A class of independent linear models is considered, where the variances of the different models are unequal but they have a common vector parameterθoccurring in their mean vectors. For testing the hypothesisH0:θ= 0, some exact test procedures are derived that combine the information from all the models. The procedures are extensions of the test suggested by Cohen and Sackrowitz for recovering interblock information in balanced incomplete block designs. Simulation results on the power of the new tests and some standard tests are reported in the context of (1) testing the hypothesis concerning the common mean of two univariate normal populations and (2) recovering interblock information in a block design that is not a balanced incomplete block design. The numerical results indicate that the new tests have excellent performance in terms of power compared to some standard tests.