Threshold models are useful in concentration-effect studies to describe regions of exposure that result in background response. These models are parameterized to estimate the background response, the concentration-effect relationship, and the join point between the two, called the threshold. If the threshold is different from zero, then it can be inferred that exposure to the chemical at regions below the threshold do not increase risk above background. When the exposure is to many chemicals, fixed-ratio ray designs can be used to assess risk to single chemicals and to specified mixtures of chemicals. This article describes the inference resulting from use of a threshold model for correlated binary data supported by a ray design. An example of the effect of three hepatotoxins in the development of rats is provided. In addition, a two-stage simultaneous optimal design criterion is described for the threshold model along rays of fixed ratios. The approach is illustrated through a simulation study of the hepatotoxin data.

Key Words
Binary data; Hepatotoxin exposure; Ray design.

Chris Gennings is Associate Professor, Department of Biostatistics, Medical College of Virginia, Virginia Commonwealth University, Richmond, VA 23298-0032. Pam Schwartz is Coordinator, Biometrics, Central Research Division, Pfizer, Inc., Groton, CT 06340.