Hi Seth, The number of meaningful PCs (i.e., those with non-zero variance) is normally determined by the number of independent variables. In geometric morphometrics that number is the number of landmark coordinates (k *2 for 2D landmarks or k*3 for 3D). Procrustes superposition removes some of the degrees of freedom from the variables, 4 for 2D landmarks (2 position or translation, 1 size, and 1 rotation) or 7 for 3D (3 position, 1 size, and 3 rotation). For 30 3D landmarks you would expect 30*3-7 = 83 PCs.
However, sample size also limits degrees of freedom if n is less than number of variables. Presuming you have 3D landmarks, your data are sample limited. Technically you expect n-1 PCs for sample limited data, 59 in your case. Zelditch et al’s text _Geometric Morphometrics for Biologists_ provides an explanation that is not overly heavy on mathematics. Dryden & Mardia’s _Statistical Analysis of Shape_ gives a more formal explanation. There are several reasons why you might have 60 instead of 59: might be rounding error and the last eigenvalue is really 0, might be the non-Euclidean Kendall shape space which leaves just a little more apparent variance than the Procrustes degrees of freedom suggests if your eigen decomposition algorithm has high precision. Note also that covariance between the landmarks also reduces the degrees of freedom. If two coordinates covary completely then you have fewer non-zero eigenvalues than expected. O’Keefe et al, 2022 discuss some of the nuances related to covariance between landmarks( On Information Rank Deficiency in Phenotypic Covariance Matrices. Syst Biol. 2022 Jun 16;71(4):810-822. doi: 10.1093/sysbio/syab088) Best wishes, David iPhone typing... [email protected] On Aug 12, 2022, at 9:12 PM, Seth Boren <[email protected]> wrote: Hello! My name is Seth Boren, and this is my first question I have ever asked here! I am working on a morphometric analysis that involves 60 specimens with 30 landmarks each. I have arranged the landmarks of these specimens into a morphologika file and then ran it through a GPA then PCA using EVAN-Toolbox. The output file for my analysis has 60 PCs listed, with 60 eigenvalues associated with them and the % explained variance broken up into 60 portions. However, shouldn't there be 30 PCs instead, equaling the number of landmarks? If anyone could help me understand what is happening you have my eternal thanks. Thank you everyone for your time and consideration, -Seth Boren -- You received this message because you are subscribed to the Google Groups "Morphmet" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]<mailto:[email protected]>. To view this discussion on the web visit https://groups.google.com/d/msgid/morphmet2/2300f593-130a-4e6d-ac1d-dd6b5dfa2181n%40googlegroups.com<https://groups.google.com/d/msgid/morphmet2/2300f593-130a-4e6d-ac1d-dd6b5dfa2181n%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Morphmet" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/morphmet2/84C06166-9DDB-4A32-B684-0EBDC7E2AAEC%40indiana.edu.
