Multiple factor analysis
Multiple factor analysis (MFA) is a factorial method devoted to the study of tables in which a group of individuals is described by a set of variables (quantitative and / or qualitative) structured in groups. It may be seen as an extension of:
Why introduce several active groups of variables in the same factorial analysis?
data
Let us consider the case of quantitative variables, that is to say, within the framework of the PCA. An example of data from ecological research provides a useful illustration. There are, for 72 stations, two types of measurements:
Three analyses are possible:
The third analysis of the introductory example implicitly assumes a balance between flora and soil. However, in this example, the mere fact that the flora is represented by 50 variables and the soil by 11 variables implies that the PCA with 61 active variables will be influenced mainly by the flora at least on the first axis). This is not desirable: there is no reason to wish one group play a more important role in the analysis.
The core of MFA is based on a factorial analysis (PCA in the case of quantitative variables, MCA in the case of qualitative variables) in which the variables are weighted. These weights are identical for the variables of the same group (and vary from one group to another). They are such that the maximum axial inertia of a group is equal to 1: in other words, by applying the PCA (or, where applicable, the MCA) to one group with this weighting, we obtain a first eigenvalue equal to 1. To get this property, MFA assigns to each variable of group a weight equal to the inverse of the first eigenvalue of the analysis (PCA or MCA according to the type of variable) of the group .
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