Investigation of excess environmental risk around putative sources: Stone's test with covariate adjustment

Stone (Statistics in Medicine, 7, 649-660 (1988)) proposed a method of testing for elevation of disease risk around a point source. Stone's test is appropriate to data consisting of counts of the numbers of cases, Yi say, in each of n regions which can be ordered in increasing distance from a point source. The test assumes that the Yi are mutually independent Poisson variates, with means mu i = Ei lambda i where the Ei are the expected numbers of cases, for example based on appropriately standardized national incidence rates, and the lambda i are relative risks. The null hypothesis that the lambda i are constant is then tested against the alternative that they are monotone non-increasing with distance from the source. We propose an extension to Stone's test which allows for covariate adjustment via a log-linear model, mu i = Ei lambda i exp (sigma jp = 1 xij beta j), where the xij are the values of each of p explanatory variables in each of the n regions, and the beta j are unknown regression parameters. Our methods are illustrated using data on the incidence of stomach cancer near two municipal incinerators.