# Understanding Correspondence Analysis: A Comprehensive Guide for 2023

This is a comprehensive guide to understanding correspondence analysis and is designed for anyone looking to gain an understanding of how correspondence analysis works and how to apply it.

In particular, this guide will be especially helpful for market researchers who want to know how to use correspondence analysis to analyze consumer preferences and brand loyalty.

However, anyone who needs to analyze categorical data and identify relationships between different variables will find this guide useful.

## Table of contents

- Introduction to Correspondence Analysis
- Correspondence Analysis vs. Other Multivariate Analysis Techniques
- Correspondence Analysis vs. Multiple Correspondence Analysis
- Applications of Correspondence Analysis in Market Research
- Conducting Correspondence Analysis
- Interpreting Correspondence Analysis Results
- Best Practices for Effective Correspondence Analysis
- Real-Life Examples of Correspondence Analysis in Action
- Advanced Correspondence Analysis Techniques
- Combining Correspondence Analysis with Other Techniques

## Introduction to Correspondence Analysis: Definition, Purpose, and Benefits

### What is correspondence analysis?

At its core, correspondence analysis is a statistical method for summarizing tables and analyzing the relationship between two or more categorical variables. It is a data visualization technique that aims to find patterns and associations between the categories of different variables.

In correspondence analysis, a contingency table is created to represent the frequencies or counts of the categories of the variables. The contingency table is then transformed into a matrix of proportions or percentages to normalize the data.

The method then calculates the principal components of the normalized matrix and maps the variables and categories onto a two-dimensional plot based on their proximity to each other. The plot displays the patterns and associations between the variables and categories, with closer proximity indicating a stronger relationship.

Correspondence analysis is commonly used in market research, social sciences, and other fields where categorical data analysis is important.

### What are the benefits of correspondence analysis?

Some benefits of correspondence analysis include:

**Visualizing patterns:**Correspondence analysis provides a way to graphically represent the relationships between categorical variables in a dataset, which can help to identify patterns and trends that might not be immediately apparent from looking at the raw data.**Simplifying complex data**: Correspondence analysis can be particularly useful when dealing with large, complex datasets with many categorical variables. By reducing the dimensionality of the data, correspondence analysis can help to simplify the analysis and make it easier to interpret.**Identifying associations:**Correspondence analysis can help to identify associations between categorical variables in a dataset. This can be useful for identifying potential causal relationships or for understanding the underlying factors that are driving patterns in the data.**Hypothesis testing:**Correspondence analysis can also be used to test hypotheses about the relationships between categorical variables in a dataset. By comparing the observed relationships to what would be expected by chance, researchers can determine whether there are significant associations between variables.

## Correspondence Analysis vs. Other Multivariate Analysis Techniques: Which One to Choose

When you are choosing a multivariate analysis technique to use, you should carefully consider the nature of your data and what you are trying to find out. It is important to weigh up the relative strengths and weaknesses of different techniques and choose one that is best suited to your needs.

As already mentioned, correspondence analysis is particularly useful for when you need to visualize patterns, simplify complex data and identify associations between variables. It is especially well-suited for visualizing and interpreting complex data sets, as it generates graphical representations of the relationships between categories.

The first thing to do is to consider your data. Correspondence analysis is useful when you have a table with the minimum of two rows and two columns and no missing data or negative values. Your data should also all have the same scale. You can also perform correspondence analysis on square tables.

However, there may also be some other situations where it may be better to use a different multivariate technique. For example, if you are working with continuous data, you may want to consider techniques such as principal component analysis (PCA) or factor analysis. Similarly, if you are interested in exploring linear relationships between variables, regression analysis may be a better choice.

## Correspondence Analysis vs. Multiple Correspondence Analysis

This section will explain the difference between correspondence analysis and multiple correspondence analysis and when you would choose each one. As already discussed, when it comes to deciding on which multivariate analysis technique to use, your decision will largely depend on your specific research questions and the number of variables you need to analyze.

However, when considering correspondence analysis, you may be tempted to use multiple correspondence analysis. Although multiple correspondence analysis sounds better than correspondence analysis because of the word ‘multiple’, it may not actually be better for your purposes. For the majority of real-world data problems, correspondence analysis is a more appropriate technique.

Correspondence analysis is a technique used to summarize relativities in tables and is based on a contingency table which shows the frequency of each combination of categories for two variables. As tables are the backbone of a lot of data analysis, it is a technique that can be used in many applications. In correspondence analysis, the categories of each variable are represented as points on a two-dimensional graph, and the distance between the points indicates the degree of association between the variables.

Multiple correspondence analysis is a technique for analyzing categorical variables and is based on a hypercube, which shows the frequency of each combination of categories for three or more variables. Hence, the word ‘multiple’ refers to the number of dimensions on the input table.

In multiple correspondence analysis, the categories are represented as points in a higher-dimensional space, and the dimensions are chosen to capture the maximum amount of variation in the data. Therefore you will want to choose multiple correspondence analysis if you are analyzing three or more categorical variables and you want a general understanding of how they are related.

## Applications of Correspondence Analysis in Market Research and Consumer Insights

If you are a market researcher, you might be wondering how to best use correspondence analysis to provide insights for your clients. Some applications of correspondence analysis in market research and consumer insights are:

**Brand positioning**: You can use correspondence analysis to identify how different brands are perceived by consumers based on certain attributes like price, quality, and features. This can help your clients better understand their brand positioning and make decisions on how to improve their brand image.**Customer segmentation:**Correspondence analysis can be used to group consumers based on their preferences or buying habits. This can help companies to develop targeted marketing strategies and to tailor their products or services to different consumer segments.**Product development:**Correspondence analysis can be used to identify which product attributes are most important to consumers and how these attributes are related to each other. This can help companies to develop products that better meet consumer needs and preferences.**Advertising effectiveness:**Correspondence analysis can be used to assess the effectiveness of advertising campaigns by identifying which messages and images resonate with different consumer segments. This can help companies to refine their advertising strategies and to improve their return on investment.**Customer satisfaction:**Correspondence analysis can be used to identify factors that contribute to customer satisfaction or dissatisfaction. This can help companies to improve their products or services and to retain customers.

## Conducting Correspondence Analysis: A Step-by-Step Guide

When it comes to conducting a correspondence analysis, there are many calculations involved. This guide will show you how you can perform the various calculations for correspondence analysis, but keep in mind that there are various software packages like Displayr available that will do the heavy lifting and calculating for you automatically - saving you time and effort.

Here is a step-by-step guide to perform a correspondence analysis:

**Step 1: Prepare the data**

The first step is to prepare the data for the analysis. The data should be in a contingency table format, which displays the frequency counts of two or more categorical variables. Each row of the table represents a level of one variable, and each column represents a level of another variable. The table should contain frequency counts or percentages, and any missing data should be appropriately handled.

**Step 2: Calculate the expected frequencies**

The expected frequencies are calculated based on the assumption of independence between the two categorical variables. This is done by multiplying the row and column marginal totals and dividing by the grand total. The expected frequency for each cell is compared to the observed frequency to calculate the residual.

**Step 3: Calculate the chi-square statistic**

The chi-square statistic is calculated as the sum of the squared residuals divided by the expected frequencies. This measures the degree of association between the two categorical variables.

**Step 4: Calculate the row and column profiles**

The row and column profiles are calculated by dividing the observed frequencies by the row or column marginal totals. These profiles represent the proportions of each category for each variable.

**Step 5: Calculate the singular value decomposition**

The singular value decomposition (SVD) is a mathematical technique used to decompose the chi-square matrix into its component parts. This results in eigenvalues and eigenvectors, which represent the dimensions of the analysis.

**Step 6: Calculate the contributions and cosines**

The contributions and cosines are calculated to measure the degree of association between the categories and the dimensions of the analysis. Contributions measure how much each category contributes to the variation in the analysis, while cosines measure the similarity between categories and dimensions.

**Step 7: Visualize the results**

The results of the correspondence analysis can be visualized using a biplot, which displays the row and column profiles as points in a two-dimensional space. The points are plotted based on their contributions and cosines to the first two dimensions of the analysis. This allows for easy interpretation of the relationships between the categories and variables.

If you’d like to avoid manually performing these steps, you can have Displayr do it for you in a few steps.

## Interpreting Correspondence Analysis Results: Understanding the Output

The output of your correspondence analysis typically includes a set of biplots, which show the relationship between the variables in your data set. Here are some steps to help interpret the results of your correspondence analysis:

**Check your conclusions against your raw data.****Interpret the Eigenvalues:**Eigenvalues represent the amount of variance explained by each dimension in the correspondence analysis. Generally, the higher the eigenvalue, the more important the corresponding dimension is in explaining the relationships between the variables. You can use eigenvalues to determine the optimal number of dimensions to include in your analysis.**Examine the biplot:**The biplot is a graphical representation of the relationships between the variables in the data set. Each variable is represented by a point on the plot, and the relationships between the variables are represented by the distance and angles between the points. The closer two points are to each other, the more similar the corresponding variables are. Conversely, if two points are far apart, the corresponding variables are dissimilar. You can also look for patterns in the biplot, such as groups of points that are clustered together, which indicates that the corresponding variables are highly correlated.**Analyze the contributions:**The contributions of each variable to each dimension are also important to examine. The contributions represent how much each variable contributes to the explanation of the corresponding dimension. Generally, variables with higher contributions are more important in explaining the relationships between the variables.**Look for outliers:**Outliers in a correspondence analysis can represent variables that do not fit well into the overall pattern of the data. Outliers can be caused by errors in data entry or other data quality issues. You should examine any outliers carefully to determine if they are valid or if they should be removed from the analysis.

## Best Practices for Effective Correspondence Analysis

Some simple best practices for effective correspondence analysis include choosing appropriate data, preparing the data and choosing the best software for your needs. Here are some other important things to get right that will make your correspondence analysis more effective.

### Normalization and Scaling

The default setting of the resulting plot from a correspondence analysis is called the normalization. This setting decides how the map should be interpreted. It is crucial to understand normalization as your choice of normalization setting could result in a misleading plot and each normalization setting has its own limitations.

Some of the main normalizations are standard (otherwise known as symmetrical), row principal, column principal or principal. Your choice of either row principal or column principal would depend on which relationship between the categories you are most interested in. You could use principal normalization which would show both row and column associations in the analysis, but has the disadvantage of misrepresenting the relationship between row and column categories.

If you select row or column principal you can also improve your normalization by scaling.

To find out more about how to use normalization, scaling, and rotation methods to get the best model, see this webinar recording.

### Adding Supplementary Points

You can also retrospectively add supplementary points to a correspondence analysis to help in the interpretation of results by providing additional context. These supplementary points do not influence the placement of your original core data points and are added after your map has been created. This additional context could include depicting changes over time or focusing on a subset of data.

### Add images to your plot

You can also add images to your plot to make the data visualization easier to understand at a glance and more visually appealing.

## Correspondence Analysis Software: Top Tools and Features

There are several statistical software packages that can help conduct correspondence analysis. These include:

**Displayr:**Displayr is a statistical software package built to handle different data types, automate many of the processes involved with correspondence analysis and make visualizing results easy.**R**is a free and open-source software that is widely used for statistical analysis, including correspondence analysis. R provides several packages such as 'FactoMineR' and 'vegan' that can be used for conducting correspondence analysis.**SPSS:**SPSS is a statistical software tool that provides a graphical user interface for conducting correspondence analysis.**Python:**Python is a programming language that is increasingly used for statistical analysis. The 'scikit-learn' package in Python provides different options for conducting correspondence analysis.

When selecting a software package to use, there are several features to consider.

- Input data format: Make sure to choose a software tool that will accept different data types (categorical, numeric, multiple response, etc.) and help you prepare your data for your analyses.
- Techniques and customization options: Some software tools provide several customization options, such as controlling the number of dimensions to be analyzed, choosing the scaling method, and selecting the statistical tests to be used. Try to select a tool that will handle all key variants, including multiple correspondence analysis, correspondence analysis of square tables and more. Make sure there are also customization options like rotation, different normalizations and adding supplementary data points.
- Visualization options: Correspondence analysis software typically provides several options for visualizing the results, such as scatter plots, biplots, and heatmaps. In Displayr, you can also use a moonplot, which is one of the easiest visualizations to interpret, particularly for brand maps.
- Output options: Correspondence analysis software should also provide options for exporting the results in various formats, such as tables, charts, and images.

Also make sure to consider your purpose for analysis. If you are a market researcher, Displayr is perfect for the analysis and presentation of brand association data and automates the creation of market structure maps. This makes it much easier to show brand associations, who competes with who, how big brands are, and the bases of their competition.

## Real-Life Examples of Correspondence Analysis in Action

Correspondence analysis is commonly used in market research to analyze the relationships between consumer preferences and product attributes. For example, a survey may ask consumers to rate different brands of soda based on attributes such as taste, aroma, and price. Correspondence analysis can be used to visualize the relationships between these variables and identify the most important factors driving consumer preferences.

It can also be used to help companies understand what makes each brand unique. For example, this brand positioning dashboard uses correspondence analysis to show how people view each soda brand. The top-right corner of the map shows all the highly caffeinated energy drinks, clustered together and owning energy-related attributes. Fanta appears in the top-left corner, representing the brand's focus on fun and appeal to kids. Meanwhile, Coke, Pepsi, and Lift sit near the middle of the map, indicating that they may not be as differentiated as other brands. Using this same example, you can see how rotating a correspondence analysis can make the patterns in the data easier to identify.

Here is another example, using European car brands. The correspondence analysis looks at which brands are mostly closely associated with attributes such as popular, luxury, sporty, family and green.

You can also use correspondence analysis to identify customer concerns and attitudes. For example, this correspondence analysis shows some American travelers’ concerns about traveling to various countries. It then uses bubble charts to show the significant relationships in residuals in the correspondence analysis.

## Advanced Correspondence Analysis Techniques: Non-Symmetric Correspondence Analysis

Non-symmetric correspondence analysis is an extension of correspondence analysis that allows for the analysis of non-symmetric contingency tables, where the rows and columns have different numbers of categories.

In correspondence analysis, the rows and columns are treated symmetrically, so the method assumes that the categories in the rows and columns are equivalent. However, in some cases, such as when analyzing data from surveys or questionnaires, the categories in the rows and columns may not be equivalent.

Non-symmetric correspondence analysis addresses this limitation by allowing the rows and columns to have different weights. Specifically, non-symmetric correspondence analysis assigns different weights to the rows and columns based on their relative importance in the analysis.

## Combining Correspondence Analysis with Other Techniques: Factor Analysis, Cluster Analysis, and Regression

You can also combine correspondence analysis with other techniques such as factor analysis, cluster analysis and regression analysis. This is typically done to allow for a more comprehensive analysis of the data by capturing both the relationship between the categorical variables and other things like continuous variables, the binary outcome variable and the similarity between observations.

To combine correspondence analysis with factor analysis, you’ll need to create a joint analysis of the categorical and continuous variables. To do so, construct a joint variance-covariance matrix that takes into account both the correlations among the continuous variables and the association among the categorical variables.

Both correspondence analysis and factor analysis are general techniques to help find patterns in data. If you are optimizing product ranges and portfolio planning, there is also a special tool called TURF (Total Unduplicated Reach and Frequency). You can find out more about when to use TURF, Factor and Correspondence Analysis here.

You can combine correspondence analysis with cluster analysis to identify groups of observations that are similar based on their categorical variables. First apply correspondence analysis to the categorical variables to create a set of components that summarizes the relationship. Then, apply cluster analysis to group the observations based on the values of the components.

Similarly, you can apply a logistic regression to a correspondence analysis in order to create a joint analysis of the categorical variables and the binary outcome variable.