Cluster analysis is a type of multivariate analysis technique that can be applied in many fields: from computer science, medicine and biology, from archaeology to marketing, whenever it is necessary to classify a large amount of information into distinguishable groups.

Cluster analysis is used to group statistical units (records) that have common characteristics and assign them to categories not defined a priori.
The groups (clusters) formed must be as homogeneous as possible inside (or even similar, intra-cluster) and heterogeneous outside (or even dissimilar, inter-cluster).

In Cluster Analysis you can use:

-qualitative variables, which present modalities (e.g. gender, level of education, marital status, etc.)

The distance matrix D is useful to know how many statistical units are different from each other, it is decisive for the choice of variables to be considered.
The distance matrix, of dimensions n×n , is a symmetric matrix that has on the greater diagonal all 0, this is because the distance between a point and itself is zero.

Before creating the distance matrix, the starting matrix must be standardized, so each variable will have the same weight as the others.

To obtain the distance matrix D it is necessary to calculate the distances between the points.
Depending on the type of variable, quantitative or qualitative, with which you are working, these distances can be calculated in different way

Thanks to the link rule we can choose the type of link that we will use to form clusters, among the following:

After choosing the most appropriate link for your analysis and the creation of groups, you can create the graphic representation: the Dendogram. It is represented according to increasing ordinates the level of aggregation of the clusters. On the x-axis there are points, on the y-axis there are distances

The distance between clusters tends to increase and for this reason we choose a stop rule that allows us to choose the number of groups we want to get.

To do this we use the tree cutting technique:

After importing the dataset into R, we start with the analysis in Cluster:

The result obtained with the Simple linkage:

The same procedure is carried out for the complete linkage and the average linkage.
You will compare the results and choose the most representative link for the analysis you are conducting.

Comparing the three links, the most suitable is the complete likage as it divides the clusters better, avoiding that there is too much internal homogeneity at the expense of external heterogeneity. It also prevents the formation of outliers, i.e. clusters composed of a single point.