See the papers by Stayton, especially the most recent, which provides statistical tests for convergence. 

Stayton, C. T. 2006. Testing hypotheses of convergence with multivariate data: Morphological and functional convergence among herbivorous lizards. Evolution 60:824-841.

Stayton, C. T. 2008. Is convergence surprising? An examination of the frequency of convergence in simulated datasets. J. Theor. Biol. 252:1-14.

Stayton, C. T. 2015. The definition, recognition, and interpretation of convergent evolution, and two new measures for quantifying and assessing the significance of convergence. Evolution 69:2140-2153.

I'm not 100% familiar with Stayton's methods, but from what I recall, it assumes that estimates of ancestral characters are somehow reliable, correct? That can be problematic in the absence of fossil specimens (if you are using BM, and calculating confidence intervals, that is).

Wouldn't that also require phylogenies? From what I understand of the question, there is no genetic information therefore there is no phylogeny. I would suggest evolutionary modeling approaches (OUCH, SLOUCH and related techniques; see for an intro Butler, M. A., & King, A. (2004). Phylogenetic comparative analysis: a modeling approach for adaptive evolution. The American Naturalist, 164(6), 683–695.) but that also requires phylogenies.

Maybe with the genetic groups you can assume that group averages are the ancestral state of each genetic cluster and apply Stayton's methods, but I'm not sure that would fly. Depending on how these species are related to each other, ancestors could be more similar to each other than group averages.

Maybe at this point you are restricted to the investigation of similarities using standard techniques, such canonical variate analysis, for example.

Dear Miriam, thank you for the references, they look promising for what I am looking for.

Dear Fabio, 

Thanks for the suggestions. Yes I thought convergence/divergence  would be hard to show on morphological data alone, but I was still hoping that someone would have some hints how to do this. We actually allready did the multivariate statistical analyses (such as PCA, DFA), and we know for a rough phylogeny of the three clades (families) we are working with (ifrom the literature), so at least we have a rough idea when convergences (similar outcomes in unrelated clades) or divergences (different outcomes within a clade) occured. Because we are aware that kind of conclusions may not be solid I was searching for a tool that could give us this kind of answer based on morphology... As it seems and I was afraid, there is no standard approach to it.  Saying differently, multivariate analyses gave us really good results (4, tentatively 5 morphogroups, for which we can think that in some the states of characters are result of convergent and in others of divergent evolutuion) , so I was wondering if there is any test to support that, at this stage of research without the phylogeny. I guess we will need to add the phylogenetic reconstruction of the analysed taxa as well.

Btw, since we know that from whole array of species from one morhogroup (morphogroups are reflecting both clade membership and are obviously of adaptive nature at the same time) occuring at the same location, only few species can timely and spatially co-occur, I believe there is also a competition among the ones with the same strategies (most probably also between the ones with different startegies since the food source for all is the same). Does anyone know if this kind of data can serve as a further support for convergences/divergences, and if this is so, how to include this data on temporal activity?

Kind regards,

Jure

Dear Jure,

I don't quite see how that would help your case. To my mind, even if you demonstrate that species from one morphotype are spatially and temporally segregated (corollarily, you would also have to show that species from different morphotypes can co-occur) you have not advanced the argument that there is evolutionary convergence at play. You only showed that there is some kind of macroecological pattern there that could, im principle, happen even if all species from one morphotype are monophyletic (i.e. that specific adaptations arose only once).

I guess that to answer an macroevolutionary question you need either a phylogeny or some ideia of the phylogenetic arrangement of groups. Since you only have broad genetic clusters i would say the best you can do is try to describe differences between those clusters with regular statistical approaches. I think Figueirido have some interesting non-completly phylogenetic approaches to these questions.

Figueirido, B., MacLeod, N., Krieger, J., Renzi, M. D., Perez-Claros, J., & PALMQVIST, P. (2011). Constraint and adaptation in the evolution of carnivoran skull shape. Paleobiology, 37(3), 490–518.

Figueirido, B., Palmqvist, P., & Pérez-Claros, J. A. (2009). Ecomorphological correlates of craniodental variation in bears and paleobiological implications for extinct taxa: an approach based on geometric morphometrics. Journal of Zoology, 277(1), 70–80. http://doi.org/10.1111/j.1469-7998.2008.00511.x

Figueirido, B., Serrano-Alarcón, F. J., Slater, G. J., & Palmqvist, P. (2010). Shape at the cross-roads: homoplasy and history in the evolution of the carnivoran skull towards herbivory. Journal of Evolutionary Biology, 23(12), 2579–2594. http://doi.org/10.1111/j.1420-9101.2010.02117.x

Figueirido, B., Tseng, Z. J., & Martín-Serra, A. (2013). Skull shape evolution in durophagous carnivorans. Evolution, 67(7), 1975–1993. http://doi.org/10.1111/evo.12059

Maybe them can provide some inspiration.

Best

May I submit to the researchers in this exchange that measuring (and quantifying) linear dimensions is not a morphometric approach -- even if it is done using multivariate statistics. There are too many ways morphologically different specimens can have matching linear dimensions.

The method of choice is to define landmarks, determine their 3D data and then use the vast toolkit of Geometric Morphometrics.

Kind regards, Hermann

Dear Hermann,

I know this is a tangent, but could you qualify that assertion? By "not a morphometric approach" what do you mean by "morphometrics"?

Because it seemed to me that the use of linear distances (or other non-landmark based measurements) is consistent with "classical" or "tradicional" morphometrics (sensu Marcus, 1990. Chapter 4. Traditional morphometrics. In Proceedings of the Michigan Morphometric Workshop. Special Publication No. 2. F. J. Rohlf and F. L. Bookstein. Ann Arbor MI, The University of Michigan Museum of Zoology: 77–122). 

Best regards,

The meaning of the concept "Morphometrics" has changed considerably. The source you quote is 1990; that is more than 1/4 of a century ago. When I started using morphometrics (together with Marcus and Bookstein, by the way), using linear measurements was already considered obsolete. (no offence, but you can easily construct -- for yourself -- two wildly different objects with comparable linear dimensions).

Already in 1991, Bookstein published a landmark (no pun intended) book "Morphometric Tools for Landmark Data", in which he demonstrates the futility of trying to reliably detect morphometric features using linear dimensions.

In a frontispiece, he says "Morphometrics is the statistical study of biological shape and shape change. ... "

Of course, in the meantime there exists a wealth of literature and software packages (including by Rohlf, whom you also cite) that use geometric morphometrics. I personally am surprised that there exist researchers still using linear dimensions, which go back to the 19th century. And we are in the 21st.

Does this help?

Yes, in a way. Thanks for that. But I still don't see the necessary connection between something being outdated (or less powerful) and it lacking the necessary requirements for being called "Morphometrics". It doesn't follow either from Bookstein's definition you gave (far from being the only one available) because shape has being investigated with other techniques that are not landmark-based. Mosimann's shape-ratios comes to mind. 

Just wanted to chime in and say that there are some great visualization tools in the R package, Geomorph that allow you to see species shape averages connected to one another in a phylogenetic tree. I'm pretty sure you can create a tree and load it in - i.e. doesn't have to be based on strict molecular data.

See page 65 in the User Guide attached below if it might work for your shape data. (Citation: Dr. Emma Sherratt wrote that function and the guide.)

That plot, a phylomorphospace, gives a very useful but low-dimensional summary of the data. Species that appear to converge in that plot may not be similar to each other on other dimensions (and might not converge at all).  Another useful visualization tool is a tanglegram (a cophylo plot in phytools). This pairs the phylogeny with a phenogram, and after rotating them to optimize the vertical matching of tips, lines are drawn to connect species' positions in the phylogeny to their position in the phenogram. The plots are complex but they have the advantage of showing convergence of shape rather than overlap on two dimensions.

I know this dangerously close to bring the whole discussion on the tangent of the "definition of morphometrics" but I just want to add that I have seen a great deal of work using landmarks just derived from 2 dimensional data (usually photographs taken specifically under the parameters for each specimen involved).

More to the actual topic:

A naive approach could be to construct a phylogeny just using the morphological data and a maximum parsimony approach and than look at the individual traits a posteori to see which ones seem to be convergently acquired.

If there is a published phylogeny of the species involved you could make a morphospace analysis of the body components in question and just look how closely the clusters come to the phylogeny o the group. Based on these plots you could use conventional statistics to figure out how different each species as a whole is and compare emerging patterns to the phlogeny.

Miriam Leah Zelditch's idea of using a tanglegram seems like a good visualization tool.

I would like to thank all of you for your thoughts, some are for sure useful. I for sure will look to the quick guide to geomorph, thank you for this one very much.

I do agree geometric morphometrics can sometimes (actually in many cases) be really powerful, especially when investigating the shape. Nevertheless, when you have to dissect the animal and you are measuring chitinous body parts interconnected with joints I find it very difficult (if not impossible) to adjust correctly whole appendage in the same position so that it would allow the use of geometric morphometrics. Moreover what would then be the best approach with using geometric morphometrics when one have many different appendages, each consisted of several different articles? Should one do per partes analyses, article by article?

Generally, I agree to use geometric morphometrics, or both (geom. morph. + trad. morph.) if I find the object suitable for both approaches (e.g. butterfly wings seem a good example to me).

I just found a study of leporid lagomorphs that tackles a similar problem and uses 3D landmarks for morphometrics and phylo-morphospace plots:

Kraatz B, Sherratt E. 2016. Evolutionary morphology of the rabbit skull. PeerJ 4:e2453 https://doi.org/10.7717/peerj.2453

https://peerj.com/articles/2453/

I agree that there seems no obvious way to overcome (tackle) the challenges posed by having to dissect an organism (or parts of it) in order to find a suitable metric for classification.

My caveat in such situations: in order to fix (in 3D) the positions of 4 points (a tetrahedron, not necessarily a regular tetrahedron), one needs to measure 6 distances (to form what is technically called a truss). for 6 points ( a general octahedron), 12 distances need to be measured; and so on ... The problems arises when not measuring all the distances needed to uniquely specify the truss: you can easily imagine what happens when you measure only 9 (say) distances of the truss specifying the octahedron. You have no way of controlling for the variation in the missing dimensions: a very weird figure (a tube-like one, say) or a regular octahedron may have the same distances you measured. So, the distances measured reveal nothing about the form being estimated by the incomplete truss. (By the way: if the truss is complete: there exist public software packages/routines that can convert the truss dimensions into 3D coordinates.)

My suggestion if you have destroyed the shape by dissecting: tabulate into categories (row: the (categorized) dimensions/lengths; columns the putative species) and then perform a correspondence analysis. A method I am developing -- in an unrelated problem: look at how the dimensions are distributed within each species and then calculate a maximum-likelihood ratio when comparing the pairwise.

Cheers, Hermann