Low-dose-rate extrapolation using the multistage model

The distribution of the maximum likelihood estimates of virtually safe levels of exposure to environmental chemicals is derived by using large-sample theory and Monte Carlo simulation according to the Armitage-Doll multistage model. Using historical dose-response we develop a set of 33 two-stage models upon which we base our conclusions. The large-sample distributions of the virtually safe dose are normal for cases in which the multistage-model parameters have nonzero expectation, and are skewed in other cases. The large-sample theory does not provide a good approximation of the distribution observed for small bioassays when Monte Carlo simulation is used. The constrained nature of the multistage-model parameters leads to bimodal distributions for small bioassays. The two modes are the direct result of estimating the linear parameter in the multistage model; the lower mode results from estimating this parameter to be nonzero, and the upper mode from estimating it to be zero. The results of this research emphasize the need for incorporation of the biological theory in the model-selection process.