## 4 Experiments
### Approach Validation  
#### 4.1.1 Dataset and Settings  
For the validation of our approach, we generated synthetic datasets to contain a known 'amount' of evidence of discrimination. To this end, we defined a simple decisional process regarding the hiring of candidates on the basis of their personal characteristics. Table 4 shows the variables characterizing the hiring process along with their domain. We consider the variable Age as a 'discriminatory variable', whereas the others are considered 'context variables'. We generate the synthetic data in two steps. First, we created the discriminated groups as groups of subjects sharing the same value of Age as well as a (randomly chosen) subset of context values; the values of the other context variables were randomly assigned from the respective domains. Discriminated subjects have a probability of 80% of being assigned to class \"N\". Second, we generated all 'non-discriminated' subjects simply by randomly selecting a value for each context variable and for the Age variable. Non-discriminated subjects have a 50% probability of being assigned to either the \"N\" or \"Y\" class.  
In total, we generated 12 datasets by varying the number of discriminated groups (i.e., 1, 2 and 3) and the complexity of the dataset to test our methodology in different situations. The complexity of a dataset is defined over two dimensions: (1) presence/absence of _noise_, intended as subjects that do not belong to any of the generated context groups but in which one or more context variables assume a value used in one of the discriminated groups; and (2) the presence/absence of _overlapping_, meaning that subjects in two or more discriminated groups share at least a context variable and its value. Combining these two dimensions, we obtain four types of datasets: (i) without noise and overlapping, which represents the simplest situation; (ii) without noise but with overlapping; (iii) with noise but without overlapping; (iv) with noise and overlapping, which represents the most difficult situation to  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Variables** & **Variable Domain** \\ \hline Education & Doctorate, Master, Bachelor, HighSchool \\ Speak Language & Y, N \\ Previous Role & Employee, Manager, Self-Employed, Unemployed \\ Country & USA, Europe, SA, China, India \\ Age & 25–50, 50+ \\ Class & Y, N \\ \hline \hline \end{tabular}
\end{table}
Table 4: Variables used for the generation of the synthetic dataset along with their domain.  
deal with, since spurious correlations can easily arise (note that the presence of overlapping only has an impact when more than one discriminated group exists). For every dataset, we generated a total of 10,000 subjects; among them, every discriminated group covered 25% of the dataset. We chose 25% to strike a balance between absolute minority (\(<\)50%) and small groups. For every dataset, we tested several configurations of support, which varied between 1% and 10% with a step of 1%, and confidence, which varied between 50% and 95% with a step of 5%. Note that when generating the regression models, we did not consider potential interactions among the variables in the experiments; namely, we used only factors of the first order when computing the regression models. While this choice can lead to lose some interesting correlation, it provides us with a good approximation of the relations characterizing the decisional process, and prevents the generation of noisy recommendations. Directionality is given by the sign of the corresponding coefficients.