In
statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...
, factor analysis of mixed data or factorial analysis of mixed data (FAMD, in the French original: ''AFDM'' or ''Analyse Factorielle de Données Mixtes''), is the
factorial method devoted to data tables in which a group of individuals is described both by quantitative and qualitative variables. It belongs to the exploratory methods developed by the French school called ''Analyse des données'' (data analysis) founded by
Jean-Paul Benzécri
Jean-Paul Benzécri was a French people, French mathematician and statistician. He studied at École Normale Supérieure and was professor at University of Rennes 1, Université de Rennes and later for most of his career at the Paris Institute of ...
.
The term ''mixed'' refers to the use of both quantitative and qualitative variables. Roughly, we can say that FAMD works as a
principal components analysis
Principal component analysis (PCA) is a popular technique for analyzing large datasets containing a high number of dimensions/features per observation, increasing the interpretability of data while preserving the maximum amount of information, and ...
(PCA) for quantitative variables and as a
multiple correspondence analysis In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Eucli ...
(MCA) for qualitative variables.
Scope
When data include both types of variables but the active variables being homogeneous, PCA or MCA can be used.
Indeed, it is easy to include supplementary quantitative variables in MCA by the
correlation coefficient
A correlation coefficient is a numerical measure of some type of correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two components ...
s between the variables and
factors
Factor, a Latin word meaning "who/which acts", may refer to:
Commerce
* Factor (agent), a person who acts for, notably a mercantile and colonial agent
* Factor (Scotland), a person or firm managing a Scottish estate
* Factors of production, suc ...
on individuals (a factor on individuals is the vector gathering the coordinates of individuals on a factorial axis); the representation obtained is a correlation circle (as in PCA).
Similarly, it is easy to include supplementary categorical variables in PCA. For this, each category is represented by the center of gravity of the individuals who have it (as MCA).
When the active variables are mixed, the usual practice is to perform discretization on the quantitative variables (e.g. usually in surveys the age is transformed in age classes). Data thus obtained can be processed by MCA.
This practice reaches its limits:
* When there are few individuals ( less than a hundred to fix ideas ) in which case the MCA is unstable ;
* When there are few qualitative variables with respect to quantitative variables (one can be reluctant to discretize twenty quantitative variables to take into account a single qualitative variable).
Criterion
The data include
quantitative variables
and
qualitative variables
.
is a quantitative variable. We note:
*
the correlation coefficient between variables
and
;
*
the squared
correlation ratio In statistics, the correlation ratio is a measure of the curvilinear relationship between the statistical dispersion within individual categories and the dispersion across the whole population or sample. The measure is defined as the ''ratio'' of t ...
between variables
and
.
In the PCA of
, we look for the function on
(a function on
assigns a value to each individual, it is the case for initial variables and principal components) the most correlated to all
variables in the following sense:
:
maximum.
In MCA of ''Q'', we look for the function on
more related to all
variables in the following sense:
:
maximum.
In FAMD
, we look for the function on
the more related to all
variables in the following sense:
:
maximum.
In this criterion, both types of variables play the same role. The contribution of each variable in this criterion is bounded by 1.
Plots
The representation of individuals is made directly from factors
.
The representation of quantitative variables is constructed as in PCA (correlation circle).
The representation of the categories of qualitative variables is as in MCA : a category is at the centroid of the individuals who possess it. Note that we take the exact centroid and not, as is customary in MCA, the centroid up to a coefficient dependent on the axis (in MCA this coefficient is equal to the inverse of the square root of the eigenvalue; it would be inadequate in FAMD).
The representation of variables is called
relationship square. The coordinate of qualitative variable
along axis
is equal to squared correlation ratio between the variable
and the factor of rank
(denoted
). The coordinates of quantitative variable
along axis
is equal to the squared correlation coefficient between the variable
and the factor of rank
(denoted
).
Aids to interpretation
The relationship indicators between the initial variables are combined in a so-called relationship matrix that contains, at the intersection of row
and column
:
* If the variables
and
are quantitative, the squared correlation coefficient between the variables
and
;
* If the variable
is qualitative and the variable
is quantitative, the squared correlation ratio between
and
;
* If the variables
and
are qualitative, the indicator
between the variables
and
.
Example
A very small data set (Table 1) illustrates the operation and outputs of the FAMD . Six individuals are described by three quantitative variables and three qualitatives variables. Data were analyzed using the R package function FAMD FactoMineR .
In the relationship matrix, the coefficients are equal to
(quantitative variables),
(qualitative variables) or
(one variable of each type).
The matrix shows an entanglement of the relationships between the two types of variables.
The representation of individuals (Figure 1) clearly shows three groups of individuals. The first axis opposes individuals 1 and 2 to all others. The second axis opposes individuals 3 and 4 to individuals 5 and 6.
The representation of variables (relationship square, Figure 2) shows that the first axis (
) is closely linked to variables
,
and
. The correlation circle (Figure 3) specifies the sign of the correlation between
,
and
; the representation of the categories (Figure 4) clarifies the nature of the relationship between
and
. Finally individuals 1 and 2, individualized by the first axis, are characterized by high values of
and
and by the categories
of
as well.
This example illustrates how the FAMD simultaneously analyses of quantitative and qualitative variables. Thus, it shows, in this example, a first dimension based on the two types of variables.
History
The FAMD's original work is due to Brigitte Escofier and Gilbert Saporta. This work was resumed in 2002 by Jérôme Pagès. The most complete presentation of FAMD in English is included in a book of Jérôme Pagès.
Software
The method is implemented in the R packag
FactoMineR The method is implemented in the Python librar
prince
References
{{reflist
Factor analysis