This paper provides some of the underlying mathematical derivations for the one-hit, multihit, multistage, Weibull, and pharmacokinetic risk models. Our purposes are to remove for the nonmathematician some of the mystery as to the derivation of the formulas for each particular risk model and to discuss some of the assumptions contained in the risk models. Confidence limits and maximum likelihood estimates of the model parameters are not discussed, since they are not pertinent to our objectives. Rai and Van Ryzin(1) have outlined these procedures in sufficient detail.
References
1.
1. Rai, K., and Van Ryzin, J. (1979). Risk assessment of toxic environmental substances using a generalized multi-hit dose-response model. In: Energy and Health, Practices of SIAM Institute. N. Brewlaw and A. Whittemore (eds.). Philadelphia: SIAM Institute.
2.
2. Hoel, D.G., Gaylor, D., et al. (1975). Estimation of risks of irreversible, delayed toxicity. J. Toxicol. Environ. Health1, 133–151.
3.
3. Cornfield, J. (1977). Carcinogenic risk assessment. Science198, 693.
4.
4. Van Ryzin, J., and Rai, K. (1980). The use of quantal response data to make predictions. In: The Scientific Basis of Toxicity Assessment. H. Whitschi (ed.). Amsterdam: Elsevier-North Holland Biomedical, pp. 273–279.
5.
5. Abbott, W.S. (1925). A method of computing the effectiveness of an insecticide. J. Econ. Ent.18, 265–267.
6.
6. Haseman, K.J., Hoel, D.G., and Jennrich, R.I. (1981). Some practical problems arising from use of the gamma multi-hit model for risk estimation. J. Toxicol. Environ. Health8, 379–386.
7.
7. Armitage, P. (1982). The assessment of low dose carcinogenicity. Biometrics38, 119–129.
8.
8. Armitage, P., and Doll, R. (1961). Stochastic models for carcinogenesis. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, Vol. 4, pp. 19–38.
9.
9. Crump, K.S., Guess, H.A., et al. (1977). Confidence intervals and tests of hypotheses concerning dose response relations inferred from animal carcinogenicity data. Biometrics33, 437–451.
10.
10. Hartley, H.O., and Sielken, R.L. (1977). Estimation of “safe doses” in carcinogenic experiments. Biometrics3, 1–30.
11.
11. Mantel, N., and Bryan, W.R. (1961). Safety testing of carcinogenic agents. J. Natl. Cancer Inst.27, 455–470.
12.
12. Weibull, W.A. (1951). Statistical distribution function of wide applicability. J. Appl. Mech.18, 293–302.
13.
13. Kovar, J., and Krewski, D.User Instructions for Risk 81. A Computer Program for Low Dose Extrapolation of Quantal Response Toxicity Data. Health and Welfare, Canada.
14.
14. Hoel, D.G., Kaplan, N.L., and Anderson, M.W. (1983). Implication of nonlinear kinetics on risk estimation in carcinogenesis. Science219, 1032.
15.
15. Krewski, D., and Van Ryzin, J. (1981). Dose–response models for quantal response toxicity data. In: Current Topics in Probability and Statistics. M. Csorgo, D. Dawson, J.M.K. Rao, and E. Saleh (eds.). New York: North-Holland.
16.
16. FOOD AND DRUG ADMINISTRATION. Advisory Committee on Protocols for Safety Evaluation. (1971). Panel on carcinogenesis report on cancer testing in the safety evaluation of food additives and pesticides. Toxicol. Appl. Pharmacol.20, 419–438.
17.
17. Gehring, P.J., and Blau, G.E. (1977). Mechanisms of carcinogenesis: Dose response. J. Environ. Pathol. Toxicol.1, 163–179.
18.
18. White, M. (1972). In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability. L.E. Le Cam, J. Neyman, and E.L. Scott (eds.). Berkeley: University of California Press, Vol. 4, pp. 287–307.
