F. Bretz (University of Hannover, Germany),

A. Genz (Washington State University, USA),

L.A. Hothorn (University of Hannover, Germany)

Studentized multiple contrast tests: distribution under the null and the alternative

We consider multiple comparison procedures with and without order restrictions. Such procedures include the tests of Williams, Hayter, Hirotsu, Rom et al., and many others. One main problem of all of these approaches is their limited numerical availability under the null and the alternative hypotheses. We show that all of the above approaches can be represented as studentized multiple contrast tests. We introduce new methods of computing the arising central and non-central multivariate t-distributions. Numerical issues are discussed and the new approaches are compared to existing ones. The results indicate that we are able to compute p-values (and hence quantiles) and power values (and hence sample sizes) robustly and reliably at moderate accuracy levels (about 4 significant digits) within a few seconds of workstation time for multiple comparison problems with up to 20 groups. A data example illustrates the application and the availability of the presented techniques.

References:

- Bretz, F. (1999) Powerful Modifications of Williams' Test on Trend. Ph.D. thesis, University of Hannover.
- Genz, A. and Bretz, F. (1999) Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361-378
- Genz, A. and Bretz, F. (2000) Methods for the Computation of Multivariate t-Probabilities. (submitted)
- Hothorn, L.A., Neuhäuser, M. and Koch, H.F. (1997) Analysis of randomised dose-finding studies: closure test modifications based on multiple contrast tests. Biometrical Journal, 39, 467-479.