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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. 2 — Acyclic graphs illustrating the multivariable regression (MVR) design for (a) a set of independent ASD-related variants G_i (ASD-MVR) and (b) a set of independent ADHD-related variants G_j (ADHD-MVR). For both MVR models, increaser marker alleles (G_i or G_j) were aligned to ASD ( $\hat{β}$ _ASD), ADHD ( $\hat{β}$ _ADHD) and EA ( $\hat{β}$ _EA) GWAS SNP estimates, capturing genetic association at the single-variant level. Using a bidirectional MVR framework (ASD-MVR; ADHD-MVR), the aggregate association effect with EA across all variants was simultaneously estimated for ASD risk ( $θ$ _ASD; $θ$ _#ASD) and ADHD risk ( $θ$ _ADHD; $θ$ _*ADHD), including a regression intercept ( $θ$ _0*; $θ$ _0#). *Estimation of polygenic risk effects using alleles that were selected to increase liability for ASD, but are shared, by position, with ADHD risk. ^#Estimation of polygenic risk effects using alleles that were selected to increase liability for ADHD, but are shared, by position, with ASD risk. (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. 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 Table 3. All tests passed the multiple-testing threshold of P < 0.0023. G_i was selected from ASD(iPSYCH, woADHD) and G_j from ADHD(iPSYCH), both at P_thr < 0.0015 and P_thr < 0.05. ASD ( $\hat{β}$ _ASD), ADHD ( $\hat{β}$ _ADHD) and EA ( $\hat{β}$ _EA) SNP estimates were extracted from ASD(iPSYCH, woADHD; N = 32,985), ADHD(iPSYCH; N = 37,076) and EA(SSGAC; N = 766,345) GWAS statistics respectively. 3D scatter plot of (d) ASD-MVR (G_i; P_thr < 0.0015) and (e) ADHD-MVR (G_j; P_thr < 0.0015), as shown in (c). The regression plane reflects the estimated MVR effects for ASD-MVR ( $\hat{θ}$ _ASD; $\hat{θ}$ _*ADHD) and ADHD-MVR ( $\hat{θ}$ _ADHD; $\hat{θ}$ _#ASD), respectively. 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.