Counting degrees of freedom in hierarchical and other richly-parameterised models

Drawing on linear model theory, we rigorously extend the notion of degrees of freedom to richly-parameterised models, including linear hierarchical and random-effect models, some smoothers and spatial models, and combinations of these. The number of degrees of freedom is often much smaller than the number of parameters. Our notion of degrees of freedom is compatible with similar ideas long associated with smoothers, but is applicable to new classes of models and can be interpreted using the projection theory of linear models. We use an example to illustrate the two applications of setting prior distributions for variances and fixing model complexity by fixing degrees of freedom.