Gaussian processes (GPs) are widely used as distributions of random effects in linear mixed models, which are fitusing the restricted likelihood or the closely related Bayesian analysis. This article addresses two problems. First, we proposetools for understanding how data determine estimates in these models, using a spectral basis approximation to the GP underwhich 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 outcomey out of the GP and error parts of the fit, we apply a linear-model diagnostic, the added variable plot (AVP), both to theoriginal observations and to projections of the data onto the spectral basis functions. The spectral- and observation-domainAVPs estimate the same coefficient for a covariate but emphasize low- and high-frequency data features respectively and thushighlight the covariate’s effect on the GP and error parts of the fit, respectively. The spectral approximation applies to dataobserved on a regular grid; for data observed at irregular locations, we propose smoothing the data to a grid before applyingour methods. The methods are illustrated using the forest-biomass data of Finley et al. (2008).
