Prediction and model comparison for areal unit data

Areal unit or discrete spatial data is customarily modeled with the goal of spatial smoothing, typically using Markov random field models. Examples include image restoration and disease mapping. Here, we focus on a different issue for such data; we consider the set of areal units as only partially observed. One application is to learn about the smoothing behavior of various Markov random field models. That is, if two different smoothing priors are used, how can we quantify the relative smoothing that each imposes? We propose to fit models of interest to a portion of the data and hold out the rest for model comparison. A second application concerns the setting where, in fact, only a portion of the areal units have been observed, and we seek prediction of the remainder. Our motivating context investigates the performance of semiconductor chips, created as dies (the areal units) within wafers within lots, yielding nested modeling structure. Multiple tests are administered to each die involving both binary and continuous measurements. In practice, only a small subset of the dies are sampled, resulting in prediction of performance for the remaining unsampled dies. Furthermore, dies in the same locations are tested on each wafer, and the manufacturing process encourages within wafer, between wafer and between lot dependence. Other missing data applications include damaged images and small area estimation with missing observations for some units. We demonstrate prediction first with an image that is observed at several rates of missingness. Then, a well-studied Ohio lung cancer dataset is used for model comparison with regard to smoothing. Finally, examination of the nested modeling for semiconductor chip data is offered.