Spatial missing imutation using a gaussian process in a bayesian hierarchical poisson random effects model : application to an air pollution health effects study

Abstract

Poisson random effects regression model is often used to analyze spatially-varying associations between a count response variable and an exposure variable when the data for multiple locations are available and multiple observations are obtained in each location (e.g. multi-site time series data). Also, when location-specific predictors that may explain the spatial variability in the exposure-response association are available, a hierarchical regression structure can be added to the Poisson random effects model in a Bayesian framework.

However, often, the location-specific predictors are not observed for all locations and missing data occur. One na"ive approach is to use only the complete case data but it is not recommendable as using the reduced sample results in a loss of power in statistical inference. Because the location-specific predictors should be spatially correlated, one may consider missing data imputation by modeling a Gaussian process (GP) with some spatial correlation structure. One caution should be made for the choice of the correlation because misspecified correlation should yield a bias both in imputation and in the statistical inference in a hierarchical Poisson regression modeling.

In this research, motivated by an air pollution health effects study data where we encounter such missing data problem, we examine the correlation misspecification bias in the spatial GP modeling for missing predictor imputation as well as in the statistical inference for a Bayesian hierarchical Poisson regression modeling. Also, we apply the spatial GP imputation to our data selecting the best correlation structure via a cross-validation and compare the results of Poisson regression modeling between using the complete case data only and using all data with imputation.