tissue specific eQTLs in more distal locations [Brown et al., 2012]. We were therefore interested to explore the case where our two traits share one causal variant, but one or both are also under the influence of additional, distinct variants. Testing a single hypothesis of colocalising versus not colocalising variants cannot capture the complexity of this situation, but it is instructive to ex- plore the test’s expected behaviour in order to allow proper interpretation with real data where the number, and sharing configuration of causal variants is unknown. Figure 4 shows that, under a tagging genotyping scenario, the proportional test tends to reject the null of colocalisation in the case of any distinct causal variants, even in the presence of an additional shared variant, although with slightly less power than when there is no shared variant. So far, also, we have assumed that it was enough to consider only the universe of two SNP models when applying our BMA approach. The motivation for this was that a two SNP model leads to a one degree of freedom test, and might therefore be expected to maximise power. We examined the effect of averaging over either all two SNP or all three SNP models in the context of the above multiple causal variant simulations. This shows that type one error rates are similar, that power is similar for two SNP models, but that power can be increased by averaging over three SNP models compared to two SNP models, particularly when there are really three or more distinct causal variants (Figure 4) Existing evidence suggests that a single locus may contain variants which predispose to any one of multiple diseases, e.g. the non-synonymous C1858T SNP in PTPN22 is associated with rheumatoid arthritis and T1D [Stahl et al., 2010; Barrett et al., 2009], or distinct variants which predispose to different diseases, e.g. distinct variants in IL2RA are associated with T1D and multiple sclerosis [Maier et al., 2009; Martin et al., 2012]. We used the proportional colocalisation approach outlined above to examine the disease signals for the autoimmune thyroid diseases HT and GD from a recent dense genotyping study [Cooper et al., 2012]. We first examined the seven regions where a significant single SNP effect has been identified in both diseases, i.e. at study-wide significant levels for GD ( p < 1 . 1 × 10 − $^{6}$, a permutation derived Application to colocalisation testing of Autoimmune Thyroid Diseases Varying the number of SNPs in Bayesian Model Averaging models 13