Multiple correspondence analysis
Encyclopedia
In statistics
, multiple correspondence analysis (MCA) is a data analysis
technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space
. The procedure thus appears to be the counterpart of principal component analysis for categorical data . MCA is an extension of simple correspondence analysis
(CA) in that it is applicable to a large set of categorical variables.
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 crosstabulations 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.
Associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). 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.
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'.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, multiple correspondence analysis (MCA) is a data analysis
Data analysis
Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making...
technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. The procedure thus appears to be the counterpart of principal component analysis for categorical data . MCA is an extension of simple correspondence analysis
Correspondence analysis
Correspondence analysis is a multivariate statistical technique proposed by Hirschfeld and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data...
(CA) in that it is applicable to a large set of categorical variables.
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 crosstabulations between the categorical variables, and has an analogy to the covariance matrix
Covariance matrix
In probability theory and statistics, a covariance matrix is a matrix whose element in the i, j position is the covariance between the i th and j th elements of a random vector...
of continuous variables. Analyzing the Burt table is a more natural generalization of simple correspondence analysis
Correspondence analysis
Correspondence analysis is a multivariate statistical technique proposed by Hirschfeld and later developed by Jean-Paul Benzécri. It is conceptually similar to principal component analysis, but applies to categorical rather than continuous data...
, and individuals or the means of groups of individuals can be added as supplementary points to the graphical display.
Associations between variables are uncovered by calculating the chi-square distance between different categories of the variables and between the individuals (or respondents). 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
Factor analysis
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...
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.
In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu
Pierre Bourdieu
Pierre Bourdieu was a French sociologist, anthropologist, and philosopher.Starting from the role of economic capital for social positioning, Bourdieu pioneered investigative frameworks and terminologies such as cultural, social, and symbolic capital, and the concepts of habitus, field or location,...
, notably in his books La Distinction
La Distinction
La Distinction is a book by French sociologist Pierre Bourdieu , based on Bourdieu's empirical research on French culture. Taken from studies conducted by Bourdieu in 1963 and concluded in 1967-68, the book was originally published in France in 1979...
, 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
Field (Bourdieu)
Field is one of the core concepts used by French social scientist Pierre Bourdieu. A field is a setting in which agents and their social positions are located. The position of each particular agent in the field is a result of interaction between the specific rules of the field, agent's habitus and...
, 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'.
External links
- Le Roux, B. and H. Rouanet (2004), Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis at Google Books: http://books.google.com/books?id=a6bDBUF58XwC&lpg=PP1&dq=geometric%20data%20analysis&hl=no&pg=PP1#v=onepage&q=&f=false
- Greenacre, Michael (2008), La Práctica del Análisis de Correspondencias, BBVA Foundation, Madrid, available for free download at the foundation's web site http://www.fbbva.es/TLFU/tlfu/esp/publicaciones/libros/fichalibro/index.jsp?codigo=300