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|author-link2=Jörg Blasius|title=Multiple Correspondence Analysis and Related Methods|url=https://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 is a [[Multivariate statistics|multivariate method]] from the field of [[Ordination (statistics)|ordination]] used to simplify [[Dimensionality reduction|multidimensional data]] structures. MFA treats all involved tables in the same way (symmetrical analysis). It may be seen as an extension of:

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

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

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'' data''

Let us considerConsider 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:

# The abundance-dominance coefficient of 50 plant species (coefficient ranging from 0 = the plant is absent, to 9 = the species covers more than three-quarters of the surface). The whole set of the 50 coefficients defines the floristic profile of a station.

# Eleven pedological measurements ([[Pedology (soil study)|Pedology]] = soil science): particle size, physical, chemistry, etc. The set of these eleven measures defines the pedological profile of a station.

Three analyses are possible:

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{| class="wikitable centre" width="60%"

|+ Table 1. MFA. Test data. A etand B (group 1) are uncorrelated. C1 and C2 (group 2) are identical.

|-

! !! <math>A</math> !! <math>B</math> !! <math>C_1</math>!! <math>C_2</math>

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The core of MFA is a weighted factorial analysis: MFA firstly provides the classical results of the factorial analyses.

1. ''Representations of individuals'' in which two individuals are muchclose closerto thaneach other if they haveexhibit similar values for allmany variables in allthe different variable groups; in practice the user particularly studies the first factorial plane.

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

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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 ==

MFA is available in two R packages ([http://factominer.free.fr FactoMineR] and [http://pbil.univ-lyon1.fr/ADE-4 ADE4]) and in many software packages, including SPAD, Uniwin, [[XLSTAT]], etc. There is also a function [http://www.ensai.fr/userfiles/AFMULT%20and%20PLOTAFM%20aout%202010.pdf SAS]{{dead link|date=February 2018 |bot=InternetArchiveBot |fix-attempted=yes }} . The graphs in this article come from the R package FactoMineR.

== References ==

<!--- See [[Wikipedia:Footnotes]] on how to create references using<ref></ref> tags, these references will then appear here automatically -->

{{Reflist}}

== External links ==

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

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