PC priors for residual correlation parameters in one-factor mixed models

Abstract

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.

Variance partitioning in spatio-temporal disease mapping models

“…PC priors enhance the marginal likelihood as a simple and effective tool for fair model comparison, when the compared models have similar structure and only differ on a particular component (Sørbye and Rue, 2018;Ventrucci et al, 2020). Assume M 1 and M 2 are the interaction type I and II models, respectively: these models are the same except for a different type of interaction.…”

“…Part of the hassle in choosing a prior arises from the difficulty to interpret variance parameters, especially for intrinsic processes, where the standard deviation is to be interpreted as a conditional one (Fong et al, 2010;Riebler et al, 2016). On top of that, in models with various terms, the tendency is to set priors independently for each precision parameter, while some authors are beginning to recognize that it might be more practical to think about total variability and how each term in the model contributes to that rather than to concentrate on single variance components separately (Wakefield, 2007;Riebler et al, 2016;Fuglstad et al, 2020;Ventrucci et al, 2020). In the context of disease mapping, Wakefield (2007) proposes using an inverse Gamma prior on the total variability, along with a Beta prior that distributes the variance between a spatially correlated random field and a spatially unstructured effect (the so called BYM model, Besag et al (1991)).…”

“…Using a similar parametrization, Riebler et al (2016) present a prior that shrinks towards no spatial effect following the penalized complexity (PC) prior approach of Simpson et al (2017). Outside the disease mapping literature, Ventrucci et al (2020) develop a PC prior in one-factor mixed models for the relative contribution of group-specific variability. In a more general context, Fuglstad et al (2020) introduce a framework for hierarchically distributing the variance in additive models, where, at each level of the total variance decomposition, ignorance or preference about the variance contribution of a term is expressed via a Dirichlet or a PC prior, respectively.…”

Variance partitioning in spatio-temporal disease mapping models

“…PC priors enhance the marginal likelihood as a simple and effective tool for fair model comparison, when the compared models have a similar structure and only differ on a particular component. 35,23 Assume scriptM 1 and scriptM 2 are the interaction type I and II models, respectively: these models are the same except for a different type of interaction. The Bayes factor 36 is defined as The scale parameter θ of the PC prior for γ, which controls the decay rate from the base model (the model with no interaction), has to be chosen for both scriptM 1 and scriptM 2.…”

Predictive Complexity Priors

“…Using an exponential distribution for π KL , Simpson et al [47] show the resulting PCP on τ is a type-II Gumbel distribution. PCPs have since been used as priors for P-splines [51], distributional regression [30], autoregressive processes [49], mixed effects models [52], and Gaussian random fields [17].…”