Cluster Analysis: an example with the Pair Group Method

Given a matrix of pairwise distances among taxa, cluster analysis attempts to represent this information in a diagram called a phenogram that expresses the overall similarities among taxa. The Pair Group Method uses the following algorithm [a repetitive process for accomplishing a task]:  (1) Identify the minimum distance between any two taxa, (2) Combine these two taxa as a single pair, (3) Re-calculate the average distance between this pair and all other taxa to form a new matrix, (4) identifies the closest pair in the new matrix, (5) and so on, until the last two clusters are joined.

Consider five taxa (A, B, C, D, E) with the following distance matrix (the data could be molecular or morphological distances):
 


A B C D E
A 0 - - - -
B 20 0 - - -
C 60 50 0 - -
D 100 90 40 0 -
E 90 80 50 30 0

A & B are closest (20 units): join them into one cluster (AB) joining at 20, and recalculate the average distance from C, D, and E to (AB). [For example, the distance from C to (AB) = (60 + 50)/2 = 55, and the distance from D to (AB) = (100 + 90)/2 = 95]. This gives:
 


(AB) C D E
(AB) 0 - - -
C 55 0 - -
D 95 40 0 -
E 85 50 30 0

D & E are closest (30 units): join them into one cluster (DE) joining at 30, and recalculate the average distances between (AB), C, and (DE). [For example, the distance from (AB) to (DE) = (95 + 85)/2 = 90]. This gives:
 


(AB) C (DE)
(AB) 0 - -
C 55 0 -
(DE) 90 45 0

C & (DE) are closest (45 units): join them into one cluster (CDE) joining at 45, and recalculate the average distance between (CDE) and (AB). This gives:
 


(AB) (CDE)
(AB) 0 -
(CDE) 72.5 0

The two clusters join at 72.5. This completes the analysis.

The method illustrated is a Weighted PGM with Averaging (WPGMA). See the commentary on calculations for the difference between weighted and unweighted analyses (WPGMA and UPGMA).

These results may be presented as a phenogram with nodes at 20, 30, 45, and 72.5 units. The phenogram can be interepreted as indicating that A & B are similar to each other, as are D & E, and that C is more similar to D & E : 

Text material © 2007 by Steven M. Carr