Integration of genome-wide association studies (GWAS) and expression quantitative trait loci (eQTL) studies is needed to improve our understanding of the biological mechanisms underlying GWAS hits, and our ability to identify therapeutic targets. Gene-level association methods such as PrediXcan can prioritize candidate targets. However, limited eQTL sample sizes and absence of relevant developmental and disease context restrict our ability to detect associations. Here we propose an efficient statistical method (MultiXcan) that leverages the substantial sharing of eQTLs across tissues and contexts to improve our ability to identify potential target genes. MultiXcan integrates evidence across multiple panels using multivariate regression, which naturally takes into account the correlation structure. We apply our method to simulated and real traits from the UK Biobank and show that, in realistic settings, we can detect a larger set of significantly associated genes than using each panel separately. To improve applicability, we developed a summary result-based extension called S-MultiXcan, which we show yields highly concordant results with the individual level version when LD is well matched. Our multivariate model-based approach allowed us to use the individual level results as a gold standard to calibrate and develop a robust implementation of the summary-based extension. Results from our analysis as well as software and necessary resources to apply our method are publicly available. We develop a new method, MultiXcan, to test the mediating role of gene expression variation on complex traits, integrating information available across multiple tissue studies. We show this approach has higher power than traditional single-tissue methods. We extend this method to use only summary-statistics from public GWAS. We apply these methods to 222 complex traits available in the UK Biobank cohort, and 109 complex traits from public GWAS and discuss the findings. Recent technological advances allow interrogation of the genome to a high level of coverage and precision, enabling experimental studies that query the effect of genotype on both complex and molecular traits. Among these, GWAS have successfully associated genetic loci to human complex traits. GWAS meta-analyses with ever increasing sample sizes allow the detection of associated variants with smaller effect sizes –3. HowevAnother approach is the study of expression quantitative trait loci (eQTLs), measuring association between genotype and gene expression. These studies provide a wealth of biological information but tend to have smaller sample sizes. A similar observation applies to QTL studies of other traits such methylation, metabolites, or protein levels.The importance of gene expression regulation in complex traits –7 has moDue to sharing of eQTLs across multiple tissues, we have shown the benefits of an agnostic scanning across all available tissues . Despitehttps://github.com/hakyimlab/MetaXcan. We apply this method to simulated and real data . We make our implementation publicly available to the research community in nk study  and 109 y is the n-dimensional phenotype vector, μ is an intercept term, ti is standardized predicted expression of the gene in tissue i, gi is its effect size, and e an error term with variance p is the number of available tissue models. We use an F-test to assess the joint significance of the regression.To integrate information across tissues, MultiXcan regresses the phenotype of interest on the predicted expression of a gene in multiple tissues as follows:rp50 = 0.56 (IQR = 0.69) between different tissue models in a given gene, across genes , and foWe applied MultiXcan to 222 traits from the UK Biobank cohort. The traits were chosen based on several criteria, such as availability of well-established literature, binary traits having enough cases, or potential interest for a phenome-wide study . We used Elastic Net prediction models trained on 44 tissues from GTEx, originally presented in .We compared three approaches for assessing the significance of a gene jointly across all tissues: 1) running PrediXcan using the most relevant tissue; 2) running PrediXcan using all tissues, one tissue at a time; 3) running MultiXcan. http://biobank.ctsu.ox.ac.uk/crystal/field.cgi?id=20002). We used 50,497 cases and 100,994 controls. After Bonferroni correction, MultiXcan was able to detect a larger number of significantly associated genes (251 detections) than PrediXcan using all tissues (196 detections) or only a single tissue . 172 genes were detected by both PrediXcan and MultiXcan. APOM [PAFAH1B2 [SLC5A6 [NOTCH4 [SORT1 is detected by both MultiXcan and PrediXcan.As an illustrative example, we examined more closely the results for self-reported high cholesterol phenotype , glucos [SLC5A6 ), and va [NOTCH4 ). The weTo evaluate MultiXcan’s performance in different known scenarios, we simulated traits as a function of different numbers of causal tissues for each gene: a single tissue, multiple tissues, all available tissues. We executed PrediXcan, MultiXcan without PCA regularization, and MultiXcan with PCA regularization. We show proper calibration under the null hypothesis of no association in −16). MultiXcan performs best in scenarios with multiple causal tissues .As expected, when there is a known single causal tissue, PrediXcan with the known tissue yields more significant associations. However, when there are multiple causal tissues, MultiXcan yields more significant associations than the best single tissue PrediXcan results. In traits simulated from a single causal tissue, PrediXcan outperforms MultiXcan in 99.9% of the cases .p number of tissues). The final expression is equivalent to the omnibus test mentioned in [We infer the joint estimates of effect sizes of predicted expression on phenotype  using thioned in , which cioned in . Fig 3-ak the number of surviving components. See details in the As with the individual level approach, the correlation between tissues leads to numerical problems (due to near singular covariance matrices that need to be inverted). We address this by using a pseudo inverse approach which, in a nutshell, uses singular value decomposition (SVD) of the covariance matrix to keep only the components of large variation. This is analogous to the PCA regularization used for the individual level approach. Thus we test for significance using A robust implementation for calculating predicted expression correlation is critical to avoid unnecessary false positive results. In principle, it is possible to simply calculate the correlation between tissues using predicted expression in a reference set. However, we found that this approach can lead to large differences between the individual level data results (our gold standard) and the summary level ones when SNPs from the reference set are missing in the GWAS results. An example of this is shown in −4 (“suspicious associations”). In other words, we keep significant associations if at least one single gene-tissue pair association is borderline significant or better . This is rather conservative since it is possible that evidence with modest significance from weakly correlated tissues can lead to very significant combined association when their effects get aggregated. For example among Bonferroni significant genes in the individual level analysis, a median of 8.3% across traits (IQR = 5.7%) have the most significant marginal (PrediXcan) p-value greater than 10−4. We list the number of such genes for each of the 222 UK Biobank traits in To reduce false positives due to LD misspecification when dealing with GWAS summary statistics, we discard any significant association result for a gene if the best single tissue result has p-value greater than 10We applied S-MultiXcan to 109 traits on publicly available GWAS, chosen with a similar criteria as UK Biobank’s traits. Like the individual level method, we observed S-MultiXcan to detect more associations than S-PrediXcan in most cases (average detection increase 10), as shown in https://doi.org/10.5281/zenodo.1402225.We display a summarized comparison between S-MultiXcan and S-PrediXcan in We examine below the biological relevance of a few of the genes detected by our new method that was missed when using one tissue at a time (S-PrediXcan).POMC  Consortium’s Body-Mass Index (BMI) study, S-MultiXcan detects three genes not significant in S-PrediXcan:  obesity ); RACGAPogenesis , cell grogenesis ); and TUisorders , suggestAS3MT , related to arsenic metabolism; interestingly, environmental and toxicological studies link arsenic exposure and AS3MT polymorphisms with cardiovascular disease [CDKN2B ; IREB2  study, S-MultiXcan detected 12 associations not significant in S-PrediXcan. The top result was  disease , 23. Ass × 10−6, ) HECTD4  × 10−6, ). Other ditions, ), and ADclerosis , atrial clerosis , and othclerosis , 30).The list of significant S-MultiXcan and S-PrediXcan results for all traits can be found in Motivated by the widespread sharing of regulatory processes across tissues , we prop−4. This is far from perfect. To take full advantage of summary results and summary-based methods, reference sets that are the closest to the study population should be used. This also stresses the need to generate representative reference LD datasets for a wide variety of populations.We found high concordance, in general, between the individual level and summary version with the latter slightly more conservative. As any method relying on a reference panel, S-MultiXcan may be inaccurate when the study population has a different LD structure than the reference panel. We attempted to address this by flagging results where none of the marginal associations reached a somewhat arbitrary threshold of 10Pcolocalized > 0.5. A similar strategy may be applied for MultiXcan by restricting the analysis to gene-tissue pairs with high colocalization probability in the marginal analysis.Via simulations, we show that MultiXcan is properly calibrated under the null hypothesis of no associations. This is reassuring, but it is possible that in real data there are hidden confounders that we did not capture in our simulations. For example, significant association results might arise due to LD contamination, i.e. when causal variants for the trait and expression are different but in LD with each other, inducing a spurious correlation between the predicted expression and the trait. This is a complex problem that we are currently working to address. In Barbeira et al , we sougIn practice, we emphasize the need to further validate the significant associations with additional replication and experimental follow-up.Importantly, we provide compelling examples where using multiple tissues rather than picking one considered to be relevant for the phenotype increases the list of candidate causal genes. In our simulations, we found that only when the single causal tissue is known and the regulatory mechanism is captured perfectly by predicted expression in that tissue, using PrediXcan with that tissue yields more significant associations than MultiXcan. This scenario is unlikely to occur in practice. Therefore, in general, we recommend jointly scanning of all tissues in addition to focusing on a few tissues selected based on prior knowledge.https://github.com/hakyimlab/MetaXcan. Prediction model weights and covariances for different tissues can be downloaded from http://predictdb.org/. A short working example can be found on the GitHub page; more extensive documentation can be found on the project’s https://github.com/hakyimlab/MetaXcan/wiki. The results of S-MultiXcan applied to the 44 human tissues and a broad set of phenotypes can be queried on http://gene2pheno.org. The data used in this paper is publicly available in https://doi.org/10.5281/zenodo.1402225.We make our software publicly available on a GitHub repository: This study uses de-identified genotype and phenotype data from public repositories including dbGaP, EGA, and UK Biobank. Our study has been determined to be non-human subject research by the University of Chicago’s IRB protocol number IRB16-0921.a); vector quantities with bold lower-case letters  and matrices with bold capital letters (e.g. A). Corresponding scalar entries will be denoted by subscripts (e.g. ai is the i-th entry for vector a).In the following, we shall denote scalar quantities by italicized lower-case letter , a secure biomedical cloud operated at FISMA moderate as IaaS with an NIH Trusted Partner status for analyzing and sharing protected datasets. We computed GWAS results using BGENIE, a program for efficient GWAS for multiple continuous traits [UK Biobank genotype data for 487, 409 individuals was downloaded and processed in the Bionimbus Protected Data Cloud . For evTo allow for uniform correction of unwanted variation, we treated all traits as quantitative and adjusted for the same covariates reported in . These cOn most continuous phenotypes, there were between 300, 000 and 400, 000 individuals with available data determined by the intersection of covariates and traits. For the case of self reported diseases, we found a number of cases ranging from a few hundreds (i.e. Acne) to 50, 000 (i.e. High Cholesterol).We have demonstrated that S-PrediXcan can accurately infer PrediXcan results from GWAS Summary Statistics and LD information from a reference panel , with thSummary-MultiXcan (S-MultiXcan) infers the individual-level MultiXcan results, using univariate S-PrediXcan results and LD information from a reference panel. It consists of the following steps:Computation of single tissue association results with S-PrediXcan.Estimation of the correlation matrix of predicted gene expression for the models using the Linkage Disequilibrium (LD) information from a reference panel Discarding components of smallest variation from this correlation matrix to avert collinearity and numerical problems .Estimation of joint effects from the univariate (single-tissue) results and expression correlation.Discarding suspicious results, suspect to be false positives arising from LD-structure mismatch.To derive the multivariate regression  effect sgj (y is centered for convenience (so that no intercept term is needed), and ϵj is the marginal regression error term with variance j).More specifically, we want to obtain the multivariate regression coefficient estimates for gj  using thγ is the vector of effect sizes γj. Please note that, since the tj are standardized, then p × p identity matrix) and First, notice that the solution to the multivariate regression in From  we get Tplace in  and obtaTo compute the variance of the estimated effect sizes  we use tTtT, using only those SNPS available in the GWAS results. To do so, notice that: ij are the elements of the covariance matrix As the genotypes from most GWAS are typically unavailable, we must use a reference panel to compute We restrict the computation to using only SNPS in the intersection between reference panel and GWAS. Failing to do so may lead to inaccurate inference of predicted expression covariance, typically underestimating correlation, leading to false positives as can be seen in TtT is often close to singular and its inverse cannot be reliably calculated for many genes. To address this problem, we compute the pseudo-inverse via Singular Value Decomposition, decomposing the correlation matrix into its principal components and removing those with small eigenvalues (SVD regularization). In other terms, we will restrict the analysis to axes of largest variation of the expression data. This is analogous to the principal components-based approach used with individual level data. We denote with Σ+ the pseudo-inverse for any matrix Σ. We use the same condition number from individual-level MultiXcan (IQR = 10), with ∼ 80% of components surviving the SVD threshold.Given the high degree of correlation among many of the prediction models, Cor(T) is the autocorrelation of T, and p-vector of marginal analysis z-scores, γj/se(γj). We have used marginal regression as approximation of the residual variance of the joint regression). This simplification is conservative, and based on our comparison to the individual multivariate results we consider the loss of efficiency acceptable.To quantify significance of the inferred multi-tissue gene-level association, we use the fact that the regression coefficient estimates follow (approximately) a multivariate normal distribution: Cor(T)+ as explained in the previous section, and a χ2-test: k the number of components surviving the SVD pseudoinverse.In practice, we will use the SVD pseudo-inverse 109 public GWAS and GWAS meta-analysis summary statistics data sets were downloaded and analyzed with S-PrediXcan and S-MultiXcan, using the 44 prediction models from GTEx tissues in release version 6p. The list of traits and their Consortium/publication information is available in https://www.wtccc.org.uk/). The individual-level data was analyzed with MultiXcan; and GWAS summary statistics were obtained using PLINK2 [A type 1 Diabetes study from the Wellcome Trust Case-Control Consortium  was acqug PLINK2 , to enabhttp://predictdb.org/ resource. These models were trained using Elastic Net as implemented in R’s package glmnet [α = 0.5, on 44 tissue studies from GTEx’ release version 6p. The underlying GTEx study data was obtained from dbGaP with accesion number phs000424.v6.p1. Please see [http://gene2pheno.org resources. The databases are open to the research community for arbitrary programmatic query.Prediction Models were obtained from e glmnet , with a ease see  for detaS1 Supplementary Note(PDF)Click here for additional data file.S1 Datasettag: trait, gene2pheno.org display name; n_predixcan_significant: Number of Bonferroni-significant PrediXcan results; n_multixcan_significant number of Bonferroni-significant results for MultiXcan; n_predixcan_only number of results only significant in PrediXcan; n_multixcan_only number of results only significant in MultiXcan.Columns are: (TXT)Click here for additional data file.S2 Datasetphenotype: trait, gene2pheno.org display name; gene: Ensembl id; gene_name: HUGO name; pvalue: p-value of the S-MultiXcan association; n_models number of prediction models available for the gene; n_used number of independent components surviving PCA selection; n_samples: number of individuals available.Columns are: (TXT)Click here for additional data file.S3 DatasetPhenotype: trait, gene2pheno.org display name; model: GTEx tissue where the model was trained; gene: Ensembl Id; gene_name: HUGO name; zscore PrediXcan association Z-score, pvalue PrediXcan association p-value; n_samples: number of individuals available.Columns are: (TXT)Click here for additional data file.S4 DatasetColumns are consortium name, study name, gene2pheno.org display name, study sample size, study population, URL of portal where data was downloaded from, link to pubmed entry if available.(TXT)Click here for additional data file.S5 Datasettag: gene2pheno.org display name; consortium: Consortium Name; name: study name; n_spredixcan_significant: Number of Bonferroni-significant S-PrediXcan results; n_smultixcan_significant number of Bonferroni-significant results for MultiXcan; n_spredixcan_only number of results only significant in S-PrediXcan; n_smultixcan_only number of results only significant in S-MultiXcan.Columns are: (TXT)Click here for additional data file.S6 Datasettag: gene2pheno.org display name; consortium: Consortium Name; name: study name; gene: Ensembl id; gene_name: HUGO name; pvalue: p-value of the S-MultiXcan association; n number of S-PrediXcan results available for the gene; n_indep number of independent components surviving SVD; p_i_best best p-value of S-PrediXcan;t_i_best tissue that presented best S-PrediXcan result; p_i_worst worst p-value of S-PrediXcan; t_i_worst tissue that presented worst S-PrediXcan result.Columns are: (TXT)Click here for additional data file.S7 DatasetS7_data.txt. Columns are: consortium: Consortium Name; name: study name; tag: gene2pheno.org display name; gene: Ensembl Id; gene_name: HUGO name; model GTEx tissue where model was trained; zscore S-PrediXcan association Z-score, pvalue S-PrediXcan association p-value.Significant results included in (TXT)Click here for additional data file.S8 Datasettrait: UK Biobank trait name and code; n_flagged: number of significant genes with best individual model p-value >10−4; n_significant: number of Bonferroni-significant genes; percent: percentage of n_flagged to n_significant. MultiXcan significance was computed with condition number 30 and the individual model effects’ significance obtained from PrediXcan.Columns are: (TXT)Click here for additional data file.S9 Datasetgene: gene’s ensemble id; pvalue: significance achieved;method: a label specifying that either MulTiXcan, S-MulTiXcan with naive covariance from predicted expression, or S-MulTiXcan with correction for missing SNPs was ran.Columns are: (TXT)Click here for additional data file.S1 FigWe observe a high degree of predicted expression correlation, in agreement with recent publications on the high degree of mechanism sharing across tissues . This be(TIF)Click here for additional data file.S2 FigThere is a satisfactory agreement between the individual-level and the summary-level versions of MultiXcan in UK Biobank traits.(TIF)Click here for additional data file.S3 FigPanel a compares the MultiXcan p-values to the expected uniform distribution. Most points (genes) lie on the identity line showing no obvious inflation or deflation.Panel b compares the distribution of p-values with and without regularization.Here we use a simulated trait, generated from a standard normal distribution as the phenotype. We perform MultiXcan, regressing the simulated phenotype on predicted expression for 17,435 genes in 1,000 individuals from the UK Biobank. As described in the Methods, we drop principal components of small variation to avoid multi collinearity. We keep the number of principal components so that the condition number of the covariance matrix of the predicted expression across tissues  is below 30.(TIF)Click here for additional data file.S4 FigPanel a compares p-value distributions for traits generated from a single tissue . In this case, PrediXcan using whole blood prediction outperforms MultiXcan as expected from the fact that MultiXcan’s statistic becomes less significant when more explanatory variables of no effect are used; both unregularized and PCA-regularized MulTiXcan are similarly affected.Panel b Uses a trait built from the combination of five brain tissues . As expected, MultiXcan using only the causal tissues performs best. MultiXcan using all tissues displays the second best performance, with the regularized version being slightly better than the unregularized version. PrediXcan (i.e. a single tissue) has the lowest performance.Panel c shows simulations when all tissues are causal (for 1000 random genes); MultiXcan with PCA regularization has slightly better performance than unregularized MultiXcan, and ‘best tissue’ PrediXcan has a significantly lower performance.For each gene, we simulate traits as different combinations of predicted expression from multiple tissues in one thousand individuals from the UK Biobank. We add a noise term from the normal distribution with variance chosen so that 1% of the total variance in the trait is explained by predicted expression. For each trait, we show results from running MultiXcan with no regularization, MultiXcan with regularization (condition number < 30), PrediXcan with ‘best’ single tissue . For a trait with specific causal tissues, we also show MultiXcan using only them.(TIF)Click here for additional data file.S5 FigFor each gene, we simulate traits as different combinations of predicted expression from multiple tissues in one thousand individuals from the UK Biobank. We add a noise term from the normal distribution with variance chosen so that 1% of the total variance in the trait is explained by predicted expression. The top panel shows traits generated from the combination of 5 brain tissues , and the bottom panel a combination of all available tissues. These traits were analyzed through MultiXcan both with PCA regularization and without regularization. The lines correspond to smoothed conditional means, and the gray area displays the confidence intervals. We observe that PCA regularization has increased power over no regularization with larger effect as the number of included tissues increases. When the number of causal tissues is small (“5 Brains”), significance decreases when more tissue models are available, and the regularized and unregularized MultiXcan perform similarly. This is expected since extra uninformative components add noise and reduce power. Conversely, when all tissues are causal, significance increases as we increase the number of included tissues. Regularized MultiXcan achieves higher significance than unregularized MultiXcan.(TIF)Click here for additional data file.S6 FigUsing simulated traits in two scenarios , we display MultiXcan’s significance distribution for different PCA regularization thresholds. In both scenarios the significance remains relatively constant for all thresholds tested. More stringent regularization thresholds achieve slightly higher significance. We consider the threshold of 30 to be a conservative choice.(TIF)Click here for additional data file.S7 FigFDR < 0.05, we observe that the number of significant associations for both PrediXcan and MultiXcan increase significantly, and their difference decreases. Using smaller FDR thresholds increases the difference, and for FDR < 10−4 we observe a similar number of detections as when performing traditional multiple-testing correction at 0.05/n. This is consistent with Bonferroni correction being overly conservative because the hypotheses are not independent.The number of FDR-significant associations are shown for PrediXcan using both a single tissue and all tissues, and MultiXcan. Using (TIF)Click here for additional data file.S8 FigA scatter plot of association significance between MultiXcan and S-MultiXcan is shown for the Wellcome Trust Case-Control Type 1 Diabetes study. The left plot uses the covariance matrix computed from predicted expression in a reference panel (GTEx). The right plot uses predicted expression covariance taking into account missing SNPs (i.e.: using only SNPs in the intersection between reference panel and the GWAS study). We observe that using expression predicted in the reference panel without correction leads to false positives and negatives, as the inferred covariance is inaccurate.(TIF)Click here for additional data file.S1 Table(PDF)Click here for additional data file.


N = 79 366), n3-PUFA (N = 24 925), and MDD  were applied to individual-level data  lifetime MDD diagnoses and circulating 25-OH-D and n3-PUFA) and summary-level data analyses. Shared genetic risk between traits was tested by polygenic risk scores (PRS). Two-sample Mendelian Randomization (2SMR) analyses tested the potential bidirectional causality between traits. In individual-level data analyses, PRS were associated with the phenotype of the same trait , but not with the other phenotypes, suggesting a lack of shared genetic effects. In summary-level data analyses, 2SMR analyses provided no evidence of a causal role on MDD of 25-OH-D (p = 0.50) or n3-PUFA (p = 0.16), or for a causal role of MDD on 25-OH-D (p = 0.25) or n3-PUFA (p = 0.66). Applying genomics tools indicated that shared genetic risk or direct causality between 25-OH-D, n3-PUFA, and MDD is unlikely: unresolved confounding may explain the associations reported in observational studies. These findings represent a cautionary tale for testing supplementation of these compounds in preventing or treating MDD.Trials testing the effect of vitamin D or omega-3 polyunsaturated fatty acid (n3-PUFA) supplementation on major depressive disorder (MDD) reported conflicting findings. These trials were inspired by epidemiological evidence suggesting an inverse association of circulating 25-hydroxyvitamin D (25-OH-D) and n3-PUFA levels with MDD. Observational associations may emerge from unresolved confounding, shared genetic risk, or direct causal relationships. We explored the nature of these associations exploiting data and statistical tools from genomics. Results from genome-wide association studies on 25-OH-D ( The rationale for these interventions was inspired from epidemiological evidence of an inverse association between depression and circulating concentrations of compounds that could be easily supplemented, such as vitamin D and omega-3 polyunsaturated fatty acids (n3-PUFA). Meta-analyses of large observational studies consistently reported cross-sectional, and to a lesser extent longitudinal, associations of 25-hydroxyvitamin-D  and n3-PUFA with depression.6 In >2000 psychiatrically well-characterized participants from the Netherlands Study of Depression and Anxiety (NESDA), we previously confirmed8 lower concentrations of 25-OH-D and n3-PUFA in patients with depression as compared to healthy controls, and inverse associations with symptom severity. Previous clinical studies attempted to translate this observational evidence in interventions: meta-analyses12 pooling results from trials testing the effect of vitamin D and omega-3 supplements on depression reported conflicting findings, with studies showing both small clinical effect and no effect. Conflicting findings have been commonly attributed to heterogeneity and limitations in study design, including small samples, inadequacy of supplementation dose, or lack of blood markers to trace the biological availability of the compounds supplemented. As a consequence, the implementation of larger and methodologically robust trials has been repeatedly advocated to overcome this impasse.In recent years a growing interest has emerged in the potential value of nutritional supplementation in mood disorders. Large-scale randomized controlled trials such as MoodFOOD13 since almost all nutritional variables are correlated with one another and with many social and behavioral factors. In the second scenario, an observational association may also be detected when both traits are epiphenomena stemming from the same etiological root, sharing part of their genetic liability. Previous studies15 reported non-significant genome-wide genetic correlations between MDD and 25-OH-D or n3-PUFA. Nevertheless, these results may have been affected by the different genetic architectures of MDD  and 25-OH-D or n3-PUFA . For instance, genome-wide significant loci for 25-OH-D14 harbor relevant genes for its synthesis and metabolism such as GC , DHCR7 , and CYP2R1 and CYP24A1 . In the third scenario, the most favorable for clinical translation, an observational association is detected because a trait (25-OH-D/n3-PUFA) is effectively causal for another (MDD). For vitamin D, hypotheses based on preclinical data suggested a potential impact of vitamin D on brain structure  and pathophysiological processes relevant for mood.16 Similarly, n3-PUFA may exert important biological effects in the nervous system, modulating membrane fluidity, serotonergic transmission, and inflammatory response.17 Finally, the fourth scenario involves reverse causation, since depression may impact on 25-OH-D and n3-PUFA levels via habits related, for instance, to sunlight exposure or diet.18Before moving to further intervention studies it is however crucial to take a step back and to obtain a clearer knowledge of the exact nature of the observational associations that inspired the supplementation interventions. Observational associations could emerge indeed from different scenarios. In the first of these scenarios, observational associations emerging in nutritional epidemiology may be the product of unresolved confounding, as recently highlighted by Ioannidis,14 n3-PUFA (24 925 samples)19, and MDD (135 458 cases and 344 901 controls).15 We estimated the degree of genetic overlap between traits using both individual-level data (in NESDA) and summary-level data analyses combining GWAS summary statistics. Using summary data, Mendelian randomization analyses explored the potential bidirectional causality between 25-OH-D, n3-PUFA, and MDD.It is not trivial to disentangle from which of the above scenarios an observational correlation arises. Genetic research has made enormous progress over the last years, and now provides unique opportunities to investigate shared risk and causality between traits applying new statistical tools and results from genome-wide association studies (GWAS). In the present study, we leveraged on results from the largest GWAS on 25-OH-D (79 366 samples),8 Briefly, NESDA is an ongoing cohort study into the long-term course and consequences of depressive and anxiety disorders. In 2004–2007, 2981 participants aged 18–65 years were recruited from the community (19%), general practice (54%), and secondary mental health care (27%), and were followed-up during biannual assessments. The research protocol was approved by the ethical committee of participating universities, and all respondents provided written informed consent.Individual-level data analyses were based on 2047 unrelated participants  of European ancestry from the NESDA. Detailed descriptions of the rationale, design, and methods for the study are given elsewhere.20  lifetime diagnosis of MDD was assessed using the Composite Interview Diagnostic Instrument version 2.1, WHO) administered by specially trained research staff at baseline, 2-year and 4-year follow-up. Healthy controls included screened participants without any lifetime psychiatric disorder. Depressive symptoms severity was measured  using the 30-item self-report Inventory of Depressive Symptoms (IDS-SR30).21 For the present analyses, scores across assessments were averaged in order to index the participant’s stable exposure to depressive symptoms.As previously reported,N = 2013) and n3-PUFA (N = 2010) were quantified in blood samples taken at baseline as previously described.8 Serum 25-OH-D was measured using isotope dilution-online solid-phase extraction liquid chromatography-tandem mass spectrometry. n3-PUFA was included in a metabolite panel measured using a high-throughput proton nuclear magnetic resonance (NMR) platform .Concentrations of 25-OH-D .Differences in main characteristics between participants with MDD and healthy controls were analyzed with 14 performed a GWAS on circulating concentrations of 25-OH-D from in 79 366 samples. Summary statistics for n3-PUFA were obtained from the MAGNETIC NMR GWAS19 examining the same metabolomics platform adopted in NESDA on up to 24 925 individuals. The Psychiatric Genomics Consortium (PGC)15 performed an overarching meta-analysis2 of all available GWAS datasets with MDD, including 135 458 cases and 344 901 controls.Summary statistics were obtained from large GWAS of international consortia. The SUNLIGHT consortium14 For n3-PUFAs, we selected seven independent genome-wide significant SNPs by processing discovery GWAS summary statistics  were calculated as the number of risk alleles weighted by their effect sizes from the relative discovery GWAS summary statistics. PRS for 25-OH-D and n3-PUFA were based on genome-wide significant SNPs, since the genetic architecture of these traits may be characterized by few biologically relevant loci, harboring relevant genes for their synthesis and metabolism. For 25-OH-D we selected the six independent SNPs reported in the discovery GWAS.30 was also estimated adjusting for sex and principal components. All analyses were performed in R.Same- and cross-trait associations of the different PRS with 25-OH-D and n3-PUFA concentrations and with MDD diagnosis were estimated using regression models (linear for 25-OH-D and n3-PUFA and binary logistic for MDD) adjusted for sex and 10 ancestry-informative genetic principal components. The proportion of phenotypic variance explained by PRS was additionally estimated  in the discovery GWAS. The new instruments were used in 2SMR analyses robust against pleiotropy-related violations potentially induced by the introduction of a large number of SNPs, including MR-Egger and MR-RAPS (robust adjust profile scores),27 a recently proposed method that could provide unbiased estimates in the presence of many weak variants. All analyses were conducted using the MR-Base R package.28Two-sample Mendelian randomization (2SMR) analysesd −0.25) and n3-PUFA (Cohen’s d −0.15) (Table p = 1.4e − 5) and by ~10% for each SD increase in n3-PUFA levels . Sex-adjusted levels of 25-OH-D and n3-PUFA were associated with depressive symptoms severity . The respective PRS explained 3.5% of trait variance for 25-OH-D level, 0.8% for n3-PUFA levels, and 1.5% of MDD liability variance. Nevertheless, no association was found between PRS of one trait and the other phenotypes, suggesting a lack of shared genetic effects across traits.Table Consistent with these results, analyses focusing on lifetime MDD cases showed that only the PRS for MDD was associated with higher average 4-year severity of depressive symptom, explaining 0.7% of their variance Fig. . PRS for19 instead of selecting only genome-wide significant SNPs. The resulting PRS were less strongly, although significantly, associated with the relative phenotype , but explained a smaller proportion of variance (0.4% for 25-OH-D and 0.3% for n3-PUFA) as compared to the PRS based on the top SNPs, confirming that in the genetic architecture of 25-OH-D and n3-PUFA few biologically relevant loci may play a relative major role.In additional analyses, we recalculated the PRS for 25-OH-D and n3-PUFA using the full polygenic signal emerging from the discovery GWASQ heterogeneity test was significant (p = 1.5e − 3), indicating the presence of at least one outlier SNP potentially exhibiting horizontal pleiotropy. Inspection of the plots  recommended to run an adequately powered MR-Egger analyses, the estimates were substantially consistent with those obtained by other analyses, confirming the lack of potential causal effect of 25-OH-D and n3-PUFA on MDD and vice versa.In the opposite direction, 2SMR analyses examining MDD as exposure consistently showed no significant evidence of a causal role on 25-OH-D and n3-PUFA levels. Additionally, analyses using MR-Egger methodp = 3.7e − 2) and accounted only for 0.4% of its liability. New instruments including larger sets of SNPs were built . Results from MR-Egger and MR-RAPS analyses confirmed the lack of significant causal estimates  = 1.13, 95% CIs = 1.01–1.26, Finally, analyses based on quantitative genetics theory  vs. 25-OH-D or n3-PUFA (few genetic loci with relatively larger effect sizes). In the present analyses, despite the use of polygenic scores for 25-OH-D or n3-PUFA built only on GWAS significant loci, including biologically relevant genes for their synthesis and metabolism, no cross-trait association was found, suggesting that the biological pathways encompassing those genes are not shared with depression.Evidence of a common genetic base consistently shared between traits was not found. Even in data from the NESDA cohort, showing phenotypic associations of 25-OH-D and n3-PUFA levels with MDD diagnosis presence and severity of depressive symptoms,Furthermore, different MR analyses provided no evidence of a direct causal effect of 25-OH-D or n3-PUFA levels on MDD risk, despite the use of strong genetic instruments and the use of 2SMR techniques leveraging data from large GWAS studies. Furthermore, a potential causal effect of MDD on 25-OH-D or n3-PUFA levels was not detected, suggesting that also reverse causation does not explain the phenotypic association between these traits.30 applying MR techniques sustains the lack of causal effect of 25-OH-D on MDD, although the study was limited by the use of data for depression from a smaller GWAS (~one-third of the cases available here) and the lack of testing the potential reverse effect of MDD on 25-OH-D. Furthermore, a first study31 applying MR in a single (low-powered) pregnancy cohort found no evidence of a significant causal effect of n3-PUFA on antenatal, perinatal, and postnatal depression.The present findings are consistent with other emerging evidence. A concomitant analysis33 Interestingly, we showed here that the top genetic variant linked to n3-PUFA levels on the FADS1 gene is also associated with a wide array of traits, among which metabolic , cardiovascular , and psychiatric  features that are, in turn, associated with MDD.18 In previous results from NESDA8, we showed that adjustment for available major sociodemographic, lifestyle, and health confounders substantially reduced the effect size, although remaining statistically significant, of the association between depression and 25-OH-D or n3-PUFA. Nevertheless, not all key confounders could be exhaustively measured, and while some were not available  other very important ones, such as actual exposure to sunlight impacting on 25-OH-D levels, could be only approximated by imperfect proxies.Findings from the present study suggest that the observational associations of vitamin D or n3-PUFA and MDD, commonly reported in epidemiological studies, may be likely explained by unresolved confounding. The relationship of blood levels of 25-OH-D and n3-PUFA with MDD may indeed be confounded by several factors, such as health-related lifestyles , personality traits , and comorbidity .7 that one-third of patients with established psychiatric diagnoses of depressive disorders had 25-OH-D blood concentrations considered insufficient and a risk for musculoskeletal complications. Normalization of vitamin D in depressed patients may therefore be important to prevent related complications and disability, in particular during late life.Despite the fact that the relationship of MDD with 25-OH-D or n2-PUFA may be confounded and not be direct, epidemiological evidence consistently showing that depressed patients have lower blood level of these biological markers still raises an important concern for other health problems. For instance, we previously reported34 on multiple sclerosis suggested the existence of sensitive periods for the protective effect of vitamin D. Nevertheless, epidemiological data show that the association of 25-OH-D and n3-PUFA with MDD risk is consistent across different ranges of ages.8 Furthermore, the lack of statistically significant causal estimates, especially in analyses considering MDD as an exposure, may have resulted from low statistical power due to weak genetic instruments. Nevertheless, sensitivity analyses with polygenic instruments including larger sets of SNPs confirmed the results of original analyses. In the future, results from larger GWAS, as compared to those currently available, could be used to further confirm the present findings. Another major limitation is that both individual- and summary-level data analyses were based on samples of European ancestry and therefore results cannot be generalized to different populations. Finally, a major limitations is represented by the heterogeneity of the MDD diagnoses, which may aggregate different sub-disorders characterized by distinct pathophysiology. We previously demonstrated36 that MDD patients could be divided based on symptom profiles in subgroups characterized by partially distinct polygenic signatures. It could be hypothesized that 25-OH-D and n3-PUFA may have a causal role only in specific subtype of MDD. Nevertheless, large GWAS are needed on potential MDD subtypes, yet unavailable, in order to be used for adequately powered MR analyses.A major strength of the present study is the application of the latest techniques developed in genetic epidemiology, with analyses performed both at the individual-level data, using information from a large and psychiatrically well-characterized cohort, and at the summary-level data, combining GWAS summary statistics from the largest international consortia. An important limitation considering MR results is that genetic variants index the average lifetime exposure to risk factors and consequently could estimate an average lifetime causal effect without being able to identify a specific time in life in which the causal effect may take place. For instance, MR studies37 or that n3-PUFA supplementation had no significant effect on coronary heart disease or major vascular events.38 More recently, results from the large VITAL39 trial testing n3-PUFA supplementation in >25 000 participants showed no significant impact on the primary prevention of cardiovascular diseases or cancer.Depression-related disability is a major health and economic burden for societies that require an urgent answer. Nevertheless, despite depression’s public health relevance, treatment options available nowadays are not optimal in terms of efficacy. Epidemiological evidences are constantly scanned in order to identify potentially actionable target that may substantially improve the prevention and treatment of depression. Recently developed genomic tools could be efficiently employed to examine the nature of observational associations emerging in epidemiology, providing some indications on the most promising associations to be prioritized in subsequent intervention studies. The premature and direct translation of observational associations in nutritional epidemiology into supplementation interventions may encounter indeed disappointing results, as emerging for different health outcomes. For instance, recent large-scale meta-analyses of randomized control studies have shown, for instance, that vitamin D supplementation does not prevent fractures, falls, or improve bone mineral density,1 or VITAL-DEP2 will tell whether this conclusion is correct. During the evaluation of the present manuscript results for the MoodFOOD trial were published,40 confirming that multinutrient supplementation (including vitamin D and omega-3) was not effective in preventing the onset of major depressive episodes.In conclusion, we believe that findings from the present study, in conjunction with previous conflicting evidence from clinical studies, represent a cautionary tale for further research testing the potential therapeutic effect of vitamin D and n3-PUFA supplementation on depression, as the expectations of a direct causal effect of these compounds on mood should be substantially reconsidered. Upcoming results from already established large-scale randomized controlled trials such as MoodFOODSupplemental Materials


Improving statistical power and computational efficiency are always the research foci in genome-wide association studies (GWAS). In this study, we proposed a fast empirical Bayes GWAS method, which is based on the linear mixed model framework. The method is called Fast-EB-LMM in short. Results from simulation studies show that the Fast-EB-LMM has the highest power for quantitative trait nucleotides (QTNs) detection, the highest computational efficiency, and the strongest robustness, as compared with the efficient mixed model association (EMMA) and empirical Bayes (EB). Application to beef cattle population also verified the effectiveness of this method. We believe that Fast-EB-LMM is a valuable additional tool for GWAS.Linear mixed model (LMM) is an efficient method for GWAS. There are numerous forms of LMM-based GWAS methods. However, improving statistical power and computing efficiency have always been the research hotspots of the LMM-based GWAS methods. Here, we proposed a fast empirical Bayes method, which is based on linear mixed models. We call it Fast-EB-LMM in short. The novelty of this method is that it uses a modified kinship matrix accounting for individual relatedness to avoid competition between the locus of interest and its counterpart in the polygene. This property has increased statistical power. We adopted two special algorithms to ease the computational burden: Eigenvalue decomposition and Woodbury matrix identity. Simulation studies showed that Fast-EB-LMM has significantly increased statistical power of marker detection and improved computational efficiency compared with two widely used GWAS methods, EMMA and EB. Real data analyses for two carcass traits in a Chinese Simmental beef cattle population showed that the significant single-nucleotide polymorphisms (SNPs) and candidate genes identified by Fast-EB-LMM are highly consistent with results of previous studies. We therefore believe that the Fast-EB-LMM method is a reliable and efficient method for GWAS. Genome wide association studies (GWAS) have been carried out in humans, plants, livestock, and other species to identify statistical associations between single-nucleotide polymorphisms (SNPs) and complex traits over the past decades . MeanwhiAmong the various GWAS methods, linear mixed models (LMM) are the most efficient in terms of correcting various forms of confounding, such as population structure and cryptic relatedness ,3. SinceIn this study, we present a novel LMM-based GWAS method by using a modified kinship matrix to account for the cryptic relatedness. When a candidate SNP is tested, we exclude all SNPs in a 1Mb window around that SNP from the genome and then use all the remaining SNPs to construct the modified kinship matrix. The size of the window is an adjustable parameter for this method. Meanwhile, we use the eigenvalue decomposition and Woodbury matrix identity to ease the computational burden. Here, we consider the SNP effect as the parameter of interest and use the variance component estimated from the null model as a prior variance. We then fit the model using an empirical Bayes method (EB) . We cally be a vector of phenotypic values for a trait of interest, which is described as:X is a design matrix capturing all non-genetic (fixed) effects, e.g., systematic environmental effects and population structure effects, β is a vector of the fixed effects, kth marker, e.g., 1A1, 1A2 and 2A2, where A1 represents the minor allele while A2 represents the major allele, k on the trait (treated as a fixed effect), ξ is a vector of polygenic (random) effects with an assumed K′ is a modified marker inferred kinship matrix (defined later), and e is a vector of residual errors with an assumed We used the standard LMM as the statistical model for marker scan. The LMM automatically controls potential population structure (Q) effects and the polygenic effect via a marker inferred kinship matrix (K). Therefore, it is also called the Q + K model . Let y bZ is genotype indicator matrix of all SNPs and d  is the average value of the diagonal elements of matrix K. To avoid “proximal contamination”, we constructed a modified kinship matrix K′ by removing the effect of candidate SNP and its flanking SNPs from the original kinship matrix K. The K′ is defined as follows:Zs is a n individuals.The original marker inferred kinship matrix is defined as K to account for the polygenic effect. We then have λ under the null model.K matrix, y is n is the sample size, r is the rank of matrix X, λ, the maximum likelihood estimates of β and σ2 areWe first applied the eigenvalue decomposition algorithm to decompose the β and σ2 in Equation (4) by λ, as shown below:H and various quadratic forms involving H−1. Now, let us rewrite matrix H by:Substituting H is:H is:The determinant of The restricted log likelihood function also involves various quadratic forms in the form of Using eigenvalue decomposition and Woodbury matrix identity, matrix inversion and determinant calculation have been simplified into simple summations, and thus, the computational speed can be substantially improved.λ can be found iteratively using the Newton algorithm:Finally, a numeric solution of λ is estimated, we use the modified kinship matrix K′ accounting for polygenic effects in the scan model. As SNPs are scanned one by one, the K′ matrix is always changing. Meanwhile, we use the mixed model equations (MME) to get the best linear unbiased estimates (BLUE) of the fixed effects, including The scanned model (1) can also be written as:β and γk .p-value is calculated as:p-value also can be converted into a The Wald test is used to test the null hypothesis http://mouse.cs.ucla.edu/emma/. EB is an empirical Bayes approach implemented by the SAS program and we compiled it into an R program  [EMMA is a single marker scanning GWAS method which also considers marker effects as fixed. The R package for EMMA can be downloaded from Austria) . A. thaliana data and the 16th QTLMAS workshop data. Each experiment was analyzed using three methods .To evaluate the statistical power, the false-positive rate and the computational efficiency of Fast-EB-LMM, we conducted two simulation experiments using the A. thaliana dataset were used to do the simulation experiment and the sample size was 199 [K is the kinship matrix. Given The first simulation dataset was downloaded from Wen et al. . As desc was 199 . In the The second simulation dataset was downloaded from the 16th QTLMAS workshop and it consisted of 4100 individuals (G0–G4). Each of the following four generations (G1–G4) consisted of 20 males and 1000 females, generated from the previous generation by randomly mating each male with 51 females. There were no overlapping generations. The genome consisted of 5 chromosomes, each of 100 Mb size and carrying 2000 equally distributed SNPs. Three traits were simulated in order to mimic milk yield, fat yield and fat content with the heritabilities set at 0.35, 0.35, and 0.50. Genetic (co)variances were generated from 50 QTLs with pleiotropic effects and the QTL positions were sampled from among the even SNPs. The GWAS was conducted using 3000 individuals, all females belonging to generations G1–G3, where phenotypic values were measured for three traits (yield deviations). Details of the simulation were described by Usai MG et al. .All animals were treated following the guidelines established by the Council of China Animal Welfare. Experimental protocols were approved by the Science Research Department of the Institute of Animal Sciences, Chinese Academy of Agricultural Sciences (CAAS), Beijing, China .The population consisted of 1301 Chinese Simmental beef cattle that were born in Ulgai, Xilingol League, Inner Mongolia, China , from 2008–2014. After weaning, the cattle were moved to the Beijing Jinweifuren fattening farm for feedlot finishing under the same feeding and management system. Individuals were measured for growth and developmental traits until they were slaughtered at the age of 16–18 months; all animals were slaughtered at roughly the same time. The beef carcasses were then separated, and carcass traits were measured. We chose carcass weight (CW) and bone weight (BW) as the target traits. Descriptive statistics for these two traits are shown in p < 10−6) [All the 1301 cattle were genotyped by Illumina Bovine HD BeadChip . The SNP chips were scanned using iScan and analyzed with the Illumina’s Genome Studio software . The marker dataset consisting of 777,962 SNPs were imputed using BEAGLE v4.1  . Quality < 10−6) . MoreoveStatistical power was used to evaluate the effectiveness of Fast-EB-LMM when compared with the other two methods (EMMA and EB). The statistical power comparison for the two simulation experiments are presented in p-value is smaller than the designated threshold). The ratio of the number of such samples to the total number of replicates (1000) represented the empirical power of this QTN. When only six QTNs were simulated in the first experiment, the statistical powers for QTN detection for Fast-EB-LMM were higher than the other two methods, except QTN2 , where m is the number of SNPs. The statistical power for each QTN was defined as the proportion of samples where the QTN was detected  curve is a plot of the statistical power against the controlled false positive rate. This curve is frequently used to compare different methods for their performance in the detection of significant signals; the higher the curve, the better the method. When six probability levels for significance, from 1E-7–1E-2, were used, the corresponding statistical powers were calculated in the second simulation experiment for the three methods. The results are shown in K′ mentioned above was used to account for polygenic background in the model. The Fast-EB-LMM method was used to analyze the beef cattle data for carcass weight (CW) and bone weight (BW) and the results are listed in C12ORF74, RIMS2 and BT.88981. Four SNPs weresignificantly associated with BW, which included BovineHD0500006528 on BTA 5, BovineHD0600010952 and BovineHD0600010956 on BTA 6, and BovineHD1400017455 on BTA 14. We also found three candidate genes for BW, which were C12ORF74, LCORL, and RIMS2. The GWAS results for the two carcass traits are also shown in We first performed a principal component analysis (PCA) using all SNPs and selected the first five principal components as covariates (fixed effects) to control the population structure effects. The fixed effects in the model also included the effects of gender, year, farm, age at slaughter, fattening days and body weights. The modified kinship matrix Linear mixed model (LMM) has become a widely accepted method in genome wide association studies (GWAS) due to its excellent performance in controlling population structure and cryptic relatedness. However, statistical power and computational efficiency are two major concerns in the LMM-based GWAS methods. Recent study has shown that the SNP being tested and its flanking SNPs should be excluded from the GRM (kinship matrix). If these SNPs are included in the kinship matrix, they will compete with the main effects (as fixed effects) and thus lower the powers of QTN detection. This phenomenon is called “proximal contamination” and is one of the important inducements leading to loss of statistical power for the LMM-based GWAS methods ,15.Motivated by the above concerns, several studies have been carried out to address the “proximal contamination” problem. Lippert et al. proposed a Fast-LMM approach using FaST-LMM sampled SNPs uniformly across the genome to construct the GRM and keep the number of SNPs used to create the GRM to be less than the sample size . MeanwhiIn this study, we present a new LMM-based method, called Fast-EB-LMM. We used a strategy that is different from FaST-LMM and GCTA-LOCO by excluding the candidate SNP and its flanking SNPs when constructing the GRM. When SNPs are scanned one at a time, the GRM is always changing. Furthermore, the eigenvalue decomposition and Woodbury matrix identity algorithm were used to ease the computational burden. To validate this method, we performed simulation studies and real beef cattle data analyses.In the first simulation experiment, there were three different structures of the phenotypic data based on the genotype data of an existing population and three methods  were used. In the first scenario of the simulation with six simulated QTNs, the Fast-EB-LMM had highest statistical power for all the six QTNs simulated, followed by EMMA and EB. In one occasion (QTN2), the EMMA method was slightly more powerful than the Fast-EB-LMM method A. In theLCORL and RIMS2 genes found in this study also appeared in Miao et al.’s study [We also applied Fast-EB-LMM to a beef cattle dataset to check for its consistency with previous findings. Two carcass trait including carcass weight and bone weight were analyzed and the results are listed in ’s study . ClearlyIn summary, we presented a novel GWAS method that is based on LMM: Fast-EB-LMM. In order to prevent loss of power caused by “proximal contamination”, a modified kinship matrix was used to account for the individual relatedness in the method. The innovation of this modified kinship matrix is that it is constructed by remaining SNPs after exclusion of the candidate SNP and its flanking SNPs. Meanwhile, we used eigenvalue decomposition and Woodbury matrix identity to ease the computational burden. Results from simulation studies show that the Fast-EB-LMM method has the highest power for QTN detection, the highest computational efficiency, and the strongest robustness, compared with the EMMA and EB methods. The beef cattle data analyses also validated the efficiency of the new method. We therefore believe that the Fast-EB-LMM method is a valuable additional tool for GWAS.


Littorina saxatilis present a model system to test these predictions. We used genome‐wide association (GWA) analysis of geometric morphometric shell traits associated with microgeographic divergence between the two L. saxatilis ecotypes within three separate sampling sites. A total of 477 snails that had individual geometric morphometric data and individual genotypes at 4,066 single nucleotide polymorphisms (SNPs) were analyzed using GWA methods that corrected for population structure among the three sites. This approach allowed dissection of the genomic architecture of shell shape divergence between ecotypes across a wide geographic range, spanning two glacial lineages. GWA revealed 216 quantitative trait loci (QTL) with shell size or shape differences between ecotypes, with most loci explaining a small proportion of phenotypic variation. We found that QTL were evenly distributed across 17 linkage groups, and exhibited elevated interchromosomal linkage, suggesting a genome‐wide response to divergent selection on shell shape between the two ecotypes. Shell shape trait‐associated loci showed partial overlap with previously identified outlier loci under divergent selection between the two ecotypes, supporting the hypothesis of diversifying selection on these genomic regions. These results suggest that divergence in shell shape between the crab‐adapted and wave‐adapted ecotypes is produced predominantly by a polygenic genomic architecture with positive linkage disequilibrium among loci of small effect.Gene flow between diverging populations experiencing dissimilar ecological conditions can theoretically constrain adaptive evolution. To minimize the effect of gene flow, alleles underlying traits essential for local adaptation are predicted to be located in linked genome regions with reduced recombination. Local reduction in gene flow caused by selection is expected to produce elevated divergence in these regions. The highly divergent crab‐adapted and wave‐adapted ecotypes of the marine snail  Samples were collected from three sites on the North and Northwest coasts of Spain  and the corresponding coordinates of each of its landmarks  containing pairwise identity‐by‐state (IBS) values estimated using all of the 2405 putatively neutral SNPs that were successfully genotyped for that pair of individuals  as covariates in a linear model predicting phenotype (Yi) with intercept μ and random error term, e (PC model), similar to the model used in Price et al.  is estimated using the polygenic function, calculated using the linear mixed model:We also conducted GWA using a two‐step linear mixed model using the genomic kinship matrix (K) to estimate the proportion of phenotypic similarity between individuals that can be attributed to their extent of shared common ancestry. This model follows the family‐based association model described in Chen and Abecasis  and Aulcmmscore to approximate the fit of a second model. This model fits calculated residual values representing trait variation not explained by genome‐wide relatedness between individuals, or included covariates, as the dependent variable, using the following equation:A score test for each genotyped SNP was then performed using mmscore model using structured association  and were then combined using a weighted average with the weight being the squared inverse of the standard error of β.χ2 distribution  calculation from one‐degree‐of‐freedom χ2 test statistics obtained from the score test using the following formula:n is the number of individuals genotyped per tested SNP.Score test values distribute approximately as a 2.6vegan 2.4‐5  and divergence  of shape variation was explained by the first two relative warps which reflected differences in shell aperture and spire height. The first relative warp accounted for 43.16% (RW1) and the second accounted for 19.63% (RW2) of shell shape variation. We found five relative warps that individually explained greater than 5% of shell shape variation and cumulatively explained 84.30% of variation between individuals, of which RW3, RW4, and RW5 each respectively explained 8.79, 7.8, and 5.31% of total shape variation. Visualizing changes in shell shape with changing relative warp values using deformation grids revealed increases in RW1 values corresponding with increase in shell aperture size and reduction in spire height . A small proportion of associations were common across pairwise comparisons between models: 15 QTL (0.108) were shared between the PCA and K:SA models, and four QTL (0.023) were shared between the RDA and K:SA models. However, only one QTL was shared between the RDA and PC models, and no QTL were shared among all three comparisons. The different methods of population structure correction used in each method likely account for these differences, as we found that individuals grouped by RW1 scores also clustered by PC3, whereas population differences were explained by PC1 and PC2  for each trait Table . Howevern = 59 SNPs), and landmarks Y3 (n = 10 SNPs) and X7 (n = 5 SNPs), supporting a role for aperture shape differences and shell shape facilitating divergence between ecotypes . In contrast to the PC model, we observed separation of ecotypes followed by glacial lineages on the first two RDA axes when constrained by shell morphology, indicating shell morphology variation plays a role in genomic differentiation between ecotypes across sites  within sampling locations and within ecotypes within locations .Alignment of RAD loci containing QTL to the 3.4FST values of QTL were elevated relative to the genome‐wide distribution for all loci  polygenic architecture of differentiated shell shape and size traits, (b) elevated linkage of QTL associated with shell variation despite a genome‐wide distribution, and (c) partial overlap of divergent FST outlier regions associated with shape variation detected during genome‐wide association, suggesting increased rates of divergence at QTL underlying shell variation. Together these results from an empirical study of an emerging model system  in our dataset may have given us low coverage in inversion regions. Coupled with the level of fragmentation of the currently available reference genome, this limitation may explain why we did not find physical linkage among our QTL. The type II inversion hypothesis that balancing selection could maintain heterokaryotypes  were found in association with landmark locations related to shell aperture differences, consistent with the identification of variation in these traits between crab and wave ecotypes in the present study, as well as in past comparisons (Butlin et al., L. saxatilis is 1.32 Gb (Vitturi, Libertini, Panozzo, & Mezzapelle, Despite the discovery of loci of both large and small effect, sample size and genomic sampling method may both play a role in biasing estimates of allele number and effect size uncovered in this study. Our small sample sizes in both individuals and markers relative to massive human GWA studies may have prevented detecting the small contributions made to shell variation by many alleles dispersed across the genome (Boyle, Yang, & Pritchard, 5L. saxatilis crab and wave ecotypes across their range in northwest Spain. We used multiple models to discover the SNP markers associated with shell traits reflecting adaptations to differences in crab predation and wave action between regions of the intertidal zone. We detected polygenic trait architectures that overlapped regions of high genomic divergence as identified previously using FST outlier tests. We also identify elevated statistical linkage of identified genomic regions associated with shell traits despite no evidence of physical linkage, indicating that coordination of polygenic loci underlies ecotype formation in L. saxatilis. Future research on this system would benefit from the availability of a less‐fragmented genome that would improve the identification physical locations of QTL. It would also be interesting to associate gene expression data (e.g., Rivas et al., Our work here, using a genome‐wide SNP dataset and individual shell traits, paired with sampling across two glacial lineages, allowed us to uncover the genomic architecture of shell divergence in None declared.T.K. designed the project, conducted laboratory work and statistical analysis, and wrote the manuscript. E.G.B. helped design the project, supervised the laboratory work, assisted with interpretation of the statistical analysis, and helped write and edit the manuscript.


Daucus carota), sampled from two distinct regions in Canada, which is 25-fold larger than datasets available in existing trait databases. Citizen-collected trait data supported analyses addressing theoretical and applied questions related to (i) intraspecific trait dimensionality, (ii) the extent and drivers of ITV, and (iii) the sampling intensity needed to derive accurate trait values. Citizen science is a viable means to enhance functional trait data coverage across terrestrial ecosystems, and in doing so, can directly support theoretical and applied trait-based analyses of plants.Trait-based ecology is greatly informed by large datasets for the analyses of inter- and intraspecific trait variation (ITV) in plants. This is especially true in trait-based agricultural research where crop ITV is high, yet crop trait data remains limited. Based on farmer-led collections, we developed and evaluated the first citizen science plant trait initiative. Here we generated a dataset of eight leaf traits for a commercially important crop species ( At the same time, trait-based research has afforded a more nuanced assessment of how variation in the characteristics of individual plants within a species – now widely referred to as intraspecific trait variability (ITV) – influence ecosystem processes at finer scales3. Very recently the application of a functional trait-based approach to agroecological research, where ITV represents an especially large proportion of total plant trait variation in an ecosystem, has gained considerable attention7, with specific functional traits correlated to key agroecosystems processes, namely leaf morphological and chemical traits that are strongly coupled with crop resource acquisition strategies5.At largest-scale of integration, analyses of trait variation among plant species have tangibly advanced our capacity to describe, predict, and manage linkages between plant diversity and ecosystem functioning8; (ii) how the functional biology of crops differs from “wild” plant species e.g.10; (iii) farm- or landscape-scale biogeochemical cycling e.g.11; (iv) crop yield e.g.14; and (v) crop responses to climate change e.g.15. Based on this growing literature, it is apparent that crop functional trait data – particularly datasets that comprehensively quantify ITV within a given crop species or genotype – greatly inform our applied and theoretical understanding of plant functional biology.Research has shown that quantifying the breadth of ITV in crops has critical implications for better understanding multiple aspects of agroecosystem structure and function including: (i) the environmental impacts of artificial selection e.g.16, crop trait data has largely been missing from large functional trait databases17. Therefore to date, crop ITV collection and analyses have largely depended on “new” samples derived from common garden experiments e.g.18 or field-based observational studies e.g.21. Although these studies have been instructive, larger and robust trait data collection initiatives are commonly pointed to being a main limitation to additional trait-based agroecological research. For example, larger and geographically broad trait data collection initiatives are critical for predicting crop production and food security under local and global environmental change6.Yet while a few meta-analyses do exist22, and citizen science initiatives (among other examples) have greatly informed plant breeding programs24 and soil amendment management treatments25, with recent work assessing the accuracy of farmer-generated data and developing approaches to ensure robust datasets26. However to date, citizen science initiatives have not been employed in any studies of functional trait ecology. While there has been some success in engaging citizen science to amass herbarium specimens27 or in plant phenological monitoring28, to our knowledge there remain no efforts to merge farmer research participation with trait data collections led by ecologists.One possible avenue for enhancing our understanding of crop traits is through citizen science initiatives. The agricultural community has always maintained strong ties to the research communityDaucus carota L.) - to (ii) consolidate morphological and chemical leaf functional traits among a range of crop varieties and farms, and in doing so, (iii) demonstrate how citizen-led trait collections can be used to inform empirical assessments of the drivers of ITV within crops. Finally, we sought to (iv) critically evaluate the efficacy of crop trait collections by determining the optimal number of farms and observations needed to accurately estimate functional trait values for, and the trait space occupied by, crop species.Here we developed and evaluated a novel citizen science approach to functional trait data collection. We specifically sought to: (i) launch a citizen science protocol across Canada focused on famer-led plant trait sample collection. We then sought to use these samples - which in our study are from five varietes of carrot  and nitrogen (N) were least variable across our dataset (CV = 4.3–17.3), in comparison to leaf morphological traits including leaf area and leaf dry mass which were most variable among samples (CV = 59.7–69.6); specific leaf area (SLA), and petiole length and diameter were intermediary in terms of overall trait variability , but did not differ systematically across varieties .Across our dataset carrot leaves differentiated across two primary and significant Principal Components Analysis (PCA) axes that accounted for 32.8 and 27.4% of the variation in traits, respectively Fig. . The firHobs) was estimated as 2.4 with centroid values for log-transformed leaf area, leaf N, and SLA of 3.6, 1.4, and 5.3, respectively ) values are likely to be overestimated at smaller sample sizes of individual leaves  ranged from 2.6–2.7; the largest differences were primarily associated with differences in the mean leaf area centroid calculated for the observed vs. randomized datasets (Table Hrand(l) ranged between 2.0–2.4 with s.d. values that become progressively smaller (i.e. where s.d. surrounding Hrand(l) = 0.5 at nl = 20, to a s.d. of 0.2 where nl = 135), and largely encompass the observed hypervolume value  ranged from 0.8–1.7  values and associated s.d. values closely approximated Hobs. Estimated s.d. values surrounding Hrand(f) values are notably lower where nf ≤ 7 where the range of s.d. is 0.15–0.3, as compared to where nf ≥ 6 where s.d. surrounding Hrand(f) range between 0.39–0.63  a given crop species . Ultimately, the lack of data derived from field-based collections remains a limitation to comprehensive, global-scale, and applied trait-based analyses in agroecosystems5.While trait database initiatives dedicated to wild plant species have been instrumental in enhancing our understanding of natural plant communities worldwide, there remains a lack of consolidated functional trait information for crops33; and (ii) the intentional and unintentional consequences of domestication through analyses of crop trait dimensionality34. From an applied perspective, such data has immediate implications for: (iii) refining process-based models of crop performance under environmental change and with agroecological practices35; (iv) better understanding genotype-by-environment responses of crops to environmental conditions36; and (v) informing farmer selection of crop genotypes in participatory plant breeding programs37.We developed and tested farmer-led sampling programs as a viable means to amass crop trait data Fig. . While c24, while the use of crop functional traits has been shown to play a central role in farmer decision-making on sustainable management practices38. Merging the fields of agronomy and ecology with robust methodologies continues to enhance management prescriptions for low input place-based agriculture.Maximizing dominant agricultural production models may increase short-term yields in particular geographies and climatic zones, but at the expense of numerous global environmental problems that comprise the ecological foundations of food systems. Efforts to integrate ecological principles to achieve the optimal management of agroecosystems are critical to offset such production models. Recently, on-farm participatory variety evaluations using crowd-sourced citizen science have been shown to markedly increase the ability to select crops to adapt to climate change under heterogeneous environmentsDaucus carota L.); all agreed  were collected six weeks post-emergence from all five varieties, resulting in n = 15 samples per farm and n = 150 individual observations in total. This collection represented a nearly 25-fold increase in trait data for this species compared to existing trait databases.The recent establishment of a national variety trialing program on organic farms across Canada provided a unique opportunity to tackle pressing questions regarding crop functional traits in agroecosystems. This network of organic farms participates in long-term vegetable crop variety trials. In the summer of 2018, we contacted 10 farms from four provinces in Canada to engage farmers in a leaf trait collection for five varieties of carrot  using a scanner and analyzed images using ImageJ software39. A low-force micrometer  was used to measure petiole diameter (mm) and a ruler to measure petiole length (cm), which are key traits related to harvestability40. Following these measurements, each leaf was dried at 60 °C to constant mass and weighed, and specific leaf area  was then calculated. Dried leaves were then ground into a fine powder using a ball mill  and analyzed for total carbon (C) and nitrogen (N) concentrations (% mass basis) using a CN 628 elemental analyzer .Farmers were instructed to pat leaf samples dry, and store flat in the labelled envelopes provided, each of which contained one small desiccant pack to maintain sample integrity during shipping . Samples were then sent directly to the University of Toronto Scarborough, Canada (approximately two days in transit). Each leaf was immediately analyzed for eight morphological and chemical traits. During shipping 14 leaves were damaged, leaving a final dataset of n = 136). This was done by first assessing whether traits were normally or log-normally distributed, using a maximum-likelihood-based approach implemented with the ‘fitdistrplus’ R package41. Where traits were best described by a normal distribution (as per maximum likelihood scores), descriptive statistics were calculated means and standard deviations (SD); where traits were best described by log-normal distributions we calculated medians and median absolute deviation (MAD) values14. For all traits we also calculated coefficients of variation (CV) as an overall estimate of ITV.To provide an initial assessment of trait variability across our sampling program, we calculated descriptive statistics for all individual traits across the entire dataset (n = 136), and coupling this with a permutation multivariate analysis of variance (PerMANOVA). Both of these tests were implemented with the ‘vegan’ R package42, with the PerMANOVA being based on Euclidean distances and n = 10,000 permutations used. We also tested the statistical significance of individual PCA axes using the ‘testdim’ function in the ‘ade4’ R package43.Our next analysis was designed to assess if leaf traits varied as a function of variety, farm, and/or a variety-by-farm interaction term. This was done by first performing a principal components analysis (PCA) on all eight traits from our entire dataset . And finally, these traits loaded strongly onto the first two PCA axes with leaf area being the trait associated most strongly with PCA axis 1 and leaf N/ SLA being the strongest associated with PCA axis 2  using the ‘hypervolume_gaussian’ function in the ‘hypervolume’ R package48. All traits were log-transformed prior to analysis, and hypervolumes were estimated using a 0.05 quantile threshold such that each hypervolume contained 95% of the total probability distribution of n = 397 random samples per points. Kernel bandwidths for each trait were estimated independently using the ‘estimate_bandwidth’ function with values of 0.249, 0.105, and 0.071 for leaf area, SLA, and leaf N, respectively.Based on these three traits, hypervolume estimates were then calculated for the whole dataset . To do so, we generated datasets of n = 5, 10, 15,…, 135 randomly selected leaves through bootstrapping without replacement. At each sample size of leaves, 100 randomized datasets were generated and Hrand values were calculated using the exact methodologies described above (inclusive of the generation of new kernel density estimates for each trait dimension). We then derived a mean and standard deviation for Hrand values at each sample size for comparison to our Hobs estimate. Second, we generated randomized datasets that contained trait information from n = 1, 2, 3, …, 9 randomly selected farms within our dataset. For these analyses, the number of randomly constructed datasets at each farm sample size was also 100 (though note this procedure necessarily entailed bootstrapping with replacement where n ≤ 4 farms). Mean Hrand and standard deviations surrounding it were then calculated from all randomized dataset at a given number of farms.We then used a randomization procedure, broadly based on the approach taken by Lamanna Hrand changes as a function of the number of leaves and/or farms sampled by fitting non-linear models. Specifically, changes in Hrand at a given sample size were modeled as a function of the number of leaves randomly sampled (nl), following a negative exponential model fit of the form:Hrand(l) represents the mean 3-dimensional trait hypervolume for a randomized dataset of a given sample size of leaves (nl), a represents the maximum mean H estimated for the lowest sample sizes (i.e. n = 5 leaves), and b represents the rate of reduction in Hrand(l) estimates as sample sizes increase.Based on a preliminary visual assessment, we then chose to describe how Hrand in response to the number of farms sample (nf) was describe using an asymptotic model of the form:Hrand(f) represents the mean 3-dimensional trait hypervolume for a randomized dataset of a given sample size of farms (nf), a represents the asymptote (i.e. maximum hypervolume), b represents the difference between the y-intercept and the asymptote, and c represents the log of the rate constant. Models and parameters in Eqs n = 100 datasets of n = 2, 3, …, 15 individual leaves, for each of the n = 10 individual farms separately. Equation a, b, and c are as in Eq. Hrand(lf) represents the mean 3-dimensional trait hypervolume for a randomized dataset for a given sample size of leaves within a given farm (nlf).Changes in mentclass2pt{minimSupplementary Information_Revised


Mendelian randomization indicated no robust evidence of a causal effect of genetic liability for schizophrenia on number of children : −0.003 to 0.009, p = 0.39) or age at first birth . We find some evidence of a positive effect of genetic liability for schizophrenia on number of sexual partners . These results suggest that increased genetic liability for schizophrenia does not confer a fitness advantage but does increase mating success.Schizophrenia is a debilitating and heritable mental disorder associated with lower reproductive success. However, the prevalence of schizophrenia is stable over populations and time, resulting in an evolutionary puzzle: how is schizophrenia maintained in the population, given its apparent fitness costs? One possibility is that increased genetic liability for schizophrenia, in the absence of the disorder itself, may confer some reproductive advantage. We assessed the correlation and causal effect of genetic liability for schizophrenia with number of children, age at first birth and number of sexual partners using data from the Psychiatric Genomics Consortium and UK Biobank. Linkage disequilibrium score regression showed little evidence of genetic correlation between genetic liability for schizophrenia and number of children ( The preAnother popular theory is that stabilizing selection operates, where the optimum fitness level for a trait is approximately at the mean of the trait and fitness declines along a normal distribution on either side of this optimum ,6,11,12.Relatives of people with schizophrenia are assumed to have an intermediate level of genetic liability for the highly heritable disorder . StudiesRecent developments in genetic epidemiology mean that it is now possible to investigate the effects of genetic liability for schizophrenia in the wider population. A genome-wide association study (GWAS) identified 128 independent genetic variants from 108 loci associated with schizophrenia that explained approximately 3.4% of the observed variation in schizophrenia risk . These vWe applied a range of methods with roots in genetic epidemiology to test part of the cliff-edge hypothesis. For our main analysis, we examine whether increasing genetic liability for schizophrenia increases reproductive success in a largely post-reproductive population-based sample which is not selected on schizophrenia status, and, therefore, includes very few cases. This linear increase is predicted for part of cliff-edge fitness where a reproductive advantage among healthy individuals with higher genetic liability for the disorder compensates for lower reproductive success of those with the disorder itself. Additionally, given suggestions from family studies that there may be a fitness decline of healthy individuals with high genetic liability for the disorder, we conducted sensitivity analyses to investigate a possible nonlinear relationship where at very high levels of genetic score for schizophrenia liability, there is decreased reproductive success in the absence of schizophrenia itself ,7,17.For our main analysis, we calculated genetic correlations using linkage disequilibrium (LD) score regression between genetic liability for schizophrenia and reproductive success, measured as number of children, age at first birth and number of sexual partners. Furthermore, we used genetic variants associated with schizophrenia within a Mendelian randomization (MR) framework to estimate the causal effect of genetic liability for schizophrenia on these measures of reproductive success. MR uses single-nucleotide polymorphisms (SNPs), which are assigned at conception and are mostly independent from other variants or environments. MR therefore overcomes some limitations of observational studies previously used to investigate this evolutionary paradox, by reducing bias from confounding and reverse causation . Finally2.2.1.p < 5 × 10−8) from the Psychiatric Genomics Consortium GWAS (N = 35 123 cases and 109 657 controls) [We used independent SNPs associated with schizophrenia (ontrols) . The 128ontrols) . The finp < 5 × 10−8) from a recent GWAS by the Social Science Genetic Association Consortium [For educational attainment, we used SNPs associated with educational attainment (nsortium . As the 2.2.The exposure-associated SNPs described above were extracted from UK Biobank to derive SNP-outcome associations for our outcome data. Extraction was done using PLINK (v. 2.00) and best guess algorithms for determining alleles (full genotyping information below).2.2.1.UK Biobank is a population-based health research resource consisting of approximately 500 000 people, aged between 38 and 73 years, who were recruited between the years 2006 and 2010 from across the UK . Particu2.2.2.N = 488 377). A total of 49 979 individuals were genotyped using the UK BiLEVE array and 438 398 using the UK Biobank axiom array. Pre-imputation QC, phasing and imputation are described elsewhere [N = 814). We restricted the sample to individuals of white British ancestry who self-report as ‘White British’ and who have very similar ancestral backgrounds according to the principal component analysis (PCA) (N = 409 703), as described by Bycroft et al. [N = 79 448). Additionally, two individuals were removed due to them relating to a very large number (greater than 200) of individuals. QC protocol is described elsewhere [The full data release contains the cohort of successfully genotyped samples  between schizophrenia liability or predicted educational attainment and number of children, age at first birth and number of sexual partners. Genome-wide associations were conducted for these outcomes using linear regression, implemented in PLINK v. 2.00 through the Medical Research Council Integrative Epidemiology Unit GWAS pipeline [We used LD score regression ,41 to capipeline . In thisQ providing a measure of any overdispersion .In MR analyses, data were harmonized to ensure that the effect of the SNP on the exposure and the SNP on the outcome corresponded to the same allele. The increasing allele for schizophrenia liability and educational attainment was used. Associations for exposure SNPs and all outcome measures were then calculated in R, fitting the same covariates as listed above. Effect sizes for number of children, age at first birth and number of sexual partners analysis are listed in electronic supplementary material, tables S1 and S2. SNP-exposure and SNP-outcome data were combined using an inverse variance weighted (IVW) approach which is analogous to a weighted regression of SNP-outcome coefficients on SNP-exposure coefficients with the intercept constrained to zero , and furF-statistics (mean of the squared SNP-exposure association divided by the squared s.e. for SNP-outcome association) to indicate the strength of instrument, and I2GX statistics to assess the suitability of MR-Egger (above 0.9 is desired) [The IVW effect estimate will only be consistent if all genetic variants in the analysis are valid. Weighted median, mode-based estimator and MR-Egger regression are complementary approaches that can be used to investigate the impact of invalid instruments on our effect estimates. The weighted median estimates a consistent effect estimate if at least 50% of the instruments are valid . The moddesired) . AnalysiMR results were multiplied by 0.693 to represent the causal estimate per doubling in odds of schizophrenia risk . For chix-axis. As sensitivity analysis to assess if there was any decline in reproductive success within our sample at very high levels of genetic liability, we conducted a series of linear regressions in Stata (v. MP 15.1) between this genetic score for schizophrenia liability and our outcomes, systematically removing cumulative centiles from the maximum. This analysis included adjustment for the top 10 principal components and was repeated after removing the few schizophrenia cases in our sample. Similarly, to further investigate a possible peak in reproductive success at high genetic liability for schizophrenia, we conducted quadratic regression analysis of the genetic score for schizophrenia liability and our outcomes . Lastly, we repeated quadratic analysis separately for each sex.As an illustration of shape of the schizophrenia liability–reproductive success relationship, we created an additive unweighted genetic score for schizophrenia liability in UK Biobank. The score was created in R (v. 3.2.0), with missing SNP data replaced with the mean value for that SNP across individuals. We then divided this score into quintiles and deciles and plotted the mean number of children, age at first birth and number of sexual partners across these categories of the genetic score. We additionally plotted these relationships with reproductive success on the Analysis scripts are available on GitHub .3.In our sample, from UK Biobank, there were more females than males, a majority had children and a minority had college or university degree qualifications . The mean age was 56.9 years (s.d.: 8.0) and the mean years of education was 13.3 (s.d.: 4.4). For our outcomes, the mean number of children was 1.8 (s.d.: 1.2), mean number of sexual partners was 5.8 (s.d.: 8.6) and the mean age at first birth was 25.4 years (s.d.: 4.5).3.1.rg = −0.35, p = 8.57 × 10−41) and a strong positive genetic correlation between educational attainment-associated variants and age at first birth   . The numF-statistic (strength of instrument) of 33.23, with above 10 indicating acceptable levels of relative bias (less than 10%) [p = 3.63 × 10−10 per year increase in educational attainment) and a positive effect on age at first birth  per year increase in educational attainment . Results for all educational attainment analysis with genotype array included as a covariate in our outcome summary statistics are presented in electronic supplementary material, tables S4 and S5.Educational attainment variants showed a mean han 10%) ,51. We atainment . We also3.2.rg) between schizophrenia liability and number of children , age at first birth  and number of sexual partners   . The numF-statistic for schizophrenia genetic liability was 35.15. There was little evidence that higher genetic liability for schizophrenia increased number of children  or decreased age at first birth  (p = 5.30 × 10−10) (N = 207) with no clear change in results. Results for these analyses with genotype array included as a covariate in our outcome summary statistics are presented in electronic supplementary material, tables S4 and S5.The mean  = 0.82) . We furt = 0.82) . We foun× 10−10) . A posit× 10−10) . We repex-axis . A series of regressions between the genetic score and age at first birth, systematically removing cumulative centiles from the maximum, suggests that the relationship is strongest at intermediate levels. It appeared that estimates became slightly stronger in the analysis with number of children, although there was little statistical support (N = 207), which did not alter these results . Regression of the genetic score for schizophrenia liability and number of children showed no clear evidence, also when including a quadratic term for genetic liability for schizophrenia and when stratified by sex (Our sensitivity analysis investigating a possible nonlinear relationship is presented in figures 1 support . This and by sex . This qu4.Our results do not indicate a genetic correlation between genetic liability for schizophrenia and reproductive success using LD score regression, or a linear causal effect on number of children and age at first birth using MR techniques. This is inconsistent with cliff-edge fitness maintaining schizophrenia in the population, which would predict an increase in fitness with increased genetic liability in the general population. These results support previous research suggesting no strong evidence of a relationship between genetic liability for schizophrenia and number of offspring ,25. We fAlso consistent with previous research, we found no clear evidence for a linear association between genetic liability of schizophrenia and age at first birth  and weakCliff-edge fitness suggests that schizophrenia prevalence is sustained because the negative reproductive effects in those with an underlying genetic liability and the disorder are offset by a reproductive advantage to those who have an underlying genetic liability but do not develop the disorder . We therThis leaves us with two further theories for how schizophrenia prevalence is maintained. One is that as schizophrenia is a highly heterogeneous disorder and exhibits a highly polygenic architecture, and effects of genetic variants are individually too weak to be under negative selection ,8,10. OuA key strength of this study is the use of MR, which can provide stronger evidence of causality than observational studies ,61. We sThe present study highlights the continued importance of investigating differential fertility and contributes to understanding the maintenance of schizophrenia, and educational attainment, in the population ,20,69. E