than for conditional testing. The two-step lasso selection defined above does reduce bias compared to independent lasso selection in the two datasets, but, perhaps counter-intuitively, leads to greater bias than simply testing the pair of most associated SNPs ( k, k $^{′}$) when only tagging genotypes are available and effect sizes are large (relative risk ∼ 1.3). This is because, in this situation, lasso may select SNPs which are apparently weakly associated (either truly or through random noise) at which, as demonstrated in Supplementary Figure 7, effect estimates are more strongly biased. Principal components When using principal components to summarize the genetic variation in a region, it is not obvious how many components are required. As more components are selected, more information about the genetic variation in a region is captured, and hence we are more likely to accurately capture the signal of any causal variants. However, successive components add decreasing amounts of information whilst still adding another degree of freedom. At some point the negative effect of increasing degrees of freedom will outweigh the positive effect of increasing information, and we were concerned that the optimal test may depend heavily on the threshold used to determine the number of components selected. Instead, power seemed broadly acceptable once components capturing 70-90% of variation were selected (Supplementary Figure 10). In our 49 test regions, 70% of the variance could be captured by selecting an average of 7 (range 2-18) or 9 (range 3-44) components under a tagging or complete genotyping approach. We found type 1 error rates were controlled across the range of thresholds explored, but show for illustrative purposes the results when we fixed the threshold at ≥ 90% of genetic variation (Figure 1). We examined power to detect departure from colocalisation using simulations in which the causal variants are distinct for two traits but placed no restrictions on the LD between these variants. We first examine the theoretical maximum power of the test by testing the two causal variants themselves, which are known in a simulation study but not in real data (Figure 2). As might be expected, power increases with sample size and effect size, but is negatively correlated with the r 2 between the causal variants, and is maximum when the two are completely unlinked ( r 2 = 0). When using PCs, the power is reduced reflecting the loss of information in not knowing these causal variants, but the loss is greatest for complete genotyping scenarios, suggesting that we may be selecting too many components in the case of complete genotyping and emphasizing the 11 Proper control of type 1 error rates