The video introduces Multiple Factor Analysis (MFA), a multiblock extension of PCA/MCA designed to analyse data tables where the same set of individuals is described by several groups of variables that may come from different sources. The presenter explains the motivation for MFA – to study the whole data set while also comparing the information contributed by each group – and outlines the four‑part course series. A concrete illustration is a sensory evaluation of ten Loire Valley white wines, assessed by three juries (experts, students, consumers) each using a different set of descriptors, plus a consumer appreciation matrix and a qualitative grape‑variety variable. The speaker describes the structure of such multi‑table data (individuals, groups, variables), highlights typical applications in genomics, social surveys, and economics, and clarifies the analytical goals: global description of individuals, assessment of shared versus group‑specific information, and detection of consensus or divergence among groups. A key methodological point introduced is the need to balance the contribution of each group, which will be achieved through appropriate weighting of variables in the MFA.
Chapters
This week, we're going to look at a method to study more complex data tables, where a group of individuals is characterized by variables structured as groups and possibly coming from different information sources. Our interest in the method is due to it being able to analyze a data table as a whole, but also its ability to compare information provided by the various information sources. There are 4 course videos for you to watch which present the main details of MFA, that is, multiple factor analysis. The outline of the videos is as follows. First, we describe the nature of the data we are working with in MFA. We will present several examples of datasets and consider the kinds of questions we want to answer using MFA. Then, we will see that MFA can be seen as a type of PCA on a weighted matrix that balances the information provided by different groups of variables. We will describe this weighting and explain why it's of interest. MFA provides results on individuals and variables, just like PCA does for quantitative variables and MCA does for qualitative variables. The most important feature of MFA is that it can take into account several groups of variables. We will see how to compare information provided by each of these groups, what information is common to several groups, and what information is specific to certain groups. Lastly, we will describe several interpretation aids that are useful for analyzing MFA results. Let's start in this video by defining what kinds of data we are working with and what the associated issues are with such data. Let's first look at the dataset which will be used to present the method. This dataset concerns a sensory evaluation of 10 white wines from the Loire Valley. During this sensory evaluation, 5 Vouvray and 5 Sauvignons were tasted and compared using sensory descriptors such as acidity, bitterness, and citrus odor. To do this, the judges were required to evaluate each wine and give a score for each descriptor. Scores vary between 0 and 10, with 0 corresponding to a very low or nonexistent flavor or odor and 10 to a very strong flavor or odor. In the picture, we see the judges hard at work, shall we say, evaluating flavors and odors. When all judges have evaluated all wines according to each descriptor, a table of averages is constructed with each row corresponding to a wine and each column to a variable, that is, descriptor. So, a given entry in the table corresponds to the average value of the given descriptor for the given wine over the panel of judges. The table shown here concerns the evaluation made by a jury of experts, oenologists. We can see to the right of the table a qualitative variable that characterizes wines: grape variety that can be either Vouvray or Sauvignon. This rectangular table with individuals as rows and variables as columns can be analyzed by PCA with a qualitative variable grape variety treated as a supplementary variable. However, the problem here is a bit more complex because we have obtained a sensory description of the same wines by different juries: the jury of experts for whom we saw an excerpt before, but also a jury of students and a jury of consumers. Also, the variables used to describe the wines are not the same from one jury to the next. The expert jury evaluated 27 sensory descriptors, whereas the student and consumer juries evaluated only 15 sensory descriptors. The problems associated with this dataset are multiple. On the one hand, to describe the 10 wines from a sensory point of view using evaluations from the 3 juries, but also to see if the juries describe the wines in a similar way, or if certain juries describe certain wines in specific ways. To deal with such problems, we will focus on the data table that brings together the data from the 3 juries. We also have an evaluation of the appreciation of each wine by a jury of 60 consumers. In this table, rows correspond to wines and columns to consumers, and each entry corresponds to the score for a given wine for a given consumer. We consider this set of variables for illustrative purposes because here we are focusing our attention on sensory descriptions only. However, we can relate sensory descriptions and wine appreciation by using these variables as supplementary ones in the analysis. Similarly, the qualitative variable corresponding to grape variety will be considered as a qualitative supplementary variable. The structure of our sample dataset is as follows. The same set of individuals, for us the 10 wines, is described by several groups of variables. In our example, a group corresponds to a jury. We consider that we have i individuals in each group that we have J groups of variables and that, in each of these groups, we have KJ variables. The value taken by individual i for variable k is denoted Xik. In each group, variables can be quantitative or qualitative. In many applications, data tables like this can be found, with the same set of individuals described by several groups of variables. In fact, the groups of variables may correspond to different sources of information. Here are some examples of different applications. In genomics, a group of patients with brain cancer can be described according to 3 different and possibly complementary sources of information: DNA measurements, corresponding to measurements on genes, expression measurements, and protein measurements. The goal is then to understand which genes are involved in the disease and which are differentially expressed according to whether the patient is ill or not and the tumor type. The aim is also to know whether the 3 sources of information are consistent or whether sources provide discordant information. Another example could be a survey of young students who are asked several groups of questions about their consumption of products such as alcohol and soft drugs, their psychological state, and sleep quality, and lastly basic details like sex, age range, etc. All questions are qualitative in this survey. The aim is to broadly understand the links between sleep quality, psychological state, and drug use. We are not interested in connections between pairs of variables but rather in more global similarities. Another example is when the different groups of variables correspond to different moments of time. This makes it possible to see overall the evolution of a set of variables over time. In economics, this could correspond to a set of economic indices from one period to another or from one year to the next. Examples of multi-tables like these are increasingly numerous because, on the one hand, it is increasingly easy to collect and store data. And on the other, increasingly, we are trying to analyze more complex problems involving multiple sources of information. In the majority of examples, the problem, or problems, will be the following: to study and describe the set of individuals using all variables, and describe the relationships between variables, in one group or from one group to another. These goals coincide with those of PCA and MCA. Here, we will also take advantage of the group structure of the variables to study globally the similarities and differences between groups. Do the variables of one group provide the same information as the variables of the other groups, or is there partly shared information and partly group-specific information provided by that group? Or even, is there no link between the information provided by that group and that provided by the other groups. To do this, we can see if a particular individual is described in the same way by all groups, or if one of the groups describes it in a particular way. For example, is wine number 2 described in the same way by the 3 juries: experts, consumers, and students? If so, we can say that there is a strong consensus between the 3 juries. Or inversely, is the wine described in a specific or different way by the jury of experts, for example? Lastly, we can compare the characteristics of individuals for each group. In the wine example, this amounts to comparing the 3 configurations of wines obtained through separate PCAs constructed from the data of each of the juries. Before we go on, a crucial point in multi-table analysis is that of equilibrium of the information. One group's information should not overwrite or mask information provided by other groups. So, we have to try to balance the information provided by each group. We'll see in the next video which criterion to use to balance the groups in the MFA, and we'll also see that this balancing act can be achieved by simply weighting the variables.
