XLSTAT: Multiple Factor Analysis (MFA) on Frequency Tables

 

Background

Multiple Factor Analysis (MFA) simultaneously analyzes several tables of variables, focusing here on frequency tables. It involves two phases:

1. Perform a Correspondence Analysis (CA) on each table and record the first eigenvalue.

2. Conduct a weighted PCA on all combined tables, using the eigenvalues to balance their influence and prevent larger tables from dominating the analysis.

 

Dataset for Example

The attached data have been collected by LAURIE E. FRIDAY., and used in the following article: [Friday, L.E. (1987) The diversity of macroinvertebrate and macrophyte communities in ponds,  Freshwater Biology , 18, 87–104.].

These data correspond to a counting of species classified into different groups (tables) on various sites. We have 62 species divided into 5 groups and the measures of counting were made on 12 different sites, the species are divided in the following way:

• Table 1 contains 6 species belonging to the group Oligochaeta.
• Table 2 contains 13 species belonging to the group Coleoptera.
• Table 3 contains 13 species belonging to the group Trichoptera.
• Table 4 contains 8 species belonging to the group Mollusca.
• Table 5 contains 22 species belonging to the group Diptera.

 

Setting Up

• Open XLSTAT.

• Click on the Marketing/ Multiblock data analysis menu and select the Multiple Factor Analysis (MFA) feature.

Note that MFA is also available under the Sensory profiling menu.

 

• After clicking the button, the MFA dialog box appears.

  In the General tab:

  • Select the data covering all variables of interest, grouped into 5 tables.

  • Specify the number of tables as 5.

  • Enter the table names: Oligochaeta, Coleoptera, Trichoptera, Mollusca, and Diptera.

  • Define the number of variables per table by selecting a range in Excel.

  • Activate Variable labels since column headers are present.

 

In the Options tab:

• Select the frequency data type as we are using frequency tables corresponding to count data of species.

 

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The first results show descriptive statistics, followed by separate Correspondence Analyses (CA) on each frequency table.

These preliminary results feed into the final phase—a weighted PCA on all tables combined. The MFA results begin with the eigenvalue analysis of this weighted PCA.

The first results display variables (color-coded by group) and observations, similar to a standard PCA.

    

MFA-specific results follow, showing table coordinates used in the tables graph. The first axis relates strongly to Trichoptera and Oligochaeta tables, while the second axis links to Mollusca and Coleoptera tables.

  

Lg coefficients measure pairwise table relationships, while RV coefficients (ranging from 0 to 1) offer a more interpretable version. Here, taste and olfaction after shaking are the closest tables, and surprisingly, Diptera and Coleoptera have the highest RV coefficient.

The partial axes graph in MFA shows correlations between axes from separate analyses and MFA axes. For example, axis 1 of Oligochaeta and Trichoptera CAs are strongly negatively correlated with MFA axis 1.

  

Examining the Trichoptera CA variable coordinates (row 517), negative values on the first axis correspond to high frequencies of species T1, T2, T3, T4, and T5.

  

The last MFA chart adds projected points to observations, connected by lines. These points represent supplementary observations based on a single table, with others set to zero, showing each table’s influence on a point’s position. For example, site 14 is made more distinct by the Mollusca table, as is site 11 at the chart’s bottom.

 

Conclusion

Multiple Factor Analysis is a powerful method for analyzing complex data sets, offering diverse graphical outputs to visualize tables, variables, and observations. Here, it enabled quick mapping and clear interpretation of site positions.