causal variants in each trait, we selected one or two “causal variants” at random, and simulated a Gaussian distributed quantitative trait for which each causal variant SNP explains a specified proportion of the variance. To reflect our expectation that this test will be applied in cases in which some nominal association to a region has already been established, we discarded datasets in which all single SNP association p > 10 − $^{4}$. We either used all SNPs or the subset of SNPs which appear on the Illumina HumanOmniExpress genotyping array to conduct colocalisation testing to reflect the scenarios of very dense targeted genotyping versus a less dense GWAS chip. All analyses were conducted in R [R Development Core Team, 2010] using the coloc package for proportional colocalisation testing. An association study of the autoimmune thyroid diseases GD and HT has recently been completed using the Immunochip for genotyping, which provides dense coverage of regions of the genome previously associated with autoimmune disease [Cooper et al., 2012]. The paper presented a total of 2285 Graves’ disease cases, 462 Hashimoto’s disease cases and 9364 controls. We split the controls randomly into two groups of size 4682, and analysed each of the 13 regions reported to be associated with one or both diseases [Cooper et al., 2012]. Missing data were rare, but regression models require complete genotyping data. We therefore imputed missing genotypes by means of multiple regression, as implemented in the R package snpStats [Clayton and Leung, 2007]. We conducted proportional colocalisation analysis using the the two alternative methods set out above. For the PCs approach, we used components that captured at least 90% of the observed genetic variation. For the BMA approach, we averaged either over the universe of all possible two SNP models or that of all three SNP models. The choice of SNPs to use for testing can induce bias for two reasons. First, selecting the “most associated” SNP on the basis that the evidence for its association is strongest amongst all SNPs tested does not guarantee either that it is the causal SNP or even the best proxy. Random variation 9 Results Naive application of colocalisation tests leads to biased inference Colocalisation testing for autoimmune thyroid disease