Comparisons between results of different statistical analyses often involve assessing how much two vectors in shape space resemble each other. For instance, it might be relevant to compare an allometric regression vector and a first principal component.
Because vectors such as PCs, PLS axes, regression vectors etc. correspond to directions in shape tangent space, a straightforward method to compare such vectirs is to compute the angle between them. This method has been used both in traditional and geometric morphometrics (e.g. Cheverud 1982; Klingenberg and Zimmermann 1992; Klingenberg and McIntyre 1998; Klingenberg and Zaklan 2000; Klingenberg and Marugán-Lobón 2013).
The angle between two column vectors a and b can be computed as arccos(a'b / sqrt(a'ab'b)), where "arccos" is the arc-cosine, "sqrt" is the square root and the apostrophe stands for the vector transpose.
Statistical assessment of the resulting angles is often done by
comparison to angles between pairs of random vectors in the
multivariate space of interest. Random vectors are drawn from a
uniform distribution on a hypersphere with the appropriate
dimensionality. These comparisons have unsually been done by
simulation, but a closed-form formula for the probability has been
published (Li 2011), which allows to compute probability levels
precisely and instantly. This method is implemented here
(additional details in Klingenberg and Marugán-Lobón 2013).
This corresponds to a comparison of the observed angle against the
null hypothesis that the two vectors are completely unrelated to
each other; if this null hypothesis is rejected because the angle
is smaller than expected, the conclusion is that there is some
degree of similarity between the two vectors (note that this is not
a test examining whether the two vectors significantly differ from
each other).
In the Project Tree, the user may select one or both analyses that are to be compared. Alternatively, the selections can be made in the dialog box.
To invoke an angular comparison, select Compare Vector Directions from the Comparison Menu. A dialog box like the following will appear:
Select the analysis or the part of the analysis (such as blocks of variables for PLS of separate blocks) in the upper and lower lists. The check box labeled "Object symmetry: only show analyses with the same type of symmetry" limits the set of analyses shown in the secod list, but does not change the analysis.
A number of special conditions apply for different analyses:
MorphoJ allows comparisons between analyses that use data that agree in the number of landmarks, dimensionality (2D vs. 3D data) and in terms of properties related to symmetry (matching vs. object symmetry and, for object symmetry, the pairing of landmarks). These restrictions will prevent many inappropriate comparisons, but it is still the user's responsibility to ensure that the different analyses are based on the same set ofd landmarks (usually based on the same Procrustes fit, so that both analyses are based on the same shape tangent space). For instance, computing angles between two vectors computed in analyses of mouse mandibles and fly wings, even though both may be based on 15 landmarks in 2D and have no object symmetry, would be clearly nonsensical.
Some special conditions apply for object symmetry. For matching symmetry, all components are in the same subspace. Therefore, angles between vectors can be calculated the same way regardless of whether they relate to symmetry or asymmetry. Only the interpreations of the angles may differ.
For object symmetry, symmetric and asymmetric components of variation reside in orthogonal subspaces of the shape tangent space. Therefore, the angles between vectors in the different subspaces are 90 degrees and the corresponding P-values are 1.0.
The text output in the Results window provides information about the analyses that generated the vectors of interest and two tables of results: one table with the angles between vectors (in degrees) and the other table with the P-values of the test against the null hypothesis that the vectors have random directions in the shape tangent space. The vectors from the first analysis are used as the rows in the table, the ones from the second analysis as the columns.
Cheverud, J. M. 1982. Relationships among ontogenetic, static, and evolutionary allometry. Am. J. Phys. Anthropol. 59:139–149.
Klingenberg C.P., and J. Marugán-Lobón. 2013. Evolutionary covariation in geometric morphometric data: analyzing integration, modularity and allometry in a phylogenetic context. Syst. Biol. 62:591–610..
Klingenberg, C. P., and G. S. McIntyre. 1998. Geometric morphometrics of developmental instability: analyzing patterns of fluctuating asymmetry with Procrustes methods. Evolution 52:1363–1375.
Klingenberg, C. P., and S. D. Zaklan. 2000. Morphological integration between developmental compartments in the Drosophila wing. Evolution 54:1273–1285.
Klingenberg, C. P., and M. Zimmermann. 1992. Static, ontogenetic, and evolutionary allometry: a multivariate comparison in nine species of water striders. Am. Nat. 140:601–620.
Li, S. 2011. Concise formulas for the area and volume of a hyperspherical cap. Asian J. Math. Statist. 4:66–70.