Annals of the Institute of Statistical Mathematics

, Volume 36, Issue 3, pp 463–473

# Semi-aligned rank tests

• Taka-Aki Shiraishi
Article

## Summary

The distribution-free test based on semi-aligned rankings for no treatment effects in a two-way layout, with unequal number of replications in each cell is considered. The asymptotic χ-square distribution of the test statistic under the null hypothesis is derived. The Pitman asymptotic relative efficiency of the test (i) based on semi-aligned rankings with respect to the test (ii) based on within-block rankings, is shown to be larger than one as the number of blocks tends to infinity. Also the asymptotic properties of linear rank statistics (i) and (ii) are investigated and the asymptotic relative efficiency of the test (i) with respect to the test (ii) is again shown to be larger than one.

## AMS 1980 subject classification

Primary 62G10 secondary 62E20

## Key words and phrases

Distribution-free test two-way layout asymptotic distribution asymptotic relative efficiency

## References

1. [1]
Anderson, R. L. (1959). Use of contingency tables in the analysis of consumer preference studies,Biometrics,15, 582–590.
2. [2]
Araki, T. and Shirahata, S. (1981). Rank tests for ordered alternatives in randomized blocks,J. Japan Statist. Soc. 11, 27–42.
3. [3]
Billingsley, P. (1968).Convergence of Probability Measures, John Wiley and Sons, New York.
4. [4]
Friedman, M. (1937). The use of ranks to avoid the assumption of normality inplicit in the analysis of variance,J. Amer. Statist. Soc.,11, 27–42.Google Scholar
5. [5]
Hájek, J. and Šidák, Z. (1967).Theory of Rank Tests, Academic Press, New York.
6. [6]
Hodge, J. L., Jr. and Lehman, E. L. (1962). Rank methods for combination of independent experiments in analysis of variance,Ann. Math. Statist.,33, 482–497.
7. [7]
Mack, G. A. and Skillings, J. H. (1980). A Friedman-type rank test for main effects in a two-factor ANOVA,J. Amer. Statist. Ass.,75, 947–951.
8. [8]
Page, E. B. (1963). Ordered hypotheses for multiple treatments; A significance test for linear ranks,J. Amer. Statist. Ass. 58, 216–230.
9. [9]
Sen, P. K. (1967). One some nonparametric generalization of Wilks' tests forH M,H VC andH MVC, I,Ann. Inst Statist. Math.,19, 451–471.
10. [10]
Sen, P. K. (1968). On a class of aligned rank order tests in two-way layouts,Ann. Math. Statist.,39, 1115–1124.
11. [11]
Shach, S. (1979). An alternative to the Friedman tests with certain optimality properties,Ann. Statist.,3, 537–550.