Riebler 2016 - Simulation Study and Sardinia IDDM Application

Summary

On the 366-district Sardinia graph, 200 datasets x 9 scenarios show BYM2 with PC priors shrinks to both base models (constant risk; iid noise) and learns spatial structure when present, with model-choice criteria (RMSE, DIC, log score) no worse than iid, Besag, Leroux, Dean. The IDDM application (619 cases) supports near-constant risk, and posterior risk maps are insensitive to the prior on $\varphi$.

Overview

Truth: $\log {\theta }_i=\mu +{\tau }_b^{-1/2}{b}_i$ with $\mu =0$. Scenarios: constant ($1/\sqrt{{\tau }_b}=0$); iid ($0.5$, $b\sim N(0,I)$); spatial ($0.5$, $b\sim N(0,Q^{\star -})$); each with $E_i\in \{15,60,200\}$. Fitted with INLA.

Main Content

Table 1 - BYM2 posterior means (SD in parentheses; PC priors for $\varphi$ and ${\tau }_b$)

Scenario	$E$	$\mu$	$1/\sqrt{\tau }$	$\varphi$
Constant ($0$, $\varphi =0$)	15	-3.78E-04 (0.01)	0.02 (0.02)	0.26 (0.05)
	60	-9.15E-04 (0.01)	0.01 (0.01)	0.26 (0.05)
	200	-4.36E-04 (0.004)	0.01 (0.01)	0.26 (0.04)
iid ($0.5$, $\varphi =0$)	15	1.62E-03 (0.03)	0.50 (0.02)	0.05 (0.03)
	60	-1.94E-03 (0.03)	0.51 (0.03)	0.04 (0.02)
	200	1.21E-03 (0.03)	0.50 (0.02)	0.04 (0.02)
Spatial ($0.5$, $\varphi =1$)	15	2.14E-03 (0.01)	0.47 (0.04)	0.90 (0.06)
	60	9.11E-06 (0.01)	0.49 (0.05)	0.93 (0.04)
	200	4.70E-05 (0.004)	0.49 (0.07)	0.93 (0.04)

Interpretation: with constant risk $\varphi$ is unidentified and the prior dominates (0.26); with uniform $\varphi$ prior it sits near 0.5 (Table 4, Appendix A); iid and spatial truths are recovered; $\varphi$ is slightly lower for $E=15$ (learning needs data).

Table 2 - Comparison, constant risk, $E=60$

Model	$\mu$	$1/\sqrt{\tau }$	$\varphi$	RMSE	DIC	LS
iid	-1.65E-03	0.05	0.00	0.11	2545	3.47
Besag	-1.26E-03	0.07	1.00	0.12	2545	3.48
Leroux	-1.51E-03	0.07	0.56 (0.08)	0.11	2545	3.48
Dean	-1.69E-03	0.06	0.60 (0.12)	0.11	2547	3.48
BYM2 (unif)	-8.39E-04	0.01	0.46 (0.05)	0.12	2540	3.47
BYM2 (PC)	-9.15E-04	0.01	0.26 (0.05)	0.12	2540	3.47

Caution: parameters across models are not comparable (different priors, unscaled precision in Leroux/Dean). Only iid and BYM2 estimates are comparable. DIC slightly favours BYM2; Besag worse under iid truth, iid worse under spatial truth (Figs. 5-6, Appendix A).

Table 3 - Sardinia IDDM posterior (366 districts, 619 cases 1989-92, ages 0-29, no covariates)

Priors: $\mathrm{Pr}(\varphi <0.5)=2/3$; ${\tau }_b$ with $U=0.2/0.31$, $\alpha =0.1$ (marginal SD 0.2).

Prior for $\varphi$	quantity	mean	sd	2.5%	median	97.5%	mode
$U=.5,\alpha =2/3$	$\mu$	0.01	0.05	-0.09	0.01	0.10	0.01
	${\tau }_b$	262.79	751.60	10.14	45.36	2549.91	17.40
	$\varphi$	0.48	0.31	0.01	0.48	0.98	1.00
$U=.5,\alpha =.5$	${\tau }_b$	233.51	686.28	10.00	42.78	2241.48	17.37
	$\varphi$	0.54	0.31	0.02	0.57	0.99	1.00
$U=.5,\alpha =.1$	${\tau }_b$	201.56	616.57	9.83	40.11	1880.89	17.27
	$\varphi$	0.63	0.29	0.03	0.69	1.00	1.00
Unif(0,1)	${\tau }_b$	189.46	591.67	9.77	39.07	1734.34	17.21
	$\varphi$	0.64	0.26	0.08	0.69	0.99	1.00

High ${\tau }_b$ supports constant relative risk (zero log-RR inside all credible intervals); posterior of $\varphi$ is nearly flat; posterior risk estimates barely change with the $\varphi$ prior (Fig. 4), consistent with MacNab et al. but MacNab found strong sensitivity in the Leroux model.

Discussion (Sec. 6)

Scaling gives interpretable, transferable priors; PC priors allow reduction to simpler models and avoid overfitting; BYM2 integrates with covariates and space-time; more work needed to distribute variance across all linear-predictor terms and for formal prior sensitivity (Roos et al.). Applicable beyond disease mapping (e.g. genes as regions).

Examples

Appendix B INLA call: f(region, model="bym2", graph=..., scale.model=TRUE, constr=TRUE, hyper=list(phi=list(prior="pc", param=c(0.5,2/3), initial=-3), prec=list(prior="pc.prec", param=c(0.2/0.31, 0.1), initial=5))) (paraphrased; priors given on logit-$\varphi$ and log-precision scales).

Connections

Spatial Models - BYM (baseline model), Riebler 2016 - Penalised Complexity Priors for BYM2.

See Also

Riebler 2016 - BYM2 Intuitive Spatial Model - Overview