Multiple factor analysis: Difference between revisions

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'''Multiple Factor Analysis (MFA)''' new article content ...

Introduction

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,

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1. ''Representations of individuals'' in which two individuals are much closer than they have similar values for all variables in all groups; in practice the user especially studies the first factorial plane.

2.''Representations of quantitative variables'' as in PCA (correlation circle).

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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''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.]]

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). On the contrary, the individual 5 is more characterized by high values for the variables of group 2 than for the variables of group 1 (for the individual 5, group 2 partial point lies further from the origin than group 1 partial point). This reading of the graph can be checked directly in the data.

6. '' Representations of groups of variables '' as such. In these graphs, each group of variables is represented by a single point. 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 <math>N_i^j</math> coincide. The coordinate of group <math>j</math> along the axis <math>s</math> is equal to the contribution of the group <math>j</math> to the inertia of MFA dimension of rank <math>s</math>. This contribution can be interpreted as an indicator of relationship (between the group <math>j</math> and the axis <math>s</math>, hence the name [[''relationship square]]'' given to this type of representation). This representation also exists in other factorial methods (MCA and FAMD in particular) in which case the groups of variable are each reduced to a single variable.

[[File:AFM fig4.jpg|center|thumb|Figure4. MFA. Test data. Representation of groups of variables.]]