Factor analysis of mixed data

Criterion

The data include ${\displaystyle K}$ quantitative variables ${\displaystyle k=1,\dots ,K}$ and ${\displaystyle Q}$ qualitative variables ${\displaystyle q=1,\dots ,Q}$ .

${\displaystyle z}$ is a quantitative variable. We note:

• ${\displaystyle r(z,k)}$ the correlation coefficient between variables ${\displaystyle k}$ and ${\displaystyle z}$ ;
• ${\displaystyle {\eta }^2(z,q)}$ the squared correlation ratio between variables ${\displaystyle z}$ and ${\displaystyle q}$ .

In the PCA of ${\displaystyle K}$, we look for the function on ${\displaystyle I}$ (a function on ${\displaystyle I}$ assigns a value to each individual, it is the case for initial variables and principal components) the most correlated to all ${\displaystyle K}$ variables in the following sense:

${\displaystyle \sum _k{r}^2(z,k)}$ maximum.

In MCA of Q, we look for the function on ${\displaystyle I}$ more related to all ${\displaystyle Q}$ variables in the following sense:

${\displaystyle \sum _q{\eta }^2(z,q)}$ maximum.

In FAMD ${\displaystyle \{K,Q\}}$, we look for the function on ${\displaystyle I}$ the more related to all ${\displaystyle K+Q}$ variables in the following sense:

${\displaystyle \sum _k{r}^2(z,k)+\sum _q{\eta }^2(z,q)}$ 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 ${\displaystyle I}$ .

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 ${\displaystyle j}$ along axis ${\displaystyle s}$ is equal to squared correlation ratio between the variable ${\displaystyle j}$ and the factor of rank ${\displaystyle s}$ (denoted ${\displaystyle {\eta }^2(j,s)}$). The coordinates of quantitative variable ${\displaystyle k}$ along axis ${\displaystyle s}$ is equal to the squared correlation coefficient between the variable ${\displaystyle k}$ and the factor of rank ${\displaystyle s}$ (denoted ${\displaystyle r^2(k,s)}$).

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 ${\displaystyle l}$ and column ${\displaystyle c}$:

• If the variables ${\displaystyle l}$ and ${\displaystyle c}$ are quantitative, the squared correlation coefficient between the variables ${\displaystyle l}$ and ${\displaystyle c}$ ;
• If the variable ${\displaystyle l}$ is qualitative and the variable ${\displaystyle c}$ is quantitative, the squared correlation ratio between ${\displaystyle l}$ and ${\displaystyle c}$;
• If the variables ${\displaystyle l}$ and ${\displaystyle c}$ are qualitative, the indicator ${\displaystyle {\varphi }^2}$ between the variables ${\displaystyle l}$ and ${\displaystyle c}$.

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 ${\displaystyle R^2}$ (quantitative variables), ${\displaystyle {\varphi }^2}$ (qualitative variables) or ${\displaystyle {\eta }^2}$ (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 (${\displaystyle F1}$) is closely linked to variables ${\displaystyle k_2}$, ${\displaystyle k_3}$ and ${\displaystyle Q_3}$ . The correlation circle (Figure 3) specifies the sign of the correlation between ${\displaystyle F1}$, ${\displaystyle k_2}$ and ${\displaystyle k_3}$; the representation of the categories (Figure 4) clarifies the nature of the relationship between ${\displaystyle F1}$ and ${\displaystyle Q_3}$. Finally individuals 1 and 2, individualized by the first axis, are characterized by high values of ${\displaystyle k_2}$ and ${\displaystyle k_3}$ and by the categories ${\displaystyle c}$ of ${\displaystyle Q_3}$ 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. A more complete presentation of FAMD in English is included in a book of Jérôme Pagès.

Software

The method is implemented in the R package FactoMineR. The method is implemented in the Python library prince.