基于复发事件间隔时间下的可加可乘危险率模型
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摘要
复发事件数据常常出现在生物学、医学、经济学、社会学等领域。由于复发事件数据结构复杂,对它的统计分析已经受到学者们广泛重视,其相应研究结果不仅丰富了生存分析,医学与生物统计的研究内容,而且为各交叉学科研究提供理论依据和实际指导。
自二十世纪八十年代开始,对复发事件数据的分析研究已经取得了丰富的成果。对复发事件数据的研究主要集中在对复发时间和间隔时间的分析。对复发事数据分析的模型包含两个方面：一方面,基于强度过程,对强度函数和危险率函数进行建模。另一方面,对复发事件数据而言,由于复发事件的均值函数比强度函数更具有解释意义,因此学者们对均值函数或比率函数进行了建模。
本文在复发事件间隔时间数据下,首先回顾了可乘危险率模型和可加危险率模型,利用估计方程的思想得到参数和基准危险率函数的估计,以及所得估计的渐近性质。接着讨论了可加可乘危险率模型,对于此危险率模型中参数和基准危险率函数,同样采用估计方程的思想将其估计出来,然后证明了估计量的大样本性质。
Recurrent event datas often appear in applied research of biology, medical, economics and sociology. Since the structure of the recurrent event data is very complicated. A lot of researchers are interested in the statistical analysis. The corresponding results are not only affludent in the study of survival analysis, but also they provide theoretical and practial guide for the study of interdisciplinary.
Since the twentieth century, eighties, there are a large number of the study results on the anal-ysis of recurrent events.The study on the recurrent event data is main concentrated on the analysis of the recurrent time and the gap time. The models about the analysis of recurrent event datas contain two sides. In one side, for intensity process, we construct the model of intensity function and hazards function. In the other side, to recurrent event datas, because the mean function of the recurrent event datas is more significant than the intensity function. The researcher construct the model on the mean function or ratio function.
In this paper, the observed datas are the gap time of recurrent event. At first, we review the parametric and the nonparametric estimators of the additive hazards model and the proportional hazards model, which are obtained through the estimating equation approaches. And we look back the asymptotic properties of the proposed estimators. Next, we discuss the additive-multiplicative hazards model, for the the parametric and the nonparametric of the model, we still get the es-timators by the estimating equation approaches. Then we present the proof of the asymptotic properties.
自二十世纪八十年代开始,对复发事件数据的分析研究已经取得了丰富的成果。对复发事件数据的研究主要集中在对复发时间和间隔时间的分析。对复发事数据分析的模型包含两个方面：一方面,基于强度过程,对强度函数和危险率函数进行建模。另一方面,对复发事件数据而言,由于复发事件的均值函数比强度函数更具有解释意义,因此学者们对均值函数或比率函数进行了建模。
本文在复发事件间隔时间数据下,首先回顾了可乘危险率模型和可加危险率模型,利用估计方程的思想得到参数和基准危险率函数的估计,以及所得估计的渐近性质。接着讨论了可加可乘危险率模型,对于此危险率模型中参数和基准危险率函数,同样采用估计方程的思想将其估计出来,然后证明了估计量的大样本性质。
Recurrent event datas often appear in applied research of biology, medical, economics and sociology. Since the structure of the recurrent event data is very complicated. A lot of researchers are interested in the statistical analysis. The corresponding results are not only affludent in the study of survival analysis, but also they provide theoretical and practial guide for the study of interdisciplinary.
Since the twentieth century, eighties, there are a large number of the study results on the anal-ysis of recurrent events.The study on the recurrent event data is main concentrated on the analysis of the recurrent time and the gap time. The models about the analysis of recurrent event datas contain two sides. In one side, for intensity process, we construct the model of intensity function and hazards function. In the other side, to recurrent event datas, because the mean function of the recurrent event datas is more significant than the intensity function. The researcher construct the model on the mean function or ratio function.
In this paper, the observed datas are the gap time of recurrent event. At first, we review the parametric and the nonparametric estimators of the additive hazards model and the proportional hazards model, which are obtained through the estimating equation approaches. And we look back the asymptotic properties of the proposed estimators. Next, we discuss the additive-multiplicative hazards model, for the the parametric and the nonparametric of the model, we still get the es-timators by the estimating equation approaches. Then we present the proof of the asymptotic properties.
引文
[1]陈家鼎.生存分析与可靠性.北京：北京大学出版社,2005.
[2]王启华.生存数据统计分析.北京：科学出版社,2006.
[3]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.
[4]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.
[5]陈家鼎.生存分析与可靠性引论.合肥：安徽教育出版社,1993.
[6]Andersen P K et al. Statistical models based counting processes. New York: Springer-Verlag, 1993.
[7]Fleming T R and Harrington D P. Counting Processes and Survival Analysis. New York: Wiley,1991.
[8]Cox D R and Okaes D. Analysis of survival data. New York: Chapman and Hall,1984.
[9]Cox D R. Regression models and life tables(with Discussion). Journal of the Royal Statistical Society (series B).1972,34,187-220.
[10]Andersen P K and Gill R D. Cox's regression model for counting processes:a large sample study. Ann. Statist.1982,10,1100-1120.
[11]Andersen P K, Borgan, Gill R D and Kending N. Statistical models based counting processes. New York: Spring-Verlag, 1993.
[12]Cox D R. Partlial likelihood. Biometrika.1975,62,269-276.
[13]Tsiatis A A. A large sample study of cox's regression model. Ann.Statist.1981,9,93-108.
[14]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.
[15]Huang Y and Chen Y Q. Marginal regression of gaps between recurrent events. Lifetime Data Anal.2003,9,293-303.
[16]Lin D Y and Ying Z. Semiparametric analysis of the additive risk model. Biometrika. 1994, 81,61-71.
[17]Sun L Q, Park D H and Sun J G. The additive hazards model for recurrent gap times. Statist Sinica.2006,16,919-932.
[18]Lin D Y and Ying Zhiliang. Semiparametric Analysis of General Additive-Multiplicative Hazard Models for Counting Processes. The Annals of Statistics.1995,23,1712-1734.
[19]Kalbfleisch J D and Prentice R L. The statistical analysis of failure time data. New York: wiley,2002.
[20]Pretice R L, Williams B J and Peterson A V. On tne regression analysis of multivariate failure time data. Biometrika.1981,68,373-379.
[21]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.
[22]Prentice R L, Williams B J and Peterson A v. On the regression analysis of multivariate failure time date. Biometrika.1981,68,373-379.
[23]Abu-Libdeh H, Turnbull B W and Clark L C. Analysis of multi-type recurrent events in longitudinal studies:Application to a skin cancer prevention trial. Biometrics.1990,46, 1017-1034.
[24]Lin D Y, Wei L J and Ying Z. Semiparametric Transformation Models for Point Processes. J. AMER. Statist. Assoc.2001,96,620-628.
[25]Maller A, Sun L and Zhou X. Estimating the expected total number of events in a precess. J.Amer. Statist. Assoc.2002,97,577-589.
[26]Sun L and Su B. A class of accelerated means regression models for recurrent event data. Lifetime Data Analysis.2008,14,357-375.
[27]Chaubel D E and Cai J. Regression methods for gap time hazard function of sequentially ordered multivariate failure time data. Biometrika.2004,91,291-303.
[28]Zeng D and Lin D Y. Effecttive estimation of semiparametric transformation models for counting processes. Biometrika.2006,93,627-640.
[29]Wang M C and Chang S H. Nonparametric estimation of a recurrent survival function. J.Amer. Statist.Assoc.1999,94,146-153.
[30]Ghosh D and Lin D Y. Nonparametric analysis of recurrent events and death. Biometrics. 2000,56,554-562.
[31]Li Q H and Lagakos S W. Use of the Wei-Lin-Weissfrld method for the analysis of a recurring and a terminating event. Stat. Med.1997,16,925-940.
[32]Prentice R L and Self S G. Asymptotic distribution theory for Cox-type regression model with general relative risk form. Ann. of Stat.1983,11,804-812.
[33]Spiekerman C F and Lin D Y. Marginal regression models for multivariate failure time data. J. Am. Stat. Assoc.1998,93,1164-1175.
[34]Wei L J, Lin D Y and Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distribution. Journal of the American Statistical Associaltion. 1989,84,1065-1073.
[35]Cook R J and Lawless J F. Marginal analysis recurrent events and a terminating event. Stat. Med.1997,16,911-924.
[36]Lee E W, Wei L J and Amato D A. Cox-type regression analysis for large numbers of small groups of correlated failure time observations. In:klein J P, Goel P K(eds) Survival analy-sis:state of the art. Dordrecht:kluwer.1992,237-247.
[37]Wang M C and Chen Y Q. Nonparametric and semiparametric trend analysis of stratified recurrence time data. Biometrics.2000,56,789-794.
[38]Chang S H and Wang M C. Conditional regression analysis for recurrence time data. Journal of the American Statistical.1999,94,1221-1230.
[39]Goffman C. Calaulus of several variables. New York: Harper and Row,1995.
[40]Sen P K and Singer J M. Large sample methods in statistics. New York: Champman and Hall,1993.
[2]王启华.生存数据统计分析.北京：科学出版社,2006.
[3]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.
[4]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.
[5]陈家鼎.生存分析与可靠性引论.合肥：安徽教育出版社,1993.
[6]Andersen P K et al. Statistical models based counting processes. New York: Springer-Verlag, 1993.
[7]Fleming T R and Harrington D P. Counting Processes and Survival Analysis. New York: Wiley,1991.
[8]Cox D R and Okaes D. Analysis of survival data. New York: Chapman and Hall,1984.
[9]Cox D R. Regression models and life tables(with Discussion). Journal of the Royal Statistical Society (series B).1972,34,187-220.
[10]Andersen P K and Gill R D. Cox's regression model for counting processes:a large sample study. Ann. Statist.1982,10,1100-1120.
[11]Andersen P K, Borgan, Gill R D and Kending N. Statistical models based counting processes. New York: Spring-Verlag, 1993.
[12]Cox D R. Partlial likelihood. Biometrika.1975,62,269-276.
[13]Tsiatis A A. A large sample study of cox's regression model. Ann.Statist.1981,9,93-108.
[14]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.
[15]Huang Y and Chen Y Q. Marginal regression of gaps between recurrent events. Lifetime Data Anal.2003,9,293-303.
[16]Lin D Y and Ying Z. Semiparametric analysis of the additive risk model. Biometrika. 1994, 81,61-71.
[17]Sun L Q, Park D H and Sun J G. The additive hazards model for recurrent gap times. Statist Sinica.2006,16,919-932.
[18]Lin D Y and Ying Zhiliang. Semiparametric Analysis of General Additive-Multiplicative Hazard Models for Counting Processes. The Annals of Statistics.1995,23,1712-1734.
[19]Kalbfleisch J D and Prentice R L. The statistical analysis of failure time data. New York: wiley,2002.
[20]Pretice R L, Williams B J and Peterson A V. On tne regression analysis of multivariate failure time data. Biometrika.1981,68,373-379.
[21]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.
[22]Prentice R L, Williams B J and Peterson A v. On the regression analysis of multivariate failure time date. Biometrika.1981,68,373-379.
[23]Abu-Libdeh H, Turnbull B W and Clark L C. Analysis of multi-type recurrent events in longitudinal studies:Application to a skin cancer prevention trial. Biometrics.1990,46, 1017-1034.
[24]Lin D Y, Wei L J and Ying Z. Semiparametric Transformation Models for Point Processes. J. AMER. Statist. Assoc.2001,96,620-628.
[25]Maller A, Sun L and Zhou X. Estimating the expected total number of events in a precess. J.Amer. Statist. Assoc.2002,97,577-589.
[26]Sun L and Su B. A class of accelerated means regression models for recurrent event data. Lifetime Data Analysis.2008,14,357-375.
[27]Chaubel D E and Cai J. Regression methods for gap time hazard function of sequentially ordered multivariate failure time data. Biometrika.2004,91,291-303.
[28]Zeng D and Lin D Y. Effecttive estimation of semiparametric transformation models for counting processes. Biometrika.2006,93,627-640.
[29]Wang M C and Chang S H. Nonparametric estimation of a recurrent survival function. J.Amer. Statist.Assoc.1999,94,146-153.
[30]Ghosh D and Lin D Y. Nonparametric analysis of recurrent events and death. Biometrics. 2000,56,554-562.
[31]Li Q H and Lagakos S W. Use of the Wei-Lin-Weissfrld method for the analysis of a recurring and a terminating event. Stat. Med.1997,16,925-940.
[32]Prentice R L and Self S G. Asymptotic distribution theory for Cox-type regression model with general relative risk form. Ann. of Stat.1983,11,804-812.
[33]Spiekerman C F and Lin D Y. Marginal regression models for multivariate failure time data. J. Am. Stat. Assoc.1998,93,1164-1175.
[34]Wei L J, Lin D Y and Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distribution. Journal of the American Statistical Associaltion. 1989,84,1065-1073.
[35]Cook R J and Lawless J F. Marginal analysis recurrent events and a terminating event. Stat. Med.1997,16,911-924.
[36]Lee E W, Wei L J and Amato D A. Cox-type regression analysis for large numbers of small groups of correlated failure time observations. In:klein J P, Goel P K(eds) Survival analy-sis:state of the art. Dordrecht:kluwer.1992,237-247.
[37]Wang M C and Chen Y Q. Nonparametric and semiparametric trend analysis of stratified recurrence time data. Biometrics.2000,56,789-794.
[38]Chang S H and Wang M C. Conditional regression analysis for recurrence time data. Journal of the American Statistical.1999,94,1221-1230.
[39]Goffman C. Calaulus of several variables. New York: Harper and Row,1995.
[40]Sen P K and Singer J M. Large sample methods in statistics. New York: Champman and Hall,1993.