Mutltiple Correspondence Analysis (Part 1/4: Data - issues)

This week, we have 4 videos for you on multiple correspondence analysis, MCA for short. We'll have a look at the main features of the method, using a specific example to guide us along the way. The videos look at the following things. First, we describe the types of data MCA can be used for. With these data in mind, we will look at what our goals are and what issues we may have. This leads us to ways to manipulate the data table. In multiple correspondence analysis, as in any principal component methods, we are going to build point clouds, including point clouds of the rows and point clouds of the columns. In the MCA context, we are going to have a point cloud of individuals and a point cloud of categories. We will see how to visualize the point cloud of individuals and how to interpret it using the categories. We'll then see how to directly visualize the point cloud of categories. We'll also show how in MCA the point cloud of individuals and that of the categories can be shown simultaneously on the same graph. This is called the simultaneous representation of the point clouds. In the final video, we will go through various classical interpretation aids in multiple correspondence analysis. A running example, a survey by INSEE on leisure in France, will be used to illustrate the course each step of the way. The data we are going to be working with is a rectangular table with I individuals each with a row and J qualitative variables as columns. Be careful to make the distinction between a qualitative variable like color, and categories of that variable, such as the color blue. The entry corresponding to the intersection of row i and column j holds a value v_ij, which is the category of the jth variable for the ith individual. For example, the most well-known example for MCA is surveys. We have talked to i people, and the survey has j multiple-choice questions. Questions like, for example, what is your socioprofessional class? Possible categories of this include working class, white-collar worker, etc. Another example question: what is your family situation? Single, married, etc. Here, the variable is family situation. The categories of this variable are single, married, and so on. We use all of these data to build the data table which we can then feed to multiple correspondence analysis software. Starting from this table, we construct an indicator matrix called the complete disjunctive table or CDT. In this indicator matrix, rows are individuals, but now the columns are the categories of the qualitative variables. So, if the jth column of the, the original table has Kj categories, there will be Kj columns for this variable in the indicator matrix, each corresponding to one category of the jth variable. The entry at the intersection of the ith row and the kth column, which we call yik, will be equal to 1 if the ith individual has category k of the jth variable, and 0 otherwise. Let's take an example. The marital status variable has Kj categories, and the individual i' is in the married category, which is the 2nd one. In the complete disjunctive table, in the ith row, in the columns corresponding to the jth variable, we see that there is a 1 in the 2nd column, because this is the category that this individual is in. Clearly, they have zeros in the other columns, for the J's variable. Here, we see why this table is called disjunctive and complete. It's disjunctive because in each block of columns, there is only one 1 and all the rest are zeros. And complete because there has to be at least one 1. The complete disjunctive table plays a central role in MCA as it's this that is analyzed by software packages. However, you, The user never actually writes it down or types it into the computer. The computer builds it for you, starting with the original table. To start to understand the structure of the complete disjunctive table, we're going to calculate its margins. Let's start with the column margin. We calculate the sum of the terms of row i, and remember that Yik equals 1 if the individual is in category k, and 0 otherwise. So, for each block of columns corresponding to one variable, we have one, and only one entry with a 1 in it. Therefore, summing over all the columns, we simply get the number of variables. There are J. The column margin is therefore a constant, the same for all individuals, and the sum of the values in the table is i times j, that is, the number of individuals multiplied by the number of variables. Now, let's calculate the row margin. As yik equals 1 if the ith individual is in category k, the sum of the entries in the kth column equals the number of individuals in category k. Another way of thinking of this number is the multiplication of i times PK, that is, the number of individuals multiplied by the proportion of individuals PK in category K. Then, when we sum the KJ terms of the part of the row margin corresponding to the Jth variable, we get I. So again, the row margin sums up to I times J. Now, let's look at the goals we have when we run multiple correspondence analysis. Imagine that we have survey data. Our individuals have filled in the survey, and our variables are simply the survey questions. First, we're going to take a closer look at the individuals, and then follow this with a study of the variables. When we look at individuals in multiple correspondence analysis, as in PCA, it's a multidimensional approach. A given individual, representing a row in the table, is analyzed based on the set of categories they belong to. This will be used when we want to compare individuals. We're going to say that individuals are similar to each other if they are in a similar set of categories. If we think about this in terms of social network profiles, for instance, 2 individuals have similar profiles and thus resemble each other if they've chosen the same categories and are different if they have not many categories in common. The set of these similarities and differences is what we call inter-individual variability. In multiple correspondence analysis, it's exactly this, the variability between individuals, that we are looking at. If they have all answered the survey in the same way, there aren't any statistics to do. But of course, usually, all the individuals don't don't answer in exactly the same way, and multiple correspondence analysis can delve into this variability using a multidimensional point of view. As we are working in the principal component method framework, the way we are going to describe this variability is by extracting principal components. We are going to try to shine a light on dimensions which separate, for example, extremely different individuals from average individuals. These dimensions will be interpreted in terms of the categories. For instance, we are not going to say things like, one dimension separates individuals 1, 3, and 4 from 10, 11, and 12. Rather, we are going to say things like, this dimension separates men and women, workers from the North from workers in the South, manual workers from white-collar workers, Etc., etc. The second part of the analysis deals with the variables. Here, these are qualitative, so we are going to be interested in associations between categories, because we say that 2 qualitative variables are linked if the categories of one have some kind of connection with the categories of the other. We'd like to get an overall visual representation of the associations between categories, leading to an overall look at the links between variables. And lastly, we are going to build synthetic variables. We're going to look for a quantitative indicator based on the qualitative variables that best summarizes them. This whole framework quickly resembles principal component analysis. Both work with a table of individuals in the rows versus variables in the columns. Though, from a technical point of view, things are quite different because in principal component analysis, we have quantitative variables, whereas in multiple correspondence analyses, we have qualitative ones. We are going to illustrate multiple correspondence analyses using the results of a survey taken in 2003 on 8,403 people aged 18 and over. The INSEE Institute ran a survey on how people construct their identity, which they called the History of Life Survey. Part of the survey looked into the leisure pursuits of the French. The dataset consists of 8,403 rows and 22 columns. The variables can be divided into 2 categories. The first 18 variables deal with various leisure activities. Typical questions were like, have you been to the cinema in the last 12 months? Without being obliged to. This is noted cinema, yes/no, in the survey. Similarly, we have variables like reading, yes/no, listening to music, yes/no, and so on. We also have the number of hours on average of television watching with 5 categories: not at all, 1 hour, around 2 hours, around 3 hours, and 4 or more hours. The next 4 variables are sex, male/female, age divided into intervals, 18 to 20, 21 to 30, 31 to 40, etc., marital status, single, married, widowed, divorced, remarried, and socio-professional status, farmer/manuel laborer, professional, senior management, employee, or other. The data table has the following structure: the 8,403 individuals as rows and the 18 activities and 4 other variables as columns. The first thing we look at in a survey like this is the number of individuals in each category. Here we show the number of individuals doing each activity sorted in decreasing order: listening to music, is the most common activity with 5,947 participants out of 8,403, followed by reading, walking, etc. At the bottom of the list, we find collecting and fishing as the least performed activities. Following these, we show the number of people watching TV for each of the 5 lengths of time. Now, if we look at the other variables, we see in particular that 1,498 people did not reply to the socioprofessional status question. What we can do is to treat no response as just another category. This means that socioprofessional status now has 8 categories. Before starting to perform multiple correspondence analysis, we have to ask ourselves what type of analysis we want to do exactly, considering that the variables naturally fall into 2 groups: the activity group and the supplementary variable group. One way to go about things is to consider the activity variables as active variables, and the other variables as supplementary ones. If we run an analysis like this, the individuals are only examined using the activity variables. In other words, an individual is an activity profile. These activity profiles exhibit a certain variability, and we will go and look for the principal dimensions of variability in these profiles. Once this has been done, we then bring back the supplementary variables for illustrative purposes, and the question becomes to find links between the dimensions of variability of the activity profiles and the supplementary variables. Another point of view, the reverse of what we've just described, is to make the supplementary variables active and introduce the activities afterwards. From this point of view, individuals are characterized by descriptive questions, so we are confronted with this set of variables in which we look for main dimensions of variability. In the end, this means studying the survey design. Then, we would move on to study the links between these dimensions of variability and the leisure activities. There is even a 3rd way to attack the problem. We could set all the variables as active, meaning that individuals are represented by heterogeneous data. Without going into details, this type of analysis requires a certain equilibrium between the 2 types of data and leads on to other methods like multiple factor analysis. We are not going to go into this anymore here, and in what follows next, we will just consider the first point of view. Thus, individuals will be represented in terms of their activity profiles. Multiple correspondence analysis works from the complete disjunctive table which has been previously constructed. A first thing to note is that in MCA, all individuals have the same weight. There is no reason to accord more importance to the responses of some individual over another. So, as the sum of the weights has to equal 1, the weight of each individual is 1/i. Remember also that in the CDT, y_ik equals 1 if the ith individual is is in category k of the jth variable. This value of 1 doesn't depend on the category k, and, in particular, doesn't depend on the number of individuals in this category. However, if an individual is in a rare category, this defines them much more than if they were in a common category. To give the most extreme case, if all of the individuals are in the same category of a certain variable, that variable is useless when it comes to characterizing the individuals. This is where the idea of dividing y_ik by p_k comes from. This gives us the value x_ik, which will be bigger whenever a category an individual is in is rare. By coding the complete disjunctive table in this way, the mean of the x_ik is equal to 1. In multiple correspondence analysis, we center the data, which in the end means that the entry x_ik will actually be y_ik over p_k minus 1. The table with the x_ik is the one on which we do the projection. To make the connection with principal component analysis, x_ik is simply the centered and standardized data. So we have now seen what the data looks like in MCA. And the kinds of questions we want to answer. In the next videos, we'll see how to construct and interpret point clouds of individuals and categories, starting from the table of the x_ik's.