Appendix A

Marker Risk Ratio in Unexposed
The frequency of haplotype D,M in unexposed subjects is Pr(D,M|X = 0). The joint probability of an affected subject and haplotype D,M isPr(Y=1,D,M|X=0)=Pr(Y=1|D,M,X=0)Pr(D,M|X=0)=RDPr(Y=1|D=0,M,X=0)Pr(D,M|X=0)where RD is the causal variant risk ratio. The risk of disease in an unexposed subject with marker M isPr(Y=1|M,X=0)=Pr(Y=1,M|X=0)/Pr(M|X=0)=∑dPr(Y=1,D=d,M|X=0)/∑dPr(D=d,M|X=0).The risk ratio for the marker is thenPr(Y=1|M=1,X=0)Pr(Y=1|M=0,X=0)=∑dRdPr(d,M=1|X=0)∑dPr(d,M=0|X=0)∑dRdPr(d,M=0|X=0)∑dPr(d,M=1|X=0).

Marker Risk Ratio in Exposed
Note thatPr(Y=1,D,M|X=1)=Pr(Y=1,X=1|D,M)Pr(D,M)/Pr(X=1)=SDPr(Y=1,X=0|D,M)Pr(D,M)/Pr(X=1)=SDPr(Y=1,D,M|X=0)Pr(X=0)/Pr(X=1)where SD is the risk ratio of D on the exposure X.
AlsoPr(D,M|X=1)=Pr(X=1|D,M)Pr(D,M)/Pr(X=1)=SDPr(X=0|D,M)Pr(D,M)/Pr(X=1)=SDPr(D,M|X=0)Pr(X=0)/Pr(X=1)So the marker risk ratio in the exposed isPr(Y=1|M=1,X=1)Pr(Y=1|M=0,X=1)=∑dSdRdPr(d,M=1|X=0)∑dSdPr(d,M=0|X=0)∑dSdRdPr(d,M=0|X=0)∑dSdPr(d,M=1|X=0)
The above calculations may be performed using Table S1 available online.