纵向数据与生存数据的半参数联合模型研究
|
摘要
在临床医学研究中经常要对一个反应变量作纵向观测,同时又对另一感兴趣的事件发生的时间作记录。一个典型的例子就是在爱滋病的研究中既有CD4+和HIV病毒数量的纵向测量,也有爱滋病发作时间和病人死亡时间记录。在科学研究和临床试验中,我们往往对纵向观测量与事件发生的时间(比如病人死亡时间)之间的关系感兴趣,这种研究需要纵向数据和生存数据两方面的理论,有一定的复杂性,既有一定的理论意义又有实际应用价值。
本文主要分为以下四个部分：
第一章,我们介绍了本文研究工作的实际背景与解决相应问题的实际意义,概述了前人的研究方法和已有的成果,并综述了本文的主要工作。
第二章,我们对纵向数据半参数回归模型采用拟高斯估计的方法,是对重复测量数据分析方法的一个推广。通常的一个广泛接受的经典方法是基于广义线性模型和拟似然估计的“广义估计方程”,但是该方法有某些理论上的缺陷。我们建议的方法是通过极大化一个工作似然函数从而避免了上述理论缺陷。在理论上,我们证明了所得估计的相合性和渐近正态性。
第三章,我们研究了生存数据具有加速危险因子的加乘危险模型。本模型包含很多常见的生存分析的模型作为其特例,比如比例危险模型、加法模型、加乘危险模型和加速危险模型等。此模型与Chen和Wang(2000)[12]的区别在于本模型中的协变量被划分为三类,除了加速危险因子、乘性危险因子外还含有加性危险因子,从而回归模型中的回归参数相应分为反映协变量作用的加速危险的效应、乘法效应和加法效应,这样在评价协变量对反应变量的效用时能给出更好的解释。在适当的正则条件下证明了所得估计的相合性和渐近正态性；对累积基准危险率函数给出了Breslow-型估计,并给出了其弱收敛性的证明。我们建立的模型对生存数据的建模分析提供了一种新的选择。
第四章研究了纵向数据与生存数据的半参数联合模型。假定纵向数据满足半参数混合效应模型,假定生存数据服从含有随机效应的比例危险模型。感兴趣的问题首先是纵向数据过程的刻画,同时也感兴趣生存时间与其他协变量之间的关系。该模型是现有很多模型的推广,对给定数据下的模型选择提供了新的方法。我们用B-样条方法将非参数项的估计转化为参数估计问题,用蒙特卡洛EM算法给出了参数的极大似然估计,并用bootstrap方法得到参数估计的标准差的估计。基于一个临床试验的实际例子说明了本模型的应用。最后,介绍了有待进一步研究的问题.
In many longitudinal clinical studies, it is common that both longitudinal mea-surements of a response variable and the time to some event of interest are recorded during follow-up. A typical example is the AIDS study where CD4count and viral load are collected longitudinally and the time to AIDS or death is also monitored. It is of scientific and clinical interest to relate such longitudinal quantities to a later time-to-event clinical endpoint such as patient survival. The research needs theories about longitudinal and survival data, which has some complications in the study. Our study has more theoretical and practical value.
This thesis consists of four parts as follows:
In Chapter1, we introduce first the background of the questions and the results which have been obtained in recent years. Following, we introduce in general the results that we obtain in this thesis.
In Chapter2, we propose a method by using quasi gaussian estimation for the semi-parametrical longitudinal data models, which develop the methods for the analysis of repeated measures. More recent methodology, based on generalized linear models and quasi-likelihood estimation, has gained generalized estimating equations. But this also has theoretical problem. The method which we proposed by maximizing a working likelihood function avoids such theoretical problem. By using the classical methods, we obtain and prove the consistency and asymptotic normality on the proposed estimator.
In Chapter3, we have studied a general class of additive-multiplicative model with accelerated hazard factor for survival data. This general class model includes some popular classes of models as subclasses. The model is different from Chen and Wang(2001)[11],the estimators for the vector of regression parameters include addi-tive effect of covariates. The resulting estimators are proven to be consistent and asymptotically normal under appropriate regularity condition. Weak convergence of the Breslow-type estimator for the cumulative baseline hazard function is also estab-lished. Our model is an extended model of Chen and Jewell(2001)[11], which may provide a tool to choose modelling more appropriate for a given data set.
In Chapter4, We study joint modeling of survival and longitudinal data. In this paper, we study a general class of semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data. The longitudinal data are assumed to follow a generalized semiparametric mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on the longitudinal data process, which is infor-matively censored, or on the hazard relationship. Our model is an extended model of many current model, which may provide a tool to choose modelling more appropriate for a given data set. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their stan-dard errors using a bootstraping method. We illustrate our approach with a concrete clinical trial example. Finally, we introduce some aspects which we can do a further study or promotion by the use of our methods in this thesis.
本文主要分为以下四个部分：
第一章,我们介绍了本文研究工作的实际背景与解决相应问题的实际意义,概述了前人的研究方法和已有的成果,并综述了本文的主要工作。
第二章,我们对纵向数据半参数回归模型采用拟高斯估计的方法,是对重复测量数据分析方法的一个推广。通常的一个广泛接受的经典方法是基于广义线性模型和拟似然估计的“广义估计方程”,但是该方法有某些理论上的缺陷。我们建议的方法是通过极大化一个工作似然函数从而避免了上述理论缺陷。在理论上,我们证明了所得估计的相合性和渐近正态性。
第三章,我们研究了生存数据具有加速危险因子的加乘危险模型。本模型包含很多常见的生存分析的模型作为其特例,比如比例危险模型、加法模型、加乘危险模型和加速危险模型等。此模型与Chen和Wang(2000)[12]的区别在于本模型中的协变量被划分为三类,除了加速危险因子、乘性危险因子外还含有加性危险因子,从而回归模型中的回归参数相应分为反映协变量作用的加速危险的效应、乘法效应和加法效应,这样在评价协变量对反应变量的效用时能给出更好的解释。在适当的正则条件下证明了所得估计的相合性和渐近正态性；对累积基准危险率函数给出了Breslow-型估计,并给出了其弱收敛性的证明。我们建立的模型对生存数据的建模分析提供了一种新的选择。
第四章研究了纵向数据与生存数据的半参数联合模型。假定纵向数据满足半参数混合效应模型,假定生存数据服从含有随机效应的比例危险模型。感兴趣的问题首先是纵向数据过程的刻画,同时也感兴趣生存时间与其他协变量之间的关系。该模型是现有很多模型的推广,对给定数据下的模型选择提供了新的方法。我们用B-样条方法将非参数项的估计转化为参数估计问题,用蒙特卡洛EM算法给出了参数的极大似然估计,并用bootstrap方法得到参数估计的标准差的估计。基于一个临床试验的实际例子说明了本模型的应用。最后,介绍了有待进一步研究的问题.
In many longitudinal clinical studies, it is common that both longitudinal mea-surements of a response variable and the time to some event of interest are recorded during follow-up. A typical example is the AIDS study where CD4count and viral load are collected longitudinally and the time to AIDS or death is also monitored. It is of scientific and clinical interest to relate such longitudinal quantities to a later time-to-event clinical endpoint such as patient survival. The research needs theories about longitudinal and survival data, which has some complications in the study. Our study has more theoretical and practical value.
This thesis consists of four parts as follows:
In Chapter1, we introduce first the background of the questions and the results which have been obtained in recent years. Following, we introduce in general the results that we obtain in this thesis.
In Chapter2, we propose a method by using quasi gaussian estimation for the semi-parametrical longitudinal data models, which develop the methods for the analysis of repeated measures. More recent methodology, based on generalized linear models and quasi-likelihood estimation, has gained generalized estimating equations. But this also has theoretical problem. The method which we proposed by maximizing a working likelihood function avoids such theoretical problem. By using the classical methods, we obtain and prove the consistency and asymptotic normality on the proposed estimator.
In Chapter3, we have studied a general class of additive-multiplicative model with accelerated hazard factor for survival data. This general class model includes some popular classes of models as subclasses. The model is different from Chen and Wang(2001)[11],the estimators for the vector of regression parameters include addi-tive effect of covariates. The resulting estimators are proven to be consistent and asymptotically normal under appropriate regularity condition. Weak convergence of the Breslow-type estimator for the cumulative baseline hazard function is also estab-lished. Our model is an extended model of Chen and Jewell(2001)[11], which may provide a tool to choose modelling more appropriate for a given data set.
In Chapter4, We study joint modeling of survival and longitudinal data. In this paper, we study a general class of semiparametric likelihood approach to joint modeling of longitudinal and time-to-event data. The longitudinal data are assumed to follow a generalized semiparametric mixed effects model, and a proportional hazards model depending on the longitudinal random effects and other covariates is assumed for the survival endpoint. Interest may focus on the longitudinal data process, which is infor-matively censored, or on the hazard relationship. Our model is an extended model of many current model, which may provide a tool to choose modelling more appropriate for a given data set. We propose to obtain the maximum likelihood estimates of the parameters by an expectation maximization (EM) algorithm and estimate their stan-dard errors using a bootstraping method. We illustrate our approach with a concrete clinical trial example. Finally, we introduce some aspects which we can do a further study or promotion by the use of our methods in this thesis.
引文
[1]Altman, N.S.. Kernel smoothing of data with correlateed error[J]. Joural of the American Statistical Association,85,749-759
[2]Andersen, P.K. and Gill, R.D.. Cox's regression model for counting process:a large sample study[J]. Annals of Statistics,1982,10:1100-1120.
[3]Andersen, P.K., Borgan, Gill, R.D. and Keiding, N.. Statistical models based on counting processes[M]. New York:Spring,1993.
[4]Bickel, P.J., Klaassen, A.J., Ritov, Y., and Wellner, J.A.. Efficient and adaptive Estimation for semiparametric models[M]. Springer,1998.
[5]Carroll, R.J.. Variances are not always nuisance parameters[J]. Biometrics,2003, 59:211-220.
[6]Carroll, R.J., Fan, J., Gijbels, I., and Wand, M. P.. Generalized partially linear single-index models[J]. Journal of the American Statistical Association,1997,92: 477-489.
[7]Carroll, R.J. and Ruppert, D.. Transformation and weighting in regression[M]. New York:Chapman and Hall,1988.
[8]Carroll, R.J., Wu, C.J., and Ruppert, D.. The effect of estimating weights in linear regression [J]. Journal of the American Statistical Association,1988,83:1045-1054.
[9]chaganty, N.R.. An alternative approach to the analysis of longitudinal data vi-a generalized estimating equations [J]. Journal of statistics planing and inference, 1997,63,39-54.
[10]Chen,M.H., Shao, Q.-M., and Xu, D.. Sufficient and necessary conditions on the propriety of posterior distributions for generalized linear mixed models [J]. Sankhya Ser A.2002,64:57-85
[11]Chen, Y.Q. and Jewell, N.P.. On a general class of semiparametric hazards regres-sion models[J]. Biometrika,2001,88:687-702.
[12]Chen, Y.Q. and Wang, M.C.. Analysis of accelerated hazard model[J]. Journal of the American Statistical Association,2000,95:608-618.
[13]Chiang,C.T.. A more flexible joint latent model for longitudinal and survival time data[J]. Metrika,2011,73:151-170.
[14]Crowder, M.. Gaussian estimation for correlated binomial data[J]. Journal of Royal Statistical Society B,1985,47:229-237.
[15]Crowder, M.. On consistency and inconsistency of estimating equations. Econo-metric Theory,1986,3:305-330.
[16]Crowder, M.. On linear and quadratic estimating functions[J]. Journal of Royal Statistical Society B,1987,74,591-597.
[17]Crowder, M.. On the use of a working correlation matrix in using generalizedlinear models for repeatedmeasures. Biometrika,1995,82:407-410.
[18]Crowder. M.. On repeated measures analysis with misspecified covariance struc-ture [J]. Journal of Royal Statistical Society B,2001,63:55-62.
[19]Cox, D.R.. Regression models and life-table(with Discussion). J. R. Statist. Soc. B,1972,34:187-220.
[20]Cox, D.R., Oakes, D.. Analysis of survival data[M]. Chapman and Hall, Lon-don,1984.
[21]Davidian, M. and Carroll, R.J.. Variance function estimation[J]. Journal of the American Statistical Association,1987,82:1079-1091.
[22]Davidian, M. and Giltinan, D.M.. Nonlinear models for repeated measurement data[M]. London:Chapman and Hall,1995.
[23]De Boor, C.. A practical guide to spline[M]. Springer-Verlag, New York.1978.
[24]Devroye, L.P. and Wagner, T.J.. Distribution free consistensy results in nonpara-metric discrimination and regression function estimation[J]. Ann. statist.1980,8: 231-239.
[25]Dempster, A.P., Laird, N.M., and Rubin, D.B.. Maximum likelihood from incom-plete data via the EM algorithm. Journal of the Royal Statistical Society, B,1977, 39,1-38.
[26]Devroye, L.P.. On the almost everywhere convergence of nonparametric regression function estimates[J]. Ann. statist.1981,9:1310-1319.
[27]Diggle, P.J.. An approach to the analysis of repeated measurement [J]. Biometrics, 1988,4,959-971.
[28]Diggle, P.J.. Testing for random droputs in repeated measurements data[J]. Bio-metrics,1989,45:1255-1258.
[29]Diggle,P.J., Liang,K.Y., and Zeger,S.L. Analysis of Longitudinal Data [J]. Oxford University Press, Oxford.U.K.1994
[30]Diggle, P.J., Heagerty, P., Liang, K.-Y., and Zeger, S.L.. Analysis of longitudinal data[M]. Oxford:Oxford University Press,2002.
[31]De Gruttola, V. and Tu, X.M.. Modeling progression of CD-4 lymphocyte count and its relationship to survival time[J]. Biometrics,1994,50:1003-1014.
[32]Ding,J.M. and Wang,J.L.. Modeling longitudinal data with nonparametric multi-plicative random effects jointly With survival data[J]. Biometrics,2008,64:546-556.
[33]Fahrmeir, L., and Kneib, T. Propriety of posteriors in structured additive regression models:theory and empirical evidence[J]. J Stat Plann Infer,2009,139:843-859
[34]Fan, J., Gijbels, I., Hu, T.C., and Huang, L.S.. A study of variable bandwidth selection for local polynomial regression[J]. Statistica Sinica,1995,6:113-127.
[35]Fan, J. and Zhang, J.T.. Two-step estimation of functional linear models with applications to longitudinal data[J]. Journal of Royal Statistics Society, Series B, 2000,62:303-322.
[36]Fan,J. and Li,R.. New estimation and model selection procedures for semipara-metric modeling in longitudinal data analysis[J]. American Statistical Association, 2004, Vol.99, No.467:710-723.
[37]Faucett, C.L. and Thomas, D.C.. Simultaneously modelling censored survival data and repeatedly mea-sured covariates:A Gibbs sampling approach[J]. Statistics in Medicine,1996,15:1663-1686.
[38]Fitzmaurice, G.M.. A caveat concerning independence estimation equations with multivariate binary data[J]. Biometrics,1995,51:309-317.
[39]Ghosh, D.. Accelerated rates regression models for recurrent failure data[J]. Life-time Data Analysis,2004,10:247-261.
[40]Hall, D.B. and Severini, T.A.. Extended generalizedestimating equations for clus-teredd data[J]. Journal of the American Statistical Association,1998,33:1365-1375.
[41]Hand,D. J. and Crowder, M.J.. Practical longitudinal data analysis[M]. London: Chapman and Hall,1996.
[42]Hanson, T., Branscum, A. and Johnson, W.O.. Predictive comparison of joint longitudinal-survival modeling:a case study illustrating competing Approaches [J]. Lifetime Data Anal,2011,17:3-28
[43]Harville, D.A.. Maximum likelihood approaches to variance component estimation and to related problems[J]. Journal ofthe American Statistical Association,1977, 72:320-338.
[44]Henderson, R., Diggle, P., and Dobson, A.. Joint mod-elling of measurements and event time data[J]. Biostatistics,2000,1:465-480.
[45]Henderson. R., Diggle, P., and Dobson. A.. Identification and efficacy of longitu-dinal markers for survival. Biostatistics,2002,3:33-50.
[46]Hoover, D.R.. Rice, J.A., Wu, C. O., and Yang. Y.. Nonparametric smoothing estimates of time-varying coefficient models With longitudinal data[J]. Biometrika, 1998.85:809-822.
[47]Wu,H.L. and Zhang,J.T.. Nonparametric regression methods for longitudinal data analysis[M]. John Wiley and Sons,2006.
[48]Jennrich, R.I. and Schluchter, M.D. Unbalanced repeated measures models with structured coveriance matrices[J]. Biometrics,1986,42,805-820.
[49]Jin, Z., Lin, D.Y., Wei, L.J., and Ying, Z.. Rank-based inference for the accelerated failure time model[J]. Biometrika,2003,90:341-353.
[50]Jin, Z., Ying, Z. and Wei, L.J.. A simple resampling method by perturbing the minimand[J]. Biometrika,2001,88:381-90.
[51]Johansen, S.. The product limit estimator as maximum likelihood estimator[J]. Scand. J.Statist.1983,5:195-199.
[52]Kalbfleisch, J.D. and Prentice, R.L.. The statistical analysis of failure time data[M]. 2nd edition. Hoboken, NJ:John Wiley and Sons,2002.
[53]Kauermann, G. and Carroll, R.J.. A note on the efficiency of sandwich covariance matrix estimation[J]. Journal of the American Statistical Association,2001,96: 1387-1396.
[54]Lai, T.L. and Ying, Z.. Rank regression methods for left-truncated and right-censored data[J]. Ann. Statist,1991,19:531-56.
[55]Lai, T.L. and Ying, Z.. Large sample theory of a modified Buckley-James estimator for regression analysis with censored data[J]. Ann. Statist.1991,19:1370-402.
[56]Laird, N.M. and Ware, J.H.. Random-effects models for longitudinal data[J]. Bio-metrics,1982,38:963-974.
[57]Lange, K.L., Little, R.J.A., and Taylor, J. M. G.. Robust statistical modeling using the t distribution [J]. Journal of the American Statistical Association,1989, 84:881-896.
[58]Li L., Hu,B.,and Greene,T.. A semiparametric joint model for longitudinal and survival data with application to hemodialysis study[J]. Biometrics,2009,65:737-745
[59]Li, Y., Lin, X., and Muller, P.. Bayesian inference in semiparametric mixed models for longitudinal data[J]. Biometrics,2009,10:1541-0420.
[60]Liang, K.Y. and Zeger, S.L.. Longitudinal data analysis using generalized linear models[J]. Biometrika,1986,73:13-22.
[61]Liang, K.Y. and McCullagh, P.. Case-studies in binary dispersion source[J]. Bio-metrics,1993,49:623-630.
[62]Liang, K.L., Zeger, S.L., and Qaqish, B.. Multivariate regression analysis for cat-egorical data[J]. Journal of Royal Statistical Society B,1992,54:3-40.
[63]Liang, H. and Hardle, W.. Asymptotic normality of parametric part in partially linear heteroscedastic regression models. DP 33, SFB 373, Humboldt Univ. Berlin. Z.1997
[64]Lin, D.Y.. Goodness of fit for the Cox regression model based on a class of param-eter estimators [J]. Journal of the American Statistical Association, vol.1991,86: 725-728
[65]Lin, D.Y. and Geyer, C.J.. Computational methods for simiparametric linear re-gression with censored data[J]. Journal of Computational and Graphical Statistics, 1992,1:77-90.
[66]Lin, D.Y., Wei, L.J., Ying, I., and Ying, Z.. Semiparametric regression for the mean and rate functions of recurrent events[J]. journal of the Royal Statistical Society Series B,2000,62:711-730.
[67]Lin, D.Y., Wei, L.J., and Ying, Z.. Accelerated failure time models for counting processes[J]. Biometrika.1998,85:605-618.
[68]Lin, D. Y., Wei, L.J., and Ying, Z.. Semiparametric tranformation models for point process[J]. Journal of the American Statistical Association,2001,96:620-628.
[69]Lin. D.Y. and Ying, Z.. Simeparametric analysis of general additive-multiplicative hazard models for counting processes[J]. Ann. Statist.1995.23:1712-1734.
[70]Lin, D.Y. and Ying, Z.. Semiparametric inference for the accelerated life model with time-dependent covariates[J]. Journal of Statistical Pianning and Inference, 1995,44:47-63.
[71]Lin, D.Y., and Ying, Z.. Semiparametric and nonparametric regression analysis of longitudinal data (with discussion) [J]. Journal of the American Statistical Associ-ation,2001,96:103-126.
[72]Lin, H., McCulloch, C.E., and Rosenheck, R.A.. Latent pattern mixture models for informative intermittent missing data in longitudinal studies[J]. Biometrics,2004, 60:295-305.
[73]Lin, H., McCulloch, C.E., Turnbull, B. W., Slate, E. H., and Clark, L.C.. A latent class mixed model for analyzing biomarker trajectories with irregularly scheduled observations[J]. Statistics in Medicine,2000,19:1303-1318.
[74]Lin, X., and Carroll, R.J.. Semiparametric regression for clustered data using gen-eralized estimating equations[J], Journal of the American Statistical Association, 2001,96:1045-1056.
[75]Lin, X., and Carroll, R.J.. Semiparametric regression for clustered data[J]. Biometrika,2001,88:1179-1865.
[76]Lin, X., Wang, N., Welsh, A., and Carroll, R.J.. Equivalent kernels of smoothing splines in nonparametric regression for clustered Data[J]. Biometrika,2004,91: 177-193.
[77]Louis, T.A.. Finding the observed information matrix when using the EM algo-rithm[J]. Journal of the Royal Statistical Society, Series B,1982,44:226-233.
[78]Martinussen, T.,and Scheike, T.H.. Dynamic regression models for survival anal-ysis[M]. Springer, New York,2006.
[79]Pan, W.. On the robust variance estimator in generalisedestimating equations [J]. Biometrika,2001,88:901-906.
[80]Parzen, M.I., Wei, L.J., and Ying, Z.. A resampling method based on pivotal estimating functions[J]. Biometrika,1994,81:341-350.
[81]Patterson, H.D. and Thompson, R..Recovery of inter-biock information when biock sizes are unequal. Biometrika.1971,58,545-554.
[82]Pepe, M.S. and Cai, J.W.. Some graphical displays and marginal regression anal-yses for recurrent failure times and time-dependent covariates[J]. Journal of the American Statistical Association,1993,88:811-820.
[83]Pepe, M.S.. and Couper, D.. Modeling partly conditional means with longitudinal data[J]. Journal of the American Statistical Association,1997,92:991-998.
[84]Prentice, R.L.. Linear rank tests with right censored data[J]. Biometricka,1978, 65:167-179.
[85]Prentice, R.L.. Correlatedbinary regression with covariates specific to each binary observation[J]. Biometrics,1988,44:1033-1048.
[86]Prentice R.L. Kalbfleisch. J.D. Hazard rate models with covariates[J]. Biometrics, 1979,35:25-39
[87]Prentice, R., and Zhao, L.P.. Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses[J]. Biometrics,1991, 47:825-839.
[88]pollard, D.. Empirical processes:theory and applications[M]. IMS:Hayward,1990.
[89]Rice, J.A. and Silverman, B.W.. Estimating the mean and covariance structure nonparametrically then the data are curves. Jourval of the Royal Statistical Society, B,53,233-243.
[90]Rice, J.A.. Functional and longitudinal data analysis:perspectives on smooth-ing[J]. Stat Sinica, 2004-14:631-647
[91]Robins, J.M., Rotnitzky. A., and Zhao. L.P.. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data[J]. Journal of the American Statistical Association,1995.90:106-121.
[92]Rotnitsky, A., Scharfstein, D., Su, T.L., and Robins, J.. Methods for conduct-ing sensitivity analysis of trials with potentially nonignorable competing causes of censoring[J]. Biometrics,2001,57:103-113.
[93]Ruppert, D.. Empirical-bias bandwidths for local polynomial nonparametric re-gression and density estimation[J]. Journal of the American Statistical Association, 1997,92:1049-1062.
[94]Ruppert, D., and Wand, M.P.. Multivariate weighted least squares regression[J]. The Annals of Statistics,1994,22:1346-1370.
[95]Schaubel, D.E., Zeng, D.L., and Cai, J.W.. A semiparametric additive rate model for recurrent event data[J]. Lifetime data Anal,2006,12:389-406.
[96]Schluchter, M.D.. Methods for the analysis of informatively censored longitudinal data[J]. Statistics in Medicine,1992,11:1861-1870.
[97]Schoop, R., Graf, E., and Schumacher,M.. Quantifying the predictive performance of prognostic models for censored survival data with time-dependent covariates[J]. Biometrics,2008,64:603-610
[98]Severini, T.A., and Staniswalis, J.G.. Quasi-Likelihood Estimation in Semiparamet-ric Models[J]. Journal of the American Statistical Association,1994,89:501-511.
[99]Song, X, and Wang, C.Y.. Semiparametric approaches for joint modeling of longitu-dinal and survival data with time-varying coefficients [J]. Biometrics,2008,64:557-566
[100]Sun, L. and Su, B.. A class of accelerated means regression models for recurreent event data[J]. Lifetime data Anal,2008,14(3):357-375.
[101]Sundaram, S.. Semiparametric inference in proportional odds model with time-dependent covariates[J]. J Stat Plann Infer,2006,136:320-334
[102]Thall, P.F. and Vail, S.C.. Some covariance models for longitudinal count data with overdispersion[J]. Biometrics,1990,6:657-671.
[103]Tian. On the accerlerated failure time model for current status and interval censored data[J]. Biometrika,2006,93:329-342.
[104]Tseng, Y.K., Hsieh, F., and Wang, J.L.. Joint modelling of accelerated failure time and longitudinal data.Biometrika,2005,92:587-603.
[105]Tsiatis, A.A.. Estimating regression parameters using linear rank tests for censored data[J]. Ann.Statist.1990,18:354-372.
[106]Tsiatis, A.A. and Davidian, M.A.. A semiparametric estimator for the proportional hazards model with lon-gitudinal covariates measured with error[J]. Biometrika, 2001,88:447-458.
[107]Tsiatis, A.A, Davidian, M.. Joint modeling of longitudinal and time-to-event data: an overview[J]. Stat Sinica,2004,14:809-834
[108]Tunnicliffe-Wilson, G..On the use of marginal likelihood in time series model estimation. Journal of the Royal Statistical Society, B,51,15-27.
[109]Van der vaart, A. and Weller, J.A.. Weak Convergence and Empirical Processes[M]. Springer:New York,1996.
[110]Wang, N.. Marginal nonparametric kernel regression accounting for within-subject correlation[J]. Biometrika,2003,90:43-52.
[111]Wang,N.,Carroll,R. and Lin,X.. Efficient semiparametrie marginal estimation for longitudinal/clustered data[J]. Journal of American Statistical Association,2005, 100:147-157.
[112]Wang, Y.G. and Carey, V.J. Working correlation structure misspecification, esti-mation and covariate design:Implications for GEE performance[J]. Biometrika, 2003,90:29-41.
[113]Wang, Y.G. and Carey, V.J.. Unbiasedestimating equations from working cor-relation models for irregularly timedrep eatedmeasures[J]. Journal of American Statistical Association,2004,99:845-853.
[114]Wang, Y.G. and Lin, X.. Effects of variance-function misspecification in analysis of longitudinal data[J]. Biometrics,2005,61:413-421.
[115]Wei, L.J. Testing goodness of fit for proportional hazard model with censored observation[J]. Journal of the American Statistical Association,1984,79:649-652.
[116]Wei, L.J., Ying, Z., and Lin, D.Y.. Linear regression analysis of censored survival data based on rank tests[J]. Biometrika,1990,77:845-851.
[117]White, H.. Maximum likelihood estimation of misspecified models. Econometrics, 1982,50:1-25.
[118]Whittle, P. Gaussian estimation in stationary time series[J]. Bulletin of the Inter-national Statistical Institute,1961,39:1-26.
[119]Wild, C.J., and Yee, T.W.. Additive extensions to generalized estimating equation methods[M]. Journal of the Royal Statistical Society, Ser. B,1996,58:711-725.
[120]Wood (nee Dobson), A.M.. Joint modelling of longi-tudinal data and time to event data[D]. Uni-versity of Lancaster, U.K,2002:
[121]C.F.. On the convergence properties of the EM algorithm[J]. The Annals of Statis-tics,1983,11,95-103.
[122]Wulfsohn, M.S. and Tsiatis, A.A.. A joint model for survival and longitudinal data measured with error[J]. Biometrics,1997,53:330-339.
[123]Xu, J. and Zeger, S.L.. The evaluation of multiple surrogate endpoints[J]. Biomet-rics,2001,57:81-87.
[124]Xu, J. and Zeger, S.L.. Joint analysis of longitudinal data comprising repeated measures and time to events[J]. Applied Statistics,2001,50:375-388
[125]Yin, G. and Cai, J.. Additive hazards model with multivariate failure time da-ta,Biometrika,2004,91:801-818.
[126]Ying, Z.. A large sample study of rank estimation for censored regression data[J]. Annals of statistics,1993,21:76-99.
[127]Zeger, S.L., Liang, K. Y.. Longitudinal data analysis for discrete and continuous outcome[J]. Biometrics,1986,42:121-130.
[128]Zeger, S.L., and Diggle, P. J.. Semi-parametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters[J]. Biometrics,1994,50: 689-699.
[129]Zeger, S.L., Liang, K. Y., and Albert, P.S.. Models for longitudinal data:A generalized estimating equation approach[J]. Biometrics,1988,44:1049-1060.
[130]Zeng, D. and Cai, J.. Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time[J]. The Annals of Statistics, 2005,33:2132-2163.
[131]Zeng, D. and Cai, J.. Simultaneous modelling of survival and longitudinal data with an application to repeated quality of life measures[J]. Lifetime Data Analysis, 2005,11:151-174.
[132]Zhang, M. and Davidian, M.. "Smooth" semiparametric regression analysis for arbitrarily censored time-to-event data[J]. Biometrics,2008,64:567-576.
[133]Zhao, L.P. and Prentice, R.L.. Correlated binary regression using a quadratic exponential model[J]. Biometrika,1990,77:642-648.
[134]孙孝前,尤进红纵向数据半参数建模中的迭代加权偏样条最小二乘估[J],中国科学.2003,33(5)：470-480
[135]陈希孺.方兆本等.非参数统计[M].上海：上海科学技术出版社,1989.
[136]赵林城.关于概率密度估计的无偏性[J],中国科技大学学报.1988.02.
[137]余松林,向惠云.重复测量资料分析方法与SAS程序[M].北京：科学出版社,2004.
[138]田萍.纵向数据半参数回归模型的估计理论[M].郑州：郑州大学出版社,2008.
[2]Andersen, P.K. and Gill, R.D.. Cox's regression model for counting process:a large sample study[J]. Annals of Statistics,1982,10:1100-1120.
[3]Andersen, P.K., Borgan, Gill, R.D. and Keiding, N.. Statistical models based on counting processes[M]. New York:Spring,1993.
[4]Bickel, P.J., Klaassen, A.J., Ritov, Y., and Wellner, J.A.. Efficient and adaptive Estimation for semiparametric models[M]. Springer,1998.
[5]Carroll, R.J.. Variances are not always nuisance parameters[J]. Biometrics,2003, 59:211-220.
[6]Carroll, R.J., Fan, J., Gijbels, I., and Wand, M. P.. Generalized partially linear single-index models[J]. Journal of the American Statistical Association,1997,92: 477-489.
[7]Carroll, R.J. and Ruppert, D.. Transformation and weighting in regression[M]. New York:Chapman and Hall,1988.
[8]Carroll, R.J., Wu, C.J., and Ruppert, D.. The effect of estimating weights in linear regression [J]. Journal of the American Statistical Association,1988,83:1045-1054.
[9]chaganty, N.R.. An alternative approach to the analysis of longitudinal data vi-a generalized estimating equations [J]. Journal of statistics planing and inference, 1997,63,39-54.
[10]Chen,M.H., Shao, Q.-M., and Xu, D.. Sufficient and necessary conditions on the propriety of posterior distributions for generalized linear mixed models [J]. Sankhya Ser A.2002,64:57-85
[11]Chen, Y.Q. and Jewell, N.P.. On a general class of semiparametric hazards regres-sion models[J]. Biometrika,2001,88:687-702.
[12]Chen, Y.Q. and Wang, M.C.. Analysis of accelerated hazard model[J]. Journal of the American Statistical Association,2000,95:608-618.
[13]Chiang,C.T.. A more flexible joint latent model for longitudinal and survival time data[J]. Metrika,2011,73:151-170.
[14]Crowder, M.. Gaussian estimation for correlated binomial data[J]. Journal of Royal Statistical Society B,1985,47:229-237.
[15]Crowder, M.. On consistency and inconsistency of estimating equations. Econo-metric Theory,1986,3:305-330.
[16]Crowder, M.. On linear and quadratic estimating functions[J]. Journal of Royal Statistical Society B,1987,74,591-597.
[17]Crowder, M.. On the use of a working correlation matrix in using generalizedlinear models for repeatedmeasures. Biometrika,1995,82:407-410.
[18]Crowder. M.. On repeated measures analysis with misspecified covariance struc-ture [J]. Journal of Royal Statistical Society B,2001,63:55-62.
[19]Cox, D.R.. Regression models and life-table(with Discussion). J. R. Statist. Soc. B,1972,34:187-220.
[20]Cox, D.R., Oakes, D.. Analysis of survival data[M]. Chapman and Hall, Lon-don,1984.
[21]Davidian, M. and Carroll, R.J.. Variance function estimation[J]. Journal of the American Statistical Association,1987,82:1079-1091.
[22]Davidian, M. and Giltinan, D.M.. Nonlinear models for repeated measurement data[M]. London:Chapman and Hall,1995.
[23]De Boor, C.. A practical guide to spline[M]. Springer-Verlag, New York.1978.
[24]Devroye, L.P. and Wagner, T.J.. Distribution free consistensy results in nonpara-metric discrimination and regression function estimation[J]. Ann. statist.1980,8: 231-239.
[25]Dempster, A.P., Laird, N.M., and Rubin, D.B.. Maximum likelihood from incom-plete data via the EM algorithm. Journal of the Royal Statistical Society, B,1977, 39,1-38.
[26]Devroye, L.P.. On the almost everywhere convergence of nonparametric regression function estimates[J]. Ann. statist.1981,9:1310-1319.
[27]Diggle, P.J.. An approach to the analysis of repeated measurement [J]. Biometrics, 1988,4,959-971.
[28]Diggle, P.J.. Testing for random droputs in repeated measurements data[J]. Bio-metrics,1989,45:1255-1258.
[29]Diggle,P.J., Liang,K.Y., and Zeger,S.L. Analysis of Longitudinal Data [J]. Oxford University Press, Oxford.U.K.1994
[30]Diggle, P.J., Heagerty, P., Liang, K.-Y., and Zeger, S.L.. Analysis of longitudinal data[M]. Oxford:Oxford University Press,2002.
[31]De Gruttola, V. and Tu, X.M.. Modeling progression of CD-4 lymphocyte count and its relationship to survival time[J]. Biometrics,1994,50:1003-1014.
[32]Ding,J.M. and Wang,J.L.. Modeling longitudinal data with nonparametric multi-plicative random effects jointly With survival data[J]. Biometrics,2008,64:546-556.
[33]Fahrmeir, L., and Kneib, T. Propriety of posteriors in structured additive regression models:theory and empirical evidence[J]. J Stat Plann Infer,2009,139:843-859
[34]Fan, J., Gijbels, I., Hu, T.C., and Huang, L.S.. A study of variable bandwidth selection for local polynomial regression[J]. Statistica Sinica,1995,6:113-127.
[35]Fan, J. and Zhang, J.T.. Two-step estimation of functional linear models with applications to longitudinal data[J]. Journal of Royal Statistics Society, Series B, 2000,62:303-322.
[36]Fan,J. and Li,R.. New estimation and model selection procedures for semipara-metric modeling in longitudinal data analysis[J]. American Statistical Association, 2004, Vol.99, No.467:710-723.
[37]Faucett, C.L. and Thomas, D.C.. Simultaneously modelling censored survival data and repeatedly mea-sured covariates:A Gibbs sampling approach[J]. Statistics in Medicine,1996,15:1663-1686.
[38]Fitzmaurice, G.M.. A caveat concerning independence estimation equations with multivariate binary data[J]. Biometrics,1995,51:309-317.
[39]Ghosh, D.. Accelerated rates regression models for recurrent failure data[J]. Life-time Data Analysis,2004,10:247-261.
[40]Hall, D.B. and Severini, T.A.. Extended generalizedestimating equations for clus-teredd data[J]. Journal of the American Statistical Association,1998,33:1365-1375.
[41]Hand,D. J. and Crowder, M.J.. Practical longitudinal data analysis[M]. London: Chapman and Hall,1996.
[42]Hanson, T., Branscum, A. and Johnson, W.O.. Predictive comparison of joint longitudinal-survival modeling:a case study illustrating competing Approaches [J]. Lifetime Data Anal,2011,17:3-28
[43]Harville, D.A.. Maximum likelihood approaches to variance component estimation and to related problems[J]. Journal ofthe American Statistical Association,1977, 72:320-338.
[44]Henderson, R., Diggle, P., and Dobson, A.. Joint mod-elling of measurements and event time data[J]. Biostatistics,2000,1:465-480.
[45]Henderson. R., Diggle, P., and Dobson. A.. Identification and efficacy of longitu-dinal markers for survival. Biostatistics,2002,3:33-50.
[46]Hoover, D.R.. Rice, J.A., Wu, C. O., and Yang. Y.. Nonparametric smoothing estimates of time-varying coefficient models With longitudinal data[J]. Biometrika, 1998.85:809-822.
[47]Wu,H.L. and Zhang,J.T.. Nonparametric regression methods for longitudinal data analysis[M]. John Wiley and Sons,2006.
[48]Jennrich, R.I. and Schluchter, M.D. Unbalanced repeated measures models with structured coveriance matrices[J]. Biometrics,1986,42,805-820.
[49]Jin, Z., Lin, D.Y., Wei, L.J., and Ying, Z.. Rank-based inference for the accelerated failure time model[J]. Biometrika,2003,90:341-353.
[50]Jin, Z., Ying, Z. and Wei, L.J.. A simple resampling method by perturbing the minimand[J]. Biometrika,2001,88:381-90.
[51]Johansen, S.. The product limit estimator as maximum likelihood estimator[J]. Scand. J.Statist.1983,5:195-199.
[52]Kalbfleisch, J.D. and Prentice, R.L.. The statistical analysis of failure time data[M]. 2nd edition. Hoboken, NJ:John Wiley and Sons,2002.
[53]Kauermann, G. and Carroll, R.J.. A note on the efficiency of sandwich covariance matrix estimation[J]. Journal of the American Statistical Association,2001,96: 1387-1396.
[54]Lai, T.L. and Ying, Z.. Rank regression methods for left-truncated and right-censored data[J]. Ann. Statist,1991,19:531-56.
[55]Lai, T.L. and Ying, Z.. Large sample theory of a modified Buckley-James estimator for regression analysis with censored data[J]. Ann. Statist.1991,19:1370-402.
[56]Laird, N.M. and Ware, J.H.. Random-effects models for longitudinal data[J]. Bio-metrics,1982,38:963-974.
[57]Lange, K.L., Little, R.J.A., and Taylor, J. M. G.. Robust statistical modeling using the t distribution [J]. Journal of the American Statistical Association,1989, 84:881-896.
[58]Li L., Hu,B.,and Greene,T.. A semiparametric joint model for longitudinal and survival data with application to hemodialysis study[J]. Biometrics,2009,65:737-745
[59]Li, Y., Lin, X., and Muller, P.. Bayesian inference in semiparametric mixed models for longitudinal data[J]. Biometrics,2009,10:1541-0420.
[60]Liang, K.Y. and Zeger, S.L.. Longitudinal data analysis using generalized linear models[J]. Biometrika,1986,73:13-22.
[61]Liang, K.Y. and McCullagh, P.. Case-studies in binary dispersion source[J]. Bio-metrics,1993,49:623-630.
[62]Liang, K.L., Zeger, S.L., and Qaqish, B.. Multivariate regression analysis for cat-egorical data[J]. Journal of Royal Statistical Society B,1992,54:3-40.
[63]Liang, H. and Hardle, W.. Asymptotic normality of parametric part in partially linear heteroscedastic regression models. DP 33, SFB 373, Humboldt Univ. Berlin. Z.1997
[64]Lin, D.Y.. Goodness of fit for the Cox regression model based on a class of param-eter estimators [J]. Journal of the American Statistical Association, vol.1991,86: 725-728
[65]Lin, D.Y. and Geyer, C.J.. Computational methods for simiparametric linear re-gression with censored data[J]. Journal of Computational and Graphical Statistics, 1992,1:77-90.
[66]Lin, D.Y., Wei, L.J., Ying, I., and Ying, Z.. Semiparametric regression for the mean and rate functions of recurrent events[J]. journal of the Royal Statistical Society Series B,2000,62:711-730.
[67]Lin, D.Y., Wei, L.J., and Ying, Z.. Accelerated failure time models for counting processes[J]. Biometrika.1998,85:605-618.
[68]Lin, D. Y., Wei, L.J., and Ying, Z.. Semiparametric tranformation models for point process[J]. Journal of the American Statistical Association,2001,96:620-628.
[69]Lin. D.Y. and Ying, Z.. Simeparametric analysis of general additive-multiplicative hazard models for counting processes[J]. Ann. Statist.1995.23:1712-1734.
[70]Lin, D.Y. and Ying, Z.. Semiparametric inference for the accelerated life model with time-dependent covariates[J]. Journal of Statistical Pianning and Inference, 1995,44:47-63.
[71]Lin, D.Y., and Ying, Z.. Semiparametric and nonparametric regression analysis of longitudinal data (with discussion) [J]. Journal of the American Statistical Associ-ation,2001,96:103-126.
[72]Lin, H., McCulloch, C.E., and Rosenheck, R.A.. Latent pattern mixture models for informative intermittent missing data in longitudinal studies[J]. Biometrics,2004, 60:295-305.
[73]Lin, H., McCulloch, C.E., Turnbull, B. W., Slate, E. H., and Clark, L.C.. A latent class mixed model for analyzing biomarker trajectories with irregularly scheduled observations[J]. Statistics in Medicine,2000,19:1303-1318.
[74]Lin, X., and Carroll, R.J.. Semiparametric regression for clustered data using gen-eralized estimating equations[J], Journal of the American Statistical Association, 2001,96:1045-1056.
[75]Lin, X., and Carroll, R.J.. Semiparametric regression for clustered data[J]. Biometrika,2001,88:1179-1865.
[76]Lin, X., Wang, N., Welsh, A., and Carroll, R.J.. Equivalent kernels of smoothing splines in nonparametric regression for clustered Data[J]. Biometrika,2004,91: 177-193.
[77]Louis, T.A.. Finding the observed information matrix when using the EM algo-rithm[J]. Journal of the Royal Statistical Society, Series B,1982,44:226-233.
[78]Martinussen, T.,and Scheike, T.H.. Dynamic regression models for survival anal-ysis[M]. Springer, New York,2006.
[79]Pan, W.. On the robust variance estimator in generalisedestimating equations [J]. Biometrika,2001,88:901-906.
[80]Parzen, M.I., Wei, L.J., and Ying, Z.. A resampling method based on pivotal estimating functions[J]. Biometrika,1994,81:341-350.
[81]Patterson, H.D. and Thompson, R..Recovery of inter-biock information when biock sizes are unequal. Biometrika.1971,58,545-554.
[82]Pepe, M.S. and Cai, J.W.. Some graphical displays and marginal regression anal-yses for recurrent failure times and time-dependent covariates[J]. Journal of the American Statistical Association,1993,88:811-820.
[83]Pepe, M.S.. and Couper, D.. Modeling partly conditional means with longitudinal data[J]. Journal of the American Statistical Association,1997,92:991-998.
[84]Prentice, R.L.. Linear rank tests with right censored data[J]. Biometricka,1978, 65:167-179.
[85]Prentice, R.L.. Correlatedbinary regression with covariates specific to each binary observation[J]. Biometrics,1988,44:1033-1048.
[86]Prentice R.L. Kalbfleisch. J.D. Hazard rate models with covariates[J]. Biometrics, 1979,35:25-39
[87]Prentice, R., and Zhao, L.P.. Estimating equations for parameters in means and covariances of multivariate discrete and continuous responses[J]. Biometrics,1991, 47:825-839.
[88]pollard, D.. Empirical processes:theory and applications[M]. IMS:Hayward,1990.
[89]Rice, J.A. and Silverman, B.W.. Estimating the mean and covariance structure nonparametrically then the data are curves. Jourval of the Royal Statistical Society, B,53,233-243.
[90]Rice, J.A.. Functional and longitudinal data analysis:perspectives on smooth-ing[J]. Stat Sinica, 2004-14:631-647
[91]Robins, J.M., Rotnitzky. A., and Zhao. L.P.. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data[J]. Journal of the American Statistical Association,1995.90:106-121.
[92]Rotnitsky, A., Scharfstein, D., Su, T.L., and Robins, J.. Methods for conduct-ing sensitivity analysis of trials with potentially nonignorable competing causes of censoring[J]. Biometrics,2001,57:103-113.
[93]Ruppert, D.. Empirical-bias bandwidths for local polynomial nonparametric re-gression and density estimation[J]. Journal of the American Statistical Association, 1997,92:1049-1062.
[94]Ruppert, D., and Wand, M.P.. Multivariate weighted least squares regression[J]. The Annals of Statistics,1994,22:1346-1370.
[95]Schaubel, D.E., Zeng, D.L., and Cai, J.W.. A semiparametric additive rate model for recurrent event data[J]. Lifetime data Anal,2006,12:389-406.
[96]Schluchter, M.D.. Methods for the analysis of informatively censored longitudinal data[J]. Statistics in Medicine,1992,11:1861-1870.
[97]Schoop, R., Graf, E., and Schumacher,M.. Quantifying the predictive performance of prognostic models for censored survival data with time-dependent covariates[J]. Biometrics,2008,64:603-610
[98]Severini, T.A., and Staniswalis, J.G.. Quasi-Likelihood Estimation in Semiparamet-ric Models[J]. Journal of the American Statistical Association,1994,89:501-511.
[99]Song, X, and Wang, C.Y.. Semiparametric approaches for joint modeling of longitu-dinal and survival data with time-varying coefficients [J]. Biometrics,2008,64:557-566
[100]Sun, L. and Su, B.. A class of accelerated means regression models for recurreent event data[J]. Lifetime data Anal,2008,14(3):357-375.
[101]Sundaram, S.. Semiparametric inference in proportional odds model with time-dependent covariates[J]. J Stat Plann Infer,2006,136:320-334
[102]Thall, P.F. and Vail, S.C.. Some covariance models for longitudinal count data with overdispersion[J]. Biometrics,1990,6:657-671.
[103]Tian. On the accerlerated failure time model for current status and interval censored data[J]. Biometrika,2006,93:329-342.
[104]Tseng, Y.K., Hsieh, F., and Wang, J.L.. Joint modelling of accelerated failure time and longitudinal data.Biometrika,2005,92:587-603.
[105]Tsiatis, A.A.. Estimating regression parameters using linear rank tests for censored data[J]. Ann.Statist.1990,18:354-372.
[106]Tsiatis, A.A. and Davidian, M.A.. A semiparametric estimator for the proportional hazards model with lon-gitudinal covariates measured with error[J]. Biometrika, 2001,88:447-458.
[107]Tsiatis, A.A, Davidian, M.. Joint modeling of longitudinal and time-to-event data: an overview[J]. Stat Sinica,2004,14:809-834
[108]Tunnicliffe-Wilson, G..On the use of marginal likelihood in time series model estimation. Journal of the Royal Statistical Society, B,51,15-27.
[109]Van der vaart, A. and Weller, J.A.. Weak Convergence and Empirical Processes[M]. Springer:New York,1996.
[110]Wang, N.. Marginal nonparametric kernel regression accounting for within-subject correlation[J]. Biometrika,2003,90:43-52.
[111]Wang,N.,Carroll,R. and Lin,X.. Efficient semiparametrie marginal estimation for longitudinal/clustered data[J]. Journal of American Statistical Association,2005, 100:147-157.
[112]Wang, Y.G. and Carey, V.J. Working correlation structure misspecification, esti-mation and covariate design:Implications for GEE performance[J]. Biometrika, 2003,90:29-41.
[113]Wang, Y.G. and Carey, V.J.. Unbiasedestimating equations from working cor-relation models for irregularly timedrep eatedmeasures[J]. Journal of American Statistical Association,2004,99:845-853.
[114]Wang, Y.G. and Lin, X.. Effects of variance-function misspecification in analysis of longitudinal data[J]. Biometrics,2005,61:413-421.
[115]Wei, L.J. Testing goodness of fit for proportional hazard model with censored observation[J]. Journal of the American Statistical Association,1984,79:649-652.
[116]Wei, L.J., Ying, Z., and Lin, D.Y.. Linear regression analysis of censored survival data based on rank tests[J]. Biometrika,1990,77:845-851.
[117]White, H.. Maximum likelihood estimation of misspecified models. Econometrics, 1982,50:1-25.
[118]Whittle, P. Gaussian estimation in stationary time series[J]. Bulletin of the Inter-national Statistical Institute,1961,39:1-26.
[119]Wild, C.J., and Yee, T.W.. Additive extensions to generalized estimating equation methods[M]. Journal of the Royal Statistical Society, Ser. B,1996,58:711-725.
[120]Wood (nee Dobson), A.M.. Joint modelling of longi-tudinal data and time to event data[D]. Uni-versity of Lancaster, U.K,2002:
[121]C.F.. On the convergence properties of the EM algorithm[J]. The Annals of Statis-tics,1983,11,95-103.
[122]Wulfsohn, M.S. and Tsiatis, A.A.. A joint model for survival and longitudinal data measured with error[J]. Biometrics,1997,53:330-339.
[123]Xu, J. and Zeger, S.L.. The evaluation of multiple surrogate endpoints[J]. Biomet-rics,2001,57:81-87.
[124]Xu, J. and Zeger, S.L.. Joint analysis of longitudinal data comprising repeated measures and time to events[J]. Applied Statistics,2001,50:375-388
[125]Yin, G. and Cai, J.. Additive hazards model with multivariate failure time da-ta,Biometrika,2004,91:801-818.
[126]Ying, Z.. A large sample study of rank estimation for censored regression data[J]. Annals of statistics,1993,21:76-99.
[127]Zeger, S.L., Liang, K. Y.. Longitudinal data analysis for discrete and continuous outcome[J]. Biometrics,1986,42:121-130.
[128]Zeger, S.L., and Diggle, P. J.. Semi-parametric models for longitudinal data with application to CD4 cell numbers in HIV seroconverters[J]. Biometrics,1994,50: 689-699.
[129]Zeger, S.L., Liang, K. Y., and Albert, P.S.. Models for longitudinal data:A generalized estimating equation approach[J]. Biometrics,1988,44:1049-1060.
[130]Zeng, D. and Cai, J.. Asymptotic results for maximum likelihood estimators in joint analysis of repeated measurements and survival time[J]. The Annals of Statistics, 2005,33:2132-2163.
[131]Zeng, D. and Cai, J.. Simultaneous modelling of survival and longitudinal data with an application to repeated quality of life measures[J]. Lifetime Data Analysis, 2005,11:151-174.
[132]Zhang, M. and Davidian, M.. "Smooth" semiparametric regression analysis for arbitrarily censored time-to-event data[J]. Biometrics,2008,64:567-576.
[133]Zhao, L.P. and Prentice, R.L.. Correlated binary regression using a quadratic exponential model[J]. Biometrika,1990,77:642-648.
[134]孙孝前,尤进红纵向数据半参数建模中的迭代加权偏样条最小二乘估[J],中国科学.2003,33(5)：470-480
[135]陈希孺.方兆本等.非参数统计[M].上海：上海科学技术出版社,1989.
[136]赵林城.关于概率密度估计的无偏性[J],中国科技大学学报.1988.02.
[137]余松林,向惠云.重复测量资料分析方法与SAS程序[M].北京：科学出版社,2004.
[138]田萍.纵向数据半参数回归模型的估计理论[M].郑州：郑州大学出版社,2008.
© 2004-2018 中国地质图书馆版权所有 京ICP备05064691号 京公网安备11010802017129号 地址：北京市海淀区学院路29号 邮编：100083 电话：办公室：(+86 10)66554848；文献借阅、咨询服务、科技查新：66554700 |