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Biometrika 2001 88(2):367-379; doi:10.1093/biomet/88.2.367
© 2001 by Biometrika Trust
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Counting degrees of freedom in hierarchical and other richly-parameterised models

James S.Hodges1 and Daniel J.Sargent2

1 Division of Biostatistics, School of Public Health, University of Minnesota, 2221 University Avenue SE, Suite 200, Minneapolis, Minnesota 55414, U.S.Ahodges{at}ccbr.umn.edu 2 Cancer Center Statistics, Plummer 4, Mayo Clinic, 200 1st Street SW, Rochester, Minnesota 55905, U.S.A.sargent{at}mayo.edu

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.

Key Words: Complexity; Degrees of freedom; Hierarchical model; Prior distribution; Random-effect model; Smoothing

Received February 1998. Revised September 2000


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