A nonlinear isobologram model with Box-Cox transformation to both sides for chemical mixtures

me linear logistical isobologram is a commonly used and powerful graphical and statistical tool for analyzing the combined effects of simple chemical mixtures. In this paper a nonlinear isobologram model is proposed to analyz e the joint action of chemical mixtures for quantitative dose-response rela tionships. This nonlinear isobologram model incorporates two additional new parameters, y(min) and y(max), to facilitate analysis of response data tha t are not constrained between 0 and 1, where parameters y(min) and y(max), represent the minimal and the maximal observed toxic response. This nonline ar isobologram model for binary mixtures can be expressed as: y = y(min) + y(max) - y(min) / 1 + e(-(beta 0+beta 1d1+beta 2d2+beta 12d1d2 )) . in addition, a Box-Cox transformation to both sides is introduced to improv e the goodness of fit and to provide a more robust model for achieving homo geneity and normality of the residuals. Finally, a confidence band is propo sed for selected isobols, e.g., the median effective dose, to facilitate gr aphical and statistical analysis of the isobologram. The versatility of thi s approach is demonstrated using published data describing the toxicity of the binary mixtures of citrinin and ochratoxin as well as a new experimenta l data from our laboratory for mixtures of mercury and cadmium.