Nonparametric estimation of benchmark doses in environmental risk assessment

Abstract

An important statistical objective in environmental risk analysis is estimation of minimum exposure levels, called benchmark doses (BMDs), that induce a pre-specified benchmark response in a dose-response experiment. In such settings, representations of the risk are traditionally based on a parametric dose-response model. It is a well-known concern, however, that if the chosen parametric form is misspecified, inaccurate and possibly unsafe low-dose inferences can result. We apply a nonparametric approach for calculating benchmark doses, based on an isotonic regression method for dose-response estimation with quantal-response data (Bhattacharya and Kong, 2007). We determine the large-sample properties of the estimator, develop bootstrap-based confidence limits on the BMDs, and explore the confidence limits' small-sample properties via a short simulation study. An example from cancer risk assessment illustrates the calculations.

Keywords: Benchmark analysis; bootstrap confidence limits; dose-response analysis; isotonic regression; pool-adjacent-violators algorithm.

Figure 1

Median empirical bias from Monte Carlo evaluations for model-independent BMD estimator ξ̃₁₀ with geometric four-dose design. Horizontal line indicates zero bias.

Figure 2

Median empirical bias from Monte Carlo evaluations for model-independent BMD estimator ξ̃₁₀ with geometric six-dose design. Horizontal line indicates zero bias.

Figure 3

Empirical coverage rates from Monte Carlo evaluations for model-independent bootstrap BMDL ξ₁₀ with geometric four-dose design. Horizontal line indicates 95% nominal coverage level.

Figure 4

Empirical coverage rates from Monte Carlo evaluations for model-independent bootstrap BMDL ξ₁₀ with geometric six-dose design. Horizontal line indicates 95% nominal coverage level.

Figure 5

Empirical coverage rates from Monte Carlo evaluations for model-independent bootstrap BMDL ξ₁₀ with modified six-dose design. Horizontal line indicates 95% nominal coverage level.

Figure 6

Probability histogram (and overlaid density estimator) of bootstrap distribution for ${\stackrel{\sim }{\xi }}_{10}^{\ast }$with formaldehyde carcinogenicity data from Table 2.

Figure 7

Empirical coverage rates from Monte Carlo evaluations for model-independent bootstrap BMDL₁₀ based on smoothed isotonic interpolator under geometric four-dose design. Horizontal line indicates 95% nominal coverage level.

Figure 8

Empirical coverage rates from Monte Carlo evaluations for model-independent bootstrap BMDL₁₀ based on smoothed isotonic interpolator under geometric six-dose design. Horizontal line indicates 95% nominal coverage level.