We discuss testing for an interaction in two-way ANOVA with just one observation per cell. The known results are reviewed and a simulation study is performed to evaluate type I and type II risks of the tests. It is shown that the J. Tukey [Biometrics 5, 232–242 (1949)] and J. Mandel [J. Am. Stat. Assoc. 56, 878–888 (1961: Zbl 0109.12405)] additivity tests have very low power in case of more general interaction schemes. A modification of Tukey’s test is developed to resolve this issue. All tests mentioned in this paper have been implemented in R package additivity tests.