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 xⁿ 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).