The function quantifies components of shape variation for a set of specimens as described by
their patterns of symmetry and asymmetry. Here, shape variation is decomposed into variation
among individuals, variation among sides (directional asymmetry), and variation due to an
individual x side interaction (fluctuating symmetry). These components are then statistically
evaluated using Procrustes ANOVA. Statistical assessment of model effects for shape variation
is accomplished using permutation procedures. Methods for both matching symmetry and object
symmetry can be implemented. Matching symmetry is when each object contains mirrored pairs of
structures (e.g., right and left hands) while object symmetry is when a single object is
symmetric about a midline (e.g., right and left sides of human faces). Details on general
approaches for the study of symmetry in geometric morphometrics may be found in: Mardia et
al. 2000; Klingenberg et al. 2002.
As input, the function receives either A 3D array (p x k x n) containing raw landmarks (requiring
GPA to be performed) or a gpagen object (if GPA has been previously performed) or a geomorphShapes
object. If one wishes to incorporate semilandmarks, GPA can either be performed first using gpagen,
or within bilat.symmetry by passing adequate GPA arguments (i.e. curves, surfaces, ProcD etc,
see gpagen. If a geomorphShapes object is provided, semilandmarks are automatically
identified and slid during GPA. For "object.sym = FALSE, landmarks should be of dimension (p x k
x 2n), as each specimen is represented by both left and right configurations.
Analyses of symmetry for matched pairs of objects is implemented when object.sym=FALSE. Here,
a 3D array [p x k x 2n] contains the landmark coordinates for all pairs of structures (2
structures for each of n specimens). Because the two sets of structures are on opposite sides,
they represent mirror images, and one set must be reflected prior to the analysis to allow
landmark correspondence. IT IS ASSUMED THAT THE USER HAS DONE THIS PRIOR TO PERFORMING THE
SYMMETRY ANALYSIS. Reflecting a set of specimens may be accomplished by multiplying one coordinate
dimension by '-1' for these structures (either the x-, the y-, or the z-dimension). A vector
containing information on individuals and sides must also be supplied. Replicates of each
specimen may also be included in the dataset, and when specified will be used as measurement
error (see Klingenberg and McIntyre 1998).
Analyses of object symmetry is implemented when object.sym=TRUE. Here, a 3D array [p x k x n]
contains the landmark coordinates for all n specimens. To obtain information about asymmetry,
the function generates a second set of objects by reflecting them about one of their coordinate
axes. The landmarks across the line of symmetry are then relabeled to obtain landmark
correspondence. The user must supply a list of landmark pairs. A vector containing information
on individuals must also be supplied. Replicates of each specimen may also be included in the
dataset, and when specified will be used as measurement error.
The function also provides individual measures of signed and unsigned asymmetry, calculated as the
Procrustes distance between the right and left element (for paired structures, as detailed in
Klingenberg and McIntyre 1998) or side of the structure (for object symmetry, following Lazi<U+0107>
et al 2015). The computational difference betwen the two approaches consists in that, for object
symmetry, only paired landmarks are considered, excluding the landmarks of the midline.
Notes for geomorph 3.0
Compared to older versions of geomorph, some results can be expected to be slightly different.
Starting with geomorph 3.0, results use only type I sums of squares (SS) with either full
randomization of raw shape values or RRPP (preferred with nested terms) for analysis of variance
(ANOVA). Older versions used a combination of parametric and non-parametric results, as well as
a combination of type I and type III SS. While analytical conclusions should be consistent
(i.e., "significance" of effects is the same), these updates maintain consistency in analytical
philosophy. This change will require longer computation time for large datasets, but the
trade-off allows users to have more flexibility and eliminates combining disparate analytical
philosophies.
Note also that significance of terms in the model are found by comparing F-values for each term
to those obtained via permutation. F-ratios and df are not strictly necessary (a ratio of SS
would suffice), but they are reported as is standard for anova tables. Additionally, users will
notice that the df reported are based on the number of observations rather than a combination
of objects * coordinates * dimensions, as is sometimes found in morphometric studies of symmetry.
However, this change has no effect on hypothesis testing, as only SS vary among permutations (df,
coordinates, and dimensions are constants).
The generic functions, print, summary, and plot all work
with bilat.symmetry.