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

Toward a diagnostic toolkit for linear models with Gaussian-process distributed random effects