will affect conditional colocalisation testing because some residual association must remain after conditioning. On the other hand, it should not bias proportional testing as the aim there is to test for proportionality of effect size rather than evidence of residual effect. This may explain the substantially higher error rates for naive conditional testing versus naive proportional testing seen in Figure 1. The bias in effect size estimates affects both methods, however. In genetics, we are familiar with “Winner’s curse”, which causes effect estimates which are examined conditional on the associated p value being below some significance threshold to be biased away from the null. Some attempts to correct this effect size bias have been made, either by modelling a selection procedure defined as a single SNP exceeding a predetermined level of significance [Zollner and Pritchard, 2007], or by bootstrapping which can in theory account for the full selection strategy [Sun et al., 2011]. We explored both approaches, but found neither led to unbiased or even nearly unbiased inference (data not shown). For Zollner and Pritchard [2007], this failure is presumably down to the discrepancy between correcting for a p value that exceeds some threshold and selecting SNPs with the minimum p values. In the case of Sun et al. [2011], it is possible that single loci do not contain sufficient information for a bootstrap based correction; the corrected estimates tended to be biased in the opposite direction, suggesting the method was over-correcting. Our proposed solution is to use proportional testing and either avoid variable selection altogether by using the PCs which capture the majority of genetic variation in the region under test, or integrate over the variable selection using BMA. Either method maintains type 1 error, and the BMA approach appears more powerful than the PC approach, although both have reduced power compared to the hypothetical scenario of being able to test the causal variants themselves. Recall from equation (4) that the ppp depends on both the posterior probabilities of individual SNP models and the degree of departure from the null under each model, p $^{∗}$( m ). If the posterior probability was spread broadly over the model space and p $^{∗}$( m ) varied considerably across models, the distribution of ppp would be far from uniform, with too few observations in the tails. The fact that we observe only mildly conservative type 1 errors, even in the weakest associations considered here (case control sample sizes of 2000, and relative risks of 1.1), probably reflects our requirement in simulating our datasets that some nominal level of association must be observed for at least one SNP in a region. This means that, for our simulated data, the posterior probabilities tend to be mainly focused on a 15