## 3 Results
### Linkage Disequilibrium Score Regression (LDSC) and K-Means Clustering
Abbreviations: ADHD, attention deficit/hyperactivity disorder; AN, anorexia nervosa; ASD, autism spectrum disorders; BD, bipolar disorder; BMI, body mass index; CAD, coronary artery disease; CRP, C-reactive protein; MDD, major depressive disorder; OCD, obsessive-compulsive disorder; SCZ, schizophrenia; PTSD, post-traumatic stress disorder; T2D, type 2 diabetes; TS, Tourette syndrome. showed a positive, while MDD, PTSD, and AN a negative correlation. Finally, SCZ and ASD showed a negative correlation with the morning person chronotype. Next, we conducted k-means clustering using 7 clusters, which was suggested to be the optimal number by the silhouette coefficient (Figure 2). A visualization of the clusters in two dimensions is shown in Figure 3.  
We observed a cluster including three psychiatric disorders (MDD, PTSD and ADHD) and insomnia. The three disorders were the ones showing largest effect size as regards to the cross-trait genetic correlation with insomnia (Table 2). Two other clusters only included psychiatric disorders (BD-SCZ, and OCD-TS-AN), while one cluster included all the cardiometabolic/inflammatory traits. Finally, three clusters included a single trait (ASD, sleep duration and chronotype). Results from k-means clustering support the relationship between insomnia and the three psychiatric disorders MDD, PTSD and ADHD. We further explored the relationship between these traits using MR. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Pschiatric Trait** & **Sleep Trait** & \(\mathbf{r_{g}}\) & **se** & **Z** & \(\mathbf{p}\) & \(\mathbf{adj\,}\mathbf{p}\) \\ \hline BD & Insomnia & \(0.11\) & \(0.03\) & \(4.23\) & \(2.4\times 10^{-5}\) & \(0.0028\) \\ BD & Sleep & duration & \(0.11\) & \(0.02\) & \(4.91\) & \(9.0\times 10^{-7}\) & \(0.0011\) \\ SCZ & Chronotype & \(-0.10\) & \(0.02\) & \(-5.38\) & \(7.6\times 10^{-8}\) & \(9.1\times 10^{-6}\) \\ SCZ & Sleep & duration & \(0.15\) & \(0.02\) & \(7.37\) & \(1.7\times 10^{-13}\) & \(2.1\times 10^{-11}\) \\ MDD & Insomnia & \(0.44\) & \(0.03\) & \(17.60\) & \(2.3\times 10^{-69}\) & \(2.8\times 10^{-67}\) \\ MDD & Sleep & \(-0.11\) & \(0.02\) & \(-4.43\) & \(9.4\times 10^{-6}\) & \(0.0011\) \\ ADHD & Insomnia & \(0.37\) & \(0.03\) & \(10.67\) & \(1.4\times 10^{-26}\) & \(1.7\times 10^{-24}\) \\ PTSD & Insomnia & \(0.48\) & \(0.07\) & \(7.33\) & \(2.3\times 10^{-13}\) & \(2.7\times 10^{-11}\) \\ PTSD & Sleep & \(-0.23\) & \(0.06\) & \(-3.86\) & \(0.0001\) & \(0.013\) \\ ASD & Chronotype & \(-0.18\) & \(0.03\) & \(-5.45\) & \(5.1\times 10^{-8}\) & \(6.1\times 10^{-6}\) \\ AN & Sleep & \(-0.12\) & \(0.03\) & \(-3.83\) & \(0.0001\) & \(0.015\) \\ \hline \hline \end{tabular} Abbreviations: ADHD, attention deficit/hyperactivity disorder; AN, anorexia nervosa; ASD, autism spectrum disorders; BD, bipolar disorder; MDD, major depressive disorder; SCZ, schizophrenia; se, standard error; PTSD, post-traumatic stress disorder. The adj p column reports \(p\)-values adjusted according to Bonferroni based on the number of conducted tests (\(n=120\)).  
\end{table} 
Table 2: Significant genetic correlations between psychiatric and sleep traits. Figure 2: Assessment of the optimal number of clusters based on the silhouette coefficient.