  This paper develops goodness of fit statistics that can be used to formally
assess Markov random field models for spatial data, when the model
distributions are discrete or continuous and potentially parametric. Test
statistics are formed from generalized spatial residuals which are collected
over groups of nonneighboring spatial observations, called concliques. Under a
hypothesized Markov model structure, spatial residuals within each conclique
are shown to be independent and identically distributed as uniform variables.
The information from a series of concliques can be then pooled into goodness of
fit statistics. Under some conditions, large sample distributions of these
statistics are explicitly derived for testing both simple and composite
hypotheses, where the latter involves additional parametric estimation steps.
The distributional results are verified through simulation, and a data example
illustrates the method for model assessment.
