PC priors for residual correlation parameters in one-factor mixed models
Lack of independence in the residuals from linear regression motivates the use of random effect models in many applied fields. We start from the one-way anova model and extend it to a general class of one-factor Bayesian mixed models, discussing several correlation structures for the within group residuals. All the considered group models are parametrized in terms of a single correlation (hyper-)parameter, controlling the shrinkage towards the case of independent residuals (iid). We derive a penalized complexity (PC) prior for the correlation parameter of a generic group model. This prior has desirable properties from a practical point of view: (i) it ensures appropriate shrinkage to the iid case; (ii) it depends on a scaling parameter whose choice only requires a prior guess on the proportion of total variance explained by the grouping factor; (iii) it is defined on a distance scale common to all group models, thus the scaling parameter can be chosen in the same manner regardless the adopted group model. We show the benefit of using these PC priors in a case study in community ecology where different group models are compared.
KeywordsBayesian mixed models Group model One-way anova INLA Intra-class correlation Within group residuals
Massimo Ventrucci and Daniela Cocchi are supported by the PRIN 2015 Grant Project No. 20154X8K23 (EPHASTAT) founded by the Italian Ministry for Education, University and Research. Gemma Burgazzi is supported by the Project PRIN NOACQUA—responses of communities and ecosystem processes in intermittent rivers a National Relevant Project funded by the Italian Ministry of Education and University (PRIN 2015, Prot. 201572HW8F). The authors thank Maria Franco Villoria and Hȧvard Rue for the stimulating comments received about this work.
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