Cure Rate Models. Dr.R. Elangovan 1 * and B. Jayakumar 2. ISSN (Print) : ISSN (Online) :

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1 Asa Pacfc Journal of Research Vol: I. Issue XXXVIII, Aprl 206 ISSN (Prnt : ISSN (Onlne : Cure Rate Models Dr.R. Elangovan * and B. Jayakumar 2 Professor, Department of Statstcs, Annamala Unversty, Annamalanagar , Tamlnadu, Inda. 2 Research Scholar, Department of Statstcs, Annamala Unversty, Annamalanagar , Tamlnadu, Inda. *Presented n the Workshop on Relablty Theory and Survval Analyss. Department of Statstcs, Punjab Unversty, Chandgarh, Inda. 4th to 6th of November 205. Abstract: models, used for modelng tme-to-event data, conssts of a cured (non-susceptble fracton and an uncured (susceptble fracton and have become ncreasngly popular n the analyss of data from clncal trals. Because of the great mprovements n the treatment of cancer and some other dseases, cure rate models have become ncreasngly popular n the analyss of data from many clncal trals. Tradtonal methods of survval analyss, ncludng the well-known Cox s regresson model, assume that no patent s cured but that all reman at rsk of death or relapse. These models are concerned wth survval only and do not accommodate the possblty of cure.. For example, certan types of cancers, lke breast cancer, leukema, a substantal proporton of patents may now be cured by treatment,.e., cured proporton. The patents who are cured are called mmunes or long-term survvors, whle the remanng patents who develop a recurrence of the dseases are termed susceptble. The populaton of nterest s thus dvded nto two groups vz., cured and noncured. models provdes satsfactory models n such cases. These models are more sutable to analyze data wth cure proportons, as n the above example. Much of work has been done by many researchers n ths area. To name a few Anderson, Balakrshnan, Chatterjee Chen, Clayton, Gray, Pal, Rodrgues and Yn. Keywords: Tme-to-Event Data, Cured Fracton, Cox s Regresson Model, Cure Rate Model.. INTRODUCTION models, used for modelng tme-to-event data, consst of a survvng fracton and have become ncreasngly popular n the analyss of data from clncal trals. Recently, the developments of new drugs and treatment regmens have resulted n patents lvng longer wth certan types of cancer and heart dsease. models have applcatons n a wde array of areas such as bomedcal studes, crmnology, fnance, demography, manufacturng, and ndustral relablty. At an ndvdual level, dagnoss of cancer s regarded as a human tragedy. At the level of socety, cancer s one of the major chronc dseases, causng a notable amount of health admnstratve costs. Prognoss and possble cure from cancer are thus mportant measures of lfetmes whch can be assessed by analyzng the survval of cancer patents. Dfferent statstcal approaches are used for analyzng the cancer survval data. The results of survval analyss for cancer patents have been wdely presented and reported for dfferent human sub populatons of the globe Woolson (98, Kardaun (983, Beadle et al., (984, Sedmak et al., (989. However, very few survval results at natonal level are avalable for the populatons of Pakstan refer to Khan et al., (2004. McGarty (974 has mentoned that for adoptng any sutable. Statstcal technque for analyzng survval data, t should be assumed that the statstcal model embodes the evaluaton of some natural process wth the beleve that the model s a useful approxmaton of the real process. Several approaches have been proposed n the lterature by Leung et al. (997 and Lttle and Rubn (2002 for analyzng the survval data. In ths context, t s proposed to develop cure rate models and analyze such models usng the nferental methods avalable n the lterature based on EM algorthm. The analytcal results are substantated through real data example. 2. EARLY DEVELOPMENTS IN CURE RATE MODELS Destructve weghted Posson cure rate models have been dscussed by Rodrgues et.al. (20. Left truncated and rght censored Webull data and lkelhood nference wth an llustraton have been dscussed by Balakrshnan et.al. (202. EM algorthm-based lkelhood estmaton for some cure rate models s dscussed by Balakrshnan and pal (202. Lognormal lfetmes and lkelhood-based nference for flexble cure rate models based on COM- Posson famly s dscussed by (203. Expectaton maxmzaton-based lkelhood nference for flexble cure rate models wth Webull lfetmes s dscussed by Balakrshnan and Pal (203. An EM algorthm for the estmaton of parameters of a flexble cure rate model wth generalzed gamma lfetme and model dscrmnaton usng lkelhood-and nformaton-based methods has been dscussed by Balakrshnan and Pal (204. Latent cure rate model under repar system and threshold effect have been dscussed by Balakrshnan, et.al. (

2 49 Dr.R. Elangovan and B. Jayakumar 3. CURE RATE MODELS: models are consstng of a cured (non- susceptble fracton and an uncured (susceptble fracton. It s assumed that the survval functon for the entre populaton S (t s a mxture of cured and uncured subjects. Hence ths model s known as a mxture cure rate model and can be wrtten as S(t = S (t + ( + S (t ( c uc Where s a proporton of patents cured on treatment and - s the proporton of patents uncured and wth respect to the cure rate model we can assume a partcular dstrbuton for the survval functon of the cured groups S c (t and survval functon of the uncured groups S uc (t. Snce those who fall nto the cured part of the equaton ( wll never experence the event of nterest, the lm t S t =, and the survval dstrbuton functon S(t for the entre populaton of patents, ths leads to a parametrc survval model. The popular dstrbuton consdered are Gamma, Webull, Extreme value and ts becomes, S (t = ( + S (t (2 uc The probablty mxture densty functon correspondng to the above equaton (2 s f(t = ( f (t (3 uc And the hazard functon of the populaton of patents from equatons (2 and (3 s h(t ( f (t = uc ( Suc ( t (4 The mxture model may be parametrc or non-parametrc dependng on whether s specfed or not. Suppose there are n patents enterng to a clncal study. Let t, =, 2,, n be the observed survval tme for the th patent and let δ be a censorng ndcator defned such that d t = f s censored 0 otherwse Lkelhood for the mxture cure rate model, whch was ntally ntroduced by De Angels et.al., (997 s gven by d { } { ( ( } n ( q ( ( d L t, = f t + S t. = uc uc (5 Where are estmated by use of a lnk functon, usually the log-log, dentty, Logt or probt functons. For a detaled study, refer to Anderson (2007, Lambert et.al.,( Bayesan Cure Rate Survval Model The survval functon for T, and the survval functon for the populaton usng equaton S(t = + ( + S uc (t. We get S (t = exp( q + ( exp( q S ( t pop Where S pop (t s standard cure rate model wth cure rate equal to φ=exp(-θ and survval functon for the non-cured populaton s S uc (t. If the covarates depend on θ through the relatonshp θ=exp (x β, where x s a p x vector of regresson coeffcent for the cured and non-cured group. For the cured group, the sgn of regresson coeffcents affects the cure fracton. Those negatve regresson co-effcent leads to a large cure fracton, when the correspondng covarate takes postve values. For the non-cured group the regresson coeffcents affect the hazards functon. Specfcally, a negatve hazard, whereas a postve regresson coeffcent leads to a smaller hazard, when the correspondng covarate takes a postve value Bayesan Cure Rate Survval Model wth Fralty In ths secton the cure rate survval model can ncorporate random effects (RE to ft lfe tme data set. We apply ths method to examne patterns of clncal trals usng survval data. Models for fralty n multvarate cure fracton models consdered by Yn (2008.Thus for tmes t j observed on subjects and events j, Yn proposed multplcatve fralty at subject level combned wth Posson regresson for n the cure fracton sex p(-θ j. One opton takes S ( t = exp( - q Z Ft ( uc j j j wth Hazard rates h uc (t j = θ j Z f (t j. In ths context usng the above two equatons n the Bayesan cure rate fralty wth canoncal lnk functon n the settng of generalzed lnear models refer to Sundaram and Venkatesan ( Estmaton of the parameters usng the EM algorthm We frst note that the ndcator varable I takes on the value f the lfetme of a subject s actually observed. However, f the lfetme of a subject s censored, the subject mght belong to the cured group or mght belong to the susceptble group. Thus, the value of the ndcator varable I s unknown f the subject s lfetme s censored. Ths actually ntroduces the mssng data and motvates uc

3 50 Cure Rate Models us to employ the EM algorthm for the determnaton of the MLEs of the model parameters. Interested readers may refer to the book by McLachlan and Krshnan (2008 for a detaled dscusson on the EM algorthm. Some nterestng result regardng EM algorthm can be seen n Balakrshnan and Pal (202, 203 and 204. Usng EM algorthm, the real data example s shown n secton REAL DATA EXAMPLES Case (: Parametrc, Non-Parametrc approaches to cure rate models usng cancer patents. Dfferent statstcal approaches are used for analyzng the cancer survval data. The results of survval analyss for cancer patents have been wdely presented and reported for dfferent human sub populatons of the globe, refer to Woolson (98, Kardaun(983, Beadle et al.(984, Sedmak et.al.(989.wth the followng objectves.. To estmate the survval functon usng the standard Kaplan-Meer Estmator, 2. To estmate the cumulatve hazard functon, usng the Nelson- Aalen estmator, 3. To ft an approprate parametrc lfetme model based on Anderson- Darlng goodness of ft test. The relevant lfetme data on the patents of cancer n accord wth the Cancer Hosptal n Chenna was selected. Ths hosptal receves ts patents from a wde area n the lmts of Southern Inda. A retrospectve smple random sample desgn was used; the lfetme data on 200 male and 40 female patents of cancer belongng to dfferent classes was selected. These 340 patents of cancer were treated n Cancer Hosptal n Chenna and other Multspecalty hosptals located n Chenna and combatore and the data were collected usng January 200 to December 204. The regstraton tme was January, 200 to December 3, Assumpton and notatons In ths study the generalzed type-i censorng was consdered. For more convenence, the censorng was due to the followng reasons: a. A patent emgrated out of the study area was mpossble to follow. b. An ndvdual survved past the end of the study perod. c. The censorng was non-nformatve. For the representaton of the data consdered n ths study, each ndvdual had ts own specfc lfetme whch was rescaled at startng tme to t 0 = 0. Accordng to Klen and Moeschberger (997.. was taken as a non-negatve random varable, the tme untl the event of nterest (death due to cancer occurred.. It was assumed to be ndependently and dentcally dstrbuted wth probablty densty functon f(t the survval functon S(t and hazard functon h(t.. C r was the fxed rght censorng tme; T and C r were assumed to be ndependent. The exact lfetme of an ndvdual was known f and only f T was less than or equal to C r. v. Pars of random varables convenently represented the data, (X,δwhere δ was the censorng ndcator and X was equal to, f the lfetme was observed, and to C r f t was censored and X=mn(T,Cr Parametrc approach Consderng the lfetme parametrc model as the useful approxmaton of the real process, three lfetme models vz., Gamma, Webull and Extreme value dstrbuton were consdered. The Anderson-Darlng test, whch makes the use of these specfc lfetme dstrbutons n calculatng crtcal values, was defned wth H 0 : The data followed a specfed parametrc lfetme model. H a : The data dd not follow the specfed lfetme model. The Anderson and Darlng (954 test statstc was (2 n 2 n A = logfy ( log( ( FYn n = + + where F the cumulatve dstrbuton functon of the specfed dstrbuton was, Y was the ordered data and n was the number of observatons. The test was a one sded test and the hypothess that the dstrbuton of a specfc form was rejected f the test statstc A was greater than the crtcal value. From the class of specfed lfetme dstrbutons, the parametrc lfetme model, whch one has the mnmum Anderson-Darlng (adjusted value, gave the better ft Nonparametrc approach Cox and Oakes (984 and Kalbflesch and Prentce (2002 presented a nonparametrc approach to estmate survval functon usng standard Kaplan Meer (KM technque (Kaplan and Meer, 958. There were D dstnct tmes wth t < t 2 <.t D,d deaths or events occurred at tme and were the number of ndvduals who were at rsk at tme t. The KM estmator was defned as for all values of t n the range where there was data: f t < t S( t = d f t t < t < t Y

4 5 Dr.R. Elangovan and B. Jayakumar It was obvous from KM estmator, for t < t, S ( t = and when then ( 0 S t = for t t Cox and Oakes (984 also establshed the varance of the KM estmator usng Greenwood s relaton as: V S ( t = S ( t 2 t ty Y d ( d Moreover, KM estmator was also used to estmate the = The Nelson Aalen (NA estmator of the cumulatve hazard rate was defned up to the largest observed tme on the study (see Aalen, 978: cumulatve hazard functon; H ( t In S ( t H ( t 0 = t t d Y f t t f t t The estmated varance of the NA estmator was gven by 2 s H ( t d =. t ty The Webull model The probablty densty functon of a Webull random varables k k x ( x k l e, f( x; l, k= l l x 0 0 x < 0, Where k > 0 s the shape parameter and λ>0 s the scale parameter of the dstrbuton. The Webull dstrbuton s related to a number of other probablty dstrbuton; n partcular, t nterpolates between the exponental dstrbuton (k = and the Raylegn dstrbuton (k = 2 and λ= 2σ f the quantty X s a tme- to- falure, the Webull dstrbuton gves a dstrbuton for whch the falure rate s proportonal to a power of tme. The shape parameter k, s that power plus one, and so ths parameter can be nterpreted drectly as follows: A value of k < ndcates that the falure rate decreases over tme. Ths happens f there s sgnfcant nfant mortalty, or defectve tems falng early and the falure rate decreasng over tme as the defectve tems are weeded out of the populaton. A value k = ndcates that the falure rate s constant over tme. Ths mght suggest random external events are causng mortalty, or falure. A value of k > ndcates that the falure rate ncreases wth tme. Ths happens f there s an agng process, or parts that are more lkely to fal as tme go on The Gamma model The two parameter Gamma dstrbutons s one of the mportant models n relablty for descrbng the falure tme dstrbuton of the tems under test. The two parameter Gamma dstrbuton wth probablty densty functon s gven by x s p F( x; s, p = e x ; x 0, s> 0; p< p ps Where σ s a scale parameter and p s a shape parameter. The hazard rate µ(t or λ(t of the gamma lfe dstrbuton are ncreasng functon of t ndcatng the agng effect. If P =, then Gamma becomes an exponental dstrbuton havng the constant hazard-rate µt ( =. s For p<, the falure rate functons are decreasng. For the ntegral value of p, the Gamma dstrbuton arses as a sum of p dentcally and ndependently dstrbuted exponental random varables. If p tems ware on test and t s assumed that the falure tme dstrbuton s exponental wth parameter σ, then total tme on test would be a Gamma varable wth parameters p and σ The Extreme value model The Dstrbuton often referred to as the Extreme value dstrbuton (Type s the lmtng dstrbuton of the mnmum of a large number of unbounded dentcally dstrbuted random varables. The probablty dstrbuton functon and cumulatve dstrbuton functon are gven by x m x m b b f( x = e exp e, < X <, b > 0. b x m F x e X b ( = exp, < <, b > 0. If the values are bounded below (as s the case wth tmes of falure then the lmtng dstrbuton s the Webull. Usng the parametrc and non-parametrc approach as dscussed n secton 4 the results were shown n table, table 2 and table 3. On the whole, t appeared from Table that the female group survved more than the male group based on the two descrptve measures. These descrptve measures dd not compare the two groups at dfferent tme ponts n

5 52 Cure Rate Models tme of follow up; however, such a vsual comparson of gender survval by usng the standard non-parametrc KM and NA methods for the group of bone tumor patents s presented n Fg. and Fg. 2, respectvely. It s apparent from Fg. that female group consstently lay above that for male group partcularly upto 75 years of age. Ths dfference ndcates that female patents are the Table : Cancer group Descrptve measures of survval tme and hazard rate Average survval tme n years (T Average hazard rate (h Males Females Males Females Bone tumor Bran tumor Lung Cancer Leukema Lver cancer Oran cancer Overall Table 2: Statstcal sgnfcance test of gender survval Test Ch-Square Degrees of Freedom P-Value Log-Rank Wlcoxon better survvors. Fg. 2 s the plot of the cumulatve hazard rate for bone tumor patents, whch also shows that female prognoss of survval were better than ther male counterparts. In a smlar fashon, the survval prognoss about the remanng types of the cancer patents can also be determned. To explore the statstcal sgnfcance of gender survval, Log-rank and Wlcoxon test were used. In Table 2, the p-values for both tests were near to zero whch provded the strong statstcal evdence that males were dyng faster than females. The emprcal results of descrptve characterstcs due to nonparametrc KM and best ftted Parametrc Webull model approaches are presented n Tables 3 and 4 respectvely. Table 3 shows that female patents had greater mean survval tme (MST of 64. years than years for males. Aldna et al. (2004 also used the Kaplan-Meer approach to estmate the mean survval tme for the esophageal cancer patents n Pakstan. They estmated the mean age of 56 years n 59 percent male and 4 percent female patents, whle n ths study, 58 percent males and 42 percent females had mean age of 42.4 and 5 years, respectvely. Table 3 also shows that the estmate of medan survval tme for males was 49 years progressng to 64 years for female patents, whch agan confrms the survval superorty of females. From the parametrc pont of vew, the Webull dstrbuton seemed to be the best ftted lfe tme model based on the lower values of adjusted Anderson- Darlng test for males and females for all classes of cancer. The parameter estmates along wth the major characterstcs of the dstrbuton are shown n Table 4. Mean survval tme (MST Table 3: Descrptve characterstcs of bone tumor patents by Kaplan-Meer procedure MST Standard Error 95% Normal CI Lower Upper Medan Q Q3 F M F M F M F M F M F M F M Parameter Table 4: Parameter estmates and major characterstcs of nterest of Webull Dstrbuton for bone tumor patents Estmate Standard error 95.0% Normal CI Male Female Male Female Lower Male Lower Female Upper Male Shape Upper Female Scale Mean Survval tme Standard devaton Medan Frst Quartle Thrd Quartle

6 53 Dr.R. Elangovan and B. Jayakumar Fg.. Comparson of survval functon of male and female bone tumor patents by usng KM estmator Fg. 2. Comparson of cumulatve hazard functon of male bone tumor and female patents by usng NA estmator of males (53.24 years was lower than years for females. The percentage devatons of MST for males and females were 0.73 and.84, whch ndcated a very close estmaton of MST by usng both approaches.e. nonparametrc and parametrc. By observng the comparatve graphs of survval functon and hazard functon based on KM estmator, NA estmator and Webull lfetme model, t was observed that at young age survval rate of tumor patents was hghest, whle as age ncreased survval rate decreased. Case ( Bayesan Cure Rate Survval Models for Pulmonary Tuberculoss Followng the approach are Anderson (2007 and Chen et.al. (999 and the data set avalable usng TRC-ICMR (2007, the data relatng to Sex, Age, Weght and Doses relatng to 360 Cancer patents are collected and Mean, SD, MCSE and Coverage probabltes based and teraton relatng to Gamma, Webull, and Extreme value cure rate models has been calculated

7 54 Cure Rate Models Table 5: Posteror summares of Pulmonary Tuberculoss data under Bayesan Gamma Cure Rate Survval model for n=360 Censored % =0 at and teratons Parameters l= ; a=. 000;DIC = 787 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75% Table 6: Posteror summares of Pulmonary Tuberculoss data under Bayesan Gamma Survval model for n=360 Censored % =0 at and teratons Parameters l= ; a= ;DIC = 784 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75% Sex(β Table 7: Posteror summares of Pulmonary Tuberculoss data under Bayesan Webull Cure Rate Survval Model for n=360 Censored % =0 at and teratons Parameters l= ; a=. 7079;DIC = 689 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75%

8 55 Dr.R. Elangovan and B. Jayakumar Table 7: Contnued Table 8: Posteror summares of Pulmonary Tuberculoss data under Bayesan Webull Cure Rate Survval Model for n= 360, Censored % =0 at and teratons Parameters l= ; a= ;DIC = 58 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75% Table 9: Posteror summares of Pulmonary Tuberculoss data under Bayesan Extreme value Cure Rate survval Model for n= 360, censored % = 0 at and teratons Parameters l= ; a= ;DIC = 842 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75%

9 56 Cure Rate Models Table 0: Posteror summares of Pulmonary Tuberculoss data under Bayesan Extreme value survval model for n = 360, censored % = 0 at and teratons Parameters l= ; a= ; DIC = 445 Covarates Mean SD MCSE Percentles Iteraton 25% 50% 75% l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. : Bayesan Cure Rate model (Intercept l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. 2: Bayesan Cure Rate model (Sex

10 57 Dr.R. Elangovan and B. Jayakumar l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. 3: Bayesan Cure Rate model (Age l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. 4: Bayesan Cure Rate model (Weght and results are shown Table 5, Table 6, Table 7, Table 8, Table 9 and Table 0. The results were also depcted n Fg, Fg 2, Fg 3, Fg 4, Fg 5, and Fg 6 respectvely. The smulatons were carred out usng SAS avalable at Tamlnadu Dr. M.G.R Medcal Unversty located at Chenna. The parametrc Bayesan Cure Rate survval model wth and wthout fralty for lfetme censored pulmonary tuberculoss data have been ftted. The parameter have been estmated (unknown shape, scale, Posteror Summares of regresson coeffcents and random effects usng MCMC technques wth the help of SAS. The detaled analyses have been carred out n Tamlnadu Dr. M. G. R Medcal Unversty, Gundy, based on to teraton. It s observed that from Table 5 to 0 the Bayesan cure rate survval model performance very well and best ft to the data based on to teraton. It s further observed that the data well ft to the rght censored pulmonary Tuberculoss data compared wth cancer data set dscussed n secton 4.

11 58 Cure Rate Models l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. 5: Bayesan Cure Rate model (Doses l=0.267 α=.0000 DIC=857 l=0.528 α= DIC=774 l=0.505 α=.7089 DIC=697 l=0.450 α= DIC=578 l=0.995 α= DIC=842 l=0.356 α=.053 DIC=445 Fg. 6: Bayesan Cure Rate model ( ACKNOWLEDGEMENT The authors ndebted to Mr. M. Shanmugam, Head, Computer secton, Tamlnadu Dr. M. G. R Medcal Unversty, Chenna, for hs encouragement and data analyss and smulaton to complete ths work. REFERENCE 3. Balakrshnan, N. and Pal (204. An EM algorthm for the estmaton of parameters of a flexble cure rate model wth generalzed gamma lfetme and model dscrmnaton usng lkelhood-and nformaton-based methods Computatonal Statstcs, 30 (, Balakrshnan, N. and Pal, S. (202. EM algorthm-based lkelhood estmaton for some cure rate models. Journal of Statstcal Theory and Practce, 6, Balakrshnan, N. and Pal, S. (203. Expectaton maxmzaton-based lkelhood nference for flexble cure rate models wth Webull lfetmes. Statstcal Methods n Medcal Research (To appear. DOI: 0.77/ Balakrshnan, N. and Pal, S. (203. Lognormal lfetmes and lkelhood-based nference for flexble cure rate models based

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