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. 2021 Nov 11;12:6534. doi: 10.1038/s41467-021-26755-1

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© The Author(s) 2021

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

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Fig. 4 — Acyclic graphs illustrating the multivariable regression (MVR) design for (a) a set of independent ASD-related variants G_i, given joint association with ADHD variant set G_j (ASD-MVR with G_i|j) and (b) a set of independent ADHD-related variants G_j, given joint association with ASD variant set G_i (ADHD-MVR with G_j|i). Here, G_i|j and G_j|i are illustrated as subsets (concentric circles) of G_i and G_j across a grid of six P-values thresholds (0.0015 < P_thr < 0.5), based on ASD(iPSYCH, woADHD) and ADHD(iPSYCH) summary statistics. (c) ASD-MVR ( $\hat{θ}$ _ASD; $\hat{θ}$ _*ADHD) and ADHD-MVR ( $\hat{θ}$ _ADHD; $\hat{θ}$ _#ASD) effects as change in years of schooling per increase in log odds of ASD or ADHD liability (for the definition of $\hat{θ}$ _*ADHD and $\hat{θ}$ _#ASD, see Fig. 2). SNP sets G_i|j and G_j|i were selected from ASD(iPSYCH, woADHD) and ADHD(iPSYCH), as shown in (a, b). SNP estimates for ASD ( $\hat{β}$ _ASD), ADHD ( $\hat{β}$ _ADHD) and EA ( $\hat{β}$ _EA) were extracted from ASD(iPSYCH,woADHD; N = 32,985), ADHD(iPSYCH; N = 37,076) and EA(SSGAC; N = 766,345) GWAS statistics respectively. Multivariate inverse-variance-weighted regression estimates and corresponding 95% confidence intervals (bars) are shown. Individual effect estimates, standard errors and corresponding P-values (t-statistic, two-sided test) are provided in Supplementary Tables 10–11. All MVR effects passed the multiple-testing threshold of P < 0.0023, except for ADHD effects estimated with ADHD-MVR (G_j|i: ADHD P_thr < 0.0015; ASD P_thr < 0.05), which were present as trend (P = 0.01). Source data are provided as a Source data file. ADHD Attention-Deficit/Hyperactivity Disorder, ASD Autism Spectrum Disorder, EA educational attainment, MVR multivariable regression; P_thr P-value threshold, SNP single-nucleotide polymorphism, woADHD without ADHD.