Detecting Departure From Additivity Along a Fixed-Ratio Mixture Ray With a Piecewise Model for Dose and Interaction Thresholds

For mixtures of many chemicals, a ray design based on a relevant, fixed mixing ratio is useful for detecting departure from additivity. Methods for detecting departure involve modeling the response as a function of total dose along the ray. For mixtures with many components, the interaction may be dose dependent. Therefore, we have developed the use of a three-segment model containing both a dose threshold and an interaction threshold. Prior to the dose threshold, the response is that of background; between the dose threshold and the interaction threshold, an additive relationship exists; the model allows for departure from additivity beyond the interaction threshold. With such a model, we can conduct a hypothesis test of additivity, as well as a test for a region of additivity. The methods are illustrated with cytotoxicity data that arise when Chinese hamster ovary cells are exposed to a mixture of nine haloacetic acids.