A^VÇÚO 1 33 ò 1 5 Ï 217 c 1 Chnese Journal of Appled Probablty and Statstcs Oct., 217, Vol. 33, No. 5, pp. 517-528 do: 1.3969/j.ssn.11-4268.217.5.8 Regresson Analyss of Clustered Falure Tme Data under the Addtve Hazards Model DI RongRong (School of Mathematcs and Statstcs, Wuhan Unversty, Wuhan, 4372, Chna) WANG ChengYong (School of Mathematcs and Computer Scence, Hube Unversty of Arts and Scence, Xangyang, 44153, Chna) Abstract: Clustered nterval-censored falure tme data often arses n medcal studes when study subjects come from the same cluster. Furthermore, the falure tme may be related to the cluster sze. A smple and common approach s to smplfy nterval-censored data due to the lack of proper nference procedures for drect analyss. For ths reason, we proposed the wthn-cluster resamplng-based method to consder the case II nterval-censored data under the addtve hazards model. Wth-cluster resamplng s smple but computatonally ntensve. A major advantage of the proposed approach s that the estmator can be easly mplemented when the cluster sze s nformatve. Asymptotc propertes and some smulaton results are provded and ndcate that the proposed approach works well. Keywords: addtve hazards model; nterval-censored; wthn-cluster resamplng; semparametrc regresson 21 Mathematcs Subject Classfcaton: 62N2 Ctaton: D R R, Wang C Y. Regresson analyss of clustered falure tme data under the addtve hazards model [J]. Chnese J. Appl. Probab. Statst., 217, 33(5): 517 528. 1. Introducton Case II nterval-censored data s commonly encountered n bomedcne where the event tme of nterest s not observed drectly but known only to le between the two montorng tmes. A few methods have been proposed for regresson analyss of the ntervalcensored data. Zeng et al. [1] dscussed regresson analyss of case II nterval-censored data, usng the addtve hazards model. Wang et al. [2] developed an approach whch s easy to mplement for case II nterval-censored data and allows that the montorng tmes The project was supported by the Natonal Natural Scence Foundaton of Chna (Grant No. 7137166). Correspondng author, E-mal: wchyxf@163.com. Receved June 2, 216. Revsed November 17, 216.
518 Chnese Journal of Appled Probablty and Statstcs Vol. 33 are random and contnuous. It assumes that the falure tme of nterest follows Cox-type models [3]. However, these methods do not take nto account the clustered data. In some cases, falure tme data certanly comes from the same cluster. For example, falure tme can be the tme to dsease occurrence for the patents n the same famly or the same clnc. L et al. [4] proposed an estmatng equaton-based approach for regresson analyss of clustered nterval-censored falure tme data generated from the addtve hazards model whch does not nvolve the estmaton of any baselne hazard functon. Another commonly used statstcal method to analyse clustered falure tme data s the gamma-fralty model, ncorporatng an unobserved random effect known as fralty nto the Cox proportonal hazards model. L et al. [5] proposed a seve estmaton procedure for fttng a Cox fralty model to clustered nterval-censored falure tme data. A two-step algorthm for parameter estmaton was developed and the asymptotc propertes of the resultng seve maxmum lkelhood estmators were establshed. Kor et al. [6] gave a method for analyzng clustered nterval-censored data based on Cox s model. As ponted out n [7] that the addtve hazards model descrbes a dfferent aspect of the assocaton between the falure tme and covarates compared wth the Cox s model and the addtve model could be more plausble than the Cox s model n many applcatons. Ths s especally the stuaton when one s nterested n the rsk dfference as often the case n epdemology and publc health [8]. In ths paper, we consder case II nterval-censored data under the addtve hazards model and the stuaton where the correlated falure tme of nterest may be related to cluster sze. We assume that there exst only two montorng tmes ndependent of the falure tme of nterest under the gven covarate process. We then use the wthn-cluster resamplng (WCR) procedure under the addtve hazards model. WCR s a method for analyzng clustered data n the presence of nformatve cluster sze when estmaton of margnal effects weghted at the cluster level s of nterest. Parameter estmaton wth WCR s based on resamplng replcate data sets, each contanng one observaton from each cluster. In the followng, we present the approach under the addtve hazards model. The rest of the paper s organzed as follows. Secton 2 proposes the model and some notatons used n ths paper. Secton 3 gves a method based on the WCR method by usng the nference procedure proposed by [2] under the addtve hazards model for case II falure tme data, and Secton 4 presents some extensve smulaton studes to assess the performance of the proposed approach. 2. Notaton and Model Suppose that there are n ndependent clusters and each cluster has n j exchangeable subjects for = 1, 2,..., n and j = 1, 2,..., n. Let U j and V j denote the two montorng
No. 5 DI R. R., WANG C. Y.: Regresson Analyss of Clustered Falure Tme Data 519 tmes for the j-th subject n the -th cluster. Let Z j (t) be the correspondng p-dmensonal vector of covarates that may depend on tme t, and T j denote the falure tme of nterest for subject j n the cluster whch s ndependent of montorng tmes U j and V j gven covarate Z j (t). For each (, j), defne δ 1j = I(T j < U j ), δ 2j = I(U j T j < V j ) and δ 3j = 1 δ 1j δ 2j. The observed data are (U j, V j, δ 1j, δ 2j, δ 3j, Z j ( )). It just as ponted out n [9], the cause for cluster szes beng nformatve can be complcated and usually unknown, and some latent varables may mplctly affect the baselne hazard for each cluster and/or covarates. For example, the margnal hazard functon may be assocated wth the cluster sze through the followng fralty mode λ j (t Z j ) = λ (t) + ω β Z j (t), where β s the unknown vector of p-dmensonal regresson coeffcent, ω s the clusterspecfc random effect to account for wthn-cluster correlaton n cluster, and λ (t) s the unknown baselne hazard functon. If cluster szes are gnorable (nonnformatve to survval), the usual margnal addtve hazards model [1] s applcable, gven by λ j (t Z j ) = λ (t) + β Z j (t). (1) Motvated by the work of [2], we model the montorng varables usng Cox-type hazard functons λ U j(t Z j ) = λ 1 (t)e γ Z j(t), (2) λ V j(t U j, Z j ) = I(t > U j )λ 2 (t)e γ Z j(t), (3) where λ 1 (t) and λ 2 (t) denote unspecfed baselne functons, γ s the unknown vector of regresson parameters. For each and j, defne N (1) j (t) = (1 δ 1j)I(U j t), and condtonal on U j, defne N (2) j (t) = δ 3jI(V j t) f t U j and f t < U j. We also defne λ (1) j (t Z j) = λ 1 (t)e Λ (t) e β Z j (t)+γ Z j(t) := λ 1 (t)e β Z j (t)+γ Z j(t) (4) and λ (2) j (t U j, Z j ) = I(t > U j )λ 2 (t)e Λ (t) e β Z j (t)+γ Z j(t) := I(t > U j )λ 2 e β Z j (t)+γ Z j(t), (5) where Z j (t) = t Z j(s)ds, Λ (t) = t λ (s)ds, λ 1 = λ 1 (t)e Λ (t) and λ 2 = λ 2 (t)e Λ (t). Clearly models (4) and (5) satsfy the Cox proportonal hazards model.
52 Chnese Journal of Appled Probablty and Statstcs Vol. 33 3. The WCR-Based Procedure When cluster szes are nformatve, to estmate the unknown parameter vectors β and γ, the estmates and nference based on equaton (2) may be ncorrect. To account for nformatve cluster szes, ths secton wll propose a method based on the wthn-cluster resamplng (WCR) technque. The basc dea behnd the WCR-based procedure s that one observaton s randomly sampled wth replacement from each of the n clusters usng the WCR approach (refer to [11]). For ths, let Q be a postve nteger, we randomly sample one subject wth replacement from each of the n clusters and suppose that the resamplng process s repeated Q tmes. study perod, defne δ q 1 = I(T q Let τ denote a known tme for the length of < V q ) and δq 3 = 1 δq 1 δq 2, the (t); = 1, 2,..., n, t τ}, < U q ), δq 2 = I(U q T q q-th resampled data set denoted by {U q, V q, δq 1, δq 2, δq 3, Zq conssts of n ndependent observatons, whch can be analyzed usng the models (4) and (5) for ndependent data set. The wthn-cluster resamplng estmate s constructed as the average of the Q resample-based estmates. For the q-th resampled data, to estmate β and γ, motvated by [2], we frst estmate γ, and for ths, for = 1, 2,..., n and q = 1, 2,..., Q, we defne Ñ (1)q (t) = I(U q t) and Ñ (2)q (t) = I(V q t) f t U q and f t < U q gven the observed U q. For j = and 1, also defne S (j) 1,γ,q (t, γ) = 1 n S (j) 2,γ,q (t, γ) = 1 n I(t U q Z q )eγ (t) (Z q (t)) j, =1 =1 I(U q < t V q )eγ Z q (t) (Z q (t)) j, where a j = 1 and a for j = and 1. We construct an estmatng functon U q γ (γ) for γ as [ =1 (Z q (t) S(1) 1,γ,q (t, γ) S () 1,γ,q (t, γ) ) dñ (1)q (t) + (Z q = n {Z q (U q ) S(1) 1,γ,q (U q, γ) } =1 S () 1,γ,q (U q, γ) + n { Z q (V q ) S(1) =1 S () (t) S(1) 2,γ,q 2,γ,q (V q 2,γ,q (V q, γ), γ) (t, γ) ) S () 2,γ,q (t, γ) }. dñ (2)q Let γ q be the soluton to Uγ q (γ) =. Next we estmate β gven γ q. For ths, we also defne N (1)q (t) = (1 δ q 1 )I(U q t), N (2)q (t) = δ q 3 I(V q t) for = 1, 2,..., n, q = 1, 2,..., Q, and for j =, 1, let S (j) = 1 n S (j) = 1 n =1 I(t U q )e β (t)+γ Z q (t) ( (t)) j, I(U q < t V q =1 )e β (t)+γ Z q (t) ( (t)) j. ] (t)
No. 5 DI R. R., WANG C. Y.: Regresson Analyss of Clustered Falure Tme Data 521 We propose the estmatng equaton U q β (β, γ q) =, where U q β (β, γ) s defned as ( (1 δ q 1 ) =1 (U q ) S(1) (U q S () (U q, β, γ), β, γ) ) + n δ q 3 =1 ( (V q ) S(1) (V q S () (V q, β, γ), β, γ) where (t) = t Zq (s)ds. Then we can estmate β by β q defned as the root of U q β (β, γ q) =. Furthermore, Wang et al. [2] showed that n( β q β ) can be asymptotcally approxmated by a normal vector wth mean zero and a covarance matrx of β q that can be consstently estmated by Σ q := (Âq β ) 1 Γq (Âq β ) 1 /n, where Âq β and Γ q wll be defned n the Appendx, thus β q s consstent. As t s known to all that sample mean can reduce the system error, after repeatng ths procedure Q tmes, the WCR estmator for β can be constructed as the average of the Q resample-based estmators, whch s β wcr = 1 Q Under some regularty condtons, t can be shown that n( β wcr β ) converges n dstrbuton to a zero-mean normal random vector, and the varance-covarance matrx of β wcr can be consstently estmated by Σ wcr = 1 Σ q 1 Q Q β q. ( β q β wcr )( β q β wcr ). The proof of ths result s sketched n the Appendx. ), 4. Smulaton An extensve smulaton study was conducted to assess the fnte sample performance of the estmates proposed n the prevous sectons. For smplcty, here only consder nonnformatve cases. In the smulaton study, the true covarate Z j generated from the Bernoull dstrbuton B(1,.5). Gven the Z j s, the falure tmes of nterest were assumed to follow model (1) wth λ (t) = 2 or λ (t) = 4, the observaton tmes U j s and V j s, generated from (2) and (3) wth λ 1 (t) = 4, λ 2 (t) = 2 or λ 1 (t) = 8, λ 2 (t) = 4. The cluster sze n was randomly generated from unform dstrbuton U{1, 2, 3, 4, 5, 6, 7}. The results gven below are based on 4 replcatons wth Q = 4 resamples and the number of clusters n = 2 or 4. Table 1 and Table 2 present the results on estmaton of (γ, β ) wth true values (γ, β ) = (, ), (,.2), (,.2), (.2, ), (.2,.2) or (.2,.2). The results nclude the estmated bases (Bas) gven by the averages of the pont estmates mnus the true values, the averages of the standard error estmates (SEE), the samplng standard errors of the
522 Chnese Journal of Appled Probablty and Statstcs Vol. 33 pont estmates (SSE) and the 95% percent emprcal coverage probabltes (CP). The results ndcate that the proposed estmate seems to be approxmately unbased and the proposed varance estmate also seems to be reasonable, and all estmates become better when the sample sze ncreases. Table 1 Smulaton results for estamton of β and γ wth λ = 2, λ 1 = 4, λ 2 = 2 n = 2 n = 4 (γ, β ) BIAS SEE SSE CP BIAS SEE SSE CP (,) γ.1.63.61.9575.1.436.427.9475 β -.15.236.2381.9475.2.174.178.9475 (,.2) γ -.29.58.62.955.37.424.43.9475 β.245.2485.2442.9375 -.2.1767.178.945 (,-.2) γ.5.619.64.9475.1.436.427.9475 β.34.2463.2283.9375.4.1642.1639.9425 (.2,) γ -.3.612.66.95 -.18.418.429.9475 β.61.2447.245.945.37.176.1734.9475 (.2,.2) γ.26.64.611.94 -.8.431.43.945 β.5.2654.2556.94.27.188.1819.945 (.2,-.2) γ -.3.612.66.95 -.8.431.43.945 β.12.2282.2316.95 -.48.1681.1667.95 Table 2 Smulaton results for estamton of β and γ wth λ = 4, λ 1 = 8, λ 2 = 4 n = 2 n = 4 (γ, β ) BIAS SEE SSE CP BIAS SEE SSE CP (,) γ -.17.635.62.94.1.436.427.9475 β -.21.5167.4735.935.41.341.3417.9475 (,.2) γ -.17.63.62.94.1.436.427.9475 β -.17.5323.4837.9375.81.3491.3492.9475 (,-.2) γ -.47.612.63.955.37.424.43.9475 β.135.487.4626.9425 -.163.334.3317.9475 (.2,) γ.26.64.611.94 -.8.436.431.945 β -.12.5143.4891.945.27.3459.348.945 (.2,.2) γ -.3.612.66.95 -.18.418.429.9475 β.172.536.4913.9425 -.22.361.3549.9425 (.2,-.2) γ -.3.612.66.95 -.8.431.432.945 β.66.4712.4721.94.11.3396.344.9475 For comparson, we also consder the correlated falure tmes model used n [4], that
No. 5 DI R. R., WANG C. Y.: Regresson Analyss of Clustered Falure Tme Data 523 s, λ j (t Z j, b ) = λ (t) + β Z j + b (6) wth λ (t) = 2. The latent varables b s were assumed to follow a normal dstrbuton wth zero mean and varance equal to 1/4. The covarates Z j s were generated from the Bernoull dstrbuton wth success probablty p =.5. The montorng varables U j s and V j s were generated from (2) and (3) wth λ 1 (t) = 4 and λ 2 (t) = 2. The cluster sze n was generated from the unform dstrbuton U{2, 3, 4} and the number of clusters n = 2. The results based on 1 replcatons and the WCR method wth Q = 4 resamples for each step. The true regresson parameter γ was taken to be.25, and β was.25, and -.25. Smulated results are lsted n Table 3, and all the results lsted below L, Wang and Sun are extracted drectly from the paper of [4]. These results ndcate that the proposed procedure actually better performance than the method gven by [4]. The proposed method seems to gve smaller bases and standard errors. Ths s because the ndvduals are related and the WCR method take nto account the correlaton compared wth the method gven by [4]. So the WCR method s more effectve. Table 3 Compared smulaton results for estamton wth λ = 2, λ 1 = 4, λ 2 = 2, n = 2 L, Wang and Sun WCR (γ, β ) BIAS SEE SSE CP BIAS SEE SSE CP (,) γ.19.116.181.948.2.62.597.951 β.315.884.8273.975 -.47.246.2298.937 (,.25) γ.73.112.165.946.2.62.597.951 β -.225.8835.833.973.3.2618.243.936 (,-.25) γ -.51.1126.11.952 -.5.59.595.95 β.172.8776.8255.938 -.55.2257.2191.941 (.25,) γ.35.1154.1126.941.7.594.63.946 β.85.961.915.94.44.2454.2393.943 (.25,.25) γ.122.1138.117.948.7.594.63.946 β.731.9634.9133.971.17.269.2526.942 (.25,-.25) γ -.36.1177.1151.941.7.594.63.946 β.992.9643.9112.963 -.13.239.2271.952 Fnally, t can be seem from the Tables 1 3 that γ seems to have smaller standard error than that of β for all the estmates. Ths s because that completely observed data can be used for the estmate of γ, whle only ncompletely observed data for the estmate of β.
524 Chnese Journal of Appled Probablty and Statstcs Vol. 33 Appendx: Proofs of the Asymptotc Normalty of β wcr Proof For = 1, 2,..., n, we frst defne M (1)q M (2)q M (1)q M (2)q whch are martngales. (t) = N (1)q (t) (t) = N (2)q (t) (t) = Ñ (1)q (t) (t) = Ñ (2)q (t) t t t t I(s U q )λ 1(s)e β Zq (s)+γ Zq (s) ds, I(U q < s V q )λ 2(s)e β Zq I(s U q )λ 1(s)e γ Zq (s) ds, I(U q < s V q )λ 2(s)e γ Zq (s) ds, (s)+γ Zq (s) ds, Snce β q s the soluton of the estmatng equaton U q β (β, γ q) =. By the Taylor s expanson, we have U q β (β, γ q ) = U q β ( β q, γ q ) U q β (β, γ q ) = U q β (β, γ q ) β ( β q β ), where β s on the lne segment between β q and β. Rewrtng the above equaton yelds that ( n( βq β ) = 1 U q β (β, γ q )) 1( 1 ) n n β U q β (β, γ q ). Note that n 1 U q β (β, γ)/ β s equal to 1 n = 1 n where Z q (1) =1 + 1 n =1 + 1 n =1 ( (2) S =1 S () ( ( (2) S S () (S (1) ) 2 ) (S () )2 (S (1) dn (1)q ) 2 ) (S () )2 (t) dn (2)q (s) Z q (1) (β, γ, t)) 2 I(U q t)e β (t)+γ Z q (t) dn (1)q (t) ( (s) Z q (2) (β, γ, t)) 2 I(U q t < V q S(1) (β, γ, t) = S (), Zq (2)(β, γ, t) = and N (2)q (t) = n 1 n =1 (t) )e β S () (t)+γ Z q (t) dn (2)q (t) S (1) S (), N (1)q 1 (t) = n S () =1, N (1)q (t), N (2)q (t). It can be easly seen that n 1 U q β (β, γ q )/ β s postve defnte and n 1 U q β (β, γ )/ β converges n probablty to a determnstc and
No. 5 DI R. R., WANG C. Y.: Regresson Analyss of Clustered Falure Tme Data 525 postve defnte matrx denoted by A β, whch can be consstently estmated by Âq β := n 1 U q β (β, γ q )/ β. Averagng over q = 1, 2,..., Q resamples, t yelds that n( βwcr β ) = 1 Q n( βq β ) = 1 Q = A 1 β ( 1 U q β (β, γ q )) 1( 1 ) n n β U q β (β, γ q ) 1 Q U q nq β (β, γ q ) + o p (1). Use the Taylor seres expansons of U q β (β, γ q ) and U q γ ( γ q ) around γ, we have U q β (β, γ q ) U q β (β, γ ) = U q β (β, γ t ) γ t ( γ q γ ), U q γ ( γ ) = U q γ ( γ q ) U q γ (γ ) = U q γ (γ s ) γ s ( γ q γ ), where both γ t and γ s are on the lne segment between γ q and γ. By the consstency of γ q and rewrtng the above equatons yelds that n 1 U q β (β, γ q ) s equal to where 1 { U q n β (β, γ ) + U q β (β, γ t ) } γ t ( γ q γ ) = 1 { U q n β (β, γ ) + 1 U q β (β, γ t )( n γ t 1 Uγ q (γ s ) ) 1 } U q n γ s γ (γ ) = 1 {U q n β (β, γ ) + A q γ(bγ) q 1 Uγ q (γ )} + o p (1) := 1 {a q n 1 (β, γ ) + a q 2 (β, γ ) + A q γ(bγ) q 1 (b q 1 (γ ) + b q 2 (γ ))} + o p (1), =1 ( a q 1 (β, γ) = ( a q 2 (β, γ) = ( b q 1 (γ) = ( b q 2 (γ) = (t) s(1) (t) s(1) ) s () (t) s(1) ) s () ) 1,γ,q (t, γ) s () 1,γ,q (t, γ) 2,γ,q (t) s(1) (t, γ) ) s () 2,γ,q (t, γ) dm (1)q dm (2)q d M (1)q d M (2)q (t), (t), (t), (t),
526 Chnese Journal of Appled Probablty and Statstcs Vol. 33 A q γ and Bγ q are lmts of  q γ(β, γ) = n 1 U q β (β, γ)/ γ and B γ(γ) q = n 1 Uγ q (γ)/ γ at (β, γ ). s (j) l,β,q and s(j) l,γ,q (t, γ) denote the lmts of S(j) l,β,q and S(j) l,γ,q (t, γ), respectvely, for l = 1, 2 and j =, 1. Note that It s easy to show that ( nq) 1 U q β (β, γ ) = n {a q 1 (β, γ ) + a q 2 (β, γ )} + o p (1), =1 Uγ q (γ ) = n {b q 1 (γ ) + b q 2 (γ )} + o p (1), =1 Q changng the order of summaton as 1 Q nq U q β (β, γ q ) = 1 Q {U q nq β (β, γ ) + A q γ(bγ) q 1 Uγ q (γ )} + o p (1) = 1 n =1 := 1 n =1 1 Q U q β (β, γ q ) converge to a normal dstrbuton as n, {a q 1, (β, γ ) + a q 2, (β, γ ) + A q γ(bγ) q 1 (b q 1, (γ ) + b q 2, (γ ))} + o p (1) U (β, γ ) + o p (1), where U (β, γ ), = 1, 2,..., n are ndependent wth mean zero and fnte varance. It thus follows from the multvarate Central Lmt Theorem that ( nq) 1 Q U q β (β, γ q ) s asymptotcally normal wth zero mean. Combnng wth Slutsky s theorem, n( β wcr β ) converges n dstrbuton to a zero-mean normal random vector and covarance matrx can be consstently estmated by n Σ wcr. Wang et al. [2] showed that n( β q β ) can be asymptotcally approxmated by a normal vector wth mean zero and a covarance matrx that can be consstently estmated by n Σ q = (Âq β ) 1 Γq (Âq β ) 1, where wth Γ q = 1 n α q ( β q, γ q )( α q ( β q, γ q )) =1 α q ( β q, γ q ) = â q 1 ( β q, γ q ) + â q 2 ( β q, γ q ) + Âq γ( β q, γ q )( B q γ( γ q )) 1 { b q 1 ( γ q) + b q 2 ( γ q)}. For each resampled data, Var ( β q ) can be consstently estmated by Σ q. Average over the Q resamples, the resultng estmator denoted by Q 1 Q Σ q s also consstent. For the consstent estmator of the covarance matrx of β wcr, smlar to [11], we frst wrte Var ( β q ) = E(Var ( β q data)) + Var (E( β q data)).
No. 5 DI R. R., WANG C. Y.: Regresson Analyss of Clustered Falure Tme Data 527 By the fact of E( β q data) = β wcr, t yelds that Var ( β wcr ) = Var ( β q ) E(Var ( β q data)). Snce ( 1 E(Var ( β q data)) = E Q ( β q β wcr )( β q β wcr ) ), t can be estmated as the covarance matrx based on the Q resamplng estmators β q, q = 1, 2,..., Q, that s Ω = 1 Q ( β q β wcr )( β q β wcr ). Thus the estmated varance-covarance matrx of β wcr s Σ wcr = 1 Σ q 1 Q Q ( β q β wcr )( β q β wcr ). To show the consstency of Σ wcr, t s easy to see that Ω E(Ω) n probablty as n. Actually, ths can be easly shown by applyng the same arguments as those n the proof of [9]. Ths completes the proof. Acknowledgements The authors gratefully acknowledge the recommendatons of the assocate edtor and the revewers that led to an mproved revson of an earler manuscrpt. References [1] Zeng D L, Ca J W, Shen Y. Semparametrc addtve rsks model for nterval-censored data [J]. Statst. Snca, 26, 16(1): 287 32. [2] Wang L M, Sun J G, Tong X W. Regresson analyss of case II nterval-censored falure tme data wth the addtve hazards model [J]. Statst. Snca, 21, 2(4): 179 1723. [3] Cox D R. Regresson models and lfe-tables (wth dscusson) [J]. J. Roy. Statst. Soc. Ser. B, 1972, 34(2): 187 22. [4] L J L, Wang C J, Sun J G. Regresson analyss of clustered nterval-censored falure tme data wth the addtve hazards model [J]. J. Nonparametr. Stat., 212, 24(4): 141 15. [5] L J L, Tong X W, Sun J G. Seve estmaton for the Cox model wth clustered nterval-censored falure tme data [J]. Statst. Bosc., 214, 6(1): 55 72. [6] Kor C T, Cheng K F, Chen Y H. A method for analyzng clustered nterval-censored data based on Cox s model [J]. Stat. Med., 213, 32(5): 822 832. [7] Ln D Y, Oakes D, Yng Z L. Addtve hazards regresson wth current status data [J]. Bometrka, 1998, 85(2): 289 298.
528 Chnese Journal of Appled Probablty and Statstcs Vol. 33 [8] Kulch M, Ln D Y. Addtve hazards regresson for case-cohort studes [J]. Bometrka, 2, 87(1): 73 87. [9] Cong X J, Yn G S, Shen Y. Margnal analyss of correlated falure tme data wth nformatve cluster szes [J]. Bometrcs, 27, 63(3): 663 672. [1] Ln D Y, Yng Z L. Semparametrc analyss of the addtve rsk model [J]. Bometrka, 1994, 81(1): 61 71. [11] Hoffman E B, Sen P K, Wenberg C R. Wthn-cluster resamplng [J]. Bometrka, 21, 88(4): 1121 1134. \{ºx.eàa«mí êâ 8 Û,JJ (ÉÇŒÆêÆ ÚOÆ, ÉÇ, 4372) ] ( næêæ OŽÅ ÆÆ, ˆ, 44153) Á : àa«mí žm~ñyušæïä ïäé 5gÓ a œ/. d, žmœu aœƒ'. du"y Û IíüL, Ïd~ {ü ªÒ {z«mí êâ. ud, JÑaS Ä {5Ä\{ºx.e II.«mí K. as Ä {{üi Œþ OŽ, ù {Ì `³3u3aŒƒ'ž, OCþ u y. ìc5ÿúü [(J?Øy T {k5. ' c: \{ºx.; «mí ; as Ä; Œëê 8 ã aò: O213.9