Computing scan statistic p values using importance sampling, with applications to genetics and medical image analysis

We present an importance sampling method for deciding, based on an observed random held. if a scan statistic provides significant evidence of increase d activity in some localized region of time or space. Our method allows con sideration of scan statistics based simultaneously on multiple scan geometr ies. Our approach yields an unbiased p value estimate whose variance is typ ically smaller than that of the naive hit-or-miss Monte Carlo technique whe n the p value is small. Furthermore, our p value estimate is often accurate for critical values that are not far enough in the tails of the null distr ibution to allow for accurate approximations via extreme value theory. The importance sampling approach unifies the analysis of various random field m odels, from (spatial) point processes to Gaussian random fields, For a scan statistic M. the method produces a p value of the form P[M greater than or equal to tau] = B rho, where B is the Bonferroni upper bound and the corre ction factor rho measures the conservativeness of this upper bound. We pres ent the application of our importance sampling estimator to multinomial seq uences (molecular genetics), spatial point processes (digital mammography), and Gaussian random fields (PET scan brain imagery).