A first step when fitting multilevel models to continuous responses is to explore the degree of clustering in the data. Researchers fit variance-component models and then report the proportion of variation in the response that is due to systematic differences between clusters. Equally they report the response correlation between units within a cluster. These statistics are popularly referred to as variance partition coefficients (VPCs) and intraclass correlation coefficients (ICCs). When fitting multilevel models to categorical (binary, ordinal, or nominal) and count responses, these statistics prove more challenging to calculate. For categorical response models, researchers appeal to their latent response formulations and report VPCs/ICCs in terms of latent continuous responses envisaged to underly the observed categorical responses. For standard count response models, however, there are no corresponding latent response formulations. More generally, there is a paucity of guidance on how to partition the variation. As a result, applied researchers are likely to avoid or inadequately report and discuss the substantive importance of clustering and cluster effects in their studies. A recent article drew attention to a little-known exact algebraic expression for the VPC/ICC for the special case of the two-level random-intercept Poisson model. In this article, we make a substantial new contribution. First, we derive exact VPC/ICC expressions for more flexible negative binomial models that allows for overdispersion, a phenomenon which often occurs in practice. Then we derive exact VPC/ICC expressions for three-level and random-coefficient extensions to these models. We illustrate our work with an application to student absenteeism. (PsycInfo Database Record (c) 2020 APA, all rights reserved).

中文翻译：

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计数数据的多级模型中的分区变化。

将多级模型拟合到连续响应的第一步是探索数据中的聚类程度。研究人员拟合方差分量模型，然后报告由于集群之间的系统差异而导致的响应中的变异比例。同样，它们报告集群内单元之间的响应相关性。这些统计数据通常被称为方差分配系数 (VPC) 和类内相关系数 (ICC)。当将多级模型拟合到分类（二进制、有序或名义）和计数响应时，这些统计数据证明计算起来更具挑战性。对于分类响应模型，研究人员利用他们的潜在响应公式，并根据潜在的连续响应来报告 VPC/ICC，这些响应是观察到的分类响应的基础。然而，对于标准计数响应模型，没有相应的潜在响应公式。更一般地说，关于如何划分变异的指导很少。因此，应用研究人员可能会避免或不充分地报告和讨论聚类和聚类效应在他们的研究中的实质性重要性。最近的一篇文章提请注意一个鲜为人知的 VPC/ICC 精确代数表达式，用于两级随机截距泊松模型的特殊情况。在本文中，我们做出了实质性的新贡献。首先，我们为更灵活的负二项式模型推导出精确的 VPC/ICC 表达式，该模型允许过度分散，这种现象在实践中经常发生。然后我们为这些模型的三级和随机系数扩展导出精确的 VPC/ICC 表达式。我们用学生旷课的应用来说明我们的工作。（PsycInfo 数据库记录 (c) 2020 APA，保留所有权利）。