Beyond bilateral symmetry: geometric morphometric methods for any type of symmetry

Table 7 Procrustes ANOVA for the coral example, with a symmetry group consisting of reflection and rotation of order 6

Source	Degrees of freedom	Sums of squares	Mean squares	F	P
Individual	343	1.510	0.0044038	0.49	> 0.999999
Rotation	39	7.6968	0.19735	20.31	< 0.000001
Reflection	7	0.008013	0.0011447	4.69	0.000049
Rotation × reflection	39	7.5323	0.19314	19.78	< 0.000001
Rotation × individual	1911	18.569	0.0097171	16.04	< 0.000001
Reflection × individual	343	0.083800	0.0002443	0.40	> 0.999999
Rotation × reflection × individual	1911	18.663	0.009766	16.11	< 0.000001
[Total FA]	4165	37.316	0.0089594
Imaging error	4600	2.7869	0.0006058	1.25	< 0.000001
Digitizing error	9200	4.4426	0.0004829

The main effect of individuals is tested against the mean square for the total fluctuating asymmetry ("Total FA": pooling sums of squares and degrees of freedom across all three subspaces with asymmetric variation: rotation × individual, reflection × individual and rotation × reflection × individual). This total asymmetry is not used otherwise in the analysis.

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