共享联合脆弱模型在含终点事件冠心病复发研究中的应用
|
摘要
医学纵向研究中,某随机事件在一个受试对象身上多次反复发生,即同一个体可能在一定时间里多次经历同一种事件,如膀胱肿瘤患者在治疗过程中的多次复发,白血病患者在骨髓移植后重复感染,称该类资料为复发事件数据(recurrent event data)或复发数据。
复发事件数据具有以下特点：(1)事件发生是有次序的；(2)个体之间有异质性heterogeneity)存在；(3)多次事件间存在非独立；(4)复发事件时间存在删失。若删失是由研究结束或者随访过程中失访而引起的,可以认为该删失时间是独立的,或者是无信息的复发事件时间。但更多的实际问题研究发现,大多数删失数据可能是由于受试者的中途退出(疾病恶化或者其它原因)或者死亡而终止的,这种删失时间(如终点事件发生时间)很可能与复发事件时间有关,称之为有信息删失的复发事件时间。当复发事件和终点事件出现时,数据具有非独立性,不满足经典Cox比例风险模型假定,无法解决该类资料分析中遇到的问题。有关复发事件数据最适分析模型的研究逐渐引起学者们的关注。
本课题针对复发事件数据特点,进一步探讨不满足经典Cox比例风险模型的替代模型,以更合理地解释医学随访研究中含终点事件的复发数据所蕴藏的信息。
文中第一章主要结合实际数据,进一步论证了复发事件数据特点,概括介绍了针对无信息删失复发时间数据而提出的基础模型理论。指出用基础模型处理含终点事件的复发事件数据时,往往是将终点事件看作是删失信息来处理的,这样会掩盖终点事件所蕴含的信息,引致参数的有偏估计。第二章主要通过介绍终点事件和复发事件的联合脆弱模型理论,采用简便的参数估计方法——高斯求积估计,运用SAS9.2Proc NLMIXED过程,实现脆弱模型及含终点复发事件脆弱模型分析。进一步证实,终点事件和复发事件的联合脆弱模型是基于Cox比例风险模型关于复发事件数据分析的拓展模型之一,模型中随机效应不仅可解释子组间的异质性,尚允许个体生存时间之间存在相关性。而简单脆弱模型虽能较好地解决异质性,但由于EM算法迭代缓慢等,参数估计方法不理想,限制了脆弱模型的应用。第三章结合心血管疾病冠心病实例分析,进行了简单脆弱模型(共享伽马脆弱模型、条件共享伽马脆弱模型等)和复发事件和终点事件联合建模对比研究。尽管简单脆弱模型考虑了个体生存时间之间的相关性,个体之间的异质性,但将终点事件看作删失信息来处理,忽略了复发事件中终点事件提供的信息,掩盖了终点事件对模型参数估计的影响。而对终点事件和复发事件进行联合建模,共享联合脆弱模型结果表明,影响318例冠心病患者复发和死亡风险的因素不同。复发事件风险和终点事件风险呈正相(gamma=1.3614, p=0.0031),有统计学意义,表明在实际问题研究中,应考虑对复发事件和终点事件进行联合建模；可见冠心病患者复发风险高的患者,死亡风险也较高。治疗方法、性别、高血压病史是影响冠心病患者复发风险的因素,而治疗方法和性别也是影响冠心病死亡风险的因素,冠心病事件中女性患者的死亡风险较男性高,但复发风险则反。由复发事件和终点事件构建的共享联合脆弱模型,可以评价不同协变量分层,在删失过程和相关水平两方面的作用,为临床应用提供了分析新思路。
总之,脆弱模型是分析非独立、有异质性存在复发生存时间资料的有效方法。共享联合脆弱模型是终点事件和复发事件数据资料的分析的最适模型。共享联合脆弱模型既可以解释个体内相关性,也可以用脆弱解释个体之间的异质性。采用高斯求积法进行参数估计,不需要特定的基线风险先验分布假设,只需要近似分段常数基线风险,可运用SAS9.2Proc NLMIXED过程实现。进一步表明,高斯求积方法估计方便易行,不仅可直接求得标准误的估计值,且可拓展应用到复杂模型。有信息删失的情况下,随机效应合并入信息删失过程,直接产生简单的联合模型,以评价协变量分层在删失过程和相关水平两方面的作用,不仅可以描述个体生存时间之间存在的相关性及个体之间的异质性,也可以描述复发事件和终点事件风险。
Recurrent event data have been increasingly important in longitudinal medical studies. If a subject repeatedly happens some random events, that is the same individuals may experience the same kind of events in some time, such as bladder cancer patients relapse by his treatment, a leukemia patients after bone marrow transplantation repeatedly suffer infection, we treat such data as recurrent event data.
Recurrent event data have some characteristics as follows:(1)There is the order of the event;(2)There exists heterogeneity among individuals;(3)The recurrent event data is independence;(4) In practice, recurrent event times are subject to censoring. If censoring is caused by the end of the study or random loss to follow-up, then the censoring time can be regarded as independent or non-informative of the recurrent event times. In many applications, especially in medical studies, the observation on recurrent events may be terminated by the subject's withdrawal from the study (because of deteriorating health or some other reasons) or the subject's death. Then the censoring time (i.e., time to the terminal event) is likely to be correlated with the recurrent event times. We call the informative censoring of the recurrent event times. When the recurrent even with a terminal event, data has the independence, and the traditional Cox proportion hazards model has been unable to solve this kind of data. This will force us to find a more suitable replacement model to analysis and explain such data more reasonable. It is of interest to many researchers to study how to select a more suitable model to analysis the recurrent event data.
This topic on the basis of recurrent event data's characteristics, discusses the alternative model which do not satisfy the classic Cox hazard model, and can make the explanation of the medical follow-up study with the recurrent events data reasonable.
The first chapter of this paper combined with the actual data to introduce the characteristics and the theory of the model with recurrent event data. We found that these models are suitable for non-informative of the recurrent event data. Using them to analysis the recurrent event with terminal event, they often treat the terminal event as censoring information, then will cause the estimation of parameters biased and cover the information of the terminal event. Chapter two introduced the theory research, and parameter estimation methods of the frailty model and the recurrent event with a terminal event. We proposed a simple parameters estimation method using Gaussian quadrature techniques to estimation in joint frailty model. This estimation method is relatively straightforward and has been implemented in SAS9.2Proc NLMIXED. Thus further confirmed that frailty model was based on Cox proportion hazard model to analysis the recurrent event data. Random effects of the model on one hand explains the heterogeneity between subgroups, on the other hand allows correlation between individual's survival time. However, simple frailty model can be a good solution to heterogeneity, as the iteration of EM algorithm is slow and PPL algorithm applies only to simple frailty model etc, the parameter estimation method is not perfect and limit its application.
Chapter three combined with the example of coronary heart disease, and make comparison with a simple frailty model (shared the gamma frailty model, the conditions shared frailty model and so on) and the joint frailty model of recurrent event and a terminal event. Although the simple frailty model considers the correlation of the survival time and heterogeneity between the individual, they treat the terminal event as censoring information and ignore the information of the terminal event. However, we conduct the shared joint model. The results show that the impact factor on recurrence and death is different of318patients who had the coronary artery disease. The risk of recurrence and death is positively (gamma=1.3614, p=0.0031), which has a statistical significance show that we should consider the recurrent event and the terminal event to conduct the joint model in the actual problems. We also found that the coronary artery disease patients who with the higher risk of relapse will have higher risk of death. Treatment, gender, and the history of hypertension are factors which impact the relapse of the coronary heart disease patients.Treatment and gender are also affect the coronary heart disease patients'death, the woman who has coronary heart disease events with higher risk of death than man, but the risk of recurrence is reversed. The shared joint frailty model of the recurrent event and a terminal event can evaluate different covariates stratification in the process of censoring and relation, and provide new ideas for the application of clinical.
In short, frailty model is the effective method to analysis independent and heterogeneity of the recurrent event data. The shared joint frailty model is the optimal model to analysis the recurrent events and a terminal event. Shared joint frailty model can explain the correlation and heterogeneity between individuals. By Gaussian quadrature techniques to parameter estimation, we do not need assumption the specific prior distribution of baseline risk, and only need approximate piecewise constant baseline risk, and can be used SAS9.2Proc NLMIXED process. Further studies indicate that using Gaussian quadrature techniques to estimate parameter is convenient and feasible, not only can be directly obtained the value of standard, but also can expand the application to the complex models. With the informative censoring of the recurrent event times, we have the random effects incorporated into the process of the censoring information to directly produce simple joint frailty model to evaluate different covariates stratification in the process of censoring and relation, not only can describe the correlation and heterogeneity between individual, but also can describe the risk of recurrence and termination.
复发事件数据具有以下特点：(1)事件发生是有次序的；(2)个体之间有异质性heterogeneity)存在；(3)多次事件间存在非独立；(4)复发事件时间存在删失。若删失是由研究结束或者随访过程中失访而引起的,可以认为该删失时间是独立的,或者是无信息的复发事件时间。但更多的实际问题研究发现,大多数删失数据可能是由于受试者的中途退出(疾病恶化或者其它原因)或者死亡而终止的,这种删失时间(如终点事件发生时间)很可能与复发事件时间有关,称之为有信息删失的复发事件时间。当复发事件和终点事件出现时,数据具有非独立性,不满足经典Cox比例风险模型假定,无法解决该类资料分析中遇到的问题。有关复发事件数据最适分析模型的研究逐渐引起学者们的关注。
本课题针对复发事件数据特点,进一步探讨不满足经典Cox比例风险模型的替代模型,以更合理地解释医学随访研究中含终点事件的复发数据所蕴藏的信息。
文中第一章主要结合实际数据,进一步论证了复发事件数据特点,概括介绍了针对无信息删失复发时间数据而提出的基础模型理论。指出用基础模型处理含终点事件的复发事件数据时,往往是将终点事件看作是删失信息来处理的,这样会掩盖终点事件所蕴含的信息,引致参数的有偏估计。第二章主要通过介绍终点事件和复发事件的联合脆弱模型理论,采用简便的参数估计方法——高斯求积估计,运用SAS9.2Proc NLMIXED过程,实现脆弱模型及含终点复发事件脆弱模型分析。进一步证实,终点事件和复发事件的联合脆弱模型是基于Cox比例风险模型关于复发事件数据分析的拓展模型之一,模型中随机效应不仅可解释子组间的异质性,尚允许个体生存时间之间存在相关性。而简单脆弱模型虽能较好地解决异质性,但由于EM算法迭代缓慢等,参数估计方法不理想,限制了脆弱模型的应用。第三章结合心血管疾病冠心病实例分析,进行了简单脆弱模型(共享伽马脆弱模型、条件共享伽马脆弱模型等)和复发事件和终点事件联合建模对比研究。尽管简单脆弱模型考虑了个体生存时间之间的相关性,个体之间的异质性,但将终点事件看作删失信息来处理,忽略了复发事件中终点事件提供的信息,掩盖了终点事件对模型参数估计的影响。而对终点事件和复发事件进行联合建模,共享联合脆弱模型结果表明,影响318例冠心病患者复发和死亡风险的因素不同。复发事件风险和终点事件风险呈正相(gamma=1.3614, p=0.0031),有统计学意义,表明在实际问题研究中,应考虑对复发事件和终点事件进行联合建模；可见冠心病患者复发风险高的患者,死亡风险也较高。治疗方法、性别、高血压病史是影响冠心病患者复发风险的因素,而治疗方法和性别也是影响冠心病死亡风险的因素,冠心病事件中女性患者的死亡风险较男性高,但复发风险则反。由复发事件和终点事件构建的共享联合脆弱模型,可以评价不同协变量分层,在删失过程和相关水平两方面的作用,为临床应用提供了分析新思路。
总之,脆弱模型是分析非独立、有异质性存在复发生存时间资料的有效方法。共享联合脆弱模型是终点事件和复发事件数据资料的分析的最适模型。共享联合脆弱模型既可以解释个体内相关性,也可以用脆弱解释个体之间的异质性。采用高斯求积法进行参数估计,不需要特定的基线风险先验分布假设,只需要近似分段常数基线风险,可运用SAS9.2Proc NLMIXED过程实现。进一步表明,高斯求积方法估计方便易行,不仅可直接求得标准误的估计值,且可拓展应用到复杂模型。有信息删失的情况下,随机效应合并入信息删失过程,直接产生简单的联合模型,以评价协变量分层在删失过程和相关水平两方面的作用,不仅可以描述个体生存时间之间存在的相关性及个体之间的异质性,也可以描述复发事件和终点事件风险。
Recurrent event data have been increasingly important in longitudinal medical studies. If a subject repeatedly happens some random events, that is the same individuals may experience the same kind of events in some time, such as bladder cancer patients relapse by his treatment, a leukemia patients after bone marrow transplantation repeatedly suffer infection, we treat such data as recurrent event data.
Recurrent event data have some characteristics as follows:(1)There is the order of the event;(2)There exists heterogeneity among individuals;(3)The recurrent event data is independence;(4) In practice, recurrent event times are subject to censoring. If censoring is caused by the end of the study or random loss to follow-up, then the censoring time can be regarded as independent or non-informative of the recurrent event times. In many applications, especially in medical studies, the observation on recurrent events may be terminated by the subject's withdrawal from the study (because of deteriorating health or some other reasons) or the subject's death. Then the censoring time (i.e., time to the terminal event) is likely to be correlated with the recurrent event times. We call the informative censoring of the recurrent event times. When the recurrent even with a terminal event, data has the independence, and the traditional Cox proportion hazards model has been unable to solve this kind of data. This will force us to find a more suitable replacement model to analysis and explain such data more reasonable. It is of interest to many researchers to study how to select a more suitable model to analysis the recurrent event data.
This topic on the basis of recurrent event data's characteristics, discusses the alternative model which do not satisfy the classic Cox hazard model, and can make the explanation of the medical follow-up study with the recurrent events data reasonable.
The first chapter of this paper combined with the actual data to introduce the characteristics and the theory of the model with recurrent event data. We found that these models are suitable for non-informative of the recurrent event data. Using them to analysis the recurrent event with terminal event, they often treat the terminal event as censoring information, then will cause the estimation of parameters biased and cover the information of the terminal event. Chapter two introduced the theory research, and parameter estimation methods of the frailty model and the recurrent event with a terminal event. We proposed a simple parameters estimation method using Gaussian quadrature techniques to estimation in joint frailty model. This estimation method is relatively straightforward and has been implemented in SAS9.2Proc NLMIXED. Thus further confirmed that frailty model was based on Cox proportion hazard model to analysis the recurrent event data. Random effects of the model on one hand explains the heterogeneity between subgroups, on the other hand allows correlation between individual's survival time. However, simple frailty model can be a good solution to heterogeneity, as the iteration of EM algorithm is slow and PPL algorithm applies only to simple frailty model etc, the parameter estimation method is not perfect and limit its application.
Chapter three combined with the example of coronary heart disease, and make comparison with a simple frailty model (shared the gamma frailty model, the conditions shared frailty model and so on) and the joint frailty model of recurrent event and a terminal event. Although the simple frailty model considers the correlation of the survival time and heterogeneity between the individual, they treat the terminal event as censoring information and ignore the information of the terminal event. However, we conduct the shared joint model. The results show that the impact factor on recurrence and death is different of318patients who had the coronary artery disease. The risk of recurrence and death is positively (gamma=1.3614, p=0.0031), which has a statistical significance show that we should consider the recurrent event and the terminal event to conduct the joint model in the actual problems. We also found that the coronary artery disease patients who with the higher risk of relapse will have higher risk of death. Treatment, gender, and the history of hypertension are factors which impact the relapse of the coronary heart disease patients.Treatment and gender are also affect the coronary heart disease patients'death, the woman who has coronary heart disease events with higher risk of death than man, but the risk of recurrence is reversed. The shared joint frailty model of the recurrent event and a terminal event can evaluate different covariates stratification in the process of censoring and relation, and provide new ideas for the application of clinical.
In short, frailty model is the effective method to analysis independent and heterogeneity of the recurrent event data. The shared joint frailty model is the optimal model to analysis the recurrent events and a terminal event. Shared joint frailty model can explain the correlation and heterogeneity between individuals. By Gaussian quadrature techniques to parameter estimation, we do not need assumption the specific prior distribution of baseline risk, and only need approximate piecewise constant baseline risk, and can be used SAS9.2Proc NLMIXED process. Further studies indicate that using Gaussian quadrature techniques to estimate parameter is convenient and feasible, not only can be directly obtained the value of standard, but also can expand the application to the complex models. With the informative censoring of the recurrent event times, we have the random effects incorporated into the process of the censoring information to directly produce simple joint frailty model to evaluate different covariates stratification in the process of censoring and relation, not only can describe the correlation and heterogeneity between individual, but also can describe the risk of recurrence and termination.
引文
[1]Cox DR.Regression Models and Life-Tables. Journal of the Royal Statistical Society B1972,34:187-220[2]Prentice RL, Williams BJ,Peterson AV.On the regression analysis of multivariate failure time data.Biometrika,1981,68:373-379[3]Andersen PK,Gill RD.Cox's regression model for counting processes:a large sample study. Annals of Statistics,1982,10:1100-1120[4]Wei LJ, Lin DY,Weissfeld L.Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of the American Statistical Association1989,84:1065-1073[5]Hougaard P. Analysis of Multivariate Survival Data. New York Springer-Verlag,2000[6]Hougaard P.A class of multivariate failure time distribution. Biometrika,1986,73:671-678[7]Hougaard P.Frailty models for survival data.Lifetime Data Analysis,1995,1:255-273.[8]Aalen O.Heterogeneity in survival analysis.Statistics in Medicine,1988,7:1121-1137.[9]Lancaster T. Econometric Methods for the Duration of Unemployment. Econometrica,1979,47:939-956.[10]Pickles A and Crouchley R. A comparison of frailty models for multivariate survival data. Statistics in Medicine,1995,14:1447-1461.[ll]Oakes D. A model for association in bivariate survival data. J.R.Statist.Soc.B,1982,44:414-422[12]Oakes D.Semiparametric inference in bivariate survival data. Biometrika,1986,73:353-361[13]Oakes D. Bivariate survival models induced by frailties. J.R.Statist.Ass,1989,26:183-214[14]Parner E.Asymptotic theory for correlated gamma-frailty model. Ann. Statist,1998,26:183-214[15]Greenwood M,Yule GU.An inquiry into the nature of frequency distributions representative of multiple happenings with particular reference to the occurrence of multiple attacks of disease or of repeated accidents. Journal of the Royal Statistical Society,1920,83:255-279[16]Vaupel JW, Manton KG,Stallard E.The Impact of Heterogeneity in Individual Frailty on the Dynamics of Mortality.Demography,1979,16:439-454[17]Clayton DG.A model for association in bivariate life tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika.1978,65:141-151[18]Liu L, Wolfe RA, Huang X. Shared frailty models for recurrent events and a terminal event. Biometrics2004,60:747-756.[19]Huang X, Liu L. A joint frailty model for survival time and gap times between recurrent events. Biometrics,2007,63:389-397.[20]Huang X, Wolfe RA. A frailty model for informative censoring. Biometrics,2002,58:510-520.[21]Nielsen G, Gill R, Andersen P, Sorensen T. A counting process approach to maximum likelihood estimation in frailty models. Scandinavian Journal of Statistics,1992,19:25-43.[22]Klein J. Semiparametric estimation of random effects using the Cox model based on the EM algorithm. Biometrics,1992,48:795-806.[23]Guo G, Rodriguez G. Estimating a multivariate proportional hazards model for clustered data using the EM algorithm, with an application to child survival in Guatemala. Journal of the American Statistical Association,1992,87:969-976.[24]Louis TA. Finding the observed information matrix when using the EM algorithm. Journal of the Royal Statistical Society, Series B,1982,44:226-233.[25]McGilchrist CA, Aisbett CW. Regression with frailty in survival analysis. Biometrics,1991,47:461-466.[26]Therneau TM, Grambsch PM. Modeling Survival Data:Extending the Cox Model. Springer: New York,2000.[27]Parner E. Asymptotic theory for the correlated Gamma-frailty model. Annals of Statistics,1998,28:183-214.[28]Lei Liu and Xueling Huang, The use of Gaussian quadrature for estimation in frailty proportional hazards models. Statistics in Medicine,2007,27(14):2665-83[29]Li Q, Lagakos S. Use of the Wei-Lin-Weissfeld method for the analysis of a recurring and a terminating event.Stat Med,1997,16:925-940[30]Cook RJ, Lawless JF.Marginal analysis of recurrent events and a terminating event. Stat Med,1997,16:911-924.[31]Ghosh D, Lin DY.Nonparametric analysis of recurrent events and death. Biometrics,2000,56:554-562.[32]Ghosh D, Lin DY.Marginal regression models for recurrent and terminal event. Stat Sinica,2002,12:663-688[33]Lancaster,A. and Intrator,O. Panel data with survival:Hospitalization of HIV-positive patients. Journalof the American Statistical Association,1998,93,46-53.[34]Wang,M.,Qin,J.,and Chiang,C. Analyzing recurrent event data with informative censoring. Journal of the American Statistical Association,2001,96,1057-1065.[35]Wang,M., and Chiang,C.Non-parametric methods for recurrent event data with informative??censoring and non-informative censoring.Statist.Med,2002,21:445-456.[36]Huang, C. Y. and Wang, M. C. Joint modeling and estimation for recurrent event processes and failure time data. Journal of the American Statistical Association,2004,99:1153-1165.[37]Ye Y,Kalbfleish JD,Schaubel DE.Semiparametric analysis of correlated recurrent and terminal events.Biometrics,2007,63:78-87.[38]王宁宁.纵列生存数据的脆弱性模型估计和应用[博士论文].中山大学,2007[39]罗天娥.非正态及非线性重复测量资料分析模型及其医学应用[博士论文].山西医科大学,2007.[40]王启华,史宁中,耿直等.现代统计研究基础.北京：科学出版社,2010.[41]王宁宁,徐淑一,方积乾.脆弱性模型的随机效应检验和推广的拟合检验.中国卫生统计,2009,26(1)：2-6.[42]Chib S, Greenberg E. Understanding the Metropolis-Hastings algorithm. The American Statistician,1995,49:327-335.[43]Therneau TM, Grambsch PM, Pankratz VS. Penalized survival models and frailty. Journal of Computational and Graphical Statistics,2003,12:156-175.[44]Ripatti S, Palmgren J. Estimation of multivariate frailty models using penalized partial likelihood. Biometrics,2000;56:1016-1022.[45]Ha ID, Lee Y, Song J-K. Hierarchical likelihood approach for frailty models. Biometrika,2001;88:233-243.[46]Ha ID, Lee Y. Estimating frailty models via Poisson hierarchical generalized linear models. Journal of Computational and Graphical Statistics,2003,12:663-681.[47]Lawless JF, Zhan M. Analysis of interval-grouped recurrent event data using piecewise constant rate function.Canadian Journal of Statistics,1998,26:549-565.[48]Feng S, Wolfe RA, Port PK. Frailty survival model analysis of the National Deceased Donor Kidney Transplant Dataset using Poisson variance structures. Journal of the American Statistical Association,2005,100:728-735.[49]Andersen PK, Klein J, Knudsen K, Palacios R. Estimation of variance in Cox's regression model with shared gamma frailties. Biometrics,1997,53:1475-1484.[50]Gray RJ. A Bayesian analysis of institutional effects in a multicenter cancer clinical trial. Biometrics,1994,50:244-253.[51]Matsuyama Y, Sakamoto J, Ohashi Y. A Bayesian hierarchical survival model for the institutional effects in a multi-center cancer clinical trial. Statistics in Medicine,1998,17:1893-1908.[52]Pinheiro JC, Bates DM. Approximations to the log-likelihood function in the nonlinear??mixed-effects model. Journal of Computational and Graphical Statistics,1995,4:12-35.[53]Liu Q, Pierce DA. A note on Gauss-Hermite quadrature. Biometrika,1994,81:624-629.[54]卫生部统计信息中心编.2008年中国卫生服务调查研究：第四次家庭健康询问调查分析报告.北京：中国协和医科大学出版社,2009.[55]Nelson WB.Recurrent events analysisi for product repairs,disease recurrences,and other application.The ASA-SIAM on Statistic and Applied Probability,2003.[56]饶克勤.卫生统计方法与应用进展.北京：人民卫生出版社,2008.