The video concludes the series on Multiple Correspondence Analysis (MCA) by presenting a suite of interpretation aids common to all principal‑component methods and detailing how to incorporate supplementary information. It explains that the inertia of an MCA dimension equals the mean of the squared correlation ratios between that dimension and the variables, which bounds inertia and contributions between 0 and 1. Because individuals are embedded in a high‑dimensional space, percentages of inertia are typically low; a rule‑of‑thumb (inertia < 1/J) helps decide how many dimensions to retain. The speaker describes how to compute contributions and quality of representation, noting that many dimensions lead to weak per‑dimension quality. Supplementary elements (individuals or categories) are introduced via transition formulas and barycentric properties, allowing their coordinates to be projected onto the active space. The lecturer demonstrates interpreting an MCA plot by first using only active variables, then adding supplementary variables, and provides concrete narrative examples (young white‑collar vs. older leisure‑oriented groups). Quantitative supplementary variables are handled through correlation circles, and statistical testing (ANOVA, F‑test, t‑test) is suggested for assessing variable‑dimension links. The Burt table is presented as a full pairwise contingency matrix; running correspondence analysis on it yields the same dimensions as MCA (with eigenvalues squared). The video emphasizes that MCA reduces qualitative data to synthetic quantitative dimensions, making it valuable for survey analysis, clustering, and as a preprocessing step. The tutorial ends with a recap of key concepts and a pointer to a practical FactoMineR demonstration.
Chapters
So far, we've seen how to construct point clouds of individuals and categories in MCA. In this video, we are going to show you some interpretation aids shared by all principal component methods and also look at how to use supplementary information, including supplementary variables, in MCA. The inertia of a dimension in MCA is special because it's equal to the mean of the squares of the correlation ratios between the dimensions and the variables. This property validates our way of interpreting the synthetic variables defined by the MCA dimensions. So these MCA factors are quantitative variables which summarize the qualitative ones. Note also that, as the squares of the correlation ratios are between 0 and 1, the contribution of a given variable to a given dimension is bounded by 1. Furthermore, the mean of the correlation ratios, and thus the inertia, is also found between 0 and 1. Now let's look at the percentages of inertia. These are generally lower than in PCA and correspondence analysis because the individuals are located in a high -dimensional space, which gets bigger and bigger as the number of categories per variable increases. The following little calculation shows what the percentages of inertia associated with given dimension tend to be small. We calculate the inertia of a dimension lambda s over the total inertia, and since the eigenvalue lambda s is less than one, we find that the percentage of inertia has to be less than j over k minus j. Multiplied by 100. So, with 10 variables and 10 categories per variable, even if the variables are strongly linked, the maximal percentage of inertia that can be in a given dimension is 11%. Another quick calculation shows that the mean inertia equals 1 over the number of variables. This value can help us decide how many dimensions to interpret in MCA. Indeed, we can forget about interpreting dimensions with inertia less than 1/J. Interpretation aids like contribution and quality of representation can be calculated as for other principal component methods like PCA and correspondence analysis. However, the projection quality tends to be weak on each dimension, which is normal because there are many dimensions. As for the categories, as a weight is associated with each of them, the most important contributors are not necessarily the most spread out points on the graph. The contribution of a given variable to the construction of a given dimension can be calculated by summing the contributions of all of its categories. This contribution is equal to the square of the correlation ratio between the dimension and the variable, divided by the number of variables. To calculate the relative contribution of each variable, we divide by the inertia of the dimension lambda_s. We can now turn to looking at how to use supplementary information to interpret the simultaneous representation graph. To do this, we're going to use the transition formulas to calculate the coordinates of the supplementary elements. By element, we mean either a row or a column of the table. So here, this means either an individual or a category. In any principal component method, an element is labeled supplementary if it hasn't been used to build the dimensions. Note that, for categories, all the categories of a given variable must have the same status. They all have to be active or all supplementary. This is why we label variables active or supplementary. Barycentric properties help us to calculate the position of a supplementary individual or category. Let's take for example a supplementary category. Once the positions of the individuals have been set in stone, we can calculate the barycenter of any subset of them. In practice, MCA almost always comes with supplementary variables, as these provide context to what we are doing, and are therefore extremely precious to us. This plot shows the categories of the supplementary variables. In order not to put too much on the same plot, we don't show the active categories or the individuals. However, individuals as well as the active or supplementary categories can be put on the same plot. We see that the supplementary variables are linked to the dimensions. For instance, the groups of edges appear in a rising fashion from bottom right up to the top and around to the left. Similarly, professions and marital status categories spread out in certain ways along the dimensions. Moving on to the interpretation of the MCA plot, we can say that towards the bottom right, We have the younger individuals, mostly white-collar workers, living alone with lots of activities like cinema, shows, and computers. At the top, the individuals are older with activities like fishing, knitting, and gardening. And lastly, to the left, we find manual laborers and blue-collar workers, relatively older with few leisure activities. These interpretations involved a 2-step process. First, we used only active variables, and then we brought in the supplementary ones. Also, in our set of interpretation aids, we include the notion of supplementary quantitative variables. In MCA, active variables are by definition qualitative, so quantitative variables must be by definition supplementary. In our example, we were able to interpret the first diagonal in terms of the number of leisure activities undertaken. From which comes the idea to literally create a quantitative variable called number of activities per individual and see what it looks like. To visually represent this variable, we're going to proceed like for PCA and use the correlation circle representation. Each variable is represented by its correlation coefficient with the dimensions. This gives us the following graph, which shows that the number of activities performed is perfectly characterized by the plane defined by the first 2 dimensions. Essentially, it corresponds to the first diagonal, which comforts our initial interpretation. Note that if we do want to make a quantitative variable active, We can discretize it into classes, making it qualitative, and it can then be used as an active variable. Like in PCA, we can characterize the dimensions in MCA using active or supplementary variables, and quantitative or qualitative ones. For quantitative variables, we can calculate the correlation coefficient between each of them and a given dimension, and then sort them by importance. Here, we see that the variable number of activities is strongly related to the first dimension. For qualitative variables, we can construct an analysis of variance model for each variable, and then describe the coordinates in a given dimension in terms of the given variable. We can also do an F-test to have a look at the overall link between the variable and that dimension, and do a Student t-test to see which categories have specific coordinates in each dimension. Here, all of the p-values are extremely small as there are a lot of individuals in this dataset. There is another way to present the data when we have a set of qualitative variables, which is called a Burt table. This table sets up the whole set of variables against themselves. In the table we show here, each block is a contingency table between 2 variables. The rows and columns of each contingency table are the categories. This means that a BERT table puts up the categories of all variables against themselves. The BERT table summarizes all of the pairwise relationships between variables. In a way, it's analogous to a correlation matrix which brings together all the pairwise correlations between variables. So what kind of analysis can we put into place to study a BERT table? One simple idea coming by analogy from contingency tables is to do the same kind of analysis but for the BERT table. When we run correspondence analysis on a BERT table, we get exactly the same dimensions as in MCA but with different eigenvalues. The eigenvalues coming from a Burt table are the squares of the eigenvalues coming from the complete disjunctive table. So, which eigenvalues should we work with? Previously, we interpreted the eigenvalues coming from the complete disjunctive table in terms of the mean of the squared correlation ratios. As this interpretation is quite useful, we're going to stick with it. Lastly, this analogy teaches us something: that in the end, MCA really only depends on the pairwise links between variables. Therefore, it's again analogous with PCA, which only depends on the correlation matrix. Well, it's time to wrap it up for multiple correspondence analysis and make some conclusions. First, we have seen that MCA is the principal component method to use when we have tables of individuals versus qualitative variables. The general strategy in MCA is to look for the principal dimensions explaining the variability between individuals and to closely examine links between variables. From this general point of view, the situation is identical to PCA. Technically speaking, the main ways of interpreting the data are quite simple: an individual is at the barycenter of its categories, and a category is at the barycenter of the individuals that are in it. This gives us a quite broadly useful methodology because these types of data tables are quite common in reality and can be easily interpreted as we have seen. In particular, multiple correspondence analysis is a particularly useful method for dealing with survey data. The possibility of interpreting the eigenvalues as the means of the squared correlation ratios is a fundamental property of MCA. This validates the interpretation of the MCA dimensions as synthetic variables. The MCA dimensions are quantitative variables which summarize, in a way, the qualitative variables. We have insisted in using the squares of these links. This representation is particularly precious to us when there are a large number of variables. Like for other principal component methods, it's useful to validate our interpretations of the results by going back to look at the data. MCA will suggest to us links between qualitative variables. And, if we want, we can then construct pairwise tables for such variable pairs and analyze and visualize them using correspondence analysis. Also, the kind of convergence we see between the analysis of the complete disjunctive table and the analysis of a BERT table is an argument in favor of the general methods we have presented. It's always interesting to see when different points of view lead to the same result. Lastly, MCA allows us to pass from a table of qualitative variables to a table of dimensions, that is quantitative variables. It's therefore possible to see the method as a preprocessing step for qualitative data in order to pass it to a clustering algorithm, for example. To finish, here are a few references related to what we have seen in the MCA videos. You have now completed all of the course videos for MCA. So if you're ready to go on, you can now have a look at the video showing how to run MCA in practice using FactoMineR. Good luck!
