含有有序变量和连续变量的多元纵向数据分析
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摘要
在生物医学研究中，常常对同一个个体的多个指标在不同时刻进行重复测量，得到的测量值一般视为多元纵向数据。因为多个指标之间具有相关性，所以对多元纵向数据进行联合建模分析，势必会比分开建模分析得到更多的信息。本文的目的在于处理当响应变量值是连续型和有序型时的情形，用一个潜在变量模型来描述此类数据，得到广义线性混合效应模型，并假设给定随机效应，观测值的分布是条件的独立的，通过数值积分的方法求出边缘似然的近似值。给出参数的极大似然估计。
In biomedical studies, multiple measures of life characteristic often are recorded over time, such measures are frequently correlated as multivariate longitudinal data. Joint analysis of the outcomes variables has several potential advantages over separate analyzes.Herein we propose models for analysis of repeated measurements on continuous and ordinal variables measuring the same latent trait over time. Assuming conditional independence given random effects,we obtain the marginal likelihood density using numerical integration,then the maximum-likelihood estimation.
In biomedical studies, multiple measures of life characteristic often are recorded over time, such measures are frequently correlated as multivariate longitudinal data. Joint analysis of the outcomes variables has several potential advantages over separate analyzes.Herein we propose models for analysis of repeated measurements on continuous and ordinal variables measuring the same latent trait over time. Assuming conditional independence given random effects,we obtain the marginal likelihood density using numerical integration,then the maximum-likelihood estimation.
引文
[1] D. Todem, K. Kim and E.Lesaffre(2007). Latent-variable models for longitudinal data with bivariate ordinal outcomes. Statistics in Medicine 26,1034-1054.
[2] G.E.Bonney(1986). Regressive logistic models for familial disease and other binary traits. Biometrics 42,611-625.
[3] Garrett M. Fitzmanrice and Nan M.Laird(1997). Regression models for mixed discrete and continuous responses with potentially missing values.Biometrics 55,110-122.
[4] G.M.Fitzmaurice,N.M.Lalrd and J.H.Ware. Applied longitudinal analysis, Wiley & Sons, 2004.
[5] Heping Zhang and Kathleen Merikangas(2000). A frailty model of segregation analysis: understanding the familial transmission of alcoholism. Biometrics 56, 815-823.
[6] Heping Zhang, Rui Feng, and Hongtu Zhu(2003). A latent variable model of segregation analysis for ordinal traits. Journal of the American Statistical Association 98, 1023-1034.
[7] Isaac Meilijson(1989). A fast improvement to the EM algorithm on its own terms.J.R.Statist.Soc.B 51, 127-138.
[8] Jason Roy and Xihong Lin(2000). Latent variable models for longitudinal data with multiple continuous outcomes. Biometrics 56, 1047-1054.
[9] Jason Roy and Xihong Lin(2002). Analysis of multivariate longitudinal outcomes with nonignorable dropouts and missing covariates: changes in methadone treatment; practice. Journal of the American Statistical Association 97, 40-52.
[10] Lin Jin-guan, Wei Bo-cheng, Fu Jue-sheng(2006). Testing for departures from nominal dispersion and power simulations in discrete generalized nonlinear models with longitudinal data. Mathmatica Applicata 19, 342-347.
[11] Margaret A. Connolly and Kung-Yee Liang (1988). Conditional logistic regression models for correlated binary data. Biometrika 75, 501-506.
[12] M. A. Tanner. Tools for statistical inference. Springer, 1996.
[13] Mary Dupuis, Sammel Louise M. Ryan and Julie M. Legler(1997). Latent variable models for mixed discrete and continuous outcomes. J. R. Statist, Soc. B 59, 667-678.
[14] Nevin L. Zhang and Shihong Yuan. Latent structure models and diagnosis in traditional Chinese medisine.http://www.cs.ust.hk/faculty/lzhang/tcm/.
[15] Perter J,Diggle,Kung-Yee Liang, and Scott L. Zeger. Analysis of longitudinal data. Oxford University Press, 2002.
[16] Qing Liu and Donald A. Pierce(1994). A note on Gauss-Hermite quadrature. Biometrika 81, 624-629.
[17] Ross L. Prentice(1988). Correlated binary regression with covariates specific to each binary observation. Biometrics 44, 1033-1048.
[18] R. V. Gueorguieva and Alan Agresti(2001). A correlated probit model for joint modelling of clustered binary and continuous responses. Journal of the American Statistical Association 96, 1102-1112.
[19] R. V. Gueorguieva and G. Sanacora(2006). Joint analysis of repeatedly observed continuous and ordinal measures of disease severity. Statistics in Medicine 25, 1307-1322.
[20] Adrian．E．Raftery．统计学在社会学中的应用-1950-2000：一个简要的回顾，2001．
[21] S．Weisberg(美)著，王静龙等译．应用线性回归．中国统计出版社 1998．3．
[22] Y．M．M．比少普(美)等著，张尧庭译．离散多元分析：理论与实践．中国统计出版社 1998．3．
[23] 高惠璇等编译．SAS系统-SAS／STAT软件使用手册．中国统计出版社，1997，9．
[24] 刘红云，孟庆茂(2003)．纵向数据分析方法．心理科学进展，11，586-592．
[25] 刘沛，王灿楠(2001)．在医学纵向数据分析中SAS内部逻辑变量的应用，18，44-45．
[26] 刘祥，张菊英(2006)．有序多分类重复测量资料的广义估计方程分析．四川大学学报(医学版)，37，798-800．
[27] 王济川，郭志刚．Logistic回归模型-方法与应用．高等教育出版社，2001．
[28] 王沫然编著．MATLAB与科学计算．电子工业出版社，2005．
[29] 王松桂，史建红，尹素菊，吴密霞编著．线性模性引论．科学出版社，2004．
[30] 熊林平，曹秀堂，徐勇勇，陆建纵向观测二分类数据的广义线性模型分析．第二军医大学学报，20，1999．7．
[31] 赵宁，俞顺章(1994)．Logistic回归分析中的一些应用问题．山西医学院学报25，26-30．
[2] G.E.Bonney(1986). Regressive logistic models for familial disease and other binary traits. Biometrics 42,611-625.
[3] Garrett M. Fitzmanrice and Nan M.Laird(1997). Regression models for mixed discrete and continuous responses with potentially missing values.Biometrics 55,110-122.
[4] G.M.Fitzmaurice,N.M.Lalrd and J.H.Ware. Applied longitudinal analysis, Wiley & Sons, 2004.
[5] Heping Zhang and Kathleen Merikangas(2000). A frailty model of segregation analysis: understanding the familial transmission of alcoholism. Biometrics 56, 815-823.
[6] Heping Zhang, Rui Feng, and Hongtu Zhu(2003). A latent variable model of segregation analysis for ordinal traits. Journal of the American Statistical Association 98, 1023-1034.
[7] Isaac Meilijson(1989). A fast improvement to the EM algorithm on its own terms.J.R.Statist.Soc.B 51, 127-138.
[8] Jason Roy and Xihong Lin(2000). Latent variable models for longitudinal data with multiple continuous outcomes. Biometrics 56, 1047-1054.
[9] Jason Roy and Xihong Lin(2002). Analysis of multivariate longitudinal outcomes with nonignorable dropouts and missing covariates: changes in methadone treatment; practice. Journal of the American Statistical Association 97, 40-52.
[10] Lin Jin-guan, Wei Bo-cheng, Fu Jue-sheng(2006). Testing for departures from nominal dispersion and power simulations in discrete generalized nonlinear models with longitudinal data. Mathmatica Applicata 19, 342-347.
[11] Margaret A. Connolly and Kung-Yee Liang (1988). Conditional logistic regression models for correlated binary data. Biometrika 75, 501-506.
[12] M. A. Tanner. Tools for statistical inference. Springer, 1996.
[13] Mary Dupuis, Sammel Louise M. Ryan and Julie M. Legler(1997). Latent variable models for mixed discrete and continuous outcomes. J. R. Statist, Soc. B 59, 667-678.
[14] Nevin L. Zhang and Shihong Yuan. Latent structure models and diagnosis in traditional Chinese medisine.http://www.cs.ust.hk/faculty/lzhang/tcm/.
[15] Perter J,Diggle,Kung-Yee Liang, and Scott L. Zeger. Analysis of longitudinal data. Oxford University Press, 2002.
[16] Qing Liu and Donald A. Pierce(1994). A note on Gauss-Hermite quadrature. Biometrika 81, 624-629.
[17] Ross L. Prentice(1988). Correlated binary regression with covariates specific to each binary observation. Biometrics 44, 1033-1048.
[18] R. V. Gueorguieva and Alan Agresti(2001). A correlated probit model for joint modelling of clustered binary and continuous responses. Journal of the American Statistical Association 96, 1102-1112.
[19] R. V. Gueorguieva and G. Sanacora(2006). Joint analysis of repeatedly observed continuous and ordinal measures of disease severity. Statistics in Medicine 25, 1307-1322.
[20] Adrian．E．Raftery．统计学在社会学中的应用-1950-2000：一个简要的回顾，2001．
[21] S．Weisberg(美)著，王静龙等译．应用线性回归．中国统计出版社 1998．3．
[22] Y．M．M．比少普(美)等著，张尧庭译．离散多元分析：理论与实践．中国统计出版社 1998．3．
[23] 高惠璇等编译．SAS系统-SAS／STAT软件使用手册．中国统计出版社，1997，9．
[24] 刘红云，孟庆茂(2003)．纵向数据分析方法．心理科学进展，11，586-592．
[25] 刘沛，王灿楠(2001)．在医学纵向数据分析中SAS内部逻辑变量的应用，18，44-45．
[26] 刘祥，张菊英(2006)．有序多分类重复测量资料的广义估计方程分析．四川大学学报(医学版)，37，798-800．
[27] 王济川，郭志刚．Logistic回归模型-方法与应用．高等教育出版社，2001．
[28] 王沫然编著．MATLAB与科学计算．电子工业出版社，2005．
[29] 王松桂，史建红，尹素菊，吴密霞编著．线性模性引论．科学出版社，2004．
[30] 熊林平，曹秀堂，徐勇勇，陆建纵向观测二分类数据的广义线性模型分析．第二军医大学学报，20，1999．7．
[31] 赵宁，俞顺章(1994)．Logistic回归分析中的一些应用问题．山西医学院学报25，26-30．