The partial least squares (PLS) method is used for examining patterns of covariation between two or more sets of variables. In geometric morphometrics, one or more of these sets contain shape data. PLS may be used to related shape data to other types of data (ecological information, experimental conditions, etc.) or to other shape variables.
Two-block PLS is widely used in geometric morphometrics (e.g. Rohlf and Corti 2000). It is based on a singular value decomposition of the matrix of covariances between the two sets of variables. The result are pairs of linear combinations that successively maximize the covariance between the sets of variables, while being mutually uncorrelated across sets. The method has been used for a long time under a variety of names (e.g. Tucker 1958).
A more recent development in geometric morphometrics is the generalization of PLS to more than two blocks (Bookstein et al. 2003). For the multi-block situation, there are different solutions for different criteria that all lead to the same solution in the two-block situation. Accordingly, the choice of criterion and the associated algorithms can influence the results and their interpretation.
Different authors have pointed out various common features of PLS with principal component analysis, factor analysis and other methods. This reflects the different ways in which the procedure and its results can be interpreted.
Selecting Partial Least Squares from the Covariation menu opens a submenu:
There are two options for PLS analysis: between two separate blocks, or between blocks within a configuration of landmarks.
The first option for PLS in MorphoJ is the two-block PLS for two separate blocks of variables. 'Separate blocks' means that the variables can be from different datasets or from different data matrices in the same dataset. For shape data, all the landmark coordinates in a dataset are included in the PLS analysis together.
To analyze parts of a single structure in this way (e.g. the face and braincase of a skull), separate datasets have to be produced first for each block (e.g. using Select Landmarks) and the PLS analysis then uses thes shape variables of these new datasets as the data in the two blocks. Because the data in each dataset have undergone separate Procrustes fits, this type of analysis does not consider any covariation between the parts that is due to variation in the relative size, position or orientation of the parts.
The second option for PLS analysis is to examine the covariation between blocks of landmarks within a single configuration. This analysis considers all the covariation between blocks, including the component that is due to the relative sizes, positions and orientations of the blocks. This method is based on a single Procrustes fit of the entire configuration.
The fact that there is a joint Procrustes fit for the block of landmarks in a configuration has consequences for statistical inference by permutation tests. The tests implemented in MorphoJ take this into account by including a Procrustes re-fitting step in every round of the permutation procedure (Klingenberg et al. 2003; Klingenberg 2009)
For studies of morphometric integration, there is often a choice between analyses based on a joint Procrustes fit or separate Procrustes fits for each block. This choice is important because it can cause a substantial difference in the results, as the relative arrangement of parts can make a significant contribution to the total patterns of covariation.
The decision depends on whether the user wants to study the covariation between the shapes of the parts each considered separately, or whether the focus is on the covariation between parts in the context of the structure as a whole. In the former case, PLS analysis of separate blocks is more appropriate, in the latter, PLS analysis within the configuration of landmarks. For further discussion, see Klingenberg (2009).
Bookstein, F. L., P. Gunz, P. Mitteroecker, H. Prossinger, K. Schaefer, and H. Seidler. 2003. Cranial integration in Homo: singular warps analysis of the midsagittal plane in ontogeny and evolution. Journal of Human Evolution 44:167–187.
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., K. Mebus, and J.-C. Auffray. 2003. Developmental integration in a complex morphological structure: how distinct are the modules in the mouse mandible? Evolution & Development 5:522–531.
Rohlf, F. J., and M. Corti. 2000. The use of two-block partial least-squares to study covariation in shape. Systematic Biology 49:740–753.
Tucker, L. R. 1958. An inter-battery method of factor analysis. Psychometrika 23:111–136.