Multiple Correspondence Analysis

Extension of correspondence analysis to more than two categorical variables

Introduction

Multiple correspondence analysis (MCA) extends CA to several categorical variables. It is the analogue of PCA for categorical data and is widely used in social science and epidemiology for exploring patterns among multiple categorical variables.

Prerequisites

Correspondence analysis.

Theory

MCA operates on a disjunctive-coded indicator matrix of all categorical variables. Burt’s approach decomposes the Burt table (cross-tabulation of all pairs). Each category becomes a point; each individual becomes a point; low-dimensional projection preserves chi-squared distances.

Assumptions

Categorical variables; no strong zero cells; sufficient sample size.

R Implementation

library(FactoMineR); library(factoextra)

data(hobbies)
mca <- MCA(hobbies[, 1:8], graph = FALSE)
fviz_mca_biplot(mca)
fviz_mca_var(mca)

summary(mca)

Output & Results

Coordinates for individuals and variable categories; inertia per axis.

Interpretation

“MCA of 8 hobby variables revealed two dimensions, corresponding to ‘active vs. sedentary’ and ‘social vs. solitary’. Age groups and gender mapped onto these gradients.”

Practical Tips

• Inertia tends to be low-per-axis in MCA; needing many dimensions for much variance is common.
• Benzecri’s correction or Greenacre’s adjusted inertia gives more meaningful variance-explained percentages.
• Supplementary variables (quali.sup) can be added without affecting the main axes.
• Use fviz_mca_var to focus on category positions.
• For mixed data (categorical + continuous), consider factor analysis of mixed data (FAMD).