Correspondence analysis is a statistical method for the analysis of multidimensional data, it is a multivariate technique that analyzes patterns of association between qualitative variables.

Qualitative variables are variables that are not represented by numbers, but by modalities, for example: gender, level of education, marital status, etc.

Since qualitative variables are used in the AC, the object of the analysis are the contingency matrices, whose elements indicate the number of times (the counts) that the characteristics of two different quantities have been detected together.

The main goal of AC is to analyze the relationships between a set of qualitative variables observed on a collective of statistical units. This is done through the identification of an "optimal" space, i.e. a small dimension that represents the synthesis of the structural information contained in the original data.

In essence, they will build a series of latent variables (or factors), a combination of the original variables, which express some concepts not directly observable in reality, but the result of the measurement of a set of variables.

In Correspondence Analysis, the variables used do not have to be independent, so the modes of one variable must influence the modes of the other.

Before carrying out a correspondence analysis it is necessary to establish the degree of interdependence between the characters considered because, if they are independent, it may not make sense to search for the correspondences between them.
For this purpose, it is necessary to apply the Chi-square test, which assesses any interdependence relationships between the qualitative variables.

The test starts of the null hypothesis that considers the two independent variables. The alternative hypothesis will be that the two variables have a certain degree of interdependence.
If the test results return a p-value < 0.05, the null hypothesis can be rejected and consequently the two variables will be considered interdependent, and you can continue with the analysis.

The contingency tables contain the joint frequencies of the variable modes. Given two qualitative variables X and Y, the relevant contingency table will contain how many times a given mode of variable X occurs with a given mode of variable Y.

The Correspondence Analysis allows to represent the phenomenon both in the space of the rows and in the space of the columns.
To do this, the row and column profile matrices must be constructed:
- dividing the absolute frequencies by the corresponding marginal rows (or column);
- dividing the relative frequencies (i.e. the absolute frequencies divided by the total number of the sample) by the respective row (or column) margins.

Row Profile Matrix	Column Profile Matrix

Finally, you have to calculate the distances between the profiles to see if the modalities are similar or not, distant or not, i.e. see if the profiles resemble each other or not.
There are two types of distances: the Euclidean distance and the Chi-square distance.

For the AC, R provides a package called FactoMineR.
First you need to install the FactoMineR package.

Given the objective of the AC, observing the inertia explained, we can see how much size the phenomenon is reduced to. We see that the first dimension alone explains about 60% of the overall variability of the data.

Joint two-dimensional graph individual-variables graphically represents how the modes of the two variables are arranged along the axes created by the newly extracted dimensions.