# Cophenetic correlation

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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.

• x(i, j) = | XiXj |, the ordinary Euclidean distance between the ith and jth observations.
• t(i, j) = the dendrogrammatic distance between the model points Ti and Tj. 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 [1] or in Python using the scipy-package [5].

## References

1. ^ Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40
2. ^ Dorthe B. Carr, Chris J. Young, Richard C. Aster, and Xioabing Zhang, Cluster Analysis for CTBT Seismic Event Monitoring (a study prepared for the U.S. Department of Energy)
3. ^ Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412 (link)
4. ^ Mathworks statistics toolbox
5. ^ "scipy.cluster.hierarchy.cophenet — SciPy v0.14.0 Reference Guide". docs.scipy.org. Retrieved 2019-07-11.