Some Nonparametric Median Procedures

Abstract

Under the nonparametric approach, various methods have been suggested to avoid the assumption of normality and homoscedasticity implicit in the analysis of variance. For the one-way classification, i.e., to decide whether $c$ samples come from the same population, Kruskal and Wallis [9] have proposed the $H$-test based on ranks; Mood and Brown [10] have proposed the $M$-test, utilizing the numbers of observations above the median of the combined sample; while the present author [2] has offered the $V$-test based on the number of $c$-plets that can be formed by choosing one observation from each sample such that the observation from the $k$th sample is the least, $k = 1, 2, \cdots, c$. For the two-way classification, Friedman [8] has made use of ranks. His statistic, to test the hypothesis that the rankings by $m$ "observers" of $n$ "objects" are independent, essentially offers a test for the two-way classification with one observation per cell. Durbin [7] has given a generalization for the balanced incomplete block design. Benard and Van Elteren [1] have generalized it still further. Mood and Brown [6, 10] also have considered the two-way classification with one observation per cell or the same number of observations per cell. In the first part of the present paper, their test has been extended to cover incomplete block situations. Mood and Brown [6, 10] have also considered some simple regression problems. In the present paper their methods are extended to discuss some additional regression problems. Next some bivariate analysis of variance problems are considered. The "step-down procedure" [11, 12] is used to reduce the problem to the univariate case with the other variate as a concomitant variate. The regression methods developed earlier are used here in these bivariate problems. The method seems to be perfectly general and could be extended to three or more variates, that is, to the multivariate situation. Most of the tests are offered on heuristic considerations. They are expected to be good for large samples.