Estimated generalized least squares in spatially misaligned regression models with Berkson error

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Abstract
In environmental studies, relationships among variables that are misaligned in space are routinely assessed. Because the data are misaligned, kriging is often used to predict the covariate at the locations where the response is observed. Using kriging predictions to estimate regression parameters in linear regression models introduces a Berkson error, which induces a covariance structure that is challenging to estimate. In addition, if the parameters associated with kriging (e.g. trend surface parameters and spatial covariance parameters) are estimated, then an additional uncertainty is introduced. We characterize the total measurement error as part of a broader class of Berkson error models and develop an estimated generalized least squares estimator using estimated covariance parameters. In working with the induced model, we fully account for the error structure and estimate the covariance parameters using likelihood-based methods. We provide insight into when it is important to fully account for the covariance structure induced from the different error sources. We assess the performance of the estimators using simulation and illustrate the methodology using publicly available data from the US Environmental Protection Agency.