## 3 Results
### Local Genetic Correlation Analysis  
We used the univariate test implemented in LAVA to assess the local heritability of 15 loci in which the SNPs found to be significant in the MR analysis were located. Local heritability was small from all loci, ranging from 0.0001 to 0.0004 for MDD and from 0.0001 to 0.0005 for insomnia. After multiple testing adjustment, four loci showed significant heritability for both phenotypes, while six and two other loci showed significant heritability exclusively for MDD and insomnia, respectively (Table 6). Figure 6: Network of interactions among proteins encoded by genes for which SNPs driving the association between MDD (exposure) and insomnia (outcome) act as eQTLs. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**SNP** & **Chr** & **Gene** & **EA** & **OA** & **b exp** & **b out** & **eQTL for Gene** & **RDB Score** \\ \hline rs77960 & 5 & Intergenic & A & G & 0.03 & 0.04 & - & 0.99 \\ rs9563152 & 13 & Intergenic & T & C & 0.04 & 0.02 & _RP11-24H2.3_ & 0.18 \\ rs6984111 & 8 & _MSRA_ & C & T & 0.04 & 0.02 & - & 0.14 \\ rs1456193 & 3 & _RP11-384F7.2_ & T & C & \(-\)0.04 & \(-\)0.02 & - & 0.18 \\ rs370771 & 6 & _LIN28B_ & G & T & \(-\)0.04 & \(-\)0.02 & (Caudate, putamen) & 0.59 \\ \multicolumn{8}{c}{_HACE1_ (Cortex)} \\ rs9576155 & 13 & _SUPT20H_ & A & G & 0.03 & 0.02 & _ALG5_ & 0.18 \\ \multicolumn{8}{c}{_CUL19_ (Caudate, cortex,} \\ rs6938026 & 6 & _CUL9_ & G & A & 0.04 & 0.02 & frontal cortex, nucleus accumbens, spinal cord) & 0.61 \\ rs77217059 & 2 & _LINC01122_ & A & T & 0.03 & 0.03 & - & 0.73 \\ \hline \end{tabular} Abbreviations: b exp, beta exposure; b out, beta outcome; Chr, chromosome; EA, effect allele; eQTL, expression quantitative trait locus; OA, other allele; RDB, regulome DB; SNP, single nucleotide polymorphism.  
\end{table} 
Table 5: SNPs driving the association between insomnia (exposure) and MDD (outcome). Of the four loci showing significant heritability for both MDD and insomnia, three showed a significant bivariate local genetic correlation (Table 7). For the locus on chr2, including the rs77217059 SNP, we observed a significant local positive genetic correlation between MDD and insomnia (rho = 0.64, r\({}^{2}\) = 0.41, adj \(p\) = 0.002) and no significant correlation between these two phenotypes and any cardiometabolic trait (Table 7). For the locus on chr 3 including the rs9831648 SNP, we observed a significant local positive genetic correlation between MDD and insomnia (rho = 0.60, r\({}^{2}\) = 0.36, adj \(p\) = 0.045), as well as significant local bivariate genetic correlation between MDD and BMI (rho = 0.48, r\({}^{2}\) = 0.23, adj p = 0.006) or between insomnia and BMI (rho = 0.69, r\({}^{2}\) = 0.47, adj \(p\) = 1.7 \(\times\) 10\({}^{-9}\)), CRP (rho = 0.66, r\({}^{2}\) = 0.44, adj \(p\) = 4.2 \(\times\) 10\({}^{-5}\)) and CAD (rho = 0.54, r\({}^{2}\) = 0.29, adj \(p\) = 0.033). Partial correlation analysis between MDD and insomnia, adjusted for BMI, CRP or both variables, suggested that these variables account for a notable proportion of the local \(r_{g}\) between MDD and insomnia. In fact, we observed that BMI and CRP levels explained a substantial proportion of the genetic variance for MDD and insomnia at this locus (e.g., BMI explained 23% and 47% of the proportion of genetic variance for MDD and insomnia, respectively, Table 8). Consistently, the correlation between MDD and insomnia was no longer significant when adjusting for these variables (Table 8). Results for partial correlation analysis between MDD and insomnia adjusting for CAD are not reported, as LAVA deemed estimates to be unreliable (estimate out of bounds). Finally, we observed a significant positive genetic correlation between MDD and insomnia at the locus on chr 13 including the two SNPs rs9536381 and rs9563152 (rho = 1, \(\mathrm{r}^{2}\) = 1, adj \(p\) = 9.9 \(\times\) 10\({}^{-5}\)).