On the Variance of Eigenvalues in PCA and MCA

International audience; In this talk, we show that, in principal component analysis (PCA) and in multiplecorrespondence analysis (MCA), the strength of the relationship between variablesfirstly determines the variance of the eigenvalues (which is an indicator of deviation fromsphericity), and secondly highlights the axes to which the variables contribute the most.In PCA on correlation matrix, given a set of numerical variables, we call linkageindex of a variable the mean of the squared correlations between this variable and theothers and average linkage index the mean of the linkage indexes.One has the two following properties:1. The variance of eigenvalues is proportional to the average linkage index.2. For each variable, the variance of eigenvalues weighted by the contributions of thisvariable to axes is proportional to the linkage index of this variable.In MCA, similar properties hold.We illustrate these properties using two classical data sets: scholar evaluations(Spearman, 1904) for PCA and physical attributes (Burt, 1950) for MCA.