Rank Order Tests for Multivariate Paired Comparisons

Abstract

The only non-parametric multivariate paired comparison tests presently available for testing the hypothesis of no difference among several treatments are (i) the Sen-David (1968) test and (ii) the Davidson-Bradley (1969) test. Both these tests are applicable to situations which involve the preferences of each individual comparison. Both these tests are the generalizations of the one-sample multivariate sign tests [1]. As such their A.R.E.'s (Asymptotic Relative Efficiencies) with respect to the normal theory $\mathfrak{F}$-test are not expected to be high. In fact the A.R.E. of the Sen-David (1968) test with respect to the normal theory $\mathfrak{F}$-test can be as low as zero (under normality). The purpose of this paper is to develop test procedures which could be considered as competitors to the Sen-David (1968) and to the Davidson-Bradley (1968) tests. The proposed procedures are based on the ranks of the observed comparison differences, and include as special cases the multivariate normal scores and the multivariate rank sum paired comparison tests. For convenience of presentation we develop the theory when the paired comparisons involve paired characteristics. Under suitable regularity conditions the limiting distributions of the proposed test statistics are derived under the null as well as non-null hypotheses, and their large sample properties are studied. It is shown that for various situations of interest the proposed procedures have considerable efficiency improvements over the Sen-David (1968) and the normal theory procedures.