# MIPART: A Partial Decision Tree-Based Method for Multiple-Instance Classification
## 4 Experimental Studies
### Statistical Results  
To evaluate the effectiveness of the proposed MIPART method, we compared its performance with several well-known multi-instance classification techniques, including Citation-KNN, MIBoost, MIEMDD, MILR, MISVM, and MITI, shown in Table 3. The comparisons were conducted using 12 diverse multi-instance datasets, described in Table 2, to ensure a thorough and reliable analysis. For statistical validation, we applied non-parametric tests, specifically the Friedman aligned ranks [46] and Wilcoxon signed-rank [47] tests. The Friedman aligned ranks test, designed to rank datasets and facilitate robust algorithm comparisons, produced a \(p\)-value of 0.00001 at a 0.05 significance level. This indicates that MIPART's performance improvements are statistically significant. The mathematical formula for the Friedman aligned ranks test is detailed in Equation (8), as follows:  
\[X_{F}^{2}=\frac{12}{Nk(k+1)}\sum_{i=1}^{k}R_{i}^{2}-3N(k+1) \tag{8}\]  
Figure 5: PRC area results for the MIPART method across twelve multi-instance datasets.  
where \(N\) is the number of datasets used in the analysis, \(k\) represents the number of algorithms being compared, and \(R_{i}\) is the sum of the ranks assigned to algorithm \(i\) across all datasets. The Friedman test statistic \(X_{F}^{2}\), is computed based on these values to determine if the differences in ranks between the algorithms are statistically significant. The Friedman aligned ranks test assigns ranks to all datasets simultaneously, avoiding the assumption of normality and making it suitable for non-parametric datasets. It is particularly effective when comparing a small number of algorithms, as in our study, and ensures comparability by considering datasets as independent.  
In addition to the Friedman test, we also performed the Wilcoxon signed-rank test to compare the performance of MIPART with its benchmark methods. The Wilcoxon signed-rank test is a non-parametric method that assesses whether the median difference between two related samples is zero, which complements the Friedman test by providing more detailed vision into algorithmic performance. The results of this comparison are presented in Table 4, where we show the Wilcoxon signed-rank \(p\)-values for each method, all of which are below the 0.05 significance threshold, confirming the superiority of MIPART in terms of predictive accuracy.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Method** & **Wilcoxon \(p\)-Value** \\ \hline Citation-KNN & 0.00222 \\ MIBoost & 0.00221 \\ MIEMD & 0.00335 \\ MILR & 0.00222 \\ MISVM & 0.00335 \\ MITI & 0.00335 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Results of the Wilcoxon signed-rank test comparing the performance of MIPART with its competing methods.  
Figure 6: ROC area results of the MIPART method on twelve multiple instance datasets.  
The mathematical formula for the Wilcoxon signed-rank [47] test is represented in Equation (9), as follows:  
\[W=\sum_{i=1}^{n}sign(d_{i})\times R_{i} \tag{9}\]  
where \(n\) represents the number of paired samples, \(d_{i}\) is the difference between the paired values for each sample, and \(sign(d_{i})\) is the sign of \(d_{i}\), indicating whether the difference is positive, negative, or zero. \(R_{i}\) is the rank assigned to the absolute value of each difference, which is used to determine the magnitude of the difference. The Wilcoxon statistic \(W\) is then used to measure whether the differences between the paired samples are statistically significant. The Wilcoxon signed-rank test orders the absolute differences between paired observations, considering the direction of the differences, namely positive or negative. It is particularly suitable when the normality assumption for differences cannot be guaranteed and provides a strong alternative to parametric tests, ensuring reliability in the presence of non-normal data distributions. Additionally, it assumes the symmetry of differences around the median.