Biology Reference
In-Depth Information
partition it into the contribution made by each taxon (the partial disparity of that taxon;
Foote, 1993a
). The additivity of variances means that the sum of partial disparities equals
the overall disparity. However, it is worth noting that the two measures weigh outliers
differently, and consequently their results can differ. Standard deviations and variances
are not linearly related, and a highly distinctive taxon has a much greater impact on a
variance than on a standard deviation.
Measuring Disparity
To measure morphological disparity (MD) by a variance, we calculate:
P
j
5
1
D
j
ðN
MD
(10.1)
5
1
Þ
2
where
D
j
is the distance of species
j
from the overall centroid (which is the grand mean
calculated over the
n
species or other groups being analyzed). We can use
Equation 10.1
to
calculate both size and shape disparity. For size data,
D
j
is the difference between the
centroid size of an individual species and the grand mean centroid size. For shape data,
D
j
is the Procrustes distance between the average shape of an individual species and the
grand mean shape. We can compute shape disparity directly by estimating those
Procrustes distances, or we can calculate the variances of coordinates obtained by a gener-
alized least squares Procrustes superimposition (GLS) or variances of partial warp scores
(including scores on the uniform component). All three approaches yield the same results
because the sum of squared coordinates obtained by GLS equals the squared Procrustes
distance to the mean, as does the sum of squared partial warp scores. In those analyses
the grand mean shape is the consensus, so if we are using partial warps we can use the
formula:
P
j
5
1
PW
j
ðN
MD
(10.2)
5
1
Þ
2
where
PW
represents the partial warp scores for an individual, so the formula tells us to
sum all the squared partial warp scores for each individual over all individuals. Because
the grand mean shape is the consensus, its partial warp scores are all zeros, so
Equation
10.2
is equivalent to
Equation 10.1
.
Both are also equivalent to:
MD
TrfSg
5
(10.3)
where
Tr
is the trace of a matrix (the sum of its diagonal elements) and
S
is the
variance
covariance matrix of the partial warp scores (including the uniform component,
and computed using the grand mean as the consensus). The diagonal elements of a
variance
covariance matrix are the variances, so this formula tells us to sum the variances
of the variables, which takes us back to the squared distances from the consensus.
To exemplify the analysis of disparity, we will measure the disparity of adult body
shape of nine species of piranhas sampled at the 16 landmarks shown in
Figure 10.8
.
