A Bayesian hierarchical model for disease mapping that accounts for scaling and heavy-tailed latent effects

Abstract

In disease mapping, the relative risk of a disease is commonly estimated across different areas within a region of interest. The number of cases in an area is often assumed to follow a Poisson distribution whose mean is decomposed as the product between an offset and the logarithm of the disease's relative risk. The log risk may be written as the sum of fixed effects and latent random effects. A modified Besag-York-Mollié (BYM2) model decomposes each latent effect into a weighted sum of independent and spatial effects. We build on the BYM2 model to allow for heavy-tailed latent effects and accommodate potentially outlying risks, after accounting for the fixed effects. We assume a scale mixture structure wherein the variance of the latent process changes across areas and allows for outlier identification. We propose two prior specifications for this scale mixture parameter. These are compared through various simulation studies and in the analysis of Zika cases from the first (2015-2016) epidemic in Rio de Janeiro city, Brazil. The simulation studies show that the proposed model always performs at least as well as an alternative available in the literature, and often better, both in terms of widely applicable information criterion, mean squared error and of outlier identification. In particular, the proposed parametrisations are more efficient, in terms of outlier detection, when outliers are neighbours. Our analysis of Zika cases finds 23 out of 160 districts of Rio as potential outliers, after accounting for the socio-development index. Our proposed model may help prioritise interventions and identify potential issues in the recording of cases.

Keywords: BYM2 model; Zika virus infection; outliers; scale mixture; spatial statistics; vector-borne disease.

Figure 1.

Map (a) and histogram (b) of the standardised morbidity ratio (SMR) distribution for the Zika counts across the 160 neighbourhoods of Rio de Janeiro, between 2015 and 2016.

Figure 2.

Left panel: French departments arbitrarily chosen to be outliers in the simulation study. Colours depict the offset category based on the empirical offset quantiles. The points represent the relative risk set to each outlying district. Right panel: Percentage of times, among the 100 replicates, that the outliers were identified by each model. The outliers are pointed out when ${\kappa }_u<1$ , where ${\kappa }_u$ is the upper bound of the posterior 95% credible interval of $\kappa$ .

Figure 3.

Top panel: WAIC across the 100 replicates for the proposed models and Congdon’s, in the simulation study. Dashed lines: mean WAIC for each model. Middle panel: box-plots of the WAIC values across the 100 simulation replicates for each model. Bottom panel: MSE over the 100 replicates for the proposed models and Congdon’s according to the true relative risk and the offset size. WAIC: widely applicable information criterion; MSE: mean squared error.

Figure 4.

Maps of the outliers indicated by each model when analysing the Zika counts. The outliers are pointed out when ${\kappa }_u<1$ , where ${\kappa }_u$ is the upper bound of the posterior 95% credible interval of $\kappa$ . The outliers on the lower tail are distinguished from the ones on the upper tail of the standardised morbidity ratios (SMR) distribution.

Figure 5.

Map highlighting some districts identified as outliers by at least one model when analysing the Zika counts. Orange: São Cristóvão; red: districts with small offsets; blue: districts whose population sizes increased significantly after the 2010 census; purple: districts combining both characteristics; and green: districts with zero cases recorded.