Correspondence Analysis (Brand Perceptual Map)

What correspondence analysis does

Correspondence analysis (CA) is a multivariate technique for visualising the associations in a two-way table of counts. It is a favourite of marketing researchers because it turns a brand-image grid, the classic survey question “Which of these brands do you associate with each of these statements?”, into a single perceptual map on which brands sit close to the attributes they are most strongly associated with. The same method maps product × usage occasion, segment × media habits, region × purchase reason or any other cross-tabulation.

CA was developed by Jean-Paul Benzécri and the French school of data analysis in the 1960s and is closely related to principal component analysis: where PCA decomposes the variance of numeric variables, CA decomposes the chi-square statistic of a contingency table. Each dimension captures part of the departure from independence, called inertia.

How it works, step by step

1. Proportions and masses. The table is divided by its grand total. Row and column totals of these proportions are called masses; they say how big each brand and each attribute is.
2. Profiles. Each row is turned into a profile (the distribution of that brand’s mentions across attributes) and compared with the average profile. CA cares about the shape of the profile, not its size, so a small brand with a distinctive image sits away from the centre while a big brand with an average image sits near it.
3. Standardised residuals. The tool computes (observed − expected) / √expected for every cell, where expected = row mass × column mass. The sum of their squares is the total inertia, equal to χ² / n.
4. Singular value decomposition. The matrix of residuals is decomposed into dimensions ordered by the inertia they explain. The squared singular values are the principal inertias (eigenvalues).
5. Coordinates. Rows and columns get principal coordinates on each dimension, and the first two dimensions form the map.

Reading the output

• χ² test: tests whether there is any association at all between rows and columns. If it is not significant, the map shows noise.
• Inertia by dimension: the percentage of the total association each dimension captures. A two-dimensional map that explains 70% or more of the inertia is usually a good summary.
• Coordinates: positions on the map. Points far from the origin have distinctive profiles; points near the origin are average.
• Contributions (Ctr): the percentage of a dimension’s inertia due to each point. Points with above-average contributions (highlighted) define the meaning of the dimension.
• cos²: the squared correlation between a point and a dimension, a measure of how well the dimension represents it. The quality column adds cos² over the two displayed dimensions; points with low quality are poorly represented on the map, so do not over-interpret their position.

Worked example with the built-in data

The example table counts associations between seven brands and six image attributes (2,141 mentions). The association is strongly significant (χ²(30) = 501.2, p < .001) and the first two dimensions capture 82.8% of the inertia. Dimension 1 (62.3%) is defined mainly by Stylish and Innovative on one side: Ember and Aurora sit there, so they own a design-led, innovative position. Dimension 2 (20.5%) contrasts Eco friendly (Grove) with Reliable and Good service (Cobalt). Brava sits near Affordable. Delta is close to the centre on dimension 1 and has a modest eco association: it has a weak, undifferentiated image. The interpretation table confirms the strongest associations, for example Ember with Stylish at more than twice the expected rate.

Symmetric and asymmetric maps

The default symmetric map plots both rows and columns in principal coordinates. It is the standard in marketing practice because it spreads both sets of points well. Strictly, distances between a row point and a column point are not defined in a symmetric map: interpret them by direction from the origin (points in the same direction are associated) and by the contributions, not by straight-line distance. If you need rigorous row-to-column interpretation, use an asymmetric map: in a row principal map, brands are in principal coordinates and attributes in standard coordinates, and a brand lies towards the attributes it is most associated with.

Preparing the data

Two layouts are accepted. A contingency table has one row per brand (with a label column) and one numeric column per attribute containing counts, exactly as many survey platforms export brand-image grids. Raw data has one row per mention or respondent with two categorical columns; the tool builds the cross-tabulation. Rows or columns with zero totals are dropped. Counts should be non-negative frequencies; percentages also work if they are on a common base, but counts are better because the χ² test uses the sample size.

Practical tips

• Have at least three rows and three columns; maps become informative with five or more of each.
• Combine rare categories, which can dominate a dimension with large but unreliable coordinates (watch for small masses with high contributions).
• Do not read brand size from the map. Report the masses or the raw counts if size matters.
• Rotate or flip mentally: the sign of each dimension is arbitrary, so left and right have no meaning beyond the labels near them.
• Compare maps over time only with care, because each analysis finds its own dimensions. A stacked table (brand × wave rows) analysed together keeps a common space.

CA, MDS and other mapping methods

Correspondence analysis is the method of choice when your data are counts or pick-any associations. When you have rating scales (each brand rated 1–10 on each attribute), multidimensional scaling with attribute vectors, or PCA on the brand means, is more natural; both are available on this site. Multiple correspondence analysis extends CA to more than two categorical variables. CA is also an excellent first look before a chi-square test of independence on the same table.

Reporting

Report the table size and total, the χ² test, the inertia of the first two dimensions, the map and the main associations, using contributions to name the dimensions. Example: “A correspondence analysis of brand × attribute associations (7 × 6, n = 2,141) showed a significant association, χ²(30) = 501.23, p < .001. The first two dimensions explained 62.3% and 20.5% of the inertia.”

Glossary of correspondence analysis terms

• Contingency table: a cross-tabulation of counts for two categorical variables.
• Mass: a row’s or column’s share of the grand total.
• Profile: a row or column divided by its total, showing its distribution across the other variable.
• Inertia: the weighted spread of profiles around the average profile, equal to χ²/n.
• Principal coordinates: coordinates scaled by the singular values, so distances between points of the same set are meaningful.
• Standard coordinates: coordinates with unit inertia, used for the reference set in asymmetric maps.
• Contribution: the share of a dimension’s inertia due to a point, used to name dimensions.
• Quality (cos²): how well a point is represented by the displayed dimensions.

Key references: Greenacre (2017), Correspondence Analysis in Practice (3rd ed.); Benzécri (1973), L’Analyse des Données; Hoffman and Franke (1986), Journal of Marketing Research.