Dear Ondra, I assume you mean an angle between shape vectors (i.e., the vectors that describe shape change between two points in shape space projected onto all or a subset of PCs), rather than between the PCs themselves (which are orthogonal). If the angle you measure describes a comparison of shape change between two groups (e.g., the shape change between males and females [sexual dimorphism] compared between two species), then I think the best way to 'test' angles is to use a randomization procedure where individual landmark configurations are assigned randomly to different group-level combinations. Then you can evaluate the observed angle as a percentile of a random distribution created by 1000s of random permutations.
Something important to realize, however, is that HOW you randomize your configurations will affect the result. Using the example above (sexual dimorphism shape vectors compared between two species), one could randomly assign (in each random permutation) configurations to one of the 4 species-sex groups, or one could first block species and assign configurations into one of 2 sex groups within species blocks, or one could block sex and assign configurations to one of the two species groups within sex blocks. These methods might yield different interpretations about the significance of observed angles. The latter two 'restricted' randomizations may have more appeal if it is known a priori, for example, that species have different shapes (i.e., species differences can be held constant). However, I would caution against this approach for comparing angles because what you effectively create is a distribution of random angles between vectors sampled from a population that has an expected value something close to uncorrelated vectors. In other words, it might not be surprising if a random distribution of angles was centered around 90 degrees. Does this mean that an observed angle of 40 degrees, found to be rare from this type of distribution, is not different than 0 degrees (i.e., parallel vectors)? Not necessarily! Alternatively, I recommend the approach we outlined in our recent paper (Collyer and Adams. 2007. Analysis of two-state multivariate phenotypic change in ecological studies. Ecology 88:683-692.). Using a two-factor linear model (e.g., shape = species + sex + species*sex), you can reduce the model by the species*sex term (meaning model parameter estimates related to species and sex effects are preserved) and use residuals as the permutable elements in a randomization test. With each random permutation, configuration residuals are randomly assigned to configurations estimated by species and sex parameters. This procedure has the nice property that the random distribution of angles is derived from a random distribution of shape variation for the species*sex term of the linear model (i.e., species and sex effects are not inadvertently randomized as well) and the observed angle can be tested against a null hypothesis of parallel vectors. A 'significant' angle is one that is greater than expected by chance based on your criterion for assigning significance (e.g., occurs less than 5% of the time). I believe a two-factor linear model (shape = A + B + A*B) has general application for this kind of problem because you need at least two levels of shape response within at least two groups to describe an angular difference. Hope that helps Mike Collyer > Dear morphometricians, > > could you advise me, how to generate random vectors? (For the purpose of > testing whether an angle between PCs is significantly more acute than > expected by chance.) From which distribution their elements should be taken? > > > Thank you in advance > > > Ondra Mikula > > Institute of Animal Physiology and Genetics > Academy of Sciences of the Czech Republic > Veveri 97, CZ-60200 Brno > Czech Republic > e-mail: [EMAIL PROTECTED] > > Michael Collyer __________________________________________________ Postdoctoral Research Associate Iowa State University Department of Ecology, Evolution, and Organismal Biology 234 Bessey Hall Ames, IA 50011 Phone: 515 294-1968 Fax: 515 294-1337 Email: [EMAIL PROTECTED] -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
