## 7.4 Multiple Correspondence Analysis

Multiple Correspondence Analysis (MCA) is the generalization of (simple) correspondence analysis to the case when we have more than two categorical variables. This analysis can also be regarded as a generalization of a normalized PCA for a data table of categorical variables.

One of the advantages of MCA is that it allows to take into account non-linear associations among variables.

**Data Table.** The analyzed (raw) data table in this case is formed by multiple categorical variables. This table can be transformed into a complete disjoint table, or into a Burt table crossing all variables two-by-two).

*For individuals*: we analyze the proximities between the row-profiles of the complete disjoint table.*For variables*: we analyze the proximities between the column-profiles of the complete disjoint table.

In both cases (rows and columns) the utilized distance is the *chi-squared* distance.

#### Interpretation

Besides a dilation coefficient:

an individual is the average point of its categories.

a category is an average point of the individuals that have chosen such category.

the global center of gravity is also the center of gravity of all the categories.

the part of the inertia due to a given category growths when its effective decreases.

it is recommended to avoid categories with very few effectives (or to project them as a supplementary category).

the inertia due to a variable growths when the number of its categories grows (it is recommended to balance the number of categories in the variables).

in the peryphery of the cloud, two categories appear close to each other if the individuals with such categories have answered in a similar way the set of active variables.

a factorial plane is interpreted in terms of how close or how far the projected categories are. The interpretation should be supported by looking at the v-test, the contributions to the axes, the cosine squares, etc.

a supplementary category is the average point of the individuals having such category.