Environmental and Ecological Statistics

, Volume 7, Issue 2, pp 135–154 | Cite as

‘A powerful alternative to Williams’ test with application to toxicological dose-response relationships of normally distributed data

  • Frank Bretz
  • Ludwig A. Hothorn


The comparison of increasing doses of a treatment to a negative control is frequently part of toxicological studies. For normally distributed data Williams (1971, 1972) introduced a maximum likelihood test under total order restriction. But until now there seems to have been no solution for the arbitrary unbalanced case. According to the idea proposed by Robertson et al. (1988) we will apply in this article the basic concept of Williams on the class of multiple contrast tests for the general unbalanced parametric set-up. Simulation results for size and power and two examples for estimating the minimal toxic dose (MTD) are given.

Williams test multiple contrast test MTD test on ordered alternatives multivariate t-distribution 


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Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Frank Bretz
    • 1
  • Ludwig A. Hothorn
    • 1
  1. 1.Department of BioinformaticsUniversity of HanoverHanoverGermany

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