# Multiple Factor Analysis Based on NIPALS Algorithm to Solve Missing Data Problems
## 2 Methods
### Nonlinear Iterative Partial Least Squares
In particular, a high correlation was obtained in periods 1 and 2 with Spanish, as well as a moderate correlation between Mathematics and Natural Sciences in the first academic period and a low correlation between subjects in the third period, which was, for example, observed with MFA with complete data. Figure 6 shows the behavior of individuals in the first factorial plane, where similar patterns of a complete data case for students 9, 21, 26, and 42 are highlighted and who highly contributed to the axis, as they are far from the gravity center. Moreover, high variability was detected, which could be produced by the generated NAs of the dataset. Figure 4: Individual factor map of MFA with complete data. Figure 3: Correlation circle of MFA with complete data. Figure 5: Correlation circle of MFA-NIPALS algorithm with 10% of NAs. Figure 6: Individual factor map of MFA-NIPALS algorithm with 10% of NAs. #### 3.1.2 Simulation Scenarios

In this section, results with complete data versus MFA-NIPALS estimates are compared. Table 2 shows the percentage of variance explained on axes 1 and 2. These percentages increased when the number of NAs rose. Moreover, the percentages of variance explained by MFA-NIPALS were higher than RIMFA ones, which could be influenced by an increment of eigenvectors on axes 1 and 2. In this comparison, RIMFA holds a percentage of variance explained close to MFA with complete data. Table 3 presents the coordinate correlations with complete data and those estimated by MFA-NIPALS and RIMFA. On axis 1, the highest correlations were detected for MFA-NIPALS and for 30% of NAs, where the RIMFA estimate differed sharply when compared to complete data. For the correlations of axis 2, it can be observed that RIMFA obtained better results than the complete data case. However, a less favorable result was obtained in the case of 30% of NAs. In summary, MFA-NIPALS performed well on axis 1 and regular on axis 2, indicating that MFA-NIPALS is a good alternative, but more simulation analysis is required to gauge the statistical and computational advantages of MFA-NIPALS versus RIMFA. Figure 7 shows the correlation between the first component of individuals of classic MFA (\(\mathbf{\psi}_{1}\)) and first component of MFA-NIPALS (\(\mathbf{t}_{1}\)). The latter analysis was made with 20 matrices randomly generated for several percentages of NAs. When the percentage of NAs increased to 20%, the median of correlations went farther from 1, indicating less correlation between classic MFA and MFA-NIPALS at 20%, 25%, and 30%. Nevertheless, correlation medians until 15% of NAs were closer to 1, indicating that MFA-NIPALS facilitated favorable results based on estimates related to component \(\mathbf{t}_{1}\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Percentage of NAs** & **MFA-NIPALS** & **RIMFA** \\ \hline
5\% & 46.89\% & 42.46\% \\
10\% & 50\% & 43.3\% \\
15\% & 51.25\% & 43.66\% \\
20\% & 56.10\% & 50.89\% \\
25\% & 60.41\% & 53.02\% \\
30\% & 59.13\% & 52.24\% \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Percentage of variance explained on axes 1 and 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Axis** & **Percentage of NAs** & **MFA-NIPALS** & **RIMFA** \\ \hline
1 & 5\% & 0.8789 & 0.9261 \\  & 10\% & 0.9395 & 0.9505 \\  & 15\% & 0.9094 & 0.9527 \\  & 20\% & 0.8920 & 0.8986 \\  & 25\% & 0.8166 & 0.7391 \\  & 30\% & 0.7813 & \(-\)0.0846 \\ \hline
2 & 5\% & \(-\)0.7918 & 0.9490 \\  & 10\% & \(-\)0 On the other hand, Figure 8 shows the correlation of the second component of classic MFA (\(\psi_{2}\)) and the second component of MFA-NIPALS (\(t_{2}\)). The correlations are close to 0.8 and in cases until 15% of NAs, whereas above 15%, the lowest correlation is observed, indicating that MFA-NIPALS provided a suitable estimation of the second component \(t_{2}\) until 15% of NAs. Figure 8: Correlation of complete data versus MFA-NIPALS with NAs on axis 2. Figure 7: Correlation of complete data versus MFA-NIPALS with NAs on axis 1.