S GENERALISED PROCRUSTES ANALYSIS

The context and details of the generalised Procrustes analysis (GPA) are covered in Section 1.2.3. However, it should be noted that all comparisons of form in this study are carried out using the GPA method of superimposition. GPA registers the landmark coordinate data, representing the form of each individual, such that all scalar, rotational and translational differences are removed, and the sum of the squared differences, the Procrustes chord distance (cf), between the equivalent landmarks of all individuals is minimised (Gower, 1975; Rohlf, 1990; Rohlf & Slice, 1990; Goodall, 1991).

d^='Z'Z(^;-x'jÿ

1=1 y=/+l

Where n specimens (individuals) are registered, the landmark coordinates of each are denoted as X. , where /= 1 are represented

\n a k X m matrix (k landmarks; m dimensions) and X \ denotes the

registered X,. .

Thus form is decomposed into size and shape components, allowing the shape of each individual to be represented in a well defined non-Euclidean shape space, Kendall’s shape space (Kendall, 1984). The statistical properties of this shape space are well understood and the shape geometry of landmark configurations is preserved throughout the analysis. Further analyses, such as principal components analysis, permutation tests for significance of angles between principal component axes, permutation tests for significance of Procrustes chord distance between means, visualisation of shape variability by warping of means and the use of thin plate spline based Cartesian transformation grids (all detailed below or previously in Section 1.2) are carried out within this shape space or rely on data obtained from this shape space. GPAs are carried out using the software package morphologika (© Paul O’Higgins & Nicholas Jones, University College London).

2.2.2.4 ANGLES BETWEEN PC AXES AND PERMUTATION TEST FOR SIGNIFICANCE

This method is used to calculate the degree and significance of the divergence between the first principal components obtained for each of two groups (whether sex, species etc.). Both groups are registered together in the same GPA and the angular value of the divergence is calculated by superimposing the two PCIs upon their origins, as

specificed by Blackith & Reyment (1971), angular values can range from 0° to 90°.

A permutation test is then used to ascertain the significance of the angle between the PCIs. A permutation test (sometimes refered to as randomisation, rerandomisation or exact tests) randomly rearranges and relabels the individuals of the groups repeatedly and recomputes the observation (the angle between the axes in this case) each time, 1000 iterations are used in this study. A distribution curve is then calculated from the 1000 permuted angles, from which the p value for the observed angle is calculated (Good, 1993). Thus “with a permutation test, you compare the observed value of the test statistic [the angle between the PCIs in this case] with the set of what-if values you obtain by rearranging and relabeling the data" (p.8. Good, 1993). Both the calculation of the angle and the permutation test for a significance value were performed using software that was written in-house (Paul O’Higgins & Nicholas Jones, University College London).

2.2.2.5 PROCRUSTES CHORD DISTANCE BETWEEN SHAPE MEANS AND PERMUTATION TEST FOR SIGNIFICANCE

The coordinate data of the individuals of both groups are GPA registered separately. The mean shape is calculated for each group as the arithmetic mean of its respective GPA registered coordinates. The coordinate data of the shape means for two groups are GPA registered together, and the distance in Kendall’s shape space between the group means is calculated (Dryden & Mardia, 1998).

A permutation test is then used to ascertain the significance of the Procrustes chord distance between the shape means. The coordinate data of all individuals of both groups are GPA registered together, the individuals are then randomly reallocated to two groups of the same size as the original groups, the mean is calculated for each group and the Procrustes chord distance between them is recorded, this is repeated 1000 times. A distribution curve is then calculated from the 1000

permuted Procrustes chord distances, and the p value for the originally observed distance between the group mean shapes is calculated from this distribution (Good, 1993). These calculations are performed using software that was written in house (Paul O’Higgins & Nicholas Jones, University College London).

2.2.2.6 PRINCIPAL COMPONENTS ANALYSIS (PCA)

Principal components analysis is a common multivariate technique used to summarise the variation of multivariate data by reducing the number of variables or dimensions. In essence, a new set of axes, principal component axes (PCs) are constructed within the multivariate data, such that the first principal component axis (PCI) is a linear combination of the original data that accounts for the maximum possible variance. This is equivalent to the longest axis of the multidimensional scatter of the data. Subsequent PCs are constructed to account for the maximal remaining variance whilst remaining orthogonal to proceeding PCs, thus the principal component has the largest variance (eigenvalue) after the n-^ principal component, and the last principal component has the lowest eigenvalue. All principal component axes pass through the mean, known as the centroid. The orthogonal relationship of all principal components relative to each other ensures that each principal component is uncorrelated. The eigenvectors of each principal component axis are analogous to the coefficient required to transform the original data into PC data. PCA can be carried out using either the covariance or correlation matrix of all variables, but in the case of shape space, the covariance matrix is used.

Thus in the case of geometric morphometric methods, as employed in this study, a PCA is used to statistically analyse the multidimensional scatter of points in Kendall’s shape space that represent the shape data of each individual after GPA registration. The application of PCA in this context is explained in more detail in Section 1.2.4. Principal components analyses were all carried out using the software

package morphologika (© Paul O’Higgins & Nicholas Jones, University College London).

22.2.7 T-TEST

The Hest is a univariate test of the equality of the means of two groups, and as such determines whether any differences are statistically significant or likely due to random variation. The t statistic is based on the assumption that the variances of the two groups are equal. However t- tests are used in this study to determine sexually dimorphic principal component axes for each species, and thus it is an unjust assumption that the variance of the PC scores of males and females, the two groups analysed, are equal. Therefore an approximate t statistic can be calculated, based on the assumption that the variances are unequal, the probability level of this t approximation is calculated from the Cochran and Cox approximation. An F-test is used to acertain whether the male and female variances are equal in each case and the appropriate t-test is used accordingly. All such f-tests were computed using the statistical software package SAS (Version 6, © SAS Institute Inc.), unless otherwise stated.