Fuzzy nonlinear regression analysis of dose-response relationships

Abstract

A general methodology for fuzzy regression is developed and illustrated by analyzing a dose-response relationship. Fuzzy regression is used as an alternative to statistical regression analysis, because the relationship between variables is imprecise, data are uncertain and sample sizes are insufficient. Three ‘goodness of fit’ criteria are used, namely, the maximum, the average vagueness and the prediction vagueness criterion. In the case of a nonlinear relationship, the calculation of fuzzy regression parameters leads to a nonlinear programming problem. This technique is illustrated by the relationship between the dose of a N-nitroso compound and the probability of tumor development in rats. This relationship is nonlinear, imprecise and based on only six data points. In the present case three common dose-response models are fitted using nonlinear fuzzy regression techniques with the six data points. Regression parameters have been obtained by minimizing a nonlinear objective function under a set of specific nonlinear constraints. The methodology is relatively simple and offers a viable alternative to statistical regression techniques.