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

Stone 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, Y(i) say, in each of n regions which can be ordered in increasing distance from a point source. The test assumes that the Y(i) are mutually independent Poisson variates, with means μ(i) = E(i)λ(i), where the E(i) are the expected numbers of cases, for example based on appropriately standardized national incidence rates, and the λ(i) are relative risks. The null hypothesis that the λ(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, μ(i) = E(i)λ(i)exp(Σ(j=1)(p) x(ij)β(j)), where the x(ij) are the values of each of p explanatory variables in each of the n regions, and the β(j) are unknown regression parameters. Our methods are illustrated using data on the incidence of stomach cancer near two municipal incinerators.