no widely accepted procedure for calculating a NOAEL in settings where exposure is measured on a relatively continuous scale. Indeed, the current definition of NOAEL involves an implicit assumption that the dose levels are grouped in some way. Grouping is common in the context of controlled animal studies, but most epidemiological studies, including the available MeHg studies, measure exposure on a continuous scale.

Problems with the NOAEL and LOAEL approach have led to increasing interest in the development of alternative approaches based on dose-response modeling techniques. The benchmark dose was defined by Crump (1984) as a lower 95% confidence limit on the dose corresponding to a moderate increase (e.g., 1%, 5%, or 10%) over the background rate. Because the benchmark dose generally occurs within the range of experimental data, Crump and others have argued that its estimation is relatively robust to model choice. In an extensive empirical comparison of NOAEL and benchmark-dose calculations, Allen et al. (1994) found that the NOAEL in a typically sized developmental toxicity study was, on average, 6 times larger than the BMDL corresponding to a 5% risk. The NOAEL was higher than even a 10% BMD, on average, by a factor of 3. Leisenring and Ryan (1992) came to somewhat similar conclusions based on analytical considerations. Crump (1984) used the abbreviation BMD to refer to the benchmark dose. Other authors, including Crump (1995), use BMD to denote the estimated dose that corresponds to a specified risk above the background risk and BMDL to denote the corresponding lower limit. This latter notation has become standard usage now and will be used throughout the remainder of this chapter.

BENCHMARK-DOSE CALCULATIONS FOR CONTINUOUS OUTCOMES

Benchmark-dose calculations for quantitative outcomes (e.g., birth weight or IQ) are more complicated than those for quantal responses, such as presence or absence of a defect. Although Crump (1984) discussed how to calculate a BMD for a quantitative outcome, Gaylor and Slikker (1992) were the first to develop the approach in any detail. Their first step is to fit a regression model characterizing the mean of the outcome of interest as a function of dose and assuming that the data are normally distributed. The next step is to specify a cutoff to define values



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