Research Article Interval Estimation for Extreme Value Parameter with Censored Data

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1 Inernaional Scholarly Research Nework ISRN Applied Mahemaics Volume 2, Aricle ID , 2 pages doi:.542/2/ Research Aricle Inerval Esimaion for Exreme Value Parameer wih Censored Daa Eun-Joo Lee, Dane Walker, David Ellio, 2 Kalyn Mahy, 2 and Seung-Hwan Lee 2 Deparmen of Mahemaics, Millikin Universiy, Decaur, IL 62522, USA 2 Deparmen of Mahemaics, Illinois Wesleyan Universiy, Bloomingon, IL 672, USA Correspondence should be addressed o Eun-Joo Lee, elee@millikin.edu Received 4 April 2; Acceped 3 April 2 Academic Ediors: D. Kuhl and H. a. Larrondo Copyrigh q 2 Eun-Joo Lee e al. This is an open access aricle disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, and reproducion in any medium, provided he original work is properly cied. The Weibull disribuion is widely used in he parameric analysis of lifeime daa. In place of he Weibull disribuion, i is ofen more convenien o work wih he equivalen exreme value disribuion, which is he logarihm of he Weibull disribuion. The main advanage in working wih he exreme value disribuion is ha unlike he Weibull disribuion, he exreme value disribuion has locaion and scale parameers. This paper is devoed o a discussion of saisical inferences for he exreme value disribuion wih censored daa. Numerical simulaions are performed o examine he finie sample behaviors of he esimaors of he parameers. These procedures are hen applied o real-world daa.. Inroducion In medical research, daa documening he ime unil he occurrence of a paricular even, such as he deah of a paien, is frequenly encounered. Such daa is called ime-o-even daa, also referred o as lifeime, survival ime, or failure ime daa, which has in general righskewed disribuion. For his reason, he Weibull disribuion is widely used. In place of he Weibull disribuion, i is ofen more convenien o work wih he equivalen exreme value disribuion in which daa are he logarihm of hose aken from he Weibull disribuion Lawless. Specifically, if Y has a Weibull disribuion wih f y ) λβ λy ) β exp [ λy ) β ], y >,. where λ>andβ> are parameers, hen T log Y has an exreme value disribuion wih b /β and u log λ. The main convenience in working wih he exreme value

2 2 ISRN Applied Mahemaics disribuion is ha unlike he Weibull disribuion, his disribuion has locaion and scale parameers. An excellen review on exreme value disribuions can be found in Coles 2 and Koz and Nadarajah 3. A common feaure of lifeime daa is ha he daa poins are possibly censored. For example, he even of ineres may no have happened o all paiens. A paien undergoing cancer herapy migh die from a road acciden. In his case, he observaion period is cu off before he even occurs. In such a case, he daa is said o be censored, and i would be incorrec o rea he ime-o-deah as lifeime. When daa are censored as in he case of he cancer paien who dies from a road acciden, convenional saisical mehods canno be direcly applied o analyze he daa. Inseady, special saisical mehods are necessary o handle such daa. Censored daa have been sudied by many auhors. Kaplan and Meier 4 proposed he esimae, he so-called produc limi esimae, of he disribuion funcion, Cox 5 inroduced he sill commonly used proporional hazard model, Buckley and James 6 invesigaed he censored regression model, and Prenice 7 sudied wo-sample censored daa. More recen works include, among ohers, Tsiais 8, Wei e al. 9, Wei and Gail, and Lee and Yang. Our objecive in his paper is o sysemaically sudy he inference procedures for he exreme value disribuion wih censored daa. Based on he crieria of he empirical coverage probabiliy and he confidence inerval lengh in he numerical sudies, we observe ha he log ransformaion of he esimae enhances resuls by he usual normal approximaion, and he likelihood raio mehod is effecive for small sample sizes when heavy censoring. Noe ha ideally, we expec ha empirical coverage probabiliies are close o he heoreical coverage probabiliy, and empirical mean lenghs are relaively shor. Similar work on he maximum likelihood esimaion in he Weibull disribuion wih censored daa can be found in Cohen 2. This paper is organized as follows: Secion 2 describes he maximum likelihood esimaes and heir confidence inervals when he lifeime daa includes some censored observaions. Numerical resuls and a graphical mehod for checking he model are presened in Secion 3. This secion also illusraes he procedures using a vaginal cancer daa se for ras. Secion 4 saes he conclusion. 2. Procedures 2.. Maximum Likelihood Esimaion The probabiliy densiy funcion for he exreme value disribuion considered here is f b e u /b exp e u /b) <<, 2. where < u < and b > are locaion and scale parameers, respecively. Suppose ha he daa follows he random censoring model. Consider a random sample consising of n observaions. Le T i, i,...,n be lifeimes ha are independen and posiive random variables wih he densiy funcion f, he disribuion funcion F, and he survival funcion S P T i.lec i,i,...,nbe censoring variables wih he survival funcion

3 ISRN Applied Mahemaics 3 G P C i ha are independen of T i. We observe T i only if T i C i, and so he available daa consiss of pairs i,δ i, i,...,n, where i min T i,c i and δ i { if i T i, if i C i. 2.2 Noice ha δ i I T i C i is he censoring indicaor ha is when T i C i and oherwise. Le f df /d and g dg /d. Then, we have P i, δ i P T i, T i C i f G, P i, δ i P C i, T i >C i g S. 2.3 The above probabiliies can be combined ino he single expression P i, δ i { f i G i } δ i { g i S i } δ i. 2.4 This yields he sampling disribuion of i,δ i, i,...,n n { f i G i } δ i { g i S i } δ i n n G i δ i g i δ i f i δ i S i δ i. 2.5 Knowing ha g and G do no conain any parameer of ineres, we have he likelihood funcion defined as L u, b n f i δ i S i δ i. 2.6 I can be easily shown ha for he exreme value disribuion, he survival funcion is S exp e u /b). 2.7 Hence, he above likelihood funcion can be wrien as L u, b n { b exp i u b )) δi )) e i u /b exp e δ i i u /b }. 2.8 To esimae u, b, we find he values of u, b ha maximize log L u, b by seing he derivaive d log L u, b /d u, b equal o zero and solving for u, b. From 2.8, we have he log likelihood funcion log L u, b { [δ i log b i u b } ] e i u /b δ i e i u /b 2.9

4 4 ISRN Applied Mahemaics which is equivalen o log L u, b r log b δ i i u b e i u /b 2. by leing r n δ i.differeniaing 2. wih respec o u and b in urn and equaing o zero, we obain he esimaing equaions log L { } e z i r, u b log L { } r n δ i z i n z i e z i, b b 2. where z i i u /b. The above equaions can be solved by some numerical echniques such as he Newon- Raphson ieraion or random search o locae he esimaes, û and b, ofu and b, respecively. In his paper, he random search was used for simpliciy, alhough i is compuaionally inensive. From 2., he maximum likelihood equaions can be wrien as r, eẑi ẑ i n δ eẑi i ẑ i r, 2.2 which are equivalen o eû r e i/ b ) b, 2.3 n ie i/ b n e i/ b b r δ i i, 2.4 respecively. Equaion 2.4 can be solved ieraively for b, hen û calculaed from 2.3. To make inferences abou u, b, we can use he fac ha, by he usual large-sample heory, he join disribuion of û and b is approximaely bivariae normal wih mean u, b and covariance marix I u, b. Symbolically, û, b ) [ ] N u, b,i u, b, 2.5

5 ISRN Applied Mahemaics 5 where I is he Fisher informaion marix, defined as ) ) E 2 log L E 2 log L u 2 u b I u, b ) ) E 2 log L E 2 log L. 2.6 u b b 2 I is ofen difficul o evaluae he expecaions in I u, b, so a naural procedure is o use he approximaion û, b ) [ ] N u, b,i, 2.7 where I is he observed informaion marix 2 log L u I 2 2 log L u b 2 log L u b 2 log L b 2. u,b û, b 2.8 Wih z i i u /b, we have 2 log L u 2 r, û, b b2 2 log L b 2 b2 r û, b 2 log L b2 ẑ u b i. eẑi û, b ẑ 2 i eẑi ), 2.9 These yield r I b2 ẑ 2 i eẑi r ẑ 2 i eẑi ẑ 2 i eẑi Inference Procedure From he usual large-sample heory, we have û, b ) [ ] u, b N,,I. 2.2

6 6 ISRN Applied Mahemaics Thus, û u d d N,, b b d22 d N,, 2.22 where he marix I d ij,i,j, 2. From he asympoic normaliy of û, b u, b and he esimaed variance, inference procedures on u, b can be easily obained. For example, possible asympoic α % confidence inervals for u and b are [û z α/2 d, û z α/2 d ], [ b zα/2 d22, b zα/2 d22 ], 2.23 respecively, where z α/2 is he α/2 -percenile of he sandard normal disribuion. However, such an inerval for he scale parameer b may conain negaive values. Recall ha <u< and b>. Similar o Lawless, his problem can be repaired by using he log ransformaion. Le ξ log b and ξ log b. A sandard argumen shows ha log b log b is asympoically equivalen o b/ b. From his, we have [û ] [ ] [ ] û u u û u ξ ξ log b ) log b b, 2.24 b b which is equivalen o û u A b b 2.25 wih A b Therefore, we have û, ξ ) u, ξ N,,AI A ), 2.27 where A is he ranspose of A. Le 2,2 h enry of AI A be m 22. Then, an asympoic α % confidence inerval for ξ is [ ξ zα/2 m22, ξ zα/2 m22 ]. 2.28

7 ISRN Applied Mahemaics 7 Hence, since b e ξ, an asympoic α % confidence inerval for b is [e ξ z α/2 m22,e ξ z α/2 m22 ] Noe ha he inerval always lies in he posiive half of he axis. The procedures based on he normal approximaion are appropriae for quie large sample sizes. An appealing alernaive is o use likelihood raio procedures. Chi-squared χ 2 disribuions, approximaing he disribuions of likelihood raio es, are ofen found o be adequae for small sample sizes. We include he confidence inerval from hese procedures for a comparaive sudy. The procedures are discussed below. Consider he es problem H : b b versus H a : b / b. The likelihood raio es, wih level α, rejecs H when log max H L u, b max Ha L u, b >χ2,α, 2.3 where max H L u, b max u L u, b is achieved a ũ maximizing log L u, b. This yields n e i/b ũ b log r ), 2.3 a which log L u ũ,b Noe ha max Ha L u, b L û, b wih probabiliy. A α % confidence inerval for b is b : log L ũ, b L û, b ) χ 2,α Similarly, a α % confidence inerval for u is L u, b ) u : log L û, b ) χ 2,α, 2.34 where max H L u, b max b L u,b is achieved a b, obained from log L/ b u, b. 3. Numerical Sudies Several experimenal simulaions were carried ou o assess he performance of he confidence inervals discussed in Secion 2. We repor he simulaion resuls based on

8 8 ISRN Applied Mahemaics Table : Simulaion resuls, empirical coverage probabiliy ECP and empirical mean lengh EML of 95% confidence inervals of u and b. ECP EML Censoring n Mehod u b u b 2% 3% 4% 5% 6% 7% LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR LR

9 ISRN Applied Mahemaics 9 Table 2: Confidence inerval for u and b, vaginal cancer daa. Mehod C.I. of u Lengh C.I. of b Lengh Normal , , Log , , LR 5.38, , he crieria of he empirical coverage probabiliy and he empirical mean lengh of he confidence inerval. For he simulaion sudy, he survival daa is aken from he exreme value disribuion wih u andb. The censoring disribuions are normal, where C i c Normal,,wihc chosen o resul in various censoring percenages in he samples. For he sample sizes of n 2, 5, and, resuls were based on 5 repeiions. For he censored daa, he censoring proporions of 2%, 3%, 4%, 5%, 6%, and 7% were used. These censoring proporions were obained from seings of c.75,.33,.8,.47,.87,.3, and.87, respecively. Simulaion resuls based on hese seings are summarized in Table. I should be noed ha alhough he normal approximaion procedures are adequae for quie large samples, he approximaions on which hey are based are raher poor for smallsize samples Lawless. One possible way o solve his problem is a ransformaion. For example, simulaion resuls presened in Table show ha he log ransformaion appears o be one way of alleviaing his problem. Tha is, reaing log b as approximaely normal is preferable o reaing b as approximaely normal. In Table, Mehod denoes he confidence inerval resuls from he large-sample normal approximaion, and Mehod 2 in parenhesis indicaes he confidence inerval resuls by he log ransformaion of he maximum likelihood esimae, b. In he Table, ECP is he empirical coverage probabiliy, and EML means he empirical mean lengh of he confidence inerval. The likelihood raio mehod is denoed by LR. I is known ha he likelihood raio mehod is ofen found o be appropriae for small sample sizes. We now look a he resuls for he censored daa case presened in Table. Overall, he empirical coverage probabiliies of he parameers by Mehod are noiceably improved by Mehod 2 and LR for all sample sizes in every censoring proporion considered. If we resric our aenion o he scale parameer v in Table, he performance of he confidence inerval obained by Mehod is improved by he log ransformaion Mehod 2 for he small sample size a all censoring proporions. I is also observed ha he LR mehod ouperforms Mehod 2 ha is superior o Mehod. The 95% confidence inervals from he 5 simulaions resul in 5 independen Bernoulli random variables, where success occurs when he rue parameer is covered by he confidence inerval and he probabiliy of success is 95%. Thus he 95% error margin for he empirical coverage probabiliy is /5. This implies ha he empirical coverage probabiliy is expeced o fall wihin he inerval.939,.969. For he sample size of n 2, Mehod fails o provide he confidence inervals ha lie in he inerval.939,.969, indicaing ha i does no achieve he heoreical coverage probabiliy of 95%. The confidence inervals by LR fall wihin he inerval.939,.969, whereas Mehod 2 is close bu sill does no achieve he heoreical coverage probabiliy. For moderae and large sample sizes, here seems o be no dominan mehod ha ouperforms he ohers. The confidence inervals by Mehods, 2 and LR approximaely achieve he heoreical coverage, probabiliy of 95%. However, LR ends o be superior o Mehod when censoring

10 ISRN Applied Mahemaics log[ log ^S)] log[ log ^S)] log[ log ^S)] 2% censoring a 4% censoring c 6% censoring e log[ log ^S)] log[ log ^S)] log[ log ^S)] 2 3% censoring b 5% censoring d 7% censoring f Figure : Plos of versus log log Ŝ, n. is heavy. As sample size increases for all censoring proporions, he confidence inerval lengh decreases. I seems ha he confidence inerval lenghs by Mehod are slighly shorer han Mehod 2 and LR bu hese differences are no subsanial. In he case of he locaion parameer u, Mehods, 2, and LR show nearly all of he same performance for all sample sizes when daa are no heavily censored. However, for small sizes wih heavily censored daa, LR appears o ouperform Mehod, achieving he heoreical coverage probabiliy of 95%. Mehod fails o achieve he nominal level in his case. The differences of he confidence inerval lenghs are no subsanial, alhough Mehod is slighly shorer han LR.

11 ISRN Applied Mahemaics versus log log^s))) Figure 2: Plo of versus log log Ŝ, Vaginal cancer daa. We also discuss a graphical mehod for checking he adequacy of he disribuion. The exreme value survival funcion saisfies log log S u /b,so u b log log S. Wih u and b, herefore, log log S is a linear funcion of, and a plo of log log S versus should be roughly linear if he exreme value disribuion is reasonable. When daa are censored, he mos widely used esimae for S is he Kaplan- Meier esimae Kaplan and Meier 4, also referred o as he produc-limi esimae of he survival funcion. The Kaplan-Meier esimae is defined as Ŝ d ) j, 3. j: j n < j where d j represens he number of lifeimes a ime j,andn j is he number of individuals uncensored before j.theŝ was used for S in his paper. Figure shows a linear relaionship beween and log log S, alhough some indisinguishable deviaions from linear rend are deeced under heavy censoring. For a sample size of n, Figure was obained from he same seings as he preceding simulaions. The procedures are applied o a real daa se. Pike 3 gives resuls of a laboraory experimen concerning vaginal cancer in female ras. In his experimen, 9 ras are pained wih he carcinogen DMBA, and he number of days unil he appearance of a carcinoma was observed. A he end of sudy, 7 ou he 9 ras had developed a carcinoma, and his indicaes ha wo of he imes are censored. The censoring proporion is 2/9.5. See Pike 3 and Lawless for deails. In order o check he adequacy of an exreme value model, we consider aplooflog log Ŝ versus. Figure 2 shows his o be roughly linear, suggesing ha he exreme value disribuion could be reasonable. Anoher graphical approach employed by Lawless for he vaginal cancer daa also confirms ha he model is plausible. The esimaion procedures under he exreme value disribuion give ha û and b.649, and he inerval esimaion resuls are summarized in Table 2. In his sudy concerning vaginal cancer for ras, sample size is small and censoring level is low. Therefore, among he

12 2 ISRN Applied Mahemaics hree confidence inervals, he inerval consruced by he likelihood raio mehod would be he mos reliable, especially for he scale parameer of he model. 4. Concluding Remarks In his paper, we have invesigaed he inference procedures for he exreme value disribuion wih censored observaions. The exreme value disribuion is a useful model in he parameric analysis of lifeime daa. Through numerical sudies, he inference procedures, based on he maximum likelihood esimaes, were examined. The usual normal approximaion procedures were enhanced by means of he log ransformaion and he likelihood raio mehod. By analysis of he empirical coverage probabiliies and he empirical mean lenghs of he confidence inervals, we have found ha he likelihood raio mehod is very effecive for small sample sizes when daa are heavily censored. A graphical mehod for checking he adequacy of he disribuion was also discussed. The procedures were hen applied o a real-world daa se. References J. F. Lawless, Saisical Models and Mehods for Lifeime Daa, John Wiley & Sons, New York, NY, USA, S. Coles, An Inroducion o Saisical Modeling of Exreme Values, Springer, New York, NY, USA, 2. 3 S. Koz and S. Nadarajah, Exreme Value Disribuions, Imperial College Press, London, UK, 2. 4 E. Kaplan and P. Meier, Nonparameric esimaion from incomplee observaions, he American Saisical Associaion, vol. 53, pp , D. R. Cox, Regression models and life-ables, he Royal Saisical Sociey, vol. 34, pp , J. Buckley and I. James, Linear regression wih censored daa, Biomerika, vol. 66, pp , R. L. Prenice, Linear rank ess wih righ censored daa, Biomerika, vol. 65, no., pp , A. A. Tsiais, Esimaing regression parameers using linear rank ess for censored daa, The Annals of Saisics, vol. 8, no., pp , L. J. Wei, Z. Ying, and D. Y. Lin, Linear regression analysis of censored survival daa based on rank ess, Biomerika, vol. 77, no. 4, pp , 99. L. J. Wei and M. H. Gail, Nonparameric esimaion for a scale-change wih censored observaions, he American Saisical Associaion, vol. 78, no. 382, pp , 983. S.-H. Lee and S. Yang, Checking he censored wo-sample acceleraed life model using inegraed cumulaive hazard difference, Lifeime Daa Analysis, vol. 3, no. 3, pp , A. Cohen, Maximum likelihood esimaion in he Weibull disribuion based on complee and on censored samples, Technomerics, vol. 7, pp , M. C. Pike, A mehod of analysis of a cerain class of experimens in carcinogenesis, Biomerics, vol. 22, pp. 42 6, 966.

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