Uncertainty of the hazardous concentration and fraction affected for normal species sensitivity distributions

Species in the environment vary according to their sensitivity to a toxican t. Because these differences in sensitivity are unique to the toxicant at c onsideration and laboratory data sets to assess this variability are very s mall due to cost, it is important to provide uncertainty estimates of (1) e nvironmental quality objectives (hazardous concentrations) derived from the se laboratory data and (2) fraction of species affected at given, or predic ted, laboratory or environmental concentrations. This article focuses on th e normal (Gaussian) distribution of species sensitivity. It examines and co mpares results of Problems (I) and (2) from two opposing statistical philos ophies, Bayesian and Classical, leading to vastly different numerical appro aches. For the normal model, both approaches lead to identical answers, num erically. Extrapolation factors for the lower, median, and upper estimates of the hazardous concentration at six levels of protection are derived. Fur thermore, upper, median, and lower estimates of the fraction affected at gi ven, standardized, logarithmic concentrations have been tabulated. This tab le can be used directly for risk assessment without reference to protection levels or hazardous concentrations. The confidence limits for hazardous co ncentration and fraction affected depend heavily on the number of species t ested and are independent of the toxic substance involved (provided the mod el is right), due to correction for the mean and standard deviation of the toxicity data. The equivalence of confidence limits for hazardous concentra tion and fraction affected is captured in the law of extrapolation: the upp er (median, lower) confidence limit for the fraction affected at the lower (median, upper) confidence limit of the hazardous concentration is equal to the fraction affected (e.g., 5%) used to define the hazardous concentratio n. The upper confidence limit for the fraction affected at the median estim ate of the hazardous concentration for 5% of the species is a fixed number depending on the sample size of the toxicity data only. It amounts to 46% a t n=3, do,vn to 20% at n=10, and still 12% at n 30. (C) 2000 Academic Press .