The video explains how to describe and interpret the dimensions (principal components) obtained from principal component methods such as PCA, MCA, or MFA using the dimdesc function from the FactoMineR R package. The presenter demonstrates the workflow on the Decathlon dataset, distinguishing active, supplementary, and categorical variables, and shows how to extract continuous variable correlations, p‑values, and the squared correlation ratio for categorical variables. By adjusting significance thresholds, the speaker illustrates how to reveal additional links (e.g., the competition variable) and interprets the meaning of positive and negative coefficients for both continuous and categorical variables. Practical advice is given on interpreting active versus supplementary variables, handling large variable sets (e.g., gene expression data), and cautions about over‑interpreting correlations.
Chapters
The purpose of this video is to show how to describe the dimensions obtained by a principal component method such as a principal component analysis or a multiple correspondence analysis or a multiple factor analysis . Every time we perform a principal component method, we want to describe and to interpret the principal components or the dimensions. So the methodology is the same: we will describe the dimensions thanks to the variables of the row To show how to use the dimdesc function that allows one to describe the dimensions, I will take a PCA example. I take the PCA example that I have used in the movie on PCA. So the PCA was performed on a dataset Decathlon. First, I load the library FactoMineR, then I load the dataset Decathlon. And I perform the PCA on this dataset. The first 10 variables correspond to the performances of athletes during a decathlon. Variables 11 and 12 are supplementary variables that correspond to the number of points and the rank of the athlete during the decathlon. And the variable 13 is a categorical variable. The PCA is performed and gives as outputs the variables graph and the individuals graph. I do not interpret these graphs here, but I will give tools to interpret the dimensions. Therefore, there is a dimdesc function of FactoMineR that can be applied on the result object. This object is obtained from any principal component method. So I just use dimdesc, and with this line of code, we try to interpret the dimensions with both continuous and categorical variables. We have a description of the first dimension by continuous variables. So the variable number of points is correlated with the first dimension. The correlation coefficient is equal to 0.956 and the correlation coefficient is significantly different from 0. This means that the variable number of points is linked to the first dimension and can be used to interpret this dimension. Similarly, the variable long jump is positively correlated and the variable 100 meters is negatively correlated to the first dimension. So on the top, we see the variables that are positively correlated to the dimension and on the bottom, the variables that are negatively correlated to the dimension. The p-values are increasing for the coefficients that are positive and then the p-values are decreasing. So on the top we have the most characterizing and positively correlated variables, and on the bottom the most characterizing variables but negatively correlated to the dimension. So this function is very useful when there is a lot of variables. So for instance, if you work on a gene dataset and you have 20,000 genes, this tool function will help you to quickly learn which genes means are the most correlated, the most linked to the dimension. So after the results for the first dimension, you have the variables that describe the second and the third dimension. So one remark: the active variables and the supplementary variables are used to characterize the dimension. It is the weighted sum of the active variables are linked to the first dimension since they participated to the construction of the dimension. So even if the correlation coefficient is significantly different from 0, the coefficient should be interpreted with caution. On the contrary, supplementary variables were not involved in the construction of the dimensions, so if they are correlated to the first dimension, we can interpret the strong link between the first dimension and the supplementary variable. This is more interesting when the variable was not used in the construction of the dimensions. There is no categorical variable that characterizes the first dimension because the categorical variables are not sufficiently linked to this dimension. As an example, I will modify this threshold and say that I want all the variables that characterize the dimensions not for the conventional threshold of 5% but for a threshold equal to 20%. So by doing this, we have again a characterization of each dimension by the continuous variables. So we recover the same results as before, but now we see that the categorical variable named competition appears to be significant. Therefore, this variable competition is linked to the first dimension. We calculated the squared correlation ratio So the squared correlation ratio is the indicator that is used in one-way analysis of variance to see if there is a link between a continuous variable, so the dimension, and a categorical variable, here the variable competition. So the link is significant at a threshold of 20% and not at 5%. This is why the variable does not appear before. So these are the results for an overall link between the dimension and the categorical variables. But we can also see the links from the categories of the categorical variable and the dimension. So we see that the category Olympic Games has a coefficient equal to 0.44, and this coefficient is the hat alpha_i coefficient in the one-way analysis of variance. With the model, the dimension explained by the competition. And the coefficient of the second category, Decastar, is estimated by -0.44. So the sum of the coefficients equals 0. We can interpret these results in the following way: the individuals who participated to the Olympic Games have higher coordinates in average on the first dimension, and those who participated to the Decastar have lower coordinates, always in average. So this is much more interesting when you have more than 2 categories. Then we have the results for the second and the third dimension. So for the second dimension, there are only continuous variables that characterize the second dimension, and for the third dimension, it is the same, there is no categorical variable. That characterize the dimension.
