Accommodating the ecological fallacy in disease mapping in the absence of individual exposures

In health exposure modeling, in particular, disease mapping, the ecological fallacy arises because the relationship between aggregated disease incidence on areal units and average exposure on those units differs from the relationship between the event of individual incidence and the associated individual exposure. This article presents a novel modeling approach to address the ecological fallacy in the least informative data setting. We assume the known population at risk with an observed incidence for a collection of areal units and, separately, environmental exposure recorded during the period of incidence at a collection of monitoring stations. We do not assume any partial individual level information or random allocation of individuals to observed exposures. We specify a conceptual incidence surface over the study region as a function of an exposure surface resulting in a stochastic integral of the block average disease incidence. The true block level incidence is an unavailable Monte Carlo integration for this stochastic integral. We propose an alternative manageable Monte Carlo integration for the integral. Modeling in this setting is immediately hierarchical, and we fit our model within a Bayesian framework. To alleviate the resulting computational burden, we offer 2 strategies for efficient model fitting: one is through modularization, the other is through sparse or dimension-reduced Gaussian processes. We illustrate the performance of our model with simulations based on a heat-related mortality dataset in Ohio and then analyze associated real data.