  Benchmark dose analysis aims to estimate the level of exposure to a toxin
that results in a clinically-significant adverse outcome and quantifies
uncertainty using the lower limit of a confidence interval for this level. We
develop a novel framework for benchmark dose analysis based on monotone
additive dose-response models. We first introduce a flexible approach for
fitting monotone additive models via penalized B-splines and
Laplace-approximate marginal likelihood. A reflective Newton method is then
developed that employs de Boor's algorithm for computing splines and their
derivatives for efficient estimation of the benchmark dose. Finally, we develop
and assess three approaches for calculating benchmark dose lower limits: a
naive one based on asymptotic normality of the estimator, one based on an
approximate pivot, and one using a Bayesian parametric bootstrap. The latter
approaches improve upon the naive method in terms of accuracy and are
guaranteed to return a positive lower limit; the approach based on an
approximate pivot is typically an order of magnitude faster than the bootstrap,
although they are both practically feasible to compute. We apply the new
methods to make inferences about the level of prenatal alcohol exposure
associated with clinically significant cognitive defects in children using data
from an NIH-funded longitudinal study. Software to reproduce the results in
this paper is available at https://github.com/awstringer1/bmd-paper-code.
