Michael Greenacre

The Geometric Interpretation of Correspondence Analysis

Abstract Correspondence analysis is an exploratory multivariate technique that converts a data matrix into a particular type of graphical display in which the rows and columns are depicted as points. The method has a long and varied history and has appeared in different forms in the psychometric and ecological literature, among others. In this article we review the geometry of correspondence analysis and its geometric interpretation. We also discuss various extensions of correspondence analysis to multivariate categorical data (multiple correspondence analysis) and a variety of other data types.

Biplots of compositional data

Summary. The singular value decomposition and its interpretation as a linear biplot have proved to be a powerful tool for analysing many forms of multivariate data. Here we adapt biplot methodology to the specific case of compositional data consisting of positive vectors each of which is constrained to have unit sum. These relative variation biplots have properties relating to the special features of compositional data: the study of ratios, subcompositions and models of compositional relationships. The methodology is applied to a data set consisting of six-part colour compositions in 22 abstract paintings, showing how the singular value decomposition can achieve an accurate biplot of the colour ratios and how possible models interrelating the colours can be diagnosed.

Correspondence Analysis in the Social Sciences.

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Correspondence analysis in medical research

Various applications of correspondence analysis to biomedical data are presented. The basic concepts of profile, mass and chi-squared distance are introduced in an initial simple example using data on the relationship between headache types and age. The main result of the correspondence analysis is a geometric map of this relationship showing how the relative frequencies of headache types change with age. A second example maps the association between personality types and various medical diagnostic groups, while a third example deals with categorical rating scales such as an efficacy scale for a medication or a scale of pain. A final example illustrates the more complex situation when several categorical variables are involved using test data on a collection of bacterial isolates, with the object of comparing bacterial types and understanding the inter-relationships of the different tests.

Clustering the rows and columns of a contingency table

A number of ways of investigating heterogeneity in a two-way contingency table are reviewed. In particular, we consider chi-square decompositions of the Pearson chi-square statistic with respect to the nodes of a hierarchical clustering of the rows and/or the columns of the table. A cut-off point which indicates “significant clustering” may be defined on the binary trees associated with the respective row and column cluster analyses. This approach provides a simple graphical procedure which is useful in interpreting a significant chi-square statistic of a contingency table.

Biplots in correspondence analysis

Conditions under which correspondence analysis maps are biplots are discussed, as well as the interpretation of such biplots. It is shown that the asymmetric map which jointly displays the profiles and the vertices which define the unit vectors in the profile space is a biplot. A number of different ways of interpreting this joint plot are discussed, some of these being dependent on the choice of the x2 metric in the profile space. Biplot axes can be defined and calibrated on the zero-to-one profile scale in the usual way to recover approximations to the individual profile elements. Finally, the biplot interpretation in the context of multiple correspondence analysis is discussed. It is pointed out that joint correspondence analysis leads to a joint display of several variables which can be calibrated in a similar fashion to recover profile elements of the subtables of the Burt matrix.

Subset correspondence analysis: Visualizing relationships among a selected set of response categories from a questionnaire survey

This study shows how correspondence analysis may be applied to a subset of response categories from a questionnaire survey (e.g., the subset of undecided responses or the subset of responses for a particular category across several questions). The idea is to maintain the original relative frequencies of the categories and not reexpress them relative to totals within the subset, as would normally be done in a regular correspondence analysis of the subset. Furthermore, the masses and chi-square distances assigned to the subset of categories are the same as those in the correspondence analysis of the whole data set, which leads to a decomposition of total variance into parts if the whole data set is subdivided into disjoint subsets. This variant of the method, called subset correspondence analysis, is illustrated on data from the International Social Survey Programme’s Family and Changing Gender Roles survey.

Power transformations in correspondence analysis

Power transformations of positive data tables, prior to applying the correspondence analysis algorithm, are shown to open up a family of methods with direct connections to the analysis of log-ratios. Two variations of this idea are illustrated. The first approach is simply to power transform the original data and perform a correspondence analysis - this method is shown to converge to unweighted log-ratio analysis as the power parameter tends to zero. The second approach is to apply the power transformation to the contingency ratios, that is, the values in the table relative to expected values based on the marginals - this method converges to weighted log-ratio analysis, or the spectral map. Two applications are described: first, a matrix of population genetic data which is inherently two-dimensional, and second, a larger cross-tabulation with higher dimensionality, from a linguistic analysis of several books.

Correspondence analysis of the Spanish National Health Survey

This report gives a comprehensive explanation of the multivariate technique called correspondence analysis, applied in the context of a large survey of a nations state of health, in this case the Spanish National Health Survey. It is first shown how correspondence analysis can be used to interpret a simple cross-tabulation by visualizing the table in the form of a map of points representing the rows and columns of the table. Combinations of variables can also be interpreted by coding the data in the appropriate way. The technique can also be used to deduce optimal scale values for the levels of a categorical variable, thus giving quantitative meaning to the categories. Multiple correspondence analysis can analyze several categorical variables simultaneously, and is analogous to factor analysis of continuous variables. Other uses of correspondence analysis are illustrated using different variables of the same Spanish database: for example, exploring patterns of missing data and visualizing trends across surveys from consecutive years.

Correspondence analysis of square asymmetric matrices

The application of correspondence analysis to square asymmetric tables is often unsuccessful because of the strong role played by the diagonal entries of the matrix, obscuring the data off the diagonal. A simple modification of the centring of the matrix, coupled with the corresponding change in row and column masses and row and column metrics, allows the table to be decomposed into symmetric and skew symmetric components, which can then be analysed separately. The symmetric and skew symmetric analyses can be performed by using a simple correspondence analysis program if the data are set up in a special block format. The methodology is demonstrated on a social mobility table from the first democratically elected Parliament in Germany, the Frankfurter Nationalversammlung. The table cross-tabulates the jobs of parliamentarians when first entering the labour market and their jobs in May 1848 when the Parliament started its first session.