## 6 Related Work
This approach, however, does not support any statistical validation of the discovered associations. To address this issue, several approaches have been proposed. Some focus on mining non-redundant rules by comparing each rule with its generalizations and discarding those rules, not showing any improvement in terms of support [44] or confidence [8]. Other approaches, instead, fall within the field of _statistical association rule mining_, i.e., they focus on mining statistically significant positive associations, ensuring that these associations are unlikely to be due to chance. Some approaches apply statistical tests to rule assessment metrics, e.g., the confidence intervals [45]. Statistical tests have also been used to validate feature-class correlations, filtering rules involving features with an insufficient/not significant level of correlations [9], or assessing possible correlations between itemsets and the class by means of hypothesis testing (e.g., [46, 47, 48]). While these approaches do improve the statistical robustness of the discovered set of rules, they come with some drawbacks. Approaches exploiting relevance metrics only indirectly assess the statistical significance of the impact of the feature values on the class values. Moreover, the use of different metrics can lead to different results, and some commonly-used metrics also come with some drawbacks. For example, it is well known that rules whose consequent has a high support tend to have a high value of confidence, without this implying a real dependency among the antecedent and the consequent [46]. Furthermore, confidence allows measuring only one direction of the impact of feature values, i.e., positive correlations. In addition, all these approaches are mainly intended as filtering mechanisms. Overall, the support provided to the statistical validation of discovered association rules is still limited and not as mature as in other techniques, such as regression analysis. Indeed, the scope of the performed evaluation is only limited to the values of the explanatory variables that occur in the rule under analysis. Little or no support is provided to explore how the impact changes with respect to the different values of the variables. The combination of associations rules and regression analysis is mostly unexplored; to date, only a few approaches have investigated potential advantages and applications. For instance, [PERSON] and [PERSON] [49] investigated the use of association rule mining to detect the set of the most interesting interactions to take into account when building a regression model. However, rule mining is only used to identify interactions among features. Moreover, only a single model is returned, which does not allow differentiating among different possible contexts in which discrimination could have occurred. It is also worth noting that a very aggressive rule filtering mechanism is adopted, so that only correlations with a strong support are considered, which is not always desirable when dealing with biases that involve small portions of the overall population. Other approaches exploit rule mining in the iterative building of the (best) regression model. For example, [PERSON] [50] defines so-called _polynomial association rules_ to determine non-linear correlations among a set of (continuous) features and the class, and use an iterative regression model-building procedure that picks the best polynomial rule at each step to determine the factor to include in the regression model. Furthermore, in this case, the output consists of a single, 'optimal' model; moreover, polynomial rules are targeted to numerical domains. Other approaches combine their predictive capabilities in a single hybrid system to enhance classification performance. For example, [PERSON] et al. [51] showed an application to determine faulty modules, where samples described by a (set of) rules are classified accordingly, while samples for which no rules are available are classified according to a regression model.