Clusters galore: insights about environmental clusters from probability theory

The posterior probability of a causal explanation given that an environmental cancer cluster is statistically significant depends on the prior probability of an environmentally caused cluster, the sensitivity of the statistical test and its specificity. The prior probability is low, because it is rare to have enough carcinogen in the general environment to cause a relative risk of cancer high enough to achieve statistical significance in a small geographic area. The sensitivity and specificity are not great. The likelihood that a census tract escapes statistically significant elevations in all 80 types of cancer can be calculated. Many of the thousands of census tracts will, by chance alone, have at least one type of cancer whose elevation is statistically significant. Actual observation from a large cancer registry confirms this probabilistic prediction. Applying the principles of Bayes' Theorem would suggest that most statistically significant environmental cancer clusters are not due to environmental carcinogens. One would have to investigate hundreds of environmental cancer clusters to find one with a true environmental cause.