Direct calculation of likelihood-based benchmark dose levels for quantitative responses

A benchmark dose (BMD) for quantitative responses is a lower confidence limit (LCL) on the effective dose corresponding to a specified risk level r. A commonly adopted method for calculating the BMD is to obtain a pointwise upper confidence curve U(d) on the risk function and then invert this relationship by solving the equation U(d)=r. The solution d is taken to be the BMD. Sciullo et al. (2000) have shown that the coverage achieved by this inversion method is at least as great as the coverage achieved by U (·) but that there is otherwise no general relationship between the two coverage probabilities. The present paper develops a method for direct calculation of the BMD based on the asymptotic distribution of the likelihood ratio statistic. It is further shown that the direct method and the inversion method are equivalent when U (·) is also based on the likelihood ratio. Since the direct method is known to be asymptotically correct, it follows that the LR-based inversion method is also asymptotically correct. However, the direct method is computationally faster and easier to program. Finally, some simulation studies are conducted to assess the small sample coverage probabilities of the direct method when responses follow either a normal or a gamma distribution.