Individual reactions to a report which identifies an excess of risk near a putative source are determined mainly by some quoted significance level. One reaction, involving a commonly used 'coincidence' argument is given a simple Bayesian explanation. It is argued that interpretations of such reports should if possible allow both for data selection and for uncertainty in the null expectations underlying the significance levels. Tests are proposed, based on the first isotonic regression estimator under an order restriction, which allow for the effects of selecting a study region in the light of the data and have a simple form. Data on cancer incidence around two nuclear plants are used to illustrate.