A pseudo-penalized quasi-likelihood approach to the spatial misalignment problem with non-normal data.

Spatially referenced datasets arising from multiple sources are routinely combined to assess relationships among various outcomes and covariates. The geographical units associated with the data, such as the geographical coordinates or areal-level administrative units, are often spatially misaligned, that is, observed at different locations or aggregated over different geographical units. As a result, the covariate is often predicted at the locations where the response is observed. The method used to align disparate datasets must be accounted for when subsequently modeling the aligned data. Here we consider the case where kriging is used to align datasets in point-to-point and point-to-areal misalignment problems when the response variable is non-normally distributed. If the relationship is modeled using generalized linear models, the additional uncertainty induced from using the kriging mean as a covariate introduces a Berkson error structure. In this article, we develop a pseudo-penalized quasi-likelihood algorithm to account for the additional uncertainty when estimating regression parameters and associated measures of uncertainty. The method is applied to a point-to-point example assessing the relationship between low-birth weights and PM2.5 levels after the onset of the largest wildfire in Florida history, the Bugaboo scrub fire. A point-to-areal misalignment problem is presented where the relationship between asthma events in Florida's counties and PM2.5 levels after the onset of the fire is assessed. Finally, the method is evaluated using a simulation study. Our results indicate the method performs well in terms of coverage for 95% confidence intervals and naive methods that ignore the additional uncertainty tend to underestimate the variability associated with parameter estimates. The underestimation is most profound in Poisson regression models.