Collaborative Research: Variable Selection for Mixed Effect Models

社会科学中的大多数研究问题涉及个人行为和社会背景之间的复杂相互作用。为此，线性和广义线性混合效应模型已被广泛使用。混合模型的社会科学应用通常从大量变量开始。通过评估每个变量的显著性，研究者选择合适的模型。因此，在社会科学应用中，变量选择是混合效应建模的一个组成部分。然而，由于大量的参数，传统的变量选择程序，如AIC和BIC，是计算上不可行的。Fan和Li（2001）提出了一类基于非凹惩罚似然（SCAD）的变量选择方法。SCAD惩罚具有Oracle属性，使得基于SCAD惩罚的估计量收敛到真实模型。本文推广了Fan-Li的思想，利用SCAD研究线性和广义线性混合效应模型的变量选择过程。这些工作不仅有助于混合效应模型的估计和计算，而且有助于对混合效应模型的理论理解。研究人员计划开发变量选择工具，供应用研究人员在混合效应模型的框架内同时选择变量和估计参数。测量学生的成绩及其决定因素一直是教育研究的核心兴趣之一。在这一领域，数据和研究问题通常本质上是分层的。教育评估中常见的典型数据结构是学生嵌套在学校中;有时涉及更多级别，例如嵌套在地理区域中的学校。重要的是要解决不同类型的课程，资源的可用性对个别学生的学习成绩等的影响，例如，是一个学校的财务计划重要吗？学校的家长参与政策是否对学生的评估起着关键作用？教育研究的这些特点使得分层线性/广义线性模型成为该领域最重要的统计工具。许多这样的模型从大量的解释变量开始（国家教育进步评估（NAEP）在教师和学校层面都有数百个变量），研究人员特别感兴趣的是找到重要的变量并估计它们的重要性。在这项建议中开发的技术将准确地回答教师和学校水平的变量是重要的，以及在多大程度上他们是如此。

Most research questions in social science involve complex interactions between individual behaviors and the social contexts. For this purpose, linear and generalized linear mixed effect models have been widely used. Social science applications of mixed models often start with a large number of variables. Through assessing the significance of each variable, researchers select the appropriate model. Hence in social science applications, variable selection is an integrate part of mixed effect modeling. However, due to the large number of parameters, the traditional variable selection procedures, such as AIC and BIC, are computationally infeasible. Fan and Li (2001) proposed a class of variable selection procedures via nonconcave penalized likelihood (SCAD). The SCAD penalty has an Oracle property such that the estimators based on the SCAD penalty converge to the true model. The investigators propose to extend the ideas of Fan-Li and study the variable selection procedures via SCAD for linear and generalized linear mixed effect models. The work not only contributes to the estimation and computation of mixed effect models, but also adds to the theoretical understanding of them. The investigators plan to develop variable selection tools for applied researchers to simultaneously select variables and estimate parameters in the framework of mixed effect models.Measuring students' achievement and its determinants has been one of the central interests in educational research. In this area, data and research questions are usually hierarchical by nature. The typical data structure often seen in educational assessment is that students are nested within schools; sometimes more levels are involved, such as schools nested within geographical areas. It is important to address the impacts of different types of curriculums, availability of resources on individual student's academic performance, etc. For example, is a school's financial program important? Does school policy for parental involvement play a critical role regarding students' assessment? Such features of educational research have made hierarchical linear/generalized linear models the most important statistical tools in this field. Many such models start with a large array of explanatory variables (the National Assessment of Educational Progress (NAEP) has hundreds of variables at both teacher and school levels) and it is of particular interest for researchers to find the significant variables and estimate how important they are. The technique developed in this proposal will answer accurately which variables of teacher and school levels areimportant and to what degree they are so.