a hypothetical two-locus model with k = 4 alleles at one (antigen) locus along with a second disease locus. Under a recessive model of inheritance we require two disease alleles, D, and among cases we may consider the probabilities of obtaining the different antigen combinations under the recessive model: Pr(AiD∕AjD∣Disease)=Pr(Disease∣AiD∕AjD)Pr(AiD∕AjD)Pr(Disease), where AiD/AjD are the possible genotypes of diseased individuals, for i, j = 1 , . . . , k. Thomson (1983) parameterized the model in terms of ki = Pr (Ai | D), and let f2 be the probability of disease given two copies of the disease allele, and pD the frequency of the disease allele. Under the recessive model two disease alleles are required and so Pr(Disease)=∑i,j,i≠jPr(Disease∣AiD∕AjD)Pr(AiD∕AjD)=f2pD2[∑i=1kki2+∑i,j=1,i≠jkkikj]=f2pD2, where the summations within the square brackets equal 1 because we have added over the set of conditional allele probabilities. Hence Pr(AiD∕AjD∣Disease)={ki2i=j2kikji≠j}, which is equivalent to HWE. Deviations from this model suggest that the recessive model does not hold. In particular we may interpret, for example, negative f34 as an excess of A3A4 genotypes among diseased individuals. The single f model is more difficult to interpret in this context but corresponds to a general excess (negative f) or deficit (positive f)
