HLA-DQA1/DRB1 and LPA). We also validate previous suggestions that APOE, CHRNA3/5, CDKN2A/B, SH2B3 and FOXO3A influence longevity. Next we show that giving up smoking, educational attainment, openness to new experience and high-density lipoprotein (HDL) cholesterol levels are most positively genetically correlated with lifespan while susceptibility to coronary artery disease (CAD), cigarettes smoked per day, lung cancer, insulin resistance and body fat are most negatively correlated. We suggest that the effect of education on lifespan is principally mediated through smoking while the effect of obesity appears to act via CAD. Using instrumental variables, we suggest that an increase of one body mass index unit reduces lifespan by 7 months while 1 year of education adds 11 months to expected lifespan.Genomic analysis of longevity offers the potential to illuminate the biology of human aging. Here, using genome-wide association meta-analysis of 606,059 parents’ survival, we discover two regions associated with longevity ( Variability in human longevity is genetically influenced. Using genetic data of parental lifespan, the authors identify associations at HLA-DQA/DRB1 and LPA and find that genetic variants that increase educational attainment have a positive effect on lifespan whereas increasing BMI negatively affects lifespan. Here we focus on a narrower question—the extent and nature of its genetic basis and how this inter-relates with that of health and disease traits. In what follows, we shall use longevity as an umbrella term. We shall also more specifically refer to lifespan (the duration of life) and long-livedness . Up to 25% of the variability in human lifespan has been estimated to be genetic6 and some technical appeal (focussing on the truly remarkable), but such studies can be limited and costly in their recruitment. We recently showed that the extension of the kin-cohort method7 to parental lifespans, beyond age 40, of genotyped subjects could be used to detect genetic associations with lifespan with some power in genomically British participants in UK Biobank (UKB)4. Here we extend that approach in a genome-wide association meta-analysis (GWAMA) to discovery across UKB European- and African-ancestry populations and 24 further population studies (LifeGen), mainly from Europe, Australia and North America, to search for further genetic variants influencing longevity. We then use those GWAMA results to measure genetic correlations and carry out Mendelian randomisation (MR) between other traits and lifespan seeking to elucidate the underlying effects of disease and socio-economic traits on longevity, in a framework less hampered by confounding and reverse causality than observational epidemiology.Prospective genomic studies of lifespan have been hampered by the fact that subject participation is often only recent, allowing insufficient follow-up time for a well-powered analysis of participant survival. On the other hand, case-control studies of long-livedness have had successIn total, 606,059 parental lifespans were available for analysis, of which 334,974 were already complete Table .Table 1SHLA-DQA1/DRB1, LPA, CHRNA3/5 and APOE, in which the lead SNPs rs34831921, rs55730499, rs8042849 and rs429358, respectively, associate with survival at genome-wide significance (p < 5 × 10−8) (Table HLA-DQA1/DRB1) and rs55730499 (LPA), both showed statistically significant, directionally consistent, evidence of association at the proxy SNPs in strongest LD in the largest  publicly available set of GWAS summary statistics for extreme long-livedness (CHARGE-EU 90+)6, with p < 0.0035 for both SNPs. As our GWAS results were of the observed effect of offspring genotype on parent phenotype and the actual effect of carrying an allele for the individual concerned (rather than their parent) is twice that observed in a parent-offspring kin-cohort study4, all reported effect sizes (and their standard errors) throughout this manuscript have been doubled to give the estimated effect size in the allele carriers themselves. The hazard ratios for one copy of the minor alleles were 0.942 and 1.074 for rs34831921 (HLA-DQA1/DRB1) and rs5573049 (LPA), respectively, corresponding to an increase/decrease in lifespan of ~ 0.6/0.7 years for a carrier of one additional copy of the minor allele.In our GWAS of 586,626 European parental lifespans, we find four regions 8) Table . The two6 summary statistics using Z-scores and equal weights for each study, reflecting their similar statistical power. We found strengthened signals, substantially at APOE  and slightly in the LPA region . No improvement of statistical significance was observed in the HLA-DQA1/DRB1 region, where there were no SNPs in strong LD with the lead LifeGen SNP, nor was there an increase in significance near CHRNA3/5. However, in this meta-analysis one further region near AKAP7/EPB41L2 on chromosome 6 just reached genome-wide significance , with a consistent direction of association in all 9 cohorts studied. No association was observed at this SNP in African mothers, or fathers and mothers of European ancestry (HR (SE) = 0.97 (0.038), 1.01 (0.007) and 1.00 (0.008), p = 0.51, 0.21 and 0.77, respectively  significant in the analysis of both parents combined. However, we found one GW significant signal , in an intergenic region of chromosome 2 associating with lifespan for fathers (HR (SE) for G allele = 1.22 (0.0354), ely Fig. .p < 0.0003, two-sided test) of association; CDKN2A/B, SH2B3 and FOXO3A ) previous study. However, for CDKN2A/B, the 95% CI for our estimate is entirely below that from the more robust of the two studies considered. No statistically significant  evidence of association was found for the other 10 genes. In all cases (with the possible exceptions of ABO and 5q33) our estimates of the odds ratio were close to 1 and our 95% CI did not include previous estimates, suggesting, at least for the remaining 8 SNPs , that our non-replication did not arise solely from lack of power.We next attempted to validate 13 candidate genes identified in previous longevity studies. In our study, only three of these genes showed statistically significant, directionally consistent evidence  of the lead SNP was 1.07 (.01) for men and 1.13 (.01) for women, whereas for CHRNA3/5 it was 1.07 (.01) for men and 1.04 (.01) for women  for ages 40−75 and 1.14 (.01) for ages 75+, whereas for CHRNA3/5 they were 1.08 (.01) for 40−75 and 1.03 (.01) for age 75+  while, conversely, the FOXO3 locus had greater effect at older ages(75+).Consistent with our previous reportsmen Fig. . Convers75+ Fig. , with si75+ Fig. , althougcts Fig. , b, with8, an on-line tool which searches 88 complex trait GWAMAs and three GWAS catalogues. For the SNP in the LPA region, associations were found with blood lipids and coronary traits. For the SNP in the HLA region, we found associations with rheumatoid arthritis and Crohn’s disease. For the CHRNA3/5 region, we found associations with traits which associate with smoking behaviour: nicotine dependence, lung cancer, chronic obstructive pulmonary disease and schizophrenia. Finally, for the APOE region, we saw associations with Alzheimer’s disease, age-related macular degeneration, blood lipids, adiposity, cardiac and cognitive ageing traits (Supplementary Data We tested the four SNPs identified in the discovery phase Table  for asso9: 46 showed meaningful genetic correlations (rg) with lifespan . The most strongly correlated with mortality were coronary artery disease (CAD) and cigarettes smoked per day, rg (SE) = 0.66 (0.05) and 0.58 (0.11), respectively. Those most negatively correlated were years of schooling and former vs. current smoker, rg (SE) = −0.47 (0.05) and −0.64 (0.09), respectively  and the strong correlations between others, we clustered the 46 traits which showed a significant and meaningful rg into nine clusters. Positive genetic correlations with mortality for the clusters ranged from 0.68 (smoking) to 0.17 (rheumatoid arthritis and breast cancer), whilst negative correlations varied from −0.50 (education) to −0.15 (age at menarche);  and C-reactive protein (CHARGE-CRP12) made available to us to perform two-sample Mendelian randomisation to investigate causal influences on lifespan. Of more than 90 tested phenotypes, seven risk factors  and six disease susceptibilities  significantly associated with mortality , followed by fasting insulin, body mass index and CAD, while years of education showed by far the most beneficial effect (4.7 years), when comparing the estimated effect of moving from the first to the third quartile of the phenotype distribution. Similarly, we estimate moving from the bottom to the top of the interquartile phenotypic range of CRP increases lifespan by 0.7 years.CHRNA3/5 and APOE and discovered two further associations, at LPA and HLA-DQA1/DRB1, with replication of the further associations in a long-livedness study. We found no evidence of our lead SNPs at the CHRNA3/5, LPA and HLA-DQA1/DRB1 loci associating with traits other than smoking behaviour, cardio-metabolism and rheumatoid arthritis, respectively, while finding more pleiotropy at APOE. We also robustly replicated previous work suggesting associations with longevity at CDKN2A/B, SH2B3/ATXN2 and FOXO3A. We found no evidence of association between lifespan and the other 10 loci previously found to suggestively associate with lifespan, despite apparent power to do so. We showed strong negative genetic correlation between CAD, smoking and type 2 diabetes and lifespan, while education and openness to experience were positively genetically correlated. Using MR, we found that moving from the 25th to 75th percentile of cigarettes per day, systolic blood pressure, fasting insulin and BMI causally reduced lifespan by 5.3, 5.2, 4.1 and 3.8 years, respectively, and similarly moving from the 25th to 75th percentile of educational attainment causally extended lifespan by 4.7 years. Strikingly, we also found that increased CRP increases lifespan, as a causal effect, the reverse of its correlation.We replicated previous findings of genome-wide significant associations between longevity and variants at 13. Variation in LPA has been extensively studied14, and found to influence cardiovascular disease15 and type 2 diabetes16. A close proxy to our lead SNP  has been strongly associated with decreased Lp(a) size and increased Lp(a) plasma concentration and is one of the strongest predictors of coronary heart disease risk with an odds ratio of 1.7 per allele, consistent across populations17, all suggesting that rs55730499 affects mortality by increasing Lp(a) levels and susceptibility to cardiovascular events.Lipoprotein(a) is a spherical lipoprotein carrying cholesterol and triglycerides in the bloodstreamHLA-DQA1/DRB1. MHC class II genes encode components of the antigen-presenting apparatus and are the most polymorphic region of the human genome. Genes within the MHC have previously been associated with many autoimmune conditions and other traits, including psoriasis18, rheumatoid arthritis19, multiple sclerosis20 and T1D21. In a recent informed GWAS of longevity, Fortney et al.22 identified, but failed to replicate, two variants close to the HLA-DRA locus22.The large major histocompatibility complex (MHC) encompasses FOXO3A locus has been repeatedly reported by other studies23 as associating with extreme longevity. Variant rs3800231, which exhibits the strongest association in our data, seems to exert its beneficial effect on people aged above 75 but may have a neutral, or deleterious effect at younger ages, supporting the consensus that FOXO3A plays a putative role in extreme longevity and general health into old age. This contrasts our findings for the CHRNA3/5, LPA, HLA-DQA1/DRB1 loci, where effects appear to be specific to disease susceptibility, rather than general ageing. The CDKN2A/B locus at 9p21 has previously been associated with CAD24, while the missense allele rs3184504-T we identified within the SH2B3/ATXN2 locus has been previously associated with increased risk for type 1 diabetes25, diastolic blood pressure26 and several autoimmune conditions29.The ABO and 5q33.3/EBF1 may be due to a combination of limited power in our study, despite its size, and a degree of winner’s curse in previous findings. However, for CAMK4, C3orf21, GRIK2, IL6, RGS7, CADM2, MINPP1 and ANKRD20A9P, our findings appear inconsistent with the previous work, suggesting those findings were either false positive associations, or differences in effects are due to the differences between the types of lives studied by us and other studies.The failure to replicate previous findings for lifespan increases at The use of different cohorts from a diverse range of countries with common shared ancestry is common in GWAMA and potentially gives rise to heterogeneity in effect sizes, whatever the trait under consideration. However, a study of lifespan is perhaps particularly susceptible to such effects, as mean lifespans vary by cohort  has a beneficial correlation with lifespan (rg = 0.24), is in line with a recent meta-analysis which has shown a life-lengthening effect of subjective well-being on lifespan32. Similarly, depression has been shown to increase mortality, and is one of the strongest quality-adjusted life expectancy losses, twice as much as better-studied risk factors such as smoking, heart disease, stroke and diabetes33. Our results thus reinforce the importance of public policy focusing not only on physical health but also on general well-being in order to increase life expectancy and quality34.The lack of observed genetic correlation between mortality and schizophrenia is perhaps surprising, given the known increased risk of early death due to schizophrenia35. If true, this pattern is somewhat analogous to findings for the N-terminal fragment of pro-BNP, which is a protective molecule, but observationally positively associates with cardiac failure and adverse cardiovascular outcomes36. Our finding that a reduction in one BMI unit leads to a 7-month extension of life expectancy, appears broadly consistent with those recently published by the Global BMI Mortality Collaboration, where great effort was made to exclude confounding and reverse causality37. We also found each year longer spent in education translates into approximately a year longer lifespan. When compared using the interquartile distance, risk factors generally exhibited stronger effects on mortality than disease susceptibility. Although both CAD and cigarette smoking show a very similar genetic correlation with lifespan, the measured effect of smoking is twice as large as that of CAD, perhaps because smoking influences mortality through multiple pathways.In general the results of the MR analyses appear consistent with those of the LD score regression estimates. This might be expected since the main difference is that MR compares two phenotypes using just a small number of SNPs which the underlying GWAS were powered to find, and LD score regression uses the whole genome. Nevertheless, as a result the latter may indicate a shared heritable confounding factor, rather than a causal effect, which appears to be the case for our CRP results, as the measured effect of CRP on lifespan is in the opposite direction to the genetic correlation. CRP’s effects per se are not well understood, but our results lead us to speculate it may have a protective function, rising in the presence of disease, rather than causing it, despite observational associations with disease and consequent attempts to develop a drug to reduce itOur results show that longevity is partly determined by the predisposition to common diseases and, to an even greater extent, by modifiable risk factors. The genetic architecture of lifespan appears complex and diverse and there appears to be no single genetic elixir of long life.As is conventional in GWAMA, analysis was carried out locally at each cohort and then meta-analysed centrally. Initial phenotype and genotype quality control were carried out in accordance with local standards, with variants imputed to 1000 Genomes . Cohort characteristics, including genotyping and imputation methods and summary statistics of the parental lives analysed are described in Supplementary Datas We conducted an association test between parental survival  and offspring genotype. To do so, survival traits were transformed into residuals, permitting analysis as quantitative traits. To facilitate standardisation across the GWAS consortium, residuals for GWAS were calculated in accordance with the analysis plan set out below using a common R protocol distributed to all groups. These residual traits were then tested for association in a GWAS over the imputed SNP panel.38,e ratio associated with X  and Z1,…., Zk the other variables fitted i.e., subject sex, and the first 10 PCs of genetic structure along with each studies’ usual further covariates, such as batch or assessment centre.Parents who died below the age of 40 were excluded. Analysis was thus of survivorship beyond the age of 40. Association testing was conducted under the following Cox Proportional Hazards ModelX). Martingale residuals39 are,δi and τi are the parent status  and age of the ith individual, Z1,…., Zk. Where the allele count, X, has an effect, 39.Rather than fit the full model in one step, we calculated Martingale residuals of the Cox model (excluding 39. This transformed trait was then tested for association with each SNP separately under the following (additive) model,X the non-reference allele count of the marker, with e being normally distributed and independent. Despite this efficient approach, runtimes in UK Biobank were still potentially onerous, so RegScan 0.240 was used there as it is ideally suited for multiple, residualised traits in large data sets.However, although these residuals are associated proportionately with the hazard ratio and thus permit statistical hypothesis testing, their relationship with the hazard ratio depends of the (parent) population structure, in particular the proportion dead. The Martingale residuals were therefore scaled up by 1/(proportion dead) separately for each parent gender, to give a residual trait with a 1:1 correspondence with the hazard ratioFor cohorts with significant relatedness, all but one subject amongst relatives with coefficient of kinship > 5% were excluded, to create a  unrelated population, in preference to conventional  mixed modelling among family based studies. This was done because the genomic relationship matrix among offspring does not precisely reflect the genetic covariance among parental traits. As an example, consider the offspring of two brothers: the correlation between genetic values of the father trait is 0.5, but of the mother trait is 0, while the Genetic Relationship Matrix (GRM) entry would be 0.25 in both cases. The GRM thus does not fully express covariance among (parent) trait values across subjects, so the smaller unrelated population was used. Exceptions to this were for CILENTO, ERF, GeneSTAR, MICROS and OGP, where it was impractical to exclude relatives, and mixed modelling was used.41, but sometimes more conservative, as follows. UK Biobank data were read into METAL42 first, standardising all subsequent input alleles to that imputation. SNPs with mismatching alleles in other GWAS were rejected. SNPs were removed from a cohort’s GWAS if the minor allele frequency for that cohort was < 0.01. As all studies had in excess of 500 lives, this meant that minor allele count exceeded 10 Alleles with an info score (observed variance in dosage/expected under HWE) < 0.3 were excluded. Each GWAS was checked for systematic errors in allele coding/frequencies and test statistics for SNPs passing QC. After QC, SNP counts were 13,689,868 for European fathers, 13,643,373 for European mothers, 20,305,364, for African fathers and 20,296,065 for African mothers.After preparing GWAS results locally, cohorts submitted these to the central team for meta-analysis. The central meta-analysis was carried out in METAL, with QC following that of Easy-QCAfrican and European ancestries were meta-analysed separately, as were the results for each parental sex, using inverse variance meta-analysis in METAL. Double genomic control was applied. The median λ for 78 GWAS was 0.998 and the maximum was 1.048, suggesting good control for stratification. The highest λ was for UK Biobank—genomically British, the most powered study. After the first level of genomic control, results were meta-analysed by inverse variance, while keeping continental ancestry separate and parental sex separate. The λ applied was 1.034, 1.023, 1.027, 1.028, for European fathers, mothers, African fathers, mothers, respectively. Finally, within continent across parent inverse variance meta-analysis was applied. As expected, due to environmental correlation among spouses, there was some inflation: λ of 1.107 and 1.094, for Europeans and Africans, respectively, giving two final combined meta-analyses (African and European) for both parents combined, subject to double genomic control.4. All reported effect sizes throughout this manuscript were therefore doubled to give the estimated effect size in the allele carriers themselves. The effect of hazard ratios on lifespan was calculated from survival curves of the Cox model by each cohort. The weighted average effect of hazard ratio on lifespan across all cohorts and both sexes was that a 1% reduction in hazard extended expected lifespan by 0.108 years. To avoid an undue sense of precision, and in accordance with an actuarial rule of thumb, where applicable, hazard ratios were converted to estimated effects on lifespan using a 10% HR: −1 year of lifespan ratio.These GWAS results were of the observed effect of offspring genotype on parent phenotype. The actual effect of carrying an allele for the individual concerned (rather than their parent) is twice that observed in a parent-offspring kin-cohort study6. None of the lead SNPs were present in that dataset, so proxy SNPs in strongest LD were chosen using LDlink43, with European populations selected. The SNP showing the highest r2 with each LifeGen lead SNP was extracted from the CHARGE GWAS. The Rotterdam study was part of both GWAMAs, but the trait measured was in different people. In our study, we considered the lifespan of parents, whereas the long-livedness analysis was in the offspring. The LifeGen and CHARGE-EU 90+ GWAMAs were then meta-analysed using p-values and direction of effect  with equal weights placed on each GWAMA, using METAL. The choice of equal weights was made, rather than weights reflecting sample size, because (i) the CHARGE extreme case-control approach is more powerful per sample than parent lifespan Cox modelling, and comparison of n is not straightforward, (ii) the Z-test statistics for rs4420638 (the most significant SNP overlapping in both studies) were similar: 9.4 and 9.5 for CHARGE and LifeGen, respectively, for the same n, indicating similar overall power.Genome-wide significant European lead SNPs at each QTL were then looked up in the largest independent GWAS of lifespan with published summary statistics, for survivorship beyond age 90 vs. younger controls (CHARGE-EU 90+)8 to search for other trait associations with our lead SNPs. Pheno Scanner settings were: rsid, Catalogue = GWAS, p-value cutoff = .001, proxies = 1000 G, r2 = 0.6.We used PhenoScanner3, Deelen et al.6, Emanuele et al.44, Flachsbart et al.45, Fortney et al.22, Malovini et al.46, Newman et al.47, Willcox et al.23 and Zeng et al.48. Variants with MAF above 1% in 1000 Genomes, from these papers were taken forward, if they exhibited genome-wide significance (p < 5 × 10−8) or they had suggestive associations that were replicated. In aggregate, 18 variants in 13 gene regions were identified, of which four were genome-wide significant in the original study, while nine were suggestive . We therefore proceeded as follows. The most significant longevity association, APOE, was used to estimate the relationship between OR observed in case-control studies and HR observed by us, as follows. For APOE variant rs4420638(G) loge OR for survival beyond age 90 has been estimated elsewhere as −0.336 and our observed HR was 0.07, giving an empirical factor of −4.7 to estimate ORs for case-control extreme long-livedness from lifespan HRs. This factor was applied to our observed HR for all the candidate SNPs in Fig. p-value and effect estimate. We inferred standard errors, assuming that a two-sided test with a normally distributed estimator had been used.Whilst comparable http://ldsc.broadinstitute.org/)49. As the parent phenotype-offspring genotype GWAS halves the genetic effects4, both the genetic covariance and sqrt(heritability) estimates are halved, resulting in 1:1 estimation of the offspring-offspring genetic correlation(rg), from parental GWAS-offspring GWAS based estimates of rg. LDHub estimates rg between one test GWAS and ~ 200 traits from metabolomics to common diseases such as cardiovascular disease and lung cancer, using LD score regression9. Given their redundancy and number, the metabolomic traits were excluded from the analysis. We added diastolic and systolic blood pressure50, C-reactive protein (CRP)12 and breast cancer11 to the traits present in LDhub49, using GWAMA summary statistics for these studies provided to us. Each of these was run through the LDHub server in order to estimate the genetic correlations with the other traits while the genetic correlation with lifespan was estimated by using a local run of LD score regression. The Benjamini and Hochberg multiple correction test procedure was applied to determine the statistical significance of the resulting genetic correlations.We estimated genetic correlations between mortality and other traits using our both parents European ancestry GWAMA summary statistics and the LDHub web portal (We then defined three categories of traits: (a) Meaningfully genetically correlated to mortality if estimated rg >  = 0.15 and FDR < 0.05; (b) Not meaningfully genetically correlated to mortality if 95% CI for rg ⊂ ; and (c) Otherwise, insufficient evidence.51. Other than to define the initial list, mortality was not included in the clustering analysis. At the same time, some highly correlated traits, which the clustering algorithm sought to combine, appeared to capture distinct clinical aspects and these were therefore kept separate. In particular, we split an education/smoking/rheumatoid arthritis group into three; we separated CAD out from the cluster for dyslipidaemia and waist-hip ratio (DL/WHR) and we separated breast cancer from age at menarche. The 12 clusters (and their constituent traits) were as follows: obesity , smoking , DL/WHR , triglycerides and waist-to-hip ratio), kidney , type 2 diabetes , blood pressure (systolic and diastolic), happiness (depressive symptoms and subjective well-being), breast cancer, CAD, educational attainment , rheumatoid arthritis and age at menarche.After subsetting to only those meaningfully genetically correlated to mortality, we estimated all genetic correlations among those traits; some pairs of traits showed very high correlations. For example, many were genetically correlated to BMI and obesity, we thus used the ICLUST clustering algorithm to cluster the most similar ones. The number of clusters was chosen empirically, by visual inspection. The ICLUST algorithm from the psych R package clusters items hierarchically based on the loading of the items on the factors from factor analysis. Two clusters are then merged together only if by their joining their internal consistency increases. As rotation matrix for the factor analysis we used “promax” which is a high efficiency algorithm which allows correlation between the different factorsThe resulting correlation matrix amongst clusters was used to estimate partial genetic correlations between the clusters using the matrix inversion method as implemented in the corpcor R package (ISBN: 978-0-470-74366-9).p < 5 × 10−8. We then performed LD clumping52  in order to prune all non-independent SNPS. Some traits had no SNPs below the significance threshold and were thus excluded.As a further step to identify which traits affect, rather than merely correlate with, mortality and to determine how much they shorten or lengthen lifespan, we performed a multiple step two-sample Mendelian randomisation (MR) study using summary statistics. We first identified a list of 96 candidate phenotypes selected amongst diseases and disease-associated risk factors (Supplementary Table MR was performed using the inverse variance method utilising each of the selected traits as exposures and mortality as outcome. Where the instrument for the trait was composed of a single SNP, we used the Wald ratio instead. We then defined as candidate traits all the phenotypes with a Benjamini and Hochberg FDR < 0.05. We also verified the absence of directional pleiotropy using MR Egger regression, but none of the candidate traits showed statistically significant evidence of pleiotropy once corrected for multiple testing (Supplementary Table Several traits associated with BMI and obesity were extremely redundant: we thus removed obesity class 1, obesity class 2, obesity class 3, overweight, extreme body mass index, hip circumference, and childhood obesity. Finally, myocardial infarction was removed since 20 of the 22 SNPs composing its instrument were also in the CAD instrument. Supplementary Table th and 75th phenotypic percentiles. For normally distributed traits this difference corresponds to 1.345 phenotypic standard deviations. For binary traits, the variance of the logistic distribution is constant, this difference instead corresponds to an interquartile distance of 2.2 times the beta coefficient estimated from the logistic regression. Thus, the difference in years between the two considered quantiles was estimated to be:For each significant trait, we estimated the difference in expected years of life between the 2512.All relevant data that support the findings of this study are available from the corresponding author upon request or from UK Biobank, LDHub, MRbase, BCAC, CHARGE-CRPDescription of additional supplementary filesSupplementary InformationSupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Data 5Supplementary Data 6Peer Review File


MixP is an implementation that uses the Pareto principle to perform genomic prediction. This study was designed to develop two new computing strategies: one strategy for nonMCMC-based MixP (FMixP), and the other one for MCMC-based MixP (MMixP). The difference is that MMixP can estimate variances of SNP effects and the probability that a SNP has a large variance, but FMixP cannot. Simulated data from an international workshop and real data on large yellow croaker were used as the materials for the study. Four Bayesian methods, BayesA, BayesCπ, MMixP and FMixP, were used to compare the predictive results. The results show that BayesCπ, MMixP and FMixP perform better than BayesA for the simulated data, but all methods have very similar predictive abilities for the large yellow croaker. However, FMixP is computationally significantly faster than the MCMC-based methods. Our research may have a potential for the future applications in genomic prediction. Recently, various implementations have been proposed for genomic prediction, such as genomic best linear unbiased prediction (GBLUP)2, ridge-regression BLUP (RRBLUP), BayesA, BayesB1, BayesCπ3, BayesLASSO5, BayesSSVS6, fast Bayesian methods8, MixP9, among others. GBLUP and RRBLUP assume a constant variance for all SNP loci, which may be an imprecise assumption if a trait is affected by a small number of QTL loci1. Bayesian methods propose more flexible prior assumptions for SNP effects (or variances). Generally, the prior distributions of Bayesian methods assume that there are large variances in some SNP loci and small or even zero variances at other loci, which seems to be more realistic. The implementations of Bayesian methods, such as BayesA, B, Cπ and LASSO, are mainly based on Markov chain Monte Carlo (MCMC) algorithms, requiring much more computation time to estimate SNP effects. To increase the computational speed, researchers have suggested some fast Bayesian methods, such as fast BayesB7 and emBayesB8. Yu and Meuwissen proposed another fast Bayesian method using the Pareto principle to perform genomic prediction9.The advent of next generation sequencing technology has accelerated the development of the theory behind quantitative molecular genetics approaches, such as quantitative trait loci (QTL) mapping, genome-wide association (GWA) studies and genomic selection. Genomic selection was first proposed by Meuwissen 10. This principle states that approximately 20% of the population possesses 80% of the wealth in a country. Similar theories have been further applied in various fields, such as in genomic prediction by Yu and Meuwissen9, resulting in the method termed MixP. The prior distribution of MixP is a mixture of two normal distributions, which assumes that x% of the SNPs cause (100 − x)% of the genetic variance, so the remaining (100 − x)% of SNPs decide the remaining x% of genetic variance. Here we assume γ = x%, and (1 − γ) = (100 − x)%. The large and small variances are proposed as follows9:σ12 and σ22 represent the large and small variance of a SNP effect, respectively; Vg is the total additive genetic variance; M is the number of SNPs; and γ ≤ 0.5. The prior for MixP assumes that all SNPs have effects, but each SNP has only two possible variances: σ12 or σ22. This is similar but not completely identical to the assumptions found in two other Bayesian methods (BayesA and BayesCπ). In BayesA, the prior also assumes that all SNPs have effects but each SNP has its own variance. The variances of the SNP effects in BayesA follow an inverse-chi-squared distribution11. The prior for BayesCπ assumes that SNPs with non-zero effects have a common variance, which is similar to the assumption of MixP, which assumes that “large” SNPs have a common variance (σ12). However, SNP effects with small variance may be shrunk to zero in BayesCπ.The Pareto principle was proposed by the economist Vilfredo Pareto at the beginning of the 20th century9. However, a multivariate normal density and an inverse matrix are included in the derivation, increasing the difficulty in understanding the derivation. In the nonMCMC-based MixP, the γ is given but not estimated, such that the optimal value of γ should be searched using a cross-validation. However, the parameter γ can be estimated using the MCMC algorithm. For the sake of convenience in distinguishing different algorithms, the MixP not based on the MCMC algorithm is termed fast MixP (FMixP), and the MixP based on the MCMC algorithm is termed MCMC-based MixP (MMixP) here.MixP is also a fast Bayesian method that is not based on a MCMC algorithmIn this study, we developed two new computing strategies for FMixP and MMixP, respectively. The first strategy used a product of univariate densities instead of the multivariate normal density to estimate SNP effects for FMixP; the second strategy attempted to use the MCMC algorithm to derive the solutions for MMixP. In addition, the strategies were used to analyse the results on simulated data from an international workshop and real data on large yellow croaker, and compared the predictive abilities with estimations by BayesA and BayesCπ.γ = 0.07). The accuracy of BayesA is lower than that of BayesCπ, MMixP, and FMixP (γ = 0.07), but higher than FMixP when γ = 0.5. BayesCπ and MMixP yield comparatively accurate estimates for π and γ, respectively. As there are 48 QTLs simulated in the genome, the true value of π (or γ) is 48/5726 ≈ 0.0084, which is very close to the values estimated by BayesCπ and MMixP. The γ estimates in the Gibbs sampling cycles are shown in Fig. The predictive results of various Bayesian methods for the simulated data are shown in Table 9 and our FMixP, and found that the two derivations could yield the same prediction accuracies. Graphs of the correlation and regression coefficients of TBV on GEBV  and b, respectively) against γ for FMixP are presented in Fig. γ. Overall, FMixP yields the highest accuracy when the value of γ is close to 0.07, but this value is higher than the true value (0.0084). The distributions of SNP effects estimated by FMixP and MMixP are shown in Fig. We compared the predictive results between MixP introduced by Yu and Meuwissenγ (the probability of a SNP with a large variance) estimated by MMixP is much higher than that estimated in the simulated data, indicating that there may be many QTLs affecting the phenotypes. The results of FMixP show that predictive abilities are optimized when the probability of a SNP with a large variance in specific traits is 0.02 or 0.05. However, these optimal points are not obvious because the predictive abilities are still very close to the best results when γ = 0.5, which is not consistent with the results from the simulated data. Figure γ for FMixP for various traits. It shows that the value of γ barely affects the predictive ability as long as γ is larger than 0.05 or even 0.02. The values of γ estimated by MMixP are 0.28, 0.32, 0.27 and 0.31 for the traits body weight, body length, body height and length/height, respectively.Table γ = 0.5 is longer than that for FMixP with γ = 0.05 in the simulated data, but this difference is not obvious in the real data. We also compared the computation time between MixP introduced by Yu and Meuwissen and our FMixP, and the results showed that the time of their MixP was approximately 20~25% longer than that of our FMixP.Table γ = 0.5, the results of FMixP are equivalent to those of GBLUP or RRBLUP, an observation which was also mentioned by Yu and Meuwissen9. Hence, the predictive result of FMixP when γ = 0.5 is the same as that of GBLUP in Shepherd et al.8, in which the same simulated data was used. Therefore, we actually compared the results of five methods  in this study. The results show that the ranking of the predictive results among the different methods is not consistent between the simulated and real data. In the simulated data, the ranking according to predictive accuracy is: BayesCπ ≈ MMixP ≈ FMixP (γ = 0.07) > BayesA > GBLUP. However, all of the methods yield almost the same result within a given trait in real data from large yellow croaker. A reasonable explanation may be that there is a small number of QTLs in the simulated data but many more QTLs in the real data. There are two reasons that support this speculation: (i) The simulated results of Yu and Meuwissen showed that accuracy was not sensitive to γ when the number of QTL loci was large, but FMixP with γ < 0.5 performed better than GBLUP if there was a small number of QTLs9. The results shown in Figs γ estimated by MMixP in simulated data are much lower than that estimated in the real data, indicating there may be many QTL loci affecting the phenotypes in large yellow croaker. Another explanation is that when the LD between markers is not strong, the accuracy may be due to the relationships captured by markers13. In this case, the GBLUP and various Bayesian methods may yield similar predictive results.In this study, we compared the predictive abilities among BayesA, BayesCπ, MMixP and FMixP. When π. A smaller π means more SNPs have zero effects and thus do not need to be sampled from the posterior normal distribution. MMixP needs more computation time than BayesA and BayesCπ, because there are more variables that need to be sampled in MMixP. For example, the SNP effect with variance equalling zero is not sampled in BayesCπ. However, all SNP effects need to be sampled in each Gibbs sampling cycle in MMixP, because each SNP may have a large or small variance. The computational speed for FMixP with γ = 0.05 is faster than for FMixP with γ = 0.5 in the simulated data. The possible reason for this is that the number of QTLs is very small in the simulated data. FMixP with γ = 0.05 is closer to the real QTL distribution, so that FMixP with γ = 0.05 has a faster convergence speed. This also suggests that there may be more QTL loci in the real data, because there is no obvious difference in computation time for FMixP with γ = 0.05 or 0.5.In addition to the predictive accuracy, computational speed is another important aspect in genomic prediction. This study shows that FMixP is significantly faster than the MCMC-based Bayesian methods. The main reason for this difference is that FMixP is not based on MCMC algorithms which are sampling processes and require many cycles to obtain a precise solution. It shows that the computation time for BayesCπ is slightly longer than BayesA in the real data, but slightly shorter in the simulated data. This is because the computational speed of BayesCπ is based on the value of 9, we used a simpler derivation to obtain the solutions in FMixP. The advantage of FMixP is the extremely fast computational speed. However, the probability of a SNP having a large variance (represented as γ) and variances of SNP effects cannot be estimated by this implementation. Instead, using the MCMC algorithm can estimate the γ and various variances, but the computational speed is significantly slower than FMixP. The two strategies may provide some references to others who want to perform genomic prediction in the future.This study proposed two new computing strategies: one strategy for FMixP and the other one for MMixP. Compared with the derivation of Yu and MeuwissenThis study and all experimental protocols were approved by the Animal Care and Use Committee of the Fisheries College of Jimei University . All methods were performed in accordance with approved guidelines.The linear model for genomic prediction was as follows:y is a vector of phenotypic records, X is the design matrix for fixed effects, and u is a vector of fixed effects. In the simulated data, X = (1 1 … 1)′ and u is overall mean, whereas in the real data, the fixed effects were the sexual effects, Xi = (1 0) for male and (0 1) for female. B is the matrix of SNP genotypes , g is a vector of SNP effects, and e is a vector of residual effects, where e ~ N. Genotypic codes were standardised using the formula: pj is the frequency of allele ‘a’ at locus j.where 9. According to the prior distribution for SNP variance, the prior for SNP effect gj can be written as a mixture of normal distributions:gj is the effect of SNP j.In this study, the prior distribution was the same as that described by Yu and Meuwissen7 to estimate the SNP effects. This algorithm estimates E(g|y) for each SNP effect in turn, where the current effects of the other SNPs are assumed to be known values. For example, if E(gj|y−j) is estimated, the current effects of all other SNPs are used to calculate the y−j, i.e.,Bk is a vector from the kth column of B. The expectation of SNP effect, E(gj|y−j), is estimated by a Bayesian model9:f is a multivariate normal density. Evaluating this multivariate density will be computationally intense because it involves calculating the determinant and inverse of variance-covariance matrix for the data y−j. However, the f, where Y = (Bj′Bj)−1Bj′y−j and σ2 = (Bj′Bj)−1et al.7). Unlike the derivation of Yu and Meuwissen9, we did not calculate the multivariate likelihood but simplified the derivation using f to replace f algorithmequation , the numThe first term in formula  can be dgj in the normal distribution with a mean Yσ12/(σ12 + σ2) and variance σ2σ12/(σ12 + σ2), so this term equals Yσ12/(σ12 + σ2). Thus, the first term of formula −1X′(y−Bĝ). We judged the convergence of solutions at the tth iteration according to the formula (Gt−Gt−1)′(Gt−Gt−1)/(Gt′Gt) < 10−8, where G = ( û′ ĝ′)′.The fixed effects are estimated in each iteration by the formula: û  = , respectively.The scale parameter δj and gj are sampled from their joint conditional distribution, because the sampling strategy of gj is dependent on the value of δj. The joint conditional distribution can be written as:g−j and δ−j represent the vectors of SNP effects and indicator variables except gj and δj, respectively, and σ2 = . Then the conditional distribution for δj can be written as:The equation . Thus, \As f is a multivariate density, the case of which is similar to that in FMixP. An efficient way is to use the product of univariate distributions of Bj′y−j instead of the distribution of y−j14. The f(Bj′y−j|Bj′Bj)2Bj′BjV1 = (Bj′Bj)2σ12 + Bj′Bjσe2 and V2 = (Bj′Bj)2Bj′Bjδj has been updated, gj is sampled as:Note that equation  can be wσ12 appears only in its own prior and the normal distribution of gj with δj = 1, the posterior distribution of σ12 can be derived as:k is the number of SNP loci with δj = 1. Similarly, the posterior distribution of m is the number of SNP loci with δj = 0.As the γ was set to 0.5, and the posterior probability is drawn from the Beta:The starting value of γ is larger than 0.5, we can switch the labels of the variance γ to 1 − γ. The posterior distributions of fixed effect u and residual variance 15.Note that if the sampling value of 1 and BayesCπ3, were used for comparison with MixP. The prior distribution of variances of SNP effects in BayesA follows an inverse-chi-squared distribution, i.e., σj2 ~ χ−211. In BayesCπ, SNPs with non-zero effects have a common variance that also follows an inverse-chi-squared distribution3. The degree of freedom (v) of the inverse-chi-squared distribution was set to 5.0. As the SNP genotypes had been standardised, parameter s2 was set without ∑2pj(1 − pj) in the denominator, which was different from the formula derived by Habier et al.3 and Gianola et al.16. In this study, s2 = [(v − 2)Vg]/(vM) in BayesA and s2 = [(v − 2)Vg]/(πvM) in BayesCπ, where π represents the probability of a SNP with a non-zero effect and is estimated by the MCMC algorithm. Vg is total additive genetic variance which is estimated using the R-package “EMMREML” (Version 3.1) that is one of packages22 used to estimate genetic parameters. Before Vg estimation, a genomic relationship matrix (G matrix) was calculated using the formula2: jth column of P is a vector of the frequency of allele ‘a’ at the jth locus, i.e., Two other Bayesian methods, BayesAet al.23 and a summary is given as follows. Through a simulation of a historic population of 50 generations, 4665 and 1200 individuals were simulated in the training and testing data sets, respectively. Six-thousand biallelic SNP loci were evenly spaced on 6 Morgan chromosomes, and 5,726 SNPs with minor allele frequencies (MAF) ≥0.05 were used for research. Forty-eight QTL loci were simulated, and the effects were sampled from a gamma distribution with a scale parameter 5.4 and a shape parameter of 0.42. The residual values were sampled to obtain a heritability value of 0.3 for the trait.Both the simulated and real data were used to compare the predictive results of various statistical methods. The simulated data had been distributed to the participants of the QTLMAS XII workshop. The data was described in detail by Lund Larimichthys crocea), which is one of the most commercially important marine fish species in southeast China and Eastern Asia24. All fish were reared in a breeding nucleus farm named ‘Jinling Aquaculture Science and Technology Co. Ltd.’ in Ningde City, Fujian Province, P.R. China. In total, 30 males and 30 females were mated randomly in a pool, and a total of 500 progenies  were randomly selected and measured in the experiment. The trial was carried out in the Key Laboratory of Healthy Mariculture for the East China Sea when the fish were two years old. Four quantitative traits, body weight, body length, body height and the length/height ratio, were selected to perform genomic prediction. Growth rate and body shape (customers prefer purchasing fish with slender bodies) are the important traits for large yellow croaker, so these four traits were selected for research. The parameters of the four traits are shown in Table The experimental materials were large yellow croaker  and IlluminaF and indexing primers. Fragments that were 200~300 bp in size were isolated using a Gel Extraction Kit (Qiagen). Then, pair-end sequencing was performed using an Illumina high-throughput sequencing platform. The raw sequencing reads were quality checked by FastQC25. The high-quality, filtered reads were mapped to the large yellow croaker reference genome sequence by BWA version 0.7.1026. The alignment files were then sorted and the duplicates marked by Picard (http://picard.sourceforge.net). Then, the GATK package27 was applied for SNP calling. As a result, 29,748 SNPs with a missing rate ≤20%, a MAF  ≥0.05 and genotypes in Hardy-Weinberg equilibrium were selected for further analysis. Beagle Version 3.3.2 software was used to impute the missing SNPs28.Fin samples from 500 individuals were collected for genotyping. The Genotyping-By-Sequencing (GBS) method was used to construct the libraries for next generation sequencing (NGS). Genomic DNA was incubated at 37 °C with γ, we varied the value of γ from 0.01 to 0.5 (50 levels were used with 0.01 as a step size).Genomic prediction by a replicated training-testing method was used to evaluate the predictive results of the real data. Cross-validation of 10 replicates was performed. All 500 individuals were randomly and evenly divided into 10 groups of 50 individuals each. In each replicate, one of the groups was selected as the testing data set while the remaining nine groups were used as the training data set. To observe the relationship between the predictive results of FMixP with r was used to measure the predictive accuracy1, where GEBV = Bĝ, ĝ is the vector of estimated SNP effects and B is the SNP genotypes; and an individual true breeding value (TBV) can be obtained by summing up all simulated QTL effects. We give a brief explanation below. If an individual GEBV is close to its TBV, the predictive accuracy is high. But if one aims to assess the predictive accuracies of a set of GEBV, one can use r, and higher r suggests higher predictive accuracy.In the simulation, the correlation coefficient between true genetic values and predicted genetic values, GEBV, and the phenotypes adjusted for the covariates , ry−Xû29. The higher correlation between them is, the higher genetic variance captured by the genetic SNPs is, leading to higher predictive ability.In the real data analysis, because the true breeding values are unknown, we used the predictive ability to measure the predictive accuracy, which is described as the correlation coefficient between All 500 individuals were added to the prediction model to estimate the computation time for various Bayesian methods. All of the calculation processes (except the REML process) were implemented in Fortran90 codes and run on the computer server of Jimei University.Raw DNA sequencing reads were deposited in NCBI with the project accession PRJNA309464 and SRA accession SRR3114179.


The MendelianRandomization package can be downloaded from the Comprehensive R Archive Network (CRAN) within R, or directly from [https://cran.r-project.org/web/packages/MendelianRandomization/]. Both R and the MendelianRandomization package are released under GNU General Public Licenses (GPL-2|GPL-3).MendelianRandomization is a software package for the R open-source software environment that performs Mendelian randomization analyses using summarized data. The core functionality is to implement the inverse-variance weighted, MR-Egger and weighted median methods for multiple genetic variants. Several options are available to the user, such as the use of robust regression, fixed- or random-effects models and the penalization of weights for genetic variants with heterogeneous causal estimates. Extensions to these methods, such as allowing for variants to be correlated, can be chosen if appropriate. Graphical commands allow summarized data to be displayed in an interactive graph, or the plotting of causal estimates from multiple methods, for comparison. Although the main method of data entry is directly by the user, there is also an option for allowing summarized data to be incorporated from the PhenoScanner database of genotype—phenotype associations. We hope to develop this feature in future versions of the package. The R software environment is available for download from [ The MendelianRandomization software package enables Mendelian randomization analyses to be undertaken using summarized data on genetic associations for a variety of previously published methods.The use of this package should lead to more reproducible and more credible causal inferences, due to the range of robust methods included in the package.However, a critical and principled approach to Mendelian randomization investigations is required for causal claims to be justified; robust methods can reveal inconsistencies in the analyses, but conclusions that rely on statistical approaches in the absence of biological understanding of the variants included in the analysis will always be speculative.,,,Mendelian randomization is the use of genetic variants as instrumental variables for assessing causal relationships from observational data.The inverse-variance weighted (IVW) method provides a consistent estimate of the causal effect of the exposure on the outcome when each of the genetic variants satisfies the assumptions of an instrumental variable.Software code is available for the implementation of each of these methods. However, this code is currently spread among the appendices of various manuscripts. The MendelianRandomization package brings the code together in a single location and makes implementation of these methods simpler and easier to reproduce, meaning that applied researchers can focus on the most important aspects of a Mendelian randomization investigation, namely the choice of genetic variants and assessment of the instrumental variable assumptions. Code for implementing all of the methods in the paper using the Mendelian Randomization package is provided in the MRdata_HDL_CHD <- mr_inputThe data entry function in the MendelianRandomization package is the mr_input function. The required inputs for this function are the beta-coefficients for associations with the exposure and their standard errors, and the beta-coefficients for associations with the outcome and their standard errors. A minimal example is:The variables hdlc, hdlcse, chdlodds and chdloddsse are provided as part of the MendelianRandomization package as example data, and represent the associations of 28 variants with high-density lipoprotein cholesterol (HDL-c) and their standard errors (hdlc and hdlcse), and the associations of the same 28 variants with coronary heart disease (CHD) risk and their standard errors (chdlodds and chdloddsse), taken from Waterworth et al.mr_ivw(MRdata_HDL_CHD)where MRdata_HDL_CHD is as defined above. By default, the mr_ivw function reports the causal estimate and its standard error, a 95% confidence interval based on a normal approximation, and the residual standard error and associated heterogeneity test statistic, indicating whether the causal estimates from the individual genetic variants are similar to each other or not.The IVW, MR-Egger and weighted median methods are then implemented by taking the MRInput object as an input, and outputting causal estimates and related statistics. For the IVW method, an example is:Similar output is provided by the mr_egger and mr_median functions, except that the MR-Egger method additionally reports an intercept parameter, representing the average pleiotropic effect of a genetic variant.A further option is the possibility of accounting for correlation between genetic variants in either the IVW or the MR-Egger method. This illustrates one benefit of the package–although the use of generalized weighted linear regression has been proposed for performing the IVW method with correlated variants,mr_allmethodsThe mr_allmethods function can be used to implement several of the methods offered in this package at once. An example is:This implements the IVW, MR-Egger, weighted and simple median methods, and puts the estimates, standard errors and confidence intervals into a table for easy comparison A.mr_plot)then the result is a static graph illustrating the various causal estimates then the result is an interactive graph of the genetic associations with the outcome against associations with the exposure. By default, error bars for the associations and the IVW estimate are also plotted. By mousing over a datapoint, an infobox appears with the name of the genetic variant and its associations with the exposure and with the outcome. This can be used for identifying genetic variants having heterogeneous associations with the outcome compared with other variants, and may be helpful in detecting variants to check for pleiotropy and possible violation of the instrumental variable assumptions.If the argument for the mr_plot function is an MRInput object, for example:P-values. The residual standard error from the weighted regression is substantially greater than 1, and the null hypothesis of the heterogeneity test is rejected, indicating that the genetic variants are not all estimating the same causal effect and suggesting the presence of pleiotropy for one or more variants. The I2 statistic quoted is a measure of weak instrument bias, and is described elsewhere.mr_plotAs an example, the output from the mr_egger function applied to the HDL-c and CHD data provided as part of the package is presented in A screenshot of the resulting interactive graph shows the possibility of identifying individual variants from the graph. The orientate = TRUE option ensures that all variants are plotted after orientation to the exposure-increasing allele, as this re-orientation is undertaken in the MR-Egger method.path.proxies <- system.file# this file has been downloaded from PhenoScanner (v1.1)extract.pheno.csv(exposure = “log(eGFR creatinine)”, pmidE = 26831199, ancestryE = “European”,outcome = “Asthma”, pmidO = 20860503, ancestryO = “European”,rsq.proxy = 0.6, file = path.proxies, snps = “all”)Finally, although the primary method for inputting data into the package is directly by the user, it is possible to incorporate data from PhenoScanner, a database of genotype—phenotype associations that can be queried using a web browser.2r greater than the threshold of 0.6 provided. Correlation estimates are provided by PhenoScanner, and can be calculated either for 1000 Genomes or HapMap participants of European descent. In this example, associations with asthma are not available for two variants; they are replaced by associations with suitable proxies. The output from the extract.pheno.csv function can then be used in the mr_ivw or similar functions for estimating a causal effect.This code takes the file provided by PhenoScanner, and outputs an MRInput object consisting of the genetic associations with the exposure “log(eGFR creatinine)” estimated in Europeans and reported by the study with PubMed ID 26831199,The eventual aim is to make this functionality seamless, so that PhenoScanner can be called directly from R as part of the MendelianRandomization package, or as a dependency to the package. Both this package and the PhenoScanner tool are still under development, and we hope that this aim will be achieved in the near future.www.mr-base.org] platforms to enable increased automation of Mendelian randomization analyses, as well as a wider range of methods such as methods for multivariable Mendelian randomization.The MendelianRandomization package makes performing Mendelian randomization analyses using summarized data relatively straightforward. The package is still under development: future additions will include greater integration with the PhenoScanner and MR-Base . It is important that Mendelian randomization is not performed in a way that avoids critical thought. In releasing this package, the hope is that it will lead to more comprehensive and more reproducible causal inferences from Mendelian randomization, and not simply add more noise to the literature.IJE online.The appendix is available as Stephen Burgess is funded by a fellowship from the Wellcome Trust (grant number 100114).Supplementary AppendixClick here for additional data file.


Weighted genetic risk scores (GRS), defined as weighted sums of risk alleles of single nucleotide polymorphisms (SNPs), are statistically powerful for detection gene-environment (GxE) interactions. To assign weights, the gold standard is to use external weights from an independent study. However, appropriate external weights are not always available. In such situations and in the presence of predominant marginal genetic effects, we have shown in a previous study that GRS with internal weights from marginal genetic effects  are a powerful and reliable alternative to single SNP approaches or the use of unweighted GRS. However, this approach might not be appropriate for detecting predominant interactions, i.e. interactions showing an effect stronger than the marginal genetic effect.In this paper, we present a weighting approach for such predominant interactions (“GRS-interaction-training”) in which parts of the data are used to estimate the weights from the interaction terms and the remaining data are used to determine the GRS. We conducted a simulation study for the detection of GxE interactions in which we evaluated power, type I error and sign-misspecification. We compared this new weighting approach to the GRS-marginal-internal approach and to GRS with external weights.N = 4465 observations).Our simulation study showed that in the absence of external weights and with predominant interaction effects, the highest power was reached with the GRS-interaction-training approach. If marginal genetic effects were predominant, the GRS-marginal-internal approach was more appropriate. Furthermore, the power to detect interactions reached by the GRS-interaction-training approach was only slightly lower than the power achieved by GRS with external weights. The power of the GRS-interaction-training approach was confirmed in a real data application to the Traffic, Asthma and Genetics (TAG) Study  GRS-interaction-training should be applied.The online version of this article (10.1186/s12863-017-0586-3) contains supplementary material, which is available to authorized users. For many diseases, genetic influences are exceedingly complex and cannot be explained by simple Mendelian modes of inheritance only. Moreover, genetic and environmental factors may jointly contribute to susceptibility clarifying the importance of analyzing gene-environment (GxE) interactions, which can be defined as “a different effect of environmental exposure in disease risk in persons with different genotypes” .en masse often access more of the heritable component of complex traits than is possible by single-variant approaches  were assigned to each child’s birth address using land use regression, atmospheric modeling, and ambient monitoring data. Gene-environment interactions between air pollution and SNPs in GSTP1 (rs1138272 and rs1695) and TNF (rs1800629) on asthma were investigated.Traffic-related air pollution, asthma, SNPs, and potential confounder data were pooled across six birth cohorts. Parents reported physician-diagnosed asthma from birth to 7–8 years of age . Individual estimates of annual average air pollution [nitrogen dioxide  and GSTP1 rs1138272  were marginally associated with asthma and a significant interaction between GSTP1 rs1138272 and NO2 on asthma was detected (Bonferroni-corrected p = 0.012) [The main findings of the pooled analyses were that NOMore information about the TAG study can be found in –25.N = 593 observations), which is one of the six birth cohorts included in the TAG study. We compared the p-values derived from weighted GRS with weights from the pooled analysis as published in [Ntest = 296), 1:2 (Ntest = 395) and 1:3 (Ntest = 444)).In our analysis, we focused on the German Infant Study on the influence of Nutritional Intervention plus environmental and genetic influences of on allergy development (GINIplus) as study sample , this marginal association was not significant in GINIplus.In Table GSTP1 rs1138272 and NO2 on asthma, which was identified in the pooled analysis [p-values were achieved by applying the GRS-marginal-internal approach and GRS with external weights, followed by the GRS-interaction-training (using 25% of the data for training and the remaining 75% as test data). The weights from the GRS-marginal-internal approach were almost identical to the univariate estimates from the pooled analysis. The GRS-interaction-training approach was the only approach that correctly identified GSTP1 rs1138272 as the only SNP that interacts with air pollution  by settIn this article, we presented a new weighting approach, called GRS-interaction-training, for GRSxE interaction studies in which parts of the study sample are used to estimate the weights and the remaining data are employed to determine the GRS.In a simulation study and a subsequent real data application, we compared the performance of this approach to weighted GRS with internal weights from the marginal genetic effects, called GRS-marginal-internal , and GRSOur simulation study has shown that the power for detecting GxE interactions reached by applying the GRS-interaction-training approach was only slightly lower than the power achieved by weighted GRS with external weights from the marginal genetic effects estimated in an independent study population that fits perfectly to our own study sample. If the external data, however, did not fit to the own study sample perfectly or the sample size of the external data was not larger than our own sample size, the power was higher when using the GRS-interaction-training approach.The sample size of the test data in the GRS-interaction-training approach is only half of the sample size from the GRS-marginal-internal approach, because in the GRS-interaction-training approach half of the data is used to determine the weights and the remaining test data to calculate the GRS and to estimate the interaction. Nevertheless, if there were no external weights available and the underlying GxE interaction effect was larger than the marginal genetic effect, the highest power was reached with the GRS-interaction-training approach. If the underlying marginal genetic effect was substantially larger than the GxE interaction effect, the GRS-marginal-internal approach was more appropriate.Motivated by the idea that the interaction itself might be more suitable to estimate the weights than the marginal genetic effect, we divided each of our datasets into a training and a test dataset and used the interaction estimates from the training data as weights for the GRS in the test data. Dudbridge 2013) evaluated a similar approach for the detection of marginal genetic effects and reported that the optimal balance of sample sizes between training and test datasets is close to one-half regardless of the proportion of noise SNPs or the p-value threshold [ evaluateOur simulation study has confirmed that the gold standard for the construction of GRS is to use external weights, e.g., from the marginal genetic effects estimated in independent study populations, if the external data fit very well to the study sample. This strong assumption means that the marginal genetic associations in the external data are the same as in our own study sample, this might but must not be reached if the phenotype is assessed in exactly the same way and that there is no ethnic or age difference between the study populations. In real data analyses, these assumptions are often not fulfilled because large scale GWAS are not published for every phenotype and sometimes only in populations with different ethnicity, sex or age range.The violation of these assumptions might lead to a decrease of power for detecting interaction effects with GRS with external weights. Therefore, in the practical analysis of real data, using internal weights from the study population itself might often be a more powerful alternative to detect GxE interactions.However, in our real data application, the power reached by GRS with external weights was similar to the power reached by the two approaches with internal weights. One reason for that might be that our study sample (GINIplus) was included in the estimation of the “external” effects. Therefore, the effect estimates from the pooled analysis might fit slightly better to the GINIplus data than they would have fitted if the GINIplus data would not have been part of the pooled analysis. Furthermore, a limitation of the GRS-interaction-training approach is that the GRSxE interaction term can only be estimated in a subset (i.e. the test data) of the original sample data which reduces the power to detect interactions.−8) marginal genetic effect in the external study population, whereas SNPs with a predominant interaction effect are usually not presented. For GxE interaction studies, this leads to a publication bias towards SNPs with predominant marginal genetic effects. To avoid this publication bias and to increase the power for detecting GxE interactions, estimates from genome wide gene-environment interaction studies might be used. However, up to now, very few genome-wide gene-environment interaction studies have been published because of the limited power to detect interactions in genome-wide analyses.A major limitation of GRS with external weights is that we can only include SNPs for which the marginal genetic effects have been published. In this regard, GRS with external weights are usually restricted to SNPs with a genome-wide significant  larger than the marginal genetic effect, the GRS-interaction-training approach was more powerful than the GRS-marginal-internal approach, particularly in the presence of noise SNPs. Furthermore, applying the GRS-marginal-internal approach in scenarios with predominant interaction effects might lead to a high number of sign-misspecifications when the MAFs of the risk SNPs are ≥0.2 and in the presence of noise.However, in scenarios with a predominant marginal genetic effect and a smaller additional interaction effect, the GRS-marginal-internal approach achieved a slightly higher power than GRS-interaction-training approach with an acceptable number of sign-misspecifications.In real data applications, the decision if the interaction or the marginal genetic effect is predominant, should be made a priori and be based on biological knowledge. If the SNPs were chosen because the underlying genes had been identified to be marginally associated with the same or a related phenotype , independently of the environmental exposure, the weights should be determined from the marginal genetic effects . Nevertheless, if the SNPs were chosen because of their potential impact on the biological mechanisms mediating the association between the environmental exposure and disease development, the weights should be determined from the interaction term (GRS-interaction-training approach). Either this knowledge might be based on mechanistic studies or on epigenome-wide association studies (EWAS). EWAS present differentially methylated probes (DMPs) and regions (DMRs) in balance to disease outcomes  was part of the TAG consortia (N = 4465), the weights from the pooled marginal genetic effects were not independent from our sample data. However, this problem does also often occur in the real data practice because large scale genome-wide meta-analyses often include all study populations that are available for the considered phenotype and thereby often include the own study sample as well.Our real data application has the limitation that we could only include the three SNPs from which we had previous knowledge about the marginal genetic and interaction effects in a large pooled analysis . HoweverIn conclusion, when no appropriate external weights are available , we recommend to use internal weights from the study population itself to construct weighted GRS for GxE interaction studies. If the SNPs were chosen because a marginal genetic effect was hypothesized, the weights should be estimated from the marginal genetic effects . If the SNPs were chosen because of their potential impact on the biological mechanisms mediating the association between the environmental exposure and disease development, the weights should be estimated from the interaction term itself in a training dataset (GRS-interaction-training approach).


Background: Smoking is the strongest environmental risk factor for reduced pulmonary function. The genetic component of various pulmonary traits has also been demonstrated, and at least 26 loci have been reproducibly associated with either FEV1 (forced expiratory volume in 1 second) or FEV1/FVC . Although the main effects of smoking and genetic loci are well established, the question of potential gene-by-smoking interaction effect remains unanswered. The aim of the present study was to assess, using a genetic risk score approach, whether the effect of these 26 loci on pulmonary function is influenced by smoking.Methods: We evaluated the interaction between smoking exposure, considered as either ever vs never or pack-years, and a 26-single nucleotide polymorphisms (SNPs) genetic risk score in relation to FEV1 or FEV1/FVC in 50 047 participants of European ancestry from the Cohorts for Heart and Aging Research in Genomic Epidemiology (CHARGE) and SpiroMeta consortia.Results: We identified an interaction  between an unweighted 26 SNP genetic risk score and smoking status (ever/never) on the FEV1/FVC ratio. In interpreting this interaction, we showed that the genetic risk of falling below the FEV1/FVC threshold used to diagnose chronic obstructive pulmonary disease is higher among ever smokers than among never smokers. A replication analysis in two independent datasets, although not statistically significant, showed a similar trend in the interaction effect.Conclusions: This study highlights the benefit of using genetic risk scores for identifying interactions missed when studying individual SNPs and shows, for the first time, that persons with the highest genetic risk for low FEV1/FVC may be more susceptible to the deleterious effects of smoking. This paper reports a genetic risk score-by-ever smoking interaction on FEV1/FVC as a result of smoking was greater among individuals who are genetically predisposed to lower FEV1/FVC ratio.In individuals of European ancestry, the reduction in FEVGenetic risk score-by-ever smoking interaction can allow the identification of subgroups in the population whose genetic background makes them more susceptible to the deleterious effects of smoking.1) or its ratio with the forced vital capacity (FEV1/FVC), form the basis of the diagnosis of chronic obstructive pulmonary disease (COPD).1 or FEV1/FVC in the general population.,,Spirometric measures of pulmonary function, such as the forced expiratory volume in 1 second -by-smoking interaction, Hancock et al.,Here, we explored gene-by-smoking interaction effects limited to genetic variants previously found to be associated with pulmonary function in standard marginal effects GWAS,et al.IJE online) provides the main characteristics of the studies included; complete details of study-specific pulmonary function testing protocols have been published.1/FVC and FEV1 measured at that visit. For studies with spirometry at more than one visit, measurements from the baseline visit or the most recent examination with spirometry data was used. Smoking history  was ascertained by questionnaire at the time of pulmonary function testing. Pack-years of smoking were calculated for current and past smokers by multiplying smoking amount (packs per day) and duration (years smoked). Approximately 2.5 million autosomal SNPs were tested for interaction with smoking status (ever smoking vs never smoking) and pack-years, for two outcomes: FEV1 and FEV1/FVC (see next section). We also used two independent datasets of individuals of European ancestry to test for replication. The first replication dataset included 8859 unrelated individuals, and the second dataset included 9457 family-based individuals. The look-up was done in the GWAS for marginal genetic effects done separately in ever and never smoker as part of a recent meta-analysis of FEV1 and FEV1/FVC.The present analysis relies on the Hancock 1 and FEV1/FVC after regressing out age, ageIJE online):Gβ and G and exposure lE, G and exposure kE, and 0β the intercept.The analysis performed in this study used summary statistics data from the aforementioned meta-analysis of 19 studies performed by Hancock et al.et al.1 and FEV1/FVC separately for ever smokers and never smokers. The residuals were normalized using a rank-based inverse normal transformation, again separately in ever smokers and never smokers. These transformed residuals were then used as the phenotype for association testing under an additive genetic model in each exposure strata. Inference of the interaction effects from the exposure-stratified analyses are described in the IJE online).Detailed description of studies used in the replication analysis can be found in Soler Artigas 1 and FEV1/FVC from Soler Artigas et al.IJE online). This weighting scheme is most powerful when the magnitude of interaction effects is proportional to the SNP marginal effects. Finally, for our third multivariate analysis, we derived a standard omnibus test of all interaction effects. This test will retain power in the presence of effects in both directions or of different magnitudes. Although there is strong correlation among the 12 tests performed , we used a stringent Bonferroni P-value correction threshold of 4 × 10–3 to account for multiple testing.First, we considered an unweighted genetic risk score-by-smoking interaction where the risk score simply sums the number of risk alleles . This unweighted genetic risk score is most powerful when the interaction effects have the same direction as marginal SNP effects . Second, we used a weighted genetic risk score where SNPs were weighted by the absolute value of their marginal effect estimates obtained from stage 1 screening of FEVGRS) is usually expressed as GRS = Σm[iw × iG], where m is the number of SNPs included in the genetic risk score and w =  are the weights attributed to each single SNP. Following previous notation, the test of interaction between the GRS and the exposure kE can be applied using the following model:γ0, GRSγ, INTγ are the intercept, the main effect of the GRS, the main effect of the exposure lE and the interaction effect between kE and the GRS, respectively. However, because individual-level data were not directly available, we performed the test of INTγ from summary statistics of interaction effects using an inverse-variance weighted sum as proposed by Aschard.INTγ was derived as follows:kE and the SNP iG obtained from iw is the weight applied to SNP i.G Under the null hypothesis of no interaction effect, When raw data are available, the weighted genetic risk score , we also derived the omnibus test using summary statistics. Under this independence assumption, the iG × kE interaction terms would also be independents,m degrees of freedom as follows:kE and the SNP iG obtained from The standard omnibus test of all interaction effects consisted of evaluating jointly 1 or FEV1/FVC increases the risk of death1/FVC < 0.70 and FEV1 < 80% of the predicted value. According to recent studies,,1/FVC < 0.70, depending on smoking characteristics and age distribution. Several studies argue for a more stringent threshold to define COPD,GRS interaction effects can further be translated in terms of risk prediction. For pulmonary function, low FEV1/FVC below a given threshold  in ever smokers vs never smokers conditional on the unweighted GRS. This quantity is defined as the joint probability of having both FEV1/FVC in the interval  and the GRS in the interval . This can be expressed as the following integral:y, e and g are FEV1/FVC, smoking status and the GRS, respectively, and f1 and f2 are the probability density function of y and g. The detailed derivation of the above integral is available as IJE online.To explore the impact of interaction effect on the risk of disease, we derived the relative risk (RR) of having FEV1 or FEV1/FVC at genome-wide significance (P < 5 × 10–8) in marginal association tests,,et al.,,P-value) with each of these phenotypes. The final list included 26 SNPs per phenotype, with only two SNPs being different between FEV1 and FEV1/FVC as previously reportedIJE online). Estimated interaction effects of these SNPs were extracted from the meta-analysis summary statistics for the four tests performed in the Hancock et al.1 and FEV1/FVC; and SNP-by-smoking pack-years interaction effect on FEV1/FVC and FEV1. As shown in IJE online), nine SNPs showed nominal significance (P < 0.05) out of the 104 tests performed; however, none remained significant after accounting for multiple testing . The minimum P-value was observed for the interaction between rs993925, near the TGFβ2 gene, and smoking status on FEV1 .We selected 26 loci previously found to be associated with FEV1 and FEV1/FVC) for a total of 12 tests. As shown in 1/FVC had a P-value below the Bonferroni significance level . The strongest signal was observed for the unweighted genetic risk score-by-smoking status interaction effect on FEV1/FVC . The Cochran’s Q test for heterogeneity of the interaction effect across studies was not significant (P = 0.97) and the forest plot of study-specific results did not display any obvious outlier (IJE online).Next, using these data, we conducted three multivariate (as opposed to single SNP) interaction analyses, testing jointly for the interaction effects between those SNPs and either smoking status or pack-years on the two phenotypes , there is an enrichment for negative interaction effects. Indeed, even a binomial test can be used to confirm the unbalanced direction of interaction effects . The genetic risk score-based interaction test exploits such enrichment by testing for the average interaction effect across all SNPs.IJE online). We summarized the contribution of the unweighted genetic risk score-by-smoking interaction on FEV1/FVC in The contrast between this significant risk score interaction and the absence of strong single SNP interaction effects can be explained by looking at the distribution of the single SNP interaction effect estimates. IJE online). Although the interaction effects were not significant, both replication samples showed consistent negative GRS-by-ever smoking interaction effect on FEV1/FVC  and a Cochran’s Q test for heterogeneity showed no significant difference in the three effect estimates (P = 0.51).We used two independent datasets, one of 8859 unrelated individuals and another of 9457 related individuals, to test for independent replication of our results  in ever smokers compared with never smokers. 1/FVC in the lowest 1% of the population distribution, whereas ever smokers with a GRS below the 20th percentile have on average an 18% excess RR of falling in that same FEV1/FVC category compared with never smokers. Applying the same approach for FEV1, we observed a similar pattern (IJE online). However, as expected, the lower magnitude of the genetic risk score-by-ever smoking interaction on FEV1 implied a lower difference in RR between ever smokers and never smokers.To quantify the impact of this result from a public health perspective, we estimated the impact of the genetic risk score-by-smoking interaction on having FEV1/FVC and FEV1), smoking and genetic data, we identified a gene-by-smoking interaction effect on FEV1/FVC by using a GRS composed of 26 SNPs identified and replicated in a prior GWAS meta-analysis of marginal genetic effects. To our knowledge, our study is the first to report a synergistic action of genes and smoking on pulmonary function . Our study also highlights the importance of developing and applying alternative strategies to evaluate interaction effects for lung phenotypes along with other complex traits and diseases. The genetic risk score-based approach enabled us to identify an interaction when the standard univariate test  failed to identify any interactions.Using the largest dataset to date of European ancestry participants from the general population with pulmonary function (FEVIJE online). Previous work has shown that combined analyses are more powerful when effects exist in both strata,N = 18 316 individuals in the combined replication population, we are underpowered. This sample size provides less than 50% power, at nominal significance of 5%, to detect interaction effects with the GRS.Replication studies showed interaction effect estimates in the same direction as the discovery study but were not significant, and the magnitude of interaction effects were substantially smaller. We acknowledge that, despite careful evaluation of the interaction effects in the discovery sample, the observed signal might be overestimated or confounded by unmeasured complex factors. However, we can a priori rule out a systematic bias of the single SNP interaction effects in the discovery study, because the genomic inflation factor λ, defined as the ratio of the median of the empirically observed distribution of the test statistic to the expected median,,,Genetic risk score-by-exposure interaction can have higher clinical value than the identification of single SNP-by-exposure interaction by capturing a wealth of information in a single measure to identify subgroups in the population whose genetic background makes them more susceptible to the deleterious effects of smoking.1/FVC. As shown in Our results may also elucidate biological mechanisms underlying the interplay between genes and smoking in pulmonary function. In particular, the higher statistical power for the genetic risk score-based interaction test points towards the potential presence of an unmeasured intermediate biomarker mediating the effect of the 26 loci on FEV1/FVC and in FEV1.This study has some limitations. The 26 selected variants together explain a relatively small proportion of the additive genetic variance in FEVIn summary, the identification of interaction effects between genetic variants and environmental exposures in human traits is recognized as extremely challenging, and this quest has been mostly unsuccessful so far. In this study, we discovered novel gene-by-smoking interactions using risk scores that were not observed at the level of individual genetic variants. This risk score analysis suggests that persons with a greater genetic predisposition to low pulmonary function are more susceptible to the deleterious effects of smoking. By extension, the use of a GRS may help predict which smokers will fall below thresholds that establish the diagnosis of COPD.IJE online.Supplementary Online Data SupplementClick here for additional data file.


Phytophthora infestans infection, plant maturity, maturity corrected resistance, tuber starch content, tuber starch yield (TSY), and tuber yield (TY) of 184 tetraploid potato clones or subsets thereof genotyped with the SolCAP 8.3k SNP array. The cross-validated prediction accuracies with GBLUP and the three Bayesian approaches for the six evaluated traits ranged from about 0.5 to about 0.8. For traits with a high expected genetic complexity, such as TSY and TY, we observed an 8% higher prediction accuracy using a model with additive and dominance effects compared with a model with additive effects only. Our results suggest that for oligogenic traits in general and when diagnostic markers are available in particular, the use of Bayesian methods for genomic prediction is highly recommended and that the diagnostic markers should be modeled as fixed effects. The evaluation of the relative performance of genomic prediction vs. phenotypic selection indicated that the former is superior, assuming cycle lengths and selection intensities that are possible to realize in commercial potato breeding programs.Genomic prediction is a routine tool in breeding programs of most major animal and plant species. However, its usefulness for potato breeding has not yet been evaluated in detail. The objectives of this study were to (i) examine the prospects of genomic prediction of key performance traits in a diversity panel of tetraploid potato modeling additive, dominance, and epistatic effects, (ii) investigate the effects of size and make up of training set, number of test environments and molecular markers on prediction accuracy, and (iii) assess the effect of including markers from candidate genes on the prediction accuracy. With genomic best linear unbiased prediction (GBLUP), BayesA, BayesCπ, and Bayesian LASSO, four different prediction methods were used for genomic prediction of relative area under disease progress curve after a  Despite the important role of potato for securing world-wide human nutrition FAOSTAT, , potato Genomic prediction provides an alternative method to use genomic information in breeding decisions  examine the prospects of genomic prediction of key performance traits in a diversity panel of tetraploid potato modeling additive, dominance, and epistatic effects, (ii) investigate the effects of size and make up of training set, number of test environments, and molecular markers on prediction accuracy, and (iii) assess the effect of including markers from candidate genes on the prediction accuracy.Phytophthora infestans infection and for plant maturity (PM). PM was scored from 1 to 9, where 1 indicates very late and 9 very early maturity. For each year*location combination, which was in the following designated as environment, trait values for rAUDPC were calculated as the ratio of each clone's AUDPC and the environmental mean. As AUDPC and rAUDPC were highly correlated, we report only results obtained for the latter. Furthermore, from AUDPC and PM, the trait maturity corrected resistance (MCR) was calculated . This assessment was performed in an experiment, which was laid out as a randomized block design, with one replication per environment, where each plot had two rows with eight plants each. As described by Mosquera et al.  analyses based on allele frequency matrices considering the allele dosage information. The number of columns of these matrices correspond to the number of marker loci and the number of rows to the number of clones. Clones were grouped by k-means clustering to analyses . The molanalyses . Bi-locu*location combination was treated as an environment:yij was the entry mean for the ith clone in the jth environment, μ was an intercept term, gi was the effect of the ith clone, lj was the effect of the jth environment, and eij was the residual. As the environments comprised two purposefully selected locations, the environmental effects lj were regarded as fixed. On the basis of best linear unbiased estimation, adjusted entry means for each clone were derived in each set of environments , applying model [1] considering the clone effects as fixed. For estimation of variance components, gi was considered as random. Heritability on an entry mean basis was calculated as Since eight clones were in common between the BNA and SaKa subsets of the PIN184 population, we performed a joint data analysis for both subsets using the following statistical model, where each yearHolland, .The genetic variance among and within clusters (Qst) , BayesA, BayesCπ, and Bayesian LASSO (BL). Details of these methods were described previously  all 6052 SolCAP SNPs, (ii) in 100 independent runs each, the SolCAP SNPs were stratified sampled in such a way that a genome-wide equal distribution of SNPs was obtained with a density of 5, 1.33, 0.67, and 0.13 Mbp−1, (iii) all SNPs from the 85 candidate genes for P. infestans resistance, and (iv) all SolCAP SNPs and the 85 candidate genes for P. infestans resistance.For each of the above mentioned statistical models, different genetic models were examined: Model M1 considered only additive effects, M2 additive and dominance effects, whereas model M3 considered additive, dominance, and the three types of epistatic effects ] as the Pearson correlation between the phenotype and the genomic estimated genotypic values divided by the square root of heritability h2 of the target trait evaluated in the respective set of environments.The genomic prediction of genotypic values is of interest in our study. Therefore, we calculated the prediction accuracy [The standard scheme for validation of genomic prediction was five-fold cross-validation. For this purpose, the clones of population PIN184 or the BNA subset were randomly subdivided into five disjoint subsets. One subset was left out for validation, whereas the other four subsets were used as training set. This procedure was replicated 20 times, yielding a total of 100 cross-validation runs.Different analyses were performed to evaluate the effect of different factors (F) on genomic prediction: (F1) the sample size of the training set was varied; (F2) the number of test environments in which the training and validation set were evaluated was varied; (F3) the effect of “reduced” relationship between training and validation set was examined: (F3A) performance of one half of the clones in one cluster was predicted based on all the clones of the remaining clusters. This procedure was replicated 20 times. In each replication, a different set of genotypes was placed into the two halves of the cluster. (F3B) performance of one half of the clones in one cluster was predicted from a combination of the clones from the remaining clusters, and the other half of the clones in the considered cluster, where the number of clones in the remaining clusters, was reduced by sampling such that the total number of clones in the training set was the same as in F3A. This procedure was repeated 20 times as described for F3A. Due to the high computational effort of the Bayesian methods, these analyses as well as the simulations of different sets of SNPs, which were described above, were performed only with the prediction method GBLUP.If not stated differently, all statistical analyses were performed using statistical software R version 3.3.2 , additive and dominance M2), as well as a model with additive, dominance, and epistatic effects (M3) were examined. Across the six traits, we observed the lowest prediction accuracy for the genetic model with additive, dominance, and epistatic effects  was three to four times higher than that calculated based on molecular marker data (Gst) Table . This caPotato is currently bred as a clonal species that is highly heterozygous, which allows dominance effects to contribute to phenotypic variation. In order to quantify that contribution, we examined genetic models for genome-wide prediction that considered additive and dominance effects. On average across all traits, the prediction accuracies observed for such a model were moderately higher than for the model considering additive effects alone M1&2, Table , Table 2Another non-additive component potentially influencing the genotypic value of genotypes is epistasis. We examined a genetic model that considered, besides additive and dominance effects, the three types of epistatic interactions. Across the six traits, we observed the lowest prediction accuracy for that model .CRX) divided by the direct selection response (RX)  and considering the h2 estimates and the prediction accuracies of our study, the maximum relative cycle length of indirect selection can be calculated for which identical selection gains are realized with direct and indirect selection. These maximum relative cycle lengths of indirect selection were 103% (MCR), 98% (rAUDPC), 85% (TSC), 73% (PM), and 60% (TSY and TY). Therefore, genomic prediction for MCR is already superior to phenotypic selection even without reducing the length of one cycle, whereas for the other traits, one cycle of genomic predictions needs to have maximally the above mentioned length relative to phenotypic selection to result in the same gain of selection. These are very promising figures, compared with the results reported for other crop species  of a standard potato breeding scheme Table . TherefoIn conclusion, our results indicate that the application of genomic prediction in breeding programs for tetraploid potato has a high potential for increasing the gain from selection.BS: Conceived and designed the study; BS and DV: Analyzed the data and wrote the paper. Both authors read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.