Multiple Factor Analysis (MFA)

When to use Multiple Factor Analysis

Multiple Factor Analysis (MFA) makes it possible to analyze several tables of variables simultaneously, and to obtain results, in particular charts, that allow studying the relationship between the observations, the variables and tables. Within a table the variables must be of the same type (quantitative or qualitative), but the tables can be of different types.

This method can be very useful to analyze surveys for which one can identify several groups of variables, or for which the same questions are asked at several time intervals.

Principles of Multiple Factor Analysis

The Multiple Factor Analysis is a synthesis of the PCA (Principal Component Analysis) and the MCA (Multiple Correspondence Analysis) that it generalizes to enable the use of quantitative and qualitative variables. The methodology of the MFA breaks up into two phases:

1. We successively carry out for each table a PCA or an MCA according to the type of the variables of the table. One stores the value of the first eigenvalue of each analysis to then weigh the various tables in the second part of the analysis.
2. One carries out a weighted PCA on the columns of all the tables, knowing that the tables of qualitative variables are transformed into complete disjunctive tables, each indicator variable having a weight that is a function of the frequency of the corresponding category. The weighting of the tables makes it possible to prevent that the tables that include more variables do not weigh too much in the analysis.

The originality of method is that it allows visualizing in a two or three dimensional space, the tables (each table being represented by a point), the variables, the principal axes of the analyses of the first phase, and the individuals. In addition, one can study the impact of the other tables on an observation by simultaneously visualizing the observation described by the all the variables and the projected observations described by the variables of only one table.

Results for Multiple Factor Analysis

Correlation/Covariance matrix

This table shows the correlations between all the quantitative variables. The type of coefficient depends on what has been chosen in the dialog box.

Results on individual tables

The results of the analyses performed on each individual table (PCA or MCA) are then displayed. These results are identical to those you would obtain after running the PCA or MCA function of XLSTAT.

Multiple Factor Analysis

Afterwards, the results of the second phase of the MFA are displayed.

• Eigenvalues: The eigenvalues and corresponding chart (scree plot) are displayed. The number of eigenvalues is equal to the number of non-null eigenvalues.
• Eigenvectors: This table shows the eigenvectors obtained from the spectral decomposition. These vectors take into account the variable weights used in the Multiple Factor Analysis.
• Coordinates of the tables: The coordinates of the tables are then displayed and used to create the plots of the tables. The latter allow to visualize the distance between the tables. The coordinates of the supplementary tables are displayed in the second part of the table.
• Contributions (%): Contributions are an interpretation aid. The tables which had the highest influence in building the axes are those whose contributions are highest.
• Squared cosines: As in other factor methods, squared cosine analysis is used to avoid interpretation errors due to projection effects. If the squared cosines associated with the axes used on a chart are low, the position of the observation or the variable in question should not be interpreted.
• Lg coefficients: The Lg coefficients of relationship between the tables allow to measure to what extend the tables are related two by two. The more variables of a first table are related to the variables of the second table, the higher the Lg coefficient.
• RV coefficients: The RV coefficients of relationship between the tables are another measure derived from the Lg coefficients. The value of the RV coefficients varies between 0 and 1.

Results for quantitative variables

The results that follow concern the quantitative variables. As for a PCA, the coordinates of the variables (factor loadings), their correlation with the axes, the contributions and the squared cosines are displayed.

The coordinates of the partial axes, and even more their correlations, allow to visualize in the new space the link between the factors obtained from the first phase of the Multiple Factor Analysis, and those obtained from the second phase.

The results that concern the observations are then displayed as they are after a PCA (coordinates, contributions in %, and squared cosines).

Last, the coordinates of the projected points in the space resulting from the Multiple Factor Analysis are displayed. The projected points correspond to projections of the observations in the spaces reduced to the dimensions of each table. The representation of the projected points superimposed with those of the complete observations makes it possible to visualize at the same time the diversity of the information brought by the various tables for a given observation, and to visualize the relative distances from two observations according to the various tables.

Tutorials

• Multiple Factor Analysis (MFA) with XLSTAT

Screenshots