Additivity is an important assumption in the analysis of data from a randomized complete block design or other two-way classifications. Statistical tests for detecting non-additivity have been proposed by Tukey and Mandel. These tests have been developed under the usual assumptions that the errors are uncorrelated and the error variances are homogeneous. Often, however, these assumptions are obviously violated with error variances being heterogeneous or errors being correlated or both. The effect of departures from the usual assumptions on Tukey's test and on the tests of concurrence, non-concurrence, and slopes proposed by Mandel are investigated in this thesis. Tukey's test and the test for concurrence are similar in that both test the same hypothesis but use a different estimate for experimental error;The effect of heterogeneous error variances on tests of non-additivity is that the actual levels of significance are different from those obtained when the error variances are homogeneous. This difference can be substantial and is not always in the same direction. The degree the tests are affected does not simply depend on the value of the variances or how the variances are arranged. Other factors, such as the number of rows, row effects, and column effects, together with the variance heterogeneity affect the tests. In general, it can be concluded that the test for concurrence is affected by these factors to a much lesser degree than for slopes and non-concurrence. Tukey's test generally performs well, but can be affected greatly for some combinations of variances and column effects;Correlated errors also affect the actual levels of significance for the tests of non-additivity. The correlation structures investigated are first order serial correlation and first order autoregressive. These were chosen to represent how repeated measures in time might affect tests of non-additivity. Correlated errors can affect the actual levels of significance substantially. The departure from the desired levels of significance can be in either direction depending on the column effects, and the magnitude and sign of the correlation parameter. It was shown that the direction of the departure can change when different column effects are used with the same correlation structure. In general, the tests for slopes and non-concurrence are affected much more than the test of concurrence. Tukey's test performs well, but overall is affected more than the test of concurrence.