Nonlinear mixed-model regression to analyze herbicide dose-response relationships

Plant responses to various doses of herbicides usually follow a sigmoid model where the potency is given by the 50% inhibition (I-50) value. To assess the potency of a herbicide under a range of environmental conditions, a series of independent bioassays are necessary to account for assay-to-assay variation. Analysis has conventionally been done by separate analysis of the individual bioassays or by simply pooling data. Analyzing the individual bioassays separately throws up relevant information on interassay variation. Such a model becomes too complex because a full set of model parameters is needed for each data set. Pooling data instead, and analyzing the bioassay jointly, inflates parameter uncertainty because of oversimplification. Such a simple model would have too few variables, and the fixed-effect estimates would be more uncertain because they would be explaining the interassay random effects. This means that the underlying statistical model is not realistic. Therefore, we propose a new technique of intermediate complexity that outperforms either technique and provides biologically realistic estimates that allow us to compare herbicide potencies. With this technique, we simultaneously analyze independent experiments by using a combination of nonlinear regression and mixed models. The case study uses a group of independently run bioassays with two photosystem II-inhibiting herbicides, diuron and bentazon, by measuring the oxygen evolution of thylakoid membranes. The introduction of random elements in the nonlinear regression parameters reduces the uncertainty in the parameters of interest. We demonstrate that it is possible to pool data from independent experiments to assess which parameters can be assigned a random element, to conduct hypothesis testing, and to calculate stable confidence limits and thus obtain a more precise interpretation of the biologically relevant parameters, such as I-50, compared with the conventional nonlinear regression models of the individual bioassays.