correlation than the essentially linear one-stage model. The lowest correlation arises from the Weibull model. This is because, unlike the one-stage or multistage models, the Weibull model allows for supralinearity at low and moderate doses, and thus admits a greater range of TD50 values.
Crouch et al. (1987) commented on the absence of observations in the upper left and lower right triangular regions in scattergrams similar to those shown here in Figure 2. The absence of points in the upper left hand region is due to the lower limit on the number of tumors observed in the exposed groups in order to demonstrate a statistically significant increase in tumor occurrence. This implies that highly toxic chemicals of weak carcinogenic potency would likely go undetected in a standard bioassay, since such agents would not yield a measurable excess of tumors at the MTD. Crouch et al. (1987) attribute the absence of points in the lower right hand region to a lack of chemicals with extremely high potency relative to their MTDs. Reith & Starr (1989a) dispute this latter conclusion on the basis that experimental design constraints preclude the observation of potencies much larger than those shown in Figure 2. (This point is explored in greater detail in section 3.2 below.) Whether or not such "supercarcinogens" exist has been recently debated by the National Research Council (1992).
3.2 Range of Possible TD50Values
Bernstein et al. (1985) noted that TD50 values calculated from bioassay data vary within a limited range about the MDT as a function of the observed tumor response. To illustrate, suppose that the probability P(d) of a tumor occurring at dose d follows the one-stage model in (2.2), and the background tumor rate P(0) = 1 - e is known to be 0.10. Suppose further that 50 animals are exposed to a single dose D = MTD and that x of these animals develop the tumor of interest. Solving the equation
leads to the estimate