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

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}}}}.

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]

References

^ Sokal, R. R. and F. J. Rohlf. 1962. The comparison of dendrograms by objective methods. Taxon, 11:33-40
^ 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)
^ Rohlf, F. J. and David L. Fisher. 1968. Test for hierarchical structure in random data sets. Systematic Zool., 17:407-412 (link)
^ Mathworks statistics toolbox
^ "Introduction to dendextend".
^ "scipy.cluster.hierarchy.cophenet — SciPy v0.14.0 Reference Guide". docs.scipy.org. Retrieved 2019-07-11.
^ "Cophenetic correlation coefficient - MATLAB cophenet".

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