Methods for the comparison of mean shapes in different samples are collected in the Comparison menu. 'Comparison' is used in two different meanings: the analyses grouped in this menu include both statistical comparisons (canonical variate analysis and discriminant functions) and phylogenetic comparative methods (included in 'Map onto phylogeny').
Canonical variate analysis (CVA) is one of the most widespread types of analysis in morphometrics. It is used to separate known groups in the data and provides an ordination that maximizes the separation of the group means relative to the variation within groups. It is usually applied to data from more than two groups. If there are only two groups, CVA is identical to a linear discriminant function analysis.
Discriminant function analysis (DFA) is a method to obtain optimal decision rules for distinguishing groups. Formally, it is the same as a CVA with just two groups. However, the emphasis tends to be on the degree of separation of the groups and not, as in CVA, on the relative arrangement of multiple groups. Accordingly, more emphasis is put on the on the probability of correct and incorrect classification of observations.
MorphoJ follows this difference. As a result, the implementations of CVA and DFA differ in some aspects. Because CVA simultaneously uses all groups in a dataset to compute axes of maximal group separation relative to a shared estimate of within-group variation, it implies the assumption that the covariance structure within all the groups is the same. For this reason, the pooled within-group covariance matrix is used throughout for CVA, even for computing Mahalanobis distances between pairs of groups (under the assumption of a shared within-group covariance structure, the pooled within-group covariance matrix is the best estimate). In contrast, the implementation of DFA conducts each pairwise comparison of groups separately, even if there are more than two groups included in a dataset and multiple comparisons are made as part of the DFA procedure. Accordingly, only the covariance matrices of the pair of groups included in a comparison is considered. As a result, Mahalanobis distances (and associated statistical tests) computed as part of the DFA prodedure can therefore differ from those obtained in conjunction with CVA. Moreover, the DFA procedure carries out a leave-one-out cross-validation to assess the reliability of classification.
Phylogenetic comparative methods are designed for the statistical analysis of data from different species, which cannot be considered as mutually independent data points because of shared ancestry among species. MorphoJ implements several approaches to approach this type of study that explicitly map shape data onto a phylogenetic tree (imported from a NEXUS file). The procedure Map Onto Phylogeny does such a mapping, which can be used to visualize reconstructed ancestral shapes or to superimpose the phylogeny onto ordination plots. Moreover, MorphoJ produces new datasets with reconstructed changes along the branches of the phylogeny and independent contrasts.
The procedure Collect Statistics on Tree Set is a specialized method for assessing the effects of phylogenetic uncertainty on comparative analyses of shape. A phylogeny tree set is a set of trees that characterize the uncertainty about an estimated phylogeny, for instance, a set of trees from a bootstrap analysis. Looking at the distribution of statistics, such as the tree length or the P-value of the permutation test for a phylogenetic signal, can provide helpful insights into the possible effects of uncertainty about the estimated tree on those statistics.
Many morphometric analyses produce vectors in shape space as part of their results: principal components (PCs), regression vectors, partial least squares (PLS) axes, etc. It is often of interest to compare two or more of those vectors, for instance, to examine whether an allometric regression vector is similar to a PC1 or whether the PLS axes for integrattion between parts of a landmark configuration correspond to the PCs of the entire structure.
This sort of comparison can be done by comparing the shape changes corresponding to the vectors of interest. Alternatively, it is possile to compute the angle between vectors in shape space. Compare Vector Directions does this sort of angular comparison and computes P-values under the null hypothesis that the vectors are random vectors drawn from a uniform distribution on a hypershpere with dimensionality corresponding to that of the shape tangent space.