What is correspondence analysis?

How can you find out if there’s a relationship between X and Y or A, B, and C? Correspondence analysis can help you find out. It’s a statistical technique used to identify and visualize the hidden patterns and connections between variables.

Correspondence analysis is an incredibly versatile method for interpreting complex data sets by visually representing how they relate. It helps even if you don’t necessarily know what you’re looking for.

The valuable insights uncovered have applications in market research, user experience (UX) design, social science, and many other fields.

This article explores the fundamentals of correspondence analysis, including how you can incorporate this statistical method into your data-driven research projects.

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Correspondence analysis transforms large amounts of data into a simplified visual representation. It shows the connections, patterns, and correspondence between different categories of variables.

It works by plotting each variable as a point on a graph. The distance between them explains the strength and nature of their connection. Closer points are more correlated, and vice versa.

The method can reveal underlying associations and allow researchers to draw meaningful conclusions. For example, a graph analyzing the performance of a new line of products might plot demographic variables for market segments as well as their buying behavior based on the products’ specific features.

Correspondence analysis is a powerful exploratory tool because you don’t need a clear hypothesis to conduct the analysis. In other words, it’s a statistical fishing expedition for discovering useful information that wouldn’t have been apparent using other techniques.

The method can handle numerical and categorical data. It has been used to inform everything from marketing strategies to product design.

Simple correspondence analysis (SCA) is also known as principal coordinates analysis or simple matching analysis.

It explores the interplay between two variables within a data set. The analysis starts with a contingency table, where data from the different categories is represented with rows and columns.

Researchers use SCA to create a two-dimensional graphical representation of the table. It calculates how the two specific binary variables (presence/absence) relate.

This analysis can condense large data tables from both variables into a visual format that makes identifying patterns easier.

Multiple correspondence analysis (MCA) incorporates three or more variables into a single analysis. It can examine complex interactions between different categories simultaneously, providing a more comprehensive overview of the data in a multivariate approach.

This method is particularly useful for analyzing very large data sets where multidimensional relationships are possible. The resulting graph reduces lots of complex information into a much more manageable form, making it easier to find and interpret hidden information.

Correspondence analysis is a valuable statistical tool for extracting actionable insights. Many industries use it.

Here are some of the method’s key applications:

• Retail—understanding store layout patterns and the performance of certain types of products

• Market research—analyzing the response to targeted campaigns based on different demographic factors

• UX design—tracking data about how users interact with interfaces to make them more intuitive and easy to navigate

• Human resources—understanding potential patterns in employee satisfaction and attrition

• Healthcare—reviewing any correlations between patient outcomes and aspects of their treatment

Conducting simple or multiple correspondence analysis is a systematic process. You start by collecting the data and finish by interpreting your results to find any potentially important connections. Each step is crucial for deriving accurate and meaningful insights from your data.

Step 1: data collection

Gather relevant data on the variables you want to explore for the objective of your analysis (if you already have one). Otherwise, gather any categorical data that could be meaningful for an aspect of your research. You can use survey results, performance metrics, or data from another source.

Step 2: data pre-processing

Before you start the analysis, you’ll need to structure the data to make it suitable for this type of statistical comparison. Pre-processing can include:

• Converting categorical data into numbers, if necessary

• Combining categories into broader groups if you have too many within a variable, which can simplify the analysis

• Reviewing the data to make sure it’s complete and high-quality enough for accurate, reliable results, e.g., by removing missing values, outliers, or inconsistencies

• Organizing the data into a contingency table, often with rows representing one variable (like demographics) and columns representing another (like product preferences)

• Standardizing the data to account for any differences in scale between the variables you’re trying to compare (also known as “normalizing”)

Step 3: computing correspondence analysis

Next, conduct the correspondence analysis. You’ll need to use various mathematical computations to help you find key relationships between variables. R and Python software packages or templates on the internet can help with these complex statistical calculations.

Here are some essential aspects of this process:

• Calculate the percentage of each category’s frequency compared to the total frequency for all the categories in the contingency table. This will help you understand your data’s overall distribution and calculate each category’s relative importance.

• Measure the similarities and differences between categories by comparing their percentage distributions. Higher similarity scores indicate a closer relationship and lower scores suggest a weaker one.

• Apply a mathematical technique called a singular value decomposition (SVD) to find the most significant patterns between your variables. This will serve as the basis for creating a two-dimensional graph (a biplot).

• Use the results from the SVD to calculate each variable’s coordinates (each row and column on your contingency table) and position them on the graph. This step lets you visualize the categories’ relationships and clearly see underlying patterns or associations.

Step 4: interpreting the results

The final step is interpreting the resulting graph and drawing meaningful conclusions about the potential connections. This is a very subjective process. It identifies any clusters of data points that might suggest a trend or association.

Correspondence analysis only offers simplified representations of the relationships between variables and doesn’t capture every detail. This method is also exploratory in nature, so you should treat your findings as the starting point for further research rather than a definitive conclusion.

Correspondence analysis is a very versatile and insightful tool for sorting through statistical data. However, it also has some important limitations that researchers should consider so that they can ensure meaningful results. These limitations include the following:

• All data has to be consistent: the data analyzed must be structured in a “clean” way to yield an accurate, meaningful visualization of potential patterns. This requires thorough pre-processing of the data.

• Analysis is influenced by outliers: the correspondence analysis is unlikely to pick up useful information if the input is not high-quality and reliable. Any extreme data points that weren’t addressed can skew the results enough to cause an incorrect interpretation.

• Coordinates on the maps are selectively scaled: using different scaling methods to create your two-dimensional graph can emphasize other relationships. This introduces a lot of subjectivity. You’ll need to become familiar with these nuances and choose a scaling method that’s appropriate for the type of research you’re doing, especially when comparing different data sets.

• The results might lack statistical significance: any relationships that correspondence analysis uncovers might be coincidental. Researchers will have to use various additional techniques to confirm the significance and validity of any findings.

Correspondence analysis is a popular statistical technique for studying connections, but it’s not the only technique that can uncover insights.

Here are some notable alternative methods that might be appropriate, depending on your project’s specific goals:

Chi-squared tests

You can use this technique to determine whether there’s a relationship between two specific sets of categorical data. However, it doesn’t offer the visualization aspect.

Conducting a chi-squared test allows you to work backward by analyzing a contingency table and calculating the expected correlation data between two completely unrelated variables.

If there’s a statistically significant difference between this expected correlation and the actual amount measured, you can conclude that there’s likely a significant relationship. Software can perform the chi-squared calculations for you.

Principal components analysis (PCA) and factor analysis (FA)

Principal components analysis and factor analysis are statistical techniques used to simplify complex data sets and find hidden patterns. However, they’re better suited for numerical data than categorical data. Correspondence analysis, on the other hand, can handle both.

Principal components analysis reduces the number of variables in your data by computing new ones (principal components) that make it easier to analyze potential relationships.

Factor analysis is very similar. It reduces multiple variables into a smaller group of the most important ones. These are referred to as factors. This process reveals hidden trends in the data.

By transforming data into a visual representation, correspondence analysis allows researchers to explore potential relationships between variables, even if they don’t yet know what the connections may be.

Despite its limitations, this statistical analysis tool is a valuable and versatile starting point for many different research projects, especially when combined with other methods to help validate findings.