Let me put some of the arguments in my own words and jargon to test my understanding of what is being said.
Jim's response (One more thought.) reminds me of an argument that came up at a meeting decades ago where some participants were arguing for the superiority of Mahalanobis D as a "truer" measure of distance between two multivariate objects than Euclidean or Manhattan distances. If one looks at the matrix equations for Mahalanobis D they reduce to the square root of the sum of independent F-tests. Some people in that era had the idea that D was more a reality because it combined dimensions with a realistic perspective, i.e. by dividing the difference component by its standard deviation. I agree that we should not go down that path in our thinking about Geometric Morphometrics and should keep our geometry as close to the real world shapes as possible. D did have its use. It was able to be tested for significance while a straight Euclidean distance could not be tested without understanding the covariance matrix leading to D. We are, if I am not mistaken, in the same relative situation with Geometric Morphometrics. In the D era there was also a strong movement to ignore the original measures. D could actually be combining more than the 3 physical dimensions and include measured qualities other than spatial making distance an abstraction in hyperspace. In that abstract hyperspace the normalization of each dimension by the variance within that dimension was logical. Belief in the "reality" of D however was misplaced. It was a statistical test metric with significance but no direction to its distance. With Geometric Morphometrics I think we have a feeling that we may be able to make sense of actual shapes and differences between shapes. We need to keep the shape aspects of our studies separate from the covariances of those shapes with interesting covariates. I am convinced that I can study the shapes within an Analysis of Dispersion framework (ie MANOVA/MANCOVA) as long as I keep my shape measures and my covariates and experimental design elements well defined and separable in my mind. The question in my opinion is whether I can objectively evaluate Fred's approach to see whether it presents a preferable departure from my comfortable approach. It would be good to set up some objective comparisons but can we when the two approaches are as different as I suspect? Joe Kunkel Biology Department UMass Amherst [EMAIL PROTECTED] On Feb 5, 2004, at 9:25 AM, [EMAIL PROTECTED] wrote: > One more thought. > Sender: [EMAIL PROTECTED] > Precedence: bulk > Reply-To: [EMAIL PROTECTED] > > In a univariate anova the distinction I made in the previous note is > less clear. The "division" in an anova is by a scalar constant (square > root of the error mean square) so that the relative differences between > pairs of means stays the same. The relative differences between pairs > of > means stays the same. > > In a MANOVA the "division" is by a matrix (the projection of the group > means onto inversely weighted within-group eigenvectors) so that > differences in some directions are stretched and differences in other > directions are compressed. Unless the within-group covariance matrix is > proportional to an identity matrix, the relative distances between > pairs > of means will change - perhaps drastically. > > Thus the distinction between these two kinds of distances is especially > important for the analysis of multivariate data and one must think > about > their relevance for the questions you wish to ask. > > ----------------------- > F. James Rohlf > State University of New York, Stony Brook, NY 11794-5245 > www: http://life.bio.sunysb.edu/ee/rohlf > == > Replies will be sent to list. > For more information see > http://life.bio.sunysb.edu/morph/morphmet.html. > > ---------------------------------------------------- Joseph G. Kunkel, Professor Biology Department University of Massachusetts Amherst Amherst, MA 01003 http://www.bio.umass.edu/biology/kunkel/ == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
