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]

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.

• ${\displaystyle x(i,j)=|X_{i}-X_{j}|}${\displaystyle x(i,j)=|X_{i}-X_{j}|}, the Euclidean distance between the ith and jth observations.
• ${\displaystyle t(i,j)}${\displaystyle t(i,j)}, the dendrogrammatic distance between the model points ${\displaystyle T_{i}}${\displaystyle T_{i}} and ${\displaystyle T_{j}}${\displaystyle T_{j}}. This distance is the height of the node at which these two points are first joined together.

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

${\displaystyle c={\frac {\sum _{i<j}[x(i,j)-{\bar {x}}][t(i,j)-{\bar {t}}]}{\sqrt {\sum _{i<j}[x(i,j)-{\bar {x}}]^{2}\sum _{i<j}[t(i,j)-{\bar {t}}]^{2}}}}.}${\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.^[5]

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

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

• Cophenetic

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

• Covariance and correlation