  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, \ldots The tests take advantage of the recent development
of the global envelope tests in the context of spatial statistics. 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 a guidance for proposing more complicated models and
hypotheses. We assess the performance of the proposed tests in a simulation
study and compare them to several well-established tests of independence.
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).
