XLSTAT: Run Generalized Procrustes Analysis (GPA) in Excel

 

Background

Generalized Procrustes Analysis (GPA), a method that is used in several domains, is used in sensory analysis before a Preference Mapping to reduce the scale effects and to obtain a consensus configuration. It also allows comparing the proximity between the terms that are used by different experts to describe products.

 

Dataset for Example

The data come from a study where ten experts rated four similar cheeses multiple times on acidity, strangeness, and hardness. The values here are the average ratings per cheese and expert.

To create a consensus configuration for external preference mapping, the data will be transformed to remove scaling effects (differences in rating scale use) and position effects (tendencies to rate higher or lower).

 

Setting Up

• Open XLSTAT.

• Click on the Marketing/ Multiblock data analysis menu and select the Generalized Procrustes Analysis feature.

• After clicking the button, the dialog box opens.

In the General tab:

• Select the data for the configurations (each configuration is one expert’s set of ratings).

• Enter 10 as the number of configurations (experts).

• Since each expert rated on the same 3 dimensions, choose the Equal option for the number of dimensions.

• If dimensions varied, you would select a column specifying each configuration’s dimensions.

• For clarity, select labels for configurations (experts) and objects (cheeses).

In the Options tab:

• Select the following options.

 

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The PANOVA table summarizes the effectiveness of each Generalized Procrustes Analysis transformation in reducing total variability. Scaling is the most effective, shown by the lowest p-value. 

The second table and chart show residuals by object after transformation. Cheese C3 has the smallest residual, indicating strong consensus among experts.

The third table and chart show residuals by configuration after transformation. Expert2 has the highest residual, indicating their ratings deviate most from the consensus.

The next table and chart show scaling factors from the Generalized Procrustes Analysis. Factors below 1 mean an expert used a wider scale; above 1 means a narrower scale. Experts 1 and 3 tend to use a narrower scale than others.

A consensus test is performed using permutations to verify if the consensus configuration is significant. It checks whether the observed Rc (variance explained by consensus) is higher than 95% of values from permuted data.

Another permutation test is used to verify how many dimensions should be retained to display the results. We see here that for the third dimension, the F value is below the 95th percentile. So we can conclude that two dimensions are enough.

The next results show the PCA on the consensus configuration (unstandardized). Although Gower’s Procrustes Analysis aligns each expert’s data to consensus via rotation, this PCA provides the optimal transformation under PCA constraints. The eigenvalues indicate that 99% of variability is captured by the first two axes, consistent across all experts.

 

Results are split between the consensus configuration and individual configurations. The consensus object coordinates can be used later in a PREFMAP analysis as product positions. The correlation circle shows "Strangeness" mainly on the negative side of axis 1, while Acidity and Hardness are mixed. Expert 6’s zero rating on Strangeness places it at the origin.

The next two charts map objects, colored by configuration and by object. Points cluster near the first axis, which explains 96% of variability. XLSTAT uses orthonormal maps to prevent misinterpretation.

To improve legibility, adjust the scale options (via Excel or XLSTAT AxesZoomer), resulting in the following clearer map:

Cheeses C1 and C3 are clearly separated on the map, showing strong expert consensus. In contrast, C2 and C4 are less distinct, indicating experts find them harder to differentiate.