XLSTAT: Multidimensional Scaling (MDS) in Excel

 

Background

Multidimensional Scaling ( MDS ) is a data analysis method which is widely used in marketing and psychometrics.

The aim of the methods is to build a mapping of a series of individuals from a proximity matrix (similarities or dissimilarities) between these individuals. In the ideal case where we have a matrix giving the distances between some points on a surface (for example the cities of a country), the Multidimensional Scaling allows rebuilding the exact map of the points (within about a symmetry and/or rotation).

To build an optimal representation, the Multidimensional Scaling algorithm minimizes a criterion called "Stress". The closer the stress is to zero, the better the representation.

 

Dataset for Example

The attached data correspond to a survey performed over 10 testers which have been asked to rate (the score ranges from 1 to 5) five chocolate bars, where only the product P1 is already available on the market.

Our aim is to show how the products position themselves on a map, given the opinion of the testers.

Review the video and text below for details.

 

Setting Up

Creating a Proximity Matrix

A proximity matrix is needed to perform a Multidimensional Scaling analysis, but here we have a individuals x products table. Therefore, we need first to compute the dissimilarities between products.

• Open XLSTAT.

• Click on the Describing data menu and select the Similarity/Dissimilarity matrices feature.

• The dialog box appears.

In the General tab:

• Select the data from the Excel sheet.

• Set Data type to Quantitative.

• Set Proximity type to Euclidean distance.

• The proximity matrix is generated by default and doesn't need to be specified as output.

• Click OK to start the computations.

You will then get the Euclidean distance matrix between products, which serves as the basis for Multidimensional Scaling analysis.

 

Setting up the Multidimensional Scaling dialog box

• Click on the Analyzing data menu and select the Multidimensional Scaling feature.

• The Multidimensional Scaling dialog appears.

In the General tab:

• Select the Euclidean distance matrix from Excel.

• Check the Dissimilarities option.

• Choose the absolute model to preserve original distances without scale effects.

• Run the analysis from 4 to 2 dimensions to evaluate distortion from dimensionality reduction.

In the Options tab:

• Select Random for Initial configuration.

• Choose Kruskal's stress for Stress.

Note: Without a specified start, initial points are random, so results may vary slightly but won't affect interpretation. To ensure a true optimum, increase repetitions, max iterations, and accuracy.

• Click on the OK button to launch the computations.

• After you have accepted to plot the map on the first two dimensions by clicking Done.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

The first table shows stress decreasing with more dimensions, with a sharp drop between 2D and 3D and stability from 3D to 4D (4D is perfect for 5 points).

A map is created on the Dim1 x Dim2 plan, for the 2-dimensional configuration.

Maps can be built for other axis pairs in 3D or 4D, but viewing higher-dimensional maps in 2D may cause misleading projection effects. Use 2D maps only for 2D configurations. To view 4D data in 2D, apply PCA first.

For better visualization, use XLSTAT-3DPlot to create 3D plots by selecting the data.

• Click on the Visualizing data menu and select the XLSTAT-3DPlot feature.

The result we obtain is as follows:

Testers clearly distinguished the products. P2, with more chocolate, and P4, with the least, are opposite on the 3D chart. Testers significantly preferred P2. Although P3 and P5 have similar average scores, they are distant in the plot, reflecting mixed opinions due to peanuts in P3, liked by some testers but not others.

 

Conclusion

In conclusion, Multidimensional Scaling maps tester-rated products, offering richer insights than basic statistics.

Note: There’s no strict statistical test for representation quality, but the Shepard diagram helps assess it. It plots observed dissimilarities (x-axis) against MDS distances (y-axis). Greater spread means lower reliability; a preserved ranking indicates reliability, and points on a line signify perfect quality.

The chart on the left corresponds, for the data used in this example, to the representation in a 4D space. The chart on the right corresponds to the representation in the 2D space. We notice a strong difference of the spread of the points between the two charts.

Note: for an absolute model the disparities are equal to the dissimilarities, which is why they are confounded with the line for the 2D space, and hidden behind the distances on the 4D Sheppard diagram.