XLSTAT: Correspondence Analysis (CA) from a Contingency Table in Excel

 

Dataset for Example

The attached data correspond to a survey asking moviegoers their opinion on a film they had just seen. The audience was also asked to give their age category.

We will use the example to set up and interpret a Correspondence Factorial Analysis. The objectives of this method are to study the association between two variables (rows and columns of a contingency table) and the similarities between the categories of each variable  (rows and columns respectively).

Setting Up

• Open XLSTAT.

• Click on the Analyzing data menu and select the Correspondence analysis feature.

• After clicking the button, the Correspondence Analysis dialog appears.

   

In the General tab:

• • Select the full table in Excel.

  • Choose Two-way table format for contingency tables, or Observations/Variables table for individuals/variables data.

  • Check Labels included if row and column category names are present.

In the Options tab:

• Select None under Advanced analysis for classic CA.

• Do not activate Non-symmetric analysis.

• Choose Chi-square for Distance.

Note:

• For NSCA: enable Non-symmetric analysis (only Chi-square allowed).

• For CA with Hellinger distance: disable Non-symmetric analysis and choose Hellinger.

• For Detrended CA: select Detrended analysis under Advanced analysis.

In the Outputs tab:

• Select the Row and column profiles and Chi-square distance.

In the Maps sub-tab of the Charts tab:

• Three alternative ways of mapping the results are available. The rows and columns symmetric map is the most commonly used. For the purpose of the tutorial, all mapping alternatives were chosen.

• Additionally, select Confidence Ellipses to display confidence ellipses that identify the categories which contribute to the dependency between the row and column categories of the contingency table.

• Click on the OK button to launch the computations.

• The results are displayed on a new Excel sheet.

 

Interpreting the Results

Before interpreting results, it's key to understand profiles—sets of relative frequencies showing how one variable's categories vary with another's.

The first result is the Chi-square test of independence. If the observed value exceeds the critical value and the p-value is below alpha, we conclude a significant association between rows and columns. In this case, differences likely exist between age groups and their appreciation profiles.

Eigenvalues represent the variance explained by each dimension. The analysis quality is assessed using the eigenvalue table or scree plot. A high-quality analysis has the first few eigenvalues summing close to the total. In this example, the first two account for 97% of total inertia, indicating strong quality.

Tables are displayed for rows and columns, showing weights, distances to the origin, squared distances, inertias, and relative inertias.

Weights are marginal proportions used in distance calculations. A larger distance to the origin indicates a profile more distinct from the average, contributing more to variable dependence. Here, age groups 25–34, 35–44, and 45–54 have the shortest distances, meaning their profiles are closest to the mean.

The row and column profiles, along with the mean profile, are then displayed. In this example, age groups 25–34, 35–44, and 45–54 have similar profiles, close to the mean—consistent with their short distances to the origin.

The distances between the rows (respectively columns) gives information about the similarity between categories . Again, the age groups 25-34, 35-44 and 45-54 appear to be similar, with distance below 0.2.

Principal and standard coordinates for rows and columns are displayed. Standard coordinates are the principal coordinates divided by the square root of the factor's eigenvalue, with their weighted sum-of-squares equal to 1 per factor.

Next, a table shows row and column contributions—the importance of each category to each factor. A contribution above 1/I (rows) or 1/J (columns) indicates significance. In this example, group 16–24 is key for factor F1, while groups 65–74 and 75+ are key for factor F2.

The next table shows the squared cosines for rows and columns, indicating the importance of each factor for each category. Their sum equals 1 per category. In this example, almost all the variance for the 16–24 group is explained by factor F1.

The maps begin with the symmetric (French) plot, the most common display. It shows row and column profiles together in principal coordinates, with equal scaling. Distances between row (or column) points approximate Chi-square distances. In this example, age groups 25–34, 35–44, and 45–54 nearly overlap, indicating similar profiles.

Note: Proximity between row and column points cannot be directly interpreted in this plot.

Confidence ellipses can be added to symmetric row or column plots. If the origin lies within a category's ellipse, that category doesn't contribute to the variable dependency. In this example, ellipses show that age groups 25–34, 35–44, and 45–54 don't contribute, while group 16–24 does.

The asymmetric row plot shows columns in the row space (columns use standard coordinates, rows use principal coordinates), while the asymmetric column plot shows rows in the column space. Distances between rows and columns are interpreted by projecting row points onto column vectors.

Interpretation depends on relevance; here, age groups are interpreted in appreciation space. The first dimension contrasts “GOOD” vs. “BAD.” The 16–24 group rated “GOOD” more than other groups, though “GOOD” wasn’t their highest overall rating. Row profiles are close to the mean, as points lie near the origin.

Contribution coordinates for rows and columns are calculated by dividing standard coordinates by the square root of each category's mass.

In the contribution biplot, rows are in contribution coordinates and columns in principal coordinates (and vice versa). Distances of row (or column) points to the origin reflect their contribution to the map. In this example, row points retain their positions from the asymmetric plot, while column points move closer to the origin.

Correspondence Analysis effectively analyzes two-way tables. For surveys with more than two categorical variables, Multiple Correspondence Analysis (MCA) is the preferred method.

The following video addresses CA theory and an implementation in XLSTAT: