Biology Reference
In-Depth Information
Current methods in geometric morphometrics rely on the theoretical foundation of shape
space as defined by
Kendall (1984)
. Now known as “Kendall's shape space,” it is non-
Euclidean (curved, nonlinear) multidimensional space where configurations of two or
more dimensions can be plotted as a single point. For the simplest shape, a triangle, Kendall's
shape space takes the form of a sphere. For shapes with more than three landmarks, the shape
space is a more complex, manifold surface in hyperspace that cannot be depicted. This means
that coordinate-based configurations do not exist in Euclidean space
,
which may confound
linear statistical methods commonly used in traditional morphometrics.
Slice (2001)
demonstrates that the coordinates for configurations fitted via a generalized
Procrustes analysis (the most popular method for more than two configurations) do not actu-
ally exist in Kendall's shape space, but are instead associated with a hemisphere that has
properties very similar to Kendall's space. Fortunately for both Kendall's shape space and
the Procrustes hemisphere, the point corresponding to the average configuration for a group
is approximately tangent to a linear vector plane. This creates a link to Euclidean space,
where the coordinates for configurations can be projected for use in linear statistical analyses.
In order to analyze Procrustes coordinates with standard, linear statistical procedures, the
tangent space can be constructed for projection of coordinates into a Euclidean plane. Alter-
natively, some linear statistical methods such as principal components analysis approximate
the curved shape space and are often used to bypass the more statistically complex projection
into tangent space. This brief overview of “Kendall's shape space” and the Procrustes hemi-
sphere attributes and implications for geometric morphometrics should be augmented by
reading
Kendall (1984)
and
Slice (2001)
as well as reviews found in
Slice (2005)
and
Mitter-
oecker and Gunz (2009)
.
Types of Data
Currently there are two primary data types common in geometric morphometric research
in biological anthropology. Since the type of data employed in a study depends on the
research question being investigated, it is useful to consider the benefits of each data type.
Modes of data acquisition for each of the data types are also presented.
Landmark Coordinates
The most common type of coordinate data employed in biological anthropology is from
anatomical landmarks, many of which are endpoints of standard craniometrics or other
linear distances on the skeleton. Typically, landmark coordinates are collected relative to arbi-
trary axes that are specific to each specimen, meaning that they are not comparable until
differences due to orientation and location have been eliminated. Not all landmarks are
created equal, as the quality of anatomical information contained in landmarks varies based
on location. It is desirable for landmarks to be homologous; in other words, they are discrete
structures or features that occur in approximately the same location on all specimens.
Recognizing that not all landmarks have the same degree of homology, Bookstein
(1991:63
e
66) defined three types of landmarks.
Type I landmarks
are defined as locations
based on distinct structures that intersect or are juxtaposed. The cranial landmark, bregma,
is a Type I landmark, as it is the intersection of the sagittal and coronal sutures in the midsag-
ittal plane. This landmark occurs in the same anatomical location on each cranium.
Type II
