Choosing between two income distribution models typically involves testing two non-tested hypotheses, that is hypotheses such that one cannot be obtained as a special or limiting case of the other. Cox (1961, 1962) proposed a classical testing procedure based on the comparison of the maximised likelihood functions for the two models. In this paper it is shown that such a procedure is not robust in that a single observation can reverse the decision. Its robustness properties as well as other properties are shown in simulated examples