Multiple Local Maxima in Restricted Likelihoods and Posterior Distributions for Mixed Linear Models

Henn, Lisa; Hodges, James S.

doi:10.1111/insr.12046

Cited by 3 publications

(1 citation statement)

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“…Given

ρ

, the approximate restricted likelihood is identical to the likelihood arising from a gamma‐errors generalized linear model with identity link, as in Hodges (, Ch. 15) and Henn and Hodges (). As such, the

v_{j}^{2}

are the data, the gamma shape parameter is

1 / 2

E false(v_{j}^{2} false) = σ_{s}^{2} a_{j} false(ρ false) + σ_{e}^{2}

, and

normalVar false(v_{j}^{2} false) = 2 {false(σ_{s}^{2} a_{j} false(ρ false) + σ_{e}^{2} false)}^{2}

.…”

Section: Approximating the Gaussian Processmentioning

confidence: 99%

Toward A Diagnostic Toolkit for Linear Models with Gaussian-Process Distributed Random Effects

2018

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Summary 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).

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“…Given