Biology Reference
In-Depth Information
GEOMETRIC MORPHOMETRIC METHODS
Originally geometric morphometrics included several methods for analyzing shape infor-
mation encoded in coordinate data that were widely used in biological anthropology. Analyt-
ical approaches such as Euclidean distance matrix analysis (EDMA) and elliptical Fourier
analysis (EFA) were included in early discussions of the suite of geometric morphometric
methods (e.g., Richstmeier et al., 2002). While EFA is useful for certain research designs
(e.g.,
Christensen, 2005; Sholts et al., 2011b
) and many geometric morphometrics programs
are capable of performing this analysis, EDMA is no longer as common as once it was.
Rohlf
(2003)
compared error and bias in estimation of mean shapes for several geometric morpho-
metric methods including generalized Procrustes analysis (described below) and EDMA.
This study demonstrated that generalized Procrustes analysis generated the least error and
no bias when used to estimate mean shape. For this review of methods, only the most
common methods currently in use are presented.
Procrustes Methods
Coordinate data, the basis for geometric morphometric methods, define a form under
investigation by encoding its geometric properties. Each configuration is defined by a set
of arbitrary axes where the coordinates are relative only to each other, meaning that multiple
configurations are not comparable to one another. Configurations composed of raw coordi-
nates also contain size information. The most popular approach to removing the differences
in location, orientation, and size among configurations is the Procrustes superimposition
(
Adams et al., 2004; Slice, 2005, 2007; Mitteroecker and Gunz, 2009
).
Procrustes superimposition uses a least-squares solution to bring multiple configurations
into a common coordinate system and align homologous landmark coordinates through
rotation (
Rohlf and Slice, 1990
). Typically, Procrustes superimposition is set to scale configu-
rations to a common Centroid Size without the use of a least-squares estimate, known as
a partial Procrustes fitting. Specifically, configurations are translated into a common coordi-
nate system by aligning the centroid of the coordinates for each configuration. Often, this is
accomplished by locating all centroids at the shared coordinate system's origin. Next, the
configurations are scaled to a common Centroid Size, which is the measure of the distances
(square root of the sum of the squared distances) from all landmark coordinates to the config-
uration centroid. For Procrustes superimposition, Centroid Size is typically set to equal one.
This removes uniform size differences among configurations. Finally, the configurations are
rotated so that sum of the squared distances is minimized between the homologous
landmark coordinates. In the case of two configurations, this is fairly straightforward, but
for more than two configurations, an iterative rotation process is needed.
The generalized Procrustes analysis (GPA) translates and scales the configuration as
described above, and then rotates all configurations to a least-squares fit with one of the
configurations from the sample. After this initial fitting process, the average of all coordinates
is calculated to generate a first-round mean configuration. The process is then continued with
each configuration again being fit to the estimated mean configuration leading to the arrival
at a new mean configuration. This iterative process continues until the sum-of-the-squared
deviations between the fitted configurations and the means meet some criterion (minimal
