Multiple factor analysis
Multiple factor analysis (MFA) is a factorial method[1] 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 is a multivariate method from the field of ordination used to simplify multidimensional data structures.
MFA treats all involved tables in the same way (symmetrical analysis).
It may be seen as an extension of: Why introduce several active groups of variables in the same factorial analysis?
data Consider the case of quantitative variables, that is to say, within the framework of the PCA.
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.
This is not desirable: there is no reason to wish one group play a more important role in the analysis.
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.
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
In this example the first axis of the PCA is almost coincident with C. Indeed, in the space of variables, there are two variables in the direction of C: group 2, with all its inertia concentrated in one direction, influences predominantly the first axis.
For its part, group 1, consisting of two orthogonal variables (= uncorrelated), has its inertia uniformly distributed in a plane (the plane generated by the two variables) and hardly weighs on the first axis.
Group 2 variables contribute to 88.95% of the inertia of the axis 1 of the PCA.
is .976; The first axis of the MFA (on Table 1 data) shows the balance between the two groups of variables: the contribution of each group to the inertia of this axis is strictly equal to 50%.
The weighting of the MFA, which makes the maximum axial inertia of each group equal to 1, plays this role.
Thus, in this example, we may want to perform a factorial analysis in which two individuals are close if they have both expressed the same opinions and the same behaviour.
Each judge scores each descriptor for each product on a scale of intensity ranging for example from 0 = null or very low to 10 = very strong.
We want to achieve a factorial analysis in which two products are similar if they were evaluated in the same way by both juries.
Conclusion: These examples show that in practice, variables are very often organized into groups.
Beyond the weighting of variables, interest in MFA lies in a series of graphics and indicators valuable in the analysis of a table whose columns are organized into groups.
Representations of individuals in which two individuals are close to each other if they exhibit similar values for many variables in the different variable groups; in practice the user particularly studies the first factorial plane.
Indicators aiding interpretation: projected inertia, contributions and quality of representation.
In the example, the contribution of individuals 1 and 5 to the inertia of the first axis is 45.7% + 31.5% = 77.2% which justifies the interpretation focussed on these two points.
): that is the cloud analysed in the separate factorial analysis (PCA or MCA) of the group
In the example (figure 3), individual 1 is characterized by a small size (i.e. small values) both in terms of group 1 and group 2 (partial points of the individual 1 have a negative coordinate and are close one another).
Two groups of variables are close one another when they define the same structure on individuals.
Extreme case: two groups of variables that define homothetic clouds of individuals
In practice, this representation is especially precious when the groups are numerous and include many variables.
These factors are represented as supplementary quantitative variables (correlation circle).
Besides balancing groups of variables and besides usual graphics of PCA (of MCA in the case of qualitative variables), the MFA provides results specific of the group structure of the set of variables, that is, in particular: The small size and simplicity of the example allow simple validation of the rules of interpretation.
are a research topic of applied mathematics laboratory Agrocampus (LMA ²) which published a book presenting basic methods of exploratory multivariate analysis.
