Correspondence Analysis

Correspondence analysis is a way of seeing the ASSOCIATION in a two-way cross-tabulation rather than measuring it. Consider, for example, the crosstabulation in Table 1, taken from the 1994 International Social Survey Programme (ISSP) on Family and Changing Gender Roles. This table shows, for 24 different countries, how many respondents think that the ideal number of children in a family is 0, 1, 2, 3, 4, or 5 or more. Response percentages for each country are given in parentheses: For example, in Australia, 833 of the 1,529 respondents, or 54.5%, considered two children to be ideal.

The usual Pearson CHI-SQUARE statistic for this 24 × 6 table has the value 6,734, and Cramér's V is 0.206, both of which are highly significant. In fact, any MEASURE OF ASSOCIATION, even with a low value, will almost always be highly significant for a table with such high frequencies. Clearly, there are significant differences between countries, but what are these differences? We can see from the table that the United States and Canada are quite similar, but how can we compare their positions on this issue with the European countries? Correspondence analysis can answer this question because it aims to show the similarities and differences between the countries in a compact, graphical way.

The correspondence analysis of Table 1 is shown in Figure 1. This is a two-dimensional “map” of the positions of the row and column points, showing the main features of the contingency table. The points representing the columns form an arch pattern, starting from the bottom right and ending at the bottom left. This indicates a monotonic ordination of the countries in terms of their opinions on this issue, from those wanting fewer children (e.g., [former] East Germany, Bulgaria, and the Czech Republic) to those wanting more children (e.g., Ireland, Israel, and the Philippines). The horizontal axis, or first principal axis, captures this [Page 204]general scaling of the countries, whereas the vertical second principal axis contrasts the extreme categories with the middle categories. Such an arch pattern is frequently encountered in correspondence analysis maps. Most of the highly developed countries are around the center of the map, which represents the average, but there are interesting differences among them. The United States, for example, lies lower down on the second axis, mostly because of its relatively high frequencies of “4 children” compared to other countries around the center. Japan, by contrast, lies high up on the second axis because of its very high frequency of “3 children.”

Each country finds its position in the map according to its profile, or set of relative frequencies, shown as percentages in Table 1. The center of the map represents the average profile—that is, 0.6%, 3.7%, 55.9%, 29.1%, 8.5%, and 2.1% for all the countries (see last row of Table 1)—and a particular country will be attracted in the direction of a category if its percentage is higher than the corresponding average value. The method can be equivalently defined as a variant of METRIC VARIABLE multidimensional scaling, in which distances between countries are measured by the so-called chi-square distances (Benzécri, 1973), and each country is weighted proportional to its respective sample size.

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