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Abstract:
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Degrees of freedom (DF) have been used in frequentist methods for significance tests. The idea has been extended to Bayesian methods to measure complexity of hierarchical models. Spiegelhalter et al. (2002) proposed the "effective number of parameters," p_D, which is easily computed but has practical problems (Lu et al. 2006). Hodges and Sargent (2001) extended linear model theory to give another complexity measure (DF) for a hierarchical model's overall fit. In special cases, H&S's DF factors naturally into effect-specific DF, but in general, it is unclear how to attribute DF to individual effects. We give a new definition of DF that naturally decomposes a fit's total DF into effect-specific DF for arbitrary normal-error linear hierarchical models. This gives a way to place DF-based priors on smoothing parameters, and to describe the complexity of individual effects.
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