The tests on the homogeneity of the columns of the coefficient matrix in a multiple multivariate linear regression with some rows of the matrix constrained by synchronized orderings, using the test statistics obtained by replacing the unknown variance–covariance matrix with its estimator in likelihood ratio test statistics, form a family of ad hoc tests. It is shown in this paper that the tests in the family share the same alpha-level critical values and follow the same distributions for computing their p-values. A sufficient condition is established for other tests to enjoy these properties, and to be more powerful. Two such more powerful tests are examined.