Temporal surveillance using scan statistics

We describe two classes of statistics for testing an arbitrary model of disease incidence over time against an alternative model involving a spike (pulse) superimposed on this background. The statistics are each based on taking the maximum of some function comparing observed and expected numbers of events in a window of width w. One approach applies p-values for scan statistics calculated for a constant background rate to this more general problem. For a fixed window, w, the approach gives a simple formula to determine p-values for retrospective analysis, or to sound an alarm for either continuous or grouped prospective data. The latter application involves a new approximation for the distribution of the maximum number of cases in w consecutive intervals. The second approach based on generalized likelihood ratio tests (GLRTs), sounds an alarm for a higher than anticipated rate of events in a scanning window of fixed length, or for window sizes that lie in a region. GLRTs are constructed for continuous observations, for grouped data, or for a sequence of trials. As for GLRTs used in retrospective evaluations, simulation is required to implement the prospective procedure. For grouped surveillance data, we compare by simulation, operating characteristics of the P-scan with fixed windows (both correctly specified and not), the fixed-window GLRT, the variable-window GLRT, and a variant of the CUSUM. The simulations demonstrate a very high correlation between the P-scan and corresponding fixed-window GLRT.