RE: Burnaby's method and discriminant analysis tolerance
That's right - I was perhaps a little too fast-and-loose when I equated Darroch and Mosimann with Burnaby. It would have been more precise to say that Darroch and Mosimann and Burnaby CAN give the same results, depending upon how you set them up. Dividing each measurement through by the geometric mean and then ln-transforming the ratios (one approach to Darroch and Mosimann) should give the same result as using Burnaby with an isometric PC1 (each eigenvector coefficient = 1 over the square root of the number of measurements), since the geometric mean is the measure of scale in PCA when using the VCV matrix of ln-transformed data, according to Jolicoeur. Also, there's a nice geometric demonstration of how these kind of things work in Bookstein et al. (Red Book), Section 2.2.2. Finally, another nice thing about the Darroch and Mosimann adjustment is that it's so easy to get a set of scale-adjusted data using NTSYS and other programs. If the rows are observations and columns are measurements, just ln-transform the data, center them on the row means, and save the new data set of logshape variables. Tim Cole At 10:29 AM 2/23/2004 -0500, you wrote: In Burnaby's method one does not divide by some measure of size. The method is closer to the idea of using residuals from regression to remove the effects of size (i.e., it uses subtraction not division operations). What the method does is to removes all variation parallel to a specified vector (or vectors) and thus reduces the dimensionality of the variation even though the number of variables stays the same. The formula used in Burnaby's method is quite general and has many other applications. In the Burnaby method, the variation in the columns (variables) of a data matrix are transformed to eliminate variation parallel to a specified linear combination of variables. The linear combination is usually PC1 but it can be any vector (actually it can be a set of vectors) that define directions in multivariate space in which you would like to remove all variation. The technique can also be applied to the rows (observations) of a data matrix. In this case it can be used to perform familiar operations such as computing deviations from the means or transforming a data matrix into a matrix of deviations from regression. This generality has made the equations used in the Burnaby method among my favorites! Jim F. James Rohlf - Dept. Ecology Evolution SUNY, Stony Brook, NY 11794-5245 -Original Message- From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED] On Behalf Of [EMAIL PROTECTED] Sent: Friday, February 13, 2004 8:01 AM To: [EMAIL PROTECTED] Subject: Re: Burnaby's method and discriminant analysis tolerance I don't really remember the details of what the Burnaby scale-adjustment does, but I think it's somewhat similar to Darroch and Mosimann's approach to scale adjustment (someone correct me if I'm wrong). With DM, the measurements are transformed by dividing through by some reasonable measure of size (for example, the geometric mean of all the distances). If Burnaby and DM are similar in what they do to the data, you're probably having problems with discriminant analysis because your variance-covariance matrices are singular (rank = number of measurements - 1) and can't be inverted. Darroch and Mosimann describe how to get the discriminant function scores out when using their brand of scale adjustment. Darroch JN and Mosimann JE (1985) Canonical and principal components of shape. Biometrika 72: 241-252. I hope this helps. Tim Cole At 12:21 PM 2/11/2004 -0500, you wrote: Have anyone had problems with the tolerance in discriminant analysis if the input variables for such anlaysis have previously been transformed by Burnaby's method? We are studing the population structure of a fish species in the North Atlantic. 17 variables (distances between landmarks) have been measured for each fish , following the truss network model and in adition we have measured some other structures as eye diameter, fin lengths, etc. We have 4391 cases, distributed in seven geographical locations. In order to eliminate the size influence, we have used two methods, i.e., residuals against standart length, and the Burnaby's method. Once we have removed the size effect, we run a discriminant analysis to observe differences between areas. We have no problem if we use the residuals as input for the discriminant analysis. But we cannot perform a discriminant analysis using as input the Burnaby's transformed variables, because we have problems with the tolerance of the variables: the matrix is ill-conditioning. The problem doesn't seem to be in a particular variable or in a group of data (data has been carefully screened for outliers). Simply, there is some redundancy. However the correlations
Re: Burnaby's method and discriminant analysis tolerance
I don't really remember the details of what the Burnaby scale-adjustment does, but I think it's somewhat similar to Darroch and Mosimann's approach to scale adjustment (someone correct me if I'm wrong). With DM, the measurements are transformed by dividing through by some reasonable measure of size (for example, the geometric mean of all the distances). If Burnaby and DM are similar in what they do to the data, you're probably having problems with discriminant analysis because your variance-covariance matrices are singular (rank = number of measurements - 1) and can't be inverted. Darroch and Mosimann describe how to get the discriminant function scores out when using their brand of scale adjustment. Darroch JN and Mosimann JE (1985) Canonical and principal components of shape. Biometrika 72: 241-252. I hope this helps. Tim Cole At 12:21 PM 2/11/2004 -0500, you wrote: Have anyone had problems with the tolerance in discriminant analysis if the input variables for such anlaysis have previously been transformed by Burnaby's method? We are studing the population structure of a fish species in the North Atlantic. 17 variables (distances between landmarks) have been measured for each fish , following the truss network model and in adition we have measured some other structures as eye diameter, fin lengths, etc. We have 4391 cases, distributed in seven geographical locations. In order to eliminate the size influence, we have used two methods, i.e., residuals against standart length, and the Burnaby's method. Once we have removed the size effect, we run a discriminant analysis to observe differences between areas. We have no problem if we use the residuals as input for the discriminant analysis. But we cannot perform a discriminant analysis using as input the Burnaby's transformed variables, because we have problems with the tolerance of the variables: the matrix is ill-conditioning. The problem doesn't seem to be in a particular variable or in a group of data (data has been carefully screened for outliers). Simply, there is some redundancy. However the correlations between variables are not particularly high. We have also study if the problem is in the data, running the Discriminant Analysis with different combinations of the seven locations we have. But the results don't give us a clue. For example, when doing the analyses with four locations (a-d), it works. But as soon, as you introuduce some of the other three (e-g), it fails. However, some combinations of e, f or g, with other locations it works. Thus, not neccessarily the problem is in the locations e-g, but when these locations are together with some other, but there is no clear pattern. The same thing occurs with the variables. We have removed the variable than enter at last step (when tolerance drops below the limit), but then is another variable which cause problems, and if removed is another one and so on. We suspect that the problem is relared with the way that burbany method estimate the transformed variables. Can anyone help us? Thanks in advance, Lola =20 Dolores Garabana Barro Institute of Fisheries Research Eduardo Cabello, 6 36208 Vigo (Spain) e-mail: [EMAIL PROTECTED] == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html. Theodore M. Cole III, Ph.D. Department of Basic Medical Science School of Medicine University of Missouri - Kansas City 2411 Holmes St. Kansas City, MO 64108 USA Phone: (816) 235 -1829 FAX: (816) 235 - 6517 e-mail: [EMAIL PROTECTED] www: http://c.faculty.umkc.edu/colet == Replies will be sent to list. For more information see http://life.bio.sunysb.edu/morph/morphmet.html.
