Cophenetic correlation

Suppose that the original data {X_i} have been modeled using a cluster method to produce a dendrogram {T_i}; that is, a simplified model in which data that are "close" have been grouped into a hierarchical tree. Define the following distance measures.

• x(i, j) = | X_i − X_j |, the ordinary Euclidean distance between the ith and jth observations.
• t(i, j) = the dendrogrammatic distance between the model points T_i and T_j. This distance is the height of the node at which these two points are first joined together.

Then, letting ${\displaystyle {\bar {x}}}$ be the average of the x(i, j), and letting ${\displaystyle {\bar {t}}}$ be the average of the t(i, j), the cophenetic correlation coefficient c is given by[4]

${\displaystyle c={\frac {\sum _{i<j}(x(i,j)-{\bar {x}})(t(i,j)-{\bar {t}})}{\sqrt {[\sum _{i<j}(x(i,j)-{\bar {x}})^{2}][\sum _{i<j}(t(i,j)-{\bar {t}})^{2}]}}}.}$

It is possible to calculate the cophenetic correlation in R using the dendextend R package or in Python using the scipy-package [5].

• Cophenetic

• Numerical example of cophenetic correlation
• Computing and displaying Cophenetic distances