Riebler 2016 - BYM2 Intuitive Spatial Model - Overview

Summary

Riebler, Sorbye, Simpson and Rue (2016, Statistical Methods in Medical Research; arXiv:1601.01180) explain and test the BYM2 parameterisation of the Besag-York-Mollie disease-mapping model: $b={\tau }_b^{-1/2}(\sqrt{1-\varphi }v+\sqrt{\varphi }{u}^{\star })$ with a scaled structured component $u^{\star }$. Then ${\tau }_b$ is a marginal precision independent of the graph, $\varphi$ is the share of variance that is spatial, and penalised-complexity (PC) priors give both interpretable, transferable hyperpriors. Only the scaling/PC part is covered here; for ICAR/BYM basics see Spatial Models - BYM.

Overview

Problem. In BYM the structured and unstructured effects are confounded and the precisions ${\tau }_u,{\tau }_v$ live on different scales; Leroux and Dean mixing parameterisations fix the allocation but leave the precision unscaled, so a prior on ${\tau }_b$ means different things on different maps, and common gamma priors overfit (no mass at the base model).

Contribution. A systematic study of BYM2 (Simpson et al. 2017 proposal): scaling the Besag precision so the geometric mean of marginal variances is 1; a sparsity-preserving augmented parameterisation; PC priors for $({\tau }_b,\varphi )$; simulation on the 366-district Sardinia graph (200 datasets x 9 scenarios) and an IDDM application (619 cases).

Motivating PyMC examples

https://redam94.github.io/pymc-challenges/examples.html: E18 - Areal models on a graph: ICAR, proper CAR, BYM2 (Scottish lip cancer, 56 districts; graph Laplacian and BYM2 scaling factor, pm.ICAR/pm.CAR, Moran’s I, exceedance maps, PSIS-LOO vs K-fold, spatial confounding).

Source Map

Paper section	Covered in
Abstract, 1 Introduction	this note
2 Disease mapping (Poisson likelihood, SMR, ${\eta }_i=\mu +z_i^{\top }\beta +b_i$)	Riebler 2016 - Classical Disease Mapping Models Besag BYM Leroux Dean
3.1 Besag, BYM, Leroux, Dean	Riebler 2016 - Classical Disease Mapping Models Besag BYM Leroux Dean
3.2 Why scaling (Fig. 1, eq. 6)	Riebler 2016 - Scaling and the BYM2 Construction
3.3-3.4 BYM2, sparse parameterisation	Riebler 2016 - Scaling and the BYM2 Construction
4 PC priors (eq. 9, Fig. 2)	Riebler 2016 - Penalised Complexity Priors for BYM2
5.1 Simulation study, Tables 1-2	Riebler 2016 - Simulation Study and Sardinia IDDM Application
5.2 Sardinia IDDM, Table 3, Figs. 3-4	Riebler 2016 - Simulation Study and Sardinia IDDM Application
6 Discussion	Riebler 2016 - Simulation Study and Sardinia IDDM Application
Appendix A (extra figs/Table 4), B (INLA code), C (PC derivation)	Riebler 2016 - Penalised Complexity Priors for BYM2, Riebler 2016 - Simulation Study and Sardinia IDDM Application

Examples

Worked scaling-factor and PC-prior code in Riebler 2016 - Scaling and the BYM2 Construction and Riebler 2016 - Penalised Complexity Priors for BYM2.

Connections

• Relevance to marketing measurement: BYM2 is the principled way to smooth noisy geo-level (DMA, state, county) outcomes or effects over adjacency while keeping an interpretable “how spatial is it” parameter $\varphi$; this is useful for geo-experiment readouts and geo-level MMM random effects (analogy; not covered by the paper). See Hierarchical Models for partial pooling and Spatial Models - BYM.
• Related field-based route: Simpson 2016 - Going Off Grid - Overview (continuous Matern fields) and Hilbert Space Gaussian Processes.

See Also

Riebler 2016 - BYM2 Intuitive Spatial Model