The lecture concludes the correspondence analysis series by presenting classical interpretation aids common to principal component methods. It explains how to assess the quality of representation of rows/columns using projected inertia and cosine of angles, and how to compute contributions of points to dimension inertia, emphasizing the balance of distance and mass. Practical guidance is given for selecting well‑represented, extreme points for interpretation, illustrated with a toy taste dataset and a more extreme example with highly unequal row marginals, showing why contributions matter when marginal sums differ. The speaker demonstrates adding supplementary elements (e.g., a Fields Medal column) via transition formulas. The concept of distributional equivalence is introduced, showing that grouping rows or columns with identical profiles yields identical CA results, a key property for textual data. Limits on the number of non‑zero inertia dimensions are derived (≤ min(columns‑1, rows‑1)), leading to a discussion of phi‑squared and Cramér’s V as measures of link intensity, with calculations for the Nobel Prize and flavor datasets. The overall conclusion recaps that CA visualizes deviation from independence, decomposes total inertia into dimensions, and simultaneously represents rows and columns, providing insight into qualitative variable relationships. The video ends by directing viewers to a practical tutorial using FactoMineR.
Chapters
Let's now move on to classical interpretation aids including the quality of the representation and contributions. These aids are common to all of the various principal component methods we see in this course. The quality of representation of a point can be measured using the same criteria we used for a cloud, that is the projected inertia of the point divided by its total inertia. This criterion shows us how much the deviation of the profile for the mean profile is represented by each dimension. If we switch back to the geometric interpretation, we have a point Mi representing profile i which is projected onto the axis Us as the point His. To calculate the quality of representation of profile i by the s-th dimension, we thus calculate the projected inertia of Mi on Us divided by the total inertia of m_i, which gives us f_i point times O_h_i,s squared divided by f_i point times O_m_i squared. The f_i points cancel, which means that the quality of representation doesn't depend on the weights. In fact, this ratio is none other than the cosine of the angle between O_m_i and u_s. This way of looking at things in terms of the cosine brings up the following question: how well is the deviation between the profile and the mean profile well represented by the dimension? Is the point M_i pointing in the same direction as the dimension? Let's look now at the quality of representation in our toy dataset on tastes in which there was confusion between sour and bitter. Look at the sour point. It has a quality of representation of 0.89 on the first dimension and 0.11 on the second. This means that it is much further away from the mean in the first dimension than in the second. And clearly, this makes sense. Along the first dimension, sourness is never associated with perceived sweetness, while on the second dimension, sourness is occasionally associated with bitterness. Thus, the deviation from the mean profile is much larger on the first dimension than on the second. In practice, the quality of representation is used in the following way. When we have lots of points, to start interpreting things, we select a few of them that are well represented and with fairly extreme coordinates, that is, far from the origin on the dimension or dimensions we are interested in. This means that we are choosing points which should be helpful for interpretation because it should be easy to find in these data the meaning of the dimensions, remembering that the deviation between the profile being looked at and the mean profile is essentially expressed with respect to these dimensions. The 2nd classical interpretation aid, found in all the principal component methods, Is the idea of contribution. To calculate the contribution of a point i to the inertia of the s dimension, we first calculate a row indicator that of the projected inertia of the point, which equals f i point times O H I s squared. This row indicator is then turned into a percentage by dividing by the total inertia of the s dimension and. And multiplying by 100. We can also add up the contributions of several elements to a given dimension so as to measure their total contribution. What is the point of these contributions? Well, they indicate how much we can say that a given dimension exists because of one or several elements. If, for example, for a given dimension, we see that one element contributes 90% of the inertia, we can simply limit the interpretation of this dimension to this one element. Which begs the question, what is our definition of a big contribution? Well, we say it's big if both the distance OHis is quite large and the mass fi point is quite large too. In this sense, we can say that the idea of what a large contribution is is a kind of compromise between distance to the origin and weight. In practice, contributions are usually used when dealing with big data tables in order to select a subset of elements to start interpreting. So we want to select those that contribute the most. Ideally, this means finding the set of elements that both contribute highly and are well represented. Let's illustrate this idea on an example. We are not going to use the examples we have seen before, because their row and column marginals are not so different. Instead, we'll use this little dataset in which the row marginals are quite varied. For instance, the marginal sums of A and D are more than 10 times smaller than those of B and C. When we perform correspondence analysis on this table, we get the following results. The first dimension dominates with 83% of the total inertia, so we are going to stop right there. It separates columns X1 and X4, where we see that X1 was mostly characterized by rows A and B, and X4 by rows C and D. On this dimension, some The coordinates of A and D are much more extreme, or further out, than those of B and C. Now, let's look at contributions. When we look at the contributions table, we see that the contributions of B and C are much larger than those of A and D, even though A and D have much more extreme coordinates. This comes from the marginal sums. B and C have clearly the smallest coordinate values, but by far the largest weights. This example thus clearly shows the importance of looking not just at the graphical output of the correspondence analysis, but also at the contributions. But before we go overboard, don't forget that this is quite an extreme example with huge differences in marginal sums of a factor of 10 or more. As a broad conclusion therefore, if we have huge differences in marginal sums, it's very important to have a look at contributions. On the other hand, if there is little difference between the marginal sums, looking at contributions won't bring much more information to the table with respect to the plot as the coordinate values in the plot also represent the contributions quite well. Like in principal component analysis, it's easy here to bring supplementary elements into the analysis. We can simply apply the transition formula to calculate, for example, the coordinates of an extra column. Here, in our Nobel example, we have added a column for the Fields Medal, which is often considered the equivalent of a Nobel Prize in mathematics. We see in the plot that mathematics is located next to the Nobel Prizes for Peace and Literature. Does this hint at the underlying connections between mathematics and philosophy? Perhaps. We also see that mathematics is close to France and Russia, which are 2 countries with a particularly strong mathematical heritage. Now let's move on to talk about a remarkable property of correspondence analysis. It's called distributional equivalence. As we will see, it's extremely useful when analyzing textual data. If 2 rows or 2 columns have exactly the same profile, we may ask the question, is it useful to group them together as one? If we do this for 2 rows, for example, we then replace them with one row in which we have summed the 2 rows together. The next question would be, which table should we analyze? Should we do correspondence analysis on the first table or on the modified one? Well, the distributional equivalence property of these tables is that both analyses lead to the same result. This is extremely important in the analysis of lexical tables and calms the debate about whether or not we should group together singular and plural versions of words, verb conjugations, synonyms, etc. Due to their equivalent distributions, we know that these words will have the same profile. It doesn't matter if we group them together or not. And of course, if they don't have the same profile, they shouldn't be grouped together because they must be representing different notions. We finish this presentation of correspondence analysis by thinking about how many dimensions with non-zero inertia there are. Let's consider this for a point cloud of rows with I points in a J-dimensional space. As there are J dimensions, We might guess that we can get j axes with nonzero inertia. In fact, not at all. Remember that we are analyzing a profile which has a specific feature: its sum is 1. What this means is that the cloud exists, in reality, in a j-1 dimensional subspace. Therefore, the maximum number of axes with nonzero inertia is less than or equal to j-1. Next, let's think a bit more about the number of points. There are I points, so if I equals 2, we can represent them using just 1 dimension. In the general case, we can perfectly represent I points with at most I minus 1 dimensions. Overall, where does this lead us? Well, the maximum number of dimensions with nonzero inertia in correspondence analysis has to be less than or equal to the minimum of these 2 numbers, that is, the number of columns minus 1 and the number of rows minus 1. So, for the Nobel Prize example, as we have a table with 8 rows and 6 columns, we can be absolutely sure that, with 5 dimensions, we will have found 100% of the inertia. So what does this mean for phi squared, which is equal to the sum of the eigenvalues? And remembering that these are all less than or equal to 1, well, it means that phi squared must be less than or equal to the number of dimensions with nonzero inertia, that is, less than or equal to the smallest of i minus 1 and j minus 1. So this gives us the idea of taking the ratio of the value of phi squared over its maximum possible value. This gives us what is called Cramer's V, a measure that is between 0 and 1. It captures the intensity of the link between pairs of qualitative variables. We therefore have 2 measures of link intensity: the phi squared and the Cramer's V. They are quite similar measures, with the advantage for Cramér's V that it's always found between 0 and 1. Let's calculate Cramér's V for the Nobel Prize dataset. We get 0.03. This is quite small, which tells us that we are far from the situation of a strong exclusive link between pairs of categories: one country, one prize category. So, what about if we calculate Cramer's V for the tables with the 3 flavors? Here, we get 0.69 and 0.52, which are much higher. This means that there is a fairly strong relationship between the 2 variables. So, what is the overall conclusion we can make about the Nobel Prize data? Well, first, it shows that, even on small datasets, Correspondence analysis can give us a nice visual representation of the deviation from independence, which can help us decrypt the table better. This will be even more true for larger data tables. In the Nobel Prize data, we have seen that essentially the deviation from independence is structured as a separation between the scientific prizes and the rest, and to a smaller degree between physics and chemistry, and the economic sciences. The position of each country in the plot shows their relationships with the prizes. The United Kingdom, right in the middle, has not specialized in any particular prize and wins them in the proportion to which they have been given out overall. France and Italy, on the other hand, specialize in peace and literature prizes while Japan and Germany tend to get them in physics and chemistry, and the US and Canada in economics. So, the overall conclusion is this: to study possible links between qualitative variables, we build a contingency table, the foundation. If there are links, they are to be found in the deviation between what the real data table looks like and what it would look like if there were independence. Correspondence analysis means constructing a point cloud for the rows and a point cloud for the columns. The total inertia of a point cloud measures the strength of the deviation from independence. This total inertia is then decomposed into a sequence of dimensions of decreasing importance, each representing some aspect of the link between the variables. Lastly, correspondence analysis provides us with a simultaneous representation of the rows and columns in which a point's position reflects its participation in the deviation from independence. If you want to go further into the details of correspondence analysis, we invite you to delve into the book, which works through the subject in a similar way. You've now arrived at the end of the course videos for correspondence analysis. Next, go and watch the video on how to use correspondence analysis in practice using FactoMineR and have a go at the suggested exercises.
