Biology Reference
In-Depth Information
TABLE 12.3
Two-Way Mixed Model Procrustes ANOVA of Fluctuating Asymmetry of the Prairie Deer
Mouse Cranium
Source
SS
D
MS
F
p
Individuals
0.15978
5980
0.0000267188
7.16
,
0.001
Sides
0.0064622
52
0.000124272
33.31
,
0.001
Individuals
3
Sides
0.0223116
5980
0.000003731
1.28
,
0.001
Measurement error
0.0689126
23608
0.0000029190
-
is in units of Mahalanobis distance (which takes the covariance structure into account). All
three can be calculated by hand (or rather, in a spreadsheet). Of course, the calculation
does not have to be done by hand
there are programs that will do it for you, but it is
easy to understand the procedure if you can implement it yourself.
The first measure is a conventional Procrustes distance between each individual's
right
left distance and the bilaterally symmetric mean shape. To calculate this in a spread-
sheet, open the file that contains the superimposed right and left sides. Subtract the coordi-
nates of one side from those of the other. This subtraction gives the difference between the
right and left sides for that individual. Then square the differences for each coordinate,
summing those squares over all the coordinates for that individual. The square root of that
sum is the measure of overall FA for each individual. The second metric based on the
Procrustes distance differs from the first only in that the average directional asymmetry is
the standard instead of the bilaterally symmetric mean. This measure of FA is calculated
just like the first except the population's average right
left difference is subtracted from
each individual's right
left difference. No such subtraction was necessary in the first
case because the average right
left difference for a bilaterally symmetric form is zero.
After subtraction, the differences are squared and summed and the square root is taken of
the sum.
The third metric is a Mahalanobis distance between the two sides (
Klingenberg and
Monteiro, 2005
). This calculation, which is more involved than the other two, can also be
done by hand. The first step is to compute the right
left differences for each individual.
But these differences are not squared or summed over the coordinates. Rather, after calcu-
lating the right
left differences for each coordinates for all individuals, the data are sub-
jected to a Principal Components Analysis (PCA) of the covariance matrix. Then the scores
for the principal components (PCs) are standardized to unit variance (which is done by
squaring them, summing the squares, taking the square root and dividing each score by
that value). The number of PCs to use in this calculation depends on the dimensionality of
the data. If the data consist solely of landmarks, and the PCA was done on the partial
warps, all the PCs should be used when computing FA. If the analysis is instead done
using the superimposed coordinates (or Procrustes residuals), the last four PCs should be
excluded from the analysis because there are 2
K
4 dimensions but 2
K
PCs. When the
data include semilandmarks, the number of PCs to use is 2
K
2
1
L
2
4 (for two-dimensional
data). The same reasoning extends to three-dimensional data
the number of PCs used in
