Analyses of Fish Species Richness with Spatial Covariate

Legislation passed in 1990 reducing the allowable sulfur dioxide emission levels in the United States is expected to reduce acidity in the Adirondack region of New York State. The number of fish species in a lake (species richness) depends on a number of physical and chemical factors, including the area, elevation, and acidity of the lake. Data on these and other factors are available for 1,166 Adirondack lakes. The data are analyzed with the goal of quantifying the effects of acid deposition on species richness after controlling for other important factors. A plot of the residuals from a standard multiple regression model against spatial location reveals a strong nonlinear relationship between species richness and lake location. With only one realization of the data, it is difficult, if not impossible, to tell whether this dependence should be modeled as deterministic spatial trend or as a spatial covariance structure. Two models are thus considered. The first is a multiple linear regression model with spatially correlated errors. As is common in geostatistics, it is assumed that the correlation between two errors is a function only of the vector separating the corresponding lakes. This is accomplished by modeling the correlation among the errors with a parametric variogram model. The parameters of the variogram are estimated via restricted maximum likelihood, after which the regression parameters are estimated using generalized least squares. The second model considered is a partial linear (or semiparametric) model containing a nonlinear term in spatial location. Locally weighted quadratic regression is used in conjunction with an estimation technique described by Speckman to estimate simultaneously the regression parameters and the "smooth" function of longitude and latitude. Finally, the ultimate goal of the 1990 legislation is a 50% reduction in sulfate deposition. Both models are used to form prediction intervals for species richness under that scenario. Extrapolation is avoided by using the criteria of Weisberg. The results suggest that a 50% reduction in sulfate deposition may result in a substantial increase in the fish species richness of many Adirondack lakes, particularly those with pH values between 4.5 and 6.5.