> [Moderator: Made public as per Igor's suggestion. Followup > to...well...follow.] ...snip...
With respect to the use of partial or relative warp scores in statistics, many common, multivariate statistical procedures are invariant to the orthogonal axes used to record specimen observations. For instance, assuming the use of a full set of basis vectors for your sample variation: Principal components analysis (PCA) - this will find the orthogonal linear combinations of your original variables ordered by the variance of the projections of the data onto them. The results (plots and eigenvalues, but not eigenvector coefficients whose values are variable-specific) will be identical (up to a reflection) regardless of whether you use partial warps, Procrustes residuals, or any other appropriately numbered set of orthogonal axes in the space. These results will be, in turn, the same as an alpha=0 relative warps analysis (which is just a PCA of partial warp scores). Canonical variates analysis (CVA) - this is just a PCA of group means AFTER factoring out the within-group covariance. This makes the within-group scatter of the transformed sample statistically circular. You'll get the same results regardles of your starting axes - you factor out noncircular variation that is only affected as to direction by different choices of axes, and you do a PCA on the results of that - see (PCA). Multivariate analysis of variance (MANOVA) - this is just a comparison of residual and "explained" variation depending on the model - e.g., within- and between-group variance for tests of group mean differences. The standard stats Wilks' lambda, Pillai's trace, etc. just use the eigenvalues (products, sums, maximum) of within and between (explained and unexplained) covariation, which is, like PCA invariant to the actual baseline. NOTE: All of the above assumes the use of either a) any complete set of basis vectors for the space under consideration or b) a complete set for the subspace "occupied" by your sample. I believe the original question that started all of this was about discriminant functions. This is a bit different. If the goal of a discriminant function analysis is the classification of new specimens, then it, in a sense, matters not what you do. If the goal is classification, then the validity of whatever is done is determined solely by its ability to correctly classify new specimens. You could use the first few PCAs, every third partial warp, neural networks, or tarot cards. That which does the best job is the best of what you have tried. If you wish to divine something about biology from the procedure, you have a bit more of a problem. You may be able to find some justification, perhaps even a good one, for a particular linear combination of variables to have biological, functional, or evolutionary meaning, but it won't be in the mathematics and having done so, you will have defined some other variable(s) that is probably more interesting and defensible anyway - relative body depths, spine length, etc. The original post was concerned with the problem of small samples and large numbers of variables. In some sense this is just "mechanical" in that statistics programs want to compute proper inverses of covariance matrices and cannot/will not execute if said covariance matrix is singular. Small n is a problem for any large suite of variables, but Procrustes residuals are guaranteed to have this problem regardless of the number of specimens used - degrees of freedom are lost due to the superimposition so that the superimposed data set occupies only a subspace of that of the original data and all covariance matrices computed in this space from the full compliment of residuals will be singular. The use of partial warp scores addresses this issue in that they, plus the uniform components, are of proper dimension. In the latter case, sample size then becomes the (somewhat) critical again. To address the mechanical problems of computing discriminant functions or anything else that wants to invert covariance matrices there are a several things one can do. First, and most simple, is to do a PCA and proceed with scores on PCs with non-zero eigenvalues. This guarantees that the space in which you are working will be at least minimally filled. If you are doing something using within-group covariance, then you must use only the number of PCs one less that your smallest sample - an occasionally very restrictive requirement. The other alternative is to "roll your own" and copy out of the statistical texts the relevant formulae and replace things like S^(-1) with S^(-) or S^(G), i.e., substitute the use of a generalized inverse for the proper inverse shown in most formulae. A good generalized inverse can be constructed from the SVD of S, where you replace the nonzero singular values with their inverses and reconstitute the matrix using onlyt the vectors associated with the nonzero singular values. For other cases like MANOVA, you can do randomization tests comparing traces of within or between broup SS with a large number of samples whose association with the relevant variables has been randomized. Another comment - even if you have enormous samples far exceeding the number of variables and these variables have not been constrained by forces like a superimposition operation, covariance matrices can still be singular. This is because organisms are not random variables with unconstrained covariance. High correlations amongst variables (that is at least partially required to produce recognizablly similar organisms) can isolate sample variation to a small subspace of variable space. That is, after all, the secret to the utility and efficacy of PCA. And a final comment on PCAs. I find it generally underappreciated (even unknown) amongst biologists that if any of the eigenvalues are identical, then the set of eigenvectors associated with that set are not unique. That is, they are just constructed bases of that subspace of "round" variation and freely rotatable. As it happens, rounding error and such seldom produce identical numerical values, so the resulting PCs may look unique, but they are so only due to the rounding error. This situation can occur at any scale of variation from the first few PCs to the last or anywhere in between. Oh my, I have gone on and written enough that at least something must surely be wrong. I am confident that others will step in and point out my errors. Best, dslice -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
