Power Transformations and Tests of Environmental Impact as Interaction Effects

Statistical analysis of the environmental impact of a pollutant point source or other human intervention often focuses on an interaction effect comparing, for example, measured pollutant levels before and after closure of a pollutant point source between regions presumed affected and unaffected by the intervention.This comparison may be meaningful only in terms of the original units of measurement of the environmental response.Statistical modeling of environmental data, however, is often most appropriately carried out on a transformed response scale.In this article we present a random effects model for rainfall acidity measured in the course of a study of the effects of closure of the ASARCO copper smelter in Tacoma, Washington.The response variables, concentrations of hydrogen, sulfate, and nitrate ions, were transformed to a cube-root scale for analysis.To assess the impact we introduce a simple approximate method to test for interaction expressed on the original scale of measurement in 2 . 2 experimental designs (such as time . region) with data analyzed in terms of one of the common power transformations.The test is an approximate version of the Wald test and hence is asymptotically equivalent to a likelihood ratio criterion.The latter is, however, more difficult to compute, requiring a numerical optimization.Both methods perform well on simulated data, even for small sample sizes.We apply this test in the context of a moderately complicated model for spatial, temporal, and random storm effects on acid deposition in the study of the impact of the ASARCO copper smelter.