XLSTAT: Partial Least Squares PLS regression in Excel 

Background

Partial Least Squares regression is going to allow us to obtain a simultaneous map of the judges, the descriptors, and the products, and then to analyze for some judges which descriptors are related to their preferences.

 

Dataset for Example

The data used in this article correspond to 6 orange juices described by 16 physico-chemical descriptors and evaluated by 96 judges. The data has been extensively analyzed in [Tenenhaus, M., Pagès, J., Ambroisine L. and & Guinot, C. (2005); PLS methodology for studying relationships between hedonic judgments and product characteristics; Food Quality and Preference. 16, 4, pp 315-325].

 

Setting Up

• Open XLSTAT.

• Click on the Modelling data menu and select the PLS Regression feature.

• Once you have clicked the button, the Partial Least Squares regression dialog box is displayed.

In the General tab:

• In the Dependent variable(s) field, select with the mouse the ratings of the 96 judges. The ratings are the "Ys" of the model as we want to explain the ratings given by the judges.

• In the Quantitative variable(s) field, select the explanatory variables, that are in our case the physicochemical descriptors.

• The names of the orange juices have also been selected as Observation labels.

In the Options tab:

• Fix the number of components at 4 in the Stop conditions.

In the Charts tab:

• The Colored labels option has been activated in order to make the reading of the charts easier.

• Uncheck the Vectors option in order not to saturate the charts.

• The computations start when you click on OK. The display of the results is stopped to allow you to select the axes for the maps.

• In the “Select Axes” box you only need to click on Done so that the charts are only displayed for the first two axes.

 

Interpreting the Results

Initial tables show basic statistics and correlations (dependent variables in blue, quantitative predictors in black).

Then, PLS regression-specific results are presented.

The first table and bar chart show regression quality by number of components.

The Q² cumulative index reflects the overall fit and predictive quality of the 96 models. Even with 4 components, Q² remains low—indicating variable model quality across judges.

However, the cumulative R²Y and R²X values are near 1 with 4 components, showing that the PLS regression effectively summarizes both X and Y variables.

The first correlation map visualizes how X and Y variables correlate with the first two components.

Some judges near the center of the correlation map show low correlations. For instance, judge J54 is mainly correlated with the fourth component, which itself is weakly linked to the explanatory variables.

Vitamin C is poorly represented on the first two dimensions, indicating it has little influence on judges’ preferences—expected since it doesn’t strongly affect taste. Strong correlations are observed between fructose and glucose, between the two pH measures, and negative correlations between pH, acidity, and titer. Judges are diverse in preferences, as shown by their wide distribution across the correlation circle.

Another map plots dependent variables on the c vectors and explanatory variables on the w* vectors, illustrating overall relationships. The w* vectors reflect variable weights in the models.

Projecting an explanatory variable onto a dependent variable’s vector (shown only if there are fewer than 50 dependent variables) gives insight into its influence on that dependent variable.

The coordinates of the orange juices in the t-component space are available in a table and displayed on a map, showing clear differentiation between products.

A new correlation map overlays the products onto the previous map. "Obs" was renamed to "Juices" by selecting a point and editing the series name via the Excel toolbar. As with most XLSTAT outputs, the charts are standard Excel charts and can be easily customized.

Tenenhaus et al. interpret this chart as revealing four distinct clusters of judges. They recommend rerunning PLS regression for each group, which improves fit. For example, R²Y rises from 0.53 (all judges) to 0.63 for the first group.

Next, results for the u and u~ components are shown, with a chart displaying juices in the u~ space.

Tables then show Q² and cumulative Q² values by component for each dependent variable. For some (e.g., J5, J6, J7), the best Q² is reached with just 1–2 components.

Optional tables (not included here) show R² values for each input variable across components.

Finally, the VIP (Variable Importance for the Projection) table highlights key explanatory variables. With one component, Vitamin C, Sweetening Power, Odor Intensity, and Taste Intensity show low influence.

The next table shows the model coefficients for each dependent variable, followed by the model equations, which can be reused for simulation or prediction.

For each model, XLSTAT provides goodness-of-fit metrics, standardized coefficients, and prediction/residuals tables. Judge J1’s model shows a good fit (R² = 0.88), but low degrees of freedom suggest possible overfitting. Wide confidence intervals for the coefficients support this. Since the cumulative Q² peaked with 2 components, a simpler model is likely better.

A new PLS regression was run for J1 with 2 components. Results on the PLS2 sheet show improved model quality, with the updated chart displaying the standardized coefficients.

Only "Smell intensity" and "Odor typicity" have coefficients significantly different from zero. The predictions and residuals table confirms that Judge 1's ratings are well reproduced by the model.

Finally, the DModX and DModY tables and charts help detect outliers. In this case, no outliers are found, as all values are below the DCritX and DCritY thresholds.