The generalization to k alleles, A1 , . . . , Ak, is relatively straightforward. In an obvious notation let pij be the frequency of genotype AiAj , and nij be the observed count, i, j = 1 , . . . , k, j ≥ i. The likelihood is again multinomial: (2)Pr(n∣p)=n!∏i,j=1,j≥iknij!∏i,j=1,j≥ikpijnij, where n = (n11, n12 , . . . , nkk), and p = (p11, p12 , . . . , pkk) are k(k + 1)/2-dimensional vectors and n=∑i,j=1,j≥iknij Under HWE pii=pi2,i=1,…,k, and pij = 2pipj , i, j = 1 , . . . , k, j ≥ i. We can parameterize the saturated model as pii=pi2+pi∑j≠ipjfij,pij=2pipj(1−fij) so that we have introduced a set of fixation indices fij (Weir, 1996; Ayres and Balding, 1998); fij = 0 for all i ≠ j recovers the HWE model. Again we may examine estimates of fij to discover the reasons for departure from HWE; a positive/negative fij indicates a deficiency/excess of heterozygotes of type AiAj. A disadvantage of this model is that the fixation indices are on awkward ranges:
