How do you interpret and visualize the results of correspondence analysis?

1 What is correspondence analysis?

Correspondence analysis is a way of transforming a contingency table, which shows the frequencies of two or more categorical variables, into a graphical display, which shows the distances and angles between the categories. The graphical display is called a biplot, and it consists of two sets of points: one for the rows and one for the columns of the table. The closer the points are, the more associated they are. The farther the points are, the more dissimilar they are. The angle between the points indicates the nature of the association: acute angles mean positive association, right angles mean independence, and obtuse angles mean negative association.

2 How to read a biplot?

A biplot is a two-dimensional representation of the correspondence analysis, but it may not capture all the information in the data. To evaluate the quality and significance of the biplot, you should look at metrics such as inertia, eigenvalues, percentages of inertia, contribution and quality. Inertia measures the total variation in the data and is equal to the chi-square statistic divided by the total frequency. Eigenvalues measure the amount of variation explained by each dimension of the biplot, while percentages of inertia measure their relative importance. Contribution measures how much each category contributes to the definition of each dimension and is equal to the squared coordinate of each category divided by the eigenvalue of each dimension, multiplied by 100. Quality measures how well each category is represented by each dimension and is equal to the squared correlation between the original frequency and the projected frequency of each category on each dimension.

3 How to interpret the dimensions?

The dimensions of the biplot are the axes that define the space where the categories are plotted. They are derived from the principal components of the correspondence analysis, which are linear combinations of the row and column variables that maximize the variation in the data. The dimensions can be interpreted as latent factors that underlie the relationships between the categories. For example, if you perform a correspondence analysis on a table that shows the preferences of different age groups for different types of music, you may find that the first dimension represents a contrast between classical and rock music, while the second dimension represents a contrast between jazz and pop music.

4 How to visualize the categories?

The categories plotted on the biplot can be labeled with their names, symbols or colors that indicate their groups. Depending on the purpose and audience of the analysis, categories can be visualized in a variety of ways. For instance, a symmetric plot displays both row and column categories on the same biplot to compare and contrast them across both variables. Alternatively, a row plot only shows the row categories on the biplot, while a column plot only shows the column categories on the biplot; these are useful for highlighting differences among the respective categories with respect to the other variable. Lastly, a joint plot can show both row and column categories on separate biplots with the same axes and scale in order to examine the correspondence between them.

5 How to enhance the biplot?

The biplot can be enriched with several features that can help you comprehend and communicate the results of the correspondence analysis. For example, confidence ellipses demonstrate the regions where the categories are likely to be located with a certain probability. This can indicate the variability and uncertainty of the categories, as well as their significance and clustering. Additionally, supplementary categories that are not part of the original table can be projected onto the biplot, providing additional information and context for the analysis. Furthermore, seriation is a way of ordering the categories along each dimension according to their coordinates. This can reveal trends and patterns in the data, such as gradients or cycles.

6 How to use correspondence analysis for survey design?

Correspondence analysis can be a useful tool for survey design, as it can help you identify and optimize the variables and categories that are relevant and meaningful for your research question. For instance, it can be used to explore the structure and dimensions of your data, evaluate different ways of coding or grouping categorical variables, test and validate assumptions and expectations about your data, and communicate findings in an attractive way with graphs and tables.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?