Choice of SNPs for proportional colocalisation testing The choice of SNPs for colocalisation testing will be shown in this paper to have a considerable influence on the type 1 error rate of colocalisation tests. For the proportional approach, two strategies have been applied. Either colocalisation has been tested using the pair of SNPs k and k ′ defined above [Plagnol et al., 2009] or a lasso approach, where SNPs are first selected in a lasso for one trait, and then additional SNPs are selected in a further lasso for the other trait [Wallace et al., 2012]. However, as shown by Miller [Miller, 1984], any variable selection method must induce bias in the estimated coefficients ( b$_{1}$ , b$_{2}$ ) if the estimation occurs in the same dataset as the selection. The aim of selecting the most informative subsets of SNPs for proportional colocalisation testing is to minimise the degrees of freedom of the test, and hence maximize power. However, unless independent data are available for variable selection, this increase in power comes at a cost to type 1 error rate control as shown above. In this paper, we propose two methods for avoiding this problem. Summary of genetic variation by principal components If the region of interest displays strong LD, a modest number of principal components (PCs) are generally required to capture most of the SNP variation (Supplementary Figure 9) and we can use a subset of the most informative components for colocalisation analysis. Because PCs are by definition uncorrelated, and because the selection is not based on their relationship to the traits of interest, the estimated coefficients at any such subset are unbiased. To allow PC analysis of two datasets, we first form a combined genotype matrix, center and scale each SNP, and then define the principal components. Colocalisation testing is performed using the projection of the data onto the transformed basis for the most important components. The optimal choice of threshold defining the “most important” components is not obvious, and we explore that in our simulations. Bayesian Model Averaging Alternatively, we may combine the ideas of Bayesian model aver- aging (BMA) [Viallefont et al., 2001] and posterior predictive p values, to treat the model describing the joint association itself as a nuisance parameter, and average the p values not just over the pos- terior for η , but also over the posterior for all SNP selection models. Analogous to equation (3), 7