Multiple Factor Analysis

Multiple factor analysis (MFA) analyzes observations described by several “blocks” or sets of variables. MFA seeks the common structures present in all or some of these sets. MFA is performed in two steps. First, a principal component analysis (PCA) is performed on each data set, which is then “normalized” by dividing all its elements by the square root of the first eigenvalue obtained from its PCA. Second, the normalized data sets are merged to form a unique matrix, and a global PCA is performed on this matrix. The individual data sets are then projected onto the global analysis to analyze communalities and discrepancies. MFA is used in very different domains such as sensory evaluation, economy, ecology, and chemistry.

MFA is used to analyze a set of observations described by several groups of variables. The number of variables in each group may differ, and the nature of the variables (nominal or quantitative) can vary from one group to the other, but the variables should be of the same nature in a given group. The analysis derives an integrated picture of the observations and of the relationships between the groups of variables.

The goal of MFA is to integrate different groups of variables describing the same observations. In order to do so, the first step is to make these groups of variables comparable. Such a step is needed because the straight-forward analysis obtained by concatenating all variables would be dominated by the group with the strongest structure. A similar problem can occur in a non-normalized PCA: Without normalization, the structure is dominated by the variables with the largest variance. For PCA, the solution is to normalize (i.e., to use z scores) each variable by dividing it by its standard deviation. The solution proposed by MFA is similar: To compare groups of variables, each group is normalized by dividing all its elements by a quantity called its first singular value, which is the matrix equivalent of the standard deviation. Practically, this step is implemented by performing a PCA on each group of variables. The first singular value is the square root of the first eigenvalue of the PCA. After normalization, the data tables are concatenated into a data table that is submitted to PCA.