Rich Strauss wrote: > At 10:24 AM 12/19/2003, you wrote: > >> Sangdon Lee wrote: >> > Dear All, >> > >> > Thanks for valuable information. Just one more question! >> > Could somebody explain to me the difference/similarity among >> > Mahalanobis distance, Hotelling's T-square and PCA ? I believe that >> > Hotelling's T-square is the sum of squares of the principal component >> > scores if all components are extracted as the number of input >> > variables. It seems to me that the Mahalanobis distance is the same as >> > the Hotelling's T-square. >> >> You are correct. T-squared from PCA with all possible components is >> exactly equal to Mahalanobis distance. If fewer than the maximum >> number of components are used, then the two quantities would NOT be >> equal. > > I think that the confusion here, which I missed before, is whether we're > dealing with one or two groups. T^2 is the multivariate test for two > groups, whereas Mahalanobis D^2 can be used with either one or two > groups. For a single group, D^2 measures the deviation of a single > observation from the group centroid. For two groups, D^2 estimates the > difference between the group centroids. Thus the relation between T^2 > and D^2 (and discriminant analysis) is for the case of two groups, while > D^2 relates to PCA only for a single sample. For one group S represents > the total sums-of-squares, while for two groups S represents the pooled > within-group sums-of-squares.
There is a PCA quantity called T-squared which is defined for 1 group. See the reference to Jackson that I mentioned earlier. In that case, T-squared equals Mahalanobis when all principal components are used. There is also a quantity called T-squared (usually attributed to Hotelling) that can be used to test the difference between two group means. My answer above relates to the 1 group case. You are correct about these quantities in the two group case. -- Paige Miller Eastman Kodak Company paige dot miller at kodak dot com http://www.kodak.com "It's nothing until I call it!" -- Bill Klem, NL Umpire "When you get the choice to sit it out or dance, I hope you dance" -- Lee Ann Womack . . ================================================================= Instructions for joining and leaving this list, remarks about the problem of INAPPROPRIATE MESSAGES, and archives are available at: . http://jse.stat.ncsu.edu/ . =================================================================
