Computational Statistics

, Volume 16, Issue 1, pp 165–171

# An algorithm for construction of multiple hypothesis testing

• Koon-Shing Kwong
Article

## Summary

Recently, the Simes method for constructing multiple hypothesis tests involving multivariate distributions of the test statistics with a particular form of positive dependence has been proved to strongly control the Type I familywise error rate. In this paper, an algorithm is provided so that distributions of ordered test statistics with certain correlation structures can be exactly and efficiently evaluated. Therefore, in some multiple hypothesis testing we can apply the algorithm to obtain tests which are more powerful than the conservative tests based on the Simes method. An example of how to apply the algorithm to the step-up multiple test procedure with a control treatment is presented.

## Keywords

Familywise error rate Multiple comparisons with a control Multivariate t distribution

## Notes

### Acknowledgments

The author is grateful to the associate editor for those helpful comments.

## References

1. Dunnett, C.W. and Tamhane, A.C. (1992), A step-up multiple test procedure. Journal of the American Statistical Association, 87, 162–170.
2. Dunnett, C.W. and Tamhane, A.C. (1993), Power comparisons of some step-up multiple test procedures. Statistics and Probability Letters, 16, 55–58.
3. Dunnett, C.W. and Tamhane, A.C. (1995), Step-up multiple testing of parameters with unequally correlated estimates. Biometrics, 51, 217–227.
4. Finner, H. and Roters, M. (1998), Asymptotic comparison of step-down and step-up multiple test procedures based on exchangeable test statistics. Annals of Statistics, 26, 505–524.
5. Hochberg, Y. (1988), A sharper Bonferroni procedure for multiple tests of significance. Biometrika, 75, 800–802.
6. Hochberg, Y. and Rom, D.M. (1995), Extensions of multiple testing procedures based on Simes’ test. Journal of Statistical Planning and Inference 48, 141–152.
7. Hochberg, Y. and Tamhane, A. (1987), Multiple Comparison Procedures. New York: Wiley.
8. Hommel, G. (1988), A stagewise rejective multiple test procedure based on a modified Bonferroni test. Biometrika, 76, 624–625.
9. Liu, W. (1997), Some results on step-up tests for comparing treatments with a control in unbalanced one-way layouts. Biometrics, 53, 1508–1512.
10. Rom, D.M. (1990), A sequentially rejective test procedure based on a modified Bonferroni inequality. Biometrika, 77, 663–665.
11. Samuel-Cahn, E. (1996), Is the Simes improved Bonferroni procedure conservative? Biometrika, 83, 928–933.
12. Sarkar, S.C. (1998), Some probability inequalities for ordered MTP2 random variables: a proof of the Simes conjecture. Annals of Statistics, 26, 494–504.
13. Sarkar, S.C. and Chang, C.K. (1997), The Simes method for multiple hypothesis testing with positively dependent test statistics. Journal of the American Statistical Association, 92, 1601–1608.
14. Simes, R. J. (1986). An improved Bonferroni procedure for multiple tests of significance. Biometrika, 73, 751–754.
15. Tamhane, A.C., Liu, W. and Dunnett, C.W. (1998), A generalized step-up-down multiple test procedure. Canadian Journal of Statistics, 20, 353–363.
16. Tong, Y.L. (1980), Probability Inequalities in Multivariate Distributions. New York: Academic Press.