## 3 Results and Discussion
### Subsumption Checking  
The proposed method provides clear and understandable information about the inconsistencies found in the rule-base and verifies every rule with another set of rules to find redundancies. To ensure that the extracted PRs is consistent, this study focused on reducing subsumption and circular rules. Regarding [27], the proper rule and its corresponding sub-antecedent have the same consequent itemset, while the proper rule and its corresponding sub-consequent have the same antecedent itemset. In these conditions, a proper rule might have subset items in other rules, which can lead to the occurrence of redundant rules. Hence, it is necessary to eliminate either the proper rule or its corresponding sub-antecedent/consequent when the certainty factor of one rule is higher than the other. For instance, Table 4 shows an example of subsumption rules detected by rule IDs 14, 90, 134, and 350 which contained subset items of its proper rule (rule 350) and have the same consequent item set (_mri_ : _Abnormal_). Regarding certainty factor values of the rules shown in the table, we can see that all the rules in the sub-antecedent group are lower that its proper rule. This implies that a single proper rule (rule 350) is sufficient to express the knowledge or the meaning of the three sub-antecedent rules. Therefore, the proper rule is more comprehensive than its sub-antecedent rules. In other words, the proper rule could allow the support and improvement of the diagnosis of abnormal MRI result when more conditions are known, such as shown in rule 350: an elderly patient with prehypertension level of systolic and normal total cholesterol level, rather than each of those sub-antecedent conditions appears alone to diagnose MRI result '_abnormal_'. The proper rule could represent better knowledge with a quite high positive dependency between the conditions presented in the antecedent part and its consequent part indicated by certainty factor value of around 72%, which is also the highest amongst its possible sub-antecedent rules. Another example can be seen in Table 5, which shows that the program detected that rules 45, 46, 315, 319, 321, and 846 are subsumption. Rule 846 is basically the proper rule and rules 46, 315, 319, and 321 are its sub-consequences because they have the same antecedent items, but the items in their consequent part are a subset of the consequent part of the proper rule. In the case of subsumption redundancy rules, Table 5 shows that all sub-consequence rules have higher CF values than its corresponding proper rule. Since sub-consequence rules imply the same meaning as the proper rule, the proper rule 846 is considered to be redundant in the presence of its sub-consequence rules. Consequently, sub-consequence rules are retained and re-evaluated with the other rules while the proper rule is discarded. Different from the previous instance, the proper rule shown in Table 5 holds the least certainty factor value, around 16.4%, in comparison with its possible sub-consequent rules found in the generated rules. \begin{table}
\begin{tabular}{c c c c c} \hline
**Rule ID** & **Antecedent** & **Consequent** & **CF** & \\ \hline
14 & A: Elderly & MRI: Abnormal & 0.631 & sub-antecedent \\
90 & [A: Elderly \(\wedge\) SBP: Prehypertension] & MRI: Abnormal & 0.593 & sub-antecedent \\
134 & [A: Elderly \(\wedge\) TC: Normal] & MRI: Abnormal & 0.662 & sub-antecedent \\
**350** & **[A: Elderly \(\wedge\) SBP: Prehypertension \(\wedge\) TC: Normal]** & **MRI: Abnormal** & **0.722** & **proper rule** \\ \hline Notes: & & & & \\ A & : Age & & & \\ SBP & : Systolic Blood Pressure & & & \\ TC & : Total Cholesterol & & & \\ MRI & : Magnetic Resonance Imaging & & & \\ \hline \end{tabular}
\end{table} 
Table 4: Example of subsumption rules (proper rule and its sub-antecedents) (1).