  Gaussian processes (GPs) are widely used as distributions of random effects
in linear mixed models, which are fit using the restricted likelihood or the
closely-related Bayesian analysis. This article addresses two problems. First,
we propose tools for understanding how data determine estimates in these
models, using a spectral basis approximation to the GP under which the
restricted likelihood is formally identical to the likelihood for a
gamma-errors GLM with identity link. Second, to examine the data's support for
a covariate and to understand how adding that covariate moves variation in the
outcome y out of the GP and error parts of the fit, we apply a linear-model
diagnostic, the added variable plot (AVP), both to the original observations
and to projections of the data onto the spectral basis functions. The spectral-
and observation-domain AVPs estimate the same coefficient for a covariate but
emphasize low- and high-frequency data features respectively and thus highlight
the covariate's effect on the GP and error parts of the fit respectively. The
spectral approximation applies to data observed on a regular grid; for data
observed at irregular locations, we propose smoothing the data to a grid before
applying our methods. The methods are illustrated using the forest-biomass data
of Finley et al.~(2008).
