We derive Neyman's locally asymptotically most powerful test criterion to test a linear hypothesis, and in particular, obtain such criteria to test linear hypotheses in binomial and multinomial experiments. We also discuss testing log-linear hypotheses in the parameters of a multinomial experiment, and give a simple criterion for testing the hypothesis of no second order interaction in ap×q×r(p, q, r, > 2) contingency table. Asymptotic tests of the hypotheses mentioned above are available in the literature, but the tests proposed in this paper are simple and are based on explicit optimality consideration.