Currently, association studies are analysed using statistical mixed models, with marker effects estimated by a linear transformation of genomic breeding values. The variances of marker effects are needed when performing the tests of association. However, approaches used to estimate the parameters rely on a prior variance or on a constant estimate of the additive variance. Alternatively, we propose a standardized test of association using the variance of each marker effect, which generally differ among each other. Random breeding values from a mixed model including fixed effects and a genomic covariance matrix are linearly transformed to estimate the marker effects.p-values are virtually identical to those calculated from tests for one marker effect at a time. Moreover, the standardized test reduces computing time and memory requirements.The standardized test was neither conservative nor liberal with respect to type I error rate , compared to a similar test using Predictor Error Variance, a method that was too conservative. Furthermore, genomic predictions are solved efficiently by the procedure, and the p-value) in each chromosome is selected, and the segment is expanded one Mb upstream and one Mb downstream of the marker. A genomic matrix is calculated using the information from those markers only, which is used as the variance-covariance of the segment effects in a model that also includes fixed effects and random genomic breeding values. The likelihood ratio is then calculated to test for the effect in every chromosome against a reduced model with fixed effects and genomic breeding values. In a case study with pigs, a significant segment from chromosome 6 explained 11% of total genetic variance.The following steps are used to locate genome segments displaying strong association. The marker with the highest − log contains supplementary material, which is available to authorized users. The availability of high density genotypes of single nucleotide polymorphism (SNP) markers for plants and livestock species, in conjunction with phenotypic data for complex traits, allows the calculation of: 1) estimates of genomic breeding values (GEBVs)  for genoThere is an increasing number of GWAS data sets analyzed by mixed models and multiple testing procedures , after fgenomic selection proposed by Meuwissen et al. , i.e. the count of the allele used as reference. In the sequel, we will use the sub index i to refer to the individual. Matrix M was standardized to matrix Z that has generic elements equal toThe genomic relationship matrix was estimated from observed and imputed high density (~44 K) SNP genotypes. Genotypes were expressed as allelic dosage , 15, sucZ are then calculated by subtracting twice the frequency of the reference allele at the jth marker (pj), to the corresponding element of M[pj(1 − pj) of each element in the column multiplied by the number of columns (m) in M. The allele frequency pj was calculated from the F0 generation . The genomic relationship matrix was finally calculated as:Elements of ent of M, and theUsing the genomic relationship matrix from equation , the ceny is the phenotypic vector containing the data on 13-week tenth rib backfat (mm), X is the incidence matrix that relates records to the fixed effects of sex in β, vector a contains the random breeding values such that e is the random error vector such that I is the identity matrix. Variance components were estimated with REML using the regress version 1.3-10 R package [where  package .Following Stranden et al.  an equivg of SNP effects. To show that (2a) and (2b) are equivalent models, we employ the fact that a = Z g. Then, the variances of a and g are related in the following manner:Every element in (2b) is defined as before except for the vector G = Z Z ' and Necessary conditions for models (2a) and (2b) to be equivalent  are that gIn this section, we describe the algorithm to calculate the variance of the estimated SNP effects lues in , 20, 21,The last step results from the fact that model equivalence involves equation   is obtaNow, we know that the predictor error variance (PEV) of So thatCaa results from inverting the coefficient matrix of the mixed model equations [Matrix quations  such thaThen, on replacing with the latter expression into m × m) with m the number of SNP. However, we only need its diagonal elements. Also notice that the first term in (5), Z ' G1−  Z, can be computed and stored to be reused for the different traits, whereas Caa has to be computed for each trait.Expression (5) results in a large matrix of dimension (The estimated SNP effects in (3) were standardized by dividing with their corresponding p-values were assessed as 1 minus the cumulative probability density of the absolute value of SNPe j, a number that was then multiplied by 2 so as to obtain:The Φ(x) is the cumulative density function of the normal distribution for the random variable x. When analyzing the trait 13 week tenth rib backfat (mm), the p-values for each SNP were plotted across the genome as –Log10  using the physical position of the SNP in Mega-bases (Mb).where jth SNP effect (3) was performed using the PEV A second standardization of the p-values and genome screening for SNPep j were assessed and plotted in the same fashion as for SNPe j.As discussed above, SNPe j and SNPep j affected the nominal size of the test for the effect to be equal to zero. Data on 928 animals with 44,055 SNP each were used for the study, and the 1018 SNP on chromosome 18 were reshuffled. Two scenarios were considered: 1) Dependency: rows of the genotype matrix were permuted for columns corresponding to SNP on chromosome 18, thus keeping Linkage Disequilibrium (LD) within chromosomes but breaking the relationship between genotypes and phenotypes for the 1018 SNP on the chromosome. 2) Independency: the genotype of any animal was permuted independently by marker  for those SNP on chromosome 18, and the relationship with the phenotype was broken too. For both scenarios model (2a) was fitted to the data, and two tests were calculated for each scenario: test1 = SNPej and test2 = SNPepj. Permutations were repeated 200 times per scenario, and in each permutation the G matrix was calculated while fitting model (2a). As a result, the heritability of the trait was similar to the original heritability due to relationships in the other 17 chromosomes being kept intact, and p-values for those SNP (that are now non-associated) on chromosome 18 were obtained for the different tests. Under the null hypothesis and assuming independence , an approach that controls for type I error appropriately [p-values follow a uniform distribution. Consequently, to estimate the empirical quantiles of the distribution for the null hypothesis, we used a uniform density U ∼  to generate 200 replicated sets for the 1018 p-values.A plasmode simulation was performed to compare how the standardized values priately , the 101p-values.The SNP effects were tested on a one by one basis. The model approach used for testing purposes is better known as “efficient mixed-model association” (EMMA) . The modp-values were selected to form SNP segments. These segments were defined by taking all SNP within one Mb upstream and one Mb downstream of the SNP with smallest p-value on each chromosome. The size of the segment was chosen using a criterion similar to the one employed by Hayes et al. [r2 = 0.2 at 1 Mb (data not shown), which essentially would imply a minimal contribution to the additive variance from markers located beyond such distance. Moreover, segment sizes about two Mb have been reported to be significant in association studies [G1 (as described in (1)) using all SNPs that belonged to the segment, whereas genomic relationship matrix G2 was built using all remaining SNPs. The model fitted can be represented as:After the genome screen using model 2a, the SNP with the smallest s et al. . The poi studies , 26–28. a1 is the vector of additive random effects associated with those SNP located in the segment, such that a2 is the vector of additive random effects associated with all SNPs except those involved with a1, such that G2 is computed by subtracting from G the columns of Z related to the segment being tested. Let Zs be a matrix having as columns those related to the segment being tested, then Gis different from segment to segment. Additionally, the calculation of G1 and where s et al.  used a ss et al.  gave sims et al. Gis diffe−05 . Hence, in order to evaluate the significance of the segments, a second p-value for the Likelihood Ratio Test  was calculated to compare against BC. This p − valueLRT was assessed as 1 minus the distribution function of a chi-square (χ2) random variable with 0.5 degrees of freedom [To adjust the level of significance for multiple comparisons, a Bonferroni Correction (BC) was performed. In this context, if the pig genome is ~2800 Mb long and the average size of the segment is 2 Mb, there are 1400 segments along the genome with corresponding multiple tests. Thus, for α = 0.05, the BC was equal to 0.05/1400 = 3.571429 freedom  as folloΩ(x) is the distribution function of a random variable having the χ2 as density, and LRT is the Likelihood Ratio Test obtained by contrasting appropriate models.where p-values of the 44055 SNP were obtained as described in the Methods section. First, the p-values for SNPej, i.e. using 13-week tenth rib backfat (mm). Large peaks  above 5 can be seen at chromosomes 6 and 3, suggesting noticeable genetic variation for the trait. On the other hand, p-values for SNPepj (i.e. standardized with prediction error variance) were very large, with a maximum − Log10 of 0.20. In essence, the pattern observed in Figure gj) − Var gj) = 2.6768. The use of the square root of the difference between those two values resulted in a practically constant denominator for the test-statistic that was equal to 2.66. Also, a look at Figure The 0 p-value of 0.20.p-value) for each case of standardization. First, the p-values for test1 (SNPej) obtained in the scenario under independent SNPs  displayed an identical distribution of p-values when obtained by the reference distribution U ∼ . In contrast, under dependency  less extreme p-values were observed, a fact that was not reflected in a uniform distribution. This is a well known fact in human genetic epidemiology [p-values from associated SNP under the assumption of independence results in tests that are too conservative. On the other hand, for test 2 (SNPepj) even p-values obtained for independent SNP  displayed a distribution that was too conservative. Furthermore, the results from the dependent scenario (LD) were even more conservative than those from the independent scenario (results not displayed in the Q-Q plot), thus indicating that the use of the square root of In order to study the type I error rate of the two proposed tests we performed a plasmode simulation . A plasmemiology , where tp-values that were almost identical ) on each chromosome were chosen, whereas no segments were tested using LRT for regions with SNPepj resulting in exceedingly low p-values. The three segments from chromosomes with the smallest p-values are displayed in Table p-value). The estimates of the variance components and the LogLikelihood obtained from model equation . On the contrary, the segments located on all other chromosomes were not significant. The proportion of variance explained by the segment from chromosome 6  = 8.02) was 11% of the total variance, a fact that was reflected in a similar reduction of the estimated additive variance Results from the LRT indicated that the segment on chromosome 6 was significant: sufficient statistics of our methodology are: the Best Linear Unbiased Prediction (BLUP) of the breeding values from an animal model, G as the covariance matrix , Z as ta et al. .SNPej. The p-values for the tests of specific genome regions were calculated with a level of significance − Log10 = 5. Additionally, Prediction Error Variance  = 0.20) and detected stronger signals (higher peaks in the Manhattan plot) for SNPepj than with SNPej. Furthermore, a simulation was carried out with the same structure of SNPs markers and animal data as in the current study, in order to compare the performance of empirical p-values of both standardized tests. The SNPs markers of chromosome 18 were reshuffled, and two scenarios were simulated: 1) Dependent genotypes (LD), and 2) Independent genotypes (LE). Neither scenario displayed a relationship with the phenotype, whereas both standardized tests were calculated at each scenario. The reference distribution for the p-values considered was the uniform. In the independent scenario (LE), standardization with p-values that resembled the uniform density, but in the dependent scenario (LD) the SNPej performed conservatively. Instead, the standardization with p-values that were closer to the simulated ones. Moreover, the performance of SNPej under LD was not too conservative, a scenario that could be extrapolated to the genotypes in the current study. In addition, the p-values calculated using the EMMA procedure [SNPej. These results suggest that SNPej behaves reasonably to control type I error rate or false positives. Also, the computing time for fitting model (2a) and then calculating (6) using expressions (3)-(5) was 2.5 to 3 times less than the computing time for the EMMA model.First, the SNP effects rocedure  were simjth marker was obtained as follows: SNPep j (7), when the estimated SNP effects In order to identify SNP with important phenotypic associations , the cal values , 9, 34 h values , 20, 21. values , so thatSNPej) presented here was that each SNP effect was scaled by its own (and different) standard deviation rather than the use of a prior variance [SNPej, involves the same variance for the same SNPs markers and animals, i.e. Z, a function of 2pj(1 − pj), takes into account this latter quantity into SNPej. Additionally, the matrix Z uses the allele frequencies from the F0 generation calculated with unrelated individuals, and a proper expected variance by marker (see Methods section). In addition, the test statistics SNPej that standardizes SNP effects produces a p-value, a result that is appealing to many researchers that are more familiar with the method of testing one SNP at the time rather than with the proportion of additive variance that is explained by a genomic region. A further advantage of the method is that detection of many false positives are avoided, and genome positions with sizeable effects are highlighted.In contrast, the advantage of the standardized test  were selected, and for each of these SNP a segment of 2 Mb long (1 Mb at each side) was created. The next step was to estimate the variance components and the Log-Likelihood from the centered animal models (2a) and (8). The latter model includes the random vector of SNP segments a1. Lastly, we compare the performance of both models. Hayes et al. [G is the same for all segments, so that it was calculated only once, and stored in memory for the calculations, whereas in the model of Hayes et al. [G has to be calculated for each segment. This implies an extended computing time and higher requirements of CPU memory to obtain similar results to those from model (8).Later in the research, genome segments that expressed higher signals were located. To this purpose, SNPs with the smallest s et al.  used a ss et al.  a differ13 week tenth rib backfat in the current population under study.To evaluate the significance of the segments, the effects of each chromosome segment were tested by the Likelihood Ratio Test. The size of the test was adjusted by the Bonferroni correction. As a result, the segment located on chromosome 6  was significant, and explained 11% of the trait total variance. Previous studies by Edwards et al.  and Choihttp://www.animalgenome.org/QTLdb/pig.html), have been reported to be associated with the trait. Furthermore, recent studies showed the importance of chromosome 6 [Additionally, forty eight markers between the physical position between 128 Mb and 139 Mb on chromosome 6  on each chromosome for trait 13-week tenth rib backfat (mm) by standardization SNPejand EMMA. The blue and red circle represents highest − Log10 on each chromosome by the standardization SNPej and efficient mixed-model association (EMMA) using rrBLUP. respectively. (PDF 29 KB)Additional file 1:Dispersion plot of − Log10 for trait 13-week tenth rib backfat (mm) by EMMA and standardization SNPej. Dispersion plot for 44055 –log10  by efficient mixed-model association (EMMA) using the rrBLUP R package in the x axis, and by the standardization SNPej in the y axis. Red straight line is the reference line 0–1. (PDF 255 KB)Additional file 2:Variance components and LogLikehood for models with or without the segment for all chromosomes. (Results for the 18 chromosomes). (PDF 181 KB)Additional file 3:


SERPINA6, encoding corticosteroid binding globulin , and SERPINA1, encoding α1-antitrypsin (which inhibits cleavage of the reactive centre loop that releases cortisol from CBG). Three partially independent signals were identified within the region, represented by common SNPs; detailed biochemical investigation in a nested sub-cohort showed all these SNPs were associated with variation in total cortisol binding activity in plasma, but some variants influenced total CBG concentrations while the top hit (rs12589136) influenced the immunoreactivity of the reactive centre loop of CBG. Exome chip and 1000 Genomes imputation analysis of this locus in the CROATIA-Korcula cohort identified missense mutations in SERPINA6 and SERPINA1 that did not account for the effects of common variants. These findings reveal a novel common genetic source of variation in binding of cortisol by CBG, and reinforce the key role of CBG in determining plasma cortisol levels. In turn this genetic variation may contribute to cortisol-associated degenerative diseases.Variation in plasma levels of cortisol, an essential hormone in the stress response, is associated in population-based studies with cardio-metabolic, inflammatory and neuro-cognitive traits and diseases. Heritability of plasma cortisol is estimated at 30–60% but no common genetic contribution has been identified. The CORtisol NETwork (CORNET) consortium undertook genome wide association meta-analysis for plasma cortisol in 12,597 Caucasian participants, replicated in 2,795 participants. The results indicate that <1% of variance in plasma cortisol is accounted for by genetic variation in a single region of chromosome 14. This locus spans  Cortisol is a steroid hormone from the adrenal glands that is essential in the response to stress. Most cortisol in blood is bound to corticosteroid binding globulin (CBG). Diseases causing cortisol deficiency (Addison's disease) or excess (Cushing's syndrome) are life-threatening. Variations in plasma cortisol have been associated with cardiovascular and psychiatric diseases and their risk factors. To dissect the genetic contribution to variation in plasma cortisol, we formed the CORtisol NETwork (CORNET) consortium and recruited collaborators with suitable samples from more than 15,000 people. The results reveal that the major genetic influence on plasma cortisol is mediated by variations in the binding capacity of CBG. This is determined by differences in the circulating concentrations of CBG and also in the immunoreactivity of its ‘reactive centre loop’, potentially influencing not only binding affinity for cortisol but also the stability of CBG and hence the tissue delivery of cortisol. These findings provide the first evidence for a common genetic effect on levels of this clinically important hormone, suggest that differences in CBG between individuals are biologically important, and pave the way for further research to dissect causality in the associations of plasma cortisol with common diseases. The adrenal steroid hormone cortisol plays a vital role in adaptation to environmental stress. In response to stressors such as starvation, infection or injury, cortisol secretion is elevated by activation of the hypothalamic-pituitary-adrenal (HPA) axis. Cortisol acts predominantly through glucocorticoid receptors to induce a wide range of physiological responses, including liberating fuel (by facilitating gluconeogenesis and lipolysis), maintaining cardiovascular homeostasis (by inducing sodium retention and vasoconstriction), altering mood and memory (in favour of focusing on ‘fight or flight’ responses), and acting as a ‘brake’ on the innate immune response (preventing bystander damage from unrestrained inflammation) Epidemiological data suggest that subtle activation of the HPA axis associates with many of these traits within the population, in people who do not harbour the tumours which cause overt Cushing's syndrome. In these studies higher plasma cortisol concentration, measured in the morning, provided a robust marker of the activation of the HPA axis which accompanies high blood pressure, hyperglycaemia and dyslipidaemia Morning plasma cortisol has a heritability of 30–60% GC = 1.005, 10P values by chromosome for age- and sex-adjusted cortisol z-scores are shown in P values, listed in We conducted a discovery meta-analysis of genome-wide association studies (GWAMA) of morning plasma cortisol levels, investigating ∼2.5 M SNPs in 12,597 men and women, aged 14–102 years, of European origin (P = 4.0×10−12 after genomic control). The effect allele frequency was 0.22 and this variation explained 0.13% of the morning plasma cortisol variance. A forest plot showed consistent directional effects in all studies, with the T allele at rs12589136 associated with higher morning plasma cortisol (2 = 0.18).There was strong evidence for associations between plasma cortisol and genetic variation found at chromosome 14q32. In an additive genetic model, the lead SNP rs12589136 reported a per minor allele effect of 0.10 cortisol z-score , rs11621961 (1 kb downstream of SERPINA6) and rs2749527 (30 kb upstream of SERPINA6) as markers representing genome-wide significant signals in this region. Individually, the beta  for minor  alleles at rs12589136, rs2749527, and rs11621961 were 0.10 , −0.08 , and −0.08 ; joint analysis showed these SNPs have partially independent effects, with beta (95%CI) 0.07 , −0.04 , and −0.03 , respectively.A recombination boundary containing cortisol . A clumpSERPINA6 gene encodes corticosteroid binding globulin (CBG). The neighbouring (upstream) gene, SERPINA1, encodes α1-antitrypsin, the inhibitor of neutrophil elastase which cleaves and inactivates CBG The SERPINA6/SERPINA1 region  showed no evident inflation of test statistics (not shown). Conditional analyses adjusting for each of the partially independent genome-wide significant variants  in a subset of the meta-analysis population reduced the significance of all other SNPs to P>1×10−5 for an association with plasma cortisol.A quantile-quantile plot after removal of the SERPINA6 and SERPINA1 were identified as gene-wide significant (P<3×10−6); both included rs12589136 in the gene boundary.We used a gene-centric approach to analyse the combined effect of all SNPs within a gene, rather than individual SNP associations, using VEGAS 10P values highlighted for SNPs in these candidate genes showed that only SERPINA6 reached genome-wide significance (SERPINA6 was associated with plasma cortisol (SERPINA1 was not included in the candidate gene list.A list of 61 candidate genes thought likely to influence plasma cortisol was collated by a panel of experts. A Manhattan plot of the −logificance . Using gcortisol . SERPINAP = 0.0002), rs11621961 (P = 0.003) and rs2749529  allele at rs12589136 was not associated with ‘total’ CBG immunoreactivity but was strongly associated with the proportion of CBG bound by a monoclonal antibody against an epitope in the reactive centre loop of CBG SERPINA6/SERPINA1 region was associated with α1-antitrypsin concentrations in blood. However, α1-antitrypsin levels were negatively correlated with plasma total cortisol ; P = 0.002) and calculated free cortisol ; P = 0.021), although they did not correlate with ratio of intact/total CBG 0.02 .We explored the associations between rs12589136, rs11621961, rs2749527 with cortisol and CBG phenotypes in more detail in 316 subjects from the CROATIA-Korcula cohort . TogetheSERPINA6/SERPINA1 region associated with plasma cortisol. From 34 variants on the exome chip in this region , 9 were polymorphic in this sample , calculated free cortisol , and the proportion of CBG bound by the reactive centre loop antibody .After removing subjects with the rare Leuven variant (ie. heterozygotes), rs12589136 remained associated with total cortisol . In part, this is determined by differences in total CBG concentrations, and in part in association with a previously unrecognised variability in the immunoreactivity of the reactive centre loop of CBG. Since the process of CBG cleavage by neutrophil elastase and resultant reconfiguration of the reactive centre loop is considered important in the release of bioavailable cortisol within target tissues These results clearly attribute inter-individual differences in morning plasma cortisol amongst Europeans to genetic variation within a region on chromosome 14 containing the SERPINA6/SERPINA1 region of chromosome 14 accounted for <1% of the variance in plasma cortisol, this signal may be obscured by substantial unmeasured confounding and measurement error and may comprise a considerable component of the estimated 30–60% heritability of plasma cortisol The diverse actions of cortisol, and the striking clinical consequences of glucocorticoid excess or deficiency, have led many investigators to propose a central role for variations in cortisol levels in determining common quantitative traits. However, cortisol has not been measured as widely in epidemiological cohort studies as many other phenotypes. This may reflect the perceived difficulty of obtaining samples at a fixed time of day and in un-stressed conditions, to avoid confounding effects. The CORNET consortium had to decline samples from many cohorts in which time of sampling was inadequately controlled, and even then there was high variability in plasma cortisol. Thus, although the variants we identified in the SERPINA6Previous investigations of the genetic determinants of plasma cortisol  associated with impaired RCL antibody binding was associated with higher total plasma cortisol and higher cortisol binding activity. These observations are inconsistent with the interpretation that impaired RCL antibody binding represents enhanced RCL cleavage Cleavage of the reactive centre loop (RCL) of CBG by neutrophil elastase and inhibition of elastase activity by α1-antitrypsin has been recognised for more than 20 years SERPINA6We found further evidence for the importance of CBG using exome chips in the genetic isolate population of Korcula in Croatia, where we discovered an unusually high prevalence of heterozygotes for the Leuven mutation in SERPINA6 and SERPINA1 in conditional analysis, and the rare variant Leuven mutation was in linkage disequilibrium with the only rare variant we identified in SERPINA1. Recent studies have identified variants in SERPINA1 that are associated with coronary artery calcification 2 = 0.35). These findings are consistent with variation in the SERPINA6/SERPINA1 locus affecting downstream actions of cortisol, but it remains unclear if an interaction exists between the variants at these two genes. Mutations in SERPINA1 cause the syndrome of α1-antitrypsin deficiency, but we are not aware of any investigations of CBG or cortisol in these patients, and their HPA axis may be disturbed anyway by un-restrained neutrophil-mediated tissue damage. Rare variants in SERPINA1 (notably rs112635299) have been associated with α1-globulin plasma protein levels, of which α1-antitrypsin is a major constituent, using GWAS with 1000 Genomes imputation SERPINA6/SERPINA1 region will be required to clarify the basis for altered interaction between the two gene products.It is possible that a combination of alterations in CBG substrate as well as in neutrophil elastase level and/or activity may determine cleavage of CBG and tissue delivery of cortisol, especially in local sites of inflammation These findings emphasise the biological importance of plasma protein binding for steroid hormones, and are analogous to recent findings that a common variant in sex hormone binding globulin contributes to variation in total testosterone levels We performed a meta-analysis of genome-wide association studies of morning plasma cortisol in 12,597 subjects from 11 western European population-based cohorts: CROATIA-Vis (n = 885), CROATIA-Korcula (n = 898), CROATIA-Split (n = 493), ORCADES (n = 886), Rotterdam Study (n = 2945), NFBC1966 (n = 1195), Helsinki Birth Cohort Study 1934–44 (n = 451), ALSPAC (n = 1567), InChianti (n = 1207), PREVEND (n = 1151), and PIVUS (n = 919). Replication was tested in 2,795 subjects from three independent cohorts: Raine Study (n = 797), ET2DS , and MrOS-Sweden (n = 929). Cortisol was measured by immunoassay in blood samples collected from study participants between 0700 and 1100 h. Inclusion criteria were adults aged 17 years or older from Caucasian populations; exclusion criteria were current glucocorticoid use, pregnant or breast feeding women, and twins (exclusion of one). Characteristics of the study populations are presented in Each study performed single marker association tests, and study-specific linear regression models which used z-scores of log-transformed cortisol, additive SNP effects, and were adjusted for age and sex (model 1); age, sex, and smoking (model 2); or age, sex, smoking and body mass index (model 3). Imputation of the gene-chip results used the HapMap CEU population, build 36. In cohorts with consanguineous populations (ORCADES and Croatia), adjustments for principal components in kinship matrices were performed using ProbABEL; for other cohorts, Identity-By-Descent coefficients were calculated using PLINK and related participants excluded. In the majority of cohorts, participants were only included if blood samples had been obtained within a 60 minute time interval, when variations in time of sampling were ignored. In a subset of cohorts, samples were obtained over a wider time interval  and time of blood sampling recorded; for these cohorts, three further models were run as above but also including time of sampling, calculated as minutes from first sampling time, as an additional covariate −8 and poor imputation quality . Quantile-quantile (QQ) plots and genomic control (lambda) were used to confirm quality control. Sex chromosomes were not analysed.Quality control was carried out on the imputed genome-wide data for all 11 studies prior to meta-analysis; this excluded all samples with a minor allele frequency (MAF) <2%, call rate <95%, Hardy-Weinberg equilibrium (HWE)<1×10GC = 1.005. The genome-wide significance threshold for the meta-analysis was P<5×10−8. Percentage variation of cortisol was calculated from meta-analysis results as **(beta2/sd2). A regional plot was generated using LocusZoom We performed fixed effects meta-analysis, which used combined allelic effects weighted by the inverse of their variance for each of the models using the GWAMA program P-value<5×10−5 in meta-analysis were used for clumping. We grouped the SNPs within 500 kb of the index SNP that have r2 >0.2 with the index SNP.To detect independent top SNPs on the basis of empirical estimates of linkage disequilibrium between the SNPs, we used the clumping function as implemented in PLINK 2 = 0.905, D′ = 1.0) in all replication cohorts. Genotypic association analysis in these studies followed the same methods as those described above for the discovery meta-analysis, adjusting for age and sex.Genes identified in the meta-analysis were evaluated in the Raine Study, MrOS-Sweden, and Edinburgh type 2 Diabetes Study (ET2DS). Raine Study and MrOS-Sweden had GWAS data so we extracted the replication SNP results, and ET2DS was genotyped at the Wellcome Trust Clinical Research Facility Genetics Core Laboratory in Edinburgh using the OpenArray genotyping platform. rs2749527 failed manufacture for the OpenArray, and was replaced with rs2749529 , rs11621961 , and rs2749527  individually. We then did fixed effects meta-analysis using GWAMA program.We performed meta-analyses of sex-specific GWAS and conditional GWAS in a subset of populations, using single marker association tests of z-scores of log-transformed cortisol. For the sex-specific analysis, each study adjusted for age in both men  and women . For the conditional analysis, each study adjusted for age, sex, and each of the −6. The overlap of SNPs included in the gene boundaries in our results indicates this is likely to be an overly conservative correction factor We used the Versatile Gene-based Association Study (VEGAS) program A list of 61 candidate genes with known biological function in the regulation of cortisol was compiled by a panel of experts in the field. All SNPs within and ±50 kb of these genes were examined in the GWAMA results, and gene-based p values were inspected in VEGAS results.SERPINA6/SERPINA1 gene region  in the CROATIA-Korcula samples (n = 898) were extracted from an Illumina Exome Chip v1 analysis. Genotypes were called in GenomeStudio (Illumina) using the CHARGE Consortium joint calling cluster file (http://www.chargeconsortium.com/main/exomechip) −6); prephasing was performed using ShapeIt v2 Genotypes for the More detailed phenotyping was undertaken in samples from 316 participants in the CROATIA-Korcula cohort, comprising 158 age- and sex-matched homozygotes at the top hit variant rs12589136 . An additional 160 randomly selected samples had CBG measured, however two samples failed genotyping resulting in an additional 47 T/G and 111 G/G.3H]-cortisol  as the labelled ligand and dextran-coated charcoal to separate the CBG-bound [3H]cortisol, as previously described Total plasma cortisol was measured with a commercial radioimmunoassay . Total CBG was also measured in CROATIA-Korcula samples using a commercial radioimmunoassay . Total cortisol binding capacity was measured using a ligand-saturation assay, with [As CROATIA-Korcula is a population isolate, we used the polygenic and mmscore functions in GenABEL Table S1Characteristics of participants in cohorts included in genome-wide association meta-analysis and replication for morning plasma cortisol.(DOCX)Click here for additional data file.Table S2Genome-wide data characteristics for discovery cohorts used in meta-analysis.(DOCX)Click here for additional data file.Table S3−5 in discovery genome wide association meta-analysis for morning plasma cortisol.All SNPs with p-values <5×10(DOCX)Click here for additional data file.Table S4Top 10 genes identified in VEGAS as associated with morning plasma cortisol from genome wide association meta-analysis with adjustment for age and sex.(DOCX)Click here for additional data file.Table S5VEGAS results for candidate genes based on genome wide association meta-analysis results adjusted for age and sex.(DOCX)Click here for additional data file.Table S6Replication results in individual cohorts and by meta-analysis for association with morning plasma cortisol of SNPs representing independent signals in the SERPINA6/SERPINA1 region discovered in genome-wide association meta-analysis.(DOCX)Click here for additional data file.Table S7Associations with plasma cortisol of variants in SERPINA6/A1 locus on chromosome 14 identified by exome chip in n = 808 subjects from CROATIA-Korcula.(DOCX)Click here for additional data file.Text S1Supporting methods.(DOCX)Click here for additional data file.


We validate our methods by using empirical data from individuals with mixed European ancestry from the Population Reference Sample study and show that our approach is able to localize their recent ancestors within an average of 470 km of the reported locations of their grandparents. Furthermore, simulations from real Population Reference Sample genotype data show that our method attains high accuracy in localizing recent ancestors of admixed individuals in Europe . We explore the limits of ancestry localization under our approach and find that performance decreases as the number of distinct ancestries and generations since admixture increases. Finally, we build a map of expected localization accuracy across admixed individuals according to the location of origin within Europe of their ancestors.Ancestry analysis from genetic data plays a critical role in studies of human disease and evolution. Recent work has introduced explicit models for the geographic distribution of genetic variation and has shown that such explicit models yield superior accuracy in ancestry inference over nonmodel-based methods. Here we extend such work to introduce a method that models admixture between ancestors from multiple sources across a geographic continuum. We devise efficient algorithms based on hidden Markov models to localize on a map the recent ancestors ( These assumptions are clearly violated when individuals have ancestors from multiple unknown geographic regions, as occurs with recently admixed populations in the Americas (such as African-Americans) and more generally, individuals who have ancestry from multiple regions within the same continent . Recent works have circumvented this issue by first inferring segments of different continental ancestry  followed by independent application of PCA only on segments of specific continental ancestry , in which case local ancestry need not be inferred as individuals are heterozygous at all loci for both ancestries  . Assigning a small number of ancestors helps maintain computationally tractability and is biologically meaningful for various important use cases. Where admixture in the recent past is suspected, there will be a particular genomic signature of repeated observations of the same ancestral locations across many unlinked portions of the genome. Modeled this way, we both uncover the nature of the admixture events as well as combine the signal from each of these loci for increased accuracy of the locations of each contributing lineage. We also take advantage of the simple observation that if one allele is inherited from a specific ancestor, then most likely the neighboring alleles are also inherited from the same ancestor. Specifically, we use a model-based framework for genetic variation in the geographical continuum  to infer patterns of variation across geography in Europe . Our method is able to localize the grandparents of the admixed individuals in POPRES data within an average of 470 km of their reported ancestry. The accuracy is dependent on the specific ancestries and ranges from 305 km for individuals with Swiss and French ancestry to 701 km for those with Spanish and Portuguese ancestry. We use simulations from the POPRES genotype data to show that the localization accuracy within Europe decreases with increased number of ancestors and with the number of generations since the admixture. We also show that inference accuracy (at the genome-wide and locus-specific level) increases as distance among ancestors increases. Finally, we provide an analysis of ancestry localization error across all pairs of countries in Europe as resource for the community interested in subcontinental ancestry in Europe.We use empirical genotype data from the Population Reference Sample (POPRES) project  whereas other variants may not vary at all with geography . We perform inference in this model to find the ancestor locations on the map that maximize the likelihood of the observed genome  (see Diploid data with admixed ancestry). As the number of generations since admixture increases, the total number of ancestors to localize increases dramatically, making the inference problem very challenging. To account for this effect we limit the number of different locations for the recent ancestors for the maternal  haplotypes to a fixed constant , see below) with varying amount of contributions to ancestry (see below). We devise efficient algorithms to jointly optimize the locations of the ancestors as well as the proportion they contribute to the genome of the admixed individual (see below). For example, in the case of three generations ago, one ancestry location may contribute 1/8th to the admixed genome if only one ancestor comes from that location and may contribute 1/2 to the admixture process if half the ancestors come from that specific location. Finally, we note that the diploid model is symmetric making M and N interchangeable.Although we presented a simple example of our framework with two ancestors on the map, the model flexibly handles diploid data with arbitrary number of generations and ancestors localized on the map . However, these functions have been shown to capture general trends in common-variant frequencies sufficiently well enough in isolation-by-distance samples to produce highly accurate spatial assignment in individuals with nonmixed ancestry  the multisite haplotype of an admixed haplotype, where L is the number of SNPs typed across the genome and ih ∈ {0, 1} encodes the number of reference alleles at SNP i. Due to the admixture process, the haplotype can be viewed as a mosaic of segments coming from ancestors from multiple locations on the map. We define variables Z =  as indicators for an allele coming from ancestry location j  and write the likelihood of the haplotype data as function of ancestry locations X. The likelihood for a given admixed haplotype data having M ancestry locations X =  where each ancestry contributes proportionally with Π =  is defined as:Z encodes the mosaic structure of the admixed haplotype  and can be modeled using a Markov chain as follows:τ = {τ1, …, Lτ−1} stand for the recombination probability (within the past g generations) between each two neighbor loci. The alleles present at a site i on a haplotype is modeled as a Bernoulli variable with a success probability given by the allele frequency a and b are estimated beforehand. Thus, they should be regarded as fixed parameters in this article. An illustration of the model is given in For simplicity, we start by introducing the model for haploid data and extend it to genotype data in the next section. Denote by M ancestral locations given data for an admixed haplotype, followed by posterior decoding in the HMM to obtain locus-specific predictions. This can be achieved by maximizing the likelihood function (2) with respect to X. By treating X as parameters and Z, Π as hidden variables, this maximization falls within the procedure of the standard expectation-maximization (EM) algorithm  and ancestral proportion Π(t):α/β are standard forward/backward HMM functions and can be efficiently calculated (see Supplementary Note).The expectation step is similar to the forward-backward algorithm for HMM, which calculates the posterior probability of hidden variables X and in Π , Π(t)) from the E step, and the shorthand iq(jx) is defined as:ijD and ijE denote constants calculated from the posterior P, Π(t)) in the E step.The maximization step alternatively optimizes the Q functions in Supporting Information, File S1.We perform the maximization by taking advantage of the convex properties of the equation and using analytical forms for the Hessian of the function. The complete derivations are given in To give an overview of the whole EM algorithm, an illustration of the aforementioned EM algorithm is given in X∗, we compute the posterior probability for Z, which leads to a locus specific assignment of ancestry at each allele in the genome. The most probable local ancestral locations are found by maximizingFile S1).Having obtained the maximum likelihood geographical locations lgorithm . To compM paternal ancestry locations X =  with proportions Π =  and N maternal ancestry locations Y =  with proportions Ω = . Denote by g =  the multisite genotype of an admixed genotype, where L is the number of SNPs typed across the genome and ig ∈ {0, 1, 2} encodes the number of reference alleles at SNP i. Then the likelihood becomes:pZ and mZ now encode the ancestry status of the paternal  alleles  and can be modeled through the same Markovian process as:if is the allele frequency function in logistic form (1). The probability P(Z) models the recombination events in paternal and maternal ancestries, and the probability We next extend the haploid model to genotypes by considering M + N ancestral locations for a given mixed individual genotype. This can be achieved by maximizing the likelihood function (3) with respect to X and Y, which, analogous to the haploid case, can be performed using the EM algorithm  and Y(t).α and β can be calculated recursively using a procedure similar to the forward-backward algorithm for HMMs.In short, the expectation step is similar with forward-backward algorithm in HMM, which calculates the posterior probability of hidden variables X, Y Π and Ω. In detail, it first optimizes the Q function in X and Y by fixing Π and Ω. Second, it optimizes the Q function in Π by fixing X and Y and Ω. Third, it optimizes the Q function in Ω by fixing X, Y and Π. These two steps are performed alternatively until the maximization converges. The Q function in X and Y in the first step can be derived as followsijkC denotes the posterior E step, and the shorthand iq is defined as:ijD and ijE denote constants calculated from the posterior E step. We omit the Q function in Ω in the third step here as it is very similar with the aforementioned function.The maximization step alternatively optimizes the Q functions in File S1. As noted in File S1, the Q function Q, Y(t), Π(t), Ω(t)) in the first step is not concave in general. However, we can still use convex optimization techniques to get a local optimal solution. In practice, we observe that the function is concave almost all the time.As in the haploid case, we leverage the convexity of the function and analytical forms for the Hessian to efficiently optimize the Q function. The complete derivations and optimization details are given in X∗ and Y∗ for each ancestry, we can compute the posterior probability for pZ and mZ, which leads to the spatial local ancestry inference. The most probable local ancestral states are obtained by maximizingE step in the algorithm (see Supplementary Note).Having obtained the maximum likelihood geographical locations lgorithm . The posi.e., the paternal haplotype and maternal haplotype are from the same set of ancestral populations. We allow for this case by setting M = N and enforce a constraint jx = jy in the M step.In the aforementioned notations we derived the general solution that allows for paternal and maternal ancestries to be different from each other, which is suitable for applications of inference of parental locations or grandparent locations. A simplifying case is when maternal and paternal ancestries are homogeneous, We applied our methods to a data set collected from European populations, which was assembled and genotyped as part of the larger POPRES project φ(id+1 − id) where id′s are the locations of each SNPs, g denotes the number of generations, and φ is the probability of one recombination per generation per base-pair  followed by adjustment of the transition rates in our model by a factor of 10−2. This adjustment factor is used to regularize the recombination rate, such that the number of inferred short ancestry windows is appropriate for the number of generations and distinct ancestries and will vary with these parameters. Results at different LD pruning levels and the corresponding adjustment factors are reported in Table S1 and Table S2.Although our approach models admixture LD, it assumes that markers are independent conditional on local ancestry (no background LD). Preliminary results (not shown) that used transition rates based on the assumed number of generations and recombination rate (g − 1)φ(id+1 − id) where id′s are the locations of each SNP in base pair, g denotes the number of generations, and φ is the probability of one recombination per generation per base pair . To account for the distance among ancestries, we also compute the relative prediction error, defined as the ancestral location prediction error divided by the distance between the true ancestry locations used in simulations. Note that we use as the “true” ancestral locations for the admixed individual the set of country centers from the M + N ancestries.We use several metrics to assess performance of SPAMIX in simulations and real data. For the ancestral location prediction, we evaluate the results by computing the average geographical distance between predicted locations and true locations in simulations  from various areas in Europe to serve as “ancestor” genomes. We used them to simulate an admixed individual and attempted to recover the original ancestral locations using the simulated genome and a set of logistic gradients inferred from the remaining unused individuals . SPAMIX attains an ancestry localization accuracy  for individuals with two recent ancestors in Europe of 550 km .We investigated the performance of our model through simulations from the POPRES data  in our approach. Eliminating loci found in strong disequilibrium with each other (LD pruning) was observed to increase accuracy even though the model had less data to use (see Table S1 and Table S2); therefore, all results in the main text are obtained after LD pruning .To test how much is gained by explicit modeling of correlations among SNPs induced by segments of recent shared ancestry (admixture LD), we also inferred the recent ancestry location using a naïve model that assumes all SNPs to be independent  compared with unphased diploid data. It is increasingly often the case that access to pedigree data allows haplotypes to be determined with high accuracy. Therefore, we quantified the gain in ancestry localization accuracy arising from having access to phased haplotype data , which we expect to continue as the number of generations increases . That is, the frequency of a given variant is allowed to change in a given direction on a map only according to a parametrized logistic function. Although this approach has been shown to provide a good approximation of common variation that leads to accurate ancestry inference, we hypothesize that the error in fitting logistic gradients to real data limits the method’s accuracy. To assess this scenario, instead of using real individual’s haplotype data, we simulated admixed haplotypes directly from the logistic gradients we inferred from POPRES data . We observe a large increase in accuracy in this idealized scenario as compared to simulations from real haplotype data .Our framework models genetic variation as function of geography by assuming a logistic gradient for the spatial distribution of genetic variations  to estimate for a given admixed individual. For a fixed number of generations (four), we varied the number of ancestry locations to estimate. The parental inference is different from two ancestry inference, as the parental inference assumes that one haplotype is from paternal ancestry and one from maternal ancestry. However, the two-ancestry inference assumes that both of the haplotypes are mosaic of two ancestries (M = N = 2). As expected, we observe decreases in performance as the number of ancestral locations increases. For example, the average prediction error increases from 550 for two ancestries to 639 km for four ancestral locations  compared with closely related ones   decreases with the distance between ancestries in Europe . This effect is presumably due to assignment errors in the local ancestry that have a much bigger impact if the ancestral locations are further apart. Although fewer local ancestry errors are being made with increased distance (see Locus-specific inference), these errors have a stronger impact on the ancestral localization due to their higher distance to the true location.It is well known that accuracy of ancestry inference correlates with genetic distance between ancestral populations. Discrete local ancestry can be inferred with very high degree of accuracy in mixtures of highly diverged populations  increases with the distance between ancestral locations . Similar to the ancestor localization, we observe that although the total number of assignment errors is reduced with increased distance, these errors have a bigger impact when averaging across all sites to compute the average allele localization error. Therefore, we observe that the average local ancestry prediction error is increased as distance between ancestral locations is increased.An advantage of our framework is that in addition to identifying the most likely locations of the recent ancestry of admixed individuals, it can also provide an assignment of each allele in the genome to each ancestry location. We observe that local ancestry prediction accuracy . e.g., Portugal, Spain, Italy), and increased performance for subcontinental admixtures from populations located geographically in the center of Europe  . This can be an effect of biased sampling in the POPRES data, that sampled more individuals from European center, but also can be an effect of having more information to localize individuals in SPAMIX. In general, we observe a prediction accuracy ranging from 411 km for admixtures from Spain and Italy to 641 km for individuals with recent ancestors from Spain and the United Kingdom.We also investigated the variance in performance according to the ancestor’s labeled origin . A total of 470 admixed individuals were analyzed using SPAMIX (see Table S3). The error distance is lower than simulated experiments likely due to the large proportion of the admixed individuals of French and Swiss ancestries, which can be accurately localized (average of 305 km). As discussed previously, we note that SPAMIX ancestor localization performance varies greatly across Europe with ancestors from pairs of countries localized at the boundary of European map being harder to localize .Finally, we investigated whether high accuracies observed in simulations also can be attained in real data. Using SPAMIX, we localized the recent ancestry of all admixed European individuals from POPRES . For example, we are working on using a logistic quadratic function to model the spatial genetic structure, which is a generalization of the original SPA model. It will further enhance the admixed ancestor inference.Table S2). Such LD adjustments have proved fruitful in improving localization accuracy for unadmixed individuals . We found it necessary to modify the transition rates used in our inference by a multiplicative factor based on the level of LD pruning applied to the SNP list  and missing data . Alternative inference approaches can be taken, for example, the likelihoods we define in Equations (2) and (3) could be used in a Bayesian Markov Chain Monte Carlo approach method to sample from the posterior distribution of the spatial prediction . Future work could improve on our framework by providing constraints to the optimization procedure or by a postoptimization adjustment . In addition, the underlying SPA approach works best in the simple isolation-by-distance settings and is not expected to work well in for complex scenarios with barriers or irregular geometries. We leave a full investigation of such modifications of our framework as ongoing work and recommend the use of cross-validation approaches with individuals of known ancestry to assess the suitability of the model prior to application.Our work provides a framework of predicting ancestral locations for admixed individuals which can be further improved. For example, due to the continuous nature of the approach, ancestral locations can be predicted to be outside of standard geographic boundaries (A direct benefit of the proposed model is that it leads to efficient optimization procedures for tackling inference problems such as localization of ancestors in the genetic-geographic map. In this work we have presented such an algorithm based on the well-known EM procedure, which leverages the HMM of the admixture process joint with the gradient representation of genetic variation as function of geography.


N = 82,315) and putamen volume , with approximately 9.3 million SNP z-scores in both cases. We show that, over a broad range of z-scores and sample sizes, the model accurately predicts expectation estimates of true effect sizes and replication probabilities in multistage GWAS designs. We assess the degree to which effect sizes are over-estimated when based on linear-regression association coefficients. We estimate the polygenicity of schizophrenia to be 0.037 and the putamen to be 0.001, while the respective sample sizes required to approach fully explaining the chip heritability are 106 and 105. The model can be extended to incorporate prior knowledge such as pleiotropy and SNP annotation. The current findings suggest that the model is applicable to a broad array of complex phenotypes and will enhance understanding of their genetic architectures.Genome-wide Association Studies (GWAS) result in millions of summary statistics (“z-scores”) for single nucleotide polymorphism (SNP) associations with phenotypes. These rich datasets afford deep insights into the nature and extent of genetic contributions to complex phenotypes such as psychiatric disorders, which are understood to have substantial genetic components that arise from very large numbers of SNPs. The complexity of the datasets, however, poses a significant challenge to maximizing their utility. This is reflected in a need for better understanding the landscape of z-scores, as such knowledge would enhance causal SNP and gene discovery, help elucidate mechanistic pathways, and inform future study design. Here we present a parsimonious methodology for modeling effect sizes and replication probabilities, relying only on summary statistics from GWAS substudies, and a scheme allowing for direct empirical validation. We show that modeling z-scores as a mixture of Gaussians is conceptually appropriate, in particular taking into account ubiquitous non-null effects that are likely in the datasets due to weak linkage disequilibrium with causal SNPs. The four-parameter model allows for estimating the degree of polygenicity of the phenotype and predicting the proportion of chip heritability explainable by genome-wide significant SNPs in future studies with larger sample sizes. We apply the model to recent GWAS of schizophrenia ( Thus, there is a need for the development of analytical methods appropriate for the many phenotypes that have, or are expected to have, high polygenicity , a quantity that has not hitherto been a focus of much research. This quantity requires knowing the full distribution of test-statistics. In line with the parsimony of the model, the parameters will be directly interpretable—for example, one gives an index of the polygenicity—and being able accurately to estimate them and their uncertainties is a central component in this study. These will be used, for example, in power calculations to predict the proportion of additive chip heritability (which in turn is the proportion of phenotypic variance explainable by additive genetic effects of common SNPs assayed by GWAS arrays) that is discoverable as a function of sample size.  Other recently developed methods that enable estimating chip heritability and proportion of variance explained are LD Scoring  In these Gaussians, it is important to incorporate the allele SNP heterozygosity . Null and non-null effects distributed throughout the genome could be captured by using only a single Gaussian; a two-groups mixture of Gaussians distribution has been used for ubiquitous null and sparse non-null effects; modifying this slightly will additionally allow for ubiquitous non-null effects: dedicating one Gaussian to ubiquitous null and non-null  effects, and the other to sparse (large) non-null effects. Intuitively, sparse effects represent SNPs that are in strong LD with causal SNPs , while the ubiquitous non-null effects largely arise from weak LD with causal SNPs, and the null effects—effects that do not replicate—arise from environmental and error contributions.It is the relatively large effects that GWAS have discovered in recent years, yet for many phenotypes it appears that very large numbers of much smaller effects remain unidentified  shows that the z-score for a given SNP can be written as a sum of genetic effect, δ, and a remainder term, ϵ, encompassing environmental and error contributions, assumed to be independent of δ: z = δ + ϵ, where H is the heterozygosity , H = 2p(1 − p), p being the allele frequency for either of the two SNP alleles . Thus, the variance of z is N · H. We introduce a four-parameter two-component Gaussian mixture model for the marginal distribution of z-scores assuming SNPs belong to a class of ubiquitous effects or to a class of sparse effects:Assuming a linear relationship between a quantitative phenotype and genotype (logistic relationship for case-control designs), a massively univariate, or marginal regression, approach with effective sample size 0 is the prior probability  that a SNP is in the ubiquitous class (π0 ≈ 1) of “small” replicating effects (described by 1 = 1−π0 is the prior probability that a SNP is in the sparse class of “large” replicating effects (described by 1 is the fraction of independent SNPs characterized by the broader (sparse) normal probability distribution function (PDF), which we denote the index of polygenicity (π1 ≪ 1); ϕ is the normal PDF with mean μ and variance σ2; all SNPs have a component that is a null effect, i.e., a non-replicating error/environmental contribution , is the probability that z corresponds to the π1 arm of Equation 1, i.e., the posterior probability that a SNP with this z-score has a sparse effect:In the usual terminology applied to two-groups mixture models Efron, , the Bayz, is a weighted sum of sparse and ubiquitous contributionsThen, from Equation 1, it is easy to show that the posterior distribution of effect sizes δ, given where,. Note that from Equations 2, 3, and 5,E denotes expectation. Alternatively, based on Equation 5, we can write δ = δu + δs for non-sparse and sparse effects, respectively, wherewhere The objective, then, is to determine the empirical distribution of z-scores and the four model parameters that best characterize the data, assess the quality of the resulting model fit, and address its implications. Below we describe how the replication probability can be measured empirically. Of particular interest will be the accuracy of the model prediction of this quantity.N, heterozygosity H, and the four model parameters: z as the adjusted effect size, with context making it clear what is meant. .As there might be confusion about the terms bias and effect size, a note of clarification is in order. A true regression coefficient β for association between genotype and phenotype  can be thought of as a true effect size. An estimate 0. Independent of correlated errors, random genotyping errors will lead to underestimation of the regression coefficient—the effect of regression dilution, or regression attenuation ) will result in smaller Wald statistics, all else equal. Genotype measurement error, however, may induce correlation among the genotypes and have a biased effect, operating in a manner similarly to population structure, resulting in a positive contribution to the null variance component σ Fuller, .We analyzed summary statistics (z-scores) from the PGC schizophrenia sample of 35,476 cases and 46,839 controls across 52 separate substudies, with 9,279,485 genotyped and imputed SNPs; restricting allele frequency to be greater than 0.005 reduced the number of SNPs by 2% . We randomly and repeatedly divide the data into complementary discovery and replication sets, and calculate the empirical expected z-score, and the expected square of the z-score, in the replication set, given a z-score in the discovery set. We also calculate empirical posterior estimates of the variance of the effect size in the replication set given a z-score in the discovery set, and the replication probability (defined below) for z-scores in the discovery set.1 and N, N. Thus, to separate out these contributions and to examine the effects of sample size on posterior expectation values, the division of the data into complementary discovery and replication sets is not just a single division, e.g., split-half. Rather, we repeatedly and randomly divide the data with 10% of the effective sample in the discovery set (90% replication), then do the same in increments of 10%, through 90% of the effective sample in the discovery set (10% in replication). In the current work we do not use raw genotype data, but summary statistics from 52 studies with uneven distribution of sample sizes. So, for example, random draws are made from the 52 studies so that the studies selected for the discovery set will comprise approximately 10% of the total effective sample size, and so on for the other percentage breakdowns. The number of random draws per each percentage breakdown was 100, sufficient to provide smooth empirical a posteriori estimates of the quantities of interest. For each random draw, the SNPs were randomly pruned: for SNPs in LD, corresponding to correlation coefficient r2 ≥ 0.8, the SNP selected was randomly chosen—not necessarily the one with largest z-score. Using a fixed correlation coefficient allows for uniform pruning such that each LD block is treated equivalently (large block not over-represented); randomly selecting the representative SNP explicitly avoids the “winner's curse”  provides an empirical estimate for the posterior expectation value of zr given zd, E(zr|zd). Note that the empirical zr, being a mean in the replication set, is a direct estimate of the effect size δr in the replication set corresponding to zd in the discovery set: E(zr|zd) = E(δr|zd). Similarly, empirical estimates for For a given discovery-replication division of the data, z-scores from the discovery set were binned in 200 equally-spaced bins between zd for some SNP, the effect can be deemed to replicate if the corresponding z-score zr in the replication sample has the same sign as zd, and its p-value from a one-tailed test, based on the standard normal cumulative distribution, is less than a chosen threshold, say pt = 0.05, corresponding to −|zr| < zt = −1.645. For binned discovery sample z-scores, the empirical replication probability for a given bin is defined as the fraction of z-scores in the bin that replicate, i.e., have replication-sample p-value pr < pt. As before, averaging these over the 100 repetitions (for a given percentage breakdown) provides an empirical estimate of the replication probabilities, R, which we also denote Pr(pr < pt).Given a discovery sample z-score Nd and a new replication sample of effective sample size Nr, and noting that effect sizes are proportional to the square root of effective sample sizes, the posterior distribution for zr given zd is given by a modification of Equation 5:For a discovery sample of effective sample size wheretdr, μu, and μs on zd and Nd, and of σu and σs on Nd, has been dropped to simplify the notation). Since z = δ + ϵ, with δ and ϵ assumed to be independent and E(ϵ) = 0, the expected effect size δr in the replication sample, given a z-score zd in the discovery sample, can be read off from Equation 14:, it also follows thatR for a given zd is the probability, in the complementary replication sample, of getting a z-score zr  more significant than a chosen threshold, zt ≤ 0 , which is simply the proportion of z-scores more significant than the threshold, which in turn is given by the cumulative distribution function (CDF) version of Equation 14:The replication rate m and variance s2, evaluated at zt.corresponding to the normal PDF with mean Pr(z|δ) is simplyThe likelihood p-value p) that will pass significance threshold pt, given the expected effect size δ, is just the CDF:Thus, the probability of having a z-score  posterior expectation values as δ and η, the cost function The four model parameters were estimated by minimizing a convex cost function that was composed of a sum of two terms: the weighted sum of the squares of the differences between empirical estimates and model estimates of effect sizes (expected z-scores) and of the expected z-scores-squared . More specifically, denoting empirical replication z-scores as i is over the 9 different percentage breakdowns of the dataset into complementary discovery-replication fractions; the sum over j is over the 100 repetitions for each breakdown; and the sum over k is over the 200 discovery sample z-score bins. For a given percentage breakdown i and repetition j, k (note again that the mean replication z-score for a given bin provides a direct estimate of the effect size), and j through variation in heterozygosity H from repetition to repetition); k, and ηijk is the corresponding model prediction. For given i and j, the weighting wijk is the number of SNPs (z-scores) in the k-th bin. With i indexing the ratio Nr ∕ Nd and k indexing zd , δijk ≡ E(zr|zd) is given by Equation 19 (E(ϵ) = 0), and Here, the sum over N, the posterior expectation of the square of the “true” effect size for that sample isIn particular, for a given sample of effective size tdr, μu, and μs on z and N, and of σu and σs on N, has been dropped to simplify the notation). Best-fit model parameters were determined by Nelder-Mead minimization of fminsearch. explained by sparse effects, S, for SNPs with −|z| < zt for some threshold zt < 0, or equivalently with p-value less than the corresponding threshold pt, can be estimated by simulating z-scores for all SNPs, whereby the unobservable null, ubiquitous, and sparse effects can explicitly be assigned to individual SNPs. From Equation 1 and the implicit decomposition z = δ + ϵ, all of the simulation SNPs i  are assigned an environmental/error component ϵi drawn from a normal distribution with mean zero and variance given by the estimated 1 of the SNPs (indexed by k)—conceptually, SNPs that are in strong LD with causal SNPs—are assigned an additional component δc, k drawn from a normal distribution with mean zero and variance k is zk = δc, k + ϵk.  component δu, j drawn from a normal distribution with mean zero and variance zj = δu, j + ϵj. Note that the total effect size for a SNP in the π1 category can be decomposed into “ubiquitous” and “sparse” components: δc, k = δu, k + δs, k.) The proportion of chip heritability explained by SNPs with sparse effects is given by the ratioFor a given sample size kth SNP, emphasizing its dependence on N . The numerator and denominator can be averaged over several repetitions for a smooth estimate of S. Alternatively, given a set of z-scores {zk}, replace N large enough so that S approaches 1. In any case, the effects of LD can increasingly be mitigated by higher levels of pruning.where p-value , and then reassessed at stage-2 in an independent replication data set, with the final assessment for significance being from the combined datasets , and let Ndr = Nd + Nr denote the sample size for the combined dataset. Define the inverse-variance weights zd, the posterior expectation for the z-score zdr in the combined dataset isIt is common practice to estimate effect sizes in a multistage design where, at stage-1, candidate SNPs are selected in a discovery dataset using a liberal dr is the effect size and E(δr|zd) is given by Equation 19. The variance is simplywhere δzr|zd) is given by Equations 20 and 24, thus allowing Rdr in the combined dataset for zd in the discovery dataset, which we can also write as Pr(pdr < pt) where pdr is the p-value for the SNP in the combined data set, is given by Equation 21, only replacing zt on the right-hand side with where var, with zt = −5.33  with . Thus, for a SNP whose discovery sample z-score is zt = −1.64 , for split-half discovery and replication samples. It is significant that in (A), in a neighborhood of approximately ±3 around zd = 0 the expected effect size is non-zero: the black line has a positive slope. This implies that there exists ubiquitous non-null “small” effects . As the neighborhood extends further, “large” effects found in the discovery sample correspond to “large” effects in the replication sample.For a range of z-scores in the discovery sample between −6 and 6, Figure The parameter estimates  for schizophrenia were:a = 0), then without changing the other parameters the fit corresponds to the green dashed line: in the ±3 neighborhood around zd = 0, small discovery effects do not replicate , while sparse “large” effects do replicate, in only approximate agreement with the empirical estimates. The alternative scenario assumes no sparse effects, σb = 0: without altering the other parameters, the model fit is shown by the blue dashed curves in Figure zd. Corresponding curves for explicit model fits with parameters (b = 0 the model is degenerate with respect to π1).. The model fit with these parameters is shown as the solid red curve in Figure For putamen volume, the parameter estimates  were:Thus, putamen volume is approximately 40-times less polygenic than schizophrenia.Discovery and replication sample sizes have a pronounced influence on the empirical estimates of effect size, variance, and replication rate, but these dramatic changes across a wide range of z-scores are remarkably well matched by the model estimates  and δs (red curve), given by Equations 12 and 13, for z-scores between -6 and 6, for an effective sample size N ≃ 34, 000. Also shown is the total effect size (black) and the posterior variance of the effect size (blue curve). The individual components can be seen to behave as expected: the green curve shows the ubiquitous non-null effects increasing away from the origin, reaching a peak, and falling to zero for large values of z; the red curve shows the sparse effects component essentially flat near the origin, indicating a lack of sparse effects in this neighborhood, and then monotonically increasing, beginning near where the ubiquitous effects start falling to zero. The peaks in the variance can be seen to arise from the regions where sparse effects are already prominent but the small effects have not yet died off. Figure r2 ≥ 0.8, we further restricted to SNPs whose total linkage disequilibrium (TLD), given by the sum of their LD r2's with neighboring SNPs, was less than 15 , reducing the number of SNPs to ≃1 million . The empirical effect sizes, for a sample breakdown of 50% discovery and 50% replication, are shown in Figure r2 ≥ 0.8 without restricting by TLD , the abscissa gives the proportion of SNPs whose p-values are at least as significant as the ordinate value; the dashed line at 45° corresponds to the null hypothesis where the distribution of SNP z-scores is assumed to follow a standard normal distribution . The model plots are given by Equation 1, replacing the PDF with the CDF, and provide a remarkably good fit to the empirical plots. The earlier deviation of the schizophrenia plot, compared with the putamen plot, from the null line is due to the higher polygenicity of schizophrenia.Figure p ≤ 5 × 10−8) for schizophrenia and putamen volume, as a function of the total number N of subjects in the samples , as given by Equation 26. The fraction of tagged variance explained by GWAS is expected to approximately equal the fraction of additive genetically-determined phenotypic variance, or narrow-sense heritability, explained. The blue asterisk indicates the sample size from current ENIGMA data ; the green asterisk, for schizophrenia, gives N = 76, 326, assuming the effective sample size from the current PGC2, Neff = 38, 163, arose from an equal number of cases and controls: N∕2 = Ncases = Ncontrols = Neff. Thus, we estimate that 15% of chip heritability for schizophrenia is currently explainable by genome-wide significant SNPs in PGC2; for these SNPs, the replication rate at pt = 0.05 for our split-half sample is 97% or higher. GWAS on approximately half a million each of cases and controls would need to be performed to explain all the chip heritability for schizophrenia. For putamen volume, 14% of chip heritability appears to be explainable by genome-wide significant SNPs given the current sample size in ENIGMA. In contrast to schizophrenia, however, approximately only 100,000 people would need to be assessed to fully explain chip heritability for putamen volume. The higher sample size requirements for schizophrenia are due partly to its higher polygenicity.Figure z, to the phenotypic variance va of the trait is usually estimated to be z2 ∕ N, since z = δ + ϵ), so that the per-allele contribution to phenotypic variance is proportional to E(δ2|z)∕N, not z2 ∕ N. Figure E(δ2|z) vs. z2  is given by the ratio of the height on the black dotted line to the height on the appropriate curve, at z2. For example, for schizophrenia with a total sample size of N=50,000 and a z-score on the threshold of genome-wide significance (z ≃ ± 5.33), va will be over-estimated by a factor of 2.2 (the height of the black dot relative to the blue dot).The per-allele contribution of a locus, with z-score N = 76, 326), where the discovery sample size Nd is 20, 50, or 90% of the total, as a function of p-value in the discovery dataset. For example, in discovery samples equal to 50 or 90% of the total, to have a probability of reaching genome-wide significance in the combined dataset approximately equal to 0.8 would require having reached genome-wide significance in the discovery dataset; for a discovery sample only 20% of the total, the same probability of replicating requires the discovery sample p-value to be at least p = 10−9, i.e., more significant than the genome-wide threshold.Figure Here we present a simple modeling framework, a scale-mixture of Gaussians that is a modification of previously published methodologies , whereas we have shown here that it is more appropriate to consider the effect sizes being described by a Gaussian mixture distribution, with non-null ubiquitous and sparse components. It is not clear how this affects the result for h2. A follow-up study, using all of the subjects and SNPs employed here, reported that 3.4% of variation on the liability scale to schizophrenia, adjusted for case-control ascertainment, is explained by genome-wide significant loci  We presented a four-parameter two-groups mixture of normals parametric model for the distribution of GWAS summary statistics. Additionally, we presented an empirical scheme for estimating replication z-scores, their variances, and replication rates. The model parameters are then estimated by minimizing a cost function that depends on the differences in model and empirical estimates of effect sizes and expected z-scores-squared. Applying the model and empirical scheme to recent large GWAS of schizophrenia and putamen volume, we showed that effect sizes, along with variances and replication rates, are accurately described by the simple model over a wide range of z-scores, and over a wide range of discovery and replication sample sizes when the relationships can change dramatically. We further showed how the model can be used to estimate the fraction of additive SNP heritability (an estimate of the fraction of chip heritability) explainable by genome-wide significant SNPs in a massive univariate setting, as a function of sample size. The model enables estimation of an index for the degree of polygenicity of a phenotype, because structurally it allows for separate distributions of the large number of ubiquitous  effects and the relatively small number of sparse (large) effects, postulating that these different classes of effects will be distributed differently. We showed that ignoring the contribution of ubiquitous non-null effects severely degrades the accuracy of the model. A variant of the model, particularly for phenotypes likely to be of low polygenicity, e.g., individual gene expression or methylation, where interpretation of the ubiquitous component as currently presented might become problematic, could incorporate a third gaussian for ubiquitous null effects, with variance parameter p-value less than some threshold—indicating a potential candidate for true association with the phenotype—an accurate estimate of its probability of replicating can be calculated for replication samples of different effective numbers of subjects. It should be noted, however, that to have reasonable probability of replicating in a multistage GWAS, it might be necessary that candidate SNPs in the discovery sample substantially exceed genome-wide significance. The model can be improved and extended by letting the model coefficients depend on SNP heterozygosity, and by incorporating additional information like SNP functional annotation category , NIH GM104400-01A, the Research Council of Norway , the South-East Norway Regional Health Authority (#2013-123) and KG Jebsen Foundation.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.


FTO) gene that may confer different risk for obesity depending on transmission from mother or father. Genome-wide genotypes and pedigree information from the Sorbs population were used. Phased genotypes among 525 individuals were generated by AlphaImpute. Subsequently, 22 SNPs within FTO introns 1 to 3 were selected and parent-of-origin specific association analyses were performed using PLINK. Interestingly, we identified several SNPs conferring different genetic effects (P≤0.05) depending on parental origin—among them, rs1861868, rs1121980 and rs9939973 . To confirm our findings, we investigated the selected variants in 705 German trios comprising an (extremely) obese child or adolescent and both parents. Again, we observed evidence for POE effects in intron 2 and 3 (P≤0.05) as indicated by the parental asymmetry test. Our results suggest that the obesity risk transmitted by several FTO variants may depend on the parental origin of the allele. Larger family-based studies are warranted to replicate our findings.Genome-Wide Association Studies (GWAS) were successfully applied to discover associations with obesity. However, the GWAS design is usually based on unrelated individuals and inheritance information on the parental origin of the alleles is missing. Taking into account parent-of-origin may provide further insights into the genetic mechanisms contributing to obesity. We hypothesized that there may be variants within the robustly replicated fat mass and obesity associated ( FTO gene which was discovered in 2007. Variants at FTO locus confer the largest genetic effect size among all obesity susceptibility loci and was robustly replicated in multiple populations and under various study designs ..N = 948)2 and for which pedigree information was available (141 families) were included in the present study. Identity-by-state among 525 individuals was 0.743 mirroring the degree of relatedness. 47 individuals out of 525 are affected with T2D. To avoid potential confounding we adjusted the analysis for T2D state. Characteristics of the analyzed individuals from the Sorbs population are summarized in The Sorbs were extensively phenotyped for a wide range of anthropometric and metabolic phenotypes including past medical and family history oGTT, T2D, weight, height, WHR and BMI as described elsewhere . 525 indFTO SNPs was performed in a GWAS data set (genotyped by Affymetrix Genome-Wide Human SNP Array 6.0) of 705 trios comprising an extremely obese child or adolescent and both biological parents – all of central European ancestry. All index cases were at least overweight (BMI ≥90th percentile), and 83.8% were extremely obese (BMI ≥99th percentile) compared to reference data from the German National Nutrition Survey I  was used for genomic DNA extraction based on its protocol. The microarray processing and genotype calling was described in detail elsewhere . We inclFTO-SNPs from intron 1 to intron 3 were selected .In total, 387,837 SNP markers  overlapping from 500K Affymetrix GeneChip and the Affymetrix Genome-Wide Human SNP Array 6.0 were included in the analyses. All analyses were standardized to the forward strand. 27 FTO were re-genotyped in the Sorbs data set using the TaqMan SNP genotyping system (Life Technologies) according to manufacturer’s protocol. Fluorescence was detected using ABI PRISM 7500 Sequence Detecting System. To guarantee genotyping reproducibility, a random ~5% of the samples were re-genotyped in all SNPs; all genotypes matched initial designated genotypes.To exclude technical artifacts not captured by the quality filtering in the GWAS the variants rs8050136 and rs8053740 within FTO introns 1 to 3 the average phasing rate per SNP was 89%. Particularly, 5 out of 27 SNPs, showed a missing rate per SNP ≥ 30% and were removed from further analysis. Missing rates per SNP are listed in The genotypes from 525 individuals were phased by the software AlphaImpute  using deTo test for parent-of-origin specific associations of the 22 SNPs with BMI in the Sorbs, three different types of association tests (each 22 independent tests) were performed . First w2) was estimated by Haploview 4.2 [–4). All P-values >7.5x10–4 but ≤ 0.05 were considered to be of nominal statistical significance. All P-values are provided uncorrected for multiple testing.Prior to statistical analysis BMI was ln-transformed to approximate a normal distribution. Linear regression analysis was applied to test for allelic associations with BMI variation. All analyses were adjusted for age, sex and T2D. Student´s t-Test was applied to test for differences between linear regression derived beta values (effect size estimates) between maternal and paternal association test results. Linkage disequilibrium . The data were analysed using the parental-asymmetry test (PAT) developed by Weinberg . In shorP ≤ 0.05) for an association with BMI variation in either standard association test, independent maternal or paternal association tests  showed suggestive evidence  which iP≤0.05). Particularly, applying a Student´s t-Test comparing linear regression derived effect estimations  between paternal and maternal association tests revealed nominal differences at these SNPs  show, compared to association tests ignoring the parental origin, stronger effects for BMI along with evidence for differences in the effect sizes between paternal and maternal transmissions.Among the 22 SNPs, 10 variants show differences in association when evaluating paternal or maternal alleles, separately. Three of the 10 variants  confer different effect directions between paternal and maternal alleles , Table 2 P≤0.05; , one of 2 values are shown from black (r2 = 1) to white (r2 = 0).LD plots were generated for the Sorbs population using Haploview 4.2. Pairwise rP>0.30; P≤0.05; We analyzed the 10 variants identified in the Sorbs data set in an independent data set of 705 trios comprising an extremely obese child or adolescent and both biological parents . Albeit FTO variants [SLC2A10 and KCNK9. In the present study we explored POE at the FTO locus. We analyzed both, BMI variability in adult Sorbs and (extreme) obesity with an early onset in German trio families.Despite the reported relatively large genetic effect size of variants  and the variants . A knownvariants . POE mayvariants  put furtFTO variants. As described earlier [The main finding of our study is that we observed indications for potential POE at several  earlier  the stro earlier  may be m earlier  turn outhttp://igc.otago.ac.nz/home.html) several published studies showing POE at loci on chromosome 16q are listed: e.g. for type 1 diabetes [FTO locus on chromosome 16q12 may be conceivable but needs to be investigated in further, larger studies including additional functional tests and wet lab techniques such as analyzing epigenetic patterns.Our data may be a hint for potential effects depending on alleles inherited from the mother or the father. The Sorbs data might represent a population-specific phenomenon but may also be of broader interest as suggested by the trio data, especially for research groups working on family-based data sets. In the catalogue of POE  obesity with an early onset, our results are limited by the small sample sizes that are only powered to detect large genetic effects. Consequently, we selected the FTO locus with the strongest polygenic obesity signal reported so far. In sum, our data need to be cautiously interpreted and larger studies are needed to confirm our signals suggesting that POE might exist at the FTO locus.However, the results of our study need to be very cautiously interpreted. None of our results withstands a correction for multiple testing. Thus, our data can only be interpreted as suggestive indications for possible POE effects in S1 Table(DOC)Click here for additional data file.