## 4 Experiments
### Approach Application to Two Real-Life Use Cases
#### 4.2.1 Use Case on Credit Risk  
**Dataset and Settings. ** The German dataset consists of 1000 records representing the assessment of credit risk (good or bad) of bank account holders [15]. The dataset encompasses 21 variables, grouped according to the following categories: personal properties (checking account status, duration, savings status, property magnitude, type of housing), properties related to past/current credits and requested credit (credit history, credit request purpose, credit request amount, installment commitment, existing credits, other parties, other payment plan), properties related to the employment status (job type, employment since, number of dependents, own telephone), and personal attributes (personal status and gender, age, resident since, foreign worker). We discretized the numeric attributes as suggested in [4]. Following [17], we considered the decisional process to be affected by discrimination if the final decision was influenced by the fact that the holder belongs to one or more of the following subgroups: non-single female, older than 52 years and foreign worker. **Results. ** Table 7 reports the descriptive statistics for the 80 experiment runs for varying levels of support (\([0.03,0.10]\) with steps of 0.01) and confidence (\([0.5,0.95]\) with steps of 0.5). We observe that the number of rules in the output is higher but resembles the same distribution as the number of derived regression models. Absolute values are, as one would expect, much higher for real datasets than for the experiment with synthetic data. The median experimental setting produces 5971.5 rules and 231.5 models. The stable ratio of models to rules between the two settings (synthetic and real) suggests that the pre-conditions for the two experiments are comparable. If we focus on the significant, unique regressors, we observe a stronger reduction; indeed, thanks to model selection, we obtain a median and a maximum of 26.5 and 32 regressors to consider (as opposed to, respectively, 5971.5 and 77,219 rules). Figure 6 reports the relation between regressors and rules across experiments. We can observe that while the results obtained using rule mining vary significantly for different support and confidence settings, regressors turn out to be quite stable, with variations of the order of a few dozens across all experiments. Furthermore, the number of rules exceeds the numbers of regressors of a factor ranges from 10 to 1000, depending on the support level. \begin{table}
\begin{tabular}{l r r r r r r r} \hline
**Metric** & **Min** & **1 st Q** & **Mean** & **Med.** & **3 rd Q** & **Max** & **sd** \\ \hline \#rul & 54.0 & 2894.0 & 14,668.1 & 5971.5 & 19,166.5 & 77,219.0 & 20,004.6 \\ \#mod & 12.0 & 117.25 & 373.1 & 231.5 & 491.25 & 1560.0 & 386.3 \\ \#regr & 12.0 & 23.8 & 25.3 & 26.5 & 28.2 & 32 & 4.7 \\ \hline \end{tabular}
\end{table} 
Table 7: Descriptive statistics of the experiment runs for the credit dataset. Figure 7 reports the (log) ratio between models and rules for each experimental setting. At every level of support and confidence, the number of regressors is, on average, notably smaller than the number of rules. The stability of the results given by our approach is consistent with what we observed in Section 4.1. Nevertheless, the reduction is especially strong at a low level of confidence and support. These results point out that the number of regressors remains manageable for being analyzed by a human policy maker, whereas the number of rules explodes. This suggests that our approach can be very effective in practice to obtain usable and statistically significant indications of biases in the data without the need to fine-tune the support/confidence levels in input to rule mining. The overhead in terms of output from low support and confidence levels is limited, whereas one is not incurred in the risk of removing potentially relevant rules.