Correspondence analysis is a statistical technique that factors the categorical variable and shows the dimensions or association of the categorical variable to each other. Correspondence analysis shows results in categories and not by their association. In the case of correspondence analysis, data may be nominal, ordinal or continuous. Correspondence analysis starts tabulating data from two-way cross-classification and generalized n way of two variables. Correspondence analysis is a similar technique to factor analysis. Factor analysis determines which variable clusters are together, and correspondence analysis determines which category values are close to each other. Correspondence analysis visualizes these category values on a map, and these map values are plotted with their axes. Correspondence analysis is based on the point distance, so it does not support significance testing.
Key concepts and terms in correspondence analysis:
Correspondence analysis: Correspondence analysis is also called corresponding mapping, perceptual mapping, correspondence factor analysis or principle component analysis of nominal data. Correspondence analysis measures the distance between nominal variables on a map, where each variable is associated with each other.
Correspondence table: In correspondence analysis, the correspondence table shows the marginal variance explained by the object. This helps the researcher to determine the relationship between the variables virtually.
Points: In correspondence analysis, the value of the variable is called “point”.
Point distance: In correspondence analysis, chi-square technique is used to measure the distance, rather than the Euclidian distance. In correspondence analysis, the distance matrix is the input of the principle component analysis.
Correspondence map: In correspondence analysis, the correspondence map is the visual representation of variable categories on a particular dimension. The principle component method is used to determine the distance between the variable categories. In correspondence analysis, if biplot is used, then the distance can be interpreted if raw and standardized data has been used. For instance, the distance can be interpreted if column one and column two have been standardized. If we are using distance, for row and column distance, then the symmetrical normalization is used.
Contribution of points to dimensions: In factor analysis, factor loading determines the dimension of the factor, while in correspondence analysis, the point determines the dimension of the factor, which shows variance explained in a particular dimension.
Eigenvalues: Eigenvalues is also called the characteristic roots. In correspondence analysis, the correspondence table shows the variance explained by each dimension. First, however, Eigenvalues shows the largest variance explained by the dimension.
Total inertia: In correspondence analysis, the sum of all the Eigenvalues is called the total inertia or the total variance explained by the dimension.
Significance testing of correspondence analysis: Chi-square test is used to test the total inertia in correspondence analysis with the associated probability.
Singular value: In correspondence analysis, singular value is interpreted as the square root of an Eigenvalue. Singular value shows the maximum canonical correlation between the categories of the variable in the analysis of a given dimension.
Correspondence analysis in SPSS: Correspondence analysis is available in SPSS in the add-on module.
Assumption in correspondence analysis:
Homogeneity: In correspondence analysis, it is assumed that there is homogeneity between the column variable of the analysis. If homogeneity is not present in the analysis, then the result will be misleading.
Distributional assumption: Correspondence analysis is a non-parametric technique that assumes distributional assumptions.
Category assumption: In correspondence analysis, it is assumed that the discrete data has many categories. If few categories are present in the data, then log linear analysis is preferred over correspondence analysis.
Negative values: In correspondence analysis, negative value is not considered.
Continuous data: In correspondence analysis, discrete data is used. If we are using continuous data, then the data must be categorized into range.
Correspondence analysis is an exploratory technique not a confirmatory technique.
Correspondence Analysis Resources
Adachi, K. (2003). Correspondence analysis, multiple correspondence analysis, and joint correspondence analysis. Japanese Psychological Review, 46(4), 547-563.
Benzecri, J. -P. (1992). Correspondence analysis handbook. New York: Marcel Dekker.
Choulakian, V. (2008). Taxicab correspondence analysis of contingency tables with one heavyweight column. Psychometrika, 73(2), 309-319.
Clausen, S. -E. (1998). Applied correspondence analysis: An introduction. Thousand Oaks, CA: Sage Publications.
Goldstein, H. (1987). The choice of constraints in correspondence analysis. Psychometrika, 52(2), 207-215.
Greenacre, M. J. (1984). Theory and applications of correspondence analysis. New York: Academic Press.
Greenacre, M. J. (1993). Correspondence analysis in practice. London: Academic Press.
Kroonenberg, P. M., & Lombardo, R. (1999). Nonsymmetric correspondence analysis: A tool for analyzing contingency tables with a dependence structure. Multivariate Behavioral Research, 34(3), 367-396.
Mullet, G. M. (1987). Correspondence analysis: A new tool for image studies. Journal of Professional Services Marketing, 2(3), 41-61.
Noma, E., & Smith, D. R. (1985). Scaling sociomatrices by optimizing an explicit function: Correspondence analysis of binary single response sociomatrices. Multivariate Behavioral Research, 20, 179-197.
Vermunt, J. K., & Anderson, C. J. (2005). Joint correspondence analysis by maximum likelihood. Methodology, 1(1), 18-26.
Weller, S. C., & Romney, A. K. (1990). Metric scaling: Correspondence analysis. Newbury Park, CA: Sage Publications.
