The joint action of two mixed stimuli with a bivariate threshold

The general mathematical statistical theory of a mixture of two stimuli in quantal bioassay or other binary response is investigated assuming the existence of thresholds. Univariate thresholds have long been studied. Different serious attempts on bivariate threshold theory have emerged since the late 1940s. Some have been based on biologically established fact mixed with analysis, while others have been more heuristic. In this paper we investigate the properties of correlations of thresholds. Joint actions are then interpreted according to those correlations and deviations from independence. Isoboles are lines, curves, planes, or surfaces of constant probability. They are also investigated, and their properties are used in proofs of important theorems. It is proved that additivity or additivism of two stimuli is present if and only if the thresholds of the two stimuli are perfectly correlated. Furthermore, additivity requires univariate dose-response functions to be everywhere concave on the nonnegative real numbers (doses). Finally, threshold theory, itself, dictates cumulative distributions of marginal thresholds that are everywhere nondecreasing and concave on the nonnegative reals. That is, they cannot have points of inflection. Such popular distributions as the normal and logistic have points of inflection at their respective medians. This does not invalidate their usefulness. They are simply modeling phenomena other than those which are the subject of this paper, namely, threshold theory.