Statistical Analyses .1 Repeatability

Observer error was assessed by comparing the Procrustes distances between paired specimens of the entire sample with those of repeated measurements retaken from 15 individuals. The effects of allometry on between-specimen variation were accounted for by multiple regressions on the logarithm of centroid size. Failure to account for allometry could obscure the effects of measurement error. Figure 3.2 A and B shows the results of the analyses before and after the effects of allometry were accounted for, respectively.

The difference between partial Procrustes superimposed original and repeated configurations were modelled on the mean shape to further evaluate measurement error.

In order to illustrate error at each landmark, a wireframe mesh was produced with 95%

28 confidence ellipses around landmark scatters (Figure 3.3). Although error is averaged among the landmarks through Procrustes superimposition methods, those with excessive error will generally stand out, given that there are enough well placed repeatable landmarks to anchor the configuration.

3.2.4.2 Preliminary analyses

Preliminary refining of the dataset was required before the main analyses could be run. This included mitigation of asymmetry, removing outliers and sectioning the main dataset into the subsets needed for the subset analyses. These procedures are described in this section.

Klingenberg et al. (102) defined two types of bilateral symmetry. The first, known as matching symmetry, occurs when mirror images of a structure occupy opposite sides of an organism (e.g. the left and right arm). The second, known as object symmetry, occurs when symmetry exists along two sides of a midline of the same structure, as is seen in the human skull. Certain degrees of developmental asymmetry occur naturally in even the most symmetric of objects (102,103). Although the study of asymmetry is the focus of many studies, it was beyond the scope of this investigation and needed to be controlled for. Failure to do so would introduce additional variation to the dataset that would obscure results and lead to spurious conclusions. Asymmetry was removed by finding unilateral landmarks, which lie in the midline of the cranium, and zeroing them.

Bilateral landmarks from the left side of the cranium were then reflected onto the right and relabelled (102). Mean configurations of a specimen and its reflection, after Procrustes superimposition, are considered symmetric. By running analyses on the symmetric component of the data, any asymmetry was eliminated together with any variation associated with it. The morphological analysis program Morphologika (104) was used for preliminary visualisation of the raw data. This included preliminary partial Procrustes superimpositions (PPS) and principal components analyses (PCA) that were used to analyse the overall distribution of the data and to identify outliers.

After the data were visually assessed in Morphologika, data and classifier files were read into the statistical software R (version 3.1.3, R Core Team, 2015). Outliers were identified in Morphologika as any points far removed from the main cluster in the

29 scatterplot of principal component one. Renderings of each specimen’s landmarks were also assessed to ensure that no errors were made during data collection. Once identified, outliers were removed and PPS was performed on the symmetric component of the remaining data. Semilandmarks, including helper landmarks, were slid using a thin-plate spline deformation (with minimum bending energy) (31) and centroid size was saved.

Thereafter, the helper landmarks were removed, resulting in a reduction in the total number of landmarks per specimen from 281 landmarks to 158. Centroid size was also recalculated based on the reduced number of landmarks.

3.2.4.3 Analysis of tooth loss

After preliminary analyses, the following set of statistical procedures was conducted using programmed algorithms and scripts in the statistical program R (version 3.1.3, R Core Team, 2015). These procedures were repeated for the skull in its entirety as well as for each of the subsets listed in Table 3.3.

Full Procrustes superimposition permits centroid sizes to vary and allows for the attainment of true minimal distances among specimens. Despite this, partial Procrustes superimposition (PPS) is often the preferred superimposition method as, after tangent space projection, it reflects the true distance between specimens better than full Procrustes superimposition (102). An uncentered correlation coefficient was therefore used to test the difference between the two techniques, and PPS was found to produce less distortion due to projection. Therefore, PPS was used to align landmarks by translation, scaling, and rotation (60,105) and the resulting data were projected orthogonally to the tangent plane. The thin plate spline algorithm outlined in Gunz (106,107) was used to allow semilandmarks to slide with each iteration, rather than minimising Procrustes distance. Helper landmarks were utilised because sliding-semilandmarks are shifted along a tangent to the curve with each iteration during Procrustes superimposition, and not along the curve itself. These helper points were then discarded prior to any statistical analysis (100). Finally, permutation tests were employed to determine whether significant size and age differences exist between the male and female groups.

30 In this investigation, the analysis of dental loss was categorical and scored as absent (0) or present (1). Although the accepted method for analysing relationships between categorical data is by multiple correspondence analysis (108), the question arose whether an ordination method, such as principal components analysis (PCA) or 2-Block partial least squares (PLS), could also be used to investigate such relationships.

Both PCA and multiple correspondence analysis were run on the binary dental data and the results were compared by means of an uncentered correlation coefficient. A correlation of 0.998 was found, suggesting that PLS is an appropriate method for assessing categorical data. PLS was thus employed to assess the effects of dental loss on the morphology of the face and basicranium.

Next, the RV coefficient (109) was used to assess whether a significant amount of covariation exists between the two blocks of data (tooth loss and cranial shape). This coefficient is superficially comparable with Pearson’s correlation coefficient in that zero represents no association and one a perfect association (the amount of covariation between X and Y expressed as a fraction of the total variation in blocks X and Y). The significance of the covariation was tested using a permutation test to assess whether the coefficient obtained is significantly greater than that resulting by chance (110). Where significant covariation existed this was explored using PLS. PLS is a relatively new dimension reducing technique that explores covariation between a set number of variables or blocks, as opposed to PCA which explores variation within a block (60,111).

Due to the fact that the dependant variable, in this case dental loss had been specified, PLS is a better technique than PCA for this study (112). Vector diagrams and colour coded polygon figures were used to visualise the effects of dental loss on craniofacial morphology.

3.3 RESULTS