XLSTAT: Multiple Factor Analysis (MFA) in Excel

 

Background

Multiple Factor Analysis (MFA) analyzes several variable tables simultaneously to explore relationships between observations, variables, and tables. Each table must contain variables of the same type (quantitative or qualitative), though table types may differ.

MFA involves two phases:

1. Individual Table Analysis:
   Perform a PCA (for quantitative) or MCA (for qualitative) on each table. Record the first eigenvalue of each.

2. Global Analysis:
   Conduct a weighted PCA on all combined tables.

   • Qualitative tables are converted to disjunctive form.

   • Variables are weighted based on category frequency.

   • Tables are weighted by the inverse of their first eigenvalue to balance influence regardless of size.

 

Dataset for Example

The attached data have been collected by Asselin C. and Morlat R. from INRA, Angers, France, and used in the following article: [ASSELIN C., PAGES J., and MORLAT R. (1992). Typologie sensorielle du Cabernet Franc et influence du terroir. Utilisation de méthodes statistiques multidimensionnelles. J. Int. Sci. Vigne Vin, 26, 3, 129-154].

The data correspond to the tasting of 21 wines from the Loire region in France by 36 of experts. The dataset comprises 21 observations and 31 dimensions. The 31 dimensions can be grouped into 6 categories:

• the first 2 qualitative variables are related to the geography (appellation and soil);

• the next 5 quantitative variables correspond to the olfaction after rest;

• the next 3 quantitative variables correspond to visual criteria;

• the next 10 quantitative variables correspond to the olfaction after shaking;

• the next 9 quantitative variables correspond to the taste;

• the last 2 quantitative variables correspond to global ratings.

With this dataset, we will analyze how the wines relate to each other, and to identify which criteria seem to agree (are redundant?) or disagree. We decide not to use the two qualitative variables and the last two quantitative variables in the first part of the study, but to only use them as supplementary variables at the end of the study: we don't want the analysis to be based on anything else but on objective tasting criteria.

 

Setting Up

• Open XLSTAT.

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

 

Once you click the button, the dialog box opens.

In the General tab:

• Select all variables of interest.

• Set the number of tables to 6.

• In Table label, select the names of the 6 tables: geog, rest, vis, shake, taste, glob.

• Specify the number of variables per table by selecting the corresponding Excel range (as counts vary).

• Check Variable labels since column headers are included.

In the Options tab:

• Select Mixed as the data type.

• Specify the column indicating variable types (use 0 for quantitative and 1 for qualitative).

 

In the Supplementary data tab:

• Select the column indicating table status.

• Use 1 for active tables and 0 for supplementary ones.

• 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, with supplementary table variables highlighted in blue.

Next, XLSTAT performs separate analyses on each table:

• PCA for quantitative tables

• MCA for qualitative tables

In this case, one MCA and five PCAs are conducted. These feed into the final Multiple Factor Analysis, a weighted PCA using the first eigenvalue of each table for weighting.

The MFA results begin with eigenvalue analysis, showing that the first two factors explain nearly 70% of the total variance.

The correlation map shows:

• Typicity and global quality are strongly linked to variables like Shake3 and Taste5, and align with Factor 1.

• Vision variables are also strongly correlated with Factor 1.

• Olfaction after shaking variables are dispersed across three quadrants, indicating diverse influence.

• Factor 2 is mainly associated with the Rest5 variable.

The observation chart with centroids shows:

• T1 and T2 wines cluster closely, influenced mainly by Factor 2 (linked to Rest5).

• 1DAM wine scores highest on Factor 1.

• 2DAM aligns with typicity and global quality, indicating it's one of the top-rated wines.

• "Ref" SOIL also aligns with high ratings.

• 1VAU and 2ING wines are in the opposite direction, indicating the poorest ratings.

The table coordinates are shown and used to create the tables map. The first factor strongly relates to the four active tables (high coordinates and contributions), while the second factor mainly associates with olfaction after rest and, to a lesser degree, olfaction after shaking.

Lg coefficients measure the pairwise relationships between tables. RV coefficients, derived from Lg, range from 0 to 1 for easier interpretation. Here, taste and olfaction after shaking are the closest tables, and surprisingly, taste and vision also show a high RV coefficient.

The partial axes map shows how factors from the first-phase separate analyses relate to the MFA factors. Here, there is a strong relationship, though this may not always occur.

The last chart adds projected points for supplementary observations—using only one table’s data while others are zeroed—and draws lines linking them to original points. This shows how each table influences an observation’s position. For example, for wine T2, the smell-at-rest data increases its distinctness from other wines. Generally, smell at rest often amplifies differences between wines.

 

Conclusion

Multiple Factor Analysis is a powerful method for analyzing complex datasets, offering diverse graphical outputs to visualize tables, variables, and observations. In this example, it effectively positioned wines on a map, enabling easy interpretation of their relationships.