An Empirical Study of Statistical Properties of Variance Partition Coefficients for Multi-Level Logistic Regression Models

Abstract

Partitioning the variance of a response by design levels is challenging for binomial and other discrete outcomes. Goldstein (2003) proposed four definitions for variance partitioning coefficients (VPC) under a two-level logistic regression model. In this study, we explicitly derived formulae for multi-level logistic regression model and subsequently studied the distributional properties of the calculated VPCs. Using simulations and a vegetation dataset, we demonstrated associations between different VPC definitions, the importance of methods for estimating VPCs (by comparing VPC obtained using Laplace and penalized quasilikehood methods), and bivariate dependence between VPCs calculated at different levels. Such an empirical study lends an immediate support to wider applications of VPC in scientific data analysis.

Keywords:

• Empirical distribution
• Laplacian approximation
• Multi-level logistic models
• Variance partition coefficients

Mathematics Subject Classification:

• 62J12
• 62E17
• 60J22

Acknowledgments

We thank Chuck Rose for helpful comments on an early draft of this manuscript, and Mark Holland for checking the derivations of the formulae in Sec. 2. We are grateful to the Associate Editor and one referee for providing many constructive suggestions.