Description

The k-sample Anderson-Darling, Kruskal-Wallis, normal score and van der Waerden score tests are used to test the hypothesis that k samples of sizes n_1, …, n_k come from a common continuous distribution F that is otherwise unspecified. They are rank tests. Average rank scores are used in case of ties. While ad.test is consistent against all alternatives, qn.test tends to be sensitive mainly to shifts between samples. The combined versions of these tests, ad.test.combined and qn.test.combined, are used to simultaneously test such hypotheses across several blocks of samples. The hypothesized common distributions and the number k of samples for each block of samples may vary from block to block.

The Jonckheere-Terpstra test addresses the same hypothesis as above but is sensitive to increasing alternatives (stochastic ordering).

Also treated is the analysis of 2 x t contingency tables using the Kruskal-Wallis criterion and its extension to blocks.

Steel's simultaneous comparison test of a common control sample with s=k-1 treatment samples using pairwise Wilcoxon tests for each control/treatment pair is provided, and also the simultaneous confidence bounds of treatment shift effects resulting from the inversion of these tests when sampling from continuous populations.

Distributional aspects are handled asymptotically in all cases, and by choice also via simulation or exact enumeration. While simulation is always an option, exact calculations are only possible for small sample sizes and only when few samples are involved. These exact calculations can be done with or without ties in the pooled samples, based on a recursively extended version of Algorithm C (Chase's sequence) in Knuth (2011), which allows the enumeration of all possible splits of the pooled data into samples of sizes of n_1, …, n_k, as appropriate under treatment randomization or random sampling, when viewing tests conditionally given the observed tie pattern.

Details

Author(s)

Fritz Scholz and Angie Zhu

Maintainer: Fritz Scholz <fscholz@u.washington.edu>

References

Hajek, J., Sidak, Z., and Sen, P.K. (1999), Theory of Rank Tests (Second Edition), Academic Press.

Knuth, D.E. (2011), The Art of Computer Programming, Volume 4A Combinatorial Algorithms Part 1, Addison-Wesley

Kruskal, W.H. (1952), A Nonparametric Test for the Several Sample Problem, The Annals of Mathematical Statistics, Vol 23, No. 4, 525-540

Kruskal, W.H. and Wallis, W.A. (1952), Use of Ranks in One-Criterion Variance Analysis, Journal of the American Statistical Association, Vol 47, No. 260, 583–621.

Lehmann, E.L. (2006), Nonparametrics, Statistical Methods Based on Ranks, Revised First Edition, Springer, New York.

Scholz, F. W. and Stephens, M. A. (1987), K-sample Anderson-Darling Tests, Journal of the American Statistical Association, Vol 82, No. 399, 918–924.