Multiple Correspondence Analysis (MCA)

Simple CA analyzes one two-way contingency table formed by exactly two categorical variables, decomposing the association between row and column categories. MCA extends this to three or more variables by operating on the indicator matrix or the derived Burt matrix, generalizing the geometric framework to handle many categorical variables at once while preserving the same chi-squared distance interpretation of proximity on the map.

Common practice applies Greenacre's adjusted inertia criterion: retain dimensions whose adjusted eigenvalue exceeds the average adjusted eigenvalue. Scree plots of adjusted inertia values also help identify an elbow. Because raw inertia values in MCA are deflated by construction, never rely on the unadjusted percentage-explained figures alone to decide how many axes are meaningful.

MCA treats all input variables as nominal by default, ignoring any natural ordering among categories. When variables are genuinely ordinal, analysts can impose order constraints through nonlinear PCA or ordinal MCA extensions, or they can verify post hoc that the recovered dimension reflects the intended ordering, which often happens naturally when the ordinal structure is strong in the data.