Testing the Rank of a Matrix with Applications to the Analysis of Interaction in ANOVA
Abstract This article develops some general theory for testing the rank of a matrix. Applications include tests of interaction in two-factor experiments that are more powerful than the usual F test for certain reasonable alternatives. The particular tests studied are (a) the likelihood ratio (LR) test of rank(M) = 0 versus rank(M) = r, where M is a matrix of expectations, and (b) the union-intersection (UI) test of rank(M) = 0 versus rank(M) ≥ r. In the two-factor application, M is the a X b matrix of interaction parameters. The UI test that the interaction has rank = 0 versus rank ≥ 1 is a simultaneous test that all product interaction contrasts are zero. It is shown that the asymptotic distributions of the UI and LR test statistics are identical. A distinct advantage of the UI test is that the small-sample null distribution of the test statistic is known and can be computed. Tables of exact percentiles of the UI test statistic for testing rank(M) = 0 against rank(M) ≥ 1 are given. The UI test is illustr...