On Rank Tests for Experimental Designs

Statistical methods based on ranking are known to be versatile and powerful techniques. Nonparametric rank tests for two-way analysis of variance have been studied by several authors, for example: rank transform (RT ) proposed by Conover and Iman (1981); asymptotic theory of RT discussed by Hora and Conover (1984), Akritas (1990) and Thompson and Ammann (1990). McKean and Vidman (1994) suggested to use a rank-based robust general linear model method. Van der Waerden (1952/1953) suggested that instead of using the ranks of the observations we use the quantiles of a standard normal distribution; we call it a normal score test. Mansouri and Chang (1995) discussed rank tests and showed good results by using a normal score test for an interaction term. In this paper, we propose using nonparametric quantile estimates of a random sample instead of the corresponding ranks or normal scores. The idea arises to propose the nonparametric quantile score, since it may use more information of the data than just ranks or normal scores. The nonparametric quantile score induces a new test statistic which is given in Section 2. Its asymptotic distribution is studied in Section 3. Comparisons of its performance with other rank tests are investigated in Section 4 by using three numerical examples The computational results show that the nonparametric quantile score test reduces the type I error rate for testing the interaction term, more than other rank-based tests do.