Statistical analysis of chemical transformation kinetics using Markov-Chain Monte Carlo methods

For the risk assessment of chemicals intentionally released into the environment, as, e.g., pesticides, it is indispensable to investigate their environmental fate. Main characteristics in this context are transformation rates and partitioning behavior. In most cases the relevant parameters are not directly measurable but are determined indirectly from experimentally determined concentrations in various environmental compartments. Usually this is done by fitting mathematical models, which are usually nonlinear, to the observed data and such deriving estimates of the parameter values. Statistical analysis is then used to judge the uncertainty of the estimates. Of particular interest in this context is the question whether degradation rates are significantly different from zero. Standard procedure is to use nonlinear least-squares methods to fit the models and to estimate the standard errors of the estimated parameters from Fisher's Information matrix and estimated level of measurement noise. This, however, frequently leads to counterintuitive results as the estimated probability distributions of the parameters based on local linearization of the optimized models are often too wide or at least differ significantly in shape from the real distribution. In this paper we identify the shortcoming of this procedure and propose a statistically valid approach based on Markov-Chain Monte Carlo sampling that is appropriate to determine the real probability distribution of model parameters. The effectiveness of this method is demonstrated on three data sets. Although it is generally applicable to different problems where model parameters are to be inferred, in the present case for simplicity we restrict the discussion to the evaluation of metabolic degradation of chemicals in soil. It is shown that the method is successfully applicable to problems of different complexity. We applied it to kinetic data from compounds with one and five metabolites. Additionally, using simulated data, it is shown that the MCMC method estimates the real probability distributions of parameters well and much better than the standard optimization approach.