It has been proposed that gene expression may be subject to both global regulatory variation which acts across multiple tissues and secondary tissue specific regulators [21]. Neither approach covers this case explicitly in its construction, but it is instructive to examine their expected behaviour. The proportional approach tends to reject a null of colocalisation, suggesting that a single distinct causal variant can be sufficient to violate the null hypothesis of proportional regression coefficients. In contrast, the Bayesian approach tends to favour the shared variant in the cases covered by our simulations (median PP4 > median PP3), and either hypotheses H3 or H4 can potentially have strong support (PP4 >0.9 in close to 50% of simulations, and PP3 >0.9 in around 25% of simulations). Of course, the ultimate goal should be to extend these tests to cover multiple causal variants, but in the meantime, it can be useful to know that a high PP4 in our proposed Bayesian analysis indicates strong support for “at least one causal variant” and that rejection of the null of proportionality of regression coefficients indicates that the two traits do not share all causal variants, not that they cannot share one.
