value directly on the basis of clinical considerations or other information. For example, it might be appropriate to define 2,500 g as the cutoff in an epidemiological study of birth weight. In that case, P0 can be expressed as a function of C as follows:
As discussed by Crump et al. (1998), there are advantages to using the approach based on specifying a fixed P0 (i.e., the first option), because the calculations simplify in this setting, particularly in the presence of covariates (see also E. Budtz-Jørgensen, Copenhagen University, N. Keiding, Copenhagen University, and P. Grandjean, University of Southern Denmark, unpublished material, May 5, 2000). Under the assumption that the error terms follow a normal distribution, it follows that the benchmark dose will be the solution, BMD, to
Notice that the estimated BMD simply corresponds to a constant divided by the dose-response slope from the regression model. That concept is important, because it provides some theoretical justification for some analyses (presented later) that are based on the inverse of the estimated benchmark doses from several MeHg studies.
Several authors have suggested variations on how to calculate BMDs for continuous outcomes. For example, Kodell and West (1993) and West and Kodell (1993) extended the Gaylor and Slikker approach to allow the model variance to depend on dose level (the calculations above assume a constant σ2). Crump (1995) developed a more general approach that relaxed the normality assumption required by previous approaches. Bosch et al. (1996) proposed a nonparametric approach that