New Developments in Estimation, Selection and Applications for Mixed Models

该奖项根据 2009 年美国复苏和再投资法案（公法 111-5）提供资金。 相关数据存在于科学和人文学科的各个领域，从计算生物学和地质学到健康和社会研究。然而，现代相关数据经常呈现额外的复杂性，例如高维性、非线性和非高斯性，为此需要更复杂的模型。 很多时候，这些模型需要大量的参数，并且参数的数量会随着样本量的增加而增加，甚至可能大于样本量。 这些在理论和计算方面都提出了重大挑战，因为很难直接应用传统的似然技术。 该提案旨在开发创新的统计程序和高效的计算算法，用于使用混合模型分析具有复杂特征的相关数据。研究人员重点研究了三类混合模型：线性混合模型、非线性混合模型和广义混合模型，并扩展了部分一致性和非凹惩罚最小二乘法的概念，以解决混合模型估计和选择中的几个具有挑战性的问题。 特别是，研究者 1) 探索了线性混合模型中部分一致性的概念，并开发了一种简单而稳健的两步估计方法； 2）开发惩罚最小二乘法来选择固定效应以及随机效应的协方差和精度矩阵； 3）将非线性混合模型估计和检验问题表述为模型选择问题，并开发了非线性混合模型的组选择方法； 4）将所提出的惩罚最小二乘法扩展到超高维变量选择； 5）将所提出的两步估计方法和惩罚最小二乘方法推广到广义线性混合模型。该提案的研究成果将极大地拓宽混合模型的应用，特别是在对不同类型数据进行联合建模方面。 例如，可以以一种统一的方式对临床和基因组数据进行联合建模，其中基因表达等基因组数据被视为随机效应并解释组间的异质性，而年龄、性别和血压等临床数据被视为固定效应并且是主要关注点。 这种联合建模方法允许人们解释基因之间的相关性，并以系统的方式而不是传统的逐个基因的方式研究基因或遗传途径。 此外，该提案的研究结果也将为分析高维和海量数据提供线索。 例如，通过将部分一致性的概念扩展到混合模型，人们将更好地理解如何探索高维数据的独特结构，以提取有价值的信息，从而对某些参数产生一致、高效和鲁棒的估计。此外，所提出的方法将通过跨学科合作工作介绍给其他领域的研究人员，并将通过开发研究生和本科生课程以及培养研究生来融入研究者的教育活动中。 实现所提议方法的开源 R 和 Matlab 代码将向公众开放。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Correlated data are seen in diverse fields of sciences and humanities, ranging from computational biology and geology to health and social studies. However, modern correlated data frequently present additional complications such as high-dimensionality, nonlinearity and nongaussianity, for which more complex models are needed. Often times, these models require a large number of parameters, and the number of the parameters can increase with the sample size and can even be greater than the sample size. These pose significant challenges on both theoretical and computational fronts, as it is difficult to directly apply traditional likelihood techniques. This proposal aims to develop innovative statistical procedures and efficient computing algorithms for analyzing correlated data with complicated features using mixed models. The investigator focuses on three classes of mixed models: linear mixed models, nonlinear mixed models and generalized mixed models, and extends the concept of partial consistency and the nonconcave penalized least squares method to address several challenging issues in mixed model estimation and selection. In particular, the investigator 1) explores the concept of partial consistency in linear mixed models and develops a simple yet robust two-step estimation method; 2) develops penalized least squares methods to select fixed effects as well as the covariance and precision matrices of random effects; 3) formulates the nonlinear mixed model estimation and testing problems as model selection problems and develops a group selection method for nonlinear mixed models; 4) extends the proposed penalized least squares method to ultra-high dimensional variable selection; and 5) generalizes the proposed two-step estimation method and penalized least squares method to generalized linear mixed models. The research findings of this proposal will greatly broaden the applications of mixed models, especially in jointly modeling different types of data. For example, one can jointly model clinical and genomics data in a unified way where genomics data such as gene expressions are treated as random effects and explain the heterogeneity among groups while clinical data such as age, gender and blood pressure are treated as fixed effects and are of primary interest. Such a joint modeling approach allows one to account for the correlation among genes and to study genes or genetic pathways in a system way rather than traditional gene-by-gene way. Moreover, the research findings of this proposal will also shred light on analyzing high-dimensional and massive data. For example, by extending the concept of partial consistency to mixed models, one will have better understandings as how to explore the unique structure of high-dimensional data in order to extract valuable information to produce consistent, efficient and robust estimates for some parameters. In addition, the proposed methodologies will be introduced to researchers in other areas through interdisciplinary collaboration work, and will also be integrated into the investigator's educational activities by developing graduate and undergraduate curriculums and by training graduate students. Open source R and Matlab codes implementing the proposed methodologies will be made available to general public.