With reference to the Yang Chang skull group query: I am a devotee of Analysis of Dispersion combined with Factorial Design of CR Rao (1965).

The model is Y = XB Y is a n by 3k+c matrix of n rows of k xyz coordinates plus c covariates such as size and weight covariates. X is a n by p design matrix of assignment multipliers to chose the B matrix factor values. -the elements of X are usually 1s, 0s and -1s. B is a matrix of p by 3k+c factors which are tested for their significance as factors. As usual the solution is B = (X’X)^-1(X’Y). 3k columns of Y are the matrix of landmarks rotated and resized by the usual landmark alignment process. And one of the c columns of the Y matrix is the resizing size value produced in the alignment. I use the Analysis of Dispersion (Rao 1965) to test the significance and obtain the crossproducts matrices that reduces the total sums of squares and crossproducts matrix Y’Y for the mean and independent factors, in your case the groups of skulls. The residual covariance matrix is used to test for the significance of the factors which at this point are not corrected for the covariance with size (for instance). Next, within Rao's Analysis of Dispersion one tests the factor crossproducts matrices of the Y columns for whether there is any additional information in one or more of the covariates (such as the size column). If there is no additional info in the size covariate one can ignore it. If one or more covariates is significant one can change the Equation to be: Y’ = ZB = [Yc | Yd | X]B Y' is the Y matrix with the covariate columns removed. Yc and Yd are the significant covariate columns appended to the design matrix X to create Z extended design. That system of matrix equations is solved the same way as previously but since c and d covariates are likely not independent of the factor design columns their sums of squares and crossproducts reduction are calculated by subtraction rather than being independently calculable. This is all quite simple if one is familiar with factorial design, just generalized to a n by 3k Y matrix of observations. It is explained in matrix algebra by Rao (1965). I have implemented it in R if you have the patience of following the rules of design matrices and choosing factors to be tested independently or by subtraction. I am not sure Jim would approve of this specific approach but I am convinced that it is correct and gives you a test of significance as well as a way of calculating the expected Y', Yhat. Yhat = ZB Which provides with the factors which allows computation of average differences between factor groups and with chosen c and d covariate effects. CR Rao (1965) Linear Statistical Inference and Its Applications.John Wiley & Sons, New York, 522pp. I have recently upgraded my R scripts to be consistent with recent versions of R. The two relevant R-scripts are Andy.R and ad info.R http://www.bio.umass.edu/biology/kunkel/pub/r_scripts/andy/ <http://www.bio.umass.edu/biology/kunkel/pub/r_scripts/andy/> The sample inputs found there are csv matrices that a simple YX matrices of appended Y and design matrix X. If you need help with their use I am available by Email. Joe -·. .· ·. .><((((º>·. .· ·. .><((((º>·. .· ·. .><((((º> .··.· >=- =º}}}}}>< Joseph G. Kunkel, Research Professor 122C/125 Pickus Center for Biomedical Research Marine Science University of New England Biddeford ME 04005 http://www.bio.umass.edu/biology/kunkel/ <http://www.bio.umass.edu/biology/kunkel/> > On Apr 11, 2019, at 5:31 AM, yang Chang <cylove1...@gmail.com> wrote: > > I want to compare skull morphology using geometric morphometrics from several > species with different diets in a same genus. But I don't know much about how > to analyze these shape data. What statistical analysis can I do? I know that > principal component analysis can be done to visualize shape variations. So > can I use the original coordinate data? Do I need to remove the effects of > size and phylogenetic relationships to do principal component analysis? If > so, do I use PGLS to do the regression of shape and size, and then use the > obtained residual as the principal component analysis? In addition to this > analysis, what other aspects can I do? Looking forward to reply.Thanks very > much! > > -- > MORPHMET may be accessed via its webpage at http://www.morphometrics.org > --- > 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 morphmet+unsubscr...@morphometrics.org. -- MORPHMET may be accessed via its webpage at http://www.morphometrics.org --- 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 morphmet+unsubscr...@morphometrics.org.