The video explains the necessity of balancing the influence of variable groups in Multiple Factor Analysis (MFA) and presents two weighting strategies. It first critiques weighting by total group inertia, showing that it can let a unidimensional group dominate the analysis. It then introduces weighting by the square root of each group's largest eigenvalue, which equalizes the main axis of each group and preserves within‑group structure while allowing multidimensional groups to contribute across several dimensions. The speaker outlines the practical steps: compute the first eigenvalue for each group, divide each variable by the square root of that eigenvalue, and run a PCA on the concatenated weighted tables. A wine‑sensory dataset with three juries (experts, students, consumers) illustrates the method; eigenvalues for each jury are reported, and the effect of weighting on the MFA dimensions and on individuals/variables plots is described. The individuals plot reveals a clear separation between two grape varieties, while the variables plot (circle of correlations) shows shared and divergent sensory descriptors across juries. The lecture emphasizes that MFA retains the group structure, provides standard PCA‑type plots, and allows supplementary elements, setting the stage for later discussion of group‑specific contributions.
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We saw in the previous video that the data to which multiple factor analysis can be applied is that where the variables are structured as groups. We've seen that it's important to balance the influence of each group of variables in the analysis. Let's now look at how to do this and how to implement it in reality. First, why is it important to balance the influence of each group of variables in the analysis? Note that we have already tried to balance the influence of variables when doing factor analysis before. Indeed, when constructing a standardized PCA, the influence of each variable is balanced to avoid that variables with the greatest variance have more influence in the analysis, that is, more influence in the calculation of distances between individuals. In a similar way, with several groups of variables, We want to balance the groups in order to avoid certain groups having a too large contribution to the calculation of distances between individuals. An initial idea, which approximates what happens with PCA, consists in balancing the influence of each group of variables by dividing by its inertia. Indeed, the inertia is a kind of variance in the multidimensional framework. We could therefore weight the groups so that each group has an equivalent inertia. To do this, it's sufficient to weight each variable by the total inertia of the group to which it belongs. With such a weighting, if 2 groups don't have the same number of variables, the group with less variables will have the same influence as the group with more. From this point of view, balance is attained. However, this weighting doesn't take into account the method used here: factor analysis. Indeed, factor analysis is sensitive to the distribution of the inertia from one group to another. In the example shown in this small diagram, a first group is composed of 8 highly correlated variables. The second group is composed of 3 orthogonal variables. Lastly, the third group is identical to the second. With the weighting by the total inertia of each group, the first dimension of group 1 will have a weight close to 1. This group is unidimensional and nearly all of its inertia is located in only one dimension, while the first dimension of groups 2 and 3 will have a weight of 1/3. If the total inertia of the group is 1, each variable represents 1/3 of the inertia. Consequently, the first dimension for the overall analysis will coincide with the first dimension of the first group. But we are looking for some kind of balance in order to find common structures from one group to another. So we would in fact like to have appear the shared structure of groups 2 and 3. So a second possible idea consists in balancing the groups no longer as a function of a total inertia, but as a function of the maximum inertia of a dimension. Thus, we can avoid a unidimensional group having too much importance in the analysis, and the richness provided by a multidimensional group, that is, a group with several axes of variability, is not missed or hidden. To construct such an equilibrium, we divide the values of a variable by the largest eigenvalue of the group to which it belongs. In fact, we divide by the square root of the largest eigenvalue. In our little example, this weighting makes it possible to see the share structure generated by groups 2 and 3. This small made-up example corresponds to a situation that is relatively common in practice: some unidimensional groups along with some multidimensional groups. The proposed weighting avoids giving too much importance to unidimensional groups which contain less rich information than the others. Here we have found a weighting that we like and it remains only to look for the main principal components as we have seen for PCA and MCA and CA. The difficulty lies in the choice of weighting, as has been explained by Jean-Paul Belzecry, who has said that doing factor analysis, from a mathematics point of view, is just matrix diagonalization, but the science and art of it is to find the right matrix to diagonalize. The weighting we have found amounts to using, for each variable, weight equal to the inverse of the first eigenvalue of the group to which it belongs. Thus, the first step of the analysis consists in calculating the first eigenvalue of each group. Then, weighting a variable by the inverse of the inertia of the first eigenvalue of the group to which it belongs amounts to dividing each variable by the square root of the first eigenvalue of its group. In the notation used here, X_j corresponds to the centered or centered and standardized table j depending on whether the variables of the group j are normalized or not. Lastly, we just have to do a PCA of the table which juxtaposes the weighted variables of all the tables. For our wine dataset, the PCAs constructed on the data of each jury give the following eigenvalues. The first eigenvalue of the jury of experts, equal to 11.74, is greater than the first eigenvalues of the student and consumer juries. This can be explained by the fact that a greater number of descriptors were evaluated by the jury of experts. We also note that the following eigenvalues, eigenvalues 2 and 3, are larger for the jury of experts. If a PCA were to be constructed on the whole dataset without first weighting the juries, the jury of experts would strongly influence the results, in the sense that the first dimensions of the analysis would be very much related to the description of the jury of experts. By balancing the influence of each jury through a weighting by each group's first eigenvalue, Each jury will make the same contribution to the construction of the MFA dimensions. The first point to notice is that the weighting chosen is the same for all variables in a group. This leads to preserving the group structure. In other words, within a group, the equilibrium between variables is preserved. The ratio lambda_2 over lambda_1 remains the same: 6.78 divided by 11.74 equals 0.58 divided by 1. Since the equilibrium within a group remains the same, it is also possible to standardize, if desired, the variables in a group depending on, on the equilibrium we want to have within the group. This won't upset the balance between groups. More precisely, thanks to the weighting, the first eigenvalue of each group is made equal to 1 and therefore the main axis of variation has the same variation from one group to the next. Thus, one group cannot generate on its own the first MFA dimension, even if it has many variables and is highly structured. One last important thing to note: this weighting sets to 1 the main dimension of variability, and the following eigenvalues are decreased proportionally. Thus, a more multidimensional group will contribute to more dimensions because its overall inertia after reweighting is greater. A more multidimensional group contains more information, so it is normal that it contributes to more dimensions. Now, is it a problem that a group has a greater total inertia than the others? In fact, no, because this total inertia is spread out over more dimensions. We rediscover here a feature that we have already seen in multiple correspondence analysis. We have seen that an MFA can be considered as a PCA on the weighted table that juxtaposes the data from all the groups. Since it's a global PCA, we'll first obtain the same plots as in PCA: an individuals plot and a variables plot. The individuals plot helps us to visualize the overall similarities between individuals taking into account information on all of the variables. The variables plot makes it possible to visualize the relationships between all the variables. And of course, these 2 plots can be studied simultaneously in order to understand the similarities and differences between individuals due to the variables. In addition to the plots, indices for quality of representation and contribution are also the same as in PCA. Also, as with any PCA, it's possible to add supplementary elements, for instance, supplementary individuals or additional variables, be they quantitative and/or qualitative. For our example, here is the individuals plot. The wines are colored according to the grape variety: red for Sauvignon and green for Vouvray. Coloring the individuals according to grape variety shows a contrast between the 2 grape varieties. The Sauvignons are in the upper left corner and the Rouvray in the lower right. Recall that the grape variety variable was not an active one and was therefore not used to construct the dimensions. This contrast between grape varieties is therefore closely related to to the sensory description variables evaluated by the juries. We also note that the Vouvray are more spread out in the plane, which can be interpreted as follows. The Vouvray have more variety from a sensory point of view as compared with the Sauvignon, which are more homogeneous. One final note, we can see several groups of wines, for example, the Aubusières-Marnies and Fontenay-Coteaux Vouvray are very similar sensorially speaking, as are Fontenay Domaine and Fontenay-Brûlés. The variables plot with the circle of correlations is shown here. The variables were colored according to the group they belong to. In red, the variables used by the experts with just the name of the descriptor. In green, the variables used by the students with the name of the descriptor followed by index S. And in blue, the variables used by the consumers, with the name of the descriptor followed by index c. Clearly, this plot is overloaded with information, and it is difficult to read because there are so many variables. But remember, we have several groups of variables, and we have represented them all. So this graph summarizes a lot of information. That's why it's so overloaded. So, in order to study this graph more easily, we can select certain variables for better visualizing. The graph is therefore the same as before, we have just masked some labels, highlighting a few of them shared by the different juries. We can see for example that the variables passion fruit odor are quite related, meaning that from one jury to the next, this variable was experienced in a similar way. The same is true for sweetness. On the other hand, the acidity variable was experienced differently. The consumers and students experienced this variable in the same way, but differently to the experts. The correlation coefficient between the acidity variable for the expert jury and the consumer jury is 0.34, while that between the acidity variable for the student jury and the consumer jury is 0.76. We cannot know who correctly understood the acidity variable, but it's possible that the consumer and student juries had more difficulty in assessing acidity when the sweetness-acidity balance changed. We are not going to go further into the interpretation of this variables plot with all its richness and which combined with the individuals plot makes it possible to have a sensory description of the wines obtained from the 3 juries. These 2 plots are entirely equivalent to those obtained using PCA, but MFA benefits from the group structure, which we have not taken advantage of before except in weighting the groups. We will see in the next video the contribution this group structure brings to the story A contribution which is specific to MFA and which provides much of its interest.
