Testing the Rank of a Matrix With Applications to the Analysis of Interaction in ANOVA
本文发展了检验矩阵秩的一般理论，并将其应用于双因素实验中的交互作用检验，提出了比传统F检验更强大的似然比和联合交检验方法，并给出了小样本精确分布表。
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 illustrated by testing the rank of a matrix of interaction parameters in a two-factor experiment.