MCA is performed by applying the CA algorithm to either an indicator matrix or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are dummy variables representing categories of the variables. Analyzing the indicator matrix allows the direct representation of individuals as points in geometric space. The Burt table is the symmetric matrix of all two-way cross-tabulations between the categorical variables, and has an analogy to the covariance matrix of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display. In the indicator matrix approach, associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals. These associations are then represented graphically as "maps", which eases the interpretation of the structures in the data. Oppositions between rows and columns are then maximized, in order to uncover the underlying dimensions best able to describe the central oppositions in the data. As in factor analysis or principal component analysis, the first axis is the most important dimension, the second axis the second most important, and so on, in terms of the amount of variance accounted for. The number of axes to be retained for analysis is determined by calculating modified eigenvalues.
Recent works and extensions
In recent years, several students of Jean-Paul Benzécri have refined MCA and incorporated it into a more general framework of data analysis known as geometric data analysis. This involves the development of direct connections between simple correspondence analysis, principal component analysis and MCA with a form of cluster analysis known as Euclidean classification. Two extensions have great practical use.
It is possible to include, as active elements in the MCA, several quantitative variables. This extension is called factor analysis of mixed data.
Very often, in questionnaires, the questions are structured in several issues. In the statistical analysis it is necessary to take into account this structure. This is the aim of multiple factor analysis which balances the different issues within a global analysis and provides, beyond the classical results of factorial analysis, several results specific of the group structure.
Application fields
In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu, notably in his books La Distinction, Homo Academicus and The State Nobility. Bourdieu argued that there was an internal link between his vision of the social as spatial and relational --– captured by the notion of field, and the geometric properties of MCA. Sociologists following Bourdieu's work most often opt for the analysis of the indicator matrix, rather than the Burt table, largely because of the central importance accorded to the analysis of the 'cloud of individuals'.
Multiple correspondence analysis and principal component analysis
MCA can also be viewed as a PCA applied to the complete disjunctive table. To do this, the CDT must be transformed as follows. Let denote the general term of the CDT. is equal to 1 if individual possesses the category and 0 if not. Let denote, the proportion of individuals possessing the category. The transformed CDT has as general term: The unstandardized PCA applied to TCDT, the column having the weight, leads to the results of MCA. This equivalence is fully explained in a book by Jérôme Pagès. It plays an important theoretical role because it opens the way to the simultaneous treatment of quantitative and qualitative variables. Two methods simultaneously analyze these two types of variables: factor analysis of mixed data and, when the active variables are partitioned in several groups: multiple factor analysis. This equivalence does not mean that MCA is a particular case of PCA as it is not a particular case of CA. Its only means that these methods are closely linked one another as they belong to the same family: the factorial methods.
Software
There are numerous software of data analysis that include MCA, such as STATA and SPSS. The R package also features MCA. This software is related to a book describing the basic methods for performing MCA.
