基于高层次结构数据的多水平模型贝叶斯推断及应用
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摘要
面对具有多层次嵌套结构的数据,构建多水平模型是统计建模的一个重要研究课题。经典的参数估计方法主要采用极大似然估计法(ML),然而当面对高层数量单位小或数据结构不平衡时,极大似然估计在估计精度上存在一定不足;而贝叶斯方法充分应用了有效的先验信息,可以弥补其不足。本文在高层次结构数据多水平模型的研究基础上,探索高层次结构数据的多水平模型贝叶斯推断理论,并以云南省红河州农户收入数据作实证分析,建立了基于县-村-户嵌套结构的农户收入影响因素多水平模型,对比分析模型参数的ML估计、经验贝叶斯(EB-ML)估计和完全贝叶斯估计,从而充分展现了高层次结构数据多水平模型的完全贝叶斯推断方法,在拟合高层数量单位小或数据不平衡时具有的特征和优势。
Building multilevel model is an important technology of statistical modeling with hierarchical nested structural data. Maximum likelihood estimation method(ML) is mainly used in the classical method of model parameter estimation,but when a small number of units at high-level or unbalanced data structure are occurred, this approach has some drawbacks within estimation accuracy. However, Bayesian estimation method has obvious advantage that makes full use of effective prior information to avoid its deficiency. Based on the research of high-level structure data of multilevel model, this paper explores Bayesian inference method for multilevel model of high-level structural data. In addition, this paper makes an empirical analysis on farmers income data of Honghe prefecture of Yunnan province, establishes a multilevel model of farmers income and studies its influence factors based on the county-village-households nested structure. Through a comparison and analysis on the model parameters estimation between the ML estimation, empirical Bayes(EB-ML) estimation and fully Bayesian estimation, we find that the fully Bayesian inference method of multilevel model with high-level structure perform best.
Building multilevel model is an important technology of statistical modeling with hierarchical nested structural data. Maximum likelihood estimation method(ML) is mainly used in the classical method of model parameter estimation,but when a small number of units at high-level or unbalanced data structure are occurred, this approach has some drawbacks within estimation accuracy. However, Bayesian estimation method has obvious advantage that makes full use of effective prior information to avoid its deficiency. Based on the research of high-level structure data of multilevel model, this paper explores Bayesian inference method for multilevel model of high-level structural data. In addition, this paper makes an empirical analysis on farmers income data of Honghe prefecture of Yunnan province, establishes a multilevel model of farmers income and studies its influence factors based on the county-village-households nested structure. Through a comparison and analysis on the model parameters estimation between the ML estimation, empirical Bayes(EB-ML) estimation and fully Bayesian estimation, we find that the fully Bayesian inference method of multilevel model with high-level structure perform best.
引文
[1] Goldstein H. Multilevel Statistical Models[M]. New York:Halsted Press, 2010.
[2] Raudenbushi S W, Bryk A S. Hierarchical Linear Models:Applications and Data Analysis Methods[M]. Thousand Oaks:Sage Publications, 2007.
[3] Dempster A P,Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. Journal of the Royal Statistical Society, 1977, 39:1-8.
[4]石磊,向其凤,陈飞.多水平模型及其在经济领域中的应用[M].北京:科学出版社,2013.
[5]张敏,鲁筠,石磊.基于高层次结构数据的多水平发展模型设计及应用[J].数量经济技术经济研究,2017,(6):134-147.
[6]石磊.多水平模型及其统计诊断[M].北京:科学出版社,2008.
[7] Lindley D V, Smith A F M, Bayes estimates for the linear model[J]. Journal of the Royal Statistical Society:Series B(Methodological), 1972,34(1):1-41.
[8] Dempster A P. Comment on Tanner and Wong[J]. Journal of the American Statistical Association,1987, 82:541.
[9] Seltzer M H, Wong W H, Bryk A S. Bayesian analysis in application of hierarchical models:Issues and methods[J]. Journal of Educational and Behavioral Statistics Summer, 1996, 21(2):131-167.
[10] Seltzer M H. Sensitivity analysis for fixed effects in the hierarchical model:A Gibbs sampling approach[J]. Journal of Educational Statistics, 1993, 18:207-235.
[11] Browne W J, Draper D. A comparison of Bayesian and likelihood-based methods for fitting multilevel models[J]. Bayesian Analysis, 2006,(3):473-514.
[12] Polson N, Scott J. On the half-Cauchy prior for a global scale parameter[J]. Bayesian Analysis, 2012:887-902.
[13] Demirhan H, Kalaylioglu Z. Joint prior distribution for variance parameters in Bayesian analysis of normal hierarchical models[J]. Journal of Multivariate Analysis, 2015:163-174.
[14] Browne W J. MCMC Estimation in MLwiN2.32[M]. United Kingdom, Centre for Multilevel Modeling University of Bristol Pressed, 2015.
[15] Sturtz S, Ligges U, Gelman A. R2WinBUGS:A package for running WinBUGS from R[J]. Journal of Statistical Software, 2005,12(3):1-16.
[16] Congdon P D. Applied Bayesian Hierarchical Methods[M]. New York:Taylor and Francis Group:CRC Pressed, 2010.
[17]石磊,向其凤,张炯.物质资本、人力资本、就业结构与西部民族地区农户收入增长[J].数理统计与管理,2011, 30(6):1030-1038.
[18]李兴绪,刘曼莉.边境民族地区农户收入影响因素的实证分析一以云南红河州农户为例[J],数理统计与管理,2011,30(4):604-613.
[19]郭志刚.分层线性模型:应用与数据分析方法[M].北京:社会科学文献出版社,2016.
[20]朱慧明,韩玉启等.贝叶斯多元统计推断理论[M].北京:科学出版社,2006.
[21]刘金山,夏强.基于MCMC算法的贝叶斯统计方法[M].北京:科学出版社,2016.
[2] Raudenbushi S W, Bryk A S. Hierarchical Linear Models:Applications and Data Analysis Methods[M]. Thousand Oaks:Sage Publications, 2007.
[3] Dempster A P,Laird N M, Rubin D B. Maximum likelihood from incomplete data via the EM algorithm[J]. Journal of the Royal Statistical Society, 1977, 39:1-8.
[4]石磊,向其凤,陈飞.多水平模型及其在经济领域中的应用[M].北京:科学出版社,2013.
[5]张敏,鲁筠,石磊.基于高层次结构数据的多水平发展模型设计及应用[J].数量经济技术经济研究,2017,(6):134-147.
[6]石磊.多水平模型及其统计诊断[M].北京:科学出版社,2008.
[7] Lindley D V, Smith A F M, Bayes estimates for the linear model[J]. Journal of the Royal Statistical Society:Series B(Methodological), 1972,34(1):1-41.
[8] Dempster A P. Comment on Tanner and Wong[J]. Journal of the American Statistical Association,1987, 82:541.
[9] Seltzer M H, Wong W H, Bryk A S. Bayesian analysis in application of hierarchical models:Issues and methods[J]. Journal of Educational and Behavioral Statistics Summer, 1996, 21(2):131-167.
[10] Seltzer M H. Sensitivity analysis for fixed effects in the hierarchical model:A Gibbs sampling approach[J]. Journal of Educational Statistics, 1993, 18:207-235.
[11] Browne W J, Draper D. A comparison of Bayesian and likelihood-based methods for fitting multilevel models[J]. Bayesian Analysis, 2006,(3):473-514.
[12] Polson N, Scott J. On the half-Cauchy prior for a global scale parameter[J]. Bayesian Analysis, 2012:887-902.
[13] Demirhan H, Kalaylioglu Z. Joint prior distribution for variance parameters in Bayesian analysis of normal hierarchical models[J]. Journal of Multivariate Analysis, 2015:163-174.
[14] Browne W J. MCMC Estimation in MLwiN2.32[M]. United Kingdom, Centre for Multilevel Modeling University of Bristol Pressed, 2015.
[15] Sturtz S, Ligges U, Gelman A. R2WinBUGS:A package for running WinBUGS from R[J]. Journal of Statistical Software, 2005,12(3):1-16.
[16] Congdon P D. Applied Bayesian Hierarchical Methods[M]. New York:Taylor and Francis Group:CRC Pressed, 2010.
[17]石磊,向其凤,张炯.物质资本、人力资本、就业结构与西部民族地区农户收入增长[J].数理统计与管理,2011, 30(6):1030-1038.
[18]李兴绪,刘曼莉.边境民族地区农户收入影响因素的实证分析一以云南红河州农户为例[J],数理统计与管理,2011,30(4):604-613.
[19]郭志刚.分层线性模型:应用与数据分析方法[M].北京:社会科学文献出版社,2016.
[20]朱慧明,韩玉启等.贝叶斯多元统计推断理论[M].北京:科学出版社,2006.
[21]刘金山,夏强.基于MCMC算法的贝叶斯统计方法[M].北京:科学出版社,2016.