Factor analysis of mixed data
In statistics, factor analysis of mixed data, or factorial analysis of mixed data, 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 founded by Jean-Paul Benzécri.
The term mixed refers to the simultaneous presence, as active elements, of quantitative and qualitative variables. Roughly, we can say that FAMD works as a principal components analysis for quantitative variables and as a multiple correspondence analysis 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 coefficients between the variables and factors on individuals ; the representation obtained is a correlation circle.
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.
When the active variables are mixed, the usual practice is to perform discretization on the quantitative variables. Data thus obtained can be processed by MCA.
This practice reaches its limits:
- When there are few individuals in which case the MCA is unstable ;
- When there are few qualitative variables with respect to quantitative variables.
Criterion
is a quantitative variable. We note:
- the correlation coefficient between variables and ;
- the squared correlation ratio between variables and .
In MCA of Q, we look for the function on more related to all variables in the following sense:
In FAMD, we look for the function on the more related to all variables in the following sense:
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.
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.
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 . The coordinates of quantitative variable along axis is equal to the squared correlation coefficient between the variable and the factor of rank .
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
In the relationship matrix, the coefficients are equal to , or . The matrix shows an entanglement of the relationships between the two types of variables. The representation of individuals 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 shows that the first axis is closely linked to variables, and . The correlation circle specifies the sign of the correlation between, and ; the representation of the categories 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. HistoryThe 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.SoftwareThe method is implemented in the R package |