UPGMA analysis

Cluster Analysis: an example

    Given a matrix of pairwise distances among taxa, cluster analysis represents 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 in the matrix between any two taxa,
    (2) Combine these two taxa as a pair,
    (3) Re-calculate the average distance between this pair and all other taxa to form a new matrix,
    (4) Return to Step 1: Identify the closest pair in the new matrix
    (5) Continue, until the last two clusters are joined.

Consider five taxa (A, B, C, D, E) with the following 5x5 distance matrix:

	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 a new 4x4 matrix with an (AB) pair:

	(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 a new 3x3 matrix with (AB) and (DE) pairs:

	(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 a 2x2 cluster, with only one value:

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

The two clusters join at 72.5. This completes the analysis. [Comment on calculations]

These results may be presented as a phenogram with nodes at 20, 30, 45, and 72.5 units. Note that these are values are the calculated pairings in the algorithm. The phenogram indicates that A & B are most similar to each other, followed by D & E, and that C is more similar to D & E. The phenogram of similarity would have the same shape as a cladogram of relationship, if and only if rates of change were absolutely constant.