多类型复发事件在间隔时间数据下的危险率模型
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摘要
在医学,生物学,经济学等研究过程中,研究的个体有时会重复地经历某一事件或者多次的失效,这种事件称为复发事件.复发事件是广泛存在的,然而复发事件数据的结构往往是很复杂的.因为这些事件之间常常存在一定的相依性和有序性.越来越多的学者正致力于解决这方面的问题,并且获得了许多有意义的结果.
复发事件数据根据研究对象的种类一般可以分为两种类型.当只对一种可反复发生的事件感兴趣时,这时得到的数据称为单类型复发事件数据.然而,有时研究的个体可能会同时经历多种不同类型的复发事件且类别之间存在相关性,这时我们必须同时研究各类事件,这样得到的数据称为多类型复发事件数据.
对于复发事件,我们常常感兴趣的是研究协变量对复发事件的影响程度.历史上,许多学者提出了各种不同的回归模型来研究协变量对单类型复发事件的影响.本文主要讨论的是多类型复发事件在间隔时间数据下的三种危险率回归模型.首先介绍复发事件的一些背景,综述了单类型复发事件在间隔时间数据下的比例危险率模型,可加危险率模型以及可乘可加危险率模型；接着我们提出多类型复发事件的比例危险率模型,同时给出模型中参数以及非参数函数的估计方法,建立这些估计的渐近性质并且进行模拟研究；其次我们提出多类型复发事件的可加危险率模型,给出了模型中参数以及非参数函数的估计方法,研究了这些估计的渐近性质；再次我们提出多类型复发事件的可乘可加危险率模型,给出了模型中参数以及非参数函数的估计方法,讨论了这些估计的渐近性质.
In many application areas, subjects may usually experience multiple events or failure over time, which have been termed recurrent events. Recurrent event date often arises in many applica-tion areas, such as, medical science, biology and economics. However, the structure of recurrent event dates is usually complex. Because there exist correlation and sequence between different events. Recently, more and more researchers exert their efforts to solve these difficulties and gain many significant results.
The recurrent event dates which record the recurrent time of each event of interest, includes two types. The first type termed single recurrent events is that only one type of recurrent events is of considered. However, in many settings, subjects may experience several types of recurrent events, and these types are dependent on each other. This case is naturally considered, in which dates have been called multiple type recurrent event.
For recurrent events, we are interested in assessing the effect of covariates on the recurrent events. There have been many regression models to be proposed. In this thesis, we mainly dis-cussed three models for multiple type recurrent gap times. In Chapter 2, the proportional hazards model has been put forward which is the generalized form of Cox model, methods and asymp-totic properties for inferences on this model are established and some results from simulation are presented. In Chapter 3, we study additive hazards model for multiple type recurrent gap times. Asymptotic properties for estimators of parameter are established. In Chapter 4, the additive-multiplicative hazards model for multiple type recurrent gap times are investigated. Asymptotic properties of the proposed parameter estimators are developed.
复发事件数据根据研究对象的种类一般可以分为两种类型.当只对一种可反复发生的事件感兴趣时,这时得到的数据称为单类型复发事件数据.然而,有时研究的个体可能会同时经历多种不同类型的复发事件且类别之间存在相关性,这时我们必须同时研究各类事件,这样得到的数据称为多类型复发事件数据.
对于复发事件,我们常常感兴趣的是研究协变量对复发事件的影响程度.历史上,许多学者提出了各种不同的回归模型来研究协变量对单类型复发事件的影响.本文主要讨论的是多类型复发事件在间隔时间数据下的三种危险率回归模型.首先介绍复发事件的一些背景,综述了单类型复发事件在间隔时间数据下的比例危险率模型,可加危险率模型以及可乘可加危险率模型；接着我们提出多类型复发事件的比例危险率模型,同时给出模型中参数以及非参数函数的估计方法,建立这些估计的渐近性质并且进行模拟研究；其次我们提出多类型复发事件的可加危险率模型,给出了模型中参数以及非参数函数的估计方法,研究了这些估计的渐近性质；再次我们提出多类型复发事件的可乘可加危险率模型,给出了模型中参数以及非参数函数的估计方法,讨论了这些估计的渐近性质.
In many application areas, subjects may usually experience multiple events or failure over time, which have been termed recurrent events. Recurrent event date often arises in many applica-tion areas, such as, medical science, biology and economics. However, the structure of recurrent event dates is usually complex. Because there exist correlation and sequence between different events. Recently, more and more researchers exert their efforts to solve these difficulties and gain many significant results.
The recurrent event dates which record the recurrent time of each event of interest, includes two types. The first type termed single recurrent events is that only one type of recurrent events is of considered. However, in many settings, subjects may experience several types of recurrent events, and these types are dependent on each other. This case is naturally considered, in which dates have been called multiple type recurrent event.
For recurrent events, we are interested in assessing the effect of covariates on the recurrent events. There have been many regression models to be proposed. In this thesis, we mainly dis-cussed three models for multiple type recurrent gap times. In Chapter 2, the proportional hazards model has been put forward which is the generalized form of Cox model, methods and asymp-totic properties for inferences on this model are established and some results from simulation are presented. In Chapter 3, we study additive hazards model for multiple type recurrent gap times. Asymptotic properties for estimators of parameter are established. In Chapter 4, the additive-multiplicative hazards model for multiple type recurrent gap times are investigated. Asymptotic properties of the proposed parameter estimators are developed.
引文
[1]Kalbfleisch J D and Prentice R L. The statistical analysis of failure time date. New York: Wiley,2002.
[2]Byar D P. The veterans administration study of chemoprophylaxis for recurrent stage I blad-der tumors:comparisons of placebo, pyridoxine, and topical thiotepa, In Bladder T umors and Other T opics in Urological Oncology, Pavone-Macaluso Ed M, Smith P H and Edsmyr F. New York:Plenum,1980,363-370.
[3]Prentice R L, Williams B J and Peterson A V. On the regression analysis of multivariate failure time data. Biometrika.1981,68,373-379.
[4]Lawless J F and Nadeau C. Some simple robust methods for the analysis of recurrent events. Technometrics.1995,37,158-168.
[5]Abu-Libdeh H, Turnbull B W, Clark L C. Analysis of multi-type recurrent events in longi-tudinal studies:application to a skin cancer prevention trial.1990,46,1017-1034.
[6]Schaubel D, Johansen H, Dutta M, Desmeules M, Becker A and Mao Y. Neonatal character-istics as risk factors for preschool asthma. J Asthma.1996,33,255-264.
[7]Andersen P K and Gill R D. Cox's regression model for counting processes:a large sample study. Ann Statist.1982,10,1100-1120.
[8]Andersen P K, Borgan, Gill R D and Kending N. Statistical models based counting pro-cesses. New York:Springer-Verlag,1993.
[9]Wang M C and Chang S H. Nonparametric estimation of a recurrent survival function. J Amer Statist Assoc.1999,94,146-153.
[10]Wang M C and Chen Y Q. Nonparametric and semiparameic trend analysis of stratified recurrence time data. Biometrics.2000,56,789-794.
[11]Wei L J, Lin D Y and Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Amer statist.1989,84,10065-1073.
[12]Cook R J and Lawless J F. Marginal analysis recurrent events and a terminating event. Stat Med.1997,16,911-924.
[13]Pepe M S and Cai J. Some graphical displays and marginal regression analysis for recurrent failure times and time dependent covariates. Journal of the American Statistical Association. 1993,88,811-820.
[14]Lin D Y, Wei J L, Yang I and Ying Z. Semiparametrix regression for the mean and rate functions of recurrent events. J Roy Statist. Soc Ser B.2000,62,711-730.
[15]Lin D Y, Wei L J and Ying Z. Semiparametric transformation models for point processes. J Amer Statist Assoc.2001,96,620-628.
[16]Cai J W and Schaubel D. Marginal means/rates Modes for multiple type recurrent events. Lifetime Data Analysis.2004,10,121-138.
[17]Cox D R and Okaes D. Analysis of survival data. New York:Chapman and Hall,1984.
[18]Cox D R. Regression Models and Life Tables Journal of the Royal Statistical Society. Ser B. 1972,34,187-220.
[19]Cox D R. Pattial likelihood. Biometrika.1975,62,269-276.
[20]Tsiatis A A. A large sample study of cox's regression model. Ann Statist.1981,993-108.
[21]Fleming T R and Harrington D P. Counting processes and survival analysis. New York: Wiley,1991.
[22]Aalen O O. A model for nonparametric regression analysis of counting processes. In lecture notes in statistcs,2Ed, Klonecki N, Kosek A and Rosinski J. New York:Springer,1980, 1-25.
[23]Breslow N E. Contribution to the discussion of Cox D R. J R Statist Soc B.1972,34,216-217.
[24]Huang Y and Chen Y Q. Marginal regression of gaps between recurrent events. Lifetime Data Analysis.2003,9,293-303.
[25]Breslow N E and Day N E. Statistical methods in cancer research. The Design and Analysis of Cohort Studies. Lyon:IARC,1987.
[26]Thomas D C. Use of auxiliary information in fitting nonpoportional hazards models. Mordern Statistical Methods in Chronic Desease Epidemiology.1986,197-210.
[27]Mckeague I W and Sasieni P D. A partly parametric additive risk model. Biometrika.1994, 81,501-514.
[28]Lin D Y and Ying Z. Semiparametric analysis of the additive risk model. Biometrika.1994, 81,61-71.
[29]Sun L Q, Park D H and Sun J G. The additive hazards model for recurrent gap times. Statist Sinica.2006,16,919-932.
[30]Lin D Y and Ying Z. Semiparametric analysis of genernal additive-multiplicative intensity models for counting processes. Technical Report 129, Dept, Biostatistics, Univ, Washington, 1993.
[31]Buckley J D. Additive and multiplicative models for relative survival rates. Biometrics.1984, 40,51-62.
[32]Lin D Y and Ying Z L. Semiparametric Analysis of General Additive-multiplicative hazard models for counting processes. The Annals of Statistics.1995,23,1712-1734.
[33]Ye Y and Liu H B. The addititive-multiplicative hazards model for recurrent gap time. Statist Sinica.2010.
[34]Pollard D. Empirical process:theory and applications. Institute of Mathmatical statistic, Hayward, California,1990.
[35]Goffman C. Calculus of several variables. New York:Harper and Row,1965.
[36]Spiekman C F and Lin D Y. Marginal regression models for multivariate failure tie data. J Amer statist.1998,93,1164-1175.
[37]Sen P K and Singer J M. Large sample methods in statistics. New York:Champman and Hall,1993.
[38]Van der Vaart A W and Wellner J A. Weak convergence and empirical processes. New York: Springer,1996.
[39]Lawless J F.数据寿命的统计模型与方法.茆诗松,濮晓龙,刘忠译.北京：中国统计出版社,1998.
[40]Lee E T.生存数据分析的统计方法.陈家鼎,戴中维译.北京：中国统计出版社,1998.
[41]陈家鼎.生存分析与可靠性.北京：北京大学出版社,2005.
[42]王启华.生存数据统计分析.北京：科学出版社,2006.
[2]Byar D P. The veterans administration study of chemoprophylaxis for recurrent stage I blad-der tumors:comparisons of placebo, pyridoxine, and topical thiotepa, In Bladder T umors and Other T opics in Urological Oncology, Pavone-Macaluso Ed M, Smith P H and Edsmyr F. New York:Plenum,1980,363-370.
[3]Prentice R L, Williams B J and Peterson A V. On the regression analysis of multivariate failure time data. Biometrika.1981,68,373-379.
[4]Lawless J F and Nadeau C. Some simple robust methods for the analysis of recurrent events. Technometrics.1995,37,158-168.
[5]Abu-Libdeh H, Turnbull B W, Clark L C. Analysis of multi-type recurrent events in longi-tudinal studies:application to a skin cancer prevention trial.1990,46,1017-1034.
[6]Schaubel D, Johansen H, Dutta M, Desmeules M, Becker A and Mao Y. Neonatal character-istics as risk factors for preschool asthma. J Asthma.1996,33,255-264.
[7]Andersen P K and Gill R D. Cox's regression model for counting processes:a large sample study. Ann Statist.1982,10,1100-1120.
[8]Andersen P K, Borgan, Gill R D and Kending N. Statistical models based counting pro-cesses. New York:Springer-Verlag,1993.
[9]Wang M C and Chang S H. Nonparametric estimation of a recurrent survival function. J Amer Statist Assoc.1999,94,146-153.
[10]Wang M C and Chen Y Q. Nonparametric and semiparameic trend analysis of stratified recurrence time data. Biometrics.2000,56,789-794.
[11]Wei L J, Lin D Y and Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Amer statist.1989,84,10065-1073.
[12]Cook R J and Lawless J F. Marginal analysis recurrent events and a terminating event. Stat Med.1997,16,911-924.
[13]Pepe M S and Cai J. Some graphical displays and marginal regression analysis for recurrent failure times and time dependent covariates. Journal of the American Statistical Association. 1993,88,811-820.
[14]Lin D Y, Wei J L, Yang I and Ying Z. Semiparametrix regression for the mean and rate functions of recurrent events. J Roy Statist. Soc Ser B.2000,62,711-730.
[15]Lin D Y, Wei L J and Ying Z. Semiparametric transformation models for point processes. J Amer Statist Assoc.2001,96,620-628.
[16]Cai J W and Schaubel D. Marginal means/rates Modes for multiple type recurrent events. Lifetime Data Analysis.2004,10,121-138.
[17]Cox D R and Okaes D. Analysis of survival data. New York:Chapman and Hall,1984.
[18]Cox D R. Regression Models and Life Tables Journal of the Royal Statistical Society. Ser B. 1972,34,187-220.
[19]Cox D R. Pattial likelihood. Biometrika.1975,62,269-276.
[20]Tsiatis A A. A large sample study of cox's regression model. Ann Statist.1981,993-108.
[21]Fleming T R and Harrington D P. Counting processes and survival analysis. New York: Wiley,1991.
[22]Aalen O O. A model for nonparametric regression analysis of counting processes. In lecture notes in statistcs,2Ed, Klonecki N, Kosek A and Rosinski J. New York:Springer,1980, 1-25.
[23]Breslow N E. Contribution to the discussion of Cox D R. J R Statist Soc B.1972,34,216-217.
[24]Huang Y and Chen Y Q. Marginal regression of gaps between recurrent events. Lifetime Data Analysis.2003,9,293-303.
[25]Breslow N E and Day N E. Statistical methods in cancer research. The Design and Analysis of Cohort Studies. Lyon:IARC,1987.
[26]Thomas D C. Use of auxiliary information in fitting nonpoportional hazards models. Mordern Statistical Methods in Chronic Desease Epidemiology.1986,197-210.
[27]Mckeague I W and Sasieni P D. A partly parametric additive risk model. Biometrika.1994, 81,501-514.
[28]Lin D Y and Ying Z. Semiparametric analysis of the additive risk model. Biometrika.1994, 81,61-71.
[29]Sun L Q, Park D H and Sun J G. The additive hazards model for recurrent gap times. Statist Sinica.2006,16,919-932.
[30]Lin D Y and Ying Z. Semiparametric analysis of genernal additive-multiplicative intensity models for counting processes. Technical Report 129, Dept, Biostatistics, Univ, Washington, 1993.
[31]Buckley J D. Additive and multiplicative models for relative survival rates. Biometrics.1984, 40,51-62.
[32]Lin D Y and Ying Z L. Semiparametric Analysis of General Additive-multiplicative hazard models for counting processes. The Annals of Statistics.1995,23,1712-1734.
[33]Ye Y and Liu H B. The addititive-multiplicative hazards model for recurrent gap time. Statist Sinica.2010.
[34]Pollard D. Empirical process:theory and applications. Institute of Mathmatical statistic, Hayward, California,1990.
[35]Goffman C. Calculus of several variables. New York:Harper and Row,1965.
[36]Spiekman C F and Lin D Y. Marginal regression models for multivariate failure tie data. J Amer statist.1998,93,1164-1175.
[37]Sen P K and Singer J M. Large sample methods in statistics. New York:Champman and Hall,1993.
[38]Van der Vaart A W and Wellner J A. Weak convergence and empirical processes. New York: Springer,1996.
[39]Lawless J F.数据寿命的统计模型与方法.茆诗松,濮晓龙,刘忠译.北京：中国统计出版社,1998.
[40]Lee E T.生存数据分析的统计方法.陈家鼎,戴中维译.北京：中国统计出版社,1998.
[41]陈家鼎.生存分析与可靠性.北京：北京大学出版社,2005.
[42]王启华.生存数据统计分析.北京：科学出版社,2006.