Beta-normal distribution in dose–response modeling and risk assessment for quantitative responses

Abstract

To establish allowable daily intakes for humans from animal bioassay experiments, benchmark doses corresponding to low levels of risk have been proposed to replace the no-observed-adverse-effect level for non-cancer endpoints. When the experimental outcomes are quantal, each animal can be classified with or without the disease. The proportion of affected animals is observed as a function of dose and calculation of the benchmark dose is relatively simple. For quantitative responses, on the other hand, one method is to convert the continuous data to quantal data and proceed with benchmark dose estimation. Another method which has found more popularity (Crump, Risk Anal 15:79–89; 1995) is to fit an appropriate dose–response model to the continuous data, and directly estimate the risk and benchmark doses. The normal distribution has often been used in the past as a dose–response model. However, for non-symmetric data, the normal distribution can lead to erroneous results. Here, we propose the use of the class of beta-normal distribution and demonstrate its application in risk assessment for quantitative responses. The most important feature of this class of distributions is its generality, encompassing a wide range of distributional shapes including the normal distribution as a special case. The properties of the model are briefly discussed and risk estimates are derived based on the asymptotic properties of the maximum likelihood estimates. An example is used for illustration.