Biology Reference
In-Depth Information
clusters, in the first step, the leaves
A
and
C
as they have the small-
est distance (
d
AC
5), and therefore produces the
wrong tree. The wrong grouping of long branches due to method-
ological problems is known as long branch attraction (LBA).
LBA is not exclusively a problem of UPGMA; any tree building
method using a too simplistic model of evolution or nonadditive
distances can be susceptible to LBA. An elaborated review of LBA
is Bergsten.
16
=
2
+
1
+
2
=
•
Least squares
(
LS
). Besides ultrametricity, UPGMA assumes input
distances that exactly reflect a tree. In practice, distances have an
estimation error; as a consequence, they cannot be mapped exactly
on a tree. The goal then is to find a tree (topology and branch
lengths) that satisfies them as best as possible according to some
optimality criterion. Let
ε
ij
be the normalized discrepancy for the
distance between leaves
i
and
j
, defined as
Td
-
ij
ij
e
=
,
ij
s
ij
where
T
ij
is the leaf-to-leaf path length on the tree;
d
ij
the input
distance; and
2
ij
, its variance. A very good optimality criterion is
to minimize the norm ||
σ
ε
||. A common choice is to use the
2
Euclidean norm ||
ij
, which leads to a weighted least
squares (WLS) tree. This has a statistical justification. Under a
model where the errors in
d
ij
are independently and normally dis-
tributed with expected value zero and variance
ε
||
2
=∑
ij
ε
2
σ
ij
, the tree with
||
2
is the maximum likelihood distance tree.
e
Maximum
likelihood is described later.
Sometimes, the variances are not known and modeled as a
function of the distances. In the Fitch and Margoliash method,
17
for instance, it is assumed that the variances are proportional to the
squared distances. If all of the variances are assumed to be equal,
the method is called ordinary least squares (OLS).
minimal ||
ε
e
This is usually not the case, as distance estimates are correlated if they share branches
on their leaf-to-leaf paths of the true tree. In this case, the optimal method is gener-
alized least squares.
