The Covariation menu contains methods for analyzing covariation of shape with other variables (which may be shape variables too).
Currently, MorphoJ implements the following methods for analyzing covariation: partial least squares (PLS), regression and a procedure for evaluating hypotheses of modularity in morphometric data.
PLS is a method in widespread use in geometric morphometrics for exploring patterns of covariation between shape and other variables (e.g. environmental variables, experimental treatments, or shapes of other structures). It identifies features of shape that have the strongest covariation with the other set of variables. The focus of the analysis is on these patterns of covariation, and to some extent on the strength of association. PLS analysis does not imply a direction in the association -- nothing is implied about whether variation in one set of variables is caused in the other set.
In contrast, the main purpose of regression is to predict variation in one set of variables from variation in the other set. Therefore, there is a clear difference between the two sets of variables, which may be due to a direct causal relationship. An additional use of regression, based on this, is that it can be used to correct for the influence of other variables. In morphometrics, this is most frequently done to correct for the effects of size on shape (allometry; Klingenberg 2016).
For some situations, it is helpful to use a regression to compute residuals or predicted values for a different dataset. For instance, an analysis of variation among several species may want to use a regression of shape on size in growth data within species to compute residuals as a way to correct for ontogenetic scaling. For that purpose, MorphoJ offers the possibility to compute residuals and predicted values from a regression and a suitable dataset (note that the data must be compatible in terms of dimensions, numbers of landmarks etc.).
An important concept related to covariation is modularity: morphological modules are parts whose components covary strongly, but which are relatively independent of other modules (e.g. Klingenberg 2008). MorphoJ implements a method to evaluate hypotheses of modularity (Klingenberg 2009). Given a hypothesis, specified a priori, that particular sets of landmarks in a configuration are modules, this method evaluates whether the covariation between these sets is weaker than would be expected for alternative subdivisions of the landmarks into subsets. This method has several options; for instance, the user can request that only spatially contiguous subsets are considered in the evaluation process, if this is appropriate in the biological context of a study (Klingenberg 2009).
Klingenberg, C. P. 2008. Morphological integration and developmental modularity. Annual Review of Ecology, Evolution and Systematics 39:115–132.
Klingenberg, C. P. 2009. Morphometric integration and modularity in configurations of landmarks: Tools for evaluating a-priori hypotheses. Evolution & Development 11:405–421.
Klingenberg, C. P. 2016. Size, shape, and form: concepts of allometry in geometric morphometrics. Development Genes and Evolution 226:113–137.