Multiple factor analysis: Difference between revisions

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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 [[Factor analysis of mixed data]] (FAMD) when the active variables belong to the two types.

==Introductory Example ==

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Let two groups of variables defined on the same set of individuals.

# The groupGroup 1 is composed of two uncorrelated variables A and B.

# The groupGroup 2 is composed of two variables {C1, C2} identical to the same variable C uncorrelated with the first two.

This example is not completely unrealistic. It is often necessary to simultaneously analyse multi-dimensional and (quite) one-dimensional groups.

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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 second axis, meanwhile, depends only on the group 1. This is natural since this group is two-dimensional while the second group, being one-dimensional, can be highly related to only one axis (here the first axis).

=== Conclusion about the balance between groups ===

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This balance must take into account that a multidimensional group influences naturally more axes than a one-dimensional group does (which may not be closely related to one axis).

The weighting of the MFA, which makes equal to 1 the maximum axial inertia of each group equal to 1, plays this role.

== Application examples ==

== Overview on few application areas ==

''Survey''

Questionnaires are always structured according to different themes. Each theme is a group of variables, for example, questions about opinions and questions about behaviour. Thus, in this example, we may want to perform a factorial analysis in which two individuals are close if they have expressed both expressed the same opinions and the same behaviour.

''Sensory analysis ''

A same set of products has been evaluated by a panel of experts and a panel of consumers. For its evaluation, each jury uses a list of descriptors (sour, bitter, etc.). 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. In the table associated with a jury, at the intersection of the row <math>i</math> and column <math>k</math>, is the average score assigned to product <math>i</math> for descriptor <math>k</math>.

Individuals are the products. Each jury is a group of variables. We want to achieve a factorial analysis in which two products are similar if they were evaluated in the same way and that by both juries.

''Multidimensional time series''

<math>K</math> variables are measured on <math>I</math> individuals. These measurements are made at <math>J</math> dates. There are many ways to analyse such data set. One of themway suggested by the MFA, is to consider each day as a group of variables in the analysis of the tables (each table corresponds to one date) juxtaposed row-wise (the table analysed thus has <math>I</math> rows and <math>J</math>x<math>K</math> columns).

'' Conclusion about these examples'': These examples show that in practice, the variables are very often organized into groups very often.

== Graphics from MFA ==

Beyond the weighting of variables, the interest of thein MFA lies in a series of graphics and indicators valuable in the analysis of a table whose columns are organized into groups.

===Graphics common to all the simple factorial analyses (PCA, MCA) ===

The core of the 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 much closer than they have similar values for all variables in all groups; in practice the user especiallyparticularly studies the first factorial plane.

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

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In the example:

* The first axis mainly opposes individuals 1 and 5 (Figure 1).

* The four variables have a positive coordinate (Figure 2): the first axis is a size effect. Thus, the individual 1 has low values for all the variables and individual 5 has high values for all the variables.

3. '' 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.

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* A representation of factors from separate analyses.

The small size and simplicity of the example allow tosimple easilyvalidation validateof 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 ==

The 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 package 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] . The graphs in this article come from the R package FactoMineR.

== References ==