Hmmm, guess I felt like writing... There is a fair degree of subtlety embedded in this question. First, to say something about "size" you must define how you measure size. In the realm of superimpositions, one must also define criteria for measuring and optimizing translation and rotation. Translation follows from using a least-squares criterion to compare the shapes of objects - translation to a comment centroid (anywhere, but we take the origin for convenience) minimizes the sum of squared coordinate distances betweeen specimens. The formulae for optimal rotational alignment follows from the least-squares criterion and the restriction to allow only rigid rotations. There is a least-squares method of size adjustment, but it is seldom used. Instead, specimens are all scaled to a common size.
There are, of course, and infinite number of other approaches that could be taken. Why not translate to have one landmark in the same location for all specimens? Why not rotate all specimens so a segment between landmarks points in the same direction? Why not scale so all specimens have a particular segment of the same length? Actually, all of these choices are "legitimate" and part of the construction of Bookstein shape coordinates (assuming the same segment is used for scale and rotation). Using medians leads to resistant fitting. There are, as I said, and inifite number of choices for the superimposition parameters including size. For Procrustes superimposition, it turns out that centroid size, the square root of the sum of squared distance of a specimens landmarks to their average location has certain convenient mathematical properties - e.g., small circular variation around landmark locations do not induce correlations between shape and centroid size (Bookstein, 1991), but it is by no means the only choice one could have made. The familiar choices for translation, rotation, and scaling do, however, allow for a more elegant and comprehensive theoretical development than would be possible for any alternative to date (e.g., the works of Kendall, Bookstein, Mardia, Kent, etc. etc.). That said, if you "eliminate" size (I much prefer the term "sequester" size coined, I believe, by Bookstein and emphasizing that size has been factored out of the original data and "sequestered" in a separate szie variable for subsequent analysis.), however you have defined it, how can there by "ghosts" of size in the shape data? This is because the shape data, by definition, do not differ with respect to the defined size measure, but the shapes themselves could be a function of the size of the original specimen. I don't know if this is a good example, but consider three persons: a tall, skinny person; a skinny person of medium height; and a short, skinny person. Their shape is either skinny, medium, or fat, and we choose height as a measure of size. If we isometrically scale all three to the be the same height, we find we have three skinny individuals - no shape difference, but varying (original) heights or size differences. Now consider three others: a tall, skinny person; a person of medium build and height; and a short, fat person. Following the same procedure as before we would have three individuals scaled to same size (height) but with three different shapes - skinny, medium, and fat. We could then refer back to our size measures and see there is a relationship between shape (even after adjustment for size) and size. We could conclude (recklessly due to the small sample) that size and shape are related - tall persons tend to be skinny, short persons tend to be fat. With the size and shape data obtained from Procrustes analysis we can do the same thing and determine, for instance, size-adjusted jaw robustness tends to increase centroid size (or diet or some other variable) or size-adjusted skulls become more elongated (or round) with increasing centroid size (or cube-root body weight or some other variable). -ds -- Dennis E. Slice, Ph.D. Department of Biomedical Engineering Division of Radiologic Sciences Wake Forest University School of Medicine Winston-Salem, North Carolina, USA 27157-1022 Phone: 336-716-5384 Fax: 336-716-2870 == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
