Dose-response modeling refers to the statistical problem of characterizing the probability of the occurrence of an event as a function of exposure. For simplicity, the discussion here will refer to the event of interest as "cancer." In practice, there are always subtleties and complicating factors to consider when performing a specific risk assessment. The purpose here is not to address all these issues but to outline the broad principles of model fitting and quantitative risk assessment. The one issue that the subcommittee does address in some detail is how to incorporate information on age-specific cancer rates into the risk-assessment process.
Suppose we have data from N individuals corresponding to their exposure and cancer status. More precisely, let yi take the value 1 if person i has cancer and 0 otherwise. Let xi be the exposure concentration for that same person. Then, if pq(x) represents the assumed dose-response model that characterizes the probability of cancer as a function of exposure through a set of unknown parameters q, the maximum likelihood estimator (MLE) of q is the value that maximizes the likelihood
where Õ denotes product. As discussed in Gart et al. (1986, Ch. 5), many dose-response models are available that can be used to describe the relationship between exposure and cancer prevalence. EPA often uses the multistage model (Holland and Sielkin 1993) in the analysis of animal data. That model, which is motivated by the idea that cancer occurs as the last of several irreversible steps, takes the form
where Q0, Q1, . . . Qk are elements of the unknown parameter vector q, which needs to be estimated using maximum likelihood. Usually, the parameters are estimated under the constraint that the Q's are non-negative. Often, K is taken to equal 1 (the 1 hit model) or 2.
In the past, the next step after fitting the dose-response model has been to estimate the exposure concentration that corresponds to an "acceptable" risk