Patterns › Dimension reduction

Correspondence analysis (biplot)

Correspondence analysis visualises a categorical contingency table in two dimensions by SVD, placing row and column points on shared axes for joint interpretation.

What is Correspondence analysis (biplot)?

CA decomposes a contingency table via SVD on the matrix of standardised residuals. The result is a 2D (or higher) biplot where row categories and column categories occupy the same coordinate space — proximity ≈ association.

Distances in CA are χ²-distances, not Euclidean — they reflect the categorical profiles' similarity in proportion. Total inertia (sum of squared singular values) = χ² / n, so CA is intimately related to the chi-square test.

When should I use Correspondence analysis (biplot)?

  • Visualising two-way categorical association (brand × attribute, region × disease, archaeology classification).
  • Exploring patterns in survey cross-tabs.

What data does it need?

Two categorical columns (row + column variables).

What does it report?

χ² test + total inertia + per-dim eigenvalues + % inertia + 2D biplot of row + column principal coordinates.

How do I interpret the result?

Row and column points pointing in the same direction from origin associate; points near origin contribute little to the pattern.

See also

References

  • Greenacre (2007). Correspondence Analysis in Practice, 2nd ed.