Appendix

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For risk variants on autosomal chromosomes, there are three main models to be considered: i) dominant, ii) recessive, and iii) multiplicative model.

In the case of a dominant model, the disease risk ratio between the risk variant carrier, “c”, and the non-carrier, “nc”, can be defined to be “r”, i.e. the model assumes that probability for the disease is r-times higher given the risk-genotype combination than the regular non-risk type:

Pr(A|c)/Pr(A|nc) = (r Pr(A|nc))/Pr(A|nc) = r

A straight-forward application of a conditional probability rule therefore gives

Pr(c|A)/Pr(nc|A) = (Pr(A|c)Pr( c )/Pr(A)) / (Pr(A|nc)Pr(nc)/Pr(A)) = r Pr( c )/Pr(nc)

Under the assumption of random population controls we have that

Pr( c ) = Pr(c|C) and Pr(nc) = Pr(nc|C)

and therefore

OR = (Pr(c|A)/Pr(nc|A)) / (Pr(c|C)/Pr(nc|C)) = r

Hence, the odds-ratio is simply the relative risk between the variant types. The calculations for the recessive model are identical. Only the definition of the carrier type and the non-carrier type differs, “c” and “nc” respectively. If for example, we assume that the autosomal locus has two allele types, a risk type “a” and a wild-type “b” (note that in case there are more than two variants possible, “b” can always be defined as the non-”a” allele) then for i) we define c = (aa or ab) and nc=bb whereas for ii) c = aa and nc = (ab or bb).

It is however most common to use the third model, i.e. not to assume dominant or recessive genotype to phenotype behavior but rather to assume a multiplicative model where the risk is the product of the risk associated with the two alleles copies. In this case the risk contribution of the two allele copies is considered independent and likewise the frequency distribution of the allele combinations is independent or in so called Hardy-Weinberg equilibrium.

Under the multiplicative model the ratios Pr(A|aa) : Pr(A|ab) : Pr(A|bb) = r² : r : 1 and the chances for the three allele combinations are Pr(aa) : Pr(ab) : Pr(bb) = p² : 2pq : q² , where Pr(a) = p and Pr(b) = q = (1-p) are the population frequency of allele “a” and “b” respectively.

By calculating the allelic odds-ratio in fractions of chromosomes with the given variant (unlike in the other models where it is the number/fraction of individuals) we get:

OR = (Pr(a|A)/Pr(b|A))/(Pr(a|C)/Pr(b|C)) = ((2p²r² + 2pqr)/(2pqr + 2q²)) / (p/q) = r

Here we are denoting the fraction of chromosomes with allele “a” in the affected group with Pr(a|A) which is equal to 2Pr(aa|A)+Pr(ab|A) and likewise for “b”. Also, we are using the assumption of random controls, i.e. that Pr(a|C)/Pr(b|C) = p/q.