Variable Selection for Survival Data under Weibull Distribution


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1 Artcle Varable Selecton for Survval Data under Webull Dstrbuton Ujjwal Das 1 Calcutta Statstcal Assocaton Bulletn 681& Calcutta Statstcal Assocaton, Kolkata SAGE Publcatons sagepub.n/home.nav DOI: / Abstract In a wde spectrum of natural and socal scences, very often one encounters a large number of predctors for tme to event data. An mportant task s to select rght ones, and thereafter carry out the analyss. The l 1 penalzed regresson, known as "least absolute shrnkage and selecton operator LASSO" has become a popular approach for predctor selecton n last two decades. The LASSO regresson nvolves a penalzng parameter commonly denoted by λ that controls the extent of penalty, and hence plays a crucal role n dentfyng the rght covarates. In ths artcle, we propose an nformaton theorybased method to determne the value of λ under the accelerated falure tme AFT model wth extreme value dstrbuton. Furthermore, an effcent algorthm s dscussed n the same context. We demonstrate the usefulness of our method through an extensve smulaton study. Keywords Bhattacharya dstance, ndex of resolvablty, Kullback Lebler measure, l 1 penalty, AFT model, tme to event data, Webull dstrbuton 1. Introducton The statstcal analyss of tme to event data s very common n several appled felds, such as bology, medcne, economcs, engneerng, and socal scences. Typcal examples of such an event may be onset of a dsease, death of a subject under study, occurrence of default of a corporate bond, malfunctonng of a system, and so on. It s very frequent to adjust the analyss of those event tmes by ncorporatng the nformaton avalable from covarates. Accelerated falure tme AFT model s one of the ways of modelng the tme to event data wth predctor varables. It relates the logtransformed survval tme lnearly to the covarates. [1,] Stated formally, an AFT model s ressed as: X log Y Z β + W, 1 1 Indan Insttute of Management Udapur, Udapur, Rajasthan, Inda. Correspondng author: Ujjwal Das, Indan Insttute of Management Udapur, Udapur, Rajasthan, Inda. Emal:
2 Das 53 where Y denote the tme to an event or survval tme wth covarate nformaton Z. To be specfc, β β 0,β 1,..., β p s a vector of regresson coeffcents wth β 0 as ntercept, p denotes the number of covarates, s a scale parameter, and W s the random error. A varety of dstrbutons are avalable for modelng W, namely normal, extreme value or logstc dstrbutons. AFT model comes out as an attractve alternatve to the Cox [3] proportonal hazard model for censored falure tme data [4] due to ts physcal nterpretaton smlar to standard regresson. Hutton and Monaghan [5] showed that AFT model s more robust to the model msspecfcaton and yeld narrower confdence nterval for regresson coeffcents due to ts loglnear transformaton. Later, Kwong and Hutton [6] appled the AFT model for analyss of an eplepsy data and cerebral palsy data, and showed that AFT model s superor under certan condtons. Matsushta Hagwara, Shota, Shmada, Kuramoto, Toyokura [7] appled Webull probablty model to understand the lfe table and agepatterns of dsease n Japan. More recently, Swndell [8] showed that AFT model wth Webull dstrbuton s a valuable tool for agng research. Helsen and Schmttlen [9] used Webull regresson model to analyze rght censored marketng tme. Throughout ths artcle, we assume that the tme to event follows a Webull dstrbuton, or alternatvely, we assume that W follows an extreme value dstrbuton. In some practcal studes such as genetcs, researchers may have a large number of covarates p from fewer number of observatons n, and they may need to select only few of those many covarates. Example ncludes typcal mcroarray data set that conssts of thousands of genes from a hundred subjects. Tradtonal selecton methods such as stepwse deleton or best subset selecton, although useful, may perform poorly n hgh dmensonal p >>n stuatons. The lmtatons of the exstng methods of model selecton are mentoned n Breman [10] and Fan and L. [11] As a unfed method of varable selecton for both low and hgh dmenson, penalzed approach has ganed ncreasng popularty n recent years. The penalzed methods wth some condtons on the penalty functons not only retan the good propertes of the old methods, but also enjoy theoretcal justfcatons. Among the convex penalty functons, the least absolute shrnkage and selecton operator LASSO, proposed by Tbshran, [1] has ganed enormous attenton from the researchers. LASSO s defned as the l 1 norm of the parameters: λ β 1, where β s the vector of regresson coeffcents and λ s the tunng parameter or penalzng parameter. The penalzng parameter plays an nfluental role for varable selecton. Larger value of λ exerts hgher penalty on regresson coeffcents, resultng n the ncluson of fewer varables n the model. Conversely, a small value of λ leads to less penalty and, hence, ncluson of many varables. Commonly, a sequence of λ values are generated, and then varables are detected for each value of the seres. Thereafter, a value of λ s chosen by kfold crossvaldaton, and correspondng set of predctors are ncluded n the model. Tbshran [13] used generalzed crossvaldaton for the Cox model. [3] More recently Smon, Fredman, Haste and Tbshran [14] developed an Rpackage for varable selecton n Cox model [3] va LASSO wth λ selected thorough crossvaldaton. Barron and Luo [15] developed the concept of nformaton theoretcally vald l 1 penalty by extendng the work of Grunwald. [16] Usng a smlar rsk analyss, Barron Huang, La, and Luo [17] and Barron and Luo [15] developed the concept of nformaton theoretcally vald l 1 norm penalty functon for lnear models. They obtaned a lower bound on the penalzng parameter whch makes the LASSO penalty nformaton theoretcally vald. In ths artcle, we ntroduce the nformaton theory for tme to event data under the model 1 and obtan the bound for λ. We wll use the lower bound as the value of the penalzng parameter. In addton to that, we propose an effcent algorthm for the AFT model under the assumpton of Webull dstrbuton to select varables followng Barron, Cohen, Dahmen, and DeVore. [18] Any software that performs constraned optmzaton, can be used to mplement the proposed algorthm. The artcle s organzed as follows. A bref descrpton on nformaton theory along wth related concepts and, the determnaton of the bound on penalzng parameter for the AFT model are gven
3 54 Calcutta Statstcal Assocaton Bulletn 681& n Secton. Secton 3 deals wth the algorthm and ts accuracy. Secton 4 ensures the usefulness of the proposed methodology through extensve smulaton studes. The results are presented n a tabular format for dfferent combnatons of n and p wth dfferent censorng proportons. We also compare the performance of our proposed λ wth the same obtaned by Bayesan nformaton crteron BIC. Fnally, some concludng remarks n Secton 5 completes the artcle.. Method Here, we develop the proposed bound on the penalzng parameter. As a measure of nformaton dscrepancy between two probablty dstrbuton functons P and Q, we use Kullback Lebler KL dvergence. [19] It s gven by DP, Q E p log p q px S log dpx, provded P s absolutely contnuous wth respect to Q on the support S. The gves the total ected redundancy for the data qx descrbed by q but governed by p or n other words, extent of data laned by the canddate q when the true dstrbuton s p. Throughout ths artcle, Bhattacharya Reny Hellnger dstance s used as the loss functon to judge the accuracy of the estmate. It helps to dscrmnate between two dstrbuton functons P and Q, and s gven by dp, Q log pxqxdx,. [0] For a thorough dscusson on nformaton measures. [1] Index of Resolvablty: Let L f be the lkelhood characterzed by f and f be the true value of f. Then, the ndex of resolvablty s defned as: { 1 R n f mn f F n D,L f + 1 } n penf, where f s a canddate to estmate unknown f, F s the set of all possble values of f and penf denotes some penalty functon. We use ths ndex to upperbound the statstcal rsk assocated wth the estmates obtaned by achevng the followng mnmzaton: { 1 mn f F n log } L f n penf. 3 The estmator obtaned from 3 s called mnmal complexty estmator. It can be shown that the resson under mnmzaton n 3 converges n probablty to ndex of resolvablty plus a constant entropy, whch ensures that the mnmzaton n 3 s equvalent wth the mnmzaton of the resolvablty ndex, R n f, n. For a detaled dscusson on ndex of resolvablty and ts connecton wth nformaton measures, one may see Barron et al. [18] and Luo, [] and the references theren. From 1, f s the lnear predctor gven by Z β. Let ˆf be the mnmal complexty estmator of f. Then, we measure the assocated rsk of ˆf by E[d,Lˆf ]. We choose the penalzng parameter of LASSO such that E d,lˆf mn β R p { D,L f + λ n β j }. 4 where d,lˆf d,lˆf /n and D,L f D,L f /n are the average Bhattacharya Reny Hellnger dstance and KL measure, respectvely, when averaged across the n ndependent subjects. Luo [] came up wth the lower bound of λ for lnear models. In the next subsecton, we provde a lower bound of λ so that the rsk bound n 4 s attaned under extreme value dstrbuton.
4 Das Determnaton of the Bound on Penalzng Parameter Let X 1, X,..., X n be..d. responses, and C 1, C,...,C n be the pont of censorng for n subjects. For the th subject, we observe ether the event tme X, f the subject erences the event, or some known tme pont C, whchever comes frst. Here, C s assumed to be nonnformatve. In short, for the th subject, we have V,δ,Z, where V mnx,c, δ s the censorng ndcator takng value 1 or 0, dependng on whether the subject erenced the event or censored respectvely, and Z s the covarate nformaton. Then, the densty functon of V s p C V p f V C P f f V, when δ 1 and P C f C when X >C s the survval functon and Z β. Hence, the jont lkelhood functon for n subjects can be wrtten as: δ 0 where p f. s the densty functon of X, P f C P f f f 1,f,..., f n wth f Z v 1,v,..., v n Z v Z [p f v Z] δ [ P f C Z] 1 δ, 5 We assume an extreme value dstrbuton wth locaton f Z β and shape. Then, p C f v p f v s a logwebull or gumbel densty, and P C f V C P f C s the survval functon of the same dstrbuton. Throughout the artcle, the covarates Z are assumed to be fxed, and henceforth, for notatonal smplcty, we wll drop the Z from lkelhood. Under the assumpton of known, we have the followng result that gves the bound for penalzng parameter. Result 1: The l 1 penalzed lkelhood estmator ˆf f β Z ˆβ obtaned by mn β { [ δ n x f + log x ] f + [ 1 δ C f ] } + λ n β 1 attans the rsk bound Ed,Lˆf mn β D,L f + λ n β j for every sample sze provded that λ [ {δ e v f δ e C f 1 } ] log p, 6 In practce, f s replaced by ˆf obtaned from 6, and s known.
5 56 Calcutta Statstcal Assocaton Bulletn 681& Proof: The proof s outlned n the appendx. In general, s unknown, and we state a smlar theorem on the bound of the penalzng parameter. Result : The l 1 penalzed lkelhood estmator ˆf obtaned by mnmzng [ δ x f + log x f + e ɛ 1 x f ] [ ] 1 δ C f + n n n + n log + λ n β 1 7 wth respect to β, attans the rsk bound 1 n Ed,,Lˆf,ˆ mn β, D,,L f, + λ β j + 1 δ n n x f + log p n + log4pn n for every sample sze provded that λ [ { δ + 1 δ e C } ɛ f log e ɛ n] p where ɛ s a small postve number chosen sutably, U f j β j and m 1 s gven n the proof of the theorem. We choose ɛ 1 where n s the sample sze. n The proofs wth all notatons are outlned n the appendx. Remark 1: All the lower bound of λ are on. Snce we wll use the lower bound as the value of λ, we note that λ 0asn whch may make t comparable wth the λ proposed n the lterature, for n example, Knght and Fu [3] for lnear model, and Johnson, [4] Ca, Huang and Tan [5] for censored data. From 6 and 8 we note that the proposed λ depends on the data lkelhood. Remark : We note that the lower bound of λ depends on parameter. The nonlnear form of the lkelhood functon makes the lower bound dependent of unknown quanttes. Durng computaton, the lower bound and, hence, the λ wll be computed n a teratve way. At a gven teraton, the λ wll be obtaned by usng the estmates of prevous teraton. Ths s dfferent from the exstng methods lke BIC or crossvaldaton where a sequence of λ s generated not depend on parameter, and for each value, a model s dentfed. 3. The Algorthm We propose an algorthm for the detecton of predctors n AFT model under the assumpton of extreme value dstrbuton. A smlar algorthm was proposed n Barron et al. [18] n the context of lnear models. For p <n we ft an AFT model wth extreme value dstrbuton to the data, and use the pont estmates as ntal estmate for the algorthm. For p >n, we begn wth β 0 0. The updaton rule from t 1 th step to t th step s: β t αβ t 1 + γi l where the parameters are: α [0, 1], γ R and I l whch s a vector of zero except for l th component whch s 1. We mnmze the objectve functon n 6 or 7 wth respect to α and γ for each l 1,,..., p for known or unknown
6 Das 57 scale parameter. The optmal α t, γ t and I l t are those for whch the value of the objectve functon s mnmum. We update those coordnates and keep others unchanged. We wrte the objectve functon as a functon of α and γ for l th coordnate n the followng. For known from 6 we have: x α Z β t 1 γz l x α Z β t 1 γz l L t α, γ, l 1 δ n + log + 1 n C α 1 δ Z β t 1 γz l + λ α n β t 1 j and for unknown from 7 we have: x α L t δ α, γ, l n + δ n + γ. 8 Z β t 1 γz l x α + log x α +e ɛ 1 1 δ n n log + λ α n Z β t 1 γz l n C α β t 1 j Z β t 1 γz l + Z β t 1 γz l + γ 9 Usng any standard software, one can mnmze 8 and 9. We adopt Rroutne constroptm wth the opton Nelder Mead method for performng the constraned mnmzaton of the nonsmooth functons n 8 and 9. Ths method s partcularly sutable for optmzaton of nonsmooth functon. We contnue the updaton procedure untl some convergence crteron s satsfed or a certan number of
7 58 Calcutta Statstcal Assocaton Bulletn 681& tmes the process s repeated. For unknown shape parameter, we estmate t for a gven ntal estmate of regresson coeffcents. The methodology that we used to estmate s dscussed n detal n Secton Accuracy of the Algorthm Let L f be the lkelhood functon wth unknown parameters or lnear combnaton of parameters f, estmated by ˆf k at k th teraton. Then, we have the followng result: Result 3: Let L ˆf be the mnmal complexty estmate of and L fˆ k be the estmate from k th teraton obtaned by our proposed algorthm. Then, { 1 n log 1 L ˆf k x + λv 1 k nf f n log 1 L f x + λv f + 4V } f, 10 k + 1 where v k p ˆβ j,k and V f p β j wth ˆβ j,k s the estmate of β j at k th teraton. Proof: The proof s gven n the appendx. 4. Numercal Studes We nvestgate the performance of the proposed λ along wth the algorthm through smulatons. We wll use the lower bound of λ as ts value for all numercal nvestgatons. Frst, we create a matrx of 100 rows and 1000 columns by randomly drawng 1000 observatons from a 100dmensonal multvarate normal dstrbuton wth mean 0 and parwse correlaton 0.1. Throughout the smulaton study, we keep ths matrx fxed and use approprate number of rows and columns as desgn matrx under four dfferent scenaros: a n 100, p 50, b n 1000, p 100, for low dmenson, and c n 50, p 100, d n 100, p 1000 for hgh dmenson. For a, we use the frst 50 columns of the matrx, for b we transpose the matrx, and for c, we consder the frst 100 columns wth ther frst 50 rows for numercal studes. Let β denote the true vector of regresson coeffcents. So, β s a vector of length 50 for a, of length 100 for b and c, and of length 1000 for d. In each case, we randomly choose seven elements of β and set them to unty, and rest of the elements are all zero. Let Z be the desgn matrx of approprate order. We generate the logtransformed survval tmes from an extremevalue dstrbuton wth the lnear predctor Z β as locaton parameter. The value of shape parameter s set to unty. Next, the varables are selected through the algorthm dscussed n Secton 3. As we have developed bound on λ separately for known and unknown, the smulaton study s also performed separately, and the results are summarzed n Tables 1 and, respectvely. We compare our proposed method wth the BIC, gven by loglkelhood + k log n, where k denotes the number of predctors selected n the model. For ths, a sequence of λ was generated, and for each of them, we determne the BIC. The optmal model s selected for whch the BIC becomes mnmum. In both tables, n represents the number of subjects, p s the number of covarates as canddate of the model, Cens. Pcnt. gves percentage of censorng, Mthd. represents the method used to select varables, TMDR s the true model detecton rate whch s the percentage of replcatons where the full model all correct seven covarates s detected, Medan and Mean are the number of correct varables detected, and Avg. Incln. s the average model sze, from the
8 Das replcatons. Our proposed nformaton theory based method s represented by InfTh and BIC by BIC n the tables. For unknown, addtonally, we estmate the scale parameter. The estmaton of n our context s dscussed n Subsecton Smulaton for Known Here, we use the lower bound from result 1 as the value of λ. The smulaton results are summarzed n Table 1. Table 1. Summary of Smulaton Results for Known Scale Parameter n p Cens. Pcnt. Mthd. TMDR Medan Mean Avg. Incln InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC Source: Author s own.
9 60 Calcutta Statstcal Assocaton Bulletn 681& We see that for n>p, and up to 40 percent, censorng the correct varable detecton rate s almost 100 percent. Addtonally, for n 1000,p 100, no false covarate s dentfed as nonzero. For n 50, p 100 wth 5 percent censorng, the TMDR s 68 percent and gradually decreases as censorng ncreases. For n 100,p 1000, the true model dentfcaton rate s more than 90 percent, up to 0 percent censorng. The performance sharply decreases for 30 percent or more censorng, though the medan number of correct varables detected s not far from the true ones. In general, the TMDR from our proposed method s hgher than the same from BIC. In all stuatons we consder here, the mean number of correct varables detected s close to average model sze. The phenomenon ndcates the ncluson of few ncorrect varables n the model. 4.. Smulaton for Unknown In realty, s unknown. We start wth ntal value of the regresson coeffcents β 0 0. We replace β by β 0 n 7 and then optmze t as a functon of. We call Roptmzaton routne, optmze, for ths purpose. Wth ths estmate, we estmate λ and then update the regresson coeffcents from 9, followng the algorthm. At the next teraton, frst we update the estmate of λ and then estmate by optmzng 7 wth β replaced by ts updated estmate from frst teraton. Ths new estmate of yelds another estmate of λ, and usng them, we update β. The whole process s repeated untl some convergence crteron s attaned or some fxed number of tmes. It should be remarked that the ntal estmate of s a type of maxmum lkelhood estmate. As before, the frequency of detectng correct nonzero varables along wth the average number of nonzero detecton are provded n Table. From Table, we get a clear pcture about the performance of the proposed algorthm as well as the bound of the penalzng parameter for extreme value dstrbuton under dfferent censorng proportons. For n 50 and p 100 wth 5 percent censorng, the proposed method detects the entre model 60 percent cases. The same for percent and percent censorng are 73. percent and 6.8 percent, respectvely. We also note that at least sx correct varables are detected more than 90 percent and 80 percent cases, respectvely. As the proporton of censorng ncreases, t detects the entre model less number of tmes. Around 30 percent censorng, the true model s detected around 0 percent tmes. For n 100 and p 50, up to 40 percent censorng, we get the detecton of full model more than 90 percent tmes. Next, for n 100 and p 1000 around 5 percent censorng, the detecton s as hgh as 93 percent. It remans almost same f the censorng ncreases to near 10 percent. For 15.5 percent censorng, the detecton s around 86 percent. And, for 0 percent censored data, the entre model s detected near 80 percent replcatons. Thereafter, the detecton goes down to 7 percent for 5 percent censorng. But, for 30 percent censorng, less than 50 percent tmes the true model s detected. It should be mentoned that for known shape parameter study, the correct model was detected 66 percent tmes for around 30 percent censorng. Here also, the TMDR from our proposed method s hgher than the same from BIC. We also notce that the TMDR as well as the the average number of correct varables detected from BIC are slghtly hgher than the same from our proposed method at the cost of hgher average model length, for hgher censorng scenaros. Addtonally, the computaton tme s longer for BIC, whch may be for consderng a long sequence of λ. Generally speakng, we fnd that the performance of correct detecton of our proposed method s excellent rrespectve of known or unknown varance up to 0 percent censorng. Around 5 percent censorng, t s workng well to detect the correct varables. But, we note that for 30 percent censorng, t s detectng at least sx correct varables more than 70 percent tmes not shown n the tables. Lastly, for n 1000 and p 100, the entre model s detected n all replcatons and for all percentages of censorng.
10 Das 61 Table. Summary of Smulaton Result for Unknown Scale Parameter n p Cens. Pcnt. Mthd. TMDR Medan Mean Avg. Incln InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC InfTh BIC Source: Author s own. 5. Dscusson The above results enable us to draw some concluson about the overall performance of the proposed method when the event tme s assumed to follow a Webull dstrbuton. Frst, we have come up wth a justfcaton for the penalzng parameter that makes the varable selecton procedure nformaton theoretcally vald. The bound of λ depends on the probablty model adopted to analyze the data, and hence, t reflects the uncertanty of the data through the model. Second, we have used a greedy algorthm for logwebull model. Generally speakng, our proposed method s able to detect the correct model frequently for both known and unknown shape parameter n all four stuatons consdered n smulaton study. As censorng ncreases, the detecton of correct varable becomes worse. Because of havng a
11 6 Calcutta Statstcal Assocaton Bulletn 681& closed form resson of the survval functon, the tme to perform the numercal erment was short. In the numercal erment, the correlaton among the predctors was 0.1. For hghly correlated predctors, LASSO may not work properly as a varable selecton tool, and one may need some other penalty functon to account for the correlaton. It mght be nterestng to determne the bound on penalzng parameter for those penalty functon and study the performance. We leave t as our future research. Acknowledgements The author s thankful to the anonymous referee and the edtor for ther valuable comments and constructve suggestons that mproved the artcle. Appendx A Here, we outlne the proof of the results 1 and, and show the convergence of the proposed algorthm. Proof of Result 1: From Barronet al., [18] the condton on penalty functon s: penf log Lf X log L f X L E f X X E L f X X + L f 11 Proof: We start wth the frst term from 11. Usng the fact that f we get: p f and andng f around f L f v 1,v,..., v n L f v 1,v,..., v n log Lf x L f x { f f + e v f { } δ e v e f e f } δ { } 1 δ e v f e C f e C f { } 1 δ e C e f e f { e v e f f f } δ { e C e f f f } 1 δ { [δ e v e f f f } { + 1 δ e C e f f f }] [ e f f f { } ] δ e v + 1 δ e C 1
12 Das 63 Consder the second term from the condton 11 L f V 1,V,..., V n E V 1,V,..., V n Lf V E L f V δ Lf V 1 Pδ 1 + E V δ 0 Pδ 0 [ { e f f C e x } e f e f + x f e x f dx e C f e C f + e C f T 1 + T. 13 The ntergral from 13, whch s denoted as T 1, can be evaluated lctly by changng the varable k e x w, say where k e f +e f.for th subject, we have: L f v T 1 v PX C C L f v v v dv C Lf v v dv { C e f +f 1 e v e f e f +f f e +f e f + e f 1 C { ke v 1 e ke C } e v dv } + e f e v dv Smlarly, for our convenence, we wrte T lctly n the followng. T { e C f e C e C f e f e C f + e f } 14
13 64 Calcutta Statstcal Assocaton Bulletn 681& So, the second term from 11 becomes: L E f E L f e f +f e f +e f e f + f e f +e f { 1 e ke C + e C { 1 e ke C + e C e f e f } + e f } e f We use the approxmaton A+B C+D A. Thereafter, andng by Taylor seres up to frst order and snce C p f f, then by the fact that for x 0,e x 1 + x + x, 15 can be wrtten as: L E f E L f e f +f e f +e f e f + f e f +e f e f f e f e f 1 e ke C 1 e ke C + e f + e f 1 C e ke 1 e ke C { 1 + f f + f f } 8 1 e ke C + f f e ke C e f + e f e f + f f e f + e f 1 e ke C {1 + f f }, 16 8 where e ke C s the frst order dervatve of e ke C, and we have used the fact that f log on both sdes of 16 we get p f. Now, takng L E f log E E log E L f L f L f log {1 + f f } f f 8 8 f f 4 17
14 Das 65 Together wth 1, 17 and the fact that f f VV f K penf [ e f f f { δ e x [ e f f f { δ e x [ f f VV f K } + 1 δ e C f f 4 } + 1 δ e C f f {δ e x f δ Luo [] the condton 11 reduces to: ] + K log p f f ± δ 4 4 e C f 1 }] + K log p ] + K log p [{δ e x f δ e C f 1 }] + K log p. 18 Followng the smlar route as shown n Luo, [] we dfferentate the rght hand sde of 18 wth respect to K, and then equatng ths wth zero we get [{ VVf K δ e x f δ e C f 1 }] log p Then, replacng the value of K and wrtng penf λv f n 18 wth V V f, we get: λv f VV f λ [ log p δ e x f δ e C f 1 ] [ log p δ e x f δ e C f 1 ] 19 And, ths completes the proof of the theorem. For unknown, we have another result. Proof of Result : We construct the f n the same way as n known varance, and the representer of denoted by s also constructed, snce s unknown. Followng Barron et al., [18] we consder the representers f, and n 11, we have L f, nstead of L f. From Barron et al., [18] we take L f, K log p + logk + 1, where K wll be determned later, and K + 1 4pn as suggested n Barron et al. [18] We choose as a logarthmc dscretzaton of whch makes ther dfference neglgble n log scale. By choosng ɛ to be a very small postve number, the representer s constructed as e ɛ. Thus, by constructon,. Next, followng the smlar route of Theorem 1 and some algebra, one can show the lower bound: λ [ { δ + 1 δ e C } ɛ f log e ɛ n] p Remark 4: In practce, we need to estmate m 1. Wth our choce of ɛ, wehave e n 1. Then, and are replaced by the estmate ˆ, as dscussed n Secton 4, whch yelds the estmate of m 1 as e n The above bound can be exactly same wth unknown bound derved by Luo [] for ɛ 1 ˆ n, where n s the number of subjects nvolved n the study.
15 66 Calcutta Statstcal Assocaton Bulletn 681& Proof of accuracy of the proposed algorthm: Proof: Let e k 1 n log L ˆf x L ˆf k x + λv k V f. Then, usng 0 we get: e k 1 n log L ˆf x L ˆf k x + λv k V f 1 p log ˆf x δ P ˆf C 1 δ + λv k V f n p ˆf k x P ˆf k C { } 1 log ˆf n δ,k ˆf + e x ˆf,k e x ˆf + 1 δ log [ { }] 1 ˆf,k ˆf log δ + e x ˆf,k e x ˆf + λv k V f n + 1 e C ˆf n 1 δ log e C ˆf,k, e C ˆf e C ˆf,k + λv k V f where ˆf Z ˆβ and ˆf,k Z ˆβ k wth ˆβ k, obtaned at k th teraton, s the estmate of β. To prove the theorem we need to show: e k 1 αe k α V f. 0 It s clear that to have the nequalty 0, we only need to tackle the rato of the survval functons from 0 snce Luo [] have provded a general proof of the theorem for any densty n the absence censorng. For the th subject, we rewrte the rato of the survval functons from 0 n the followng way: log e C ˆf e C ˆf,k e C ˆf e C ˆf,k ᾱ log + log e C ˆf e C ˆf,k 1 e C ˆf ᾱ { e C ˆf,k 1 { } α { e C ˆf e C ˆf e C ˆf,k } α { e C ˆf,k e C ˆf,k 1 }ᾱ e C ˆf,k 1 }ᾱ
16 Das 67 So, to prove 0 we need to show { e C ˆf } α { e C ˆf,k e C ˆf,k 1 }ᾱ 1. 1 Frst we replace ˆf,k by the algorthm that ˆf,k Z β k ᾱ ˆf,k 1 + γz l and then we pck some p customary value for the optmum α t and γ t n such a way that γz l Z β. Usng these, we rewrte the left hand sde of 1 n the followng way: { } α { }ᾱ e C ˆf e C ˆf,k 1 e C ˆf,k { e C ˆf { }ᾱ { e C ˆf ᾱe C ˆf e C ˆf αe C ˆf,k 1 } { } e C ˆf,k 1 } e C ˆf α,k 1 e C ˆf,k e C ˆf,k 1 e C ᾱ ˆf,k 1 αf e C ˆf e C ˆf,k 1 ᾱe C ˆf αe C ˆf,k 1 e C ᾱ ˆf,k 1 αf Let D e denote the denomnator of. We show that D e has maxmum for 0 <α<1. We note that for α 0 or 1, the rato n reduces to 1. For smplcty, we work wth log D e. It can be shown that and log D e α C ˆf,k 1 + C ˆf log D e ˆf,k 1 ˆf α ˆf,k 1 ˆf C ᾱ ˆf,k 1 α ˆf C ᾱ ˆf,k 1 α ˆf 3 4 respectvely. We see from 4 that log D e < 0 for α 0, 1. So, we conclude that log D α e and, hence, D e attans maxmum for α n the open nterval 0, 1. Also, D e cannot have ts maxmum for α 0 or1 snce n that case, to be consstent wth our fndng n 4, D e has to be a constant. As a result, the rato n s less than or equal to 1. Ths completes the proof of the theorem.
17 68 Calcutta Statstcal Assocaton Bulletn 681& References 1. Kalbflesch JK, Prentce RL. Statstcal Analyss of Falure Tme Data nd edton. New York, USA: John Wley & Sons Klen JP, Moeschberger ML. Survval Analyss: Technque for Censored and Truncated Data. London, UK: Sprnger Cox, DR. Regresson Models and Lfe Tables wth Dscussons. J R Stat Soc B. 197; 34: We LJ. The Accelerated Falure Tme Model: A Useful Alternatve to the Cox Regresson Model n Survval Analyss. Stat Med. 199; 11: Hutton JL, Monaghan PF. Choce of Parametrc Accelerated Lfe and Proportonal Hazard Models for Survval Data: Asymptotc Results. Lfetme Data Anal. 00; 8: Kwong GPS, Hutton JL. Choce of Parametrc Models n Survval Analyss: Applcatons to Monotherapy for Eplepsy and Cerebral Palsy. Appl Stat. 003; 5: Matsushta S, Hagwara K, Shota T, Shmada H, Kuramoto K, Toyokura Y. Lfetme Data Analyss of Dsease and Agng by the Webull Probablty Dstrbuton. J Cln Epdemol. 199; 4510: Swndell WR. Accelerated Falure Tme Models Provde a Useful Statstcal Framework for Agng Research. Exp Gerontol. 009; 443: Helsen K, Schmttlen DC. Analyzng Duraton Tmes n Marketng: Evdence for the Effectveness of Hazard Rate Models. Marketg Sc. 1993; 14: Breman L. Heurstcs of Instablty and Stablzaton n Model Selecton. Ann Stat. 1996; 4: Fan J, L, R. Varable Selecton va Nonconcave Penalzed Lkelhood and ts Oracle Propertes. J Amer Stat Assoc. 001; 456: Tbshran R. Regresson Shrnkage and Selecton va the Lasso. J R Stat Soc B. 1996; 58: The LASSO Method for Varable Selecton n the Cox Model. Stat Med. 1997; 16: Smon N, Fredman J, Haste, T, Tbshran, R. Regularzaton Paths for Cox s Proportonal Hazards Model va Coordnate Descent. J Stat Softw. 011; 395: Barron AR, Luo X. MDL Procedures wth l 1 Penalty and Ther Statstcal Rsk. Proceedngs Workshop on Informaton Theoretc Methods n Scence and Engneerng. Tampere, Fnland: Tampere Unversty of Technology; Grunwald P. The Mnmum Descrpton Length Prncple. Cambrdge, MA: MIT Press Barron AR, Huang C, L JQ, Luo X. The MDL Prncple, Penalzed Lkelhoods, and Statstcal Rsk. In: Grunwald P, Myllymak P. Tabus I. Wenberger M, Yu B, edtors. Festschrft for Jorma Rssanen, Tampere Unversty Press. 008a, pp Barron AR, Cohen A, Dahmen W, DeVore R. Approxmatons and Learnng by Greedy Algorthms. Ann Stat. 008; 36: Kullback S. Informaton Theory and Statstcs. New York: Wley reprnted n 1968 by Dover. 0. Bhattacharya A. On a Measure of Dvergence between Two Statstcal Populatons Defned by Probablty Dstrbutons. Bull Calcutta Math Soc. 1943; 35: Ebrahm N, Soof ES, Soyer R. Informaton Measures n Perspectve. Intern Stat Rev. 010; 78: Luo X. Penalzed Lkelhoods: Fast Algorthms and Rsk Bounds. PhD. Thess, Statstcs Department, Yale Unversty; Knght K, Fu W. Asymptotcs for LASSO Type Estmator. Ann Stat. 000; 8: Johnson BA. On LASSO for Censored Data. Elec J Stat. 009; 3: Ca T, Huang J, Tan L. Regularzed Estmaton for the Accelerated Falure Tme Model. Bometrcs. 009; 65:
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