The post-hoc detection of dependence

The concept of independence plays a crucial role in probability theory and has been the subject of extensive research in recent years. Numerous approaches have been proposed to validate this dependency, but most of them address the problem only at a global level. From a practical perspective, it is important not only to determine whether the data is dependent, but also to identify where this dependence occurs and how strong it is. We introduce a new method for testing statistical independence using the quantile dependence function. Rather than assessing whether the value of the test statistic exceeds a single critical threshold and subsequently deciding whether to reject the independence hypothesis, we use so-called critical surfaces that guarantee locally equal probability of exceeding it under independence. This approach enables a detailed examination of local discrepancies and an assessment of their statistical significance while preserving the overall significance level of the test.