Multidimensional Scaling (MDS) Perceptual Map

What multidimensional scaling is

Multidimensional scaling (MDS) is a family of methods that place objects, such as brands, products, stores or political candidates, as points in a low-dimensional space so that the distances between the points reflect how different the objects are. Objects perceived as similar end up close together; objects perceived as different end up far apart. In marketing, the result is called a perceptual map or positioning map and is used to see which brands compete most directly, where the white spaces in a market lie and how a repositioning moves a brand relative to competitors.

MDS starts from a matrix of dissimilarities. These can come directly from respondents (“How similar are brand A and brand B on a scale of 1 to 9?”), from substitution or switching data, or, as is most common, from computing the distance between brands’ attribute profiles in a rating survey. This tool supports all three: attribute ratings, a dissimilarity matrix or a similarity matrix.

Classical and non-metric MDS

• Classical (Torgerson) metric MDS treats the dissimilarities as actual distances. It double-centres the matrix of squared distances and takes its eigen-decomposition, which gives the coordinates directly, with no iterations. The eigenvalues show how much of the structure each dimension captures. With Euclidean distances computed from ratings, classical MDS is equivalent to principal component analysis of the profiles.
• Non-metric (Kruskal) MDS uses only the rank order of the dissimilarities: it looks for a configuration in which larger dissimilarities correspond to larger map distances, without requiring proportionality. This is appropriate for judgements on rating scales, whose intervals are not truly metric. The tool uses the SMACOF majorisation algorithm with monotone (isotonic) regression, starting from the classical solution, and minimises Kruskal’s stress-1.

Stress and the Shepard diagram

Stress-1 measures how badly the map distances fit the (transformed) dissimilarities: 0 is a perfect fit. Kruskal’s classic guidelines are: below .025 excellent, below .05 good, below .10 fair, below .20 poor, above .20 very poor. These guidelines depend on the number of objects; with only five or six objects even random data fit well in two dimensions, so judge stress with the number of objects in mind. The tool also reports the correlation between input dissimilarities and map distances.

The Shepard diagram plots every pair’s input dissimilarity against its distance on the map. For non-metric MDS, the stepped line is the monotone fit (the disparities); points close to it fit well. Pairs far from the line are badly represented, so be cautious when interpreting their distance on the map.

Naming the dimensions with attribute vectors

MDS axes have no inherent meaning and their orientation is arbitrary. When the map is built from attribute ratings, the tool fits each attribute into the space (property fitting): it regresses the brands’ mean ratings on the two coordinates and draws an arrow in the direction in which the attribute increases fastest. The arrow’s length is proportional to √R², so long arrows are attributes that the map represents well and short arrows are attributes that do not align with the main dimensions. To read a brand’s score on an attribute, project the brand onto the arrow’s line: brands further in the arrow’s direction are rated higher.

Worked example with the built-in data

The example contains ratings of eight brands on six attributes by 60 respondents (480 rows). The tool averages each brand’s ratings, standardises the attributes and computes Euclidean distances between brands. The non-metric solution has a stress of about .08 (fair) and the map distances correlate .93 with the input distances. Aurora and Ember sit close together, with the Innovation and Design arrows pointing towards them, so they compete for the same design-led, innovative position. Cobalt and Fjord lie in the direction of Reliability and Service, Brava towards Value for money and Grove towards Sustainability. Design (R² = .97) and Reliability (R² = .96) are best represented; Sustainability is weakly represented (R² = .38), meaning it varies across brands in a way the two dimensions do not capture.

Preparing the data

• Attribute ratings: long format with one row per respondent × brand, a brand label column and one numeric column per attribute. Several respondents per brand are averaged; a single row per brand also works.
• Dissimilarity matrix: one row per object with a label column and one numeric column per object in the same order, so the selected columns form a square matrix. The tool symmetrises the matrix and ignores the diagonal.
• Similarity matrix: as above, but larger values mean more similar; the tool converts it to dissimilarities.

Use at least four objects, and ideally eight or more for a meaningful two-dimensional map.

MDS versus correspondence analysis and PCA

Use correspondence analysis when the data are counts or pick-any associations (brand × attribute ticks). Use MDS when you have overall similarity judgements, or rating profiles and want a map that preserves distances between brands, especially with the non-metric option for ordinal ratings. PCA of brand-mean ratings gives a map very similar to classical MDS and adds loadings, but it treats ratings as metric. All three are descriptive maps; they show perceptions, not preferences. Preference mapping adds ideal points or vectors from liking data.

Interpreting perceptual maps well

• Interpret distances between objects, and directions of the attribute arrows, not the axes themselves.
• Only the relative positions matter: the map can be rotated or reflected without changing its meaning.
• Check stress and the quality of attribute fit before drawing strong conclusions.
• Empty regions of the map are gaps in perception, not necessarily attractive opportunities; combine the map with preference or demand data.
• If stress is poor in two dimensions, the perceptions are more complex; consider looking at the attribute table directly or segmenting respondents.

Reporting

Report the method, the source of the dissimilarities, the number of objects, the number of dimensions, stress, and the map with attribute vectors and their R². Example: “Non-metric multidimensional scaling of eight brands, based on standardised mean ratings on six attributes, produced a two-dimensional solution with stress-1 = .079.”

Glossary of MDS terms

• Dissimilarity: a number expressing how different two objects are; larger means more different.
• Configuration: the coordinates of all objects in the low-dimensional map.
• Disparities: the monotone transformation of dissimilarities that non-metric MDS fits.
• Stress-1: Kruskal’s measure of badness of fit between map distances and disparities.
• Shepard diagram: a scatter plot of input dissimilarities against map distances.
• Property fitting: projecting attribute ratings into the map as vectors to interpret its dimensions.
• Ideal point: in preference mapping, the location of a respondent’s most preferred combination of attributes.
• Torgerson scaling: the classical, closed-form solution of metric MDS based on an eigen-decomposition.

Key references: Kruskal (1964), Psychometrika; Borg and Groenen (2005), Modern Multidimensional Scaling; Hauser and Koppelman (1979), Journal of Marketing Research.