Multiple factor analysis: Difference between revisions

Revision as of 14:05, 27 May 2014

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The Multiple Factor Analysis 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:

•* the [[Principal component analysis]] (PCA) when variables are quantitative,

•* the [[Multiple correspondence analysis]] (MCA) when variables are qualitative,

•* the Factorial Analysis of Mixed Data (FAMD) when the active variables belong to the two types.

==Introductory Example ==

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===Methodology ===

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 <math>j</math> 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 <math>j</math>.

Formally, noting <math>\lambda_1^j </math> the first eigenvalue of the factorial analysis of one group <math>j</math>, the MFA assigns weight <math>1/\lambda_1^j </math> for each variable of the group <math>j</math>.

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=== Graphics specific to this kind of multiple table ===

5. ''Superimposed representations of individuals'' « seen » by each group. An individual considered from the point of view of a single group is called « ''partial individual »'' (in parallel, an individual considered from the point of view of all variables is said ''mean individual'' because it lies at the center of gravity of its partial points). Partial cloud <math>N_i^j</math> gathers the <math>I</math> individuals from the perspective of the single group <math>j</ math> (ie <math>{i^j, j = 1, J}</math>): that is the cloud analysed in the separate factorial analysis (PCA or MCA) of the group <math>j</math>. The superimposed representation of the <math> N_i^j</math> provided by the MFA is similar in its purpose to that provided by the [[Procrustes analysis]].

[[File:AFM fig3.jpg|center|thumb|Figure 3. MFA. Test data. Superimposed representation of mean and partial clouds.]]