-------- Original Message -------- Subject: Re: Analyzing ontogenetic trajectory angles Date: Wed, 20 Jul 2011 14:31:32 -0400 From: Dean Adams <[email protected]> To: [email protected] <[email protected]> David, Over the years there have been quite a few approaches for comparing ontogenetic (or other types of phenotypic) trajectories. Note first that we are discussing the angle between two multivariate regressions of shape~logCS (not simply PC1 vs. logCS). The regression line is simply a vector, and so the angle between two vectors can be obtained as the inner product of the vectors. Here I will summarize a few methods from the literature, as they relate to your questions. 1: Klingenberg and Leamy 2001 (Evol.) compared angles between estimated G and P matrices from morphometric data. Here they obtained the angle between vectors, and compared them to a distribution of angles obtained from sets of random vectors. One could adapt this approach to compare the angles between ontogenetic trajectories. 2: Zelditch et al. 2000 (Evol.) performed multivariate regressions of shape~logCS for each species. They then obtained the angles between these vectors, and used a bootstrap of individuals to obtain CI to evaluate these angles for biological importance. Quite a few variants on this approach have been proposed over the years. Both of these methods allow one to evaluate whether or not ontogenetic trajectories are oriented similarly in size-shape morphospace. However, ontogenetic trajectories can still vary in other distinct ways (see Mitteroecker et al. 2005: Evol. Devel). Thus, there may be some additional interesting patterns that one may wish to identify. 3: Mike Colllyer and I developed an approach for comparing multivariate phenotypic trajectories: Collyer and Adams 2007 (Ecol), Adams and Collyer 2007 (Evol), Adams and Collyer 2009 (Evol). In short, one obtains the trajectories of interest, and then compares their magnitudes, their directions, and their shapes (if applicable) to determine how they may vary. For ontogenetic trajectories, one fits species-specific regressions of shape~logCS. Angles between these are obtained, and assessed statistically using residual randomization of individuals (see our papers for why this resampling scheme is appropriate for such designs). The lengths of the ontogenetic trajectories could also be compared by adapting the magnitude comparisons, such that they assess magnitudes as defined by the smallest and largest individual within each species. This might be useful for identifying heterochronic changes that affect the 'length' of ontogenetic trajectories, as might be observed under progenesis or hypermorphosis. 4: As Paolo mentioned in his post, approach #3 was further extended to test yet another hypothesis. Here ontogenetic convergence, divergence, and parallelism can be assessed by comparing the distance between ontogenetic trajectories at their 'starting points' vs. the distances between trajectories at their 'ending points'. (Piras et al. 2010: Evol. Devel.; also Adams and Nistri 2010: BMC Evol. Biol.). Convergence is found when larger individuals are more similar in shape (e.g., Adams and Nistri 2010). 5: Mitteroecker et al. 2005 (Evol. Devel.) used slightly different test statistics, but followed the same general logic that extends through methods 2-4. Here they fit species-specific regressions of shape~logCS. Then then test for 'identical regressions' by obtaining SSresid for all trajectories, permuting species assignments, and repeating. Next, they alter their test statistic somewhat for evaluating overlapping trajectories (see their). The general idea here is to go through several such tests sequentially to determine whether ontogenetic trajectories differ in their direction, their length, or other attributes. Again, this is NOT an exhaustive list of approaches. Rather it is intended to provide a flavor of what has been done, and the commonalities of the approaches (both in implementation and in hypothesis testing). Hope this is helpful. Best, Dean -- Dr. Dean C. Adams Associate Professor Department of Ecology, Evolution, and Organismal Biology Department of Statistics Iowa State University Ames, Iowa 50011 www.public.iastate.edu/~dcadams/ phone: 515-294-3834 On 7/19/2011 12:28 PM, morphmet wrote:
-------- Original Message -------- Subject: Analyzing ontogenetic trajectory angles Date: Mon, 18 Jul 2011 14:46:19 -0400 From: David Katz<[email protected]> To: [email protected] Hello, I have read several morphometrics papers which test for significant differences in ontogenetic trajectory between two groups (species, subspecies, etc) by calculating the "angle" between their growth trajectories. However, parts of (or even lots of) the method remain unclear to me. First, it seems that calculation of the angle requires calculation of two simple regression lines, one for each group, with the angle being the arc subtended by the two lines. One axis for these regression plots/calculations is the distribution of specimens along a PC which is significantly correlated with size or age (usually the first PC). But it is not clear to me what the second axis is. Log centroid size? Second, the significance of the angle is tested by randomly permuting specimens between the two groups 1000 or more times, then calculating a new angle for each permutation. Significance is then tested against the distribution of permutation-generated angles. But what is the permutation procedure? Do we only permute a single specimen each time? Third, if I'm understanding the method correctly, then absent an additional scaling step, the two lines from which the angle is calculated are not likely to be of the same length. How is this accounted for, if at all? Any help, or a reference to a good, explicit journal or book explanation, would be very much appreciated. Thanks. David Katz Doctoral Candidate Department of Anthropology--Evolutionary Wing University of California, Davis Young Hall 204 /--Trying to focus on one distraction at a time/--
