Correspondence Analysis (CA) is an exploratory multivariate statistical technique designed to analyze and visualize relationships in categorical data. It is most often applied to contingency tables—tables that cross-classify observations into categories—making it particularly useful for survey data, market research, linguistics, ecology, and the social sciences.
The main idea of CA is to represent rows and columns of a contingency table as points in a low-dimensional space (usually two dimensions). This spatial map provides an intuitive visualization of the associations: categories that are closely related appear near each other, while those that are more distinct are farther apart.
From a mathematical perspective, correspondence analysis is based on a decomposition of the chi-square statistic associated with the contingency table. It extracts latent dimensions that explain the structure of association between rows and columns. These dimensions serve as the axes of the map, allowing us to summarize complex, high-dimensional relationships in a simple graphical form. In practice, correspondence analysis provides a powerful way to uncover patterns and associations hidden in categorical data, offering insights that go beyond simple frequency counts or cross-tabulations. For mathematical definitions see for example [1] (where you can also find interpretation on example used below).
Correspondence analysis and Multiple Correspondence Analysis is available in BESH stat starting from version 0.23.
Example
Example is taken from [2] page 66. A survey of 193 staff members of a company, in order to formulate a smoking policy. The staff members are cross-tabulated according to their rank (five levels) and a categorization of their smoking habits (four groups).
| None | Light | Medium | Heavy | |
| Senior Managers | 4 | 2 | 3 | 2 |
| Junior Managers | 4 | 3 | 7 | 4 |
| Senior Employees | 25 | 10 | 12 | 4 |
| Junior Employees | 18 | 24 | 33 | 13 |
| Secretaries | 10 | 6 | 7 | 2 |
To analyse the data in BESH stat copy table above to excel sheet and in the BESH stat menu select either <Contingency table analysis> → Correspondence Analysis or <Multivariate> → Correspondence Analysis. In the input section select whole contingency table including labels (and select the “Labels Selected” checkbox).
Results
First, it appears that, with a single dimension, 87.76% of the inertia can be explained, that is, the relative frequency values that can be reconstructed from a single dimension can reproduce 87.76% of the total Chi-square value (and, thus, of the inertia) for this two-way table; two dimensions allow you to explain 99.51%. Note: the maximum number of eigenvalues that can be extracted from a two-way table is equal to the minimum of the number of columns minus 1, and the number of rows minus 1.
Interpretation
The purpose of correspondence analysis is to reproduce the distances between the row and column points in a two-way table in a lower-dimensional display. What is important are the distances of the points in the two-dimensional display, which are informative in that row points that are close to each other are similar with regard to the pattern of relative frequencies across the columns. Notice that the three smoking groups (red points) on the left will project very close together on the first (horizontal) axis, a long way from the non-smoking point on the right. This is the greatest single feature in the data (considering the first axis accounts for 87% of information).
The second (vertical) axis pulls apart the three smoking levels. The profiles do not differ as much vertically as horizontally, as indicated by the much lower percentage of inertia on the second axis. Nevertheless, we can conclude that the profile of JE has relatively more light smokers than heavy smokers compared to that of JM, even though both these groups have similar percentages of smokers as seen by their similar positions on the horizontal axis.
Looking only at the row categories on the CA plot, we can see that the groups furthest apart are Junior Employees (JE) and Junior Managers (JM) on the left-hand side, opposed to Senior Employees (SE) on the right-hand side — hence the greatest differences in smoking habits are between these extremes.
For further discussion and interpretation see for example Statistica documentation or [1].
References
- Correspondence analysis. NCSS. https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Correspondence_Analysis.pdf
- Michael Greenacre. CORRESPONDENCE ANALYSIS in PRACTICE. 2ed. 2007