BAYESIAN ESTIMATOR OF A CHANGE POINT IN THE HAZARD FUNCTION

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1 Mathematcal ad Computatoal Applcatos, Vol. 7, No., pp , 202 BAYESIAN ESTIMATOR OF A CHANGE POINT IN THE HAZARD FUNCTION Durdu Karasoy Departmet of Statstcs, Hacettepe Uversty, Beytepe, Akara, Turkey. durdu@hacettepe.edu.tr Abstract- Ths artcle presets a ew approach for obtag the chage pot the hazard fucto. The proposed approach s developed wth the Bayesa estmator. Usg a smulato study, mea value ad mea square error (MSE) of proposed estmator are obtaed ad compared wth the mea ad MSE of tradtoal estmators. It s showed that the proposed estmator s more effcet tha the tradtoal estmators may cases. Furthermore, a umercal example s dscussed to demostrate the practce of the proposed estmator. Key Words- Bayesa Estmator, Chage Pot, Costat Hazard, Survval Aalyss.. INTRODUCTION Let T deote a depedet detcally dsrtrbuted radom varable of survval tmes. The hazard model of T s gve by h t 0 t t where >0, >0, >0, ad are hazard rates, ad s the chage pot. Hece, the hazard fucto, h(t), s assumed to have a costat value utl tme, ad a costat value after tme. Therefore, obtag a correct estmate of the chage pot plays a mportat role medcal ad bologcal researches. Some recet studes these felds are gve as Gupta et al. [], Tabak et al. [2], Faucett et al. [3], Gjbels ad Gurler [4], L [5], Dael ad Nader [6], Karasoy ad Kadlar [7], etc. It s well kow that the probablty desty fucto ad survval fucto of a radom varable T are gve by () αt e f(t) = (αβ)τ βt e ad αt e S(t) = (αβ)τ βt e 0 t τ t τ 0 t τ t τ (2) (3) respectvely. Note that f(t) ad S(t) have a jump pot at [8,9,0]. Ths artcle s orgazed as follows. Secto 2 troduces the avalable tradtoal estmates of. I Secto 3, my proposed estmator s preseted. A smulato study s

2 30 D. Karasoy performed ad the results of ths smulato are dscussed Secto 4. I Secto 5, I preset a umercal example to demostrate the applcato of my proposed estmator. 2. TRADITIONAL ESTIMATORS Let T,, T be a radom sample from (). From ths pot owards, wthout loss of geeralty, assume that T T, other words, after makg sutable rearragemet the order statstcs, T,, T, ca always be observed. Nguye et al. [] obtaed a cosstet estmator of the model () as follows: X where t t R t T T2... T T log R t Et R t Rt T log R t s the mea of the sample; Rt I t left-had porto of the sample; Et R t ad t Rt T 2 T =R(t)+ T E t 2, s the umber of are the mea -R t ad varace of the rgh-had porto of the sample, respectvely. Here I s a dcator fucto ad the survval tmes, T,,T, are ordered as T T. A value of t for whch X (t) s close to 0 s a caddate for a estmate of. Ths method wll be abbrevated as NRW the rest of the artcle. Basu et al. [2] preseted two estmators for the chage pot, amely ˆ BGJ = f{t0: y (t + h ) y (t ) h ˆ + }, (4) ˆ t) + }, ˆ BGJ 2 = f{t0: y (t) log( p 0 ) ˆ ( p0 where ˆ ad ˆp are the estmates of ad 0 p, respectvely; p ; 0 0 c y t logs t ; log ; h. Here p s the p 0 0 -th populato quartle ad c s a costat. Note that as BGJ the rest of ths artcle. 4 () for ths method, whch s abbrevated The motvato for these estmates are as follows. Cosder the hypothess testg problem: H 0 : h(t) = H : h(t) > If hˆ ( t) s a cosstet estmator of h(t) the oe ca costruct tests by acceptg H 0 f hˆ ( t) βˆ. The the smallest t for whch H 0 s accepted would be a caddate for a

3 Bayesa Estmator of a Chage Pot the Hazard Fucto 3 estmate of. For ˆ BGJ, take ˆ y ( t) log( p0 ) h ( t). ( ˆ t) p0 hˆ ( t) y ( t h ) y ( t) ad for BGJ 2 h ˆ, take I addto, Basu et al. [2] foud ˆ BGJ to be more effcet tha ˆ BGJ 2 by smulato, so I take oly ˆ BGJ for the smulato study ths artcle. Ghosh ad Josh [3] also vestgated the asymptotc dstrbutos of BGJ ad BGJ2. Ghosh et al. [4] cosdered the followg lkelhood fucto: R t Q t R t Ttot Q t L(,, t \ D) = e e, (5) where D deotes the data {T,,T } wth T 0 = 0 ad T + = ; A(t) = T I T t s the sum of survval tmes for the left-had porto of the sample; Q ( t) A( t) { R( t)} t ad T tot T. Ths method wll be abbrevated as GJM the rest of the artcle. I GJM, the o-formatve pror dstrbuto s gve by (,, t) =. (6) Multplyg (5) ad (6), the posteror dstrbuto s obtaed as (,, t \ D) Rt Qt e Rt T Qt e ad usg (7), the margal posteror dstrbuto for the chage pot was gve by (t D) j -! T T tot 0 tot 0 e Q t j0 k 0 k! tot k ( j )! Qt j! T j tot j (7). (8) A value of t whch maxmzes (8) s a caddate for the estmate of. Note that > () for ths method. Ghosh et al. [5] proved that the posteror dstrbuto of, ad t were asymptotcally depedet of each other. Achcar ad Lobel [8] defed a uform pror dstrbuto for ad (6) as (,, t) = αβ, (9) ad showed that the choce of formatve pror destes gave very accurate ferece results o the data set for a medcal research study.

4 32 D. Karasoy Karasoy ad Kadlar [6] used the least square estmates of ad gve by Gjbels ad Gurler [4] place of ad (6). They obtaed the posteror dstrbuto usg L t,, fucto gve by Gjbels ad Gurler [4]. Smlar to the estmator GJM, a value of t, whch maxmzes the posteror dstrbuto, s a caddate for the estmate of. Ths method wll be abbrevated as KK the rest of the artcle. 3. PROPOSED ESTIMATOR Cosderg the defto of the pror dstrbuto, (6), as GJM, ad aother pror dstrbuto, (9), as suggested by Achcar ad Lobel [8], I decde to use a pror dstrbuto defed KK wth the least square estmates of ad gve by Gjbels ad Gurler [4] place of ad (6) as follows: where Here π( α, ˆ β, ˆ t) = ˆ t ˆ t t Y ˆ t tj, (0) ˆ, () j ˆ t ˆ t ˆ t. (2) Y t j Y t j, ad t j tj Y tj Y ˆ j j. 2 2 tj tj j j (for detals, see Gjbels ad Gurler [4]). ˆ t ad It s clear that ˆ t are the least squares estmates of ad, respectvely, for each t ; t j = t / t j for t j > t, j =,2,,. I obta the posteror dstrbuto for the chage pot () by followg the path descrbed Secto 2, as follows: where (t D) ˆ t ˆ t L t, (3) L t s a lkelhood fucto (5). Note that Karasoy ad Kadlar [6] used L,, fucto gve by Gjbels ad Gurler [4] for the posteror dstrbuto (3). t

5 Bayesa Estmator of a Chage Pot the Hazard Fucto 33 Smlar to the GJM ad KK methods, a value of t, whch maxmzes (3), s a caddate for the estmate of. However, the codto () s ot requred for ths method. 4. SIMULATION I ths secto, I try to fd out whch estmator has the smallest mea square error (MSE) uder dfferet codtos. I ths smulato study, I take 0 samples of szes = 25,, ad varous values for the parameters (), as show Table, by codg a program Vsual Basc 6.0. The computed mea ad MSE values of the tradtoal estmators ad the proposed estmator are also gve Table. From Table, I observe that the proposed estmator geerally has a smaller MSE tha the other estmators, except for a few cases. Therefore, I ca fer that the proposed estmator s geerally more effcet tha the tradtoal estmators. For all cases, the proposed estmator s more effcet tha the NRW estmator, ad gves more accurate estmates cosderg mea values. For all cases where α s bgger tha, the proposed estmator s more effcet tha the KK estmator. Also, the proposed estmator s more effcet tha the GJM estmator, except for the case =, α = 2, = 0.5, =. I addto, whe the dfferece betwee ad gets hgher for the large sample szes, the proposed estmator s more effcet tha the BGJ estmator. I ca clam that the proposed estmator s more effcet tha the BGJ estmator whe α gets bgger for the varats of ad. Note that the BGJ estmator s uable to fd a sutable t satsfyg the codto (4) some cases. Also, the BGJ ad the GJM estmators ca ot be appled for < the hazard fucto. These cases are show by a dash Table.

6 34 D. Karasoy Table. Estmates of the Chage Pot ad Ther MSE Values. NRW BGJ GJM KK Proposed (6.0465) (2.833) (.9628) (.8473) (0.8969) (6.497) (6.208) (8.755) (8.8795) (8.942) (0.9207) (0.9604) (0.9808) (0.9595) (0.9792) (0.9895) 0,0207 (2.888) 0.06 (2.266) (2.2276) (0.9595) 0.00 (0.9799) (0.9904) 3.54 (.56) (0.989) (3.4582) (2.4368) (0.8929).7257 (.4249).3679 (0.337).690 (0.0607) (3.2920) (2.4097) (0.8922).358 (0.3589).839 (0.0843).0845 (0.052).737 (.805).370 (.3042).08 (3.6649).737 (3.384).370 (2.6832) (0.2258) (0.822) (0.0529) (0.8952) (0.8454) (0.6708) (0.989) (0.822) (0.0534).6625 (.6493).7044 (.602).5559 (2.8597).8298 (2.3346).9403 (2.2485).6759 (0.7833) (.4207) (2.3687) 0.46 (0.3579) 0. (0.349) (0.2826) (0.822) (0.7569) (0.742) (0.3032) (0.3084) (0.270) (0.5939) (0.6809) 2.08 (.542) (0.769) (0.5622).7234 (0.986) (0.9566) (0.9790) (0.422).079 (0.99).2333 (0.437).06 (0.375).327 (0.746).5492 (0.72) (0.583).55 (0.602).3206 (0.865)

7 Bayesa Estmator of a Chage Pot the Hazard Fucto 35 Table (cotued) NRW BGJ GJM KK Proposed (2.888) (0.8646) (0.962) (0.749) (0.3786) 0.00 (2.299) (2.236).9263 (0.6092).7655 (0.2232) (0.8454) (0.5253) (0.5836) (0.562).3624 (0.792).63 (0.406) (2.888) (0.5564).089 (2.882) 0.00 (2.299).2592 (0.336).0398 (2.290) (0.236) (0.976).9069 (3.262) 0.38 (0.394).6072 (0.253) (0.3885).0095 (0.23) (0.670) (0.9787).8836 (2.4047) (0.3757) (0.3746) (0.0785) (0.9726) (2.2070).539 (0.892).5867 (.8467) (0.2276) (.3007) (0.3776) (0.9747) (0.0460).5459 (0.666) 0.00 (2.226) (2.723) (.2344) (0.7829).9827 (0.356) (2.2074) (0.9728) (3.6855).475 (0.8338) (0.9742) (0.4020) (0.7234) (0.3427).260 (0.456) (0.645) (0.9859).434 (0.6042) (0.3757) (0.3059) (0.0783) (0.9928) (2.2089).2657 (0.223).3728 (0.65) (0.298) (.3042) (0.2969) (0.9566).058 (0.0559) (0.6622) (2.2299).6690 (0.7453) (.2345) (0.7398).803 (0.353) (2.240).9346 (0.949) (0.9757) (0.6302),279 (0.452)

8 36 D. Karasoy Table (cotued) NRW BGJ GJM KK Proposed (0.9728) (0.2269) (0.403) (0.2968) (0.744) (0.9866) (0.994) 0.08 (2.248) (2.238) (2.247) 0.02 (0.978) (0.9770) (0.9453) (2.277) (2.2307) (2.267) (8.938) (8.969) (8.986) (0.9795) (0.9888) (0.999).2093 (0.54).326 (0.0559).38 (0.58).3839 (0.87).6456 (0.6774).4556 (2.4423).826 (3.72) (3.235).0305 (0.90).296 (.29).64 (.7733).393 (0.6597).3648 (0.7972).4905 (0.8074) (0.3757) (0.2355) (.4238) (.3602) (.2307) (0.529) (0.2) (0.3765) (.553) (.4587) (.2439) (7.4547) (7.3285) (6.8368) (0.5333) (0.2) (0.3787) (0.2302) (0.229) 0.55 (.0686) (0.8345) (0.6603) (0.449) (0.3544) (0.3634) (.769) (0.9482) (6.5820) (5.9659) 0.75 (5.3287) 0.40 (0.400) (0.3264) (0.309) 0.90 (0.0855).0942 (0.0545) (0.8052) (0.4880).0644 (0.2435) (0.249) (0.27) (0.0482) (0.935).3458 (0.69) (0.3608) (5.9883).7472 (5.260).9560 (4.242) (0.2489) (0.96) (0.047) * Estmated value for the chage pot (mea of the estmates for the chage pot). The value parethess s the MSE value of the estmato. The bold umber represets the smallest MSE value.

9 Bayesa Estmator of a Chage Pot the Hazard Fucto APPLICATION I ths secto I apply the proposed estmator to data set about the survval tmes for 24 breast-cacer patets (44 of whch are cesored) obtaed from the Ocology Departmet Hacettepe Uversty Hosptal [7]. The data are reported Karasoy ad Kadlar [6]. Applyg the proposed method to ths data set, I obta the followg estmator of the hazard fucto h t t 53 t 53 where, of course, 53 s a estmate of the chage pot. Note that Karasoy ad Kadlar [6] estmated the chage pot as 48 for ths data set. 6. CONCLUSION I ths artcle, I have developed a ew Bayesa estmator for the chage pot the hazard fucto. Ths estmator has bee compared wth the exstg estmators. Smulato results show that the proposed estmator ca be used to obta the most accurate estmate of the chage pot the hazard fucto. 7. REFERENCES. R.C. Gupta, N. Kaa ad A. Raychaudhur, Aalyss of logormal survval data, Mathematcal Bosceces 39, 03-5, F. Tabak, H.G. Muller, J.L. Wag, J.M. Chou ad R.K.P. Su, A chage-pot model for reportg delays uder chage of AIDS case defto, Europea Joural of Epdemology 6, 35-4, C.L. Faucett, N. Scheker ad J.M.G. Taylor, Survval aalyss usg auxlary varables va multple mputato wth applcato to AIDS clcal tral data, Bometrcs 58, 37-47, I. Gjbels ad U. Gurler, Estmato of a chage pot a hazard fucto based o cesored data, Lfetme Data Aalyss 9, 395-4, J. L, A two-stage falure model for bayesa chage pot aalyss, IEEE Trasactos o Relablty 57, , F. Dael ad E. Nader, Parametrc estmato of chage-pots for actual evet data recurret evets models, Computatoal Statstcs ad Data Aalyss 53, , D. Karasoy ad C. Kadlar, Modfed estmators for the chage pot hazard fucto, Joural of Computatoal ad Appled Mathematcs 229, 52-57, J.A. Achcar ad S. Lobel, Costat hazard fucto models wth a chage pot: A Bayesa aalyss usg markov cha Mote Carlo methods, Bometrcal Joural 40, , 998.

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