Linkage index of variables and its relationship with variance of eigenvalues in PCA and MCA

Abstract

In the present article, we show that, in principal component analysis (PCA) on correlation matrix as well as in multiple correspondence analysis (MCA), the strength of the relationship between variables is linked to the variance of the eigenvalues, and indicates the axes to which the variables contribute the most. In PCA, we define the linkage index of a variable as the mean of the squared correlations between this variable and the others. We prove that the variance of eigenvalues is proportional to the mean linkage index and that, for each variable, the variance of eigenvalues weighted by the contributions of the variable to axes is proportional to the linkage index of the variable. In MCA, similar properties are proven regarding both categorical variables and categories. We illustrate these properties using two datasets coming from classical articles by Spearman (1904) for PCA and Burt (1950) for MCA.