What is the meaning of non-linear dose-response relationships between blood lead concentrations and IQ?

Recent literature [e.g. Canfield RL, Henderson CR, Cory-Slechta DA, Cox C, Jusko TA, Lanphear BP. Intellectual impairment in children with blood lead concentrations below 10mg per deciliter. New Engl J Med 2003;348(16):1517-1526; Lanphear BP, Hornung R, Khoury J, Yolton K, Baghurst P, Bellinger DC, Canfield RL, Dietrich KN, Bornschein R, Greene T, Rothenberg SJ, Needleman HL, Schnaas L, Wasserman G, Graziano J, Roberts R. Low-level environmental lead exposure and children's intellectual function: an international pooled analysis. Environ Health Perspect 2005;113(7):894-899] has suggested the existence of a supra-linear dose-response relationship between environmental measures such as blood lead concentrations and IQ. This communication explores the mathematical requirements placed on such dose-response relationships when the environmental measure, or independent variable, is lognormally distributed and the effect, or dependent variable, is normally distributed. Results of the analyses show that a supra-linear slope is a required outcome of correlations between data distributions where one is lognormally distributed and the other is normally distributed. The analysis shows that caution should be taken in assigning biological significance to supra-linear dose-response relationships in these instances. Detailed analyses of such data sets should be conducted to determine if the magnitude of supra-linear slopes are more or less than mathematically required, and from there to consider biological significance.