Summary: We investigate the effect of positive dependence (in particular, of positive correlation) between observed data on the tests of simple and composite hypotheses based on disparities between empirical and theoretical probabilities of fixed bins in the observation space. We show that the experimental as well as theoretical conclusions achieved previously for the Pearson chi squared tests remain valid in the whole class of disparity tests under consideration. In the case of positively correlated Gaussian observations we investigated by the Monte Carlo method the effect of dependence on the power divergence tests, where the test of power 3 emerged as the most resistant to this undesirable effect.