Bayesian measures of explained variance and pooling in multilevel (hierarchical) models
Bayesian measures of explained variance and pooling in multilevel (hierarchical) models
复制标题
DOI：
10.1198/004017005000000517
复制
发表时间：
2006-05-01
期刊：
影响因子：
2.5
通讯作者：
Pardoe, Lain
中科院分区：
文献类型：
--
作者：
Gelman, Andrew;Pardoe, Lain
解释方差（R-2）是线性回归拟合的常见总结，并已以各种方式推广到多水平（分层）模型。我们在本文中考虑的多级模型的特点是分层数据结构，其中个体被分组为单元（这些单元本身可能被进一步分组为更大的单元），变量在个体和每个分组单元上进行测量。这些模型基于不同水平的回归关系，第一个水平对应于个体数据，随后的水平对应于个体预测因子对分组单位变量影响的组间回归。我们提出了一种方法来定义R-2在每个级别的多层次模型。而不是试图创建适合的单个概括度量。我们的方法是基于比较方差在一个单一的拟合模型，而不是一个空模型。在简单回归中，我们的措施推广了经典的调整R-2。我们还讨论了相关方差比较，以总结基于特定水平回归关系将模型每个水平的估计值汇总在一起而不是单独估计的程度。该合并因子与简单分层模型中的收缩概念有关。我们说明的方法在一个数据集上的氡在县内的房屋使用一系列的模型，从一个简单的线性回归模型到一个多层次的变截距。变斜率模型
Explained variance (R-2) is a familiar summary of the fit of a linear regression and has been generalized in various ways to multilevel (hierarchical) models. The multilevel models that we consider in this article are characterized by hierarchical data structures in which individuals are grouped into units (which themselves might be further grouped into larger units), and variables are measured on individuals and each grouping unit. The models are based on regression relationships at different levels, with the first level corresponding to the individual data and subsequent levels corresponding to between-group regressions of individual predictor effects on grouping unit variables. We present an approach to defining R-2 at each level of the multilevel model. rather than attempting to create a single summary measure of fit. Our method is based on comparing variances in a single fitted model rather than with a null model. In simple regression, our measure generalizes the classical adjusted R-2. We also discuss a related variance comparison to summarize the degree to which estimates at each level of the model are pooled together based on the level-specific regression relationship, rather than estimated separately. This pooling factor is related to the concept of shrinkage in simple hierarchical models. We illustrate the methods on a dataset of radon in houses within counties using a series of models ranging from a simple linear regression model to a multilevel varying-intercept. varying-slope model.