Methods settings

We mainly compared the performance of CAVIARBF with PAINTOR, which can also incorporate functional annotations and showed very competitive performance over other methods (Kichaev et al. 2014). Because PAINTOR does not provide a direct way to simultaneously incorporate all annotations, several different strategies of selecting informative annotations were tested. We also compared our method with fgwas, assuming one causal variant and focusing on the different annotation selection scheme, where the Bayes factors are almost the same (they are exactly the same under certain settings). We implemented the forward-backward annotation selection proposed in Pickrell (2014) for both fgwas and CAVIARBF for the comparison. For the forward-backward annotation selection, we defined the significance for individual annotations using the P-value threshold 0.05. This may result in false inclusion of irrelevant annotations at the first step simply due to randomness given the large number of annotations. The later backward exclusion might reduce this effect. However, we still see obvious inflation of PIP and decreased performance under the null annotations using this search scheme. Because FM-QTL only considers as few annotations as PAINTOR, it is not compared here. All tested analysis methods are described in Table 1.

Without any specific change of settings, here is the default setting for CAVIARBF: We used fivefold CV, when CV is used for penalty parameter selection. We set the maximal number of causal variants in each locus to 3, and the parameter σa=0.1. For CAVIARBF_ENET_CV, the grid search set for λ and α was set to {2⁻¹⁵, 2⁻¹³, …,  2⁵, 10², 10³, 10⁴, 10⁵, 10⁶} × {0, 0.2, 0.3, 0.5, 0.7, 0.8, 1} for simulated data sets. For real data analysis, λ was searched from {2⁻²,  2⁻¹, …,  2¹⁰} for a finer grid search. For CAVIARBF_L2 _CV and CAVIARBF_L1 _CV, the search set for λ was the same as that in CAVIARBF_ENET_CV. PAINTOR version 2.1 was used in all simulations and we set the maximal number of causal variants to 3. For lipid data analysis, we assumed the genotype effect sizes follow a normal distribution on the standardized genotypes (mean, 0; variance, 1), to be consistent with the assumption in PAINTOR (Chen et al. 2015). But for the eQTL analysis, we assumed the genotype effect size follows a normal distribution on the original scale of genotypes, i.e., the counts of the specified allele.