Appendix 2: Formulae used in calculation of genetic distances.

1) Freeman Tukey transformation

    theta (θ)  = (0.5 *arcsin(l-(2*k)/(N+l)))+(0.5 *arcsin((l-(2*k+l )/(N+1)))

where k is the number of teeth showing the trait and N is the number of teeth observed

2) Anscombe transformation

    theta  (θ) = arcsin[l-(2*(k+0. 375)/(N+0.75))]

3) mean measure of divergence

     MMD =  l/r*Σ[(-₂)²-V]

where r is the number of traits analyzed and

4) V = (1/(N₁+0.5))+(1/(N₂+0.5))

5) the variance of the MMD =  2/r²*Σ(V²)

6) Z = √(2*S)-√((2*r)-1) where S = Σ[(θ₁-θ₂)²/V]