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Correspondence analysis
Definition
Correspondence analysis is an exploratory method for studying relationships in a table of nonnegative counts, especially a contingency table. It represents row and column patterns in a smaller number of dimensions to help reveal departures from independence.
What it means
Suppose a table counts how different customer groups choose among service channels. Raw counts mix two things: how large each group is and how its pattern of choices differs. Correspondence analysis focuses on profiles and their association structure.
The resulting map can make a large table easier to discuss. Rows with similar profiles can appear near one another under the appropriate scaling, and axes summarize important patterns in the table.
The map is not a causal explanation or a direct ranking. Its geometry depends on the scaling, and a two-dimensional display may omit substantial information. A manager should inspect the underlying counts before interpreting an eye-catching point.
How it works
The method converts the table into proportions and compares observed values with those expected under independence of rows and columns. It decomposes the standardized departures into axes, often using singular value decomposition.
Row and column masses determine weighting. The analysis reports inertia, which describes the association represented by the dimensions, along with contributions and representation quality.
Plot interpretation requires care. Distances within row points or within column points have specific meanings under suitable scaling; distances between a row point and a column point should not be read as ordinary universal association measures. Check the software's display convention.
Three practical examples
- A service team explores how channel choices differ across customer categories using a count table.
- A survey analyst examines the relationship between response categories and participant groups.
- An editor explores associations between document categories and term counts before choosing a more detailed text-analysis method.
Think of it this way
Think of comparing several baskets by the proportions of items they contain, rather than only by basket size.
The analogy has limits because correspondence analysis uses a particular weighted geometry. A map is not simply a picture of percentages, and rare categories can have large coordinates without dominating the overall association.
Worked example
Separating size from profile
Imagine a fictional table with two customer groups and two service channels. Group A has 80 email contacts and 20 phone contacts. Group B has 20 email contacts and 80 phone contacts.
Each group contains 100 contacts, and each channel contains 100. If group and channel were independent, the expected count in every cell would be 100 times 100 divided by 200, or 50.
The observed table departs from that baseline: Group A favors email relative to the overall pattern, and Group B favors phone. Correspondence analysis would summarize this association. With only two rows and two columns, there is at most one nontrivial dimension.
This does not show why preferences differ. Perhaps the groups have different issue types or access needs. The next step is to inspect those possibilities, not to treat a position on the map as a causal customer trait.
Illustrative case study
A fictional membership organization maps relationships between member types and event formats. A very rare format appears far from the origin, and the team initially assumes it is the most important finding.
The analyst checks its count, contribution and representation quality. The position reflects a distinctive profile based on few observations, so the team treats it as a question for follow-up rather than a major strategic conclusion.
They present the map alongside a compact count table and the proportion of inertia shown. The case illustrates how a visual summary becomes more useful when readers can see the evidence and the information omitted by the display.
Common misconceptions and mistakes
- Using raw counts alone to infer preference: group size and profile are different.
- Reading proximity between any two plotted points identically: row-column geometry depends on scaling.
- Treating far-from-origin points as automatically most important: mass and contribution also matter.
- Ignoring omitted dimensions: a two-axis map may not represent every category well.
Business use
Correspondence analysis can help teams explore categorical relationships and communicate patterns in a large contingency table. It is especially useful as a starting point for questions rather than as a final decision rule.
Ask for the original counts, axis information and notes on rare categories. Check whether the table contains comparable observations and whether repeated observations require special interpretation. If decisions depend on a particular association, follow the exploratory map with an analysis appropriate to that question and data collection design.
People also ask
Is correspondence analysis the same as principal component analysis?
They are related dimension-reduction approaches but use different data representations and geometry.
Does it require a target label?
No. It is generally exploratory.
Is multiple correspondence analysis identical?
No. Multiple correspondence analysis extends related ideas to data involving several categorical variables.
Check your understanding
A rare response category lies far from the origin on a map. Does that alone prove it is the most important business finding?
Answer: No. Inspect its count, contribution, representation quality and the original table. A distinctive position based on few observations may be unstable or operationally minor.
Related terms
- Contingency table
- Singular value decomposition
- Multiple correspondence analysis
- chi-square distance
- inertia
Learn next
Read contingency table. Calculate row percentages for a small count table before looking at its correspondence-analysis map, then compare the two views.
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Sources
FactoMineR documentation, Correspondence Analysis: https://rdrr.io/cran/FactoMineR/man/CA.html, checked for contingency-table input, dimensions, contributions and inertia outputs. Package research, Multiple Factor Analysis for Contingency Tables: https://journal.r-project.org/articles/RJ-2013-003/RJ-2013-003.pdf, introductory CA geometry discussion consulted; its main subject is the multiple-table extension. Examples are original.
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