Tests of simple and composite hypothesis for multinomial distributions are considered. It is assumed that the size $\alpha_N$ of a test tends to 0 as the sample size $N$ increases. The main concern of this paper is to substantiate the following proposition: If a given test of size $\alpha_N$ is "sufficiently different" from a likelihood ratio test then there is a likelihood ratio test of size $\leqq\alpha_N$ which is considerably more powerful than the given test at "most" points in the set of alternatives when $N$ is large enough, provided that $\alpha_N \rightarrow 0$ at a suitable rate. In particular, it is shown that chi-square tests of simple and of some composite hypotheses are inferior, in the sense described, to the corresponding likelihood ratio tests. Certain Bayes tests are shown to share the above-mentioned property of a likelihood ratio test.