Multiple Correspondence Analysis (MCA) in R: Visualize Categorical Tables

Multiple correspondence analysis (MCA) is correspondence analysis applied to many categorical variables at once. In R, FactoMineR::MCA() plus factoextra turns a wide table of factors into a 2D factor map where categories that co-occur sit close together.

How does MCA extend correspondence analysis to many categorical variables?

Correspondence analysis maps two categorical variables. Real survey data has dozens. The fix is to recode every variable into 0/1 dummies, stack them into one big indicator matrix, and run CA on that. The result is a single factor map that places every individual and every category of every variable on the same coordinate system. We'll fit it on the tea survey from FactoMineR, then peek at the theory.

Before any math, let's see the payoff. The block below loads FactoMineR and factoextra, fits MCA on the first 18 columns of tea (300 respondents, 18 questions about how they drink tea), and draws the variable-category biplot.

The plot puts each category of each variable on the map. Convenient categories like tea bag, lunch, and alone cluster on one side of Dim 1. Slow-ritual categories like unpackaged, evening, and friends cluster on the other. MCA recovered a "fast vs ceremonial tea drinker" axis from 18 unrelated questions without you writing a single comparison.

Figure 1: MCA recodes a categorical table into either an indicator matrix or a Burt matrix, then runs CA on it.

How does MCA encode categorical variables into a coordinate space?

Here's what MCA() actually does internally. Step one: build an indicator matrix. That's a wide table where every category becomes its own 0/1 dummy column. Step two: run the same chi-square-distance machinery as plain correspondence analysis on that wide table. The factor map you saw above is just CA applied to this expanded representation.

The block below pulls out the indicator matrix for the first three variables of tea_active so you can see the dummy encoding directly. tab.disjonctif() is FactoMineR's helper for this.

Each row is a respondent. Each column is one category, holding 1 if the respondent picked it and 0 otherwise. Row 1 didn't drink tea at breakfast, tea time, evening, or lunch. Every dummy is 0 for the chosen category and 1 for the "Not" complement. This 0/1 expansion is what MCA feeds into the CA engine.

How do you read the MCA factor map?

The MCA biplot crams a lot onto one canvas. To read it cleanly, plot categories and individuals on separate maps first. The categories map answers "which response patterns hang together?". The individuals map answers "which respondents look alike?".

The block below draws the categories-only map. Each point is a category, positioned by how its respondents differ from the average tea drinker.

Categories close together get picked by similar respondents. unpackaged, tearoom, and friends cluster on the right, the slow-ritual end. tea bag, chain store, and lunch cluster on the left, the convenience end. The opposition along Dim 1 is the dominant signal in the survey.

Now switch to the individuals map and color by the Tea variable (black, green, or Earl Grey).

The three ellipses overlap heavily, which means tea type alone doesn't define a respondent. But the centroids do drift: Earl Grey drinkers lean left (convenience side), green tea drinkers lean up (Dim 2 difference). MCA gives you a single coordinate system to make these soft pattern statements.

How do you decide how many dimensions matter (eigenvalues + Benzécri correction)?

MCA gives you up to (number of categories − number of variables) dimensions. You don't want all of them. The eigenvalue table tells you how much variance each dimension captures, but raw MCA eigenvalues are pessimistic, every additional dummy column adds a small amount of trivial noise. Benzécri's correction removes that noise.

Start by inspecting the raw eigenvalues and the scree plot.

The raw scree shows Dim 1 keeps 12.6% and Dim 2 keeps 7.4%. That looks low. It is low because the indicator matrix inflates the total inertia with structurally meaningless variation. Benzécri's correction adjusts every eigenvalue above the threshold $1/Q$ (where $Q$ is the number of active variables) and discards the rest.

The corrected formula is:

$$\lambda_s^{*} = \left(\frac{Q}{Q - 1}\right)^2 \left(\lambda_s - \frac{1}{Q}\right)^2 \quad \text{for } \lambda_s > \frac{1}{Q}$$

Where:

• $\lambda_s^{*}$ = corrected eigenvalue for dimension $s$
• $\lambda_s$ = raw MCA eigenvalue for dimension $s$
• $Q$ = number of active categorical variables

Now compute the corrected eigenvalues for the dimensions that pass the threshold.

After correction, Dim 1 alone captures roughly 47% of the meaningful variance, not 12.6%. Dim 2 drops to about 10%. The first two dimensions together explain ~57% of the structure, a much fairer summary of how much MCA actually compressed the survey.

Which categories drive the dimensions (contribution and cos2)?

Eigenvalues tell you how big each dimension is. Contribution and cos2 tell you which categories build it. Contribution is the share of a dimension's variance that a category accounts for, top contributors are the categories you cite when you name the axis. Cos2 is the share of a category's own variance that the dimension explains, high cos2 means the category sits cleanly on that axis.

The block below ranks the top 15 contributors to Dim 1 and recolors the categories map by cos2 on the first two dimensions.

The contribution plot pins Dim 1 down: it's mostly built by where you buy your tea (tearoom, unpackaged, chain store) and how it's packaged (tea bag). Categories above the dashed reference line contribute more than uniform expectation. The cos2 map then tells you which categories the first two dimensions actually represent, unpackaged is both a top contributor and well-represented (red), so it's a safe label for Dim 1.

How do supplementary variables sharpen interpretation?

Supplementary variables don't influence the geometry. They get projected onto the factor map after the fit, so you can ask "does this variable line up with my axes?" without contaminating the active fit. MCA() accepts categorical supplementary variables via quali.sup and quantitative ones via quanti.sup.

The block below refits MCA on tea with Tea, price, and where as supplementary categorical variables (columns 19, 20, 21 in the original tea table) and uses dimdesc() to list which categories associate strongest with Dim 1.

dimdesc() runs an ANOVA between each supplementary variable and the dimension. where (where you buy tea) has the largest $R^2$ on Dim 1, confirming the convenience-vs-ritual reading. The category-level table picks out specific labels: chain store+tea shop and tearoom sit on opposite ends, and unpackaged swings the dimension hardest in the negative direction.

Practice Exercises

Exercise 1: MCA on the food-poisoning survey

The poison dataset (also in FactoMineR) records 55 children, the food they ate at a school dinner, and the symptoms they reported. Fit MCA on columns 5-15 of poison (the food/symptom dummies, all factors), and identify the top contributor to Dim 1. Save it to my_top_contrib.

Exercise 2: Compare raw vs Benzécri-corrected variance on tea

For res.mca (the tea MCA fit from Section 1), compute the corrected percentage of variance for Dim 1 using the Benzécri formula, and confirm it is larger than the raw eig[1, "percentage of variance"]. Save the corrected percentage to my_corr_pct.

Complete Example

Here's a self-contained MCA on the poison dataset from start to interpretation. The pattern is the same for any survey: pick active variables, fit, check eigenvalues, plot, and add supplementary variables for confirmation.

Dim 1 separates sick from healthy children (Sick has the highest supplementary $R^2$). The top contributors are vomiting, abdominal pain, and the foods most associated with the outbreak. With ~25% raw variance on Dim 1 + Dim 2, this two-dimensional summary already pins down the food–symptom pattern.

Summary

The full MCA workflow boils down to five steps. The diagram below shows how they connect, and the table beneath maps each step to its R function.

Figure 2: Five-step MCA workflow from data to interpretation.

Three things to remember: every category is a point (not every variable); raw eigenvalues underreport the structure (use Benzécri); contribution and cos2 must be read together to name an axis.

References

1. Husson, F., Lê, S., and Pagès, J. Exploratory Multivariate Analysis by Example Using R, 2nd Edition. CRC Press (2017). Chapter 4: Multiple Correspondence Analysis.
2. FactoMineR. MCA() function reference. Link
3. factoextra. fviz_mca documentation. Link
4. Kassambara, A. STHDA: MCA - Multiple Correspondence Analysis in R: Essentials. Link
5. Benzécri, J.-P. (1979). Sur le calcul des taux d'inertie dans l'analyse d'un questionnaire. Cahiers de l'Analyse des Données, 4(3), 377-378.
6. Greenacre, M. (2017). Correspondence Analysis in Practice, 3rd Edition. CRC Press. Chapter 19: Multiple Correspondence Analysis.

Continue Learning

• Correspondence Analysis in R. The two-variable parent of MCA; read this first if biplots are new.
• PCA in R. The numeric-data counterpart of MCA; same factor-map idea, different distance metric.
• Cluster Analysis in R. Pair MCA coordinates with k-means or HCPC to label respondent groups.