Exact statistics for detecting disease outbreaks in small populations

Detecting outbreaks, structured clusters of events, is a common epidemiological problem, but traditional statistical tests are only valid for asymptotically large sample sizes and are therefore not applicable to rare events in small populations. Simulation studies have shown that the rate of false positives for traditional tests such as the chi-square test or the log-likelihood ratio are routinely much larger than their nominal rates when applied to small data sets. This results in declaring outbreaks more often than warranted, causing both undo alarm and economic losses from unnecessary closures and other interventions. In some cases, however, traditional cluster tests can fail to detect clusters that can be shown by other methods to be statistically significant, and it is difficult to anticipate whether the traditional test will overestimate or underestimate the probability of clustering for a particular situation. We describe statistical methods that can be used to detect outbreaks of rare events in small delimited populations. These tests use combinatorial formulations that yield exact p-values and that cannot violate their nominal Type I error rates. As a result, these tests are reliable whatever the size of the data set, and are especially useful when data sets are extremely small. Different statistics are sensitive to different types of structure, so that comparisons among these tests can provide evidence about the possible processes that generated the clustering. Use of these statistics can help decision-makers take defensible actions to prevent disease outbreaks and determine the causes of disease clusters.