Previous studies have shown an increased risk for mental health problems in children born to both younger and older parents compared to children of average-aged parents. We previously used a novel design to reveal a latent mechanism of genetic association between schizophrenia and age at first birth in women (AFB). Here, we use independent data from the UK Biobank  to replicate the finding of an association between predicted genetic risk of schizophrenia and AFB in women, and to estimate the genetic correlation between schizophrenia and AFB in women stratified into younger and older groups. We find evidence for an association between predicted genetic risk of schizophrenia and AFB in women , and we show genetic heterogeneity between younger and older AFB groups . The genetic correlation between schizophrenia and AFB in the younger AFB group is −0.16 (SE = 0.04) while that between schizophrenia and AFB in the older AFB group is 0.14 (SE = 0.08). Our results suggest that early, and perhaps also late, age at first birth in women is associated with increased genetic risk for schizophrenia in the UK Biobank sample. These findings contribute new insights into factors contributing to the complex bio-social risk architecture underpinning the association between parental age and offspring mental health. A recent comprehensive analysis using family data extracted from the Danish Psychiatric Central Register reported a relationship between mother’s age and risk of SCZ in her offspring13. They showed that there was higher risk in children of younger and older mothers compared to those of intermediate age , but it was unclear if this was due to psychosocial factors associated with maternal age or if mothers at higher genetic risk for SCZ tend to have children at an earlier or later age. Moreover, the very high correlation in spousal ages makes paternal and maternal contributions to this relationship difficult to disentangle. A number of possible latent mechanisms behind these epidemiological observations have been proposed14, including shared genetic risk factors between parents and offspring16  and risk of SCZ17 using the UK Biobank data in which community samples of women have been measured for AFB. First, we confirm that SCZ polygenic risk score (PRS) for women in the UK Biobank with a record of AFB  significantly predicts the U-shaped relationship found in McGrath et al.13, thereby replicating the results in Mehta et al.17. Second, we test if there is a genetic heterogeneity for AFB between younger and older AFB groups. Third, we estimate the genetic correlation between SCZ and AFB in younger and older AFB groups.In this study, we replicate and extend our earlier findingset al.17. For the UK Biobank sample, 38,892 women were used. The distribution of AFB, age at interview, and year of birth for the UK Biobank data after QC are shown in Fig. In total, 41,630 SCZ GWAS samples including 18,957 cases and 22,673 controls from 30 cohorts were used in this study Table , which w18 (PRS-score) and genomic best linear unbiased prediction method (PRS-GBLUP). We assessed the U-shaped relationship between AFB and SCZ PRS for the UK Biobank sample. We emphasize that in this novel design, it was not necessary to measure SCZ risk in offspring, and in our strategy potential confounding due to a correlation between paternal and maternal age was mostly removed.We estimated SCZ PRS for each individual in the UK Biobank sample, using the SCZ GWAS as a reference data set (see Methods). We used both the genetic profile score approachSubsequently, we estimated SNP-heritability and genetic correlation between AFB and SCZ. Because of the non-linear relationship (U-shape), we divided the UK Biobank sample into two groups with younger and older AFB. We assessed if the younger and older AFB groups were genetically heterogeneous, and if there is any significant genetic correlation between SCZ and each of the younger and older AFB groups.et al.13 and replicating the findings in Mehta et al.17, a U-shaped relationship was observed between AFB and SCZ PRS-GBLUP  or conventional profile scoring based on GWAS summary statistics from the SCZ GWAS data (PRS-score)  Fig. . We alsore) Fig. , althouget al.17, we tested if SCZ PRS could predict the response variable (see Methods) that described the relationship between SCZ risk in offspring and maternal age derived in McGrath et al.13. Figure Following Mehta Education level, income level, smoking and alcohol drinker status were additionally used to adjust the response variable in the linear prediction to test if those factors diminish the signals. Even with this conservative model, our results for the group with full range of AFB and the subgroup with AFB younger than 26 remained significant Fig. , albeit 19. For individuals born after 1945, PRS-GBLUP significantly predicted the response variable for the group with the full range of AFB and the subgroup with AFB younger than 26, even after adjusting for socioeconomic status, and smoking and alcohol drinker status, but not for the subgroup with older AFB. For individuals born before 1945, PRS-GBLUP did not significantly predict the response variable for the group with the full range of AFB  nor the subgroup with AFB younger or older than 26   Table . Our res2 = 4.96E-04 in Table Given that the AFB for women in the UK Biobank data was significantly predicted by PRS-GBLUP , 0.10 (SE = 0.01), and 0.20 SE = 0.004), for older AFB, younger AFB, and SCZ, respectively. The SNP-heritability for the older AFB group was not significantly different from 0  analyses to see if those factors change the estimates. Figure 21 was applied to estimate the genetic correlation between SCZ and younger and older AFB FB Table . As reco23, but it is well known that traditional epidemiological study designs, based on data measured for parental age and SCZ status in their offspring, have limitations with respect to disentangling genetic effects from non-genetic confounding effects such as common and residual environmental effects , although this cannot explain increased risk in offspring of younger parents. This hypothesis is biologically plausible28, but Gratten et al.29 have shown using theory and simulations that paternal age-related de novo mutations are unlikely to be the major causal factor responsible for increased risk of SCZ in offspring. Instead, they argued that increased risk of SCZ in offspring of older fathers could be due to genetic overlap between risk of SCZ and delayed parenthood. This finding is consistent with epidemiological studies showing that paternal age at first child, as opposed to paternal age at conception, accounts for the increased risk of SCZ in the children of older fathers 16. Notably, this mechanism of genetic overlap between parental age and SCZ applies equally well to offspring SCZ risk associated with early parenthood.Parental age has been consistently associated with an increased risk of SCZ in offspringcts Fig. . The eleet al.17 used a novel design to investigate the genetic relationship between SCZ and AFB in women that is free of many of the potential confounders present in epidemiological study designs . Specifically, they used SNP effects obtained using SCZ case-control data to estimate genetic risk of SCZ in a general community sample of women measured for AFB. As expected, psychiatric disorder including SCZ was not enriched in this community sample . Therefore, there is hardly phenotypic assortative mating for the psychiatric disorder in the sample. In addition, the analyses were restricted to individuals with pairwise genomic relationship <0.05 across samples to make sure that SCZ GWAS and AFB samples were independent. Thus, it was unlikely that they shared the common environmental factors. In this analysis, we used the novel design and replicated the finding in a much larger and independent community sample, the UK Biobank study. We confirmed the U-shaped relationship between AFB and SCZ PRS reported by Mehta et al.17 , reflects the very stringent QC we applied to the data, including on relatedness.In linear prediction, results from PRS-score were similar to or more significant than those from PRS-scorePGC Table . This isIn this study, we could not test the significance of association between SCZ PRS and SCZ status for the UK Biobank sample because there was no information about SCZ outcome for the sample. However, we conducted an association test between the SCZ PRS and bipolar disorder (that is highly correlated with SCZ) for the UK Biobank . This showed that the SCZ PRS was a reliable predictor of the true underlying SCZ liability for the UK Biobank sample. We also did not adjust for the assortative mating effects because the partner information is not available. However, it is not likely that such genetic assortative mating effects are substantial although a further study is required to test this hypothesis.et al.21 that standard errors are sacrificed to achieve unbiased genetic correlation and the availability of individual-level genotype data was the most preferable scenario31. Nevertheless, the estimated genetic correlations between SCZ, younger AFB and older AFB were consistent using two different approaches, i.e. GREML and LDSC  and that there is a significant genetic correlation between SCZ and AFB. Conducting parallel analyses for AFB in men is of great interest but these data are less easily available and AFB has not been recorded for men in the UK Biobank. Our results suggest that early, and perhaps also late, age at first birth in women is associated with increased genetic risk for SCZ in the UK Biobank samples. These findings contribute new insights into factors contributing to the complex bio-social risk architecture underpinning the association between parental age and offspring mental health.Informed consent was obtained from every participant in each study. Research Ethics approval was obtained from University of South Australia Human Research Ethics Committee (HREC). All methods were carried out in accordance with the relevant guidelines and regulations.32 with quality control (QC) applied as described in Mehta et al.17. In summary, before imputation, PGC group performed a QC for the raw genotypes within each cohort: SNPs with call rate <0.95 or Hardy-Weinberg equilibrium p-value < 10E-6 in controls or p-value < 10E-10 in cases were excluded. Individuals with call rate <0.98 were excluded. Then, the QCed raw genotype data were imputed with IMPUTE2/SHAPEIT34 using the full 1000 Genomes Project dataset35 as the reference set. Post-imputation quality control was performed in each cohort as: A best-guess genotype was called if its posterior probability >0.8, otherwise treated as missing. SNPs with an imputation r-squared <0.1 or MAF < 0.005, or call rate <0.98 were excluded. Out of 39 cohorts, eight cohorts were excluded because the number of SNPs passing the QC process was too small and one cohort was excluded because essential covariate information was not available. After the post-imputation QC, we combined the genotype data across all cohorts32. As in Mehta et al.17, we used HapMap3 SNPs that were reliable in estimating (shared) genetic architecture between complex traits38. In the combined dataset, SNPs with call rate <0.9, individuals with call rate <0.9 were excluded, and one individual in a pair with genomic relationship >0.05 was excluded. This less stringent QC for call rate was because SNP and individual QC had been already done by the PGC. After this QC, 688,145 SNPs (best-guess genotypes) and 41,630 individuals were remained and used to combine with the UK Biobank sample.Genome-wide association data were available from 18,987 SCZ cases and 22,673 controls from the second phase of the Psychiatric Genomics Consortium (PGC)39 data set, 80,702 female  out of 152,249 genotyped individuals were available from a community sample, in which psychiatric disorders were not enriched, with 72,355,667 imputed SNPs available. Out of all imputed SNPs, 1,242,190 HapMap3 SNPs were identified, which were filtered through the following QC filtering criteria: SNPs with imputation INFO < 0.617, minor allele frequency (MAF) <0.01, call rate <0.95, and Hardy-Weinberg equilibrium P-value < 10–717 were excluded. In addition, ambiguous strand SNPs were excluded. After this QC, 930,841 SNPs (best guess genotypes) remained and they were used to merge with the SCZ genotypic data. For individual level QC, only Caucasian females were used who clustered within 6 standard deviation from the mean of the EUR reference sample40 for the first and second genetic relationship principal components. Individuals were further excluded due to call rate <0.95. In addition, one in a pair of individuals was randomly removed if their genomic relationship coefficient was more than 0.0517. An important reason for removing closely related individuals was to reduce the possibility that the similarity between the phenotypes of those individuals could be caused by non-genetic effects (e.g. common environment effects)41. Furthermore, UK Biobank samples were excluded if their genomic relationship with any individual in the SCZ or AFB datasets used in Mehta et al.17 was >0.05, in order to ensure the independence of the UK Biobank sample. The AFB sample in Mehta et al.17 included 12,247 genotyped women measured for AFB, who were from four cohorts: Estonia, the Netherlands, Sweden, and the United Kingdom. After QC, we used 80,522 individuals  and 518,992 SNPs in the main analyses.In the first version of UK Biobanky is a vector of phenotypic data , b represents vectors of fixed effects including sex, cohort and 20 ancestry principal components (PCs)42, u is the vector of SCZ PRS, and e is the vector of residuals. X and Z are design matrices to allocate effects to phenotypic data. It is assumed that u is normally distributed as G is the genomic relationship matrix constructed as described in Yang et al.43 and e is normally distributed as I is an identify matrix and 43 or MTG245 so that a subset of u for the UK Biobank sample could be inferred based on the phenotypes of the SCZ sample and the genomic relationships between the two data sets. Mean SCZ PRS in the UK Biobank individuals grouped by their AFB was estimated to assess the previously reported U-shaped relationship17. GBLUP provides more accurate PRS (PRS-GBLUP) under a polygenic genetic architecture than the more standard genetic profile score approach18, but for comparison we also calculated PRS by the standard method (PRS-score). To estimate SCZ risk SNP effects, an association test was conducted with PLINK 1.946 using the same SCZ GWAS data used in the GBLUP analyses, with phenotypes adjusted for sex, cohort and 20 PCs. PRS-scores for individuals in the UK Biobank sample were generated by summing the count of SCZ risk alleles weighted by the SNP effects estimated from the association test. In addition to PRS-score, we used the estimated SNP effects from the full PGC SCZ GWAS study 32 to calculate a further profile risk score in the UK Biobank sample (PRS-scorePGC). However, we cannot exclude the possibility of sample overlap or relatedness between the UK Biobank and full PGC SCZ data, because we only have a permission to access to the genotypes of 30 cohorts out of 52. Note that we used all of the SNPs across the genome (after QC) to calculate SCZ PRS for the UK biobank sample.We used a GBLUP model to generate SCZ PRS for each individual in the UK Biobank sample accounting for the genetic relationship between the UK Biobank sample and the SCZ case-control sample. The GBLUP model can be written as13 that maternal ages are associated with SCZ risk in offspring, using the novel design consisting of general community sample (UK Biobank) measured for AFB and SCZ PRS. Any significant association between AFB and SCZ PRS from a linear prediction gives evidence that the U-shape relationship between maternal ages and risk in offspring  can be explained by genetic factors. Using 2,894,688 records from the National Danish Registry, McGrath et al.13 reported a U-shaped relationship between risk of SCZ in children and maternal age at birth. The resulting equation from the U-shaped relationship (z = 2.7214 − 0.1105X + 0.0018X2) can be applied to data of age at first birth of women to generate predictors of risk of SCZ in their children17. We did not consider the second model in Mehta et al.17 that was adjusted for partner’s ages because the model was shown to be over-corrected17. We calculated the response variable (z) in the UK Biobank sample from the recorded age at first birth (X) and this was used as the y-variable in analyses regressing on either PRS-GBLUP, PRS-score or PRS-scorePGC and including age at interview, assessment center at which participant consented, genotype batch, year of birth and the first 20 PCs as covariates. Socioeconomic status (i.e. education and income level)47 or smoking and alcohol drinker status were additionally used to adjust the response variable in sensitivity analyses  and P-value against the null hypothesis (i.e. SCZ PRS of women is not a predictor of AFB) were estimated.It is of interest to replicate the findings from the epidemiological observationses Fig. . Linear 17 found that younger and older AFB have reciprocal relationship with risk of SCZ (i.e. a U-shaped relationship), it is of interest to test if AFB in women is a genetically heterogeneous trait, for instance by estimating the genetic correlation between younger and older AFB groups. If the genetic correlation is significantly different from 1, it would imply that the causal variants and/or their effect sizes differ between younger and older AFB. Moreover, it would be interesting to estimate genetic correlations between SCZ case-control data and younger or older AFB groups. The UK Biobank data were divided into two groups by younger (<26) and older AFB (≥26). Then, three-variate linear mixed model analysis was conducted to estimate genetic variance and covariance between SCZ case-control data, and younger and older AFB groups. The model can be written asy are three column vectors of phenotypic observations . For SCZ case-control data, pre-adjusted phenotypes corrected for sex, cohort and 20 PCs were used. For the UK Biobank sample, the AFB phenotypes were pre-adjusted for sex, age at interview, year of birth, assessment center, genotype batch and 20 PCs. Again, other possible confounding factors such as socioeconomic status, and smoking and alcohol drinker status were additionally controlled in sensitivity analyses to check if the estimates were substantially changed. The GREML analyses were conducted with GCTA43 or MTG245 to estimate pairwise genetic correlations among the three data sets; SCZ data, and younger AFB and older AFB. We tested if the genetic correlation between SCZ and younger or older AFB was significantly different from zero. We also tested if genetic correlation between younger and older AFB was significantly different from 1 to assess heterogeneity between the two groups48. We used a Wald test to obtain p-values for the test.Since previous studiesSupplementary file


Correlations with reproductive traits imply an effect on fitness, consistent with an estimated linear selection gradient of -0.018 per MPB standard deviation. Overall, we provide genetic insights into MPB: a phenotype of interest in its own right, with value as a model sex-limited, complex trait.Male pattern baldness (MPB) is a sex-limited, age-related, complex trait. We study MPB genetics in 205,327 European males from the UK Biobank. Here we show that MPB is strongly heritable and polygenic, with pedigree-heritability of 0.62 (SE = 0.03) estimated from close relatives, and SNP-heritability of 0.39 (SE = 0.01) from conventionally-unrelated males. We detect 624 near-independent genome-wide loci, contributing SNP-heritability of 0.25 (SE = 0.01), of which 26 X-chromosome loci explain 11.6%. Autosomal genetic variance is enriched for common variants and regions of lower linkage disequilibrium. We identify plausible genetic correlations between MPB and multiple sex-limited markers of earlier puberty, increased bone mineral density  is a polygenic trait that affects the majority of European men. Here, Yap et al. estimate heritability, partitioned by autosomes and the X-chromosome, of MPB in the UK Biobank cohort, perform GWAS for MPB and find genetic correlation with other sex-specific traits. The balding process is highly patterned, suggesting that MPB progression is induced by some ageing-associated program: from initial frontotemporal hairline recession to a more severe occipital horseshoe stage1. MPB has epidemiological links with conditions including prostate cancer5, cardiometabolic disease6 and Parkinson’s Disease3. However, despite the prevalence of MPB, the underlying biology is not well understood.Male pattern baldness (MPB), or androgenetic alopecia, is an age-related condition characterised by hair thinning, miniaturisation and lossh2) on a scale of liability to be 0.817 (95%CI: 0.77–0.85). Estimates of variance explained by genome-wide autosomal SNPs (h2SNP) range from 0.34 to 0.678. (See BoxAR) was the first locus to be linked to MPB9, and was replicated across different populations10, in linkage analysis11 and in genome-wide association studies (GWAS)14. The androgen receptor binds testosterone, and increased sensitivity is associated with more severe MPB15. Three large-scale MPB GWAS studies were published in 2017, significantly increasing the number of candidate loci. Heilmann-Heimbach et al.4 performed a GWAS case-control meta-analysis on over 22,000 males, finding 63 independent MPB-associated loci, with six on the X-chromosome. Three X-chromosome loci had plausible candidate genes, including AR (on Xq12), TSR2 and KLF8 (both on Xp11.21). Hagenaars et al.8 used an even larger dataset from the UK Biobank (UKB) pilot phase (UKBv1) of over 52,000 males; they treated MPB as a ordered categorical trait with four levels, and identified 287 independent MPB-associated variants: 247 autosomal, and 40 on the X-chromosome8. In a case-control study using the UKBv1 dataset for discovery , Pirastu et al. identified and replicated 107 independent SNPs representing 71 MPB-associated loci (two on the X-chromosome); 30 loci were reported to be novel16. All three studies use different definitions for distinct loci, so the reported numbers are difficult to compare.Current evidence suggests that the genetic contribution to MPB is strong and polygenic. Early twin studies estimated the narrow-sense heritability . Since the X-chromosome contains 5% of all human genes17, and many complex diseases  are sexually dimorphic with respect to their lifetime risk, age of onset and symptoms19, it is important to investigate how the X-chromosome contributes to complex traits. Thus, MPB is an interesting trait to study due to the known X-chromosome effect, its sex-limited expression, its complex, polygenic basis, and its manifestation late in life.It is not uncommon for published GWAS to neglect analyses of the X-chromosome, as differential dosage between the sexes presents unique challengesh2 as high as 0.62. Our GWAS shows that MPB is highly polygenic, and we also find associations with other sex-limited traits  and bone mineral density that suggest pleiotropy.Here, we investigate the genetic basis of MPB, drawing upon data from 205,327 adult males of European ancestry from the UKB. We quantify and dissect genetic variation on the autosomes and the X-chromosome, finding Box 1: Definitions of heritabilityh2).Heritability is a measure of the proportion of phenotypic variance attributed to genetic factors; here, we consider only additive genetic factors  using information from close relatives (e.g. third-degree and closer). However, these estimates may be inflated as relatives may share environmental exposures and non-additive genetic effects. Genetic variance can be measured directly from DNA markers such as SNPs: so-called SNP-based heritability (h2SNP). The expected magnitude of h2SNP depends on the SNP set used in its estimation. h2SNP is estimated based on the small genetic relationships between conventionally unrelated individuals . In this method, relatedness coefficients are calculated based on similarities of alleles between pairs of individuals. Here, we used the GREML method implemented in the GCTA software21. Large sample size is required to obtain an estimate of h2SNP with a relatively small standard error because of the small variation in the genetic relatedness between unrelated individuals. The estimate of h2SNP tends to be lower than h2ped because the former is less likely to include variation due to shared environmental effects and rare variants. The estimates of h2SNP usually rely on similarity of common SNPs, which underestimates h2 if the genetic architecture is dominated by rare variants.h2SNP may be divided into further components70. Here, we separated the autosomal and X-chromosome genetic components. We also further compartmentalised the autosomal contribution into (a) low versus high LD and (b) rare versus common SNPs in our GCTA GREML-LDMS analysis23.The estimates of h2SNP and h2ped using data from both close and distant relatives, in the so-called ‘big K small K’ analysis22.It is also possible to simultaneously estimate P < 2e-16); that is on average, each year confers a 0.02 increase in MPB score. Including age-squared in the regression did not explain significantly more variance. The estimate of repeatability from longitudinal self-report data from 9603 men was 0.88 (95%CI: 0.87–0.88)  using first-degree relatives determined by UKB kinship coefficients, including 4428 full-brothers and 789 father–son pairs. Our estimate was 0.589 (SE = 0.026) for all first-degree relatives, and 0.619 (SE = 0.028) from only brother–brother pairs. In contrast, the estimate from 789 father–son pairs  .First, we estimated pedigree-based h2 (h2SNP) using pairs of unrelated individuals and GCTA software21. Due to computational constraints, we defined a sample of 87,957 individuals, which included the maximum number of related pairs (genomic relationship matrix (GRM) coefficient of relationship, rel > 0.05). From this sample, there was a maximum subset of 75,654 unrelated individuals (rel < 0.05) , partitioned as 0.359 (SE = 0.006) from the autosomes and 0.034 (SE = 0.002) from the X-chromosome  based on autosomal data to estimate h2ped and h2SNP simultaneously. This gave h2ped of 0.610 (SE = 0.030); of this, 0.349 (SE = 0.005) was explained by common autosomal SNPs (MAF > 0.01) among unrelated individuals (i.e. h2SNP) and 0.261 (SE = 0.030) was explained by close relatives (big K GRM threshold replacing rel < 0.05 with 0) (Supplementary Data h2ped was 0.604 (SE = 0.029), partitioned as 0.359 from common autosomal SNPs , 0.211 (SE = 0.030) from the pedigree not captured by common SNPs and 0.033 (SE = 0.002) from the X-chromosome  and the autosomes, we then added rare SNPs (1e-3 < MAF ≤ 0.01) and stratified the variants across MAF and LD bins using GCTA GREML-LDMS23. The analysis accounts for uneven distribution of variance explained by SNPs with different MAF and LD scores23. The total h2SNP,LDMS estimate was 0.415   SNPs , histone modifications , super enhancers (P = 3.5e-11), DNAse hypersensitivity sites , digital genomic footprint  and transcription factor binding sites  .We also performed partitioned 26 to investigate how the MPB GWAS results relate to gene function and expression, finding 850 autosomal genes that survived Bonferroni correction  demonstrated enrichment for expression in skin and vaginal tissue among these genes  with MPB was significant (Bonferroni-corrected threshold P < 0.006): age of menarche  , and rg for age at voice breaking was also significant . Taken together, the negative rg with ages at facial hair onset, voice breaking and menarche , suggest genetic covariance between earlier puberty onset in both sexes and increased MPB severity  Table . Among m30 to estimate rg between MPB and over 800 other traits (thresholding for h2SNP < 0.05 and FDR-adjusted P < 0.05), finding additional statistically-significant rg between MPB and bone mineral density  and 201532 , as well as the UKB heel BMD30 ), age of menarche 2014 GWAS33 , difference in height between childhood and adulthood at age 834 , and HOMA-B35  , voice breaking  and menarche  . The regression coefficient of the number of children fathered against age-adjusted MPB was −0.030 , which corresponds to 0.09 fewer children between men with MPB score of 4 versus 1. This also equates to −0.033 children per standard deviation of MPB. Given that the mean number of children fathered is 1.81, and assuming that relative lifetime reproductive success (number of children standardised by the mean) is a proxy for fitness, we infer a linear selection gradient of −0.018 per one standard deviation increase in MPB36.The trait ‘number of children fathered’ had a low h2 analysis exploiting first-degree relationship pairs in the UKB gave estimates as high as 0.62 . The lower father–son correlation compared to brother–brother pairs  may reflect sampling variance, non-additive effects, and common environment shared by brothers. Our big K small K analyses also showed that the pedigree-based estimate captured X-chromosome h2SNP as the two-GRM and three-GRM (with X-chromosome) estimates were both around 0.60  due to differences in trait measurement and analysis method. In addition to sampling variation, the difference in measurement scale and therefore measurement error is probably the main source of differences between the estimates. Gianola (1979)37 derived the heritability on the observed scale , and V(Yo) = wj taking on values 1, 2, 3 and 4. Therefore, it follows that:h2 is 1.18 × 0.62 = 0.73.MPB h2 of liability to baldness of 1 and claimed that this was driven by common variants16. The authors created a binary score from UKBv1 data, defining ‘cases’ (affected) as individuals with scores 3 and 4 and ‘controls’ (unaffected) as those with a score of 1. They then transformed the h2 estimate on this 0–1 scale to that of liability, using the proportion of affected individuals (i.e. those with a score of 3 or 4) in the sample  as the population prevalence. However, excluding individuals with a score of 2 creates an upward bias, and this ascertainment is unaccounted for. If liability to MPB is linear in the scores 1 to 4 (as we have found38), then the reported estimate of 1 by Pirastu et al.16 is biased by about one-third, so the actual estimate is consistent with our results.A recent MPB genetic analysis using the UKB and other datasets reported a very high h2SNP estimates for traits such as height39 and childhood intelligence40. Our GREML-LDMS analysis on the autosomes showed that common SNPs (MAF > 0.01) essentially explained all h2SNP , whereas rare SNPs (1.5e-5 < MAF ≤ 0.01) made a very minor contribution. There was also enrichment in lower LD regions  . Altogether, these SNPs explained 25% (SE = 1%) of phenotypic variance  were common (MAF > 0.01) 39, our MPB analysis has comparable sample size, and we detect a similar number of autosomal associated loci (MPB: 598 versus height: 697). However, the top MPB loci tend to have larger effect size, and explain more variance than the top height loci , we replicated 219/287 SNPs within our GWS SNPs  from a UKBv1 case-control design, that they had replicated in separate cohorts at P < 0.05. Of these, we replicated 71/103 autosomal and all four X-chromosome SNPs, corresponding to 61/71 loci  in our GWS results , of which X-chromosome conditionally-independent GWS SNPs explained 0.029 (SE = 0.008), or 11.6% (0.029/0.252). In comparison, all common X-chromosome variants explained 8.8% (0.034/0.393) of h2SNP due to common variation . Here, 624 conditionally independent GWS SNPs accounted for  SE = 0.01, of whiDownstream analyses highlighted interesting and plausible biological pathways that may provide clues for molecular analyses. These are discussed in detail in Supplementary Note 42, we looked for relationships with other sex-limited traits. We analysed eight female-specific traits with a large sample size , after Bonferroni correction (P < 0.006). In males, we observed significant rg with earlier facial hair appearance  and earlier voice breaking . These results indicate a novel association between MPB and earlier onset of puberty in both sexes. These three traits had h2SNP > 0.07 (estimated by LDSC), which is reassuring as rg can be hard to interpret when h2SNP is small , and reduced difference in height between childhood (aged 8) and adulthood34  18. Hence, rg may be underestimated if the genetic basis of earlier puberty onset also has X-chromosome components.As MPB has a androgenetic basis, and because there is sexual dimorphism in androgenetic regulationze Table  in searc43. Although the testes are the major source of androgens in males from puberty, the adrenal glands are an important source in both sexes during the earliest detectable stage of sexual maturation: adrenarche. Adrenarche occurs around the age of six, precedes pubarche (pubic hair development)44, correlates with puberty onset in both sexes45, and may initiate puberty via hypothalamic-pituitary-gonadal axis activation46. Excess adrenal androgens may induce precocious puberty, manifested in congenital adrenal hyperplasia (CAH)46. Both androgen sources converge upon the androgen signalling pathway, in which the X-linked AR gene is central.The association between early-onset puberty and hair biology may be reconciled. Androgens regulate hair growth, and simultaneously promote facial hair while causing scalp hair regression in older men46, which is consistent with the X-linked AR locus having recently been under selection11. Interestingly, mammalian patterned baldness may also have a recent evolutionary history as it is confined to a few primate species47. However, the autosomal LDSC partitioned h2SNP analysis found enrichment among conserved regions of the genome  of lumbar spine  and heel  . Similarly to early-onset puberty, androgen production may underpin both MPB and BMD50. This is supported by multiple lines of evidence: androgen-deprivation therapy for prostate cancer decreases BMD51, whereas hypogonadal men have low BMD that normalises with testosterone therapy52. Further, CAH patients with higher adrenal androgens (such as DHEAS), tend to have higher BMD53, as do women with high testosterone levels due to polycystic ovarian syndrome (PCOS)54. Interestingly, strontium ranelate (an osteoporosis therapy) induces alopecia in some patients55.LDHub also showed statistically-significant positive 4) Table . Early prg between MPB and HOMA-B (indicating increased pancreatic β-cell function) (Table 56 (insulin resistance), but not HOMA-B. Interestingly, genetic loci contributing to the difference in height between childhood and adulthood (a marker for early puberty) have been associated with age of menarche as well as adiposity34 . The association of MPB with all of these traits adds to the case for pleiotropy. This is biologically plausible as adipose tissue can store androgens and metabolise them into oestrogens. Further, testosterone may increase metabolic dysfunction such as insulin resistance by decreasing the hormone adiponectin57.We also observed significant n) Table . There arg between MPB and previously described epidemiological associations , and the h2SNP estimate for type 2 diabetes in this analysis was too low for the estimate to be reliable .LDHub analyses did not find rg between MPB and facial hair appearance, voice breaking, age of menarche and difference in height between childhood (aged 8) and adulthood  and gonads . Androgen signalling is associated with puberty onset, which explains od Table . The rg 58 and prior identification of MPB susceptibility loci within haplotypes associated with reduced fertility3. In males, we took the number of children fathered as a male fertility trait, finding negative phenotypic correlation with the number of children fathered . More concretely, men with MPB score of 4 on average had 0.09 fewer children than those with MPB score of 1, after adjusting for age. Furthermore, the significant linear selection gradient of −0.018 (P < 2.2e-16) in the contemporary UK population is similar in magnitude to that reported for other traits in the UKB36. With regards to MPB and natural selection, it has been suggested that MPB is a sign of social maturity (independent of age) and increased appeasement, which may compensate for decreased attractiveness in reproductive success59. Genetically, the jointly-independent autosomal SNP rs199441  has recently undergone positive selection60.The genetic correlation analyses may also show evidence of weak negative selection in the contemporary UK population or alternatively, pleiotropic selection. Our search was motivated by previous associations of altered androgen regulation with infertilityP < 0.05) rg between increased MPB severity and fewer number of live births  and increased age of first birth  (Table 41. These two traits show opposing directions of effect between (a) lifetime reproductive success (LRS)  and (b) rg with MPB , which is consistent with MPB being negatively correlated with LRS and fertility. The enrichment for low LD variants in our h2 analyses support a case for weak negative selection in MPB, as do downstream analyses indicating relationships between MPB COJO SNPs and fertility-related genes  Table , suggest61 and sarcopenia62). Biologically, such traits share a basis in sex-hormone regulation, so it will be interesting to investigate pleiotropy and similarities in genetic architecture.Given our results linking MPB to BMD, MPB may be a useful model trait for other late-onset diseases that are biased towards one sex  reference panel provided by the UKB66. The UKB-provided genotype probabilities were used to hard-call the genotypes for variants with an imputation info score > 0.3. We used the hard-call-threshold 0.1, setting the genotypes with P ≤ 0.9 as missing. For the autosomal GWAS analysis, we retained the hard-called variants with minor allele frequency (MAF) > 0.0001, missing genotype rate < 0.05 and Hardy–Weinberg equilibrium (HWE) P > 1e-6 as computed in the sample of unrelated Europeans, such that 18,065,087 autosomal SNPs remained. For the heritability analyses, 1,123,347 HapMap3 variants passed our quality control filters  in the set of 456,419 male and female Europeans; these were used to estimate a GRM for 205,327 males.Supplementary Fig. https://github.com/CNSGenomics/impute-pipe. The genotyped SNPs used for imputation were those used for X-chromosome UKB phasing input, passing additional QC filters  in the set of unrelated Europeans, excluding those participants that were (1) heterozygosity or missingness outliers, (2) had inconsistent reported and biological sex, (3) were aneuploid, (4) if they had been excluded from kinship inference by the UKB or (5) if they had withdrawn their consent. After the imputation, we retained markers with imputation info score > 0.3 and again estimated QC metrics in the set of unrelated Europeans. For the non-pseudo-autosomal region, we kept the markers with MAF > 0.0001 in both males and females, and those with HWE P > 1e-6. The same thresholds applied for PAR1, without stratifying by sex. This left 1,064,602 X-chromosome SNPs  for use in GWAS. For the heritability analyses, we used a subset of 262,588  X-chromosome markers with MAF > 0.01, estimated in the 205,327 males.Imputed X-chromosome genotypes were not provided in the full UKB release from 2017. We imputed the non-pseudo-autosomal and pseudo-autosomal region (PAR1) to the 1000 Genomes phase 3 mixed population reference panel using IMPUTE2. The imputation pipeline followed is available at r2 > 0.9, window size = 1000 SNPs, step size = 100) set of common autosomal HapMap3 variants (MAF > 0.01) to calibrate the GWAS analysis, as required by the BOLT-LMM software24.In addition, we created an LD pruned (LD https://www.ukbiobank.ac.uk/the-ethics-and-governance-council/).The UK Biobank study protocol has ethics approval from the North West Multi-centre Research Ethics Committee (covering the UK), the Research Tissue Bank, and all participants provided written informed consent , number of live births and number of stillbirths (Supplementary Data H2)67. We estimated repeatability by calculating the correlation between MPB measurements taken from 2006–2010, to those taken from the same individuals during 2012–2013 (n = 9603).To benchmark all estimates of variance explained, we estimated repeatability: the correlation between repeated observations on the same individuals. Repeatability can be due to genetic and permanent environmental effects and is an upper limit of a trait’s broad sense heritability  with a first-degree relative. Of these, a subset of 1482 males were in a father–son relationship (789 pairs), and 8010 males were in a brother–brother relationship (4428 pairs). For each of these three sets of individuals, we estimated variance components using REML to simulate a true pedigree analysis, with first-degree relationships set to 0.5 and all others set to 0, using GCTA software21.UKB-provided kinship estimates20. Given constraints on computational resources, we maximised the number of individuals with relatives (estimated relatedness from SNP data > 0.05) so as to estimate both variance components with good precision. Specifically, the maximum dataset  used in our h2 analyses included 47,957 individuals with relatives in the data, and a random sample of 40,000 singletons  : the big K GRM in which relatedness coefficients < 0.05 based on autosomes were set to 0, and the conventional GRM constructed from common autosomal  and X-chromosome markers . This sample of 87,957 contains a maximum set of 75,654 unrelated males, using a rel < 0.05 threshold on a genetic relationship matrix (GRM) calculated using 1,123,347 common autosomal SNPs. We used these individuals in REML analyses to estimate the h2SNP attributable to common autosomal SNPs alone, common X-chromosome SNPs alone, and both common autosomal and X-chromosome SNPs.We performed a number of variance component analyses to estimate and partition phenotypic variation associated with common SNPs in both closely-related and conventionally-unrelated individuals. These analyses help dissect the contribution from common versus rare variants and shared environmental effects, since rare variants are not well tagged by common SNPs, and close relatives may share common environmental effects23 (with the no constraint setting) on all autosomal markers  from the unrelated subset . We divided SNPs into three bins based on MAF  and four quartiles based on LD score in a genome-wide manner, giving a total of 12 LDMS components  score, we performed a GREML-LDMS analysish2SNP enrichment in the GREML-LDMS components for low versus high LD and rare versus common variants. We define MC as the heritability estimate and the number of SNPs for category C respectively. Then the fold-enrichment can be calculated by the ratio of per SNP h2, as in Eq.  was performed using the autosomal and X-chromosome genotype data, with adjusted MPB scores as the dependent quantitative trait. BOLT-LMM accounts for genetic relationships between individuals, so we used the full set of MPB males . We used the LD-pruned autosomal SNPs (described in Genotype data section) to calibrate the analysis via the –modelSnps command. We selected genome-wide significant (GWS) SNPs using a significance threshold P < 5e-8, under the infinitesimal mixed model. Note that males are haploid for the X-chromosome. Hence, although the actual number of alleles is 0/1, BOLT-LMM encodes males as 0/2. Thus, the per-allele effect size is half the effect size reported in the summary statistics output for X-chromosome SNPs.An additive mixed linear model association analysis  analysis, analysis of the X-chromosome separately from autosomes overcomes this issue.Classical LD-clumping to identify GWS loci is based on a user-specified LD We opted for the summary statistics option as the –cojo-actual-geno method  is not built to analyse the X-chromosome.We quantified the variance explained by the COJO-selected independent GWS SNPs, by creating a GRM from these SNPs on the set of 75,654 unrelated males and estimating the variance explained using REML. Since the discovery and test sets are not independent, we expect a (slight) upward bias in this estimate, even if our criteria for inclusion of SNPs are stringent.26, partitioned h2SNP analyses implemented in LDSC27, and ICSNPathway28.We used three tools to investigate the top GWS SNPs: FUMA GWAS 26, using p-values from the non-infinitesimal mixed model. We include here results from the SNP2GENE tool , as well as from the GENE2FUNC tool . Note that these analyses only consider autosomal data.We input the GWAS summary statistics into FUMA27. We used 1000 Genomes Phase 3 functional annotation categories, which included coding, 3′-UTR, 5′-UTR, promoters, enhancers, introns, histone modifications, DNAse I hypersensitivity sites (DHS), protein binding sites (CTCF and transcription factors) and conserved regions , as described by Finucane et al.27. Ten cell-type-groups were included in the analysis, including adrenal/pancreas, cardiovascular, central nervous system (CNS), connective tissue and bone, gastrointestinal, haematopoietic, kidney, liver, ‘other’  and skeletal muscle.We used LDSC to partition heritability on the autosomes by functional annotation, and for cell-group-type analysis28 to look for functional associations of the GWS SNPs. For this analysis, we used default settings , with the exception of the gene window (looking for genes within 100 kb of the uploaded SNPs).We uploaded COJO SNPs to ICSNPathway18, and androgen signalling is associated with male fertility58, we investigated whether MPB is related to reproductive success and hormone-associated traits in both males and females. For genetic correlation analysis, we selected eight female traits, as described in Female data above, and for males, we selected relative age of first facial hair , and relative age of voice breaking . Both traits are scored as 1: ‘younger than average’, 2: ‘average’, and 3: ‘older than average’. We also included prostate cancer in our genetic correlation analyses . Details of coding, exclusion criteria and UK Biobank Field Codes are provided in Supplementary Data As the androgenetic pathway is a major contributor to MPB69 to calculate genetic correlations between these traits and MPB, based on autosomal data only. LD scores were calculated using the European UK Biobank population. LDSC standard errors are calculated using a block jackknife approach, and p-values are calculated by jackknifing the regression numerator and applying a Z-test27.We used the UK Biobank data and HapMap3 SNPs to generate GWAS summary statistics for each of the eight female traits, and three male traits  . Individuals who reported fathering > 20 children were excluded from the analysis.We also determined phenotypic correlations between the male traits, as well as between age-adjusted MPB with the trait ‘number of children fathered’ (http://www.ukbiobank.ac.uk/. All analyses were performed using publicly available software: PLINK 1.9 at https://www.cog-genomics.org/plink2; GCTA at http://cnsgenomics.com/software/gcta/; BOLT-LMM at https://data.broadinstitute.org/alkesgroup/BOLT-LMM/; FUMA at http://fuma.ctglab.nl/; LDSC at https://github.com/bulik/ldsc; LDHub at http://ldsc.broadinstitute.org/; and ICSNPathway at http://icsnpathway.psych.ac.cn.Data was obtained from the UK Biobank at Supplementary InformationPeer Review FileReporting SummarySupplementary Data 1Supplementary Data 2Supplementary Data 3Supplementary Data 4Supplementary Data 5Supplementary Data 6Supplementary Data 7Supplementary Data 8Supplementary Data 9Supplementary Data 10Supplementary Data 11Supplementary Data 12Supplementary Data 13Supplementary Data 14Supplementary Data 15


In genome-wide association studies (GWASs), meta-analysis has been widely used to improve statistical power by combining the results of different studies. Meta-analysis can detect phenotype associated variants that are failed to be detected in single studies. Especially, in biomedical sciences, meta-analysis is often necessary not only for improving statistical power, but also for reducing unavoidable limitation in data collection. As next-generation sequencing (NGS) technology has been developed, meta-analysis of rare variants is proceeding briskly along with meta-analysis of common variants in GWASs. However, meta-analysis on a single variant that is commonly used in common variant association test is improper for rare variants. A sparse signal of rare variant undermines the association signal and its large number causes multiple testing problem. To over-come these problems, we propose a meta-analysis method at the gene-level rather than variant level.Among many methods that have been developed, we used the unified quadratic tests (Q-tests); Q-test is more powerful than or as powerful as other tests such as Sequence Kernel Association Tests (SKAT). Since there are three different versions of Q-test , each assumes different relationships among multiple rare variants, we extended them into meta-study accordingly. For meta-analysis, we consider two types of approaches, the one is to combine regression coefficients and the other is to combine test statistics from each single study. We extend the Q-test for meta-analysis, proposing Meta Quadratic Test (Meta-Qtest). Meta Q-test avoids the limitations of MetaSKAT. It does not only consider genetic heterogeneity among studies as MetaSKAT but also reflects diverse real situations; since we extend three different Q-tests into meta-analysis respectively, flexible Meta Q-test suggests way to deal with gene-level rare variant meta-analysis efficiently From the results of real data analysis of blood pressure trait, our meta-analysis could successfully discovered genes, KCNA5 and CABIN1 that are already well known for relevance with hypertension disease and they are not detected in MetaSKAT.As exemplified by an application to T2D Genes projects data set, Meta-Qtest more effectively identified genes associated with hypertension disease than MetaSKAT did. Genome-wide association studies (GWASs) have identified many loci that contributed to human complex traits. As genotyping technologies such as next-generation sequencing (NGS) technology evolve, we have been able to gain larger data and more accurate information on human genetics. Discovery of rare variants is one of the most valuable crops of the NGS technologies . The subPersisting on same methods in common variant analyses is not appropriate for dealing with the rare variants . Due to P value is computed using permutation that requires intensive computation and covariate adjust is not available. The method proposed to solve these problems is SKAT. SKAT is variance component score test implemented in a regression framework. To test the null hypothesis of zero regression coefficients effect sizes of genetic variants in a gene, SKAT assumes that each regression coefficient follows arbitrary distribution with zero mean and the variance, product of weight and variance component. Then, testing original null hypothesis becomes the same as testing whether each variance component is zero or not. Variance component score statistic is employed in this process. Since SKAT is derived from regression model, it can include covariate terms easily. P value is calculated analytically and diverse kernel functions that can explain genetic similarity between individuals are introduced. Furthermore, optimal version of SKAT, SKAT-O is proposed to achieve robust power regardless of directions of variants. SKAT-O is combination of burden test and SKAT, weighted average of burden and SKAT. SKAT-O searches the optimal weight that is the weight obtaining the minimum p value of test statistics. However, even though robust power is substantially attained by SKAT-O test, there is still no uniformly most powerful test in all situations [Over the past few years, various statistical methods for gene-level rare variant association test are developed. From collapsing based methods such as Combined Multivariate and Collapsing (CMC) and variable thresholds test (VT test) to variance component tests such as C-alpha and Sequence Kernel Association Tests (SKAT or SKAT-O), each method performs well in different situations –9. CMC mtuations .Gene-level Q-test is another powerful rare variant association test . It alsoMeta-analysis is a popular approach to increase the power in GWASs . AggregaThe recently proposed MetaSKAT is extended SKAT for meta-analysis for gene-level rare variants. MetaSKAT aggregates the score statistics of each variants in a gene came from SKAT. Depending on the assumption of genetic effect, homogeneity or heterogeneity of genetic effects across studies, it aggregates the summary score statistics. When genetic effects are homogeneous, summary statistics are combined across the studies first and then combined across the variants in a gene, but when genetic effects are heterogeneous, the combining order is reversed. Optimal MetaSKAT that is weighted sum of test statistic of MetaSKAT and its burden for meta-analysis is also proposed to embrace the merits of burden and non-burden test together. However, simulation results of MetaSKAT show that type I error rates are somewhat uncontrolled. There is another limitation of MetaSKAT. When cohort specific genes are detected by MetaSKAT, it does not report that which cohort is highly associated with a phenotype .In this paper, we extend the Q-test for meta-analysis, proposing Meta Quadratic Test (Meta-Qtest). Meta Q-test avoids the limitations of MetaSKAT. It does not only consider genetic heterogeneity among studies as MetaSKAT but also reflects diverse real situations; since we extend three different Q-tests into meta-analysis respectively, flexible Meta Q-test suggests way to deal with gene-level rare variant meta-analysis efficiently.Q-test is established in a multiple linear regression framework for a quantitative trait.ki is the phenotype of the ith individual in the kth study, Skji is the genotype value coded 0, 1 or 2 under an additive genotype model , βk = , is the vector of regression coefficients for genetic effects of m SNPs, Zki is the covariate value, and γ is the corresponding vector of regression coefficients. After fitting model, using the estimated regression coefficients, βk, QTest1 creates the new variable, pooled coefficients, βpooled, k. Then, the null hypothesis of interest is to test whether pooled genetic effect exists or not:where yβk, rather than pooled coefficient to allow for considering bidirectional variants effect. Corresponding null hypothesis isHowever, for QTest2 we test the vector of regression coefficients, To test the hypothesis, Q-test constructs a Wald-type statistic which has the form of quadratic statistics. Depending on the different assumptions needed to collapse the estimates of effects size parameters, there are three versions of Q-test: QTest1, QTest2, and QTest3.1 is a burden type of test. A basic assumption is that all of SNPs in a gene have the same direction of effects on the phenotype. That is, we assume that all variants within a region are all deleterious or protective. If the assumption is true, the power of the burden test becomes higher by aggregated effects of each variant. When combining variants, Qtest1 uses the inverse variance weighting method that gives more weight to SNPs that have smaller variances,QTest1 introduces MAF based weight,Wk is the diagonal matrix of each variant’s MAF based weight,Also, like SKAT, QTestBrowning . By usin1 is given by1 would work poorly. Thus, QTest2 assumes that some of variants are protective, and the others are negative. Instead of aggregating the effects of variants, it directly constructs Wald statistics for In case of mixed variants in direction of effect, Qtest freedom .\documenThe final form of Qtest2 statistics is3 is an optimal method, weighted average of burden type (Qtest1) and non-burden type (Qtest2) statistics. To combine two statistics that follow chi square distributions with different degrees of freedoms, there is a step for making Q2 has the same degree of freedom as Q1 (df = 1). To summarize the steps, first we define the new parameter, 2 statistics, Q2 ∣ 1). Using final two statistics, Q1 and Q2 ∣ 1, we can get the optimal Qtest3 statistics through a grid search of weight. Final p value of Q3 is calculated by empirically. Pre-calculated empirical distributions are employed in the final step, so the computational burden is reduced.Qtest3 could accommodate both scenarios that assume same or different direction of SNPs in a gene, its result is usually robust. The followings are detailed steps of Qtest3.Since QtestQ2 ∣ 1 which follows p2 ∣ 1 is obtained from In step 1 in algorithm 1, we compute P value and z-score have been conventional summary statistics that could combine the results across study, but in GWAS meta-analysis, the methods using p value and z-score are generally inferior to the model based meta-analysis; those methods cannot efficiently take account of the between-data-set heterogeneity [Meta-analysis requires summary statistics from each single study. ogeneity . In modeogeneity . Under togeneity . MetaQ-t1 is burden type and Qtest2 is non-burden type test. This fact is also applied to MetaQ-test. MetaQtest1 is a burden and MetaQtest2 is a non-burden type test. MetaQ-test keeps not only type of statistics but also the process derived the statistics. By doing that, it can consistently maximize merits of test statistics.Another significant feature in MetaQ-test is it is extended keeping the original statistical model structure, structure of QTest. QtestSummary statistics used and the way to synthesize them are essence in meta-analysis . Dependi1, weighted sum of effect sizes of all variants. When we suppose that there are K studies, the estimated pooled effect size of regression coefficients is α is the column vector that includes αj s  as components, and W is the diagonal matrix with wj s as diagonal elements.First, we extend Qtest1 to meta-analysis maintaining burden type test. We assume that all variants in a gene across studies are from one single study. In Qtest1, to aggregate the effects of variants, H0 : βpooled = 0. Since under the null assumption, αTWVWα, the Wald type of test statistics isIt is natural that we collapse the variants in the same locus with appropriate weights, such as sample size. Testing hypothesis in this case is Since the statistics for meta-analysis is constructed based on the preexisting result of individual level data analysis, Qhom − meta − q1 follows chi-square distribution with 1 degree of freedom, because it is the square of single standard normal random variable,Qhom − meta − q1, we need estimated We can easily check that Assuming a variant may have a different effect across studies, we can consider the case of heterogeneous effects over studies. This assumption is in accordance with a meta model with random effects. Since we allow the heterogeneity of effect, we can use the results of each study’s regression fit. The model fittings were carried out separately, so the test statistics for association are made of different estimated regression coefficients. Accordingly, the outcome of model fitting in each study that takes account of heterogeneity is appropriate for summary statistics for meta-analysis.1 follows the chi-square distribution also makes us combine Q1 s easily. Thus, test statistics of each study would be a naïve and handy summary statistic for meta-analysisThe fact that Qhet − meta − q1 is a statistic for burden test in respect of sum of single burden statistics. Wald type of statistics for a single variant is distributed as the chi-square distribution with one degrees of freedom. When we assume the independence between studies, then the sum of test statistics also follows a chi-square distribution with K degree freedom, where K is the number of studies in meta-analysis, het − meta − q1, we only require the test statistics. Therefore, we call this approach statistics based meta.However, QWe extend Qtest2 to meta-analysis in the same manner as the Meta-Qtest1: fixed(homo) and random(hetero) version of meta. However, the main difference between Meta-Qtest1 is that we do not combine the effects of different variants, we dose only collapse the same variants. The basic idea of Meta-Qtest2 is that we regard SNPs on the same locus as the single SNP. For this reason, this approach is different with burden meta-analysis.It is possible to collapse variants of same locus, because of the assumption of homogeneity in genetic effects size throughout studies. In order to make representative genetic effects size that combines SNPs on the same locus, we use weighted sum. Weights are given as proportional to sample size of study. Representative of regression coefficient of jth SNP is expressed as below,m = m1 = m2 = ⋯ = mK.For the sake of simplicity, we assume that the number of variants in a gene is the same across the study, 2, we also need variance-covariance matrix of estimated regression coefficients. Under the independence assumption of studies, we derive the variance-covariance matrix, V, analogous to Since all populations do not share the same variants commonly, some variants exit only in a specific population. We thus put an indicator function in front of index of sample size. To maintain the framework of QtestUΛUT, we construct the Wald type statistic,The following step is identical to QTest2. Using eigen-decomposed V = For rare variant analysis, overlapped variants with other studies are not many. Thus, the approach of homo-meta-Q2 that aggregates the overlapped SNPs in the same locus could not be appropriate for the rare variant analysis; the estimate of variance-covariance matrix is unstable. An easy alternative way is to assume not homogeneity but heterogeneity of genetic effects. To satisfy this assumption and to accommodate different directional effects among variants in a region, we use the result of single study.p value is conventional, but this method has a limitation upon achievement of heterogeneity among studies; it just assumes independence among studies. Instead of p-value based method, we combine test statistics of each study for Qtest2. Q2 follows a mixture of chi-square distributions and the summation of the Q2 then also follows a mixture of chi-square distributions.Combining single study results using a determines the shape of the mixture of chi-square distribution and this value depends on the gene size (number of SNPs in a gene). Therefore, the choice of a should be careful.In Homo/Hetero-meta-Q2, the value of We develop an optimal quadratic meta-analysis for robust power gain. A burden type test is known to be more powerful when most variants in a region are causal and their effect directions are the same, but a non-burden type is opposite to this case. Therefore, applying one of the two tests can lose the power to detect the associated variants with a phenotype, if every single gene dose not satisfy the same assumption. A unified approach is to use a weighted average of burden and non-burden test. Since this approach allows two conflicted assumptions about direction among variants (burden or non-burden), an optimal test usually achieves the robust power regardless of directional assumptions.Qhom − meta − q3 that is weighted average of Qhom − meta − q1 and Qhom − meta − q2. We build a test for meta with an adjustment for homogeneous case. The following are steps for constructing Qhom − meta − q3. The steps are identical to Qtest3 but for substituting collapsed variables,  instead of vector of variables from each study, . The collapsed variables are defined in homo MetaQ-test. Like Qtest3, the empirical distribution of suggested test statistic is calculated from pre-calculated distribution in the step 5.When studies are homogeneous, we suggest Meta-analysis Assuming Heterogeneous Genetic Effects across Studies: Het-meta-Q3.Qhet − meta − q3 that is weighted average of Qhet − meta − q1 and Qhet − meta − q2. The following are steps for Qhet − meta − q3. The steps are similar to Qtest3 and MetaQTest3 but, in step 2 we use the sum of Wald type statistics, When studies are heterogeneous, we suggest To evaluate the properties of the proposed methods, we performed the simulation study. We generated the genotype data using COSI program that implements coalescent model, and we obtained both length 200 kb and 10,000 European-like haplotypes and African-American-like haplotypes . By randFor type I error simulations, we generated 100 phenotypes for 100 each gene under the null hypothesis; there is no association between a gene and a phenotype.Xk1i is the covariate taking 0 or 1 value with equal probability 0.5. The rest of covariates, from Xk2i to qk indicates the number of covariates for kth study. Since we generated 100 genes and 100 phenotypes, we carried out 10,000 times association tests, and the level of significance, we gave α = 0.01 and 0.001.As the type I error simulation of MetaSKAT, Unlike type I error simulations, we generated 1000 genotype datasets and made phenotypes using the below model for power simulations,Gki, causal is a vector of causal variants in a gene, and βk, causal is their effect size. For the proportion of causal variants, we set four cases, 10, 20, 30 and 50% of variants are causal like in MetaSKAT. To illustrate the effect of burden or non-burden type test, we also assumed that all variants are positive or 80% are positive and rest of 20% are negative. Since the number of causal variants or positive variants could not be integer in some cases because of small number of variants in a gene, we generated number of causal variants in Bernoulli distribution. The regression coefficient of genetic effect is given same as the MetaSKAT, βk, causal = c|log10(MAF)|. However, we used study specific MAF rather than population MAF, because in reality we hardly get the population MAF even in Meta-analysis. Defining the coefficient in this way reflects the assumption of rare variant study; the rarer SNPs have the larger effects on the phenotype. We set c = 0.475 in 5% of causal variants, 0.375 for 10%, 0.25 for 20%, and 0.175 for 50% of causal variants respectively. Since the effect size of regression coefficient depends on the MAF, the case for different MAF across the studies assumes heterogeneous effect and the case for the same MAF assumes homogeneous effect. These assumptions rely on the belief that different populations could have different MAF, and this phenomenon is called population stratification. In the simulation work, there are some differences with MetaSKAT. First, we used haplotypes that have different distribution of allele frequencies with that of MetaSKAT and second, we used study specific MAF rather than population MAF that was used in simulation of MetaSKAT. Finally, we used Bernoulli distribution to assign causal variants rather than exact given proportion percentages. We expected that the differences might cause the slightly different power results of MetaSKAT calculated here with MetaSKAT paper.We analyzed whole exome sequencing data from Type 2 Diabetes Genetic Exploration by Next-generation sequencing in multi-Ethnic Samples (T2D-GENES) consortium. Sequencing was performed at the Broad Institute with Agilent v2 capture reagent on HiSeq platform. The consortia have exome sequencing data of 13,000 individuals from 5 different ancestry groups: African Americans , American Hispanics , East Asian , South Asians , and European . The words in parenthesis stand for the populations in the ancestry group. For meta-analysis, we selected East Asian ancestry group, EK from Korean samples and ES from Singaporean samples. Total number of samples in EK is 1086 and in ES is 1078. The primary goal of two consortia is to identify novel type 2 diabetes (T2D) related genetic factors, but data also includes several diabetes-related quantitative traits such as lipid and blood pressure traits. We applied the proposed meta methods to lipid and blood pressure traits: total cholesterol (CHOL), HDL cholesterol (HDL), LDL cholesterol (LDL), triglycerides (TG), systolic blood pressure (SBP) and diastolic blood pressure (DBP). Subjects having medication that might affect the traits were excluded for the analysis. Sample sizes considered missing count and medication are described in Table Empirical type I error rates under the setting of Scenario 1 and 2 are calculated. Since first scenario consider homogeneous case and second scenario considers heterogeneous case, we only calculate type I error in these two scenarios. The rates are calculated under each significant level, 0.01 and 0.001. Empirical type I error rates under the setting of Scenario 1 are given in Table To compare the power, we performed the simulation work. First, we compared power of our meta method with those of joint analysis to check efficiency of meta method. When meta-analysis yields highly comparable results with joint analysis, the meta method serves well as an alternative of joint analysis. Since joint-analysis cannot handle heterogeneity between studies such as different number of covariates directly, we only performed joint-analysis in homogeneous scenario settings scenario 1 and 4). In Fig.  and 4. IFigure Figure p value = 4.75e-05) at the bonferroni’s significant level (α = 5.37e-06), but this gene is also not known to be related with blood pressure. According to GO annotations, it is involved in calcium ion binding. However, the gene that has the second smallest p value in both Het-meta-Q2  and Het-meta-Q3  is KCNA5 and this gene is already known for having strong relevance to hypertension [p values of genes that were detected at the threshold, 1.00e-04 have the smallest value in MetaQ-test. Moreover, we compared our meta method results with that of single QTest analysis to verify the improved power of meta methods. Table p value  is smaller than in Het-meta-Q2. However, its p value in ES single study analysis has very small value, so p value from MetaQ-test is the compromised these two p values from EK and ES single QTest analysis. But, p values of KCNA5 from each single QTest are both larger than p value from MetaQ-test. This fact can say that KCNA5 which is known for one of causal genes of hypertension arterial was only detected by MetaQ-test and MetaQ-test improved the statistical power of QTest.We applied suggested methods to the real lipid and blood pressure data. All considered meta methods did not detect any significant genes for lipid traits. But, for blood pressure traits we gained the different association results with lipid traits. Before analysis, we filtered common SNPs. Thus, remaining number of genes for the analysis of SBP is 9303 and SNPs in each gene have MAF < 0.05 and MAC > 4. Unlike MetaSKAT methods that failed to detect any significant genes as well in the lipid traits analysis , MetaQ-test detected PCDHA9 gene in Het-meta-Q2 . Table p value = 1.17e-06) and PCCA was detected in Hom-meta-SKAT-O . DTYMK, however, is only found in EK samples, thus MetaQ-test result is only stemmed from EK QTest result . When a gene exists in a single study, then our MetaQ-test result reflects the result of single study. DTYMK is not currently identified as related with blood pressure, but as related with thymidylate kinase activity. Another detected gene in MetaSKAT, PCCA is not known to be associated with blood pressure function, but it is related with propionic academia and pcca-related propionic academia.For the analysis of DBP, after filtering common variants, the number of genes is 10,528 and SNPs in each gene have MAF < 0.05 and MAC > 3. In the meta-analysis of DBP, two genes, DTYMK and PCCA were detected at the Bonferroni’s significant level α = 4.5e-06. Tap value = 2.76e-06, Table PCCA gene was also detected using QTest1 of ES sample analysis  is known to be associated hypertension arterial and purpura thrombotic thrombocytopenic that is closely related blood pressure. Table Although CABIN1 gene was not discovered at the Bonferroni’s significant level, but it is discovered at the FDR adjusted We propose MetaQ-test for meta-analysis of gene-level rare variant association studies. The basis of MetaQ-test is preserving the prior phase of association studies that is Q-test. MetaQ-test retains the relationship among multiple variants in a region. Further, MetaQ-test considers whether the genetic effects on the phenotype are same across studies or not. Assuming same effects corresponds to fixed effect meta-analysis model or else random effect model. By considering direction of variant effects and their equivalence in effect size at the same time, MetaQ-test can cover broad range of realistic meta-analysis cases.p values and we thought that most of them are false positive. Thus, in the result of real data analysis, we excluded the Hom-meta-Q results. Therefore, we expect that the prior test of heterogeneity of genetic effects can help us to determine appropriate model. In real data analysis, we have shown that Het-meta-Q searched out some known genes associated blood pressure trait. However, it also discovered some novel genes that are not at known to be associated with blood pressure at least for now, thus to validate biological relationship between them, more research and experiment in biology field are needed. For the future research, we can combine adaptive test that can determine degree of heterogeneity of genetic effects to improve the power further.We investigated the performance of MetaQ-test through simulation and real data analysis. Simulation studies showed that type I error rates were controlled well and MetaQ-test, particularly Het-meta-Q achieved the higher than or as powerful as MetaSKAT in various scenarios. However, when causal variants are over than 50%, then our methods are not powerful as MetaSKAT. Thus, when there are small percentage of causal variants, our method is more powerful in all scenario settings. Since Hom-meta test uses estimating regression coefficients, satisfaction of model assumption affects the power of MetaQ-test sensitively. If there are no many overlapped variants in a gene across single studies, the assumption of Hom-meta is broken and Hom-meta-Q hardly perform well. For this reason, in the result of real data analysis , Hom-meta-Q have inflated


To evaluate the performance of this statistic for various situations, we simulated multiple sets of sample size and frequencies to compute statistical power. Furthermore, we applied our method to several real datasets (psoriasis genome-wide association datasets and schizophrenia genome-wide association dataset) to evaluate the performance. Careful analyses indicated that our newly developed statistic outperformed several previously developed GWAS applications. Importantly, this statistic is more robust than naive merging method in the presence of small control-reference differentiation, therefore likely to detect more association signals.Genome-wide association studies (GWASs) have identified abundant genetic susceptibility loci, GWAS of small sample size are far less from meeting the previous expectations due to low statistical power and false positive results. Effective statistical methods are required to further improve the analyses of massive GWAS data. Here we presented a new statistic (Robust Reference Powered Association Test Genome-wide association studies (GWASs) have been widely applied with the goals to detect genetic variants which contribute to complex traits in the past decade . In geneAlthough GWASs have led to abundant significant findings , a few pAvailability of large public datasets contain large amount of useful genetic information. Utilizing large public datasets as reference may increase the sample size and ease the low power brought by insufficient sample size. However, population stratification and genotyping platform differences  between p-value of 2 × 2 Fisher exact test ))-ln(P(co-re)) in which P(ca- (co + re)) is act test  of the Sact test  of the S2) has been made available.In this manuscript, we present a method to use public datasets as reference in the association analysis. The results of simulation and real data application showed that our new method could increase statistical power, particularly for small GWAS researches in real application. And the online tool , reference size . And allele frequencies were set in the reference , in control  and in case . Totally, there were 128,000 different combinations , control size . Fisher exact test was used to compute the p-value of allele frequency for each SNP. The threshold of p-value was set as 1.2 × 10-7 by Bonferroni Correction. Then we selected first 100 SNPs of lowest p-values for further analysis. Two different large public datasets, gnomad.genome. NFE  (ref1)  (ref2)  (ref1)  and gnom) (ref2) , were seodel see . Specialp-values in the whole level, we drew the Manhattan plot of GRU group  became more positive due to the added genetic information of reference. Also, our method rescued a few SNPs which turned from negative to positive  = 8.09E-17)], located 30 kb centromeric of HLA-B and 16 kb telomeric of MICA (MIM: 600169), had been previously reported to be associated with psoriasis (rs7192 [P(T) = 8.66E-15] and rs20541 [P(T) = 1.13E-13] were candidate causal SNPs at leukocyte antigen (HLA) loci (MIM: 142395) which played an important roles in pathways of psoriasis (rs1051792 [P(T) = 4.77E-13] in the MICA gene (rs1051792) had also been suggested to be specific for purely cutaneous manifestations of psoriasis (rs2442719 [P(T) = 4.47E-11], located only 1 kb from the telomeric end of HLA-B (MIM: 142830), also exhibited significant association with psoriasis (rs3130573 [P(T) = 1.70E-10] was located in PSORS1C1 (MIM: 613525) gene which was a major susceptibility locus for psoriasis . Fisher exact test was used to compute the p-value of allele frequency for each SNP. There was no statistically significant SNP with the threshold as 7.14 × 10-8 by Bonferroni Correction. The large public datasets, gnomad.genome.NFE  (ref1)  (ref1)  was seleodel see .p-value below 1e-5 in the bottom figure. And after performing our method, plenty of SNPs came to the fore with p-value between 1e-5 and 1e-8. In addition, there were significant SNPs with p-value below 7e-8, some of which had been reported in previous studies. The SNP rs12140791 is located in NOS1AP (MIM: 605551) gene which is essential for brain development and function and of potential relevance to schizophrenia  was also reported by a previous large-scale GWAS  difference.A simplistic way to alleviate such concern is to perform a control (co) vs. reference (re) Fisher exact test . If a p-p-value of 2 × 2 Fisher exact test . We define a test statistic: T = ln(P(ca - (co + re))) - ln(P(co - re)) (Denote the act test  of the So - re)) . T will p-value of T is as follow:Our null hypothesis is that the allele frequency of the SNP is equal among cases, controls and references. Under the null hypothesis, both P(ca - (co + re)) and P(co - re) are independent and uniformly distributed on . Under the null hypothesis, -ln(P(ca - (co + re))) and -ln(P(ca - co)) are exponentially distributed with parameter 1 . And T iFrom the above formula one can easily see that the P(co - re) acts as a penalizer/normalizer against P(ca- (co + re)). Therefore, our new statistic T is more robust than the simple P(ca - (co + re)) method in the presence of small control-reference differentiation. However, in the presence of strong population differentiation or genotyping platform difference, even our correction may not be effective, we therefore need to restrict P(co - re) to be not significant with a suggested threshold of 0.01.We obtained the psoriasis datasets , as a paThe schizophrenia dataset  came froAssociations between SNPs and complex traits were detected by comparing frequencies of alleles in case and control group in GWAS . SeveralOur simulations focused on several representative cases to evaluate the performance of the new model. When the allele frequencies of case, control and reference were equal, the power of T was higher than power of ca-(co+re), while both less than false positive rate. False positive results would be produced in the presence of small population genetic differentiation between control and reference. However, as the control size increased, the false positive rate of P(T) would be reduced . The falConsidering that allele frequency divergence of multiple variants due to different platforms is more serious than slight population stratification in practical application, we designed the statistic based on single variant rather than multiple variants. Allowing slight population stratification between control and reference, the simulation has shown the large sample size of control would suppress the false positive results. Moreover the statistical power and utility of our method were also elevated.rs1343708  could provide more effective information for SNPs associated with the traits.In the aforementioned application of our method to the psoriasis GWAS genetic datasets , we selep-value of some false negative genetic loci would decrease significantly down to the threshold. And our test statistic T is more robust than P(ca - (co + re)) as it takes the control-reference difference into account by penalization of P(co - re).Novel SNPs with weak positive signals could be discovered when the sample size of case and control are insufficient in GWAS . With thWe presented a new statistic T to use large public database as reference to reduce concern of potential population stratification. And the new statistic proposed here is effective to discover novel genome-wide significant loci with both small and large sample sizes.YW conceived the idea and developed the software. YL, MH, and XL contributed data analysis, generating tables and figures, and manuscript writing. YW, YL, MH, XL, YS, and LJ contributed the theoretical analysis and manuscript revision. MX helped support the GWAS datasets. YW, YL, MH, XL, MZ, JW, and MX contributed to scientific discussion. All authors contributed to final revision of the 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.


Association studies using a single type of omics data have been successful in identifying disease-associated genetic markers, but the underlying mechanisms are unaddressed. To provide a possible explanation of how these genetic factors affect the disease phenotype, integration of multiple omics data is needed.We propose a novel method, LIPID , that uses both single nucleotide polymorphism (SNP) and DNA methylation data jointly to analyze the association between a trait and SNPs. The total effect of SNPs is decomposed into direct and indirect effects, where the indirect effects are the focus of our investigation. Simulation studies show that LIPID performs better in various scenarios than existing methods. Application to the GAW20 data also leads to encouraging results, as the genes identified appear to be biologically relevant to the phenotype studied.The proposed LIPID method is shown to be meritorious in extensive simulations and in real-data analyses. In complex disease studies, genome-wide association studies (GWAS)  and epigMost existing methods use additional information to filter out nonsignificant loci and reduce the total number of tests, which, in return, improve power . On the In this paper, we propose a novel method, LIPID , to use both SNP and DNA methylation data to test whether there is an indirect effect of SNPs on a phenotype. The indirect effect and its variance–covariance matrix are derived, and a Wald test is conducted. An extensive simulation study was done to evaluate the properties and the performance of LIPID, which was also applied to analyze the GAW20 real data.n independent subjects, and for each subject, its SNP, DNA methylation, covariates, and phenotype are measured. Specifically, let Y = T be the vector of observed phenotypes; X =  be the n × k matrix denoting the observed values of k (nongenetic) covariates, including intercept; S =  and M =  be the n × r and n × p matrices regarding the genotypes of r SNPs and methylation levels of p cytosine-phosphate-guanine (CpG) sites, respectively. Assuming that phenotype Y is a continuous variable, we can use a set of linear models to capture the relationship among Y, X, S, and M as follows:εY ∼ N , vec(εM) ∼ N (ΣM⊗In), vec(·) is the vectorization operation; ⊗ is the Kronecker product; and εY and εM are independent. Note that here βS and βX are r × p and k × p matrices respectively, and ΣM is a p × p positive definite matrix. Model (1) characterizes the relationship between phenotype and SNPs, DNA methylation, and covariates, while model (2) depicts the relationship between DNA methylation (as a response variable) and SNPs and covariates. It is concluded from models (1) and (2) that the direct effect of SNPs on the phenotype is αS and the indirect effect is γ = βSαM.Suppose there are For linear models (1) and (2), we have the following maximum likelihood estimates11 represents the first r × r diagonal submatrix. As αM and βS are unavailable, their estimates Their corresponding block matrices are the variance–covariance matrices for M and Covr degrees of freedom.So an unbiased estimate for the variance–covariance matrix of b is the random effect, with mean 0 and variance–covariance matrix b + εY ∼ αM and its variance–covariance matrix are derived similarly, and the LIPID statistic has the same form.For subjects with correlation between each other, linear model with mixed effect is utilized. So model (1) is changed toS, M, and Y are all univariate. In addition, the direct effect that we are not interested in does not exist in simulation. The simulated data are generated as follows. First, SNP S is generated with a minor allele frequency (MAF) under Hardy-Weinberg equilibrium. Then, DNA methylation M is generated from a normal distribution with mean SβS and variance Y is generated from a normal distribution with mean MαM and variance Y and variance of M are fixed to 1 for simplicity. In scenario 1, the coefficients βS and αM are both 0; in scenarios 2 and 3, βS is nonzero while αM is 0; in scenarios 4 and 5, αM is nonzero with βS equal to 0. In these scenarios, the indirect effect is nonexistent, so we change the coefficient of one parameter to measure the Type I error rates under different situations. Under Ha, the indirect effect is βSαM ≠ 0. The coefficients are chosen so that different methods have moderate powers. From scenario 1 to scenario 5, the βS and αM are increased. The MAF ranges from 0.1 to 0.4.To evaluate the performance of LIPID, simulation under various scenarios are conducted. For simplicity, we assume there are no covariates, and that mentclass2pt{minimβS and αM are 0, the Type I error rate is conservative; for scenarios with 1 coefficient that is not 0 but relatively small, the Type I error rates are still conservative; for scenarios with 1 large coefficient, the Type I error rates are better controlled. Compared to o-eSNP [Table o o-eSNP , the TypFAT1  and DCTN6 , while o-eSNP fails to find any significant genes . We further use the BIOS QTL (quantitative trait locus) browser [FAT1 is a cis-meQTL (methylation quantitative trait locus) with a p value of 3.8E-07, but we did not find any meQTL on gene DCTN6. FAT1 is associated with cholesterol in DAVID , whereas DCTN6 is involved in lipid metabolism [The GAW20 real data package contains genomic, DNA methylation, and phenotypic data for more than 1000 individuals from 188 pedigrees. The phenotypic data include metabolic indices, lipoproteins, and triglyceride. Dense genome-wide SNP markers make up the genomic data. DNA methylation levels are also available on CpG sites genome-wide before and after individuals are treated with fenofibrate. The level of triglycerides and the methylation level before treatment are made use of. The covariates include gender, age, smoking status, high-density lipoprotein, metabolic disorder, and center. We use SOLAR  [ p value .4E-7 andM or MαM on S, but the variance–covariance estimates are different, and LIPID has a less-biased variance–covariance estimate, which leads to the improvement of performance of LIPID. Furthermore, we also adapt o-eSNP and LIPID to correlated subjects.Compared to o-eSNP , LIPID cApplication of LIPID to the GAW20 real data indicates that LIPID is capable of detecting genes with indirect effect. The computation is efficient, and the process takes 30 min to analyze the whole GAW20 data set on a personal computer with an Intel Core i3–4150 CPU. The genes identified appear to be functionally relevant to the trait being considered, thereby substantiating the importance of these findings and leading to confidence of genes found being true discoveries.For complex diseases, we propose a novel method to detect indirect effect of SNPs on a phenotype via methylation, and we demonstrate its superiority compared to 2 existing methods. LIPID is single-step and does not require multiple tests, compared to traditional mediation analysis; at the same time, multiple genetic loci can be used simultaneously to test indirect effect.