• fitting model;
  • hierarchical model;
  • informative covariate;
  • intensity;
  • sampling model;
  • spatial point pattern

The choice of the sampling locations in a spatial network is often guided by practical demands. In particular, many locations are preferentially chosen to capture high values of a response, for example, air pollution levels in environmental monitoring. Then, model estimation and prediction of the exposure surface become biased because of the selective sampling. As prediction is often the main utility of the modeling, we suggest that the effect of preferential sampling lies more importantly in the resulting predictive surface than in parameter estimation. We take demonstration of this effect as our focus.

In particular, our contribution is to offer a direct simulation-based approach to assessing the effects of preferential sampling. We compare two predictive surfaces over the study region, one originating from the notion of an “operating” intensity, driving the selection of monitoring sites, the other under complete spatial randomness. We can consider a range of response models. They may reflect the operating intensity, introduce alternative informative covariates, or just propose a flexible spatial model. Then, we can generate data under the given model. Upon fitting the model and interpolating (kriging), we will obtain two predictive surfaces to compare with the known truth. It is important to note that we need suitable metrics to compare the surfaces and that the predictive surfaces are random, so we need to make expected comparisons. We also present an examination of real data using ozone exposures. Here, what we can show is that, within a given network, there can be substantial differences in the spatial prediction using preferentially chosen locations versus roughly randomly selected locations and that the latter provide much improved predictive validation. Copyright © 2012 John Wiley & Sons, Ltd.