Evaluating marginal likelihood approximations of dose–response relationship models in Bayesian benchmark dose methods for risk assessment

Benchmark dose (BMD; a dose associated with a specified change in response) is used to determine the point of departure for the acceptable daily intake of substances for humans. Multiple dose–response relationship models are considered in the BMD method. The Bayesian model averaging (BMA) method is commonly used, where several models are averaged based on their posterior probabilities, which are determined by calculating the marginal likelihood (ML). Several ML approximation methods are employed in standard software packages, such as BBMD, ToxicR, and the EFSA platform for the BMD method, because the ML cannot be analytically calculated. Although ML values differ among approximation methods, resulting in BMD estimates, this phenomenon is neither widely recognized nor quantitatively evaluated. In this study, we evaluated the agreement of BMD estimates among five ML approximation methods in the BMA method. The five ML approximation methods are (1) maximum likelihood estimation (MLE)-based Schwarz criterion, (2) Markov chain Monte Carlo (MCMC)-based Schwarz criterion, (3) Laplace approximation, (4) density estimation, and (5) bridge sampling. We used eight dose–response relationship models and three prior distributions used in BBMD and ToxicR for 518 experimental datasets. The agreement among the approximation methods tended to be low in the non-informative prior distribution. Although the agreements tended to be high in the informative prior distribution, they were low in some approximation methods. Since the approximation method and the prior distribution affect the agreement, their selection should be carefully considered when implementing BMD methods.