Multiple Contrast Tests for Testing Dose–Response Relationships Under Order-Restricted Alternatives

  • Dan LinEmail author
  • Ludwig A. Hothorn
  • Gemechis D. Djira
  • Frank Bretz
Part of the Use R! book series (USE R)


In Chap. 3, 7, and 8, we discussed five test statistics that can be used for testing the null hypothesis of homogeneity of means against order-restricted alternatives. A rejection of the null hypothesis implies a significant monotone trend of gene expression with respect to dose. In this chapter, we employ an alternative method to find genes with monotonic trends, namely, the multiple contrast test (MCT). We dicuss the method for both monotone and non monotone alternatives.


  1. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T., & Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. The Annals of Mathematical Statistics, 26, 641–647.Google Scholar
  2. Barlow, R. E., Bartholomew, D. J., Bremner, M. J., & Brunk, H. D. (1972). Statistical inference under order restriction. New York: Wiley.Google Scholar
  3. Bretz, F. (1999). Powerful modifications of Williams’ test on trend. Ph.D. dissertation. Vom Fachbereich Gartenbau der Universit{a}t Hannover, Hannover,
  4. Bretz, F., & Hothorn, L. A. (2001). Testing dose-response relationship with a priori unknown, possibly non monotone shapes. Journal of Biopharmaceutical Statistics, 11, 193–207.Google Scholar
  5. Bretz, F., & Hothorn, L. A. (2003). Statistical analysis of monotone and non-monotone dose-response data from in vitro toxicological assays. Alternatives to Laboratory Animals (ALTA) (Suppl. 1), 31, 81–90.Google Scholar
  6. Bretz, F. (2006). An extension of the Williams trend test to general unbalanced linear models. Computational Statistics & Data Analysis, 50, 1735–1748.Google Scholar
  7. Bretz, F., Hothorn, T., & Westfall P. (2010). Multiple comparisons using R. Boca Raton: CRC press.Google Scholar
  8. Dunnett, C. W. (1955). A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Society (JASA), 50, 1096–1121.Google Scholar
  9. Dunnett, C. W. (1964). New tables for multiple comparisons with a control. Biometrics, 20, 482–491.Google Scholar
  10. Genz, A., & Bretz, F. (2009) Computation of multivariate normal and t probabilities. Lecture Notes in Statistics, 195 (Springer).Google Scholar
  11. Hothorn, L. A. (2006). Multiple comparisons and multiple contrasts in randomized dose-response trials—Confidence interval orient approaches. Journal of Biopharmaceutical Statistics, 16, 711–731.Google Scholar
  12. Marcus, R. (1976). The powers of some tests of the quality of normal means against an ordered alternative. Biometrika, 63, 177–83.Google Scholar
  13. Mukerjee, H., Roberston, T., & Wright, F. T. (1986). Multiple contrast tests for testing against a simple tree ordering. In R. Dykstra (Ed.), Advances in order restricted statistical inference (pp. 203–230). Berlin: Springer.Google Scholar
  14. Mukerjee, H., Roberston, T., & Wright, F. T. (1987). Comparison of several treatments with a control using multiple contrasts. Journal of the American Statistical Association, 82, 902–910.Google Scholar
  15. Robertson, T., Wright, F. T., & Dykstra, R. L. (1988). Order restricted statistical inference. New York: Wiley.Google Scholar
  16. Somerville, P. (1997). Multiple testing and simultaneous confidence intervals: calculation of constants. Computational Statistics & Data Analysis, 25, 217–233.Google Scholar
  17. Somerville, P. (1999). Critical values for multiple testing and comparisons: One step and step down procedures. Journal of Statistical Planning and Inference, 82(1), 129–138(10).Google Scholar
  18. Tukey, J. W. (1953). The problem of multiple comparisons. Unpublished manuscript. In The collected works of John W. Tukey VIII. Multiple comparisons, 1948–1983 (pp. 1–300). New York: Chapman & Hall.Google Scholar
  19. Williams, D. A. (1971). A test for differences between treatment means when several dose levels are compared with a zero dose control. Biometrics, 27, 103–117.Google Scholar
  20. Williams, D. A. (1972). The comparision of several dose levels with a zero dose control. Biometrics, 28, 519–531.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Dan Lin
    • 1
    Email author
  • Ludwig A. Hothorn
    • 2
  • Gemechis D. Djira
    • 3
  • Frank Bretz
    • 4
  1. 1.Veterinary Medicine Research and DevelopmentPfizer Animal HealthZaventemBelgium
  2. 2.Institute of BiostatisticsLeibniz University HannoverHannoverGermany
  3. 3.Department of Mathematics and StatisticsSouth Dakota State UniversityBrookingsUSA
  4. 4.Integrated Information SciencesNovartis Pharma AGBaselSwitzerland

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