A Bayesian Approach to Nonlinear Calibration Problems

Abstract

In bioassays, enzyme assays, or radioimmunoassays the concentration response relationships are mostly nonlinear. Very often because of physical restrictions or for biological reasons, the prediction samples cannot be diluted and the determination of very low concentrations is of interest. Thus one cannot be restricted to the “almost linear” part of the response curve and standard linear calibration methods are not applicable. The nonlinear calibration problem is of great importance in these types of applications. To implement the Bayesian paradigm in nonlinear calibration problems, a substantial amount of numerical integration is necessary. In practice, when concentrations of hundreds of samples have to be determined routinely, even with a highly efficient integration method (Naylor and Smith 1982), the numerical effort becomes prohibitive. An approximation method is proposed, therefore, to reduce the calculation. The precision of the approximation method is examined by validation subsamples and comparison with an integration routine based on the Gaussian quadrature strategy. An example from agricultural research is used to illustrate the underlying problem. To determine the concentrations of an agrochemical present in soil samples taken at different time points from the field, two different bioassays were performed. The plant used in one assay is extremely sensitive at low concentrations, whereas the plant used in the other assay reacts at higher concentrations. Combination of results from the two assays is also demonstrated.

Key Words:

• Bioassay
• Calibration experiment
• Combination of assays
• Prediction experiment