Four common morphometric methodologies were used: 1) canonical variate analysis; 2) eigenshape analysis; 3) relative warps analysis; and 4) the thin plate spline method. In the following section, a brief description of these techniques is introduced.

Canonical variate analysis is one of the more interesting morphometric applications of multivariate statistics. The technique is used to examine the interrelationships between a number of populations simultaneously with a goal of objectively representing the interrelationships graphically in few dimensions (ideally only two or three dimensions). The axes of variation are chosen to maximize the separation between the populations relative to the variation within each of the populations.

In algebraic terms, the first canonical variate is the linear combination of variables that maximizes the ratio of between-groups sums of squares to the within-groups sums of squares for a one-way multivariate analysis of variance of the canonical variate scores.

For *k *groups (characters) and *p* variables (populations), the canonical variate scores are obtained from the following equation:

where c^{T} is the transpose of the canonical vectors and
*x _{ij}*

where *B * is the between-groups matrix of sums of squares and cross products and
*W* is the within-groups matrix of sums of squares and cross products.

The canonical vectors *c* ** **and the canonical roots
*
f* satisfy the following equations:

and

|B -The canonical vectors are usually scaled so that

c
where *N _{w}* is the within-groups degrees of freedom.

For more information see Blackith and Reyment (1971), Reyment et al. (1984), and Reyment and Savazzi (1999).

In 1983, Lohmann developed a Q-mode principal component technique for analyzing changes in the shape of an organism that he called eigenshape analysis. The observations consist of coordinate pairs determined at definite points around the circumference of the shell. In the present study, 11 landmarks were selected around the outline of each studied ostracod specimen to express these points.

The technique is a simple ordinating procedure that has as its starting point the
*x*, *y* coordinates of a set of *p* points along outlines of
*N* objects of interest (here outline of the ostracod carapace). A transformation procedure is followed in the same manner as principal component analysis.

Rohlf (1996) stated that there are two methods of comparison of shapes in geometric morphometrics. One of them is based on the least square method and is most efficient if overall similarity depends largely on few landmarks. The other method is thin plate spline analysis, which works best if the similarity depends on many landmarks.

The thin plate spline method is based on analogy of a two-dimensional morphological object to a thin homogeneous deformable metallic plate
(Bookstein 1989,
1991); thus one specimen is fitted to another by stretching, and the numerical estimate of degree of such a smooth deformation is the
*bending energy* coefficient.

The shape variation encompasses two components, an affine (uniform) part and non-affine (non-uniform) part (Bookstein 1991). In the affine change, the orthogonality of principal axes is preserved, and parallel lines remain parallel, like the deformation of a square into a parallelogram or a circle into an ellipse. The non-affine change is represented by the residual of size-free change that remains after the difference due to any affine change has been subtracted from the total change in shape. An example is when an initially flat object is twisted or warped. For some examples of the method, see Reyment (1993, 1995a, 1995b, and 1997), Reyment and Bookstein (1993), and Reyment and Elewa (2002).