M-estimation in regression models for censored data

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1 Journal of Saisical Planning and Inference 37 (2007) M-esimaion in regression models for censored daa Zhezhen Jin Deparmen of Biosaisics, Mailman School of Public Healh, Columbia Universiy, 722 W 68h Sree, New York, NY 0032, USA Received 30 December 2005; received in revised form 7 Sepember 2006; acceped 6 April 2007 Available online 4 May 2007 Absrac In his paper, we sudy M-esimaors of regression parameers in semiparameric linear models for censored daa. A class of consisen and asympoically normal M-esimaors is consruced. A resampling mehod is developed for he esimaion of he asympoic covariance marix of he esimaors Elsevier B.V. All righs reserved. Keywords: Acceleraed failure ime model; M-esimaor; Rank esimaor; Resampling; Semiparameric model; Survival daa; Variance esimaion. Inroducion A way o sudy he covariae effecs on he response is o use he semiparameric linear regression model in which he response is sum of a linear combinaion of covariaes and error erm whose disribuion is compleely unspecified. For uncensored daa, i is well known ha he use of M-esimaion mehods for he model can overcome some of he robusness limiaions of he leas squares approach (Huber, 98; Hampel e al., 986). For righ censored daa, Riov (990), Zhou (992) and Lai and Ying (994) developed M-esimaors for he model. The approach of Zhou (992) is based on a weighed daa approach similar o ha in Koul e al. (98), which requires ha he censoring ime is independen of covariaes. The approaches of Riov (990) and Lai and Ying (994) are exensions of he esimaor of Buckley and James (979) o general M-esimaors. Under cerain condiions, boh Riov (990) and Lai and Ying (994) rigorously proved ha heir esimaing equaion has a soluion ha is n-consisen and asympoically normal. Bu hese resuls do no guaranee consisency of all soluions. They were no able o show ha he esimaing equaion has a unique soluion, nor were hey able o devise an algorihm ha could be used in finding a consisen esimaor. Also, he variance of resuling esimaor is difficul o esimae well because i involves he unknown densiy funcion of error erm and is derivaive. In his noe, we develop a new M-esimaion procedure for he semiparameric linear model wih righ censored daa. The mehod is easy o implemen and yields a class of esimaors which are consisen and asympoically normal. We also develop a resampling mehod for he esimaion of he limiing covariance marix. In he nex secion, he semiparameric linear regression model is inroduced and he new M-esimaion mehod is presened. Simulaion sudies are presened in Secion 3 and some remark is given in Secion 4. Proofs are oulined in Appendix A. Tel.: ; fax: addresses: zj7@columbia.edu, zjin@biosa.columbia.edu (Z. Jin) /$ - see fron maer 2007 Elsevier B.V. All righs reserved. doi:0.06/j.jspi

2 2. Model and M-esimaion Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Le (Y i,x i, δ i ), i =,...,n, be independen and idenically disribued observed daa, where X i is a p covariae, Y i = min{t i,c i wih T i being failure ime and C i being he censoring ime, δ i = {T i C i is he failure indicaor. Throughou he paper, i is assumed ha T i and C i are independen condiional on X i. 2.. Acceleraed failure ime model The semiparameric linear regression model is of form T i = X T i β 0 + ε i, () where β 0 is he unknown rue p parameer of ineres and ε i (i =,...,n) are unobservable independen random errors wih a common bu compleely unspecified disribuion funcion F( ). (Thus, he mean of ε is no necessarily 0). In survival analysis in which he response is he logarihm ransformaion of he survival ime, he model () is he acceleraed failure ime (AFT) model M-esimaion Riov (990), Lai and Ying (994) considered he following esimaing equaion Ψ(β; s)= 0 wih s( ) being a known funcion and Ψ(β; s) = (X i X) δ is(y i Xi T β) + ( δ Y i Xi i) Tβ s(u) d ˆF n,β (u) ˆF n,β (Y i Xi Tβ), (2) where X = (/n) n X i, and ˆF n,b () is he Kaplan Meier esimaor based on daa {(Y i X T i b, δ i), i =,...,n. Specifically, ˆF n,b () = i:e i (b)< [ ] δ i nj=, (3) {e j (b) e i (b) where e i (b) = Y i Xi T b, i =,...,n. Noice ha, if s(u) = u, hen (2) becomes he Buckley James esimaing funcion based on leas squares principle (Buckley and James, 979); if s(u) = u r wih r>, hen (2) becomes esimaing funcion of L r regression; if s(u) = sign(u), hen (2) becomes esimaing funcion of leas absolue deviaion; and (2) becomes he maximum likelihood esimaing funcion when s(u) = f (u)/f (u), where f (u) is he densiy funcion of ε and f (u) is is derivaive. Recenly, Jin e al. (2006a) sudied he leas squares approach wih s(u) = u for he model (2) along he line of Buckley and James (979). Below we exend heir approach for general funcion s(u). Specifically, wih an iniial esimaor ˆβ, which is n consisen, obain a new esimaor ˆβ () by he one-sep Newon Raphson approximaion: ˆβ () = ˆβ U(β; ˆβ,s) β β=ˆβ U(ˆβ ; ˆβ,s), (4) where { [ e U(β; b, s) = (X i X) s(e i (β)) + ( δ i ) i (b) s(u) d ] ˆF n,b (u) s(e i (b)). (5) ˆF n,b (e i (b))

3 3896 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Coninuing his process yields a class of esimaors ˆβ (m) : ˆβ (m) = ˆβ (m ) U(β; ˆβ (m ),s) β β=ˆβ (m ) U(ˆβ (m ) ; ˆβ (m ),s) (6) for m. Under regulariy condiions, i can be shown ha, if a consisen esimaor of β 0 is chosen as he iniial value, hen, for any fixed m, ˆβ (m) should also be consisen. In addiion, ˆβ (m) is asympoically normal if he iniial esimaor is asympoically normal. Acually, i can be shown ha ˆβ (m) is a linear combinaion of iniial esimaor and argeed esimaor of he esimaing equaion U(β; β,s)= 0, see Appendix A. A consisen and asympoically normal iniial esimaor of β 0 can be obained by he rank-based mehod of Jin e al. (2003) or by he leas squares based mehod of Jin e al. (2006a). In paricular, he Gehan-ype rank esimaor ˆβ G can be used as he iniial esimaor ˆβ, which can be calculaed by minimizing he following convex funcion: δ i [e i (β) e j (β)], j= where a = {a<0 a. The minimizaion can be done wih linear programming mehod (Jin e al., 2003). Throughou he sequel, we assume ha he iniial esimaor is ˆβ G. The following assumpions will be made: (C) s( ) is wice coninuously differeniable and s2 () df()<. (C2) X i B for all i for some nonrandom consan B. (C3) F( ) has a wice coninuously differeniable densify funcion f( ) such ha [ ] 2 2 f() df()<, df()<, f() sup { f( + h) + f( + h) d < for some h 0 > 0. h h 0 (C4) D s = lim n n n (X i X)Xi Tṡ(Y i Xi Tβ 0) exiss and is nonsingular, where ṡ( ) is he firs derivaive of s( ). (C5) lim n n n (X i X) j P(C i Xi Tβ 0 X i ) = Γ j () exiss for j = 0,, 2, where a 0 is he ideniy marix, a = a and a 2 = aa T. In addiion, he marix A s is nonsingular, where [ { A s = Γ 2 () Γ ] 0 ()Γ 2 () { F(v)ṡ(v)dv dλ(). (C6) The marix A G is nonsingular, where A G = lim n n 2 j= {e j (β 0 ) (X i X j ) 2 { λ()/λ() dn i (β 0,), λ() = f ()/() and N i (β 0,)= δ i {e j (β 0 ). The condiion (C2) is a echnical assumpion which may be relaxed o max i X i =O(n α ) for any α > 0. The condiion (C3) is assumed o obain linear expansion of U(β; b, s). The condiions (C4) and (C5) hold if he vecor of covariaes does no lie in a lower-dimensional hyperplane. The condiion (C6) is assumed o obain he asympoic resuls of he iniial esimae ˆβ G.

4 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Theorem. Under he condiions (C) (C6), ˆβ (m) is consisen and n(ˆβ (m) β 0 ) is asympoically normally disribued as n for each m. The proof of Theorem is given in Appendix A. The limiing covariance marices of boh ˆβ G and ˆβ (m), however, involve he unknown hazard funcion λ( ) of error erm ε. To esimae he limiing covariance marices, we develop a resampling procedure o approximae he disribuion of ˆβ (m) Variance esimaion Le ˆβ G be a minimizer of Z i Z j δ i [e i (β) e j (β)], j= where Z i (i =,...,n) saisfy following condiion (C7): (C7) Z i (i =,...,n) are independen and idenically disribued posiive random variables wih a known disribuion saisfying E(Z i ) = var(z i ) =. Furhermore, define U (β; b, s) = Z i (X i X)[s(e i (β)) + ( δ i )ξ is (b) ξ s (b)], (7) where ξ is (b) = e i (b) s(u) d ˆF n,b (u) ˆF n,b (e i(b)) s(e i (b)) wih ˆF n,b () = i:e i (b)< [ ] Z i δ i nj=, (8) Z j {e j (b) e i (b) and ξ s (b) = n n [s(e i (b) + ( δ i )ξ is (b)]. Then define he ieraive sequence: ˆβ = ˆβ G and ˆβ (m) = ˆβ (m ) U (β; ˆβ (m ),s) β β=ˆβ (m ) for m. Le F n denoe he σ-field generaed by he original daa (Y i, δ i,x i )(i=,...,n). Theorem 2. Under he condiions (C) (C7), ˆβ (m) is consisen and P( n(ˆβ (m) ˆβ (m) ) G F n ) P( n(ˆβ (m) β 0 ) G) 0 in probabiliy as n for any finie p- dimensional recangular G. U (ˆβ (m ) ; ˆβ (m ),s) (9) The proof of Theorem 2 is given inappendixa. Theorem 2 shows ha he condiional disribuion of n(ˆβ (m) ˆβ (m) ) given he daa (Y i, δ i,x i )(i=,...,n)is asympoically equivalen o he asympoic disribuion of n(ˆβ (m) β 0 ). Consequenly, he disribuion of ˆβ (m) can be approximaed by a large number of realizaions of ˆβ (m) afer repeaedly

5 3898 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Table Summary saisics for he simulaion sudies Error Censoring (%) L r esimaors r =.5 r = 2.5 Bias SE SEE CP Bias SE SEE CP Normal β β Exremevalue β β Bias, bias of he parameer esimaor; SE, sandard error of he parameer esimaor; SEE, mean of he sandard error esimaor; CP, coverage probabiliy of he 95% Wald-ype confidence inerval. generaing he random variables (Z,...,Z n ). The empirical disribuion of ˆβ (m) can hen be used o approximae he disribuion of ˆβ (m). The empirical perceniles of ˆβ (m) or Wald mehod can be used o obain confidence inervals for individual componens of β 0. A ypical way of generaing Z i is o use he sandard exponenial disribuion which has mean and variance. 3. Simulaion sudies We conduced simulaion sudies o examine he finie sample performance of he proposed mehods. The failure imes were generaed from he model log T = 2 + X + X 2 + ε, where X was generaed from he Bernoulli disribuion wih 0.5 success probabiliy, X 2 was generaed from normal disribuion wih mean 0 and sandard deviaion 0.5, and ε had he sandard normal, exreme-value disribuion. The censoring imes were generaed from he uniform disribuion U[0,c], where c was chosen o yield a desired level of censoring. We considered a sample size of 00. The L r norm wih r =.5 and 2.5 was considered for he M- esimaion. We esimaed β and β 2 wih hree ieraions wih he resampling being based on 000 realizaions of sandard exponenial random variables. The resuls based on 000 simulaed daa ses were summarized in Table. The parameer esimaor is virually unbiased. The resampling procedure esimaed sandard errors accuraely. Mos of he coverage probabiliies of confidence inervals were close o he 0.95 confidence coefficien. The resuls also showed ha he proposed mehods perform slighly beer for he exreme-value error case han he sandard normal error case. 4. Remark In his noe, we propose a new class of M-esimaion procedure for he semiparameric linear regression model for censored daa. The M-esimaor ˆβ saisfying Ψ(ˆβ; s) = 0 can be obained if he convergence of ˆβ (m) is reached as m. I is also noed ha he proposed approach can be exended o he analysis of muliple evens daa including clusered failure ime daa along he line of Jin e al. (2006a, b).

6 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Acknowledgmens I hank Dr. Melas and wo anonymous referees for heir helpful consrucive commens. This research was suppored in par by he Naional Science Foundaion career award DMS Appendix A. Proofs of asympoic resuls We firs prove he following Lemma. I will be used in he proof of Theorems and 2. Lemma. Under he condiions (C) and (C3), δ i s(e i (β 0 )) + ( δ i ) e i (β 0 ) { = Es(ε i ) + s() dm i (), (A.) where M i () = N i (β 0,) 0 I{e i(β 0 ) uλ(u) du. Proof. Noice ha { s() = δ i s(e i (β 0 )) δ i By inegraion by par, { ei (β 0 ) s() dm i () e i (β 0 ) ei (β 0 ) λ() d = = { s() e i (β 0 ) λ() d. + e i (β 0 ) + Es(ε i ). Consequenly, (A.) holds. Proof of Theorem. I was shown in Jin e al. (2003) ha ˆβ = ˆβ G is consisen and n(ˆβ G β 0 ) is asympoically normally disribued. We firs prove he heorem when m =, hen prove when m> by he inducion mehod. (i) Consisency of ˆβ (). Observe ha U(β; ˆβ,s) = (X i X)X β i T ṡ(e i(ˆβ )). β=ˆβ By assumpions (C) (C4) and Slusky s Theorem, U(β; ˆβ,s) D s n β β=ˆβ (A.2) as n. Wrie U(b) = U(b; b, s). Wih he argumens similar o hose in Lai and Ying (994) we can show ha, uniformly in b β 0 n /3, U(b) = U(β 0 ) na s (b β 0 ) + o(n /2 + n b β 0 ) (A.3)

7 3900 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) almos surely. In paricular, since n(ˆβ β 0 ) = O P (), U(ˆβ ) = U(β 0 ) na s (ˆβ β 0 ) + o P (n /2 ). Therefore, (/n)u(ˆβ ) = o P (). This ogeher wih (A.2) and expression (4), he consisency of ˆβ () follows. (ii) Convergence of n(ˆβ () β 0 ). By he expression (4) and (A.2) (A.3), n(ˆβ() β 0 ) = [ ] n(ˆβ β 0 ) + Ds n U(β ) A s n(ˆβ β 0 ) + o P () = (I D s A s ) n( β G β 0 ) + D U(β n 0 ) + o P (). s (A.4) Noe ha U(β 0 ) = = + (X i X)( δ i ) { e (X i X) δ i s(e i (β 0 )) + ( δ i ) i (β 0 ) s(u) d F β0 (u) F β0 (e i (β 0 )) { e (X i X) δ i s(e i (β 0 )) + ( δ i ) i (β 0 ) F(e i (β 0 )) { e i (β 0 ) s(u) d F β0 (u) F β0 (e i (β 0 )) e i (β 0 ). (A.5) I follows from he maringale represenaion of he Kaplan Meier esimaor (Fleming and Harringon, 99, p. 98) ha, almos surely, e i (β 0 ) s(u) d F β0 (u) e i (β 0 ) = n F β0 (e i (β 0 )) i () dm j () + o(n /2 ) (A.6) j= for some random process i (). By (A.) in Lemma and (A.5) (A.6), [ { ] U(β 0 ) = (X i X) s() + x () dm i () + o(n /2 ), (A.7) where x () is he limi of n n (X i X)( δ i ) i (). As shown by Jin e al. (2003, 2006b), n( βg β 0 ) = A G n where he η i () s are nonrandom funcions. Consequenly, by (A.4) and (A.7) (A.8), n( β() β 0 ) = n H i + o P (), η i () dm i () + o P (), where ( [ { H i = (I Ds A s )A G η i() + Ds (X i X) s() ]) + x () dm i (). (A.8) (A.9) The convergence of n( β () β 0 ) follows from he mulivariae cenral limi heorem. Is asympoic variance is he limi of (/n) n Hi 2 as n.

8 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) (iii) Consisency of ˆβ (m) and he convergence of n(ˆβ (m) β 0 ) for m>. The proof can be obained by he inducion mehod wih he use of he similar argumens in (i) and (ii) repeaedly. In fac, i can be shown ha n( β(m) β 0 ) = (I Ds A s ) n( β (m ) β 0 ) + Ds U(β n 0 ) + o P (). (A.0) Consequenly, by he inducion mehod, he consisency of ˆβ (m) follows and n( β(m) β 0 ) = n H im + o P (), where ( H im = (I Ds A s ) m A G η i() [ { { + I (I Ds A s ) m A s (X i X) s() ]) + x () dm i () (A.) for each fixed m>. Thus, n( β (m) β 0 ) is asympoically normal as n wih he variance as he limi of (/n) n Him 2 as n. This complees he proof of Theorem. Proof of Theorem 2. The consisency of ˆβ = ˆβ G and he convergence of n(ˆβ ˆβ ) were shown in Jin e al. (2006b). We firs prove he heorem when m =, hen prove when m> by he inducion mehod. (i) Consisency of ˆβ () and convergence of he condiional disribuion of n(ˆβ () ˆβ () ) given F n. Noice ha U (β; ˆβ (m ),s) = β Z i (X i X)Xi T ṡ(e i(β)). Since EZ i =, by he srong law of large number, as n, (Z i )(X i X)Xi T n ṡ(e i(β)) 0. If ˆβ (m ) is consisen, by Slusky s heorem, as n, n (X i X)Xi T ṡ(e i(ˆβ (m ) )) D s, which implies ha, as n, Z i (X i X)Xi T n ṡ(e i(ˆβ (m ) )) D s. (A.2) Since ˆβ is consisen, (A.2) holds for ˆβ. Thus, as n, U (β; ˆβ,s) D s. n β β=ˆβ Wrie U (b) = U (b; b, s). By incorporaing he random weighs Z i ino he derivaion of he asympoic lineariy in Lai and Ying (994), we can esablish he following linear approximaion analogous o (A.3): U ( β ) = U ( β ) na s ( β β ) + o(n /2 + n β β 0 +n β β 0 ) (A.3)

9 3902 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) almos surely. The slope marix A s remains he same as in (A.2) since EZ i = and A s is he limi of he gradien of n EU (b) a b = β 0. Observe ha where U ( β ) U( β ) = I + II, I = II = (Z i )(X i X)[s(e i ( β )) + ( δ i )ξ is (ˆβ ) ξ s (ˆβ )], (X i X)( δ i ) ξ is (ˆβ ) e i ( β ) s(u) d F β (u). F β (e i ( β )) Noice ha e i ( β ) s(u) d F β (u) s(e i ( β )) + ( δ i )ξ is (ˆβ ) = δ i s(e i ( β )) + ( δ i ) F β (e i ( β )). Wih expansion a β 0, i can be shown ha { s(e i ( β )) + ( δ i )ξ is (ˆβ ) ξ e s (ˆβ ) = δ i s(e i (β 0 )) + ( δ i ) i (β 0 ) Since E(Z i F) = 0, we have { I = (Z i ) (X i X) s() Es(ε i ) + o P (n /2 ). dm i () + o P (n /2 ). As in Jin e al. (2006a), by approximaing F β F β wih a weighed sum of Z i, we can show ha II = = e i ( β ) s(u) d F β (u) (X i X)( δ i ) F β (e i ( β )) (Z i ) x () dm i () + o P (n /2 ). e i ( β ) s(u) d F β (u) F β (e i ( β )) By (A.3) (A.6), we obain [ { ] U ( β ) = U( β ) + (Z i ) (X i X) s() + x () dm i () na s ( β β ) + o(n /2 + n β β 0 +n β β 0 ). Because of (A.2) and (A.7), he difference beween he expressions (9) and (6) for m = leads o n( β () β () ) = [ { ] D n s (Z i ) (X i X) s() + x () dm i () + (I Ds A s ) n( β β ) + o( + n β β 0 + n β β 0 ). I was shown by Jin e al. (2003, 2006b) ha n( β β ) = A n G (Z i ) η i () dm i () + o P (). (A.4) (A.5) (A.6) (A.7) (A.8)

10 Z. Jin / Journal of Saisical Planning and Inference 37 (2007) Since n( β β 0) = O P () and n( β β 0 ) = O P (), wehave n( β () β () ) = n (Z i )H i + o P (), (A.9) where H i is he same as ha in (A.9). Noe ha condiional on F n, we have ha (Z i )H i are independen wih mean 0 and covariance Hi 2. Consequenly, ˆβ () is consisen and he condiional disribuion of n( β () β () ) given F n converges o he asympoic disribuion of n( β () β 0 ) in probabiliy. (ii) Consisency of ˆβ (m) and convergence of he condiional disribuion of n(ˆβ (m) ˆβ (m) ) given F n for m>. By replacing ˆβ and ˆβ wih ˆβ (m ) and ˆβ (m ), respecively, in he par () and wih similar argumens herein, we can show ha β (m) β (m) = n D s By inducion, (Z i ) [ { (X i X) s() + (I D s A s )( β (m ) β (m ) ) + o P (n /2 ). ] + x () dm i () β (m) β (m) = (I Ds A s ) m ( β β ) + n {I (I Ds A s ) m A s [ { ] (Z i ) (X i X) s() + x () dm i () + o P (n /2 ). Combined wih (A.8), we have n( β (m) β (m) ) = n (Z i )H im + o P (). (A.20) Consequenly, β (m) is consisen and he condiional disribuion of n( β (m) β (m) ) given F n converges o he limiing disribuion of n( β (m) β 0 ) in probabiliy. This complees he proof of Theorem 2. References Buckley, J., James, I., 979. Linear regression wih censored daa. Biomerika 66, Fleming, T.R., Harringon, D.P., 99. Couning Processes and Survival Analysis. Wiley, New York. Hampel, F.R., Ronchei, E.M., Rousseeuw, P.J., Sahel, W.J., 986. Robus Saisics, he Approach Based on Influence Funcions. Wiley, NewYork. Huber, P.J., 98. Robus Saisics. Wiley, New York. Jin, Z., Lin, D.Y., Wei, L.J., Ying, Z., Rank-based inference for he acceleraed failure ime model. Biomerika 90, Jin, Z., Lin, D.Y., Ying, Z., 2006a. On leas-squares regression wih censored daa. Biomerika 93, Jin, Z., Lin, D.Y., Ying, Z., 2006b. Rank regression analysis of mulivariae failure ime daa based on marginal linear models. Scand. J. Sais. 33, 23. Koul, H., Susarla, V., Van Ryzin, J., 98. Regression analysis wih randomly righ-censored daa. Ann. Sais. 9, Lai, T.L., Ying, Z., 994. A missing informaion principle and M-esimaors in regression analysis wih censored and runcaed daa. Ann. Sais. 2, Riov, Y., 990. Esimaion in a linear regression model wih censored daa. Ann. Sais. 8, Zhou, M., 992. M-esimaion in censored linear models. Biomerika 79,