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.
"Detection of spatial clustering with average likelihood ratio test statistics." Ann. Statist. 37 (6B) 3985 - 4010, December 2009. https://doi.org/10.1214/09-AOS701