  An important statistical task in disease mapping problems is to identify
divergent regions with unusually high or low risk of disease. Leave-one-out
cross-validatory (LOOCV) model assessment is the gold standard for estimating
predictive p-values that can flag such divergent regions. However, actual LOOCV
is time-consuming because one needs to rerun a Markov chain Monte Carlo
analysis for each posterior distribution in which an observation is held out as
a test case. This paper introduces a new method, called integrated importance
sampling (iIS), for estimating LOOCV predictive p-values with only Markov chain
samples drawn from the posterior based on a full data set. The key step in iIS
is that we integrate away the latent variables associated the test observation
with respect to their conditional distribution \textit{without} reference to
the actual observation. By following the general theory for importance
sampling, the formula used by iIS can be proved to be equivalent to the LOOCV
predictive p-value. We compare iIS and other three existing methods in the
literature with two disease mapping datasets. Our empirical results show that
the predictive p-values estimated with iIS are almost identical to the
predictive p-values estimated with actual LOOCV, and outperform those given by
the existing three methods, namely, the posterior predictive checking, the
ordinary importance sampling, and the ghosting method by Marshall and
Spiegelhalter (2003).
