I recently responded privately to a message at interpretation of individual PCs but I thought my response might be of interest to the group.
The problem with highlighting and interpreting the "most important" PCs is that it implies that the individual PCs are likely to have special biological meaning. There is no reason for that to be true (although people can and commonly do find interpretations of almost any pattern). One should remember that PCA is a mathematical/statistical tool to project as much as possible of the variation in a multivariate space down onto as few dimensions as possible. Because it is a projection it is essential that the scales of the axes are the same when plotting PC1 vs. PC2, etc.). No biological model is involved in the definition of a PCA. It is just an elegant mathematical partitioning of the overall variance into orthogonal (uncorrelated, not independent) components. There undoubtedly are interesting directions in that space but for biological applications it does not seem likely that the underlying dimensions should happen to be perfectly uncorrelated. A user would have to justify that assumption before they should try to interpret individual PCs. It is often done but that does not make it correct. A related thing to remember is that within the plot of the low-dimensional space is that the interesting directions to interpret need not be parallel to the axes. Especially true for data with samples within and among species. The PCs may not align to either the main directions of variation within species or among species. They will be some compromise and thus even the first axis may not be that interpretable except for relatively homogeneous samples. I saw an interesting example of that a few years ago in a study in which the variation within one group of species was parallel to PC2 but variation within the other major group of species was equally along PC1 and PC2 (i.e., the major dimension of variation in the 2nd group was at an angle of about 45 degrees to the PC1 and PC2 axes. ---------------------- F. James Rohlf New email: f.james.ro...@stonybrook.edu Distinguished Professor, Emeritus. Dept. of Ecol. & Evol. & Research Professor. Dept. of Anthropology Stony Brook University 11794-4364 The much revised 4th editions of Biometry and Statistical Tables are now available: <http://www.whfreeman.com/Catalog/product/biometry-fourthedition-sokal> http://www.whfreeman.com/Catalog/product/biometry-fourthedition-sokal <http://www.whfreeman.com/Catalog/product/statisticaltables-fourthedition-ro hlf> http://www.whfreeman.com/Catalog/product/statisticaltables-fourthedition-roh lf P Please consider the environment before printing this email -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org To unsubscribe from this group and stop receiving emails from it, send an email to morphmet+unsubscr...@morphometrics.org.
