For alpha =0.055, the participation rate of UKBB, three frequency difference ratios of an SNP (indicated in the figure) with participation effects are displayed as functions of the sibling recurrence participation rate ratio, lambda_{S} . The results are from simulations under a liability-threshold model where the participation of an individual is determined by its liability score and the participation rate alpha. Given alpha, lambda_{S} is a function of the correlation between the liability scores of two siblings ( Methods ). In particular, for alpha =0.055, a correlation of 0.193 between the siblings’ liability scores leads to lambda_{S}=2, the reported enrichment of sib-pairs in the UKBB data. We simulated 500 replications from a population of 5*10^{7} sib-pairs. Allele 1 of the SNP is assumed to have a population frequency of 0.5 and the effect of the SNP is assumed to account for 0.1% of the variance of the liability score. The simulated averages of the two ratios, (F_{IBD1S}-F_{IBD1NS})/(F_{samp}-f_{pop}) and (F_{IBD2}-F_{IBD0})/(F_{samp}-f_{pop}), shown with red hollow squares and blue solid triangles respectively, are virtually indistinguishable from each other; they are roughly 1 when lambda_{S} is close to 1 and decrease gradually as lambda_{S} increases, but are always positive. The simulated average of the third ratio (F_{SIBS}-F_{SING})/(F_{samp}-f_{pop}) , where F_{SIBS} is the allele frequency in the participating sibling pairs and F_{SING} is the allele frequency in the participating individuals whose sibling does not participate, is shown with gray solid circles. For lambda_{S}=2, the first two ratios are around 0.86 and the third ratio is around 0.32.
