Suppose that we are utilizingkdifferent psychological tests, each having one subtest in common. Of particular concern is the hypothesis that these tests are parallel with respect to their means and/or covariances. A complete hierarchy of hypotheses for this situation has been developed. For example,Hm'veis the hypothesis that the tests are parallel only with respect to the means of the common test, but with respect to the covariances of both tests. (The prime indicates equality for the common test only.) This hypothesis might be tested againstHvc, the hypothesis of parallelism with respect to the covariances. Other hypotheses considered areHmvc,Hm'vc,Hm'vcandHvc. Maximum‐likelihood estimators under the various models (and under the assumption of normally distributed test scores) have been obtained, as well as the related likelihood‐ratio statistics. Approximate distributions of the likelihood‐ratio statistics are worked out, so that the tests can be applied, and an example of their use is pro