Model Uncertainty and Bayesian Model Averaged Benchmark Dose Estimation for Continuous Data


  • Kan Shao,

    Corresponding author
    1. ORISE Postdoctoral Fellow, National Center for Environmental Assessment, U.S. Environmental Protection Agency
    • Address correspondence to Kan Shao, Ph.D., NCEA B243–01, U.S. EPA, 109 T.W. Alexander Drive, Research Triangle Park, NC 27711, USA; tel: 919-541-0440; fax: 919-541-0245;

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  • Jeffrey S. Gift

    1. National Center for Environmental Assessment, U.S. Environmental Protection Agency, NC, USA
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  • [Correction added after online publication on June 11, 2013: Equation 17 has been altered.]


The benchmark dose (BMD) approach has gained acceptance as a valuable risk assessment tool, but risk assessors still face significant challenges associated with selecting an appropriate BMD/BMDL estimate from the results of a set of acceptable dose-response models. Current approaches do not explicitly address model uncertainty, and there is an existing need to more fully inform health risk assessors in this regard. In this study, a Bayesian model averaging (BMA) BMD estimation method taking model uncertainty into account is proposed as an alternative to current BMD estimation approaches for continuous data. Using the “hybrid” method proposed by Crump, two strategies of BMA, including both “maximum likelihood estimation based” and “Markov Chain Monte Carlo based” methods, are first applied as a demonstration to calculate model averaged BMD estimates from real continuous dose-response data. The outcomes from the example data sets examined suggest that the BMA BMD estimates have higher reliability than the estimates from the individual models with highest posterior weight in terms of higher BMDL and smaller 90th percentile intervals. In addition, a simulation study is performed to evaluate the accuracy of the BMA BMD estimator. The results from the simulation study recommend that the BMA BMD estimates have smaller bias than the BMDs selected using other criteria. To further validate the BMA method, some technical issues, including the selection of models and the use of bootstrap methods for BMDL derivation, need further investigation over a more extensive, representative set of dose-response data.