A Unified Approach to Rank Tests for Multivariate and Repeated Measures Designs

A unified approach to asymptotic rank tests is presented for a wide class of univariate and multivariate models, including repeated measures designs without compound symmetry. The models are assumed to be balanced and complete with more than one replication per cell. The proposed rank tests are constructed by ranking all of the observations together regardless of row, column, or component membership. This method of ranking offers increased power over the method of n-rankings. The resulting test statistic is a quadratic form in linear rank statistics. Asymptotic distributions are determined under Pitman alternatives that allow for both scale and location alternatives. The resulting statistics include tests for factor effects (both scale and location differences) in one-, two-, and higher-way layouts with repeated measures on one or several factors without assuming equicorrelation. Also included are tests for the multivariate two- and k-sample problem, as well as multivariate versions of the tests for multiway layouts and repeated measures designs. For many of the repeated measures designs without equicorrelation, no other rank based statistics have been previously studied. The results also include the asymptotic distributions for many possible rank transform tests for univariate and multivariate models, as well as a rich class of aligned rank tests.