Random Coefficient Model for Chlorophyll Nutrient Relationships in Lakes

Abstract

Statistical models of nutrient-phytoplankton relationships in lakes are usually fitted on cross-sectional data sets and thus describe multi-lake behavior using a single set of global model parameters. As a result, prediction of individual lake response reflects behavior among lakes, increasing prediction error by the among-lake variance. Alternatively, a random coefficient model may be applied to the same cross-sectional data set to yield lake-specific parameters as opposed to a single set of global parameters. The random coefficients are based on classical estimators that reflect both global behavior and lake-specific response. Under normality, known variance, and a noninformative prior, the mean of the posterior Bayesian distribution on the lake-specific parameters is equivalent to the estimated random coefficients. In this study, the random coefficient model was applied to a cross-sectional data set on chlorophyll a, nitrogen, and phosphorus in North Carolina lakes. Both global and lake-specific parameters were estimated for these data. A second data set representing different years of observations for the same lakes was used to compare the two modeling approaches on the basis of prediction variance; the result was a 12.7% reduction in mean squared prediction error for the random coefficient model. Other applications of the random coefficient model in empirical and mechanistic modeling are suggested in a concluding discussion