Dear Ondra and Mike, A resampling procedure randomly permuting group assignment and recalculating the angle between two vectors tests the null-hypothesis that both vectors are parallel – not "whether an angle between PCs is significantly more acute than expected by chance." The expected angle between two random vectors depends on the dimension of the vectors (the more variables the larger is the expected angle). For data with many variables the expected angle might indeed be close to 90 degrees.
When permuting group assignment over multiple groups, the null-hypothesis is that the vectors are of the same direction in _all_ groups. For any group design, of course, groups have to be mean-centered before permutation in order to compare the angle between within-group vectors. One cannot permute cases from populations with different group averages. Centering the groups is identical to using of residuals from a linear model as outlined by Mike. For permutation tests whether two vectors are identical (not only parallel) or overlapping see, e.g., Mitteroecker P, Gunz P, and Bookstein FL. 2005. Heterochrony and geometric morphometrics: A comparison of cranial growth in Pan paniscus versus Pan troglodytes. Evolution & Development 7:244-258. However, I would caution against comparing PCs among groups beyond the first one. While the first PC is often close to allometry, the subsequent dimensions are difficult to interpret and are not necessarily correspondent between groups. Especially when the eigenvalues are similar and do not drop off clearly, the PCs are not well defined and a comparison of them is not meaningful. Best, Philipp Mitteroecker On Mi, 30.05.2007, 13:24, morphmet wrote: > 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 > > > -- Replies will be sent to the list. For more information visit http://www.morphometrics.org
