Partitioning degrees of freedom in hierarchical and other Richly parameterized models

Hodges and Sargent (2001) have developed a measure of a hierarchical model's complexity, degrees of freedom (DF), that is consistent with definitions for scatterplot smoothers, is interpretable in terms of simple models, and enables control of a fit's complexity by means of a prior distribution on complexity. But although DF describes the complexity of the whole fitted model, in general it remains unclear how to allocate DF to individual effects. Here we present a new definition of DF for arbitrary normal-error linear hierarchical models, consistent with that of Hodges and Sargent, that naturally partitions the n observations into DF for individual effects and for error. The new conception of an effect's DF is the ratio of the effect's modeled variance matrix to the total variance matrix. This provides a way to describe the sizes of different parts of a model (e.g., spatial clustering vs. heterogeneity), to place DF-based priors on smoothing parameters, and to describe how a smoothed effect competes with other effects. It also avoids difficulties with the most common definition of DF for residuals. We conclude by comparing DF with the effective number of parameters, pD, of Spiegelhalter et al. (2002). Technical appendices and a data set are available online as supplemental materials.