Statistical Approach to Identify Threshold and Point of Departure in Dose–Response Data

The study presents an integrated, rigorous statistical approach to define the likelihood of a threshold and point of departure (POD) based on dose–response data using nested family of bent‐hyperbola models. The family includes four models: the full bent‐hyperbola model, which allows for transition between two linear regiments with various levels of smoothness; a bent‐hyperbola model reduced to a spline model, where the transition is fixed to a knot; a bent‐hyperbola model with a restricted negative asymptote slope of zero, named hockey‐stick with arc (HS‐Arc); and spline model reduced further to a hockey‐stick type model (HS), where the first linear segment has a slope of zero. A likelihood‐ratio test is used to discriminate between the models and determine if the more flexible versions of the model provide better or significantly better fit than a hockey‐stick type model. The full bent‐hyperbola model can accommodate both threshold and nonthreshold behavior, can take on concave up and concave down shapes with various levels of curvature, can approximate the biochemically relevant Michaelis–Menten model, and even be reduced to a straight line. Therefore, with the use of this model, the presence or absence of a threshold may even become irrelevant and the best fit of the full bent‐hyperbola model be used to characterize the dose–response behavior and risk levels, with no need for mode of action (MOA) information. Point of departure (POD), characterized by exposure level at which some predetermined response is reached, can be defined using the full model or one of the better fitting reduced models.