A number of statistical tests are proposed for testing whether an observed covariance matrix is consistent with an expected covariance matrix deduced from an assumed composition of the variates. This aspect of multivariate analysis has no univariate analogue and as yet has not received proper treatment in the statistical literature. The general method proposed has been called ‘structural analysis’ and its role in multivariate hypothesis‐testing is discussed.Two different types of patterned covariance matrices arising out of various linear models are considered. The ‘reducible pattern’, composed of covariance matrices which can be diagonalized by pre‐ and post‐multiplication by a known matrix, and covariance matrices which can be transformed to a tridiagonal form (such as the Guttman quasi‐simplex, and the ‘aristocratic’ matrices), are discussed and illustrated with empirical data. Likelihood ratio tests for both classes of patterns have been constructed, and in each case the estimation procedure has also been tackled.