Correspondence Analysis

How do brand preferences differ by age group? Do customers in different occupations use different services? Can you grasp at a glance the relationships between several groups and response options in a survey?

For questions like these, the method that visually analyzes the relationships between categorical variables is correspondence analysis.

Whereas crosstab analysis shows counts and proportions by category in a table, and the chi-square test tests the statistical association between two variables, correspondence analysis represents the relationships among categories in a two- or three-dimensional space to help you grasp the overall structure of the data.

It is especially useful in research that must compare relationships among many categories, such as surveys, market research, brand positioning and consumer behavior analysis. In this article, we look at correspondence analysis from its basic concepts through how it works, a real example, how to interpret a correspondence map, how it differs from crosstab analysis, and cautions for the analysis.

For example, suppose you surveyed customers in their 20s through 50s about their most preferred brand. If you divide age into four groups and brands into four, you can build a 4×4 contingency table.

The table alone shows preference proportions by age group, but as the number of categories grows, it becomes harder to see intuitively which age groups and brands are related.

Correspondence analysis represents these relationships on coordinates. Each age group and brand is shown as a point, revealing where it sits within the overall response distribution.

The important point is that correspondence analysis is not simply a method that places frequent categories close together. It compares each category's relative response pattern to analyze what characterizes it within the overall distribution.

For example, comparing four age groups and six brands produces 24 combinations of categories. The contingency table shows counts and proportions, but it is not easy to see at a glance which age groups have similar brand preference patterns or which brands stand apart from the others.

Correspondence analysis summarizes these complex relationships in a few main dimensions and visualizes them. This lets you explore the similarity of response patterns across categories, categories that depart from the overall average distribution, and categories that drive the main differences.

It is especially valuable in situations such as brand positioning, where the relationships between many groups and items need to be shown in a single picture.

The key is to compare how much each category's response distribution differs from the overall distribution. The typical analysis process is as follows.

① Build the contingency table

Count the frequency of each combination of the two categorical variables. For example, you can put age groups in the rows and brands in the columns.

② Compute response profiles by category

Compute proportions based on each row or column total. These are called the row profiles and column profiles.

③ Compute differences between categories

Use the chi-squared distance to calculate how much the response patterns of categories differ.

④ Extract the main dimensions

Extract the main dimensions that can explain the association structure in the contingency table. Correspondence analysis computes these dimensions using methods such as singular value decomposition (SVD).

⑤ Draw the correspondence map

Use the extracted dimensions as coordinate axes to visualize the positions of the categories.

Respondents are in their 20s, 30s, 40s and 50s, and there are four brands: A, B, C and D. Each respondent chose one most-preferred brand. The research question is as follows.

Crosstab results

The table shows that people in their 20s chose Brand A most often and those in their 30s chose Brand B most often. In the 40s, Brand C had the most responses, and in the 50s, Brand D was chosen most.

So the contingency table does show differences by age group. But to understand how similar the brand preference patterns of the age groups are to each other, and which categories contribute most to the overall association structure, additional analysis helps. Correspondence analysis summarizes this information and represents the relationships between categories visually.

A correspondence map is a graph that uses the main dimensions extracted in the analysis as the horizontal and vertical axes and shows each category's relative position as a point. For example, analyzing age group and preferred brand places the 20s, 30s, 40s and 50s and Brands A, B, C and D in a single coordinate space.

When interpreting a correspondence map, check the following.

Categories with similar response patterns

If row categories are located close together, they likely show similar overall response patterns across the column categories. For example, if the 20s and 30s are close on the map, you can interpret this as the two age groups having relatively similar brand preference distributions.

You can compare the positions of brands with each other in the same way. In that case, however, check how the row and column coordinates were scaled.

Categories far from the origin

On a correspondence map, the origin is a reference point related to the overall average profile. If a category is far from the origin, it likely has a response pattern distinct from the overall average distribution. Conversely, a category near the origin may show a pattern relatively similar to the overall average distribution.

However, you should also check how accurately that category is represented on the map.

Relative relationships between row and column categories

A correspondence map shows the categories of two different variables together. For example, if the 20s and Brand A lie in a similar direction, you can examine the possibility of a relative association between the 20s and Brand A.

So check not only the correspondence map but also the proportions in the original contingency table and the contributions of each category.

These dimensions represent the main directions of the association structure in the data. For example, suppose the analysis shows that the first dimension explains 70% of the total inertia and the second explains 25%.

In this case, a two-dimensional correspondence map using the first and second dimensions represents about 95% of the inertia that makes up the overall association structure. Here, inertia is a value that indicates how far the contingency table departs from independence of rows and columns. It is not the same concept as the coefficient of determination (R²) used in ordinary regression.

So the fact that the first two dimensions explain 95% of the inertia should not be interpreted as meaning they accurately predict 95% of all responses. The higher the explained inertia, the more faithfully the main association structure can be shown in a low-dimensional space, but you should also check how well each category's position is represented.

Contribution

Contribution shows how much a particular category influenced the formation of each dimension. For example, if the 20s and 50s have high contributions on Dimension 1, the response patterns of these two age groups can be seen as playing an important role in creating the main difference along that dimension.

Quality of representation (Cos²)

Quality of representation is an indicator of how well each category is represented in the chosen dimensions. For example, if a particular brand's Cos² value is low, be careful about interpreting that brand's relationships from its position on the 2D correspondence map alone.

So in correspondence analysis it is important not just to look at the positions of points, but also to examine explained inertia by dimension, category contributions and quality of representation together.

For example, when analyzing the relationship between age group and brand preference, you can use all three methods together.

Use crosstab analysis to check brand preference proportions by age group, and the chi-square test to test whether there is a statistically significant association between age group and brand preference. Then correspondence analysis lets you examine the relative relationships among the age groups and brands in map form.

The three methods do not replace one another; they can be used together depending on the purpose of the analysis.

But the two methods differ mainly in the data they analyze and how they compute the dimensions.

For example, to visualize the relationship between age group and preferred brand, you might consider correspondence analysis. To summarize several numeric variables per customer, such as purchase amount, number of visits and average purchase cycle, you might consider PCA instead.

Both methods share the common ground of dimension reduction, but you should choose based on the data structure and the purpose of the analysis.

Check the sample size and frequencies in the contingency table.

If some categories have too few observations, their positions can be unstable. In particular, rare categories can stand out far from the origin on the map, but whether they are actually important categories must be examined separately. So check the frequency of each category and the sample size.

Do not interpret every distance on a correspondence map the same way.

Distances between categories of the same kind must be distinguished from distances between categories of different kinds. For example, distances among age groups and distances among brands can each be used to understand differences in response profiles. The distance between an age group and a brand, on the other hand, may have limited direct meaning depending on how the coordinates are displayed.

So do not draw conclusions solely from whether points on the graph are close.

Check the explained inertia of the dimensions.

If the first and second dimensions explain only part of the overall association structure, it is hard to understand all the relationships in the data from the 2D graph alone. For example, if the two dimensions' cumulative explained inertia is 55%, the remaining 45% of the inertia lies in other dimensions.

In this case, it is good to examine additional dimensions or check the original response distributions of individual categories as well.

How categories are defined affects the results.

Dividing age into 20s, 30s, 40s and 50s produces a different contingency table from dividing it into 20–39 and 40–59. So the results of correspondence analysis can also change depending on how the categories are divided. Where possible, set categories based on the research purpose or real-world meaning.

Correspondence analysis alone cannot establish causation.

Even if correspondence analysis shows a relative association between a particular age group and a brand, you cannot conclude that age causes people to prefer that brand. Correspondence analysis is a method for exploring and visualizing relationships between variables, not for testing causation.

But sometimes you need to analyze three or more categorical variables together. In such cases you can consider multiple correspondence analysis (MCA).

Multiple correspondence analysis (MCA)

Multiple correspondence analysis analyzes the relationships among several categorical variables together and visualizes them in a low-dimensional space. For example, suppose you analyze several categorical variables at once, such as gender, age group, occupation, preferred brand and main purchase channel.

Simple correspondence analysis mainly uses a contingency table of two variables, whereas multiple correspondence analysis can explore how many categories are connected in data made up of three or more categorical variables. However, as the number of variables and categories grows, the results can become complex, so check explained inertia by dimension and category contributions carefully.

For example, suppose that in the earlier survey of brand preference by age group, the first and second dimensions explained 95% of the total inertia.

When reporting actual results, describe the specific relationships after checking each category's position on the correspondence map, its contribution and its quality of representation. Depending on the purpose of the analysis, also presenting the chi-square test result helps readers understand both the overall association between the two variables and the specific association structure.

Marketing and brand analysis

Analyze relationships among age groups, consumer segments and preferred brands, and explore each brand's relative positioning.

Market research and surveys

Visualize relationships among several categorical variables, such as respondent characteristics and product preferences, service choices and brand perceptions.

Social science research

Analyze patterns of attitudes or choices by occupation, education level, social characteristics and so on.

Education research

Examine the association structure among categorical responses such as grade level, type of education and preferred learning style.

Health and medical research

Exploratorily analyze relationships among categories such as age group, lifestyle habits and health behaviors.

Consumer behavior research

Identify product preference patterns by customer group and visually compare similarities and differences between groups. In this way, correspondence analysis is useful not just for checking which item was chosen most, but for understanding what overall structure many categories form.

Whereas crosstab analysis shows frequencies and proportions by category and the chi-square test tests independence between variables, correspondence analysis focuses on exploring what relative relationships many categories have with one another.

Especially when there are many categories and a complex contingency table is hard to understand, a correspondence map lets you see the main patterns in the data more intuitively. But you should not draw firm conclusions about relationships just from the distances between points on the map. It is important to check explained inertia by dimension, category contributions, quality of representation and the proportions in the original contingency table together.