# Cophenetic correlation

In statistics, and especially in biostatistics, cophenetic correlation[1] (more precisely, the cophenetic correlation coefficient) is a measure of how faithfully a dendrogram preserves the pairwise distances between the original unmodeled data points. Although it has been most widely applied in the field of biostatistics (typically to assess cluster-based models of DNA sequences, or other taxonomic models), it can also be used in other fields of inquiry where raw data tend to occur in clumps, or clusters.[2] This coefficient has also been proposed for use as a test for nested clusters.[3]

## Calculating the cophenetic correlation coefficient

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

• ${\displaystyle x(i,j)=|X_{i}-X_{j}|}$, the Euclidean distance between the ith and jth observations.
• ${\displaystyle t(i,j)}$, the dendrogrammatic distance between the model points ${\displaystyle T_{i}}$ and ${\displaystyle T_{j}}$. 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

## Software implementation

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

In Python, the SciPy package also has an implementation.[6]

In MATLAB, the Statistic and Machine Learning toolbox contains an implementation.[7]