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:10.1186/1471214811280 