Jurnal Teknologi HYPOTHESIS TESTING FOR THE PARAMETERS OF LOG-LOGISTIC REGRESSION MODEL WITH LEFT- TRUNCATED AND RIGHT-CENSORED SURVIVAL DATA

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1 Jural Tkolog HYPOTHESIS TESTING FOR THE PARAMETERS OF LOG-LOGISTIC REGRESSION MODEL WITH LEFT- TRUNCATED AND RIGHT-CENSORED SURVIVAL DATA Wa Nur Atkah Wa Mohd Ada *, Jayath Arasa Dpartmt of Mathmatcs, Faculty of Scc, Uvrst Putra Malaysa, UPM Srdag, Slagor, Malaysa Full Papr Artcl hstory Rcvd 3 August 07 Rcvd rvsd form 6 Novmbr 07 Accptd 5 Jauary 08 Publshd ol Aprl 08 *Corrspodg author waatkah@upm.du.my Graphcal abstract Abstract Lft-trucato ad rght-csorg (LTRC) ars aturally lftm data. Data may b lft-trucatd du to a lmtato th study dsg. Falur of a ut s obsrvd oly f t fals aftr a crta prod. Usually, th uts udr study may ot b followd utl all of thm hav fald but th study has to b stoppd at a crta tm. Ths troducs th rght csorg to th survval data. Log-logstc modl s xtdd to accommodat th lft-trucatd ad rght-csord survval data. Th bas, stadard rror (SE), ad root ma squar rror (RMSE) of th paramtr stmats ar computd to valuat th prformac of th modl at dffrt sampl ss, csorg proporto (CP), ad trucato lvl (TL). Th rsults show that th SE of th paramtr stmats cras as th trucato lvl (TL) ad csorg proporto (CP) cras. Havg low ad hgh TL (5% ad 5%) th data, th graphs clarly show that th mprcal powr of both tsts crass wth th cras of TL for paramtr ad. Th SE ad RMSE also dcras as th sampl s crass. Followg that, powr aalyss s coductd va smulato to compar th prformac of hypothss tsts basd o th Wald ad Lklhood Rato (LR) for th paramtrs. Th rsults clarly dcat that th Wald prforms slghtly bttr tha th LR wh dalg wth th proposd modl. Kywords: Log-logstc Rgrsso Modl, Lft-Trucatd ad Rght-Csord, Wald, Lklhood Rato, Emprcal Powr Abstrak Pmagkasa kr da papsa kaa (LTRC) tmbul scara smula jad dalam data sumur hdup. Data mugk dpsahka olh sbab had dalam rka btuk ssuatu kaja. Dalam ks, kgagala ut yag dprhatka adalah jka a gagal slpas tmpoh masa trttu. Basaya, ut d bawah kaja tdak bolh dkut shgga gagal kraa kaja tu prlu dhtka pada masa yag trttu. Olh tu, data yag mgadug tapsa kaa dprkalka. Modl Log-logstk dprluaska utuk mampug data madra yag dpagkas kr da dtaps kaa. Ralat, ralat pawa (RP), da puca m ralat prsg (PMRP) darpada aggara paramtr modl dkra utuk mla prstas modl pada sa sampl yag brba, bahaga tapsa kaa yag brba da tahap pmagkasa kr yag brba. Kputusa mujukka bahawa ralat pawa mgkat apabla kadar tagkasa kr da kadar tapsa kaa mgkat. Mmpuya TL rdah (5% da 5%) dalam data, graf jlas mujukka bahawa kuasa mprkal kdua-dua uja mgkat dga pgkata TL utuk paramtr da da SE da RMSE juga brkuraga apabla sa sampl 80:3 (08) ISSN 80 37

2 74 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) brtambah. Maka, aalss kuasa tlah djalaka mlalu kadah smulas utuk mmbadgka prstas uja hpotss brdasarka Wald da Nsbah Kmugka (LR) bag paramtr-paramtr dalam modl. Kputusa jlas mujukka bahawa Wald adalah lbh bak darpada LR apabla brurusa dga modl rgrs yag dcadagka. Kata kuc: Modl Rgrs Log-logstk, Pmagkasa Kr da Tapsa Kaa, Nsbah Kmugka, Kuasa Emprk 08 Prbt UTM Prss. All rghts rsrvd.0 INTRODUCTION Log-logstc s o of th dstrbutos that s wdly usd to modl lftm data spcally mdcal fld. It s oft usd survval aalyss as a paramtrc modl bcaus t has a o-mootoc haard rat; th haard rat that crass to a maxmum ad latr dcrass. Bt (983) dalt wth Log-logstc dstrbuto (LLD) to aaly survval data du to ts o-mootoc haard fucto, ad cocludd that t ca b a good rplacmt for th othr dstrbuto that has smlar haard fucto such as th Wbull or Log-ormal dstrbuto. Othr authors who hav do sgfcat work usg ths modl ar Cox ad Lws (966), Cox t al. (98). Th modl ca b asly xtdd to accommodat th covarats, trucatd data ad all typs of csord obsrvatos such as lft, rght ad trval. Arasa ad Adam (04) agrd that th log logstc modl was chos as ts popularty most cacr studs ad ts ablty to ft both fxd ad tm dpdt covarats asly. Lawlss (98) has mor dscusso o th trucatd data th survval fld. Th paramtrs of ths modl had uqu maxmum lklhood stmats ad ths has b provd by Gupta t al. (999). Grally, trucato ad csorg rduc th formato about th survval data. Csord data ars wh a dvdual's lf lgth s kow to occur oly wth a crta prod of tm. Lftm data ar somtms trucatd du to th study dsg. Igorg lft trucato may lad to adquacy of statstcal procdurs as dscrbd by Luo ad Tsa (009). Emura ad Wag (0) agrd that th trucatd data hav som formato about th lftm thrfor t s mportat to clud t th aalyss of survval. Th obsrvatos may tr th study at radom tm or dlayd try as th startg tm mght b radom for ach dvduals (Sh (009)). Emprcal lklhood rato mthod was frst usd by Thomas ad Grukmr (975). Thy proposd a oparamtrc lklhood rato mthod for trval stmato of survval probablts for radomly csord data. Thr mthod basd o cofdc trvals has bttr prformac as compard to ormal approxmato mthod. Howvr, th computato of th mprcal lklhood ratos wth csord or trucatd data s o-trval. Turbull (976) usd a modfd slf-cosstcy/ EM algorthm to comput a class of arbtrarly csord or trucatd mprcal lklhood ratos. L (995) dscussd mor o mprcal lklhood of trucatd data. Arasa ad Maohara (03) focusd o assssg th prformac of th log-ormal dstrbuto wth lft trucatd ad rght csord (LTRC) survval data. Som of th obsrvatos o th lft-tald of th log-ormal dstrbuto wll b dsrgardd cosqutly wth th cras of th xstg skwss of th dstrbuto wh fttd to LTRC data as dmostratd by Ca t al., 0. Th LR mthod has b show to work wth all typs of csord survval data. Ths has b hghlghtd by authors Pa ad Zhou (999) ad Murphy ad Va dr Vaart (997). Pa ad Zhou showd for rght-csord data, th mprcal LR has a ch-squar lmt (Wlks Thorm). Dogaaksoy ad Schm (993) showd that th lklhood rato mthod prformd bttr wth paramtrs of th log-ormal dstrbuto compard to th Wald mthod. For skwd data, th assumpto of ormalty oft fals as t ca't fully captur th samplg dstrbuto of th sampl statstcs bg studd, subsqutly rsultg th poor prformac of th Wald ad LR mthods costructg th cofdc trvals for th paramtr stmats spcfcally wh hghr proporto of trucato ad csorg s prst th data (Maohara, Arasa, Md & Adam (05)). No of th rsarchrs had do th prformac aalyss for th LLD wth LTRC data. O ths bass, th smulato study s coductd to accss th prformac of proposd modl wth LTRC data usg hypothss tstg basd o th Wald ad LR tsts.

3 75 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) METHODOLOGY. Log-logstcs Rgrsso Modl wth Rght- Csord ad Lft-Trucatd Survval Data T 0 b th radom varabl rprstg Lt survval tm or lftm wth dsty fucto P( t T t t) ft ( ) lm. () t0 t Th probablty dsty fucto (PDF) of LLD s gv by - ( t) f( t), t 0, 0, 0. () ( ( t) ) LLD has paramtrs whch ar ad rprst scal ad shap rspctvly. I ths papr, th LLD s xtdd to accommodat th covarat ffcts by allowg whr X ( x, x,, x ) s th vctor of X ' covarat valus, ' 0 p ar ukow paramtrs whr (,,, ' 0 p x 0 ad ) p dots th umbr of covarats th modl. Suppos w hav dpdt radom varabls. If x 0, whr s a,,, sgl fxd covarat for subjcts, th th dsty ad survval fucto of LLD ar gv () ad () rspctvly as follows, 0 x 0 x y f y, x,,, (3) 0x y Sy, x,,. ( 4) 0 x y If th csorg dcator, c th s dfd as f th obsrvato s ucsord c 0 f th obsrvato s rght csord ad x t s th obsrvd survval tm for th subjcts, th th lklhood fucto cosstg rghtcsord ad lft-trucatd s dfd as follows: L f S t u c S S t U th c c u u c (5) whr y 0 x ad y x tms., u u 0 x, y y u s th falur tms s th lft-trucato s th covarats. Followg that, th loglklhood fucto of LLD for rght-csord ad lfttrucatd s obtad as follows: { c l c c l u l l }. (6). Lklhood Equatos ad Estmato Th frst ad scod ordr partal drvatvs of th log-lklhood fucto wth rspct to paramtrs, 0 ad would b as follows: u c c u c u u c c 0 u u c x x x c x u u c c c u u u u u c c u u 0 u u c x cx x x u x u x u u (7) (8) (9) (0) () ()

4 76 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) c u c x x u x u u u u x u u u c c c u 0 u u u u u u cx c x cx x x x u x xu u u c x c x x 0 x x (3) (4) (5) Th vrs of th obsrvd formato matrx, whch ca b drvd from th scod partal drvatvs of th log-lklhood fucto valuatd at, 0,, provds us wth th stmats for th varac ad covarac of var, 0, ( 6) Th Maxmum Lklhood Estmato (MLE) of th paramtrs ca b obtad by usg th Nwto- Raphso tratv procdur..3 Ivrs Trasform Mthod Ivrs trasform mthod s o of th mthods usd to smulat th lftms, t of LLD. By sttg th survval fucto St U, whr U ~ U 0,, th radom umbrs gratd ar covrtd from uform dstrbuto to radom varabls whch ar log-logcally dstrbutd...4 Hypothss Tstg Procdurs.4. Wald Tst Lt St ( ) ( 0 x ) t 0 ( t x ) U U t 0 x (7) b th maxmum lklhood stmator for paramtr ad th log lklhood fucto of Udr mld rgularty codtos, ormally dstrbutd wth ma matrx, whr s asymptotcally ad covarac s th Fshr formato matrx, valuatd at th tru valu of th s ot always avalabl paramtr. Th matrx ad ca b rplacd by th obsrvd formato matrx, whch ca b obtad from th log lklhood fucto valuatd at as gv blow:. (8) Th Wald tst statstcs for tstg th ull hypothss, H 0 : 0 vrsus th altratv hypothss, H : 0 s gv as: whr 0 (9) s th s. whch ca b obta from th dagoal lmts of th vrs of th obsrvd Fshr formato matrx or kow as gatv Hssa. Followg that, th H 0 wll b rjctd f th tst statstcs, > whr s th quatl of th stadard ormal dstrbuto..4. Lklhood Rato Tst Th LR compars th maxmd lklhood of two std modls, th full modl ad th rducd modl. Th rducd modl s rstrctd by crta

5 77 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) codtos gv H 0. Th LR tstg th ull hypothss, : 0 H 0 : 0 H s gv as follows: whr vrsus altratv hypothss, ~ ~ 0,, (0) s th log lklhood fucto, s th vctor of usac paramtrs, lklhood stmator of,, ad maxmum lklhood stmator of partcular, th LR tst udr th s th maxmum ~ s th rstrctd udr H 0 H 0. I has asymptotcally a ch squar dstrbuto wth dgr of frdom (dof). I th dcso rul, th k H 0 wll b rjctd f th tst statstcs, > k,k whr s th total umbr of paramtrs full modl subtract th total umbr of paramtrs rstrctd modl. 3.0 RESULTS AND DISCUSSIONS A smulato study o lft-trucatd ad rghtcsord survval data (LTRC) survval data proposd by Mtra (03) s adoptd. Th stmats obtad from lug cacr data by Ta t.al (003) ar usd as th tru paramtrs valus for th smulato study whch ar, 0, =(. 655, , ). Th wk of trucato of th study, amly, s fxd throughout th smulato. Th bgg tm pot of th study or th trucato tm s 70 wks. A st of radom umbr of wks whch bascally rprsts th patt's wks of lug cacr dagoss s smulatd wth qual probablty wth rplacmt; for bfor, y th wk y bk ad aftr, aj of trucato, whr ad j,,, y k,,,. I th smulato study, th total obsrvato s dotd by N ad th trucato lvl (TL) s fxd at 0 % ad 0 %. Th lftms, t ar gratd by: 0 x t U () Th covarat valus ar smulatd from th x ~ N 0,. Th stadard ormal dstrbuto, lftm, tar addd to rsultg falur tm lss tha ybk ad y aj ad f th y whch s th trucato wk, ths obsrvatos ar rmovd ad rplacd by a w smulatd st of y, bk, yaj, t U ad y x y y. Th trucato tms y u ar c ar obtad as u bk. Th csorg tms, smulatd from th xpotal dstrbuto wth paramtr, c ~ xp, whr th valu of could b adjustd ordr to yld th dsrd approxmat csorg proporto, CP th data. I ths study, th two lvls of approxmat CP, 0.05 ad 0.5 ar chos to rprst low ad hgh lvls of CP, rspctvly. Th smulato study s coductd to assss th accuracy ad ffccy of th paramtr stmats usg =000 sampls of s N,, tru paramtrs, 0 =0,30,50,00. Th = (.655, , ) ar usd to obta th Bas, Stadard Error (SE) ad Root Ma Squar Error (RMSE) of th paramtr stmats. Powr of a statstcal tst s th total umbr of possblts that th tst wll rjct th ull hypothss, H 0 wh a spcfc altratv, hypothss tsts th paramtr vrsus H : H 0 vrsus H 0 0 : H at s tru. Th H ad tsts th paramtr : : 0 at. Data ar smulatd from Log-logstcs rgrsso modl udr H powr, H 0 ad for svral ffct ss to stmat th mprcal. Effct s rfrs to th dffrc btw th tru valu for paramtrs ad whch ar.655 ad ad th valu spcfd H. Thrfor, thr ffct ss valu ar usd for paramtrs ad whch ar (.5,.0,.5) ad (-0.,-0.,-0.5) rspctvly. Th smulatd data ar adjustd to mt th dsrd trucato lvls, TL = 0%, 0% ad 0% ad csorg proportos, CP = 5% ad 5%. Th, th Wald ad LR tstd for paramtrs ad ar ru for svral ffct ss,, TL ad CP. Th mprcal powr for th Wald ad LR ar stmatd by th proporto of rjctos for a tst statstc at two valus of sgfcac lvls whch ar 0. 0 ad Th proportos of rjcto ar calculatd by coutg th umbr of tms th hypothss tsts rjct th ull hypothss, H 0 ad th umbr of rjctos ar dvdd wth N =5000. Tabls, ad 3 show th rsults of th bas, SE ad RMSE for th paramtr stmats of th modl,, 0, at svral, TL ad CP. Both bas ad SE cotrbut to th avrag rror s of a stmator, thus th RMSE Bas SE s usd to masur th avrag ovrall rror of th paramtr stmats.

6 78 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) Th rsults Tabls, ad 3 show that th valus of Bas, SE, ad Root Ma Squar Error (RMSE) ar rlatvly low wth 0 ad havg hghr valus tha. Th trd for th bas s cosstt ad th valus for th bas ar rlatvly low for all th paramtr stmats. Ths s du to th shap paramtr whch tds to produc hgh valus of bas, SE ad RMSE. As crass, th SE ad RMSE valus clarly dcras. From th tabls, t ca b cocludd that th SE ad RMSE valus ar crasg as th CP crass but th valus show o clar trd wh th TL crass for all th paramtr stmats. I gral, w ca coclud that th stmato procdur s workg wll wth th proposd rgrsso modl. Th summary of smulato rsults, comparg th prformac of th Wald ad LR, s gv Tabls 4 through. Ths tabls dsplay th mprcal powr, gratd by ach of ths mthods at svral ffct ss, TL, CP, ad for th Wald ad LR for paramtrs ad rspctvly. Fgurs through 4 gv a graphcal vw of th mprcal powr for paramtrs ad for th Wald ad LR hypothss tsts. All th graphs show that th mprcal powr for both paramtrs crass wth th cras for both tsts. Bsds that, th powr of both tsts also cras as th ffct s whch s th dstac btw th paramtr valus udr H 0 & H bcoms largr for paramtr ad. Apart from that, th powr of both tsts crass wth th cras th CP for both paramtrs volvd. Havg low ad hgh TL (5% ad 5%) th data, th graphs clarly show that th mprcal powr of both tsts crass wth th cras of TL for paramtr. Howvr, th mprcal powr shows o clar chags for paramtr for both tsts. From Fgur 4, th rsults show th mprcal powr of both tsts s hgh at hghr valus of. By comparg ths two tsts, w ca coclud that th mprcal powr of th Wald s slghtly bttr as compard wth LR Tabl Bas, SE ad RMSE for Trucato (%) Csorg (%) Bas SE RMSE Bas SE RMSE Bas SE RMSE Bas SE RMSE Tabl Bas, SE ad RMSE for Trucato (%) Csorg (%) Bas SE RMSE Bas SE RMSE Bas SE RMSE Bas SE RMSE

7 79 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) Tabl 3 Bas, SE ad RMSE for Trucato (%) Csorg (%) Bas SE RMSE Bas SE RMSE Bas SE RMSE Bas SE RMSE Tabl 4 Emprcal Powr of Wald Tst for Paramtr at 0.0 Tst WALD Trucato (%) Csorg (%) Tabl 5 Emprcal Powr of Lklhood Rato Tst for Paramtr at 0. 0 Tst LIKELIHOOD RATIO Trucato (%) Csorg (%)

8 80 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) Tabl 6 Emprcal Powr of Wald Tst for Paramtr at 0.0 Tst WALD Trucato (%) Csorg (%) Tabl 7 Emprcal Powr of Lklhood Rato Tst for Paramtr at 0.0 Tst LIKELIHOOD RATIO Trucato (%) Csorg (%) Tabl 8 Emprcal Powr of Wald Tst for Paramtr at Tst WALD Trucato (%) Csorg (%)

9 8 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) Tabl 9 Emprcal Powr of Lklhood Rato Tst for Paramtr at Tst LIKELIHOOD RATIO Trucato (%) Csorg (%) Tabl 0 Emprcal Powr of Wald Tst for Paramtr at 0.05 Tst WALD Trucato (%) Csorg (%) Tabl Emprcal Powr of Lklhood Rato Tst for Paramtr at Tst LIKELIHOOD RATIO Trucato (%) Csorg (%)

10 8 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) Fgur Powr vs for dffrt paramtr valus udr for paramtrs ad H 0 ad H Fgur 3 Powr vs for dffrt CP for paramtrs ad Fgur Powr vs for dffrt TL for paramtrs ad Fgur 4 Powr vs sampl s, for dffrt sgfcat lvl, for paramtrs ad

11 83 Atkah & Jayath / Jural Tkolog (Sccs & Egrg) 80:3 (08) CONCLUSION Ovrall, th Wald clarly grats mprcal powr whch s slghtly hghr tha th LR. Th ovrall rsults of th Wald ar bttr tha th LR for small ad larg sampl s. It should b th prfrrd mthod wh dalg wth csord ad trucatd data. I gral, w ca coclud that th xstc of th trucato ad csorg th survval data sm to affct th Wald mor tha th LR. I futur, othr hypothss tsts ca b aalyd spcally thos basd o computr tsv tchqus such as th bootstrap or jackkf. Th Scor tst could b valuatd as a addtoal asymptotc tst. Ackowldgmt W would lk to xtd our grattud to Uvrsty Putra Malaysa ad Dr. Patrca Ta, Uvrsty of Saskatchwa, Saskatchtoo, Caada. Rfrcs [] Ca, K. C., Harlow, S. D., Lttl, R. J., Na, B., Yosf, M., Taff, J. R., ad Ellott, M. R. 0. Bas du to Lft Trucato ad Lft Csorg Logtudal Studs of Dvlopmtal ad Dsas Procsss. Amrca Joural of Epdmology. 73: [] Cox, D. R., Lws, P. A Statstcal Aalyss of Srs of Evts. Lodo: Mthu. [3] Dogaaksoy, N., ad Schm J Comparso of Approxmat Cofdc Itrval Dstrbutos usd Lfdata Aalyss. Tchomtrcs. : [4] Emura, T. & Wag, W. 0. Noparamtrc Maxmum Lklhood for Dpdt Trucato Data basd o Copulas. Joural of Multvarat Aalyss. 0: [5] Gupta, R. C., Akma, O., & Lv, S A Study of Loglogstc Modl Survval Aalyss. Bomtrcal Joural. 4(4): [6] Jayath Arasa & Mohd B. Adam. 04. Doubl Bootstrap Cofdc Itrval Estmats wth Csord ad Trucatd Data. Joural of Modr Appld Statstcal Mthods. 3(): Artcl. [7] Lawlss, J. F. 98. Statstcal Modl ad Mthods for Lftm Data. Wly, Nw York. [8] L, G Noparamtrc Lklhood Rato Estmato of Probablts for Trucatd Data. JASA. 90: [9] Mtra, Dbaja. 03. Lklhood Ifrc for Lft- Trucatd & Rght Csord Lftm Data. McMastr Uvrsty Lbrary. Papr 7599:-6. [0] O'Qugly, J. & Struthrs, L. 98. Survval Modls Basd upo th Logstc & Log-Logstcs Dstrbuto. Computr Programm Bomdc. 5(): 3-. [] Pa, X, R. & Zhou, M Usg O Paramtr Subfamly of Dstrbutos Emprcal Lklhood wth Csord Data. J.Statst, Plag ad Ifr. 75(): [] S. A. Murphy & A. W. Va Dr Vaart Smparamtrc Lklhood Rato Ifrc. Th Aals of Statstcs. 5(4): [3] Sh, P Smparamtrc Aalyss of Survval Data wth Lft Trucato ad Rght Csorg. Computatoal Statstcs ad Data Aalyss. 53: [4] Stv Bt Log-Logstc Rgrsso Modls for Survval Data. Joural of th Royal Statstcal Socty. 3(): [5] Ta, P., Tota, J., Yu, E., & Skarsgard, D Twty-yar Follow-up Study of Log Trm Survval of lmtd-stag Small-Cll Lug Cacr ad Ovrvw of Progostc ad Tratmt Factors. It J. Radato Ocology Bo/Phys. 56: [6] Thruath Maohara, Jayath Arasa, Habshah Md & Mohd Bakr Adam. 05. A Covrag Probablty o th Paramtrs of th Log-Normal Dstrbuto th prsc of Lft-Trucatd ad Rght Csord Survval Data. Malaysa Joural of Mathmatcal Sccs. 9(): 5-43 [7] Thruath Maohara & Jayath Arasa. 03. Evaluatg th Prformac of th Log-Normal Survval Modl wth Lft-Trucatd & Rght Csord Survval Data. Dpartmts of Mathmatcs, Faculty of Scc, Uvrsty Putra Malaysa. [8] Thomas, D. R. & G. L. Grukmr Cofdc Itrval Estmato of Survval Probablts for Csord Data, J. Amr. Statst. Assoc. 70: [9] Turbull, B Th Emprcal Dstrbuto Fucto wth Arbtrarly Groupd, Csord ad Trucatd Data. JRSS B [0] X. Luo, W. Y. Tsa Noparamtrc Estmato for Rght- Csord Lgth Basd Data: A Psudo-partal Lklhood Approach. Bomtrka. 96: