The use of nonregular fractional factorial designs in combination toxicity studies

When there is interest to study n chemicals using x dose levels each, factorial designs that require x n treatment groups have been put forward as one of the valuable statistical approaches for hazard assessment of chemical mixtures. Exemplary applications and cost-efficiency comparisons of full factorial designs and regular fractional factorial designs in toxicity studies can be found in Nesnow et al. [Nesnow, S., Ross, J.A., Stoner, G.D., Mass, M.J., 1995. Mechanistic linkage between DNA adducts, mutations in oncogenes and tumorigenesis of carcinogenic environmental polycyclic aromatic hydrocarbons in strain A/J mice. Toxicology, 105, 403–413] , Narotsky et al. [Narotsky, M.G., Weller, E.A., Chinchilli, V.M., Kevlock, R.J., 1995. Non-additive developmental toxicity in mixtures of trichloroethylene, di(2-ethylhexyl)phthalate and heptachlor in a 5 × 5 × 5 design. Fundamental and Applied Toxicology, 27, 203–216], and Groten et al. [Groten, J.P., Schoen, E.D., Feron, V.J., 1996. Use of factorial designs in combination toxicity studies. Food and Chemical Toxicology, 34, 1083–1089], Groten et al. [Groten, J.P., Schoen, E.D., Kuper, C.F., van Bladeren, P.J., Van Zorge, J.A., Feron, V.J., 1997. Subacute toxicity of a mixture of nine chemicals in rats: detecting interactive effects with a fractionated two-level factorial design. Fundamental and Applied Toxicology, 36, 13–29]. We introduce nonregular fractional factorial designs and show their benefits using two studies reported in Groten et al. (1996). Study 1 shows nonregular designs can provide the same amount of information using 75% of the experimental costs required in a regular design. Study 2 demonstrates nonregular designs can additionally estimate some partially aliased effects, which cannot be done using regular designs. We also provide a statistical method to evaluate the quality of an assumption made by experts in Study 2 of Groten et al. (1996).