Correspondence Analysis

One of the goals of correspondence analysis is to describe the relationships between two nominal variables in a correspondence table in a low-dimensional space, while simultaneously describing the relationships between the categories for each variable. For each variable, the distances between category points in a plot reflect the relationships between the categories with similar categories plotted close to each other. Projecting points for one variable on the vector from the origin to a category point for the other variable describe the relationship between the variables.

An analysis of contingency tables often includes examining row and column profiles and testing for independence via the chi-square statistic. However, the number of profiles can be quite large, and the chi-square test does not reveal the dependence structure. The Crosstabs procedure offers several measures of association and tests of association but cannot graphically represent any relationships between the variables.

Factor analysis is a standard technique for describing relationships between variables in a low-dimensional space. However, factor analysis requires interval data, and the number of observations should be five times the number of variables. Correspondence analysis, on the other hand, assumes nominal variables and can describe the relationships between categories of each variable, as well as the relationship between the variables. In addition, correspondence analysis can be used to analyze any table of positive correspondence measures.

Example. Correspondence analysis could be used to graphically display the relationship between staff category and smoking habits. You might find that with regard to smoking, junior managers differ from secretaries, but secretaries do not differ from senior managers. You might also find that heavy smoking is associated with junior managers, whereas light smoking is associated with secretaries.

Statistics and plots. Correspondence measures, row and column profiles, singular values, row and column scores, inertia, mass, row and column score confidence statistics, singular value confidence statistics, transformation plots, row point plots, column point plots, and biplots.

Correspondence Analysis Data Considerations

Data. Categorical variables to be analyzed are scaled nominally. For aggregated data or for a correspondence measure other than frequencies, use a weighting variable with positive similarity values. Alternatively, for table data, use syntax to read the table.

Assumptions. The maximum number of dimensions used in the procedure depends on the number of active rows and column categories and the number of equality constraints. If no equality constraints are used and all categories are active, the maximum dimensionality is one fewer than the number of categories for the variable with the fewest categories. For example, if one variable has five categories and the other has four, the maximum number of dimensions is three. Supplementary categories are not active. For example, if one variable has five categories, two of which are supplementary, and the other variable has four categories, the maximum number of dimensions is two. Treat all sets of categories that are constrained to be equal as one category. For example, if a variable has five categories, three of which are constrained to be equal, that variable should be treated as having three categories when determining the maximum dimensionality. Two of the categories are unconstrained, and the third category corresponds to the three constrained categories. If you specify a number of dimensions greater than the maximum, the maximum value is used.

Related procedures. If more than two variables are involved, use multiple correspondence analysis. If the variables should be scaled ordinally, use categorical principal components analysis.

To Obtain a Correspondence Analysis

This feature requires the Categories option.

  1. From the menus choose:

    Analyze > Dimension Reduction > Correspondence Analysis...

  2. Select a row variable.
  3. Select a column variable.
  4. Define the ranges for the variables.
  5. Click OK.

This procedure pastes CORRESPONDENCE command syntax.