Graphical tests of independence for general distributions

We propose two model-free, permutation-based tests of independence between a pair of random variables. The tests can be applied to samples from any bivariate distribution: continuous, discrete, or mixture of those, with light tails or heavy tails. Apart from the broad applicability of the tests, their main benefit lies in the graphical interpretation of the test outcome: in case of rejection of the null hypothesis of independence, the combinations of quantiles in the two marginals are indicated for which the deviation from independence is significant. This information can be used to gain more insight into the properties of the observed data and as guidance for proposing more complicated models and hypotheses. We assess the performance of the proposed tests in a simulation study and compare them with several well-established tests of independence. We observe that for monotone dependence structures, the proposed tests are competitive with most benchmark methods. In contrast, for non-monotone dependence structures, the proposed tests usually outperform the benchmark tests. Furthermore, we illustrate the use of the tests and the interpretation of the test outcome in two real datasets consisting of meteorological reports (daily mean temperature and total daily precipitation, having an atomic component at 0 millimeters) and road accidents reports (type of road and the weather conditions, both variables having categorical distribution).