Abstract
Fixed-ratio ray designs have been used for detecting and characterizing interactions of large numbers of chemicals in combination. Single-chemical dose-response data are used to predict an “additivity curve” along an environmentally relevant ray. A “mixture curve” is estimated from the mixture dose-response data along the ray. A test of additivity is equivalent to a test of coincidence of these two curves, which is based on the traditional hypothesis testing framework that assumes additivity in the null hypothesis and rejects with evidence of interaction. However, failure to reject may be due to lack of statistical power, making the claim of additivity problematic. As a solution we have developed rigorous methodology to test for additivity using statistical equivalence testing logic in which additivity is claimed based on pre-specified biologically important additivity margins, if the data support such a claim. Using the principle of confidence interval inclusion, a confidence region about the difference of meaningful functions of model parameters from the mixture model and that predicted under additivity is computed. When the confidence region is completely contained within the additivity margins then additivity is claimed with a Type I error rate chosen a priori to be some acceptably small value. The method is illustrated using an environmentally relevant fixed-ratio mixture of nine haloacetic acids where cytotoxic response is measured.
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References
Anderson, T. W. (1958), An Introduction to Multivariate Statistical Analysis, New York: Wiley.
Berenbaum, M. C. (1981), “Criteria for Analyzing Interactions Between Biologically Active Agents,” Advances in Cancer Research, 35, 269–335.
Blackwelder, W. C. (1982), “Proving the Null Hypothesis’ in Clinical Trials,” Controlled Clinical Trials, 3, 345–353.
Berger, R. L., and Hsu, J. C. (1996), “Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets,” Statistical Sciences, 11, 283–319.
Carter, W. H., Jr., Chinchilli, V. M., Myers, R. H., and Campbell, E. D. (1986), “Confidence Intervals and an Improved Ridge Analysis of Response Surfaces,” Technometrics, 28, 339–346.
Carter, W. H., Jr., Gennings, C., Staniswalis, J. G., Campbell, E. D., and White, K. L., Jr. (1988), “A Statistical Approach to the Construction and Analysis of Isobolograms,” Journal of the American College of Toxicology, 7, 963–973.
Casey, M., Gennings, C., Carter, W. H., Jr., Moser, V. C., and Simmons, J. E. (2004), “Detecting Interaction(s) and Assessing the Impact of Component Subsets in a Chemical Mixture Using Fixed-Ratio Mixture Ray Designs,” Journal of Agricultural, Biological, and Environmental Statistics, 9, 339–361.
Cox, C. (1987), “Threshold Dose-Response Models in Toxicology,” Biometrics, 43, 511–523.
Crofton, K. M., Craft, E. S., Hedge, J. M., Gennings, C., Simmons, J. E., Carchman, R. A., Carter, W. H., Jr., and DeVito, M. J. (2005), “Thyroid-Homrone-Disrupting Chemicals: Evidence for Dose-Dependent Additivity or Synergism,” Environmental Health Perspectives, 113, 1549–1554.
EPA (1986), “Guidelines for the Health Risk Assessment of Chemical Mixtures,” Federal Register, 51 (185), 34014–34025, Washington, DC: U.S. Environmental Protection Agency.
— (2000), “Supplementary Guidance for Conducting Health Risk Assessment of Chemical Mixtures,” Risk Assessment Forum, EPA/630/R-00/002, Washington, DC: U.S. Environmental Protection Agency.
Gennings, C., Carter, W. H., Jr, Campain, J. A., Bae, D., and Yang, R. S. H. (2002), “Statistical Analysis of Interactive Cytotoxicity in Human Epidermal Keratinocytes Following Exposure to a Mixture of Four Metals,” Journal of Agricultural, Biological, and Environmental Statistics, 17, 58–73.
Gennings, C., Carter, W. H., Jr., Carchman, R. A., Teuschler, L. K., Simmons, J. E., and Carney, E. W. (2005), “A Unifying Concept for Assessing Toxicological Interactions: Changes in Slope,” Toxicological Sciences, 88, 287–297.
Gennings, C., Carter, W. H., Jr., Carney, E. W., Charles, G. D., Gollapudi, B., and Carchman, R. A. (2004), “A Novel Flexible Approach for Evaluating Fixed Ratio Mixtures of full and Partial Agonists,” Toxicological Sciences, 80, 134–150.
Liu, J., Hsueh, H., Hsieh, E., and Chen, J. J. (2002), “Tests for Equivalence or Non-Inferiority for Paired Binary Data,” Statistics in Medicine, 21, 231–245.
Loewe, S. (1953), “The Problem of Synergism and Antagonism of Combined Drugs,” Arzneimittle Forshung, 3, 285–290.
Loewe, S., and Muischnek, H. (1926), “Uber Kombinationswirkunger. I. Mitteilung: Hiltsmittel der Gragstellung. Naunyn-Schmiedegergs,” Archives of Pharmacology, 114, 313–326.
McCullagh, P. (1983), “Quasi-Likelihood Functions,” The Annals of Statistics, 11, 59–67.
McCullagh, P., and Nelder, J. A. (1989), Generalized Linear Models (2nd ed.), New York: Chapman and Hall.
Meadows, S., Gennings, C., Carter, W. H., Jr., and Bae, D. (2002), “Experimental Designs for Mixtures of Chemicals Along Fixed Ratio Rays,” Environmental Health Perspectives, 110, (Suppl. 6), 979–983.
Plewa, M. J., Kargalioglu, Y., Vankerk, D., Minear, R. A., and Wagner, E. D. (2002), “Mammalian Cell Cytotoxicity and Genotoxicity Analysis of Drinking Water Disinfection By-Products,” Environmental and Molecular Mutagenesis, 40, 134–142.
Plewa, M. J., Wagner, E. D., Richardson, S. D., Thruston, A. D., Jr., Woo, Y.-T., and McKague, A. B. (2004), “Chemical and Biological Characterization of Newly Discovered Iodo-Acid Drinking Water Disinfection Byproducts,” Environmental Science and Technology, 38, 4713–4722.
Rao, C. R. (1973), Linear Statistical Inference and its Application, New York: Wiley.
SAS Software Version 9.1. (2002–2003), SAS Institute Inc., Cary, NC.
Scheffé, H. (1959), The Analysis of Variance, New York: Wiley.
Schwartz, P. F., Gennings, C., and Chinchilli, V. M. (1995), “Threshold Models for Combination Data from Reproductive and Developmental Experiments,” Journal of the American Statistical Association, 90, 862–870.
Seber, G. A. F., and Wild, C. J. (1989), Nonlinear Regression, New York: Wiley.
Simmons, J. E., Richardson, S. D., Speth, T. F., Miltner, R. J., Rice, G., Schenck, K. M., Hunter, E. S., III, and Teuschler, L. K. (2002), “Development of a Research Strategy for Integrated Technology-Based Toxicological and Chemical Evaluation of Complex Mixtures of Drinking Water Disinfection Byproducts,” Environmental Health Perspectives, 110 (Suppl. 6), 1013–1024.
Spjøtvoll, E. (1972), “Multiple Comparison of Regression Function,” Annals of Mathematical Statistics, 43, 1076–1088.
Stork, L. G. (2005), “A Statistical Equivalence Testing Approach to Testing for Additivity and Determining Sufficiently Similar Mixtures of Many Chemicals Based on Dose-Response,” Ph.D. dissertation, Department of Biostatistics, Virginia Commonwealth University, Richmond, Virginia.
Stork, L. G., Gennings, C., Carchman, R. A., Carter, W. H., Jr., Pounds, J., and Mumtaz, M. (2006), “Testing for Additivity at Select Mixture Groups of Interest Based on Statistical Equivalence Testing Methods,” Risk Analysis, 26, 1601–1612.
Tsutakawa, R. K., and Hewett, J. E. (1978), “Comparison of Two Regression Lines Over a Finite Interval,” Biometrics, 34, 391–398.
Ulm, K. (1991), “A Statistical Method for Assessing a Threshold in Epidemiological Studies,” Statistics in Medicine, 10, 341–349.
Wedderburn, R. W. M. (1974), “Quasi-Likelihood Functions, Generalized Linear Models, and the Gauss-Newton Method,” Biometrika, 61, 439–447.
Weisel, C. P., Kim, H., Haltmeier, P., and Klotz, J. B. (1999), “Exposure Estimates to Disinfection By-Products of Chlorinated Drinking Water,” Environmental Health Perspectives, 107, 103–110.
Wellek, S. (2003), Testing Statistical Hypotheses of Equivalence, Boca Raton, FL: Chapman and Hall/CRC.
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Stork, L.G., Gennings, C., Carter, W.H. et al. Testing for additivity in chemical mixtures using a fixed-ratio ray design and statistical equivalence testing methods. JABES 12, 514–533 (2007). https://doi.org/10.1198/108571107X249816
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DOI: https://doi.org/10.1198/108571107X249816