Change-Point Analysis of Survival Data with Application in Clinical Trials

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1 Ope Joural of Statstcs, 4, 4, Publshed Ole October 4 ScRes. Chage-Pot Aalyss of Survval Data wth Applcato Clcal Trals Xua Che, Mchael Baro,3 Departmet of Bostatstcs ad Programmg, Saof, Bejg, Cha Departmet of Mathematcs ad Statstcs, Amerca Uversty, Washgto DC, USA 3 Departmet of Mathematcal Sceces, Uversty of Texas at Dallas, Rchardso, USA Emal: Xua.Che@saof.com, baro@amerca.edu, mbaro@utdallas.edu Receved 7 July 4; revsed 5 August 4; accepted 6 September 4 Copyrght 4 by authors ad Scetfc Research Publshg Ic. Ths work s lcesed uder the Creatve Commos Attrbuto Iteratoal Lcese (CC BY). Abstract Effects of may medcal procedures appear after a tme lag, whe a sgfcat chage occurs subjects falure rate. Ths paper focuses o the detecto ad estmato of such chages whch s mportat for the evaluato ad comparso of treatmets ad predcto of ther effects. Ulke the classcal chage-pot model, measuremets may stll be detcally dstrbuted, ad the chage pot s a parameter of ther commo survval fucto. Some of the classcal chage-pot detecto techques ca stll be used but the results are dfferet. Cotrary to the classcal model, the maxmum lkelhood estmator of a chage pot appears cosstet, eve presece of usace parameters. However, a more effcet procedure ca be derved from Kapla-Meer estmato of the survval fucto followed by the least-squares estmato of the chage pot. Strog cosstecy of these estmato schemes s proved. The fte-sample propertes are examed by a Mote Carlo study. Proposed methods are appled to a recet clcal tral of the treatmet program for strog drug depedece. Keywords Chage-Pot Problem, Falure Rate, Kapla-Meer Estmato, Least Squares Estmato, Maxmum Lkelhood Estmato, Strog Cosstecy, Survval Fucto. Itroducto Chage-pot models studed clcal research usually refer to chages the falure rate. May artcles ad clcal reports descrbe stuatos whe after a certa survval perod, the falure rate s expected to chage due to the treatmet or durg the after-treatmet recovery. Detecto of such chages, ther estmato, ad ther How to cte ths paper: Che, X. ad Baro, M. (4) Chage-Pot Aalyss of Survval Data wth Applcato Clcal Trals. Ope Joural of Statstcs, 4,

2 X. Che, M. Baro comparso betwee dfferet groups of patets (the treatmet arm ad the placebo arm s the classcal example) s mportat uderstadg the treatmet s effect ad for the evaluato of the treatmet s success. For example, durg the zoster pa resoluto tral [], the treatmet lghtes pa from acute to subacute ad the to chroc, resultg three dfferet falure rates. As aother example, [] descrbes aalyss of the Physca s Health Study for testg the effect of beta-carotee o cacer cdece. New tumors eed tme to become detectable whle the treatmet does ot affect pre-exstg tumors. Thus, there s a approxmately two-year watg perod before the effect of the treatmet s otceable. Survval tmes ths example have a hgher tal falure rate ad a lower falure rate afterwards. Smlar examples are foud [3]-[9]. Survval data wth a chage pot are descrbed by two models for the falure rate, amely, oe model before the chage pot ad the other model after the chage pot. Whe a subject passes the chage pot, the falure rate typcally reduces, ad the probablty of the overall survval creases. Ths stuato s coceptually ad mathematcally dfferet from the classcal chage-pot model, see e.g. []-[4], where observatos follow oe dstrbuto before the chage pot ad aother dstrbuto after t. I the descrbed scearo, wth oe or several chages the falure rate, all the subjects are assumed to have the same dstrbuto. Each chage pot s uderstood as a parameter of ths dstrbuto that separates two patters, two dfferet models for the falure rate, ad typcally, t s the momet of a clcally sgfcat reducto of the falure rate. Despte the fudametal devato from the classcal chage-pot model, we wll show that classcal methods for the stadard chage-pot aalyss ca be to a certa extet appled to the survval data. Developg these methods, we ca also accout for the rght cesorg that s typcal for survval data. The goal of ths paper s to fd effcet chage-pot detecto methods for the pecewse costat falure rate models [5] [6] [8] [5] wth ukow pre-chage ad post-chage parameters. Maxmum lkelhood estmato of the chage pot presece of usace parameters s revewed; t appears cosstet uder certa codtos. A ew alteratve estmato procedure s proposed based o Kapla-Meer estmato of the survval fucto [6] followed by the least-squares estmato of the chage pot. For ths scheme, strog cosstecy of all the estmators s establshed. Ths s a rather costtutve dstcto from the classcal chage-pot models where cosstet estmato of the chage-pot parameter s ot possble. Developed methodologes are appled to the recet clcal tral of the treatmet program for methamphetame depedece coducted by Research Across Amerca Dallas TX [7]. Partcpats of ths tral were characterzed by strog addcto to methamphetame, ad the crtcal measure of effcacy was ther tme utl relapse. Proposed methods show sgfcat chage pots the survval fucto for both cotrol ad treatmet groups although the chage the treatmet group occurs earler, about two weeks after recevg the treatmet. I smple words, t appears that f a regular user of methamphetame stays away from the drug for two weeks after startg the treatmet program, the probablty of relapse o ay day thereafter reduces sgfcatly. Ths fdg has a rather sgfcat clcal meag. The rest of the paper s orgazed as follows. The falure rate chage-pot model s troduced Secto. I Secto 3, we gve a bref revew of maxmum lkelhood estmate ad ts propertes. We propose a alteratve least square estmator, fd ts covergece rate, ad prove ts strog cosstecy Secto 4. I Secto 5, we exted the strog cosstecy of the least square estmator to a more geeral model, Cox proportoal hazard model wth a chage pot. We compare the two estmato procedures by meas of a smulato study Secto 6. Secto 7 shows applcato of these methods to the Prometa clcal tral. Cocluso s gve Secto 8. Proofs of theorems, lemmas, ad corollares are the Appedx secto.. Survval Models wth Chage Pots We assume a costat falure rate fucto λ( x) λ ad λ ( x) shfts to a ew value λ( ) = λ x + λ = utl a ukow tme. Chage occurs at tme, λ ad remas at t thereafter. Thus, x x> () where λ >, λ >, ad s the chage pot, the ma parameter of terest. Cosder a sample of depedet subjects wth the falure rate fucto λ ( x). Let X deote the survval tme of subject. Survval data are ofte subject to radom rght-cesorg. If the survval tme X s cesored at tme C, the varable 664

3 { } X = m X, C X. Che, M. Baro s observed stead of X. I practcal clcal studes, rght-cesored survval tmes are rather commo due to the early termato of the observato perod or due to patets wthdrawals from the clcal tral. The dcator varable f X C δ = = X C f X > C X δ for =,,,. Ce- sorg varables C are assumed to be depedet of X. Matthews (98) ad Worsley (988) dscuss the effect of radom cesorshp. Throughout the paper, deotes the true value of the chage pot; Λ ( ) s the log-lkelhood rato gve the occurrece of a chage pot at ; ˆ, ˆλ, ad ˆλ are the maxmum lkelhood estmators of, λ, ad λ, respectvely; ad, λ, ad λ are the proposed least squares estmators of, λ, ad λ, respectvely. wll show whether the th survval tme s cesored. The, we observe pars (, ) 3. Maxmum Lkelhood Estmato Uder model (), the lkelhood fucto of X,, X s { } x ( λ λ ) = λ( ) λ( ) λ x F x,, x,, x exp t dt whch yelds the log-lkelhood rato where ( ) ( ) ( ) λ λ x λ x λ λ x λ x λ x > x x > δ = δ = ( ) = e + e e + e, λ ( ) Λ = δ log ( x )( λ λ) x ( )( )( ) > + δ x λ λ x y > = () λ λ log + ( λ λ)( x ) for < x ad δ = λ y = ( λ λ)( x ) for < x ad δ = otherwse Whe λ ad λ are kow, Λ s lear betwee ay two cosecutvely observed survval tmes, ad thus, ts maxmum s attaed at some observed survval tme X, whch equals, say, the k th ordered survval tme, X ( k ). For all j k, the value of y ( j) correspodg to the order statstc X ( j) s. Hece the maxmum lkelhood estmator for the chage pot s y s lear, ( ) Whe λ ad λ are ukow, [8] shows that ˆ = X ˆ ( ˆ, where k = arg max y ) ( ). k (4) k = k+ { X( ) X( ) X( ) X ( ) X( ) X ( ) } ˆ,,,,...,,, where ˆ = X ( ) meas that the maxmum lkelhood s attaed as approaches X ( r) from below, ad also proves that ˆ, ˆ λ ˆ, λ are cosstet. The effect of radom cesorshp has bee studed by may authors. [6] have suggested that moderate cesorshp has lttle mpact o the ull dstrbuto of the lkelhood rato, based o smulato results for type I cesorg. [5] have proved that the exact dstrbutos of test statstcs uder the ull hypothess rema uchaged for type II cesorg. For other forms of oformatve cesorg [9] have show that the asymptotc ull dstrbutos of lkelhood rato statstcs geeral rema uchaged. (3) 665

4 X. Che, M. Baro 4. Least Squares Method Based o Kapla-Meer Estmato I ths secto, we troduce a dfferet chage-pot estmato procedure whch s based o Kapla-Meer estmator of the survval fucto. Sce the Kapla-Meer method s oparametrc, the chage-pot estmato scheme proposed here ca be easly exteded to a wde varety of survval models wth chage pots arsg clcal trals ad other applcatos. Kapla ad Meer (958) proposed a famous estmator for the survval fucto S( t ) : x( j) x δ( j) j S ( x) =. (5) j+ Ths s a step fucto wth jumps at observatos X for whch δ =. It s a oparametrc estmator of the survval fucto, ad t ca be appled presece of cesorg. No assumptos are requred for the probablty dstrbuto other tha the depedece betwee the survval ad cesorg varables. Kapla-Meer estmator (5) has the followg propertes: ) It s the oparametrc maxmum lkelhood estmator of the true survval fucto S( x ). ) It has a asymptotcally ormal dstrbuto for ay x where S( x ) s cotuous. 3) It coverges almost surely to S( x ) uformly x, ad for each >, there exsts c >, such that S x S x P > for suffcetly large. Refer to [] for detals. ( ( ) ( ) ) e c 4) If o cesorg occurs or all varables are cesored at the same tme, the the Kapla-Meer estmator reduces to the usual emprcal dstrbuto fucto. 4.. Least Squares Estmato ad Strog Cosstecy Uder the pecewse costat falure rate model () wth a chage pot, the logarthm of the survval fucto at the tme x s gve as Let θ ( λ, λ, ) ( λ λ) = ( ) = λ ( λ + λ( ) ) L,, log S x x x. x x> = deote the vector of parameters. Its least squares estmator θ = (, λ, λ) those values of, λ, ad λ that mmze the error sum of squares where ( θ) = ( y ( ) ( )) x L θ = cossts of ESS, (6) j y ( x ) = logs ( x ) = δ log. (7) j+ ( j) x( j) x Lemma. At θ = θ, the error sum of squares compoets satsfy the strog law of large umbers; that s, ESS ( ) θ coverges to almost surely, as. The proof ca be foud the Appedx. To prove the strog cosstecy of the vector of least squares estmators, λ, λ terms of the resdual ( x) ESS α, ( θ) = ( y ( x) + λx) + ( y ( x) + λ + λ( x ) ) x x> ( ( )) ESS θ, we express ( ) ( ( ) ( ) ( )) = λ x + λ x + α x + λ λ x + λ+ λ x + α x x m (, ) x max (, ) < x < x ( ) ( λox λ λ( x ) α( x) ) λ o λ( x ) + λx + α( x) = A + B + C + D (8) 666

5 X. Che, M. Baro where ( λ λ ) A = x + x X m (, ) ( λ λ ( ) λ λ ( ) ) B = x + + x X max (, ) ( λ λ λ( ) ) C = x + + x < X The uform covergece of ( x) ( λ λ( ) λ ) D = x + x < X, α ad the strog law of large umbers [] mply drectly that,., A, (9) B, () C, () D. () Sce we assume that there s deed a chage-pot, t s reasoable to make the followg assumpto. Assumpto (A): There exst kow < m< M such that [ mm, ]. Assumpto (A) s a classcal assumpto the case whe a chage pot s estmated presece of usace parameters, ad t esures that samples of a suffcet sze are used to estmate the usace parameters. Uder Assumpto (A), the least squares estmator s defed as the mmzer of ESS( θ ) over the gve terval, therefore, [ mm, ]. Theorem. λ s strogly cosstet for λ uder Assumpto (A). The proof ca be foud the Appedx. Theorem. s strogly cosstet for uder Assumpto (A). Proof. ) We wll prove P ( > ) = ths part. We prove by cotradcto. Suppose for ay >, there exst N such that From Theorem ad (), we get From (3), we have for all N( ) Also, >. Hece, for suffcetly large, whch cotradcts (4). δ > ad ( ) ( > ε > ) > δ for all > N( ). P (3) P ( x ) ( λ λ). < X > > δ ( x ) ( x ) < X < X < X < X ( x ) ( X ) E >. ( x ) ( x ) ( ) X, X X P > > E < > > δ < < X (4) 667

6 X. Che, M. Baro ) We wll prove P ( ) = ths part. We also prove ths by cotradcto. Suppose for ay >, there exst for all N( ) The >. for all N( ) Also, Hece, >. ( ) δ > ad ( ) N such that P > > δ (5) ( x ) ( x ) X P > + > δ < < X ( x ) ( X ) + E + < X >. < X From () ad Theorem, we ca get Hece whereas Hece ( ) P x > > δ for. < X ( ) P λ λ > δ for. ( ) ( ( ) ) P ESS θ λ λ x + λx > δ, = ( λ λ( x ) + λx) = ( x ) ( λ + λ) = = ( λ λ ) ( x ) + E >. X > ( ) ( ( ) ) P ESS θ > λ λ x + λx > > δ for suffcetly large, whch cotradcts Theorem. Combg ) ad ) gves ( ) = Theorem 3. λ s strogly cosstet for λ uder Assumpto (A). The proof ca be foud the Appedx. 4.. Covergece Rate of the Least Squares Estmator P. Now let us vestgate the covergece rate of for kow λ ad λ. We wll aalyze the probablty that ESS( ) s less tha ESS( ) for outsde of the -eghborhood of, where s the true value of the chage pot. Theorem 4. For ay >, there exsts c >, such that for suffcetly large. P : > ε ESS ESS < e c { ( ) ( ) } 668

7 X. Che, M. Baro The proof ca be foud the Appedx. Corollary. The chage-pot estmator s strogly cosstet; almost surely as. I partcular, for ay >, there exsts c > such that ( ) P > e c for suffcetly large. Proof. Accordg to Theorem 4, for ay arbtrary sequece j >, j c such that ( j ) P( α j ) e. Hece c( j ) ( ) e { ESS( ) ESS( ) } c j P < e = <. c( j ) = : > j = e as j, there exsts c ( ) > Sce the sum of probabltes coverges, by the Borel-Catell lemma, wth probablty oe, ESS( ) ESS( ) for all : for suffcetly large. Therefore,, the mmzer of ESS( ), belogs to the j -eghborhood of almost surely ad all suffcetly large. It remas to let j go to zero over a coutable set (e.g., j = j ). For each j, we obta that j almost surely. Therefore, a.s., as. 5. Least Squares Method for the Cox Proportoal Hazard Model wth a Chage Pot Geeralzg the prevous results, ths secto we develop chage-pot estmato techques for a more geeral model, Cox proportoal hazard model wth a chage pot. Uder ths model, the hazard rate fucto has the form, h xz = h x exp β Z + h x exp β Z (6) where Z s a vector of covarates ( z z ) ( ) ( ) ( ) x ( ) ( ) x> are vectors of coeffcets, ad ( ), ( ),, k, β, β h x h x are basele hazard rates. Clearly, a model wth covarates allows to study effects of umercal ad categorcal factors o the occurrece of a chage pot ad to compare chage pots betwee subpopulatos. It s well kow that Cox proportoal hazard model s semparametrc. Ideed, t puts o assumptos o the form of basele hazard rates ( ) x (oparametrc part of model) but assumes a parametrc form of the effect of covarates o the hazard. Itroduce the followg otatos: λ( xz) = h( x) exp( β Z) s the hazard fucto before the chage pot; λ( xz) = h( x) exp( β Z) s the hazard fucto after the chage pot; F( x,, x Z) s the jot lkelhood fucto uder model (6); Λ ( Z ) s log-lkelhood rato uder model (6); S( xz ) s survval fucto uder model (6); Θ= (, β, β) s the ukow parameter vector; Θ s the least squares estmator of Θ whch, smlarly to Secto 4., mmzes the error sum of squares based o the dffereces betwee the log-survval fuctos obtaed from model (6) ad from the Kapla- Meer estmator (5). Uder model (6), the survval fucto s expressed as so that h x ad h ( ) x ( λ ) ( λ( ) λ( ) ) x ( ) = ( ) x + S xz exp sz ds exp sz ds sz d s, x> x ( ) ( ( ) ( ) ) x ( ) = ( ) = ( ) + The least squares estmator Θ= (, β, β) of the chage pot as the mmzer L Z logs x Z λ sz ds λ sz ds λ sz d s. x x> ( Z ) Θ= arg m ESS Θ Θ ad slopes β ad β s the defed j 669

8 X. Che, M. Baro of the error sum of squares where compoets y ( x ) are defed (7). ( Θ Z) = ( y ( ) ( )) x L Z ESS, (7) Strog Cosstecy ad Covergece Rate of the Least Squares Estmator = Exteto of the results of Secto 4 o the strog cosstecy of the chage pot estmator ad estmators of the usace parameters to Cox proportoal hazard model s straghtforward. Ideed, the uform strog cosstecy of the Kapla-Meer estmator holds for ay type of the uderlyg dstrbuto of survval tmes. Therefore, the error sum of squares ca be splt to four parts as (8), wth almost sure covergece holdg for each part. Alog the same les as the costat hazard rate model, we obta the followg results. Lemma. At Θ=Θ, compoets of the error sum of squares (7) satsfy the strog law of large umbers; that s, ESS ( Θ Z ) coverges almost surely to as. Theorem 5. Wth kow β ad β, the chage-pot estmator s strogly cosstet. It coverges to the true chage pot at the same rate as the costat hazard rate model;.e., for ay >, P > e c ( ) for some c > ad all suffcetly large. Proof. The proof s smlar to the proof of Theorem 4.5 ad Corollary 4.6 of Secto 4.. The followg results show that the strog cosstecy of holds eve wthout the assumpto of kow slopes β ad β. Theorem 6. The estmated slopes β ad β are strogly cosstet for β ad β uder Assumpto (A). Theorem 7. Uder ukow slope parameters β ad β, the chage-pot estmator s strogly cosstet uder Assumpto (A). Strog cosstecy of ad Secto 4. ad essetally alog the same les. For detals, see [], chapter Comparso of Estmators β presece of usace parameters s proved by the techques developed I classcal cases, uder the usual regularty assumptos, the maxmum lkelhood estmator s asymptotcally the uformly mmum varace ubased estmator. Chage-pot models volate the regularty codtos because of the dscotuty of the lkelhood fucto at the chage-pot parameter. As a result, the maxmum lkelhood estmator may o loger be optmal. I ths secto, we compare the maxmum lkelhood estmator ad the least squares estmator by meas of the followg Mote Carlo smulato study. Geeratg samples from model () s qute smple. We geerate a ( ), replace the geerated varable wth Exp( λ ) Exp λ sample, ad for those varables that exceed +. The memoryless property of Expoetal dstrbuto esures that the resultg varable has the dstrbuto accordg to (). Samples are geerated wth the chage pot = 5, cesorg tme t =, ad falure rates ( λ, λ ) take to be (.,.5 ), (.5,.5 ), ad (.3,. ). Clearly, t should be easer to detect the chage pot f the dfferece betwee λ ad λ s larger. Samples szes from to 3 are cosdered each wth Mote Carlo rus. A example of ESS, a pecewse polyomal fucto, s depcted Fgure. Table lsts the estmates of, λ, ad λ for dfferet sample sze ad dfferet actual falure rates. Table lsts the mea square errors for estmates of, λ, ad λ. These estmates ad mea square errors lead to the followg coclusos: ) Both MLE ad LSE of, λ, ad λ coverge to the true chage pot ad hazard rates as the sample sze creases. ) Both MLE ad LSE become more accurate whe the dfferece betwee λ ad λ s creased, holdg the sample sze costat. 67

9 X. Che, M. Baro Fgure. Error sum of squares ad the least squares estmator of the chage-pot. Table. Estmates of, λ, ad λ from smulated data. λ λ Sample sze MLE LSE λ λ λ λ Table. Mea squared errors of estmates of, λ, ad λ. λ λ Sample sze MSE for MLE MSE for LSE λ λ λ λ

10 X. Che, M. Baro 3) The LSE of has a lower bas tha the MLE for the same sample sze ad the same falure rates. The mea squared error of the LSE of s larger tha that of the MLE, for the same sample sze ad same falure rates, however, the hazard rates are estmated by the LSE method wth the same or lower mea square error. 7. Example: Prometa Clcal Tral I ths secto, we apply both the maxmum lkelhood method ad the least squares method to a recet clcal tral for treatg methamphetame-depedet patets coducted by Research Across Amerca, a outpatet clcal research ceter Dallas, Texas [7]. Ffty patets partcpated a ope-label study over the tme frame of 84 days. I ths study, all of the partcpats were log-term users of methamphetame. After the screeg vst o day, patets receved fve fusos durg the frst three weeks ad coducted 4 follow-up vsts. Later, a double-bld, placebo-cotrolled study was coducted to better evaluate the effect of treatmet. I the double-bld study, ether the partcpats or the clcas kew whch patets belog to whch treatmet arm. The reaso for bldg ad placebo cotrols s to determe (as much as possble) whether the effects observed the study are due to the treatmet tself ad ot other factors. For each partcpat, the survval tme s the tme to relapse, whch s the durato of tme wthout the use of drugs. Our goal here s to detect the after-treatmet effect of Prometa, whch results a sgfcat reducto of falure rate some tme after the frst three fusos. We detect such chages wth both the maxmum lkelhood method ad the least squares method. Results are lsted Table 3 ad Table 4. Frst, we estmate the chage pot for the 5-subject ope-label study. ) Usg the maxmum lkelhood method, day 3 maxmzes the log-lkelhood rato Fgure, left. The lkelhood rato test provdes a p-value of.567, whch s low eough to reject the ull hypothess there s o chage pot. O the day of the chage, the falure rate drops from.4 to.5. Thus, we coclude that the falure rate after takg the drugs reduces sgfcatly from.4 to.5 f the patets do ot use drugs for 3 days followg the treatmet. ) Usg the least squares method, the estmate for chage pot s ad the falure rate drops from.8 to.4, whch are very close to the results from maxmum lkelhood estmate. The graph of error sum of squares s Fgure, rght. Chage pots for the female ad male groups are compared to see whether occurrece of a chage pot depeds o geder. ) Usg the method of maxmum lkelhood, the estmated chage pots for males ad females are 8 ad 7 from Fgure 3, left. However, the lkelhood rato test fals to detect a sgfcat dfferece betwee the geders wth the p-value of.33,.e., there s o evdece that there are ay sgfcatly dfferet chage pots for males ad females. The falure rate reduces from.649 to. for males ad from.387 to practcally for females. ) Usg the least squares method, the chage-pot estmator for males s about day 4 ad the falure rate reduces from.494 to almost, whle the chage-pot estmator for females s 3 ad the falure rate reduces from.495 to almost. We ca see that there s almost o dfferece betwee male group ad female group chage-pot estmators from graph 3, rght. Table 3. Estmates of, λ, λ for ope-label study. Ope-label study Male group Female group λ λ λ λ λ λ MLE LSE Table 4. Estmates of, λ, λ for two-armed double-bld study. Prometa group Placebo group λ λ λ λ MLE LSE

11 X. Che, M. Baro Fally, we estmated the chage pots for the radomzed double-bld placebo-cotrolled study. Chage pots are estmated separately for the actve treatmet group ad for the placebo group. ) The graph of log-lkelhood ratos s Fgure 4, left. The estmated chage pot for the treatmet group s 3, ad the falure rate reduces from.78 to.39. For the placebo group, the chage-pot estmate s 8, ad the falure rate reduces from.45 to.53. The lkelhood rato test shows that these two groups have sgfcatly dfferet chage pots wth p-value.98. ) Wth the least squares method, the chage-pot estmator for the treatmet group s aroud day 7 ad the falure rate reduces from.7 to almost, whle the chage-pot estmator for Placebo s aroud 4 ad the falure rate reduces from.55 to.6. The graph for error sum of squares s 4, rght. Fgure. Least squares estmate of chage-pot for ope-label study. Fgure 3. Least squares estmate of chage-pot for female ad male groups. Fgure 4. Least squares estmate of chage-pot for Prometa ad Placebo groups. 673

12 X. Che, M. Baro As a result, besdes statstcal sgfcace, exstece of chage-pots the survval curves for both treatmet groups has a mportat clcal sgfcace. It shows a drop the rsk of relapse after a certa perod of abstece. Although the MLE ad LSE methods slghtly dsagree o the exact locato of chage-pots the two treatmet groups, both methods show that the after-chage falure rate s sgfcatly lower for the actve treatmet groups. Essetally, a patet has to absta from methamphetame for two weeks after recevg the treatmet, ad the the falure rate reduces sgfcatly. 8. Cocluso Detecto of chage-pots survval curves ad estmato of ther locato fds mportat applcato clcal research. Ths problem s coceptually dfferet from the stadard chage-pot aalyss, where the dstrbuto of data chages at ukow tmes. Nevertheless, smlar statstcal techques ca be used. The maxmum lkelhood approach yelds a tractable chage-pot estmator, however, a more effcet procedure ca be obtaed by the Kapla-Meer estmator of the survval fucto coupled wth the method of least squares. Ulke the stadard chage-pot problems, here both methods result strogly cosstet estmators. Ackowledgemets We thak the Edtor ad the referee for ther commets. Research of M. Baro s fuded by the Natoal Scece Foudato grat DMS Ths support s greatly apprecated. Refereces [] Desmod, R.A., Wess, H.L., Ara, R.B., Soog, S.-J., Wood, M.J., Fdda, P., Ga, J. ad Whtley, R.J. () Clcal Applcatos for Chage-Pot Aalyss of Herpes Zoster Pa. Joural of Pa ad Symptom Maagemet, 3, [] Zucker, D.M. ad Lakatos, E. (99) Weghted Log Rak Type Statstcs for Comparg Survval Curves Whe There Is a Tme Lag the Effectveess of Treatmet. Bometrka, 77, [3] Goodma, M.S., L, Y. ad Twar, R.C. () Detectg Multple Chage Pots Pecewse Costat Hazard Fuctos. Joural of Appled Statstcs, 38, [4] He, P., Kog, G. ad Su, Z. (3) Estmatg the Survval Fuctos for Rght-Cesored ad Iterval-Cesored Data wth Pecewse Costat Hazard Fuctos. Cotemporary Clcal Trals, 36, [5] Loader, C.R. (99) Iferece for a Hazard Rate Chage Pot. Bometrka, 78, [6] Matthews, D.E. ad Farewell, V.T. (98) O Testg for a Costat Hazard agast a Chage-Pot Alteratve. Bometrcs, 38, [7] Muller, H.G. ad Wag, J.L. (99) Noparametrc Aalyss of Chages Hazard Rates for Cesored Survval Data: A Alteratve to Chage-Pot Models. Bometrka, 77, [8] Nguye, H.T., Rogers, G.S. ad Walker, E.A. (984) Estmato Chage-Pot Hazard Rate Models. Bometrka, 7, [9] Sertkaya, D. ad Sözer, M.T. (3) A Bayesa Approach to the Costat Hazard Model wth a Chage Pot ad a Applcato to Breast Cacer Data. Hacettepe Joural of Mathematcs ad Statstcs, 3, [] Ahsaullah, M., Rukh, A.L. ad Sha, B. (995) Appled Chage Pot Problems Statstcs. Nova Scece Publshers, New York. [] Bassevlle, M. ad Nkforov, I.V. (993) Detecto of Abrupt Chages: Theory ad Applcato. PTR Pretce-Hall, Ic., Eglewood Clffs. [] Bhattacharya, P.K. (995) Some Aspects of Chage-Pot Aalyss. I: Chage-Pot Problems, IMS Lecture Notes- Moograph Seres (Vol. 3), [3] Che, J. ad Gupta, A.K. () Parametrc Statstcal Chage Pot Aalyss: Wth Applcatos to Geetcs, Medce, ad Face. Brkhäuser, Bosto. [4] Poor, H.V. ad Hadjlads, O. (9) Quckest Detecto. Cambrdge Uversty Press, Cambrdge. [5] Worsley, K.J. (988) Exact Percetage Pots of the Lkelhood-Rato Test for a Chage-Pot Hazard-Rate Model. Bometrcs, 44,

13 X. Che, M. Baro [6] Kapla, E.L. ad Meer, P. (958) Noparametrc Estmato from Icomplete Observatos. Joural of the Amerca Statstcal Assocato, 53, [7] Urschel, H.C., Haselka, L.L., Gromov, I., Whte, L. ad Baro, M. (7) Ope-Label Study of a Propretary Treatmet Program Targetg Type a γ-amobutyrc Acd Receptor Dysregulato Methamphetame Depedece. Mayo Clc Proceedgs, 8, [8] Yao, Y.-C. (987) Approxmatg the Dstrbuto of the Maxmum Lkelhood Estmate of the Chage-Pot a Sequece of Idepedet Radom Varables. Aals of Statstcs, 5, [9] Bardorff-Nelse, O.E. ad Cox, D.R. (984) Bartlett Adjustmets to the Lkelhood Rato Statstc ad the Dstrbuto of the Maxmum Lkelhood Estmator. Joural of the Royal Statstcal Socety: Seres B, 46, [] Dwoode, I.H. (993) Large Devatos for Cesored Data. Aals of Statstcs,, [] Bllgsley, P. (995) Probablty ad Measure. 3rd Edto, Wley, New York. [] Che, X. (9) Chage-Pot Aalyss of Survval Data wth Applcato Clcal Trals. Ph.D. Dssertato, The Uversty of Texas at Dallas, Rchardso. 675

14 X. Che, M. Baro Appedx Proof of Lemma Proof. Express y ( x) Defe α sup α ( x) the followg form, ( ) = ( ) + α ( ) = λ ( λ + λ( ) ) > + α ( ) y x log S x x x x x. x x = x. Accordg to the metoed uform covergece of the Kapla-Meer estmator of the survval fucto, ad for ay >, there exsts α P >. Hece α c > such that ( ) e c Sce θ mmzes ESS( θ ), we always have Hece ESS( θ) = α α. = = ( θ) ( θ) ESS ESS ( ).. ESS as θ. Proof of Theorem Proof. From (9), we have O the other had, ( λ λ) x X m (, ) >.. x x E X X m X m (, ) X m Hece we have λ λ. Proof of Theorem 3 Proof. From Theorems,, ad (), we obta O the other had, x > max (, ) ( x ) ( λ λ). >. ( x ) ( x ) E( X ) X> M x > max (, ) x > M Hece λ λ. Proof of Theorem 4 Proof. Frst, express ESS( ) ad ( ) ESS terms of the resdual α. From (8), ESS ( ) ( ( )) α x = = 676

15 X. Che, M. Baro ad ( ) ( ( x )) (( )( ) ( x )) (( )( ) ( x )) x m (, ) x max (, ) < x < x ESS = α + λ λ + α + λ λ + α Hece Let the followg equalty, (( )( λ λ) α( x) ) + + ( ) ( ). {( ) ( ) ( )( )( )} {( x) ( λ λ ) α ( )( )( )} x x λ λ < x {( x ) ( λ λ ) α ( )( )( )} x x λ λ x x max (, ) ESS ESS = λ λ + α λ λ < x N be the umber of x max (, ), ad N be the umber of < x. For ay > we have P { ESS( ) ESS( ) < } mm ( λ λ) N+ α( x)( )( N+ N) < P : > > ad the theorem s proved. P ( λ λ ) < α ( + ) < > e P α > mm N N N ( λ λ) ( + ) N P α > N N t c e, λt ( λ λ ) t 677

16