Benchmark Calculations in Risk Assessment Using Continuous Dose-Response Information: The Influence of Variance and the Determination of a Cut-Off Value


*Address correspondence to Salomon Sand, Institute of Environmental Medicine, P.O. Box 210, 17177 Stockholm, Sweden;


A benchmark dose (BMD) is the dose of a chemical that corresponds to a predetermined increase in the response (the benchmark response, BMR) of a health effect. In this article, a method (the hybrid approach) for benchmark calculations from continuous dose-response information is investigated. In the formulation of the methodology, a cut-off value for an adverse health effect has to be determined. It is shown that the influence of variance on the hybrid model depends on the choice of determination of the cut-off point. If the cut-off value is determined as corresponding to a specified tail proportion of the control distribution, P(0), the BMD becomes biased upward when the variance is biased upward. On the contrary, if the cut-off value is directly determined to some level of the continuous response variable, the BMD becomes biased upward when the variance is biased downward. A simulation study was also performed in which the accuracy and precision of the BMD was compared for the two ways of determining the cut-off value. In general, considering BMRs of 1, 5, and 10% (additional risk) the precision of the BMD became higher when the cut-off value was estimated by specifying P(0), relative to the case with a direct determination. Use of the square-root of the maximum-likelihood estimator of the variance in BMD estimation may provide a bias that is reflected by the cut-off formulation (downward bias if specifying P(0), and upward bias if specifying the cut-off, c, directly). This feature may be reduced if an unbiased estimator of the standard deviation is used in the calculations.