In this chapter, we consider the problem of testing the goodness-offitof either one of two location-scale families of density when these parametersare unknown. We derive an O(n-l) approximation to the densities of the maximalinvariant on which the most powerful invariant test is based. The resultingtest, which we call quasi most powerful invariant, can be applied to many situations.The power of the new procedure is studied for some particular cases.