This is the culmination of what I first wrote about back in September, the field of geometric morphometrics. You may remember that this involved measuring the shape of an object (in this case biological objects) by assigning certain landmarks a set of co-ordinates and carrying out a load of fancy calculations and statistics. Although I’ll cover it here too, and it’s probably a good deal more accurate than last time I wrote about it.

In this study I looked at whether the angle of people’s faces relative to their brain case (this is an accepted term, I’m not trying to patronise anyone!) has any influence of the overall shape of their face. Shape face includes the relative heights of different parts of the face and how square the jaw is when viewed from the side.

Figure 1 helps show the differences. Part a) explains the different parts of the face that are considered to describe face shape. If the difference between anterior face height (UFH + MFH + LFH) and posterior face height then the face shape is long-faced (Fig. 1b). If anterior face height and posterior face height are more similar it is described as short-faced (Fig. 1c). You can also see how the jaw is much squarer in Fig. 1c than Fig. 1b.

Another type of recognised variation is UFH relative to posterior facieal height. If UFH is relatively high it is described as klinorhynchy, a smaller UFH relative to posterior facial height is called airorhynchy. I’m afraid I can’t find any nicer words than those, I just know that they refer to the nose (rhynchy).

To measure these features I was given a load of side-on x-rays of people’s skulls called the Denver growth series, collected as part of a big study carried out by the American Association of Orthodontists Foundation started in the mid-20th century. On 100 of these I placed a configuration of 10 landmarks based on previous studies of facial shape in chimpanzees. Figure 3 shows these landmarks highlighted on an outline diagram, as I can’t show you a Denver Growth Series x-ray for patient confidentiality reasons.

The landmarks connected by green lines were used to calculate the angle of the face relative to the brain case. I excluded landmark 1 from the face shape analysis itself because I was looking for co-variation between shape and angle. I had to keep landmark 1 separate because it is the only measure of the brain case I have. If I included it in measurement of the ‘face’ shape then face shape would show artificially high co-variation with facial orientation*.*

The problem with landmarks is trying to make sure that they are comparable or equivalent between different specimens. To try and ensure that you are measuring the same things on each specimen a few definitions of landmarks exist. Type 1 landmarks are identified by the coming together of tissues of structures, so the intersection of 2 bony ridges or something similar. Type 2 are defined as the tip or most extreme part of a given structure, for example, landmark 1 is the uppermost point on the inner cranial surface. Type 3 landmarks can, at best be placed somewhere on a curve making them the least reliable. I didn’t use any type 3 landmarks because I knew my ability to recognise where to place them would introduce enough error to make the rest of this project almost completely uninformative anyway.

Once the landmarks for are placed and their co-ordinates recorded, just like on a map, all of them except number 1 are subjected to some maths which essentially makes all the faces the same size so that the raw features of shape can be compared between specimens. To do this, the co-ordinates for each face are averaged to find the central point, or centroid. All 100 landmark configurations are then digitally slid about until all their centroids are in the same place.

Secondly, they are scaled to the same ‘centroid size’. This involves measuring the distance between the centroid and each landmark on the face, squaring it, adding them all together, finding the square root of that and altering the original distances so that this square root equals 1. Mathematically, √(Σ[distance from each landmark to centroid]^{2}) = 1. (O’Higgins, 2000).

Finally, the landmark configurations are rotated relative to another so that the squared distance between landmark 1 on each specimen is as low as possible, and for landmark 2, 3, 4… etc.

This is known as generalised Procrustes analysis. (Procrustes was a Greek demi-God – Wikipedia).

The advantage of all this is that the resulting ‘Procrustes co-ordinates’ allow each configuration of 9 landmarks to be represented by just one point on an abstract curved graph known as Kendall’s shape space, where the axes are measured in ‘shape units’. The shape of the scatter made by these points describes the actual shape variation present in your original sample of specimens.

Fortunately, the software lets you visualise these shape variations quite easily as wireframe projections (figure 4).

The main difference that this shows is variation in the depth of the upper face compared to the lower face. On the left, the upper face is relatively taller and deeper, as shown by upward stretching of the grid in that area. At the same time we can see that the lower face is relatively shorter. The opposite is true on the left, and the decrease in depth of the upper face is clear here as the sideways compression of the grid squares.

However, these wireframes describe a portion of the overall face shape variation, and we can’t tell from these what role the angle of the face plays. To do that, you need to perform either regression or an analysis called partial least squares analysis (PLS), which each compare shape change with another variable (in this case the angle) in a different way.

Starting with regression, this assumes that angle is completely independent of any other influences and varies almost ‘at will’. Regression also assumes that face shape depends directly on facial angle. These are not necessarily realistic assumptions when considering biological processes. Nevertheless, according to regression, face shape does vary with angle (figure 5).

From these wireframes, it seems that the principal components of shape variation in relation to facial angle include variation in the relative upper facial height (airorhynchy-klinorhynchy), and changes in depth of the face as a whole.

Partial least squares analysis is generally believed to be a truer representation of how two variables are related as it analyses how they co-vary according to *shared underlying influences*, and does not assume that one depends on the other. This more accurately represents how biological processes interact and is likely to give a more accurate representation of face shape variation in relation to facial angle. The wireframes from PLS are shown in figure 6.

On the face of it, the shape variations described by the wireframes from regression and PLS are very similar. The numbers that are spat out by the software say that facial orientation is associated with up to 8.8% of the variation in face shape, where the rest of the variation is down to other factors. These factors could even include what sort of food you eat and how well you chew it. PLS also tells us that the strength of co-variation between angle and face shape is 0.1676 (maximum is 1), far lower than that generally accepted in the scientific work to prove that two factors co-vary. Normally you’d look for numbers higher than 0.6.

Additionally, the shape variations are exaggerated 50 times in the regression wireframes, whereas they are only doubled in the PLS wireframes. I *think* this reflects the facts that PLS detects a greater degree of co-variation between angle and face shape than regression does, which means that the variation will be more obvious in the images in produces.

In short, facial orientation seems to account for some of the variation in airorhynchy vs. klinorhynchy (the relative height of the upper face) but not to that great an extent. Maybe if my landmarking accuracy were better there would be some higher numbers and I could actually draw some reasonable conclusions, but the point of the exercise was to show I could use appropriate geometric morphometric techniques.

I hope that I’ve kind of shown that it isn’t too impenetrable.

– James.

###### References

Bastir, M., & Rosas, A. (2004). Facial heights: evolutionary relevance of postnatal ontogeny for facial orientation and skull morphology in humans and chimpanzees. *Journal of human evolution*, *47*(5), 359–81. doi:10.1016/j.jhevol.2004.08.009

Bastir, M., Rosas, A., & Sheets, H. D. (2005). The morphological integration of the hominoid skull: a partial least squares analysis and PC analysis with implications for European Middle Pleistocene mandibular variation. In D. E. Slice (Ed.), *Modern Morphometrics in Physical Anthropology* (1st ed., pp. 265–84). New York: Kluwer Academic / Plenum Publishers.

Lewis, V. 2008. ArtyNess Kids Crafts. avaiable from: www.artyness.co.uk [Accessed 2.43pm 12/12/13].

O’Higgins, P. (2000). The study of morphological variation in the hominid fossil record: biology, landmarks and geometry. *Journal of anatomy*, *197*(1), 103–20.