Biology Reference
In-Depth Information
FIGURE 10.3 Independent contrasts. Contrasts are differences
between sister taxa, as indicated by the gray arrow. For tips that
share a common ancestor (a), they are the difference between sam-
ple means. For unpaired tips (b), they are the difference between
that tip and the MRCA of its sister clade. For internal nodes (c),
they are the difference between the respective MRCAs.
of shape change inferred from contrasts was not visibly different from that inferred from
the species means.
You may have noticed that the regression line for contrasts passes through the origin
(the y -intercept is zero). This is because contrasts are computed from the differences
between taxa; so if there is no change in one trait, there should be no change in the other,
whatever the slope happens to be (
0). For the same reason, all values of the inde-
pendent variable, size, are positive. There is no reason to choose between subtracting
taxon A from taxon B or the reverse. What is important is computing the contrast in the
same direction for both traits (e.g. A
0
β3
5
B for independent and dependent variables) to
preserve the positive or negative slope of the relationship between traits.
Unlike many previous studies, our contrast analysis found a somewhat steeper slope
than the analysis based on trait values (0.0124 vs 0.0094), but as Rohlf (2006) points out
there is no apriori reason to expect a difference in one direction rather than the other.
Regression on untransformed species means and regression on contrasts are both unbi-
ased estimators of the sample slope and correlation. This does not mean the two produce
exactly the same result, only that underestimating and overestimating the true value
are equally likely. The purpose of using contrasts is to judge correctly whether the
slope can be considered different from zero or some other reference value, which is
achieved by correcting the numbers of degrees of freedom and accounting for the
expected covariance resulting from differences in relatedness. This may be clearer for the
next method.
The most commonly used alternative to phylogenetic independent contrasts (PIC) is
phylogenetic generalized least squares (PGLS; Martins and Hansen, 1997 ; see also
Rohlf, 2001 ). PGLS also uses the expectation that change is proportionate to time under
Brownian motion drift to weight observations, but the difference is that the transformation
is applied directly to the data in PGLS, not to contrasts computed from the data as in PIC.
Recall the regression formula, y
2
b X
, in which
is the error term. In ordinary least
5
ε
squares regression, the elements of
are expected to be independent and normally distrib-
uted with a mean of zero but, in comparative data,
ε
is expected to exhibit variances and
covariances that are predicted by the phylogenetic variance covariance matrix that is,
the heights of tips and internal nodes above the root. Factoring that covariance matrix
out of the equation, thus incorporating it in the computation of b, yields a corrected
error matrix, more accurate estimates of p , and better type I error rates.
ε
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