Multiple factor analysis: Difference between revisions

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'''Multiple Factor Analysis''' (MFA) is a [[Factorial experiment|factorial]] method<ref name="GreenacreBlasius2006">{{cite book|last1=Greenacre|first1=Michael|last2=Blasius|first2=Jorg|title=Multiple Correspondence Analysis and Related Methods|url=http://books.google.com/books?id=ZvYV1lfU5zIC&pg=PA352|accessdate=11 June 2014|date=2006-06-23|publisher=CRC Press|isbn=9781420011319|pages=352–}}</ref> 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:

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

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Table 2 summarizes the inertia of the first two axes of the PCA and of the MFA applied to Table 1.

Group 2 variables contribute to 88.95 % of the inertia of the axis 1 of the PCA. The first axis (<math>F_1</math>) is almost coincident with C: the correlation between C and <math>F_1</math> 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%.

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7. ''Representations of factors of separate analyses'' of the different groups. These factors are represented as supplementary quantitative variables (correlation circle).

[[File:AFM fig5.jpg|center|thumb|Figure 5. MFA. Test data. Representation of the principal components of separate PCA of each group.]]

In the example (figure 5), the first axis of the MFA is relatively strongly correlated (r = .80) to the first component of the group 2. This group, consisting of two identical variables, possesses only one principal component (confounded with the variable). The group 1 consists of two orthogonal variables: any direction of the subspace generated by these two variables has the same inertia (equal to 1). So there is uncertainty in the choice of principal components and there is no reason to be interested in one of them in particular. However, the two components provided by the program are well represented: the plane of the MFA is close to the plane spanned by the two variables of group 1.

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The small size and simplicity of the example allow simple validation of the rules of interpretation. But the method will be more valuable when the data set is large and complex.

Other methods suitable for this type of data are available. Procrustes analysis is compared to the MFA in.<ref>Pagès Jérôme (2014). Multiple Factor Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. 272p </ref>

== History ==

MFA was developed by Brigitte Escofier and Jérôme Pagès in the 1980s. It is at the heart of two books written by these authors:<ref> ''Ibidem''</ref> and .<ref>Escofier Brigitte & Pagès Jérôme (2008). Analyses factorielles simples et multiples ; objectifs, méthodes et interprétation. Dunod, Paris. 318 p. isbn=978-2-10-051932-3</ref>. The MFA and its extensions (hierarchical MFA, MFA on contingency tables, etc.) are a research topic of applied mathematics laboratory Agrocampus ([http://math.agrocampus-ouest.fr LMA ²]) which published a book presenting basic methods of exploratory multivariate analysis.<ref> Husson F., Lê S. & Pagès J. (2009). Exploratory Multivariate Analysis by Example Using R. Chapman & Hall/CRC The R Series, London. isbn=978-2-7535-0938-2</ref>

== Software ==

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== External links ==

* [http://factominer.free.fr/ FactoMineR] A R software devoted to exploratory data analysis.

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