Tests for dimensionality and interactions of mean vectors under general and reducible covariance structures

Likelihood ratio tests are derived for testing the structure of mean values in a two-way classification. The most general hypothesis considered is when the mean values are subject to row and column effects and interaction has a given complexity. The observations corresponding to a row or a column classification are assumed to have an unknown dispersion (variance covariance) matrix. Two types of dispersion matrices are considered, one with a general and another with a reducible structure. Some special cases are considered. The results of the paper provide generalizations of tests on dimensionality and interactions in a two-way array of mean values considered by Fisher, Anderson, Fujikoshi, Mandel, and Rao.