DIMENSION REDUCTION FOR CENSORED REGRESSION DATA

Transcription

1 The Annals f Statstcs 1999, Vl. 27, N. 1, 1 23 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA BY KER-CHAU LI, 1 JANE-LING WANG 2 AND CHUN-HOUH CHEN Unversty f Calfrna, Ls Angeles, Unversty f Calfrna, Davs and Academa Snca, Tawan Wthut parametrc assumptns, hgh-dmensnal regressn analyss s already cmplex. Ths s made even harder when data are subject t censrng. In ths artcle, we seek ways f reducng the dmensnalty f the regressr befre applyng nnparametrc smthng technques. If the censrng tme s ndependent f the lfetme, then the methd f slced nverse regressn can be appled drectly. Otherwse, mdfcatn s needed t adjust fr the censrng bas. A key dentty leadng t the bas crrectn s derved and the rt-n cnsstency f the mdfed estmate s establshed. Patterns f censrng can als be studed under a smlar dmensn reductn framewrk. Sme smulatn results and an applcatn t a real data set are reprted. 1. Intrductn. Survval data are ften subject t censrng. When ths ccurs, the ncmpleteness f the bserved data may nduce a substantal bas n the sample. Several appraches have been suggested t vercme the asscated dffcultes n regressn, ncludng the accelerated falure tme mdel, censred lnear regressn, the Cx prprtnal hazard mdel and many thers. Survval analyss becmes even mre ntrcate when the dmensn f the regressr ncreases. T apply any f the afrementned methds, users are requred t specfy a functnal frm whch relates the utcme varables t the nput nes. Hwever, n realty, knwledge needed fr an apprprate mdel specfcatn s ften nadequate. As a matter f fact, the acqustn f such nfrmatn may well turn ut t be ne f the prmary gals f the study tself. Under such crcumstances, t seems preferable t have explratry tls that rely less n such mdel specfcatn. Ths s the ssue t be addressed n ths artcle. The dmensn reductn apprach f L Ž wll be extended t settngs whch allw fr censrng n the data. We shall ffer methds f fndng lw-dmensnal prjectns f the data fr vsually examnng the censrng pattern. We shall shw hw censred regressn data can stll be analyzed wthut assumng the functnal frm a prr. Receved January 1997; revsed September Supprted n part by an NSF grant and a Guggenhem Fellwshp. Part f the research was carred ut whle vstng the Insttute f Statstcal Scence, Academa Snca, Tawan, n 1994 wth supprt frm the Natnal Scence Cuncl, Tawan, R.O.C. 2 Supprted n part by NSF Grant DMS AMS 1991 subject classfcatns. 62G05, 62J20. Key wrds and phrases. accelated falure tme mdel, censred lnear regressn, Cx mdel, curse f dmensnalty, hazard functn, Kaplan-Meer estmate, regressn graphcs, slced nverse regressn, survval analyss 1

2 2 K.-C. LI, J.-L. WANG AND C.-H. CHEN Dmensnalty sets a severe lmtatn even n the explratry stage f data analyss. Ths s true even wthut the presence f censrng. Fr example, when the dmensn s ne r tw, a tw-dmensnal r threedmensnal scatterplt f the respnse varable aganst the regressr s helpful n btanng general deas abut the shape f the regressn functn, the pattern f hetergenety and ther valuable structural nfrmatn. Hwever, as the dmensn ncreases, the ttal number f tw-dmensnal r three dmensnal scatterplts escalates quckly. Very sn ths task culd turn nt an extremely labrus exercse. Wthut prper gudance, t may nt be easy fr us t put tgether a clear glbal pcture abut the data frm varus plts. Hw t bypass the curse f dmensnalty has been an mprtant ssue; see, fr example, Huber Ž L s framewrk fr dmensn reductn n regressn begns wth the fllwng frmulatn: Ž 1.1. Y g Ž x,..., 1 k x,.. The man feature f Ž 1.1. s that g s cmpletely unknwn and s s the dstrbutn f, whch s ndependent f the p-dmensnal regressr x. When k s smaller than p, Ž 1.1. mpses a dmensn reductn structure by clamng that the dependence f Y n the p-dmensnal x nly cmes frm the k varates, x,..., 1 k x, but the functnal frm f the dependence structure s nt specfed. The k-dmensnal space spanned by the k vectrs s called the e.d.r. Ž effectve dmensn reductn. space and any vectr n ths space s referred t as an e.d.r. drectn. The prmary gal f L s apprach s t estmate the e.d.r. drectns s that we can plt y aganst the e.d.r. varates fr vsually explrng the structure f the regressn and fr mre effectvely applyng varus lw-dmensnal regressn technques t the reduced space. The ntn f e.d.r. space and ts rle n regressn graphcs are further explred n Ck Ž and Ck and Wesberg Ž T ncrprate censrng nt the dmensn reductn framewrk, let Y the true Ž unbservable. lfetme, C the censrng tme, the censrng ndcatr; 1, f Y C and 0, therwse, Y mn Y, C 4, the bserved tme. We assume that Ž 1.2. Y fllws mdel Ž 1.1.; Ž 1.3. Cndtnal n x, C s ndependent f Y. The bserved sample cnssts f n..d. bservatns, Ž Y, x,., 1,..., n frm the dstrbutn f Y, x,. The cntnuus randm varables, Y, C, are nt bserved. Cndtn Ž 1.3. s the usual ndependence assumptn t ensure dentfablty under the randm censrng scheme. If Ž 1.3. s vlated, then ne needs mre nfrmatn n the censrng mechansm t buld an apprprate mdel. Ths s nt cnsdered n ths paper.

3 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 3 Fr k 1, ur frmulatn may nclude the generalzed lnear mdel McCullagh and Nelder Ž and the lnear transfrmatn mdel Dksum Ž as specal cases. The latter als ncludes several survval analyss mdels such as the accelerated falure tme mdel, the prprtnal hazard mdel, the prprtnal dds mdel and the lgt and prbt mdels Dksum and Gask Ž Wthut censrng, slced nverse regressn Ž SIR. s a smple methd fr fndng the e.d.r. space. Instead f drectly estmatng EY Ž x.,a p-dmensnal surface, the rles f x and Y are reversed the fcus turns t the p nverse regressn E x Y, whch s a curve n R. Under apprprate cndtns Lemma 3.1 f L Ž 1991., the nverse regressn curve s shwn t fall nt a k-dmensnal subspace. In partcular, when the regressr dstrbutn has mean zer and wth the dentty cvarance, ths k-dmensnal subspace cncdes wth the e.d.r. space. Explrng ths cnnectn, SIR begns wth a smple estmate f the nverse regressn curve by parttnng the data nt several slces accrdng t the Y values and cmputng the mean f x wthn each slce; ths s the slcng step. It s then fllwed by an egenvalue decmpstn step a prncpal cmpnent type f analyss ntended t lcate the subspace cntanng the nverse regressn curve. See L Ž fr further detals. Prpertes f SIR have been studed n several places: Carrll and L Ž 1992, 1995., Chen and L Ž 1998., Ck and Wesberg Ž 1991, 1994., Duan and L Ž 1991., Hsng and Carrll Ž 1992., Schtt Ž 1994., Zhu and Ng Ž Hw des censrng affect SIR? Ths depends n the relatnshp between the censrng tme C and the regressr x. Sectn 2 cnsders the ndependence case n whch Ž 1.4. C s ndependent f x and Y. We shw that the general thery f SIR s applcable wthut mdfcatn and the drectns fund by SIR are stll cnsstent. Thus fr the ndependence case, censrng des nt ntrduce bas t the SIR estmates. Hwever, SIR wll be affected by ther censrng mechansms that d nt fllw Ž In Sectn 3, we ntrduce a general strategy t vercme ths dffculty. The prpsed apprach s t ntrduce a sutable weght functn fr the censred bservatns fr ffsettng bas n estmatng the slce means. The weght functn can be estmated by nnparametrc estmatn technques fr cndtnal survval functns. Fr smplcty, the kernel methd s used and we establsh the rt-n cnsstency fr the mdfed SIR. In Sectn 4, we brng ut a dmensn reductn settng fr studyng the pattern f censrng when the ndependent censrng cndtn Ž 1.4. s vlated. We argue fr the mprtance f vsualzng the heavy censrng regn, a nntrval task n the hgh-dmensnal stuatn. Data analysts have t recgnze ths regn because heavy censrng sets severe lmtatns n fndng the structure f regressn. The dmensn reductn assumptn n C s a natural cunterpart f Ž 1.2., Ž 1.5. C hž x,..., x,.. 1 c

4 4 K.-C. LI, J.-L. WANG AND C.-H. CHEN Then Ž 1.2. and Ž 1.5. tgether allw us t treat the survval tme and censrng tme equvalently. But t avd cnfusn, we shall refer t s and ther lnear cmbnatns as e.d.r. censrng drectns. In cntrast, the e.d.r. drectns fr Y wll be called e.d.r. lfetme drectns. Bth censrng and lfetme drectns as well as ther lnear cmbnatns wll be called jnt e.d.r. drectns. We shw hw t estmate the jnt e.d.r. drectns thrugh a duble slcng prcedure. Frm the jnt e.d.r. drectns, we can recver the e.d.r. lfetme drectns by further applyng the mdfed SIR strategy f Sectn 3. Ths s llustrated n Sectn 5. The perfrmance f the prcedure s examned thrugh tw smulatn studes. We apply ur methd t a data set abut the study f prmary blary crrhss Ž PBC. at the May Clnc. Sectn 6 cncludes ths artcle by summarzng ur fndngs. Sme questns are rased fr further study. 2. SIR under the ndependence assumptn ( 1.4 ). Dente the uncen Ž. Ž sred nverse regressn curve by y E x Y y.. Wthut censrng Ž.e., Y Y., the ppulatn versn f SIR s based n the fllwng egenvalue decmpstn: Ž 2.1. b b, x, 1 p where Ž 2.2. cv EŽ x Y. and x cvž x.. The justfcatn fr usng the frst k egenvectrs b wth nnzer egen- values t estmate the e.d.r. Ž lfetme. drectns fllws frm Lemma 3.1 fl Ž 1991., whch can be stated as fllws. LEMMA 2.1. Assume that the dmensn reductn assumptn Ž 1.2. hlds. Then fr any y, Ž Ž y. EŽ x.. falls nt the e.d.r. Ž lfetme. x space under the cndtn that Ž 2.3. fr any vectr b, EŽ b x x,..., 1 k x. s lnear. Desgn cndtn Ž 2.3. has been dscussed n several places. The perfrmance f SIR s nt very senstve t ths cndtn; see the dscussn and the rejnder f L Ž 1991., Ck Ž 1994., Ck and Wesberg Ž 1991, Carrll and L Ž In vew f the fact that mst lw-dmensnal prjectns f hgh-dmensnal data ften appear lke nrmal dstrbutns D- acns and Freedman Ž 1984., Hall and L Ž argue fr the generalty f ths cndtn n hgh-dmensnal stuatns. On the ther hand, reweghtng and subsamplng methds can als be appled t acheve Ž 2.3.: Brllnger Ž 1991., Ck and Nachtshem Ž Further dscussn n ths cndtn can be fund n L Ž

5 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 5 Censrng alters the dstrbutn f the bserved tme Y. Its effect n SIR can be studed by cmparng the censred nverse regressn curve Ž y. EŽ x Y y. wth the uncensred ne Ž y.. By cndtnng, we have Ž 2.4. EŽ x Y y. EŽ EŽ x Y, C. Y y.. Ž. Ž. Under 1.4, E x Y, C s equal t E x Y, mplyng that Ž 2.5. EŽ x Y y. EŽ Ž Y. Y y.. Ž. Snce Lemma 2.1 apples t y, the fllwng result s btaned. LEMMA 2.2. Assume that 1.2 and 1.4 hld. Then Ž Ž y. EŽ x.. x falls nt the e.d.r lfetme space under Ž T mplement SIR n the data Ž Y, x., 1,..., n, we fllw L Ž Frst we parttn y s nt H ntervals, I h, h 1,..., H. Then fr each nterval, we cmpute the parttn slce mean x by averagng where n h 1 x x, h Ý n h x I h s the number f cases fallng nt I. Then the cvarance matrx H n h Ž x x.ž x x. ˆ n Ý h h h s frmed. Fnally we cnduct the egenvalue decmpstn ˆ ˆ ˆ ˆ b b, x ˆ ˆ. 1 p Wth Lemma 2.2 and fllwng the argument n L Ž 1991., we btan the rt-n cnsstency f SIR estmates ˆb fr fndng e.d.r. lfetme drectns. Thus censrng des nt ntrduce bas t SIR. Hwever, ths s true nly when the censrng tme s ndependent f the regressrs and the true lfetme. Wthut Ž 1.4., ths appealng result vanshes and substantal bas may be nduced by censrng under the mre general cndtn Ž A strategy fr mdfyng SIR under ( 1.3 ). An deal way f bypassng the dffcultes caused by general censrng Ž 1.3. s t slce the true survval tme Y. At frst sght, ths des nt appear feasble because under censrng, Y s unbservable. The prmse cmes frm an dentty derved n Sectn 3.1, whch relates the cndtnal expectatn f x n each slce t the bserved tme Y and the censred ndcatr. Ths leads t a mdfed slcng step by a sutable weghtng scheme fr ffsettng the censrng bas n estmatng the slce means. The cnsstency f ths new prcedure s dscussed n Sectn 3.2. h h

6 6 K.-C. LI, J.-L. WANG AND C.-H. CHEN 3.1. An dentty. Let 0 t1 t2 th th be a parttn n the survval tme. The expected value f x n a slce, m j E x Y t, t.4, can be wrtten as j j ½ PY t j, tj. 4 E1Ž Y tj. 4 E1Ž Y tj. 4 E x1 Y t, t j j. E x1 Y tj E x1 Y tj Ž 3.1. m j, where 1Ž. s the ndcatr functn. The tw numeratr terms take the same Ž frm, whch nvlves the unbservable ndcatr 1 Y t.. They can be cnverted nt terms wth Y and va the dentty, Ž 3.2. Ex1Ž Y t. 4 Ex1Ž Y t. 4 Ex1Ž Y t, 0. wž Y, t, x. 4, where fr t t, S Ž t x. Ž 3.3. wž t, t, x., S Ž t x. Ž 3.4. S Ž t x. P Y t x4 cndtnal survval functn fr Y, gven x. Cnsder the plane f varables Y and C n Fgure 1. The ntegratn regn Y t s decmpsed nt tw parts. The frst regn Žarea I n Fg-. ure 1 wth Y t, C t r equvalently, Y t, cntrbutes t the frst term n the rght sde f 3.2. The secnd regn wth Y t, C t, falls nt the censred area, 0. It s cntaned n the larger regn Ždashed area II n Fgure 1. wth Y t and 0. The secnd term n the rght sde f Ž 3.2. cmes frm ntegratn ver ths larger regn wth the weght adjustment FIG. 1. Integratn regns.

7 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 7 w,,. Cndtnng s the key t justfy ths term: 4 4 Ex1Ž Y t, 0. E 1Ž Y t. Y, 0, x 4 Ex1Ž Y t, 0. E 1Ž Y t. C, Y C, x 4 Ex1Ž Y t, 0. wž C, t, x. 4 Ex1Ž Y t, 0. wž Y, t, x. 4. E x1ž Y t, C t. E x1ž Y t, 01Y. Ž t. Here the next t the last equalty s due t the cndtnal ndependence assumptn Ž 1.3., whch assures that cndtnal n x, the prbablty fr the true survval tme Y t exceed t gven C t and Y t s equal t the cndtnal prbablty gven by Ž By a smlar argument, the denmnatr terms can be cnverted va the dentty Ž 3.5. E1Ž Y t. 4 E1Ž Y t. 4 E1Ž Y t, 0. wž Y, t, x. 4. The weght functn Ž 3.3. can be further expressed as Ž 3.6. wž t, t, x. exp Ž t, t x. 4, where 1 Ž t Y t, 1 t, t x E. x, S Ž Y x. ½ 5 Y SY Ž x. the cndtnal survval functn f Y cndtnal n x. Then Ž 3.6. fllws frm the well-knwn relatnshp between survval functns and cumulated hazards; fr a prf, see the Appendx. The term Ž t, t x. s smply the ntegrated cndtnal hazard Ž gven x. functn ver the nterval t, t Estmatn. T cnstruct an estmate fr m, we replace each expecj tatn term n 3.2 and 3.5 by the crrespndng frst sample mment, ˆ ˆ E x1ž Y tj. E x1ž Y tj. Ž 3.7. mˆ j, ˆ ˆ PY t PY t j j n n Ý Ý ˆ : Y t : Y t, 0 Ž 3.8. Eˆ x1ž Y t. 4 n x n x wž Y, t, x., ˆ 4 n 4 Ý ˆ : Y t, 0 Ž 3.9. P Y t : Y t n n wž Y, t, x., where wˆ Ž,,. dentes an estmate f the weght functn Ž 3.3. t be dscussed later.

8 8 K.-C. LI, J.-L. WANG AND C.-H. CHEN After estmatng each slce mean by Ž 3.7., we can frm the cvarance matrx f the slce means n the usual way; Ý ˆ mˆ x mˆ x p ˆ, j j j j 4 4 ˆ ˆ p PY t PY t. ˆj j j Fnally, we may cnduct the egenvalue decmpstn as befre t fnd the SIR drectns Ž ˆ ˆ b ˆˆ ˆ b, x ˆ ˆ. 1 p Smthng s needed n estmatng wt, Ž t, x.. There are several ways t prceed. Fr example, we can apply Beran s estmates fr cndtnal survval functns and ther varants. Under apprprate cndtns, Beran Ž and Dabrwska Ž 1987, establshed the cnsstency f ther estmates at cnvergence rates Ž slwer than the rt n rate. smlar t thse cmmnly fund n nnparametrc regressn. These cnsstency results lead t the cnsstency f mˆ h as an estmate f m h. It s easy t see that ˆ s als cnsstent fr the cvarance matrx f the slce means m s. As n L Ž h, we can apply Lemma 2.1 t establsh the cnsstency f ˆb as estmates f e.d.r. lfetme drectns. Despte the slw rate f cnvergence n estmatng cndtnal survval functns hence the weght Ž 3.3., t s stll pssble t establsh the rt n cnvergence fr m ˆ h. We nly cnsder the kernel smthng methd here fr Ž. p smplcty. Let K p be a kernel functn n R and hn be the bandwdth n p each crdnate. We shall assume that hn 1 and nhn tends t nfnty. Further cnstrants wll be mpsed later. It s cmmn fr K Ž. p t take a prduct frm, K Ž x,..., x. KŽ x. KŽ x. p 1 p 1 p, fr sme ne-dmensnal kernel functn KŽ.. Our kernel estmate f Ž 3.6. s defned by settng n ˆ p n Ý: t Y t, SYŽ Y x. hn Kp hn Ž x x. Ž ˆ Ž t, t x. ˆf Ž x. n Ý n j: Y Y h p n Kp h n Ž x j x. j Ž Sˆ YŽ Y x., ˆf Ž x. n p Ý n p n Ž fˆ Ž x. n h K h Ž x x.. A sketch prf f the fllwng clam tgether wth the regularty cndtns needed s gven n the Appendx. LEMMA 3.1. Under the regularty cndtns Ž B.1., Ž B.3., Ž B.5. and Ž B.8., gven n the Appendx, m s a rt-n cnsstent estmate fr m, h 1,..., H. ˆ h h,

9 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 9 We can use ths lemma and fllw the same argument as n L 1991 t shw that the mdfed SIR s rt-n cnsstent. Ths s stated n the fllwng therem. The prf s mtted. THEOREM 3.2. Under the assumptn f Lemma 3.1, each ˆb h s a rt-n cnsstent estmate fr an e.d.r. drectn. Assume that fž. and SŽ x. are d-tmes cntnuusly dfferentable and that the kernel functn satsfed the mment cndtns Hx K Ž x. p dx 0 fr d 1,..., d 1, and Hx K Ž x. p dx s nnzer. Then the regularty assump- tns n Lemma 3.1 can be satsfed wth bandwdth H n 2 d prvded p d. What we have presented s far n ths sectn s a general strategy fr ffsettng the bas due t censrng. The theretcal result f Therem 3.2, hwever, may nt help much n practce. The prblem s that kernel smthng nly wrks well n the lw-dmensnal case. Thus, befre applyng the kernel methd n estmatng the weght functn, we may want t reduce the dmensnalty frst. Ths s t be dscussed n the next tw sectns. 4. Dmensn reductn mdel fr censrng tme. Analyzng the censrng pattern s an mprtant step n studyng the censred data. It helps the recgntn f the nfrmatn-pr regn n x, the regn where censrng s heavy and the regressn structure s thus harder t explre. Smetmes such an analyss may even becme a prmary part f the study. In sme ndustral applcatns, Y may be the ptental yeld f a prductn prcess and censrng C may ccur because f machne malfunctnng, fr example. In addtn t learnng hw varus nput varables x may affect the ptental yeld, qualty cntrl engneers may equally be nterested n hw they affect the censrng rate; they need such knwledge t prevent machne malfunctnng as much as pssble. Lke ts cunterpart Y, we nw assume that the censrng tme C als has a dmensn reductn structure gven by Ž Agan, the functnal frm f h and the dstrbutnal frm f are bth unspecfed. Ths mdel suggests nly that the dmensn f the regressr can be reduced frm p t c. The relatnshp between the e.d.r. space fr the censrng tme and the e.d.r. space fr the true lfetme s arbtrary. They can be ether dentcal, partly verlapped, r dsjnt. Lnear cmbnatns f ther elements frm a space whch wll be called the jnt e.d.r. space. If Y and C were used fr slcng, then by the same argument used n dervng Lemma 2.1, t s easy t see that Ž 4.1. x Ž EŽ x Y, C. EŽ x.. falls nt the jnt e.d.r. space. Ž Hwever, nstead f Y, C., we can nly bserve Y and. Ths suggests that Y and can be used smultaneusly fr slcng. Let Ž Y,0. EŽ d x Y y, 0., and Ž Y,1. EŽ x Y y, 1.. We may replace Ž 2.1. wth Ž 4.2. d Cv dž Y,. b x b, p.

10 10 K.-C. LI, J.-L. WANG AND C.-H. CHEN Ž Ž By cndtnng, E x Y, EEx Y, C. Y,... Thus frm Ž 4.1., we see that Ž y,. EŽ x. falls nt the jnt e.d.r. space. x d Ths justfes the use f egenvectrs frm Ž 4.2. t estmate the jnt e.d.r. space. The sample versn f Ž 4.2. s easy t carry ut. Dente the number f slces fr the uncensred Ž 1. bservatns by H 1. Let I 1j, j 1,..., H1 be a parttn f the pstve real lne nt nnverlappng ntervals. Smlarly, dente the number f slces fr the censred Ž 0. bservatns by H 0, and let I 0 j, j 1,..., H0 be anther parttn f the pstve real lne. We frst frm the ndvdual slce means by takng n lj Ž ˆlj. Ý Ž lj. x np x 1 l, Y I, where pˆlj s the prprtn f cases wth l fallng nt nterval I lj. Then we cmpute the cvarance matrx fr the slce means, ˆ ÝÝ p Ž d l jˆlj x lj x.ž x x.. Fnally we cnduct the egenvalue decmpstn lj Ž 4.3. ˆ ˆb ˆ ˆ ˆb, d d d x d ˆ ˆ. d1 L 1991 prpsed a ch-squared test fr determnng the number f sgnfcant e.d.r. drectns btaned by SIR. It shuld be clear that we can use the same test fr the duble slcng case. S far we have nly lcated the jnt e.d.r. drectns. We shall shw n the next sectn hw t use the prcedure n Sectn 3 t recver the e.d.r. lfetme drectns. Lkewse, we can als recver the e.d.r. drectns fr censrng tme by exchangng the rles f censrng tme and lfetme. Befre we prceed, an example s gven belw t llustrate the duble slcng prcedure dscussed n ths sectn. EXAMPLE 4.1. Take p 6 and let x Ž x,..., x. 1 6 be generated frm the standard nrmal dstrbutn. Suppse Y 4 x 1 1, 1 1 C fr x1 0, x2 x3 0, 10 therwse, where Here 1, 2 are nrmal randm varables. Generate 300 cases. Sxty-sx bservatns n the data set are censred. Nw apply duble slcng wth the number f slces equal t 5 and 10, respectvely, fr the censred and the uncensred grups. The egenvalues f SIR are fund t be 0.76, 0.35, 0.08, 0.06,..., ndcatng that the frst tw egenvectrs, ˆb d1 Ž 1.14, 0.05, 0.03, 0.00, 0.04, and ˆb Ž d2 0.06, 0.69, 0.74, 0.02, 0.10, are mprtant. Ths s cnfrmed by the ch-squared test n L Ž dp

11 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 11 The jnt e.d.r. drectns Ž 1, 0, 0, 0, 0, 0. and Ž 1 2.Ž 0,1,1,0,0,0. are captured successfully by ˆb and ˆ d1 b d2. The censred cases are fund t cluster n the frst quadrant n the plt f the frst tw SIR varates; see Fgure 2Ž. c. Statstcal nfrmatn abut the behavr f the true lfetme n that regn s very sparse. 5. Implementatn f mdfed SIR. The drectns fund by duble slcng can be used t releve the dffcultes encuntered n Sectn 3 when kernel smthng s t be appled fr estmatng the weght functn Ž Under the dmensn reductn assumptns fr bth the true lfetme and the censrng tme, Ž 1.2. and Ž 1.5., t s easy t see that the dependence f the weght functn Ž 3.3. nx s nly thrugh jnt e.d.r. varates. Ths suggests the fllwng tw-stage prcedure: 1. Apply duble slcng n Ž Y,. and fnd the jnt e.d.r. drectns, ˆb d. Let Bˆ Ž ˆb,..., ˆb. r d1 dr be the matrx frmed by the frst r sgnfcant drec- tns. 2. Apply r-dmensnal kernel smthng n Bˆ rx, t btan the weght functn w, ˆ 4 Ž 5.1. wˆ Ž t, t, x. exp ˆ Ž t, t x., where ' Ž 5.2. ˆ Ž t, t x. ž / n ˆ r n Ý S Y x h K h Bˆ : t Y t, YŽ. n r n rž x x. ˆf Ž x., FIG. 2. Three-dmensnal scatterplt f Y aganst the frst SIR varate Ž x-axs. and the secnd SIR varate Ž z-axs. fund by duble slcng. The hghlghted pnts are censred.

12 12 K.-C. LI, J.-L. WANG AND C.-H. CHEN n r ž ž ˆ // 5 j: Y j Y n r n r j ½ n Ý h K h B Ž x x. Ž 5.3. Sˆ YŽ Y x. max, c, ˆf Ž x. n r Ý n r n ˆ r Ž 5.4. fˆž x. n h K h B Ž Ž x x... Nte that a small pstve number c Ž set t 0.05 n ur examples. s used t bund Sˆ Ž Y x. Y away frm zer. Ths s needed n rder t ncrease the ˆ stablty f the factr S Y x n Ž 5.2. Y. After estmatng the weght functn, we can apply Ž 3.7. Ž 3.9. and then carry ut the egenvalue decmpstn Ž t btan estmates f e.d.r. lfetme drectns. We frst reprt tw smulatn studes t llustrate hw ths strategy wrks. Then we apply ur methd t a data set cncernng a study f prmary blary crrhss n the lver Ž PBC.. EXAMPLE 5.1. We take p 6 and generate x Ž x,..., x. 1 6 frm the standard nrmal dstrbutn. The true survval tme Y and the censrng tme C are generated frm lg lg 2 Y ; C, x1 x 2 where, are ndependent unfrm randm varables frm 0, Cndtnal n x, Y and C are seen t fllw the expnental dstrbutns wth the natural parameters, lnkng t x va x 1 ; x , respectvely. We btan 300 ndependent bservatns f Ž Y,.; amng them, 138 cases are censred. We prceed wth the SIR analyss. Frst, the methd f duble slcng n Y and as descrbed by Ž 4.3. gves egenvalues 0.34, 0.27, 0.05,.... The frst tw egenvectrs, ˆb Ž 0.67, 0.70, 0.08, 0.06, 0.11, d1 and ˆb Ž 0.69, 0.73, 0.12, 0.04, 0.10, d2, are clse t the jnt e.d.r. space fr Y and C. We use these tw drectns t reduce the x dmensn befre estmatng the weght functn wž,,.. Wth the weght adjustment gven by Ž 5.1., we perfrm SIR as descrbed by Ž 3.7. Ž t fnd the e.d.r. lfetme drectns. The egenvalues are 0.40, 0.10, 0.03,..., and the leadng egenvectr s ˆ b Ž 0.92, 0.12, 0.21, 0.08, 0.25, We see that ˆ 1 b1 s qute clse t the true e.d.r. lfetme drectn Ž 1, 0,...,0.. Fr cmparsn, we als carry ut the SIR analyss n Y wthut weght adjustment as f the censrng were ndependent f x. The frst drectn Ž 0.68, 0.69, 0.058, 0.07, 0.13, des have a substantal bas. Therefre the weght adjustment s crucal n ths example. We used the bvarate nrmal kernel functn here and the bandwdth s set at The senstvty t the bandwdth chce seems mld. EXAMPLE 5.2. Imprtant prgnstc varables affectng the hazard rate may be dfferent at dfferent survval stages. In ths example, we assume that the true survval tme Y fllws an expnental dstrbutn wth the

13 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 13 TABLE 1 The frst three egenvectrs and egenvalues f SIR fr Example 5.2 wth the duble slcng prcedure Frst vectr Ž 0.93, 0.11, 0.03, 0.03, 0.04, Secnd vectr Ž 0.09, 0.76, 0.60, 0.01, 0.03, Thrd vectr Ž 0.10, 0.55, 0.73, 0.03, 0.02, Egenvalues Ž 0.52, 0.21, 0.15, 0.03, 0.01, natural parameter equal t 2 x 1 untl tme lg 2. Frm tme n, the addtnal survval tme fllws the expnental dstrbutn wth the natural parameter 3 x 2. Mre specfcally, we assume Y * expnental wth parameter 2 x 1, Y ** expnental wth parameter 3 x 2, Y Y *1Ž Y *. Ž Y **. 1Ž Y *.. The censrng tme C fllws an expnental dstrbutn wth parameter equal t x 3. Agan 300 ndependent bservatns f Ž Y,. are btaned. Amng them, 98 cases are censred. The utput f the duble slcng prcedure s gven n Table 1. The frst three egenvectrs, whch have relatvely larger egenvalues cmpared t the rest, are then used n estmatng the weght functn fr fndng the true e.d.r. lfetme drectns. After the weght adjustment, the fnal utput f SIR s gven n Table 2. Nw we see that nly the frst tw egenvectrs stand ut and the mprtant varables x1 and x2 can be dentfed. EXAMPLE 5.3. The PBC data set cllected at the May Clnc between 1974 and 1986 has been analyzed n the lterature. The data set and a detaled descrptn can be fund n Flemng and Harrngtn Ž There are rgnally seventeen regressrs. Flemng and Harrngtn selected fve f them n ther fnal equatn fr fttng a Cx prprtnal mdel. These fve regressrs plus anther varable, the platelet cunt Ž x belw. 5, wll be used n ths llustratn: Y number f days between regstratn and the earler f death r censrng; 1f Y s due t death; 0 therwse; x age n years; 1 x presence f edema; 2 x serum blrubn, n mg dl; 3 x albumn, n gm dl; 4

14 14 K.-C. LI, J.-L. WANG AND C.-H. CHEN TABLE 2 The frst tw egenvectrs and egenvalues f SIR fr fnal result f Example 5.2 wth weght adjustment Frst vectr Ž 0.97, 0.15, 0.10, 0.04, 0.10, Secnd vectr Ž 0.16, 0.95, 0.18, 0.02, 0.06, Egenvalues Ž 0.66, 0.34, 0.05, 0.04, 0.02, x platelet cunt; 5 x prthrmbn tme. 6 Cases wth mssng values are gnred and there are 308 cases remanng. We frst apply duble censrng wth slce numbers H1 H0 10. The frst tw drectns are sgnfcant, as judged frm the sequence f utput egenvalues 0.55, 0.15, 0.05, 0.0, 0.0, 0.0. Fgure 3 shws the scatterplt f the frst tw SIR varates. Tw utlers labeled as 104 and 276 are fund frm the three-dmensnal plt Ž nt shwn here. f Y aganst the frst tw SIR varates. They are remved. We apply duble slcng agan t the remanng 306 cases. The SIR utput essentally remans the same. Ths suggests that the dmensn f the jnt e.d.r. space s tw. We prceed t fnd the true e.d.r. lfetme drectns. We take r 2 and use the tw SIR drectns reprted n Table 3 t reduce the x dmensn befre estmatng the weght functn. The kernel functn and the bandwdth are the same as n Example 5.1. The utput f the weghted SIR s gven n Table 4. Judgng frm the egenvalue sequence, the frst drectn ˆ b s 1 FIG. 3. Scatterplt f the frst tw SIR varates fund by duble slcng. bserved, square censred cases.

15 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 15 TABLE 3 The frst tw egenvectrs and egenvalues f SIR fr the PBC data n Example 5.3 Frst vectr Ž 0.02, 1.04, 0.10, 0.50, 0.00, Secnd vectr Ž 0.02,.62, 0.17, 0.97, 0.00, Egenvalues Ž 0.54, 0.16, 0.05, 0.01, 0.00, clearly mprtant. The secnd drectn s als wrth further examnatn. Fgure 4Ž a. and Ž b. shw the scatterplts f Y aganst ˆ b1x and aganst ˆ b2x. Earler analyss n Flemng and Harrngtn Ž yelds that the true lfetme depends n x thrugh the varate Q x x lg x lg x lg x 6. Ths varate turns ut hghly ˆ crrelated wth the frst SIR varate b x; the crrelatn ceffcent s ' The crrelatn between Q and 1.3b ˆ x 0.25b ˆ x s equal t ' Varable x5 makes very lttle cntrbutn t the frst tw SIR varates, wth a squared multple crrelatn f nly Ths s cnsstent wth Flemng and Harrngtn s fndng that platelet cunt s nt mprtant. Fnally, we estmate the censrng e.d.r. drectns by reversng the rles f censrng tme and the true lfetme. Ths amunts t replacng wth 1 thrughut ur estmatn prcedure. The utput s gven by Table 5 and Fgure 5. The assumptn f ndependent censrng Ž 1.4. s seen t be nvald fr ths data set. We further ntce that the frst censrng tme drectn s qute clse t the frst lfetme drectn. The crrelatn ceffcent between the frst lfetme SIR varate and the frst censrng SIR varate turns ut t be ' Sme cautn needs t be taken regardng the desgn cndtn. Of specal cncern s the secnd regressr Ž presence f edema. whch s dscrete and takes nly three values Ž 0, 0.5, 1.. Nevertheless, the crrespndng regressn ceffcent frm Table 4 s 0.90, whch s qute clse t the ceffcent based n the Cx prprtnal hazard mdel. A further study wuld be t carry ut anther SIR analyss by fcusng n the grup wth x2 0. The ther grups have nly 29 and 19 cases and thus t s nt feasble t carry ut separate analyses fr them. TABLE 4 The frst tw egenvectrs and egenvalues f the lfetme SIR drectns fr the PBC data n Example 5.3 Frst vectr Ž 0.02, 0.90, 0.09, 0.62, 0.00, Secnd vectr Ž 0.03, 2.3, 0.20, 0.28, 0.00, Egenvalues 0.54, 0.16, 0.05, 0.02, 0.01, 0.00

16 16 K.-C. LI, J.-L. WANG AND C.-H. CHEN FIG. 4. cases. Scatterplt f Y aganst the frst tw lfetme SIR varates. bserved, dt censred REMARK 5.1. In bth f ur smulatn examples, we take p 6. As the regressr dmensn p gets larger, the prblem certanly gets harder and ne mght expect the perfrmance f ur prcedure t deterrate as well. T study ths effect, we vary p frm 6 t 10, 15 and 20. The sample sze s kept the same, n 300. Fr each smulatn run, we cmpute an R-squared term fr evaluatng hw clse t the true e.d.r. lfetme drectns the estmated drectns are. Fr the set-up f Example 5.1, whch has nly ne true e.d.r. lfetme drectn, the R-squared term s smply the squared crrelatn ceffcent between ˆ b1 x and x. Snce x x 1, the R-squared term s equal t the square f the frst crdnate f ˆ b. Table 6 Ž left sde panel. 1 gves a summary f the R-squared values fr 100 smulatn runs n each case. Fr cmparsn, the R-squared values fr the SIR estmate wthut the weght adjustment are gven n the rght sde panel. We can see that the mprvement fr the mdfed SIR prcedure s stll substantal fr p as large as 20. The set-up f Example 5.2 has tw true e.d.r. lfetme drectns. Fr the frst mdfed SIR drectn, the R-squared term s just the R-squared value fr regressng ˆ b1 x aganst x and 2x lnearly. Ths s equal t the sum f the square f the frst tw ceffcents n ˆb. The R-squared value fr the secnd mdfed SIR drectn s defned smlarly. A summary fr 100 smulatn runs s gven n Table 7. TABLE 5 The frst tw egenvectrs and egenvalues f the censrng tme SIR varates fr the PBC data n Example 5.3 Frst vectr Ž 0.01, 1.43, 0.05, 0.42, 0.00, Secnd vectr Ž 0.02, 0.38, 0.15, 1.22, 0.00, Egenvalues 0.39, 0.22, 0.05, 0.03, 0.01, 0.00

17 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 17 FIG. 5. Scatterplt f Y aganst the frst tw censrng tme SIR varates. bserved, dt censred bservatns. 6. Cnclusn. We have demnstrated hw t extend the dmensn reductn methd f slced nverse regressn Ž SIR. t censred data. The extensn s straghtfrward f censrng tme s ndependent f the regressr. SIR can be appled t the bserved data Ž Y, x. drectly. Hwever, f censr- ng tme depends n the regressr, then SIR needs t be mdfed. We TABLE 6 Perfrmance f mdfed SIR as the number f regressrsž p. ncreases under the settng f Example 5.1 wth 100 runs Mean ( standard devatn) fr R p Mdfed SIR Orgnal SIR Ž Ž Ž Ž Ž Ž

18 18 K.-C. LI, J.-L. WANG AND C.-H. CHEN TABLE 7 Perfrmance f mdfed SIR as the number f regressrsž p. ncreases under the settng f Example 5.2 wth 100 runs Mean ( standard devatn) fr R p Frst mdfed SIR drectn Secnd mdfed SIR drectn Ž Ž Ž Ž Ž Ž Ž Ž ntrduce a weght functn n estmatng the slce means. The estmatn f the weght functn requres nnparametrc smthng. There are tw ptns. The frst ne s t apply the kernel smthng methd f Sectn 3. Ths s feasble nly f the number f regressrs s small Ž e.g., p 3. rfthe sample sze s substantally large. The ther ptn, whch seems mre realstc, s the tw-stage prcedure f Sectn 5. We cnduct a duble slcng SIR frst t reduce the dmensn f x befre applyng kernel smthng. Ths tw-stage prcedure reles n cndtn Ž 1.5., whch assumes that the censrng varable als has a dmensn reductn structure wth respect t the regressr. Ths assumptn appears reasnable and t ffers the pssblty f examnng the censrng pattern vsually. The man feature that dstngushes ur apprach frm mst ther methds n survval analyss s that t des nt requre the estmatn f g at the dmensn reductn stage f data analyss. Instead, after the dmensn s reduced, the estmatn f g can be pursued by applyng any lw-dmensnal smthng methds. Furthermre, ur apprach can be used t check f a ppular survval mdel s apprprate by examnng the egenvalues and the lw-dmensnal plts generated by SIR. These plts prvde valuable nfrmatn abut the general pattern f censrng, pssble presence f utlers and the shape f the regressn surface. Imputatn s a pwerful way f dealng wth the ncmplete censred bservatn. We can mpute the censred Y bservatn frst and then apply the SIR methd n L Ž drectly t the mputed data. One pssble mputatn methd s gven n Fan and Gjbels Ž Whle ther methd s effectve fr ne r tw regressrs, t s nt apprprate n the hgher-dmensnal stuatn. A feasble alternatve s frst t apply the dmensn reductn methd as utlned n ths artcle and then apply mputatn t the reduced varables. Ths prspect merts further study. The prf f rt n cnsstency as utlned n the Appendx can perhaps be mprved wth less strenuus assumptns. Whle ths requres further theretcal nvestgatn, t shuld nt affect the applcablty f the prcedure prpsed here.

19 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 19 APPENDIX ( ) A. Dervatn f 3.6. It suffces t shw that 1Ž t Y, 1. A.1 S t x exp E x. S Ž Y x. ½ 5 Y Frst, the cndtnal ndependence assumptn Ž 1.3. mples that S Ž y x. Y S Ž y x. S Ž y x., where S Ž y x. PC y x 4 C C. Usng ths relatnshp, the expectatn term n Ž A.1. can be wrtten as 1 Ž t Y, 1. 1 Ž t Y. 1 Ž Y C ½ 5 ½. S Ž Y x. S Ž Y x. S Ž Y x. 5 Y C E x E x ½ 5 C 1Ž t Y. E E Ž 1Ž Y C. x, Y. x S Ž Y x. S Ž Y x. Ž Ž.. Ž By 1.3 agan, we have E 1 Y C x, Y S Y x. C. The last expressn s seen t becme 1 Ž t Y E½. x S Ž Y x. 5 The rest f the dervatn s straghtfrward frm the relatnshp between the hazard and the survval functns. B. Prf f Lemma 3.1. T btan the rt n cnsstency fr m gven by Ž 3.7. usng the kernel estmates Ž Ž 3.13., sme regularty cndtns wll be mpsed. Let w h p K h Ž x x., j n p j 4 u w Ew x. j j j j ˆ We frst requre that the bas term f f x s f the rt n, 4 Ž B.1. Ew x f Ž x. O Ž n 2.. j j j p The trade-ff fr mpsng a smaller bas s the ncreasng f the varance, but by averagng ut many pnt estmates ver an nterval, the varance wll eventually reman small. The bas term, n the ther hand, s harder t cancel ut. T ensure Ž B1., we need t use a bandwdth smaller than the usual ptmal rate. Wth Ž B.1., we can wrte ˆ Ý k p 2 k Ž B.2. f Ž x. f Ž x. n u O Ž n.. The rate f cnvergence fr the term cntrbutng t the varance s mre flexble. We need nly assume that Ž B.3. n u O Ž n 4.. Ý k k p ˆ h

20 20 K.-C. LI, J.-L. WANG AND C.-H. CHEN Next we als assume that the bas fr the kernel estmate f S als has the rt n rate 2 B.4 E 1 Yk Yj wkj x j, Yj SY Yj x j f x j Op n. Y Ž t x. fž x. Typcally wth sutable smthness cndtns n S Ž t x. Y, the same bandwdth used t acheve Ž B.1. may als mply Ž B.4.. Dente vkj 1Ž Yk Yj. wkj E 1Ž Yk Yj. wkj x j, Y j. Smlarly t B.3, we assume Ý kj k Ž B.5. n v O Ž n 4.. By Taylr s expansn, we can fnd the leadng terms fr the term Sˆ Ž Y Yj x., j Ž B.6. Sˆ Ž Y x. fˆž x. S Ž Y x. f Ž x. n Ý v O Ž n. 2 Y j j j Y j j j kj p k p ž / ž fˆ x f x S Y x Ž j. Ž j. YŽ j j j YŽ j j. Ý kj p / k f Ž x. S Y x n v O Ž n. 2 YŽ j j. Ž j. YŽ j j. Ý kj k S Y x f x S Y x n v Ý 2 j Y j j kj p k f Ž x. S Y x n u O Ž n.. The last expressn s btaned frm Ž B.2. and the assumptns Ž B.3. and Ž B.5.. T prceed, let us smplfy the ntatn by takng f f Ž x., SYŽ Yj x j. S j,1jž Y Yj t, j 1., Ž Y, t x., ˆ ˆŽ Y, t x.. Nw apply Ž B.2. and Ž B.6. and expand the term ˆ : Ž B.7. ˆ fˆž x. n 1 Sˆ Ž Y x. w Ý j Y j j j j f n Ý 1 S w j j j j Ý Ý f n 1 w f S 2 n v j j j j kj j k

21 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 21 Ý Ý f n 1 w f S n u j j j j kj j k ž Ý /ž Ý / f 2 n u n 1 S w O Ž n 2.. k j j j p k j The frst term wll cnverge t the cumulatve hazard. Agan, under sutable smthness and bundedness cndtns n the hazard functn, the same bandwdth used befre shuld gve a bas term at the rt n rate. We shall assume that 2 B.8 E 1jSj wj x, Y f Op n. Ž Let 1 S w E 1 S w x, Y. j j j j j j j, and dente the secnd, thrd and furth terms n the rght sde f Ž B.7. as Ž I., Ž II., Ž III., respectvely. By Ž B.8., we can rewrte Ž B.7. as ˆ 2 Ý j p j f n Ž I. Ž II. Ž III. O Ž n.. Dente 1 1 Y t, 0,w. We can expand the secnd term n the rght sde f equatn Ž 3.8. t Ž B.9. Ý Ý ˆ ˆ 4 n 1 x w Y, t, x n 1 x exp Ý Ý n x 1 w n 2 x 1 wf j, j n Ý x 1w Ž I. Ž II. Ž III.. Ž 2 It remans t shw that the secnd and thrd terms are O n. p. Abbrevate the cndtnal expectatn E x, Y by E. Nte that we Ž p have E 0, E 0, var Oh. j j j n. The secnd term takes the frm f n 2 Ý a wth a x 1 wf, j j. T evaluate ts varance, we frst bserve that EŽ a ja j. 0 f j j OŽ 1. f j j, O h p n f, j j. Frm ths, a straghtfrward calculatn leads t ž / B p var n Ý a j n n O 1 n OŽ h n. O n., j The varance fr the thrd term n Ž B.9. can als be evaluated smlarly. We 2 can rewrte n Ý x 1 w I as n Ý av, wth where j, k j kj a j n Ý x 1 wf 1 f S 2 j j j w j, E v j 0, var v O h p. kj kj n Frm ths expressn, we can calculate ts varance and btan a result Ž 2. Ž smlar t B.10 : var n Ý av On.. The calculatn fr the var- j, k j kj

22 22 K.-C. LI, J.-L. WANG AND C.-H. CHEN ance f n Ý x 1 w Ž II. can be carred ut n exactly the same way. Fnally, t deal wth the term n Ý x 1 w Ž III., we express t as n 2 Ý, kau k wth 2 a x 1wf n Ý j1jsj w j. Then agan, by the same argument, the var- ance s shwn t have the rder f n. We have nw cmpleted the prf fr the rt n cnvergence fr Ž The prf fr Ž 3.9. s the same. Therefre, mˆ h s rt n cnsstent, as clamed n Lemma 3.1. REFERENCES BERAN, R. Ž Nnparametrc regressn wth randmly censred survval data. Techncal reprt, Unv. Calfrna, Berkeley. BRILLINGER. Ž Cmment n Slced nverse regressn fr dmensn reductn, by K. C. L. J. Amer. Statst. Assc CARROLL, R. J. and LI, K. C. Ž Measurement errr regressn wth unknwn lnk: dmensn reductn and data vsualzatn. J. Amer. Statst. Assc CARROLL, R. J. and LI, K. C. Ž Bnary regressrs n dmensn reductn mdels: a new lk at treatment cmparsns. Statst. Snca CHEN, C. H. and LI, K. C. Ž Can SIR be as ppular as multple lnear regressn? Statst. Snca COOK, R. D. Ž On the nterpretatn f regressn plts. J. Amer. Statst. Assc COOK, R. D. and NACHTSHEIM, C. J. Ž Re-weghtng t acheve ellptcally cntured cvarates n regressn. J. Amer. Statst. Assc COOK, R. D. and WEISBERG, S. Ž Cmment n Slced nverse regressn fr dmensn reductn, by K. C. L. J. Amer. Statst. Assc COOK, R. D. and WEISBERG, S. Ž An Intrductn t Regressn Graphcs. Wley, New Yrk. DABROWSKA, D. M. Ž Nn-parametrc regressn wth censred survval tme data. Scand. J. Statst DABROWSKA, D. M. Ž Varable bandwdth cndtnal Kaplan Meer estmate. Scand. J. Statst DIACONIS, P. and FREEDMAN, D. Ž Asympttcs f graphcal prjectn pursut. Ann. Statst DOKSUM, K. A. Ž An extensn f partal lkelhd methds fr prprtnal hazard mdels t general transfrmatn mdels. Ann. Statst DOKSUM, K. A. and GASKO, M. Ž On a crrespndence between mdels n bnary regressn analyss and n survval analyss. Internat. Statst. Rev DUAN, N. and LI, K. C. Ž Slcng regressn: a lnk-free regressn methd. Ann. Statst FAN, J. and GIJBELS, I. Ž Censred regressn: lcal lnear apprxmatns and ther applcatns, J. Amer. Statst. Assc FLEMING, T. R. and HARRINGTON, D. P. Ž Cuntng Prcesses and Survval Analyss. Wley, New Yrk. HALL, P. and LI, K. C. Ž On almst lnearty f lw dmensnal prjectn frm hgh dmensnal data. Ann. Statst HSING, T. and CARROLL, R. J. Ž An asympttc thery fr slced nverse regressn. Ann. Statst HUBER, P. Ž Prjectn pursut Ž wth dscussn.. Ann. Statst LI, K. C. Ž Slced nverse regressn fr dmensn reductn Ž wth dscussn.. J. Amer. Statst. Assc LI, K. C. Ž Uncertanty analyss fr mathematcal mdels wth SIR. In Prbablty and Statstcs Ž Z. P. Jang, S. H. Yan, P. Cheng and R. Wu, eds Wrld Scentfc Press, Sngapre.

23 DIMENSION REDUCTION FOR CENSORED REGRESSION DATA 23 LI, K. C. Ž Nnlnear cnfundng n hgh dmensnal regressn. Ann. Statst MCCULLAGH, P. and NELDER, J. A. Ž Generalzed Lnear Mdels, 2nd. ed. Chapman and Hall, Lndn. SCHOTT, J. R. Ž Determnng the dmensnalty n slced nverse regressn. J. Amer. Statst. Assc ZHU, L. X. and NG, K. W. Ž Asympttcs f slced nverse regressn. Statst. Snca K.-C. LI DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA LOS ANGELES, CALIFORNIA kcl@math.ucla.edu J.-L. WANG DIVISION OF STATISTICS UNIVERSITY OF CALIFORNIA DAVIS, CALIFORNIA C.-H. CHEN INSTITUTE OF STATISTICAL SCIENCE ACADEMIA SINICA TAIWAN