# Cophenetic correlation

﻿
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.

• 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 x be the average of the x(i, j), and letting t be the average of the t(i, j), the cophenetic correlation coefficient c is given by[4]

$c = \frac {\sum_{i

## 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
4. ^ Mathworks statistics toolbox

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Cophenetic — In the clustering of biological information such as data from microarray experiments, the cophenetic similarity or cophenetic distance of two objects is a measure of how similar those two objects have to be in order to be grouped into the same… …   Wikipedia

• Correlation and dependence — This article is about correlation and dependence in statistical data. For other uses, see correlation (disambiguation). In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation …   Wikipedia

• List of statistics topics — Please add any Wikipedia articles related to statistics that are not already on this list.The Related changes link in the margin of this page (below search) leads to a list of the most recent changes to the articles listed below. To see the most… …   Wikipedia

• List of mathematics articles (C) — NOTOC C C closed subgroup C minimal theory C normal subgroup C number C semiring C space C symmetry C* algebra C0 semigroup CA group Cabal (set theory) Cabibbo Kobayashi Maskawa matrix Cabinet projection Cable knot Cabri Geometry Cabtaxi number… …   Wikipedia