometmes eternl censorng lso hppens due to subects wthdrwl or the end-of-stud effect Here we llow tht s subect to rght-censorng b nother vrble C Hence one observes C nd I C subect to where b mn b nd I s the ndctor functon eft-truncted nd rght-censored dt consst of { n} replctons of Truncton often occurs when subect cn be recruted ccordng to certn smplng crteron [3] For emple n the stud of trnsfuson-relted AIDS dscussed n gkos Brr nd De Gruttol [] nfected people could be ncluded n the smple onl f the developed AIDS wthn the stud perod Accordngl the ncubton tme ws subect to rght-truncton b the lpse tme mesured from nfecton to the recrutment tme In ths desgn subect wth the ncubton tme eceedng the lpse tme would never be observed Another emple s the survvl nlss for resdents n the Chnnng House retrement communt n Plo Alto Clforn [6 7 8] Ths smple cn not represent the generl populton snce onl those who hd lved long enough to enter the retrement center could be observed Hence the lfetme ws left-truncted b the entr ge Notce tht truncted subect wth s completel mssng nd even ts estence s unknown An sttstcl nlss for dt subect to truncton requres mkng some ssumpton bout the ssocton between nd Independence between nd - -

4 s the most common ssumpton [ ] Ths ssumpton hs been reled b Ts [8] to weker condton of qus-ndependence whch cn be formulted s follows: H F S / : c where Pr nd F nd S re rbtrr rght-contnuous dstrbuton nd survvl functons respectvel nd c s the constnt stsfng c df ds The ont functon s defned n the upper wedge nd under H t cn be fctorzed nto the product of two mrgnl functons F nd S Snce the behvor of these functons n the lower wedge s not specfed F nd S m not be equl to the true dstrbuton nd survvl functons of nd respectvel [] The ssumpton of qus-ndependence n s weker thn ndependence Thus reecton of H mples reecton of ndependence between nd but not vce vers Mn nonprmetrc methods for truncton dt re stll vld under H If nd re trul ndependent H holds nd then F Pr Pr nd c Pr S Unlke ndependent censorshp whch cn not be verfed qus-ndependence s testble ssumpton [8] Ts [8] proposed the frst test on H b defnng condtonl verson of Kendll s tu nd then usng ts emprcl estmtor s the test sttstcs Mrtn nd Betensk [] etended Ts s de to more complcted truncton structures n whch the propertes of U-sttstcs re ppled n vrnce estmton nd lrge-smple nlss Chen Ts nd Cho [3] constructed ther test bsed on condtonl verson of Person correlton coeffcent In ths rtcle we propose dfferent methods for testng H Specfcll bsed on seres of tbles sutble for descrbng truncted dt we construct weghted log-rnk tpe tests e lso show tht the tests of Ts [8] nd Mrtn nd Betensk [] - 3 -

5 cn be vewed s our specl cses wth dfferent forms of weght To choose good weght tht leds to more powerful test we propose score test tht utlzes some dstrbutonl propertes of the tbles In prtculr the odds rto functon proposed b Cheb vest nd Abdous [] s dopted to model the dependence structure under the lterntve hpothess The estng testng procedures lso dffer n the w of estmtng the vrnce of the correspondng test sttstc Here we dopt the functonl delt method whch cn hndle fleble weght functons nd hence s more powerful tool thn the technques bsed on rnk-sttstcs or U-sttstcs The pper s orgnzed s follows In Secton we propose the mn methodolog b temporrl gnorng censorng In Secton 3 we derve the score test nd suggest model selecton method rge smple propertes re emned n Secton 4 Modfctons of ll the results to ccount for the presence of rght-censorng re presented n Secton 5 Secton 6 contns numercl nlss ncludng dt nlss nd smulton studes Concludng remrks re gven n Secton 7 THE POPOSED METHOD ITHOUT CENSOING To llustrte the mn de we temporrl gnore rght-censorng b lettng C Observed dt cn be epressed s { : n} subect to Constructng the Test Sttstcs bsed on Two-b-two Tbles Adpt to the nture of truncton we cn construct the followng tble t n observed flure pont for N d d N d N d Tble : tble for truncted dt wthout censorng The cell counts nd mrgnl counts n Tble re defned s - 4 -

6 N d d I N d I N d I I Under H nd condtonl on the mrgnl counts the cell count N d follows d the hper-geometrc dstrbuton wth N d N d E N d d N N To test qus-ndependence we propose the followng weghted log-rnk tpe sttstcs: N d N d 3 N d d where s weght functon Motvted b the G clss dscussed n Hrrngton nd Flemng [ 3] we consder sub-clss of wth prtculr form of whch cn be wrtten s N d N d ˆ N d d 4 where ˆ / n nd s pre-specfed constnt The sttstcs s nonprmetrc n the sense tht no dstrbutonl ssumpton bout the ont dstrbuton of s mde However such nformton would be helpful for choosng n pproprte weght or the vlue of n 4 whch m led to more powerful test In Secton 3 we derve score test tht utlzes the nformton of the underlng ssocton structure provded b the tbles eltonshp wth Estng Tests The tests proposed b Ts [8] nd Mrtn nd Betensk [] re both relted to condtonl verson of Kendll s tu defned s E{sgn A } where sgn s defned to be - or f or respectvel - 5 -

7 A I } { nd Note tht when the event A occurs s locted n the observble regon { : } nd hence s well-defned under the truncton settng Under qus-ndependence Ts [8] showed tht An emprcl estmtor of nd Mrtn nd Betensk [] both consdered the sttstcs cn be used for testng H Specfcll Ts [8] K I{ A }sgn{ } 5 but proposed dfferent ws of clcultng the vrnce of K For emple n bsence of tes b wrtng K s the sum of condtonll ndependent rnk vrbles Ts [8] ws ble to utlze rnk-bsed results to derve the condtonl vrnce of K eplctl Mrtn nd Betensk [] recognze the fct tht K s U-sttstc nd then derve more generl vrnce formul whch cn hndle ted dt The sttstc K hs been etended to ccount for censorng [8] or even more complcted dt structures [] Now we compre the proposed test sttstcs n 3 wth K n 5 To smplf the nlss ssume tht the dt hve no tes so tht the vlues of re ll dstnct In such cse N d N d for ll n n tbles of nterest nd the epected vlue n becomes / It cn be shown tht I{ A} sgn{ } 6 The proof of the bove equton s gven n Append C under more generl settng tht ncludes rght-censorng B settng / n we get N d N N d d n d K 7 n

8 Equton 6 mples tht s lso U-sttstc f / s determnstc functon However f we prefer fleble weght functon tht m led to more powerful test the technque of U-sttstcs s no longer pplcble for vrnce estmton nd lrge smple nlss Accordngl n Secton 4 we wll use the functonl delt method to estblsh smptotc propertes of 3 CONDITIONA SCOE TEST 3 Constructon of Condtonl kelhood As mentoned bove the weght functon n 3 ffects the power of whch depends on the dependence structure under the lterntve hpothess The Clton s model [4] chrcterzed b the constnt odds rto functon [3 4] s perhps the most populr choce for descrbng bvrte lfetme vrbles The clss of Archmeden copul AC models whch nclude the Clton s model nd the bvrte frlt fml [4] s specl cses provde sstemtc frmework to descrbe the dependence for multvrte rndom vrbles [] These concepts re modfed b Cheb vest nd Abdous [] n nlss of truncted dt Here we lso dopt ther proposl e ssume tht Pr s dfferentble nd hence the dt hve no tes Cheb vest nd Abdous [] modfed the odds rto functon sutble for truncted dt s follows: / / / Under qus-ndependence for ll It should be noted tht the cse of mples postve ssocton whle mples negtve ssocton between the two truncted vrbles The nformton of s contned n the summr sttstcs of Tble Gven the mrgnl counts N d follows Bernoull dstrbuton wth d Pr N d d N N r r

9 Ths dstrbutonl result cn be further utlzed to construct score test under lterntve hpotheses Here we ssume tht cn be formulted s follows: The odds rto functon cn be prmeterzed s { } where s prmeter nd s n unspecfed nusnce functon For ech fed s contnuousl dfferentble functon of nd lm where s the prmeter vlue under qus-ndependence Suppose tht cn be estmted b ˆ Under workng ssumpton of ndependence mong dfferent tbles of nd gnorng the dstrbutons of the mrgnl counts we cn construct the followng condtonl lkelhood functon: N d d N d d { ˆ } 8 { ˆ } { ˆ } The correspondng score functon becomes log {ˆ } {ˆ } N d N d N d d {ˆ } {ˆ } 9 where v v / Note tht equton 8 ws motvted b Clton [4] nd Okes [3] who consdered the Clton model for bvrte censored dt B settng the score test sttstc cn be obtned bsed on equton 9 Specfcll snce lm { } wth the weght functon the proposed score sttstcs hs the form of lm {ˆ } Equton provdes cler gudelne for choosng the weght functon for when the ssumptons on stted n nd re stsfed The level of power mprovement depends on whether s correctl specfed nd how ccurte cn be estmted e wll dscuss these ssues v specfc emples n Secton 3-8 -

10 3 Sem-survvl Archmeden Copul Models For dependent truncton dt Cheb vest nd Abdous [] proposed sem-survvl Archmeden copul AC models of the form Pr [{ { F } { S }] / c where c s normlzng constnt stsfng d AC models re chrcterzed b the genertng functon : [ ] [ ] where t t / t nd t t / t Furthermore the showed tht under the odds rto functon cn be wrtten s { c } where Hence AC models stsf ssumpton such tht c The cse of qus-ndependence corresponds to t log t n After pproprte prmeterzton for we m ssume tht t log for so tht ssumpton holds t An estmtor of c m be obtned usng the proposl b Cheb vest nd Abdous [] Alterntvel consderng tht c n s evluted t t suffces to estmte c c the vlue under H He nd ng [4] proposed to estmte Pr under ndependence between nd Although n the present cse c s not necessr equvlent to Pr ther de cn be modfed Specfcll one cn set c F / n under the ssumpton of S where mn B pplng the nonprmetrc estmtor Fˆ of ng Jewell nd Ts [3] we hve c Fˆ / ˆ Note tht the sme ˆ estmtor ĉ cn lso be obtned s soluton of equton of Cheb vest nd - 9 -

11 Abdous [] b settng nd t Now we derve the suggested form of weght n for selected AC models Emple Clton copul Clton s model [4] hs the genertng functon t t / for nd t log when It follows tht nd hence t lm { } whch corresponds to specl cse of n 4 Notce tht no nusnce prmeter s nvolved n the weght functon Emple Frnk copul For Frnk s model [] the genertng functon hs the form t t log{ / } for nd t log for log Snce log /{ e } nd we hve lm log h lm log h h h e Thus the suggested weght functon s gven b lm c { } t If we estmte b ˆ the resultng score test becomes n 7 whch s equvlent to K consdered b Ts [8] nd Mrtn nd Betensk [] Ths mples tht these two tests re sutble for Frnk s lterntve Emple 3 Gumbel copul For the Gumbel model the genertng functon equls { log for t t} nd t log for Under the Gumbel model onl permt t - -

12 negtve ssocton Snce / log t follows tht lm { } / log{ c } B pluggng n the estmtors of nd c n the suggested weght we denote the correspondng test s nvlog whch however s not member of n 4 In prctce there m be severl model choces under consderton e suggest heurstc pproch b choosng the model tht elds the hghest vlue of ˆ where ˆ mmzes over the correspondng prmeter spce The nfluence of weght on the power of the correspondng test wll be evluted lter v smultons 4 Asmptotc Normlt 4 ASMPTOTIC ANASIS In ths secton we stte the mn theoretcl results e ssume tht the underlng dstrbuton s bsolutel contnuous under the null hpothess n Consder clss of weghted log-rnk tpe sttstcs of the form w N d N d w{ ˆ } N d d where w v s known contnuousl dfferentble functon of v Theorem : Under H the sttstcs n / w converges n dstrbuton to men-zero norml rndom vrble The specl cse n / hs smptotc U / vrnce E{ U } where { I { I }sgn{ } d d d } d sgn{ } 3 - -

13 Sketch of the proofs re gven n Append A nd A nd more complete dscussons cn be found n Emur nd ng [9] 4 Vrnce Estmton Equton 7 shows tht n bsence of tes s equvlent to K Vrnce estmton of K hs been dscussed n Ts [8] nd Mrtn nd Betensk [] Here we propose dfferent pproch Bsed on the formul n 3 we cn estmte b pplng the method of moment nd the plug-n prncple The rguments n Append A eld the followng vrnce formul for : n ˆ n n k kl I{ A I{ A kl k } ˆ } ˆ kl k kl k sgn{ sgn{ k l k k k } n } I l kl kl 4 Ths estmtor ncorportes the vrblt of estmtng the nusnce functon hen censorng s present nltc epressons of become complcted nd not trctble Under generl stutons the ckknfe method provdes convenent tool for vrnce estmton For n rbtrr weght functon the vrnce of cn be estmted b the followng ckknfe estmtor: ˆ n / n Jck where s the sttstcs gnorng the th observton nd / n Emur nd ng [9] provde smulton results whch compre the two vrnce estmtors under sttstcs It s found tht lthough the nltc estmtor sometmes hs better performnce n vrnce estmton b producng smller men squred errors t tends to eld less ccurte tpe-i probblt compred wth the ckknfe estmtor It seems tht the hgher-order terms omtted n the lner epresson of stll pl some - -

14 role n estmtng the vrnce for fnte smples The vldt of the ckknfe estmtor s closel relted to the smoothness of the wth respect to the emprcl process ˆ I / n Ths propert requres strngent smoothness condton on the correspondng sttstcl functonl The followng theorems cn be proved b checkng suffcent condton of contnuous Gteu dfferentblt [5] for consstenc of the ckknfe method Theorem : Under H the smptotc vrnces estmted b the ckknfe method The detled proof s gven n Emur nd ng [9] 43 Asmptotc Effcenc of the Score Test of cn be consstentl The condtonl lkelhood constructed n Secton 3 s not the true lkelhood snce t gnores the dependence mong dfferent tbles nd nvolves etr estmton of the nusnce prmeter Here we nvestgte ts smptotc effcenc Under ssumptons nd of Secton 3 Tlor seres epnson of round leds to the contguous lterntve H n : n / { } n / / { } o n Under the sequence of lterntves t cn be shown tht the sttstcs n / converges n dstrbuton to the norml dstrbuton wth men ˆ lm N d N n n { ˆ } d nd vrnce The smptotc effcenc of cn be studed b comprng the noncentrlt prmeter of the ch-squre test defned s / Stndrd Cuch-Schwrz tpe rgument cn not be ppled to obtn the optml choce - 3 -

15 of due to the complcted vrnce functon tht nvolves the nusnce prmeter estmtes Note tht not onl depends on the lterntve structure but t lso functonll depends on the mrgnl dstrbutons To nvestgte the effcenc of we compute when the ont dstrbuton of follows Clton nd Frnk AC fmles wth selected mrgnl dstrbutons nmel eponentl unform nd ch-squred dstrbuton The results re depcted n Fgure For rnge of [ ] the noncentrlt prmeter s mmzed t under the Clton model nd under the Frnk model for ll the chosen mrgnl dstrbutons These results ndcte tht mong ll members of the test the score tests nd re locll most powerful under the Clton nd Frnk lterntves respectvel 5 MODIFICATION FO IGHT CENSOING In ths secton we modf the proposed tests to dust for rght-censorng whch rses when the process of observton hs to be termnted before the event of nterest occurs Consder stuton tht the lfetme vrble s rght-censored b C In presence of truncton how to formulte the censorng mechnsm deserves some dscussons e present two dfferent ws to nclude the censorng mechnsm Both settngs hve been consdered n the lterture Cse A The censorng vrble C s lso subect to the truncton crter Indvduls stsfng C re ncluded n the smple nd otherwse truncted Cse B Censorng onl ffects the ndvduls who stsf Accordngl t s ssumed tht Pr C Independent censorshp mens tht the censorng event s not relted to the dsese process In presence of truncton how to formulte the ssumpton of ndependent censorng depends on whch censorng mechnsm s dopted Now we dscuss the - 4 -

16 ssumpton for ech settng Cheb vest nd Abdous [] consdered the stuton n Cse A nd then mde the followng ssumpton: Assumpton A: C s ndependent of In Cse B however C nd cnnot be ndependent due to the mthemtcl restrcton C For ths cse defne C C where C refers to the resdul censorng tme A more proper ssumpton s gven b Assumpton B: C s ndependent of gven Note tht n bsence of truncton wth probblt one both cses reduce to the usul ndependent censorshp model In the followng subsectons we dscuss modfcton of the proposed tests under the two censorng mechnsms 5 The Proposed Test Sttstc under Censorng Under both censorng frmeworks observed dt cn be epressed s { : n} where C s rndom replcton of C C nd I C subect to Tble s modfcton of Tble such tht denotes n uncensored flure pont stsfng To smplf the presentton we use the sme nottons for the counts s before but modf ther defntons s follows N d d I N d I N d I nd I N d d N d N d Tble : tble for left truncted nd rght censored dt In Append B we show tht under H the populton odds rto of Tble s stll one under both censorng settngs Accordngl the modfed log-rnk sttstcs hs the sme form gven below - 5 -

17 N d N d 5 N d d Defne the sttstc s N d N d vˆ N d d 6 where s constnt nd v ˆ s n estmtor of Note tht the two censorng cses eld dfferent consstent estmtors of such tht vˆ /{ nsˆ du /{ nsˆ C C } u } under AssumptonA under AssumptonB where Sˆ C s the product-lmt estmtor for Pr C S bsed on dt { n} [9] nd ˆ C s the usul Kpln-Meer estmtor for S C Pr C S bsed on dt { C : n} [3] In bsence of C censorng v ˆ reduces to ˆ for both cses Nottons A nd B wll be used when v ˆ s defned under Assumpton A nd B respectvel Emur nd ng 8 showed tht under Assumpton A cn be wrtten s Hdmrd dfferentble functon of H ˆ c I C c n : / Hˆ n { H ; c c } ˆ cc H{ c c c c } sgn{ } dh? c dh c ˆ where ˆ Hˆ ; s lso Hdmrd dfferentble functon of Ĥ Asmptotc normlt of cn be estblshed b pplng the functonl delt method nd the fct tht n / Hˆ H converges wekl to men-zero Gussn process - 6 -

18 Emur nd ng 8 lso showed tht the smptotc vrnce of cn be consstentl estmted b the Jckknfe estmtor ˆ Jck Etenson of these results under Assumpton B follows essentll the sme rguments b modfng the defnton of H; Therefore the test of qus-ndependence cn be bsed on / ˆ Jck b pplng the smptotc normlt result The defnton of hs lso been modfed to ccount for censorng Usng the fct tht the order of pr s known for certn f the smller one s observed Mrtn nd Betensk [] defne the event B { } { & & & & } whch s condton for the prs beng comprble nd orderble The modfed condtonl Kendll s tu denoted s b hs the sme form s wth A beng replced b B Under qus-ndependence t cn be shown tht under both settngs E[sgn{ } B ] b The proof s essentll qute smlr s n Append B nd hence s omtted In Append C we show tht I{ B} sgn{ } 7 B settng / n reduces to the emprcl estmtor of b n I{ B }sgn{ Kb } n Note tht K b no longer belongs to the clss n 6 when dt re censored For vrnce estmton eplct vrnce formul for K b ws proposed b Ts [8] bsed on propertes of rnk sttstcs Mrtn nd Betensk [] stll ppl propertes of - 7 -

19 U-sttstcs to obtn the smptotc vrnce of K b 5 Condtonl Score Test under Censorng Now we etend the nlss n Secton 3 to the two censorng settngs Etenson under Assumpton A s frst dscussed snce t s more strghtforwrd Under the lterntve hpothess the populton odds rto of Tble s { } nd the rguments n Secton 3 cn be stll ppled bsed on the modfed counts defned n Secton 5 The condtonl score tests s specl cse of 5 wth the weght functon lm {ˆ } Consder the sem-survvl AC models n whch { } cn be wrtten s / nd c To estmte the nusnce prmeter we rewrte t s c c where v Pr / S The nusnce prmeter s estmted b C ˆ cˆ ˆ where ĉ s n estmtor of c Under H the constnt c s estmted b cˆ ˆ F / ˆ where Fˆ s the estmtor of ng Jewell nd Ts [3] bsed on truncted dt { : n} subect to The suggested weght functon under ech AC model s the sme s those presented n Secton 3 ecept tht the method of estmtng nusnce prmeter hs to be modfed It turns out tht nd A n 6 re the condtonl score tests when follows Clton nd Frnk AC models respectvel Dervton of the score test under Assumpton B becomes more complcted snce the populton odds rto of Tble s no longer { } Bsed on B of Append B the odds rto equls / / { { u / u} SC u du u / u} S u du Ths s not equl to { } unless C u for u Development of S C

20 the condtonl score test under Assumpton B wll be left s our future work Nevertheless the choce wth lm {ˆ } n 5 s stll vld test even t m not cheve the sme level of power mprovement If censorng s lght so tht { } s good ppromton of the true rto the resultng test wll stll be good choce 6 Dt Anlss 6 NUMEICA ANASIS e ppl the proposed methods to the forementoned AIDS dt nd Chnnng House dt nd compre our results wth estng nlses The frst dt contns no censored observtons gkos Brr nd De Gruttol [] dvded the AIDS dt nto two ge groups of chldren 37 subects nd dults 58 subects nd ssumed ndependence between the ncubton tme nd the lpse tme The -vlues nd p-vlues of fve tests re reported n Tble 3 Specfcll the proposed log-rnk sttstcs bsed on nd nvlog utlze the ckknfe method for vrnce estmton The tests proposed b Ts [8] nd Mrtn nd Betensk [] hve the form of or K but use ther own vrnce estmtors n the stndrdzton For the dult group ll the results show sgnfcnt devton from qus-ndependence The sgn of the -vlues ndctes postve ssocton between nd Ths mples tht people nfected n erler chroncle tme tended to hve longer length of ncubton Although smlr pttern of ssocton ws lso dscovered n the chldren group 7 t dd not rech 5% level of sttstcl sgnfcnce probbl becuse the smple sze s not lrge enough Nevertheless H s stll reected b the tests of nd Mrtn nd Betensk [] t % sgnfcnce level - 9 -

21 To determne the best weght for w we compre vlues of the ftted lkelhood under the three models nmel the Clton Frnk nd Gumbel fmles In Tble 3 log ˆ denotes the log of condtonl lkelhood when ˆ s the mmzed vlue of over the prmeter spce of the model For both covrte groups the Clton model s the best ftted one mong the compettors nd hence s recommended For the Chnnng House emple qus-ndependence between resdent s lfetme nd hs/her entr ge to the communt s emned under the two censorng mechnsms whch dffer n whether censorng could occur to truncted subect S tests re compred n Tble 4 whch nclude the tests proposed b Ts [8] nd Mrtn nd Betensk [] nd four proposed tests The score tests A nd nvlog use the suggested weghts for the three AC models respectvel wth v ˆ defned under Assumpton A The B test dopts Assumpton B to defne v ˆ the tests re vld All The frst nlss uses the dt provded n Hde [7] whch contns men nd 365 women subects Among them 86 people wthdrew from the communt eldng the censorng proporton 6 Bsed on the frst hlf of Tble 4 the -vlue of ech test ndctes slghtl postve ssocton between nd b = 88 The four tests nmel A B Ts s test nd Mrtn nd Betensk s test rech the % sgnfcnce level In fct the lkelhood nlss fvors the Frnk model under whch the score test s A ecll tht n presence of censorng Ts nd Mrtn nd Betensk s tests use the weght / n whle A nd B dopt the weght /{ nsˆ C } nd du /{ nsˆ u } respectvel Hence the re no longer equvlent C - -

22 The second nlss uses the dt n Hde [6] where onl the 97 men were studed wth 5 subects beng censored Ths subset lso revels postve ssocton between nd b =99 Bsed on the second hlf of Tble 4 the three score tests fl to reect qus-ndependence The vlues of mmzed log-lkelhood stll fvor the Frnk lterntve n whch the score test s A wth the p-vlue 68 In contrst three tests B Ts s test nd Mrtn nd Betensk s test suggest reectng qus-ndependence t 5% level p-vlues: nd 4 respectvel Now we dscuss the results of Chnnng House dt n more detl Frstl the methods of vrnce estmton seem to hve not much effect In fct f we tested the second dtset usng wth / n nd the ckknfe vrnce estmtor the correspondng -vlue becomes -33 p-vlue: 4 whch s ver close to the results bsed on the two competng tests Therefore the test result seems to be mostl ffected b the chosen weght functon Note tht the functon / n ssgns hgher weght to erl flure tme thn /{ nsˆ C } n A e suspect tht the ssocton t erler tme perod s hgher for the subset of men thn t s for the whole smple of 46 subects Notce tht for the mle group the B test reects H p-vlue: 48 whle the A test does not p-vlue: 68 To determne whch censorng ssumpton s more sutble one m further emne whether the censorng event cn lso be truncted or not For Chnnng House dt mong 86 censored subects 7 subects wthdrew from the stud nd the remnng 59 subects survved untl the end of stud The reson of wthdrw mght be due to fnncl nsecurt Assumpton B e Pr C s more plusble f the end-of-stud effect ws the prmr reson of censorng However f the fnncl ssue stll ffected person s decson on the enrollment of the communt Assumpton A m be proper - -

23 choce In ddton the defnton of the trget populton s crucl The resercher mght dopt Assumpton B b ecludng those who were not rch enough to enroll from the trget populton 6 Smulton Studes Fnte-smple performnces of the proposed test nd ther compettors re evluted v smultons ndom prs of were generted from three well-known sem-survvl AC models nmel the Clton Frnk nd Gumbel fmles dscussed n Secton 3 The level of ssocton for n AC model cn be descrbed n terms of pre-truncted Kendll s tu defned s E sgn ] whch s [ ndependent of the mrgnl dstrbutons Snce the mor gol of the smultons s to see the power mprovement n the suggested weght n we dopt Assumpton A for the censorng mechnsm under whch the score tests re derved Accordngl the censorng vrble C ws generted ndependentl from The mrgnls of C follow eponentl dstrbutons wth the hzrd rtes eldng the trgeted levels of c P e 667% 5% nd 333% for the uncensored cse nd of c P e 667% 5% nd 333% for the 5% censored cse P C 5 respectvel For ech settng we provde the vlue of condtonl Kendll s tu or b e consder three proposed tests nmel nd nvlog usng the ckknfe method for vrnce estmton For the Clton Frnk nd Gumbel lterntves the score tests correspond to nd nvlog respectvel The tests proposed b Ts [8] nd Mrtn nd Betensk [] re lso evluted In bsence of censorng these two tests constructed bsed on K re equvlent to ecept tht dfferent vrnce estmtors re used Performnces of the fve tests t n nd - -

24 re studed Tble 5 nd 6 summrze the results bsed on 5 replctons when follow the Clton model Under qus-ndependence the reecton probblt for ll tests re close to the nomnl 5% level nd s epected the power of ech test ncreses s the level of ssocton deprts from qus-ndependence In ll the cses the proposed score test s unforml more powerful thn the other tests The test nd two relted tests proposed b Ts [8] nd Mrtn nd Betensk [] hve smlr nd sometmes unstsfctor performnces Also the power of ech test mproves when the censorng rte decreses The results for the Frnk model under dfferent levels of ssocton re summrzed n Tble 7 n nd Tble 8 n As mentoned erler the score test bsed on nd the tests proposed b Ts [8] nd Mrtn nd Betensk [] use the sme weght functon when dt re not censored Under the Frnk model the three tests hve shown hgher power thn both nd nvlog s epected but cler-cut domnnce mong the three s not found Compred wth the Clton s cse the mgntude of power mprovement reduces lttle bt Ths m be due to the effect of estmtng the nusnce functon of n the suggested weght for the Frnk model Tble 9 contns the results under the Gumbel model wth nd 4 snce the sem-survvl Gumbel model onl permts negtve ssocton In contrst to the Clton nd Frnk models the dscrepnc for the power curves of dfferent tests becomes less cler Nevertheless for the uncensored cse wth n the proposed score test bsed on nvlog stll performs slghtl better thn the competng tests e suspect tht the gn b usng the suggested form of weght /log{ c } m be somewht offset b estmtng two nusnce prmeters c nd - 3 -

25 Interestngl the level of truncted proporton hs cler mpct on the power performnce f the dt follow the Frnk or Gumbel models whle t does not under the Clton model Now we provde some heurstc eplntons Under these two models the odds rto functon s monotone functon of c P or c P It turns out tht the power of ll tests ncreses s c or c gets lrger In contrst under the Clton model nd ths m epln wh the power of the tests s not much ffected b c or c In generl the smulton results confrm tht the suggested weght n cn mprove the power when the lterntve s correctl specfed On the other hnd wrong choce of weght m result n loss of power The results of the smulton studes re consstent wth the effcenc stud n Secton 43 7 CONCUDING EMAKS A relted re of reserch s testng ndependence for bvrte flure tmes nk-bsed procedures were proposed b Cuzck [5 6] nd Dbrowsk [8] Okes [] suggested concordnce test bsed on n estmte of Kendll's tu whch keeps the nformton of rnks nd hs nce epresson s U-sttstc Shh nd ous [6 7] utlzed the covrnce process of mrtngle resduls to constructs test sttstcs Hsu nd Prentce [5] generlzed the de of Mntel-Henszel sttstcs to test ndependence for rght censored dt Smlr de hs been etended to bvrte current sttus dt b Dng nd ng [7] bsed on nother formulton of tbles Ths rtcle consders left-truncted dt n presence of rght-censorng A modfed verson of Kendll s tu ws proposed b Ts [8] nd then used s the bss for testng qus-ndependence b both Ts [8] nd Mrtn nd Betensk [] Alterntvel we ppl the de of log-rnk tpe sttstcs bsed on tbles desgned for descrbng truncton dt B permttng fleble weght functon the proposed sttstcs form generl clss of tests A nce equvlence propert between the log-rnk tpe sttstcs nd - 4 -

26 the Kendll s tu sttstcs hs been estblshed Ths reltonshp llows us to compre dfferent tpes of tests under unfed frmework nd t turns out tht the weght functon pls crucl role The dstrbutonl propertes of the tbles shed some lght on the underlng lkelhood structure Accordngl motvted b the ppers of Clton [4] nd Okes [3] we derve score test when the dependence structure under the lterntve hpothess cn be modeled v the odds rto functon Compred wth the condtonl Kendll s tu mesures s better ssocton mesure snce t s ndependent of the mrgnl dstrbutons nd cn be ccurtel estmted n presence of censorng The proposed score test hs the log-rnk tpe epresson wth the weght functon chosen to ft the lterntve hpothess nd hence hs good power when the true model s ssumed The functonl delt method s ppled to derve lrge-smple propertes for the proposed test sttstcs wth fleble weght functons whch m contn nusnce prmeters Consstenc of the ckknfe vrnce estmtor s lso ustfed To fnd the score test heurstc model selecton procedure s proposed b comprng the vlues of the condtonl lkelhood functons under dfferent model choces Alterntvel Beudon nd khl-cheb [] proposed dfferent method for model selecton The lso suggested fttng the AIDS dt b Clton s model nd Chnnng House dt b the Frnk model In nlss of the Chnnng House dt we dscuss the ssue of choosng sutble ssumpton on censorng In summr one should check whether the reson of censorng cn occur to those wth Ths ssumpton lso depends on how the trget populton s defned For nlzng more complcted truncton nd censorng structures Mrtn nd Betensk [] consdered severl etended versons of Kendll s tu nd utlzed propertes of U-sttstcs n vrnce estmton nd lrge-smple nlss It would be - 5 -

27 nterestng to ppl the de of log-rnk tests to these dt settngs Ths etenson s not trvl snce the formulton of pproprte rsk sets n the constructon of tbles s not strghtforwrd e wll leve ths problem s future reserch topc APPENDI A: ASMPTOTIC ANASIS et D {[ } be the collecton of ll rght-contnuous functons wth left-sde lmt defned on [ whose norm s defned b f sup f for f D{[ } e ssume tht the functon F S c s bsolutel contnuous The emprcl process on the plne s defned s: n ˆ I n / The functonl delt method s ppled bsed on the wek convergence result of n / ˆ to men Gussn process V on D {[ } wth the covrnce structure gven b cov{ V V } for n [ A Proof of Theorem After some lgebrc mnpultons bsed on 6 we cn rewrte t s w n Ths llows us to rewrte the sttstcs s w n n ˆ w{ ˆ } I{ A} sgn{ ˆ w{ ˆ } sgn{ ˆ } } d ˆ d ˆ where the defnton of the functonl : D {[ } s w{ } sgn{ } d d B settng the rgument s Pr nd vewng the - 6 -

28 - 7 - bove ntegrl s n epectton we hve : } sgn{ } { } { w A I E } } {sgn{ } { } { E w A I E B drect clcultons we cn show the Hdmrd dfferentblt of The dfferentl mp of t } {[ D wth drecton } {[ D h s } sgn{ } { } sgn{ } { } sgn{ } { d dh w d d h w d d h w h Applng the functonl delt method [3] we obtn the smptotc epresson ˆ / / / P w o n n n where I It s es to see tht the sequences U for n re d rndom vrbles wth men-zero From the centrl lmt theorem w n / converges to men-zero norml dstrbuton wth the vrnce ] [ E U A Anltc vrnce estmtor for the G clss ecll tht the clss s sub-fml of w For ths clss one cn obtn the eplct formul of U gven n 3 Accordngl t s not dffcult to obtn n nltc estmtor of bsed on 3 s follows: The dervtve mp s gven b

29 - 8 - } sgn{ } sgn{ / d dh d d h h Hence the smptotc vrnce of cn be estmted b ˆ ˆ where ˆ ˆ } sgn{ ˆ ˆ ˆ } sgn{ ˆ ˆ / ˆ ˆ d d I d d I } sgn{ } ˆ { } sgn{ } ˆ { l k kl kl l k l k kl kl kl k k k k k k I A I n n A I n Bsed on the bove epresson one cn estmte the smptotc vrnce n AVr b equton 4 APPENDI B: ODDS ATIO OF TABE Assume tht ll the tme vrbles re contnuous Under H nd Assumpton A ll entres n Tble s observed under the condtonng event Thus the populton odds rto of Tble cn be wrtten s Under Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr H C C C C Under H nd Assumpton B ll entres n Tble s observed under the condtonng event snce Pr C holds Thus

30 - 9 - under }Pr { Pr } { B Pr Pr Pr Pr Pr Pr Pr Pr Pr Pr H u C ds u df u C S u df S df ds df u C u u C u u u u u APPENDI C: DEIVATIONS OF EQUIVAENT EPESSIONS In ths secton we prove equtons 6 nd 7 Note tht equton 6 s the uncensored cse wth C n 7 For mthemtcl convenence we defne the dscordnt ndctor } { I To smplf the nottons let nd One cn wrte w d d N : : I I Usng the fct tht I t follows tht I : : The ndctor equls zero for pr wth Therefore I : : B pplng smlr lgebrc mnpultons t follows tht

31 I Combnng I nd I we obtn : : : I{ } : For pr wth the followng equton holds: I{ & & & & } Thus we obtn the equton 7 s follows: : I{ B I{ B } } sgn{ I{ B } } The second equton follows from the permutton smmetr of ech term wth respect to rguments EFEENCES [] Beudon D nd khl-cheb 8 Archmeden Copul Model Selecton Under Dependent Truncton Sttstcs n Medcne [] Cheb vest -P nd Abdous B 6 Estmtng Survvl Under Dependent Truncton Bometrk [3] Chen C -H Ts - nd Cho -H 996 The Product-moment Correlton Coeffcent nd ner egresson for Truncted Dt Journl of the Amercn Sttstcl Assocton [4] Clton D G 978 A Model for Assocton n Bvrte fe Tbles nd Its Applcton n Epdemologcl Studes of Fmll Tendenc n Chronc Dsese - 3 -

32 Incdence Bometrk [5] Cuzck J 98 nk Tests for Assocton th ght Censored Dt Bometrk [6] Cuzck J 985 Asmptotc Propertes of Censored ner nk Tests The Annls of Sttstcs [7] Dng A A nd ng 4 Testng Independence for Bvrte Current Sttus Dt Journl of the Amercn Sttstcl Assocton [8] Dbrosk D M 986 nk Tests for Independence for Bvrte Censored Dt The Annls of Sttstcs [9] Emur T nd ng 8 Asmptotc Anlss nd Vrnce Estmton for Testng Qus-ndependence under Truncton Techncl eports of Mthemtcl Scences Chb Unverst Volume 4 No Avlble t: [] Flemng T nd Hrrngton D P 99 Countng Process nd Survvl Anlss New ork: John le nd Sons [] Genest C 987 Frnk s Fml of Bvrte Dstrbutons Bometrk [] Genest C nd Mck J 986 The o of Copuls: Bvrte dstrbutons wth unform mrgnls The Amercn Sttstcn [3] Hrrngton D P nd Flemng T 98 A Clss of nk Test Procedures for Censored Survvl Dt Bometrk [4] He S nd ng G 998 Estmton of the Truncton Probblt n the ndom Truncton Model Annls of Sttstcs 6-7 [5] Hsu nd Prentce 996 A Generlston of the Mntel-Henszel Test to Bvrte Flure Tme Dt Bometrk [6] Hde J 977 Testng Survvl Under ght Censorng nd eft Truncton - 3 -

33 Bometrk [7] Hde J 98 Survvl Anlss wth Incomplete Observtons In Bosttstcs Csebook G Mller B Efron B Brown nd E Moses eds New ork; John le nd Son 3-46 [8] Klen J P nd Moeschberger M 3 Survvl Anlss: Technques for Censored nd Truncted Dt New ork: Sprnger [9] nden-bell D 97 A Method of Allowng for Known Observtonl Selecton n Smll Smples Appled to 3C Qusrs Mon Nt Astr Soc [] gkos S Brr M nd De Gruttol V 988 Non-prmetrc Anlss of Truncted Survvl Dt th Applcton to AIDS Bometrk [] Mrtn E C nd Betensk A 5 Testng Qus-ndependence of Flure nd Truncton v Condtonl Kendll s Tu Journl of the Amercn Sttstcl Assocton [] Okes D 98 A concordnce Test for Independence n the Presence of Censorng Bometrcs [3] Okes D 986 Sem-prmetrc Inference n Model for Assocton n Bvrte Survvl Dt Bometrk [4] Okes D 989 Bvrte Survvl Models Induced b Frltes Journl of the Amercn Sttstcl Assocton [5] Sho J 993 Dfferentblt of Sttstcl Functonls nd Consstenc of the Jckknfe The Annls of Sttstcs 6-75 [6] Shh J H nd ous T A 996 Tests of Independence for Bvrte Survvl Dt Bometrcs [7] Shh J H nd ous T A 995 Inference on the Assocton Prmeter n Copul Models for Bvrte Survvl Dt Bometrcs [8] Ts - 99 Testng the Assumpton of Independence of Truncton Tme - 3 -

34 nd Flure Tme Bometrk [9] Ts Jewell N P nd ng M C 987 A Note on the Product-lmt Estmtor Under ght Censorng nd eft Truncton Bometrk [3] Vn Der Vrt A 998 Asmptotc Sttstcs Cmbrdge Seres n Sttstcs nd Probblstc Mthemtcs Cmbrdge: Cmbrdge Unverst Press [3] ng M C Jewell N P nd Ts 986 Asmptotc Propertes of the Product-lmt Estmte nd ght Censored Dt Annls of Sttstc [3] ng M C 99 Nonprmetrc Estmton From Cross-Sectonl Survvl Dt Journl of the Amercn Sttstcl Assocton [33] oodroofe M 985 Estmtng Dstrbuton Functon th Truncted Dt The Annls of Sttstcs Tble 3 Tests of qus-ndependence for the AIDS dt nvlog Ts M & B Adult -vlue P-vlue log ˆ Undefned Undefned Chldren -vlue P-vlue log ˆ Undefned Undefned

35 Tble 4 Tests of qus-ndependence for the Chnnng House dt A nvlog B Ts M & B 46 subects -vlue P-vlue log ˆ Undefned Undefned Undefned 97 men subset of -vlue P-vlue log ˆ Undefned Undefned Undefned Note: A uses the weght functon vˆ /{ nsˆ C } nd uses the weght functon vˆ du /{ nsˆ u } C B

36 Tble 5 Emprcl reecton probbltes of the proposed tests nd nvlog nd two competng tests Ts s nd Mrtn nd Betensk s tests t level 5 bsed on 5 replctons when under Clton s model wth smple sze c Pr c Pr Uncensored / b nvlog Ts M & B c c c % Censored c c c NOTE: For ech run fve test sttstcs re clculted bsed on the sme dtset

37 Tble 6 Emprcl reecton probbltes of the proposed tests nd nvlog nd two competng tests Ts s nd Mrtn nd Betensk s tests t level 5 bsed on 5 replctons when under Clton s model wth smple sze c Pr c Pr Uncensored / b nvlog Ts M & B c c c % Censored c c c NOTE: For ech run fve test sttstcs re clculted bsed on the sme dtset

38 Tble 7 Emprcl reecton probbltes of the proposed tests nd nvlog nd two competng tests Ts s nd Mrtn nd Betensk s tests t level 5 bsed on 5 replctons when under Frnk s model wth smple sze c Pr c Pr Uncensored / b nvlog Ts M & B c c c % Censored c c c NOTE: For ech run fve test sttstcs re clculted bsed on the sme dtset

39 Tble 8 Emprcl reecton probbltes of the proposed tests nd nvlog nd two competng tests Ts s nd Mrtn nd Betensk s tests t level 5 bsed on 5 replctons when under Frnk s model wth smple sze c Pr c Pr Uncensored / b nvlog Ts M & B c c c % Censored c c c NOTE: For ech run fve test sttstcs re clculted bsed on the sme dtset

40 Tble 9 Emprcl reecton probbltes of three proposed tests nd nvlog nd two competng tests Ts s nd Mrtn nd Betensk s tests t level 5 bsed on 5 replctons when under Gumbel s model wth smple szes nd c Pr c Pr / b nvlog Ts M & B n uncensored c c c n 5% censored c c c n uncensored c c c n 5% censored c c c NOTE: For ech run fve test sttstcs re clculted bsed on the sme dtset

41 Fgure Effcenc comprson of test wth [] under selected mrgnl dstrbutons : men-zero eponentl ; : unform on []; : ch-squred wth one degree of freedom - 4 -