StratumWhat Is Correspondence Analysis?
A big categorical cross-tab is a wall of counts you can't read. Correspondence analysis turns it into one map you can.
Runs in Stratum via Analyze ▸ Correspondence Analysis — the same menu path on Mac and Windows.
Suppose you ran a survey. You asked 1,186 people how they feel about fourteen Star Wars characters, on a scale from "Very favorably" to "Unfamiliar." Cross-tabulate character against rating and you get a 14 × 6 contingency table — 84 cells of counts, more than eleven thousand ratings in all. Somewhere in that grid is the whole shape of public opinion. But you cannot see it. Eighty-four numbers is a wall, not a picture.
Correspondence analysis is the tool that turns that wall into a picture. Think of it as PCA for categorical data: it takes a single cross-tab and lays its rows and its columns out on one scatter plot, positioned so that categories which go together sit close together. Read the map and the survey reads back at a glance.
The problem: a cross-tab too big to eyeball
A contingency table is the natural home for two categorical variables. Count how many respondents fall into each combination and you have the raw material. For a 2 × 2 table you can eyeball the pattern; a chi-squared test tells you whether the two variables are related, and Cramér's V tells you how strongly. On this survey the association is unmistakable — χ² = 4900.7 on 65 degrees of freedom, p < 0.0001, with Cramér's V = 0.29. Character and favorability are strongly linked.
But "strongly linked" is a single number. It confirms there is structure without showing you what the structure looks like — which characters cluster, which stands apart, which end of the rating scale each one leans toward. For that you need to see the table, not summarize it, and a 14 × 6 grid of counts is exactly the size where seeing it becomes impossible by hand.
The idea: PCA for categories
Here is the move. Principal component analysis takes many numeric columns and rotates them into a few directions that capture most of the variation, so you can plot high-dimensional data in two dimensions. Correspondence analysis does the same thing for a contingency table, with two twists: it works on counts rather than measurements, and it maps both the rows and the columns onto the same plot.
The quantity it decomposes is called inertia — the table's total departure from what you'd expect if the rows and columns were unrelated. (It is literally χ² divided by the sample size; here total inertia is 0.425.) Correspondence analysis splits that inertia into a handful of dimensions, ranked by how much of it each one explains, exactly the way PCA ranks components by variance. Keep the top two and you have a 2-D map.
Reading the map
Once the map is drawn, it reads like any scatter plot, with one rule: near means associated, far means distinctive. Two characters close together were rated similarly across the whole survey. A character sitting next to a rating category leans toward that rating. A point alone in a corner has a profile unlike anything else in the table.
On this survey the map tells its story instantly. The first dimension runs from loved to hated: the beloved originals — Han, Luke, Leia, Obi-Wan, Yoda, R2-D2 — pack so tightly that they all fall within about 0.05 of each other, while Darth Vader and the Emperor sit at the opposite end. The second dimension separates the familiar from the obscure. And out on his own, far from everyone, is Jar Jar Binks at roughly (0.86, 0.55) — the single most distinctive rating profile in the entire survey. You didn't compute that; you just looked.
How much fits in one picture?
The obvious worry with any 2-D projection is how much you throw away. Correspondence analysis answers that directly: every dimension reports the share of total inertia it carries, so you know exactly how faithful the map is.
Here the numbers are generous. Dimension 1 accounts for 76.3% of the inertia and Dimension 2 for 18.6%, so the flat two-dimensional map you're reading holds about 94.9% of everything in the original 84-cell table. Ninety-five percent of a survey, in a single scatter plot. The remaining structure is a rounding error you can safely ignore.
When to reach for it
Correspondence analysis earns its place the moment a cross-tab or a mosaic plot gets too big to read. A chi-squared test tells you two categorical variables are related; Cramér's V tells you how strongly. Neither shows you the shape of the relationship — and once a table grows past a handful of rows and columns, shape is exactly what you're missing. Correspondence analysis fills that gap: it is the picture that a large contingency table is trying, and failing, to be.
You'll find it wherever categorical data piles up — survey responses cross-tabulated against demographics, brands against attributes in market research, symptoms against diagnoses, words against documents. Anywhere you have a big table of counts and a nagging sense there's a pattern you can't quite see, this is the map that shows it.
In Stratum, it's a menu selection. Stack your categories into a two-column table, choose Analyze ▸ Correspondence Analysis, and the app decomposes the inertia, reports each dimension's share, and drills down to the map — rows and columns on one scatter, ready to read. No R, no code. This piece teaches the idea in a few minutes; if you'd like to watch the full survey worked end to end — reshape, run, and the Jar Jar reveal — that's the star-wars-survey case study this explainer feeds into.
Related: How to Run PCA → · Understanding PCA Loadings & Scores →