Those of you who have read my recent articles will probably have noticed the phrase ‘geometric morphometrics’ a few times. When mentioned to people, the usual reaction is to melt or run away screaming satanic verse and tearing chunks of hair out (*pers. obs*). This is largely due to the pretty intense mathematical basis behind the huge variety of implementable statistical procedures, which range from simple linear regressions to more complex 3D extended-eigensurface analyses. Each of these essentially provides a quantitative method of analysis of biological structures that can be interpreted in terms of biological function, a pretty crucial aspect of both zoology and palaeontology. To get to grips with the necessary analytical tools, it’s not really important to dig into the fundamentals such as how to construct a covariance matrix – these are well-defined mathematical concepts. The aim of the following few posts is to break down what is one of the most powerful yet under-used tools available for bio-structural analysis. What I’d also like to achieve is some kind of informal discussion about ideas in which geometric morphometrics can be applied to simple ‘pilot’ analyses based on freely available information, such as photographs from Morphbank. This first post will deal with the initial concepts, and future articles will provide examples of the different methods and tools available. Hopefully you will find this useful, and begin to openly develop and understand the processes involved.

Geometric morphometrics, as you might infer from the name, is the statistical analysis of form using geometric co-ordinates, or Cartesian landmarks. Form is defined here as the total dimensionality resulting from both shape and size. Size is the totality of spatial dimensions within a form, and shape is defined as the aspect of a form’s geometry that remains after scale (i.e., size), position (translation) and rotation have been normalised. Shape is essentially a localised metric for describing variation of spatial dimensions. Distinguishing between these is actually pretty crucial, as typically in an analysis you will want to differentiate between size and shape. Recently, the field has accelerated in strength due to the ability obtain three-dimensional structures such as skulls using techniques such as computer-tomography (CT) scanning. This considerably increases the information available for geometric morphometricians, and has led to numerous concurrent methodological adaptations in order to rigorously process available data. Using 3D techniques is kind of like a ‘total evidence’ approach to form analysis.

My personal opinion is that geometric morphometrics completely out-strips traditional morphometrics in terms of theoretical strength, methodology, and explanatory power. Consider simple linear measurements for starters, in for example, describing a lateral view of any mammalian hind-limb. You can imagine all sorts of bisecting, parallel, oblique and orthogonal measurements that would aid reconstruction of the form. Collecting these measurements and the relative angles would be time-consuming however, especially if you were looking for example, at sexual dimorphism of the femoral head in an antelope population. The world’s supply of coffee would be extinct before completion. However, with a simple photograph and the right software, breaking down a femur into a geometric outline or surface that you can use for all sorts of morphometric wizardry takes seconds (**not** including the months it takes until you are granted access to specimens).

Ratios are also a statistical over-simplification. The combination of measurements that can produce the same ratio is constrained only by the size of an object, and furthermore, ratios are a gross under-estimate of the potential geometric complexity of an object; try and imagine modelling a sine-wave with a linear ratio (or as complex a morphological structure as you want). Not going to happen is it.

The core of geometric morphometrics revolves around the assessment of allometry. Allometry is an ubiquitous aspect of nature, describing how organisms change their form. Discovering and interpreting allometry is the proximate target of most investigations, with the null hypothesis being isometry: no shape variation with respect to size. Typical investigable targets include detection of heterochronic trends, called paedomorphosis or peramorphosis, relating to the timing of acquisition of certain structures in an organism’s or species’ history (essential for evo-devo analyses).

Landmarks form the principal units of analysis for geometric morphometrics. Landmarks are formally known as Bookstein shape co-ordinates, with a defined Cartesian geometric position (i.e., *x*, *y*, *z *variables). Landmarks represent a subset of possible locations or distances, based on the nature of sampling. The only problems with this approach include difficulties in recognising landmarks, missing data, and possible redundance of data due to over-lapping inter-landmark spacings.

REALLY IMPORTANT: Landmarks are NOT homologous *sensu stricto*. They represent topographically correspondent ‘characters’ – you should be able to write down the exact location in an unambiguous manner. This is really important when it comes to the biological interpretation of data.

There are three types of Cartesian landmark. Although not necessarily that important, it can provide an idea of how geometrically faithful a representation of an object you have.

*Type 1*: These represent the juxtaposition of biological components, such as sutures.

*Type 2*: These represent geometric aspects of form, such as local maxima or minima of curvature. These and type 1 landmarks are typically used in structure-based analyses.

*Type 3*: These are co-ordinate dependant equidistant interpolations, such as mid-points between two type 1 landmarks. They are also known as semi-landmarks, and are typically used in refining shapes such as profile outlines.

Landmarks form the cornerstone of all geometric morphometric analyses. What they provide you with is an unambiguous and quantitative dataset, that most importantly is highly informative in terms of biological structure. With landmark data, the wealth of potential modes of analysis at your disposal is phenomenal, as are the available software packages. One I would highly recommend is the *tps* series that can be found here, as well as a rather comprehensive overview of all things geometric.

In the mean-time, I wish everyone here an awesome 2012, and try not to get apocalypsed/raptured. I’ve popped a few references at the bottom here regarding the recent application of morphometrics in the field of vertebrate zoology, definitely worth reading a few just to get to grips with how scientists are currently using the techniques. I also strongly recommend the PalaeoMath series by Norm MacLeod, freely available here through the Palaeontological Association. It’s good stuff, and includes data so you can try your own analyses!

**Next time**: Principal Components Analysis, Principal Co-ordinates Analysis, and *Procrustes* superimposition.

Barden, H. E. and Maidment, S. C. R. (2011) Evidence for sexual dimorphism in the stegosaurian dinosaur *Kentrosaurus aethiopicus *from the Upper Triassic of Tanzania, *Journal of Vertebrate Palaeontology*, **31(3)**, 641-651

Brusatte *et al*. (2011) The evolution of cranial form and function in theropod dinosaurs: insights from geometric morphometrics, *Journal of Evolutionary Biology*, DOI: 10.1111/j.1420-9101.2011.02427.x

Goswami, A., Milne, N. and Wroe, S. (2010) Biting through constraints: cranial morphology, disparity and convergence across living and fossil carnivorous mammals, *Proceedings of the Royal Society B*, doi:10.1098/rspb.2010.2031

Hadley, C., Milne, N. and Schmitt, L. H. (2009) A three-dimensional geometric morphometric analysis of variation in cranial size and shape in tammar wallaby (*Macropus eugenii*) populations, *Australian Journal of Zoology*, **57**, 337-345