# Multiple Correspondence Analysis

Multiple Correspondence Analysis tries to produce a solution in which objects within the same category are plotted close together and objects in different categories are plotted far apart. Each object is as close as possible to the category points of categories that apply to the object. In this way, the categories divide the objects into homogeneous subgroups. Variables are considered homogeneous when they classify objects in the same categories into the same subgroups.

For a one-dimensional solution, multiple correspondence analysis assigns optimal scale values (category quantifications) to each category of each variable in such a way that overall, on average, the categories have maximum spread. For a two-dimensional solution, multiple correspondence analysis finds a second set of quantifications of the categories of each variable unrelated to the first set, attempting again to maximize spread, and so on. Because categories of a variable receive as many scorings as there are dimensions, the variables in the analysis are assumed to be multiple nominal in optimal scaling level.

Multiple correspondence analysis also assigns scores to the objects in the analysis in such a way that the category quantifications are the averages, or centroids, of the object scores of objects in that category.

**Relation to other
Categories procedures. **Multiple correspondence analysis
is also known as homogeneity analysis or dual scaling. It gives comparable,
but not identical, results to correspondence analysis when there are
only two variables. Correspondence analysis produces unique output
summarizing the fit and quality of representation of the solution,
including stability information. Thus, correspondence analysis is
usually preferable to multiple correspondence analysis in the two-variable
case. Another difference between the two procedures is that the input
to multiple correspondence analysis is a data matrix, where the rows
are objects and the columns are variables, while the input to correspondence
analysis can be the same data matrix, a general proximity matrix,
or a joint contingency table, which is an aggregated matrix in which
both the rows and columns represent categories of variables. Multiple
correspondence analysis can also be thought of as principal components
analysis of data scaled at the multiple nominal level.

**Relation to standard
techniques. **Multiple correspondence analysis can be thought
of as the analysis of a multiway contingency table. Multiway contingency
tables can also be analyzed with the Crosstabs procedure, but Crosstabs
gives separate summary statistics for each category of each control
variable. With multiple correspondence analysis, it is often possible
to summarize the relationship between all of the variables with a
single two-dimensional plot. An advanced use of multiple correspondence
analysis is to replace the original category values with the optimal
scale values from the first dimension and perform a secondary multivariate
analysis. Since multiple correspondence analysis replaces category
labels with numerical scale values, many different procedures that
require numerical data can be applied after the multiple correspondence
analysis. For example, the Factor Analysis procedure produces a first
principal component that is equivalent to the first dimension of multiple
correspondence analysis. The component scores in the first dimension
are equal to the object scores, and the squared component loadings
are equal to the discrimination measures. The second multiple correspondence
analysis dimension, however, is not equal to the second dimension
of factor analysis.