Table 3

Degrees of freedom in the Procrustes ANOVA for object symmetry with rotation only in two and three dimensions

Effect

2D

3D


Individual

(n - 1) × (2k - 2)

(n - 1) × (3k + c - 3)

Rotation

2k(o - 1) + 2c - 2

3k(o - 1) + 2c - 4

Rotation × individual

(n - 1) × (2k(o - 1) + 2c - 2)

(n - 1) × (3k(o - 1) + 2c - 4)

Measurement error

(r - 1) × n × (2ko + 2c - 4)

(r - 1) × n × (3ko + 3c - 7)


Notation: For rotational symmetry of order o, the complete landmark configuration can be subdivided into o different sectors (Figure 4). Each sector contains k landmarks. In addition, there are c landmarks on the centre or axis of rotation (for 2D data, c is 0 or 1; for 3D data, c is 0 or greater). The sample consists of n individuals (specimens), and each specimen has been digitized r times.

Savriama and Klingenberg BMC Evolutionary Biology 2011 11:280   doi:10.1186/1471-2148-11-280

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