6.10 Multiple Factor Analysis (MFA)
6.10.1 Introduction
Yet another approach to the symmetric analysis of a data set described by k (usually
k > 2) subsets or groups of variables is multiple factor analysis (MFA; Escofier and
Pagès 1994; Abdi et al. 2013). This analysis is correlative; it does not involve any
hypothesis of causal influence of a data set on another. The variables must belong to
the same mathematical type (quantitative or qualitative) within each subset. If all
variables are quantitative, then MFA is basically a PCA applied to the whole set of
variables in which each subset is weighted. Do not confuse it with multiple correspondence analysis (MCA, Sect. 5.4.5) where a single matrix of qualitative variables
is submitted to ordination and other matrices may be added as supplementary
(passive) information.
MFA computation consists in the following steps:
• a PCA is computed for each (centred and optionally standardized) subset of
quantitative variables. PCA is replaced by MCA for subsets of qualitative variables. Each centred table is then weighted so that all receive equal weights in the
global analysis, accounting for different variances among the groups. This is done
by dividing all variables of each centred table by the first singular value (i.e., the
square root of the first eigenvalue) obtained from its PCA (or MCA for qualitative
variables) (Abdi et al. 2013);
• the k weighted data sets are regrouped using cbind() in R. The resulting table
is submitted to a global PCA;
• the different subsets of variables are then projected on the global result; common
structures and differences are assessed for objects and variables.
The pairwise similarities between the geometric representations derived from the
k groups of variables are measured by the RV coefficient described in Sect. 6.9.2. RV
coefficients, which vary between 0 and 1, can be tested by permutations (Josse et al.
2008).
MFA has been mainly used in economics, sensory evaluation (e.g. wine tasting)
and chemistry so far, but the potential for ecological applications is promising, as
evidenced by a few recent contributions (Beamud et al. 2010; Carlson et al. 2010;
Lamentowicz et al. 2010). Indeed, this method is useful to explore the complex
relationships among several ecologically meaningful groups of descriptors, whatever their number and type.
MFA can be computed by function mfa() of the package ade4. In this case, a
data frame comprising all blocks of variables must first be assembled and set into
class ktab by function ktab.data.frame(). Here we shall use function
MFA() of the package FactoMineR, which is more straightforward and offers
more options.
An extension of MFA to the case where the data are hierarchically organized
(e.g. in regions and sub-regions, or questionnaires structured in topics and
282
6 Canonical Ordination
6.10.1 Introduction
Yet another approach to the symmetric analysis of a data set described by k (usually
k > 2) subsets or groups of variables is multiple factor analysis (MFA; Escofier and
Pagès 1994; Abdi et al. 2013). This analysis is correlative; it does not involve any
hypothesis of causal influence of a data set on another. The variables must belong to
the same mathematical type (quantitative or qualitative) within each subset. If all
variables are quantitative, then MFA is basically a PCA applied to the whole set of
variables in which each subset is weighted. Do not confuse it with multiple correspondence analysis (MCA, Sect. 5.4.5) where a single matrix of qualitative variables
is submitted to ordination and other matrices may be added as supplementary
(passive) information.
MFA computation consists in the following steps:
• a PCA is computed for each (centred and optionally standardized) subset of
quantitative variables. PCA is replaced by MCA for subsets of qualitative variables. Each centred table is then weighted so that all receive equal weights in the
global analysis, accounting for different variances among the groups. This is done
by dividing all variables of each centred table by the first singular value (i.e., the
square root of the first eigenvalue) obtained from its PCA (or MCA for qualitative
variables) (Abdi et al. 2013);
• the k weighted data sets are regrouped using cbind() in R. The resulting table
is submitted to a global PCA;
• the different subsets of variables are then projected on the global result; common
structures and differences are assessed for objects and variables.
The pairwise similarities between the geometric representations derived from the
k groups of variables are measured by the RV coefficient described in Sect. 6.9.2. RV
coefficients, which vary between 0 and 1, can be tested by permutations (Josse et al.
2008).
MFA has been mainly used in economics, sensory evaluation (e.g. wine tasting)
and chemistry so far, but the potential for ecological applications is promising, as
evidenced by a few recent contributions (Beamud et al. 2010; Carlson et al. 2010;
Lamentowicz et al. 2010). Indeed, this method is useful to explore the complex
relationships among several ecologically meaningful groups of descriptors, whatever their number and type.
MFA can be computed by function mfa() of the package ade4. In this case, a
data frame comprising all blocks of variables must first be assembled and set into
class ktab by function ktab.data.frame(). Here we shall use function
MFA() of the package FactoMineR, which is more straightforward and offers
more options.
An extension of MFA to the case where the data are hierarchically organized
(e.g. in regions and sub-regions, or questionnaires structured in topics and
282
6 Canonical Ordination