  Generalized likelihood ratio (GLR) test statistics are often used in the
detection of spatial clustering in case-control and case-population datasets to
check for a significantly large proportion of cases within some scanning
window. The traditional spatial scan test statistic takes the supremum GLR
value over all windows, whereas the average likelihood ratio (ALR) test
statistic that we consider here takes an average of the GLR values. Numerical
experiments in the literature and in this paper show that the ALR test
statistic has more power compared to the spatial scan statistic. We develop in
this paper accurate tail probability approximations of the ALR test statistic
that allow us to by-pass computer intensive Monte Carlo procedures to estimate
$p$-values. In models that adjust for covariates, these Monte Carlo evaluations
require an initial fitting of parameters that can result in very biased
$p$-value estimates.
