Emerging evidence suggests that donor/recipient matching in non-HLA (human leukocyte antigen) regions of the genome may impact transplant outcomes and recognizing these matching effects may increase the power of transplant genetics studies. Most available matching scores account for either single-nucleotide polymorphism (SNP) matching only or sum these SNP matching scores across multiple gene-coding regions, which makes it challenging to interpret the association findings. We propose a multi-marker Joint Score Test (JST) to jointly test for association between recipient genotype SNP effects and a gene-based matching score with transplant outcomes. This method utilizes Eigen decomposition as a dimension reduction technique to potentially increase statistical power by decreasing the degrees of freedom for the test. In addition, JST allows for the matching effect and the recipient genotype effect to follow different biological mechanisms, which is not the case for other multi-marker methods. Extensive simulation studies show that JST is competitive when compared with existing methods, such as the sequence kernel association test (SKAT), especially under scenarios where associated SNPs are in low linkage disequilibrium with non-associated SNPs or in gene regions containing a large number of SNPs. Applying the method to paired donor/recipient genetic data from kidney transplant studies yields various gene regions that are potentially associated with incidence of acute rejection after transplant. Transplant matching usually focuses on non-genetic factors related to the donor, the recipient, or the graft itself, such as recipient age, donor sex, or organ size. Genetic matching in transplant has been limited to the human leukocyte antigen (HLA) region of the genome in the past, as this region codes for immune related genes that may lead to the recipient recognizing the allograft as non-self and mounting an immune response against it . AlthougSeveral studies have found evidence suggesting that donor and recipient genetic mismatch in these non-HLA regions could impact transplant outcomes. p-values from tests of individual markers  is based on the hypothesis that observing the coding regions of both the recipient and donor genomes can help identify the number of potentially incompatible amino acids between the pair . The AMSHere we propose a new method called the “Joint Score Test (JST).” JST is built upon marginal likelihood scores for testing multiple SNP effects and a gene-based D/R matching score. JST uses only the most informative linear combinations of single recipient SNP marginal likelihood scores to allow for an increase in statistical power. JST also allows for flexible adjustment for covariates and therefore it can maintain nominal type-I error rates in the presence of population stratification.We organize our paper as follows. First, we discuss the underlying model and the construction of JST. Then we present the results of extensive simulation studies. Third, we utilize JST in an association analysis of kidney transplant data. Finally, we discuss the benefits and potential limitations of the method.JST jointly tests whether recipient genotype SNPs or a gene-based D/R matching score are associated with the transplant phenotype of interest.m SNPs in a genomic region of interest are available for n transplant D/R pairs. Let jth marker for the ith recipient or donor, respectively kth covariate of D/R pair i. We consider a regression setting with continuous or categorical outcomes, Yi, and our generalized linear model (GLM) is of the formg  is the link function, K covariates for D/R pair i with regression coefficients m SNPs for recipient i with regression coefficients i with regression coefficient Suppose that genotype data for In general, each gene-based score can be written in the formThe first score, the identity-by-state (IBS) mismatch distance function, is defined asion test  and in aion test .The second score considered, the incompatibility distance function, is calculated asThis distance metric has been utilized in a kidney transplant study, where single SNP incompatibility and a genome-wide sum of this measure were found to be associated with antibody-mediated rejection . In addia denotes alleles of a genotype  distance function, is defined asgenotype . The undThe fourth score, the binary mismatch score, is based off a simplification of the AMS which assigned a score of 1 for all SNPs where the donor genotype contained an allele that was not present in the recipient genotype and a score of 0 otherwise . The sinIn this section we will focus on deriving our JST test statistic for a binary outcome, jth SNP genotype for recipient i from a weighted linear regression model i from a weighted linear regression model i or D/R pair i. As derived in the Online Resource, the likelihood score for testing the marginal association with the jth recipient SNP is equivalent toHere Similarly, the likelihood score for testing the marginal association with the gene-based D/R SNP genetic matching score is equivalent tom recipient genotype SNPs and the gene-based genetic matching score as U can be written into the matrix form,V at position T2 statistic can be constructed as T2 statistic has low power when Denote the vector of scores for all in power . Along tomponent , which tHere we propose a new statistic as follows. pth column being the pth eigenvector of s (s < s is discussed below). Let s by s identity matrix.Our statistic is based on Eigen decomposition of the sample variance-covariance matrix s+1 degrees of freedom under the null. Therefore, s+1 degrees of freedom and non-centrality parameter that is equal to its value.We can show that this statistic is asymptotically distributed as a Chi-squared random variable with s) is always a difficult task in principal component analysis. One common method is to choose the number of PCs to keep based on a predefined percentage of total variance explained. We utilize this method in our simulation studies, looking at a range of 65–99% total variance explained by the retained PCs.Choosing the number of PCs to retain .For power analyses, phenotypes were generated using the model R v4.0, . Code ton x (m+1) matrix of combined recipient genotype SNPs and gene-based D/R matching score as input as was used for JST. For our standard GLM, we fit separate models under the null and alternative hypotheses respectively, and then calculated the likelihood ratio test (LRT) statistic using the lrtest function from the lmtest  were determined with the AFR-AMR Axiom chip  (n = 404 donor-recipient pairs) was performed on a custom exome-plus Affymetrix TxArray SNP chip  Genomics Study and Genomics of Kidney Transplantation  study. Genotypes from the DeKAF cohort  , which cSNP chip , which cSNP chip . Non-Caus = 85% variance explained and SKAT with an unweighted IBS kernel. Covariates were included for recipients’ age, gender, PRA status, prior non-kidney transplantation, and an indicator for the cohort membership. p-values were adjusted for multiple comparisons using the Benjamini-Hochberg procedure to control false discovery rate (FDR).The common SNPs in the two cohorts were grouped by physical locations within 23,062 genome-wide genes (GRCh38. p13) and then analyzed using both JST with NAT2 with 500 D/R pairs, and all figures are based on 1000 D/R pairs, using data from either NAT2 or CHI3L2.A subset of the total simulation results is presented for both type I error testing and power testing. In all scenarios, results were similar for all combinations of gene, sample size, and any additional variables tested. s values between 65 and 90%, with some conservative error rates seen with s = 99% for lower outcome prevalence. Similar results can be seen for the SKAT method using either the linear or IBS kernel. The standard score test based on generalized linear model tends to have inflated type I error values when the outcome is binary, ranging from around 5–11%. When the outcome is continuous, the type I error is slightly inflated, at 6%.s = 95% reaches close to 90% power but SKAT does not reach 80% power. When the SNPs associated with the outcome are in high LD, SKAT and the JST method can have similar power levels. This can be seen in the s = 65–80% reach around 85% power. When the size of the gene is increased, as in panels C and D, we see that JST tends to have higher power than SKAT in both scenarios. Additionally, when a continuous outcome is examined, JST with any value of s tends to have much higher power than SKAT using either kernel . When the IBS gene-based score or the AMS is truly associated (s values leading to over 80% power. The Binary mismatch score being truly associated (sociated , power tsociated  tends tosociated .s value of 85% and the top five genes from SKAT analysis of the combined data sets, using an unweighted IBS kernel. Gene ranking is based on the smallest p-value for all four score models. p-values for the four different models are relatively similar for each of the five genes. The number of genotyped SNPs in the genes ranges from 3 to 15 for the JST results and from 3 to 104 in the SKAT results. Only one of the top genes found using SKAT analyses was also found in the top five for JST analysis. All five of the genes from JST analysis are significant using a FDR cutoff of less than 0.10, while none of the genes found by SKAT are significant using this method.We propose a multi-marker test statistic designed for use with paired genetic data in transplantation. The joint score test, or JST, can be used for testing whether specific gene regions are associated with transplant outcomes, either through the recipient genotype or D/R genetic matching. Compared to other statistical tests in this setting, the JST has the potential for increased power and reproducibility as compared to single SNP tests, as well as increased interpretability as compared to multi-SNP methods that sum across large regions of the genome.The Eigen decomposition of the recipient genotype covariance matrix allows for a potential increase in power for JST as compared to a standard GLM likelihood ratio test. By transforming the covariance matrix into its principal components, we are able to select only those components that are most likely to be associated and discard the remaining components. We chose to keep principal components with large eigenvalues in accordance with previous theoretical and empirical work by Our simulations showed that JST is a competitive method as compared to standard GLM, and SKAT. Type I error rates were conserved for both JST and SKAT but were inflated for GLM. This inflation could be due to the larger number of covariates being fit in this model. In power simulations, JST tended to outperform SKAT when recipient genotype SNPs, in low LD with all other SNPs in the region, were truly associated with outcome. When the recipient SNPs were in high LD, SKAT and JST performed similarly for the smallest gene but increasing the gene size while keeping the percentage of associated SNPs constant led to JST surpassing SKAT in power. These findings agree with those found by JST and SKAT behaved similarly in some of the simulation scenarios but the construction of these two test statistics is different. Using the linear kernel, the SKAT statistic is equivalent to summing the squared score statistics of the genetic data . Under tTo the best of our knowledge, SKAT has not previously been evaluated under the scenario of jointly testing for an association between a set of SNPs and a gene-based score and has not been utilized with paired transplant genetics data. Our evaluation of SKAT under these circumstances found that the method works well and is robust. We determined that the choice of kernel often impacted power levels, however, such as when the SKAT method using the linear kernel had minimal power as compared to the SKAT method using the IBS kernel under the scenario where R genotype SNPs were associated with outcome. Investigation into this phenomenon found that scaling the gene-based scores from between 0 and 1, before running SKAT with a linear kernel leads to an improvement in power . The IBSs that leads to the highest power in all scenarios. When gene-based score was associated with the outcome, the smallest s value we considered tended to have the highest power, but power tended to be similar for s values corresponding to between 65 and 80% variance explained by the PCs. When recipient genotype SNPs were associated with the outcome, changing the value of s did not tend to drastically change the power levels. Since the true association mechanism is unknown for real data analysis, it may be beneficial to run models using a few different s values, although this will increase the number of tests. An alternative is to choose a middling value of s, around 80 or 85%, which tends to have high power under either scenario of association.Based on the simulation studies, there is no clear value of IFNA5, Z98752.3, and SGK2, have been found to be associated with the immune system or specific types of immune cells which could attack a transplanted kidney if the graft is recognized as non-self (IFNA5, for example, is known to be involved in differentiation and proliferation of B and T cells, as well as being involved in the adaptive immune response, which involves the creation of antibodies that may attack a donor organ (SGK2 were found to be associated with leukocyte and monocyte counts which are directly associated with immune response (Our analysis of kidney transplant data found five genes to be statistically significant after accounting for multiple comparisons at FDR<0.10. These include genes that could plausibly lead to AR after kidney transplant. Three of the five genes, non-self . IFNA5, or organ . In prevresponse . ReplicaThe JST method does have some limitations. Transplant data analysis was limited to data from paired kidney transplants. The method can be applied to other organ data as well, as long as genotype data is available for both the donor and the recipient. For our simulation studies, we only focused on unrelated D/R pairs, but it is possible that the degree of relatedness between a donor and a recipient may impact whether the recipient experiences acute rejection. We were able to look at related versus non-related pairs in our combined kidney data sets and found that these two groups had no overlap in their top five potentially associated genes. Based on these results, we believe it is important to account for relatedness between a D/R pair in analyses. We restricted our analyses to only include common SNPs, but rare variants can be used in JST analyses. If there is interest in the association of rare variants, it is possible to construct a weighted version of the distance function as,Then a simple weighting option that will help upweight rarer minor allele frequencies is the use of j=1MAFj . AdditioIn summary, the JST is a powerful method that can be used for the analysis of paired genetic data. Use of this method could lead to the discovery of gene regions potentially important to transplant outcome, which could be further studied to try and determine the biological mechanisms behind acute rejection.


The main aim was to study the influence and interaction of different models and marker densities on GP accuracy. The GP accuracy of the six traits was found to be between 0.28 and 0.60, as demonstrated by the cross-validation results. We showed that the accuracy of GP could be improved by increasing the marker density, which is closely related to the model adopted and the heritability level of the trait. Moreover, based on two different marker densities, it was derived that the prediction effect of GBLUP model for traits with low heritability was better; while with the increase of heritability level, the advantage of Bayesian Alphabet would be more obvious, therefore, different models of GP are appropriate in different traits. These findings indicated the significance of applying appropriate models for GP which would assist in further exploring the optimization of GP.The marker density, the heritability level of trait and the statistical models adopted are critical to the accuracy of genomic prediction (GP) or selection (GS). If the potential of GP is to be fully utilized to optimize the effect of breeding and selection, in addition to incorporating the above factors into simulated data for analysis, it is essential to incorporate these factors into real data for understanding their impact on GP accuracy, more clearly and intuitively. Herein, we studied the GP of six wool traits of sheep by two different models, including Bayesian Alphabet (BayesA, BayesB, BayesC The advancement in the field of quantitative genetics and molecular biology has improved the selection and breeding methods of domestic animals , goats , aquaticFor the first time, P method . AnotherAs an important domestic animal, sheep is one of the earliest domestic animals reared by humans  and high (630 K) were applied to estimate the genetic variance components of the Alpine Merino sheep datasets. Further, based upon the SNP genotypes data, different models were adopted for GP and cross-validated to compare the accuracy of different GP methods. The main purpose of this study is to investigate the impact of different densities of SNP genotypes and different GP methods (Bayesian Alphabet and GBLUP) on the accuracy and optimization methods of GP in Alpine Merino sheep populations.All animal work carried out in this study was performed per the guidelines for the care and use of laboratory animals promulgated by the State Council of the People’s Republic of China. The study was approved (License Number: 2019-008) by the Animal Management and Ethics Committee of Lanzhou, Institute of Animal Husbandry and Veterinary Sciences, Chinese Academy of Agricultural Sciences.The original phenotypic dataset was obtained from the Sheep Breeding Technology Extension Station of Gansu Province. These datasets consisted of 11,500 individuals based on 7 different herds with information such as region (herd), sex, and date of birth. The individuals in this study included 821 Alpine Merino sheep (563 ewes and 258 rams) from HuangCheng pasture in Gansu Province, China, the pasture was under the jurisdiction of the Gansu Sheep Breeding Technology Extension Station which has a rigorously standardized system of breeding and management, to ensure that all the individuals have uniform feeding and management conditions. The average age of each individual with phenotypic data was about 12–14 months. The wool traits involved in this study were staple length (SL), clean fleece weight rate (CFWR), average FD, coefficient of variation of average FD (FD_CV), staple strength (SS), and fleece extension rate (FER). The wool from individuals was collected and evaluated according to the Agricultural Industry Standards of the People’s Republic of China (NO. NY/T 1236-2006). Wool samples (∼250–300 g) collected from the abdomen of each individual, were weighed and stored in ziplock bags . Within one week, the samples were sent to the National Animal and Rural Ministry of Animal and Fur Quality Supervision and Inspection Center  for weighing, screening, and quality identification of wool. Blood samples (∼5 ml) were also collected from each sheep from the jugular vein and immediately transferred to the vacutainer blood collection tube . Blood samples were stored at −20°C for further genotyping  was used to impute the sporadic missing alleles  for the Alpine Merino sheep. The genotyping platform for analysis was based on the array plate processing workflow of GeneTitan system  from Thermo Fisher (Affymetrix). The sites in the Illumina Ovine SNP 50K microarray were screened out from the Affymetrix HD 630K microarray and used as the datasets of low-density SNP genotypes (L-datasets). The H- and L-datasets were preprocessed using PLINK v1.9b4 software prior to the statistical analysis and variance component estimation  and GBLUP. Six wool traits from 821 samples were used to first, estimate the variance of each component, including the additive and residual variance; second, five different models were adopted to perform GP, and its accuracy was compared via fivefold cross-validation, and all these models were evaluated in SNP datasets of H- and L-datasets. Replicate measurements were not available for the individuals so that the effects of permanent environmental were not modeled. The samples involved were from different herds and sex. These factors altered the phenotype in a fixed pattern, and hence the system environmental effects were added to the framework.The statistical methods of Bayesian Alphabet involved can be written as:et al. and Habier et al G=WaFivefold cross-validation was performed to compare the accuracy of different methods of GP. During K-fold cross validation, the population should be divided randomly  . It calcAccording to the above mentioned five models, the cross-validation was performed based on two types of genotypic data (H- and L-datasets), with different densities and the BVs of the validation group (candidate population) were predicted. In addition, the above cross-validation was performed in triplicates in order to ensure the randomness of individuals in the validation group. Finally, the GP accuracy values were calculated for each validation, averaged, and then recorded as the final accuracy.https://figshare.com/projects/Evaluation_about_GP_for_821_AMS/112614). The script adopted in this study can be obtained on GitHub (https://github.com/gdlc/BGLR-R).Genotype and phenotype data are available, they could be obtained at Figshare  of each trait based on the ratio of the additive variance to the total phenotypic variance (Va/Vp). For L-datasets, heritability ranged from 0.37 (FER) to 0.70 (SL); and for H-datasets, heritability ranged from 0.29 (FER) to 0.68 (SL). The estimated results of heritability (expressed as the proportion of additive variance in phenotypic variance) shown in Estimate the phenotypic variation and additive variation of the six wool traits based on the L- and H-datasets, and calculates the heritability  of markers was much larger than the number (821) of individuals. Although many advanced statistical methods , respectively. The statistical mean of SS and SL was 33.82 ± 9.82, 80.93 ± 13.06, respectively , and their corresponding accuracy of GP was also the highest, which ranged from 0.53–0.60 to 0.45–0.55, respectively. While for two traits with the lowest heritability, SS (h2 = 0.33) and FER (h2 = 0.28), the accuracy was 0.29–0.38 and 0.28–0.36, respectively, which was lower than SL and FD_CV. For those traits with lower heritability, the correlation between phenotypic value and genetic value will be lower, the effect value of markers distributed across the genome may be estimated with lower accuracy , adopting Bayesian Alphabet  and GBLUP model to perform the GP. The heritability of six wool traits of Alpine Merino sheep was estimated, and the accuracy of the BVs prediction of these traits under different conditions was evaluated through fivefold cross-validation. To the best of our knowledge, this was the first study of optimization of GP which has been applied to the domesticated Alpine Merino sheep populations. We have observed that for traits with low heritability (SS and FER), increasing the density of markers could improves the GP accuracy, but it has little impact on traits with high heritability (SL), and even decreases the accuracy (FD). The accuracy of the GBLUP model is generally higher than that of the Bayesian Alphabet model for SS and FER, while with the improvement of heritability, the advantage of GBLUP is no longer obvious. Therefore, from this study, we conclude that different GP models are applicable to different traits: GBLUP is more suitable for traits with lower heritability (FER and SS), and for Bayesian Alphabet, especially BayesB and Bayesian LASSO, have better GP effects for traits with high heritability (FD and SL).


There are strong parallels between the 2 models in the components of fitness variance but they differ in their rates of evolution measured in terms of ∆p, gene frequency change, or of ∆W, change in mean fitness. The Price Equation is used to partition ∆W into 2 components, one owing to the genetic variance in fitness by natural selection and a second owing to change in environment. Adaptive evolution is faster in the 2-locus model with G × Gmaternal with free recombination, than it is in the 1-locus model with G × E, because in the former the maternal genetic environment coevolves with the zygotic phenotype adapting to it. I discuss the relevance of these findings for the evolution of genes with indirect genetic effects.Maternal-zygotic co-evolution is one of the most common examples of indirect genetic effects. I investigate how maternal-zygotic gene interactions affect rates of evolution and adaptation. Using comparably parameterized population genetic models, I compare evolution to an abiotic environment with genotype-by-environment interaction (G × E) to evolution to a maternal environment with offspring genotype-by-maternal environment interaction (G × G Maternal-zygotic co-evolution is one of the best examples of why so many features of adaptations involve and are dependent upon indirect genetic effects. I would argue that there are no aspects of adaptive evolution of complex phenotypes that do not involve maternal indirect genetic effects owing to their critical involvement in the earliest stages of development. Mothers are an ubiquitous, essential component of the capacity of living systems to reproduce. Whether we are considering the reproductive fissioning of a mother cell giving rise to daughter cells in a single celled protist or the more complex, sex-specific multi-cellular reproductive process of a eukaryote, maternal genetic information is an essential feature of the reproductive process.both maternal and zygotic transcripts, but some, like genes biocid in flies and mater in the mouse, are genes with strictly maternal effects acting on the zygotic phenotype. Selection on strictly maternal effect genes depends upon offspring fitness rather than maternal fitness. As a result, death of an offspring caused by a gene in the maternal genome is less effective at causing genetic change than is the death of an offspring caused by a gene with a direct effect in its own genome  and adaptation to a maternal environment with offspring genotype-by-maternal environment interaction ? A similar question could be asked in terms of adaptation of a host to its symbionts when there is symbiont-by-host genotype environment interaction (GHost × GSymbiont). In this paper, I will use the classical 1 locus-2 allele model of G × E and contrast its features with those of the 2-locus population genetic model of G × Gmaternal from My central question is this: how do maternal-zygotic interactions affect evolution and adaptation? And in particular, I am interested in the question: In outline, I first introduce the basic model and its parameterization. Second, I partition the total variance in fitness into its 3 components for both models: the genetic variance, the environmental variance and the interaction variance. This will facilitate model comparison.maternal. Whether measured by ∆p or ∆W, adaptation to a biotic environment with G × Gmaternal can be much more rapid than adaptation to an abiotic environment with G × E ; let Wi,j be the fitness of the i-th genotype in the j-th environment ; and, let fj be the frequency of environment j. If genotype, AA, is genotype 1, its average fitness across both environments is W1,. = f1W1,1 + f2W1,2. The fitness of every genotype is a function of the frequency of the selective environment. The selection coefficient, s, is positive as depicted. The mean fitness of AA homozygotes is (1 + 2f1s), that of Aa heterozygotes is (1 + f1s), and that of aa homozygotes (1). The selection coefficient (f1s) is a function of the frequency of the selective environment E1. The effect of the A allele on fitness is additive, incrementing genotypic fitness by (f1s) for each A allele in an individual’s genotype.The average fitness for each genotype is its mean fitness across the 2 different environments. Let Gmaternal in a way comparable to that of the G × E model , and cell entries to calculate the total variance in fitness, VTotal. The components are calculated using the grand mean and the genotypic means (rows) for VG, the grand mean and the environmental means (columns) for VE, and the genetic correlation across environments for VGxE. When we do this, we find the following:Using the values in whereE = VGB. Because the offspring genotypic fitness means differ, there is genetic variance, VGA. Lastly, because there is an interaction between offspring genotype fitness and maternal genotypic environment , and cell entries to calculate the total variance in fitness, VTotal. The components are calculated using grand mean and the genotypic means (rows) for VGA, the grand mean and the environmental means (columns) for VGB, and the remainder is VGxG. When we do this, we find the following:Using the values in where1, in Equation [2] has been replaced by pB, the frequency of the maternal B allele. In addition, the environmental variation term, (f1f2), in Equation [2] has been replaced by the maternal genic variation term, 2pBqb, in Equation [4]. I will now turn to the expressions for gene frequency change, ∆pA, and mean fitness change, ∆W, for these 2 similarly parameterized models.When we compare Equations [2] and [4], it is clear that the frequency of the selective environment, fThe rate of gene frequency change for the G × E model of GxE equals (1 + 2sf1pA) and there is complete gene flow among environments at the start of each generation. The genic effect on fitness is (sf1) and the genic variance is (pAqa) or ½ of the genotypic variance . With G × E, the genic and environmental effects are inter-dependent, with one being a frequency of the abundance of the other.where mean fitness, W1 near 0) because selection is weak. Conversely, evolution is relatively faster when E1 is common (f1 near 1). Note that, it is assumed that f1, the frequency of the selective environment, remains constant throughout the period of adaptive evolution.From Equation [5], it is clear that the rate of adaptation depends on whether E1, the selective environment, is rare or common . EvolutiHere, I assume that the A and B loci are unlinked and, that selection is sufficiently weak that it does not create lasting linkage disequilibrium (LD), so that I can treat LD as 0. This assumption will be relaxed below. The rates of gene frequency change for the G × G model of GxG equals (1 + 2spBpA). If there were no maternal-zygotic epistasis, then W would equal (1 + 2spA + 2smpB), where s and sm were the independent effects of the offspring allele A and the maternal allele B on offspring viability. Without epistasis, the term (spB) in Equation [6a] would be simply s, the effect of the A allele independent of maternal genotype. Similarly, without epistasis, the term (spA) in Equation [6b] would be a constant sm, the effect of the maternal B allele on offspring viability. The factor ½ is important in Equation [6b] because the change in the frequency of allele B caused by the death of an offspring changes via the maternal-offspring genetic regression, which is ½ with random mating. Change in A allele frequency depends on the strength of the genic effect on fitness which is clearly a function of pB, the frequency of the selective maternal genetic environment. The environmental effect here is the effect of the maternal genetic background which equals (spB). With G × G, the genic and environmental effects are also inter-dependent, with one being a frequency of the abundance of the other. Unlike the G × E case, where ∆f1 = 0, here the evolutionary description is incomplete without an additional equation for the genetic covariance between loci A and B, the linkage disequilibrium (LD). The value of LD is increased by selection on AB gene combinations and decreased by recombination. We will investigate these effects on LD in the sections below.where mean fitness, W1 , Equation [6b] guarantees that it will increase as allele A increases. With G × G, the environment itself evolves. Compare the rate of evolution of the A allele (starting at pA = 0.05) when E1 has an initial frequency of 0.05 with the rate of evolution of the A allele when the maternal genetic environment has an initial starting point of pB = 0.05. The trajectory of pA with G × G is faster than it is with G × E taken from comparable starting points  to (s[pB + ∆pB]) and, for the type of epistasis represented here, ∆pB > 0. Thus, the strength of selection on the A allele increases at every generation owing to its co-evolving maternal genetic environment.From Equation [6], it is clear that the rate of adaptation of the zygotic allele, A, depends on whether the B allele is rare or common in mothers . However1 . Here, aEvolution can also be viewed as ∆W, change in the mean fitness of a population before (W) and after (W’) selection, instead of or in addition to change in gene frequency. The Price Equation  is a genafter selection with the environment held constant, while (W’|E’) is the mean fitness after selection accounting for any change in the environment (E’) that might attend selection.where (W’|E) is mean fitness The partitioning of ∆W in Equation [7b] has a straightforward interpretation. The first term on the righthand side, (W’|E - W|E), is the change in mean fitness owing to natural selection. The second term, (W’|E’ – W’|E), is the change in mean fitness owing to change in the environment. I now apply this partitioned Price Equation to the 2 models.1pA). After selection, the mean fitness changes because the allele frequencies change, so that W’ now equals (1 + 2sf1pA’).  We can substitute (pA + ∆pA) for pA’ and, using Equations [2a] and [5] above, we find that Equation [7a] becomesThe definition of mean fitness for the G × E model is W = /WGxE which is VG/(WGxE)2, or the variance in relative fitness. This illustrates the direct connection between the Price Equation, Fisher’s Fundamental Theorem and the Opportunity for Selection.Fisher’s Fundamental Theorem is often interpreted as saying that “…fitness” , p. 164,election  equals (BpA). After selection, the mean fitness changes because the allele frequencies at both loci change, so that W’ now equals (1 + 2pB’pA’).We can substitute (pA + ∆pA) for pA’ and (pB + ∆pB) for pB’, using Equations [4a] and [6] above, we find that Equation [7a] becomesThe definition of mean fitness for the G × G model is W =  VGB/2WGxG, the rate of change of the environment which equals the rate of change of the maternal B allele; and, (3) (VGxG)/(WGxG)2, the co-variance between offspring fitness and maternal environment.The change in W is different when the maternal genotypic environment co-evolves. The total change in mean fitness contains 3 terms: (1) Vm, the change in W would have contributions from the additive offspring variance owing to the A allele and to the additive maternal variance owing to the B allele; that is, the additive effects would contribute to the first term. This may be best seen in a single-gene model with both additive and maternal effects [2pA])] and VG = (s + [sm/2])2 (2pAqa).) and (3) are positive, evolution with this type of synergistic epistatic selection involving a genetic environment is B) for the A locus and (spA) for the B locus, tend to increase over time because each is a function of an increasing gene frequency (A different interpretation of Equation [9d], based on Equations [6a] and [6b] is that the genic effects, (sprequency . Moreovethe most common and important examples of indirect genetic effects. Previous models have studied 1-locus 2 allele models with indirect maternal effects and direct zygotic or offspring effects (e.g., maternal, and with a comparably parameterized model of G × E. Since a great deal more theoretical and empirical attention has been devoted to G × E, it provides a well-known basis for comparison. Moreover, I have modeled one of the simplest types of G × E, scale differences in genotypic fitness across environments. This type of G × E is often not a matter for practical concern because it can be eliminated, statistically, by changing the scale on which a phenotype was measured (Indirect genetic effects as heritable environments have important evolutionary consequences. Maternal genetic effects are likely ts e.g., . Drown ameasured . As showmaternal can be much faster than the rate with G × E. The main reason for this difference is that the maternal environment is heritable and can co-evolve with an offspring gene adapting to it. In contrast, in G × E models, the underlying frequency of selective environments remains fixed throughout adaptive evolution. There is a positively synergistic interaction between maternal and zygotic genes that causes the offspring genic effect on fitness, spB, to change at each generation as the B allele in the maternal background increases in frequency. Differently put, whatever the statistical merit of transforming away scale type G × E when the environment is fixed, it is much more difficult to ignore or minimize the importance of such interactions when the scale effect itself is evolving and changing at every generation.The primary finding is that the rate of adaptive evolution with G × GContrasting views of the genetic architecture underlying fitness-related traits have polarized evolutionists since Darwin’s time” (about 90 per cent of all amino-acid substitutions have a neutral or beneficial impact only in the genetic backgrounds in which they occur, and must therefore be deleterious in a different background of other species.” It is one of the interesting features of additive × additive epistasis for fitness that the sign of an allele’s effect on fitness changes as the genetic background at another locus changes (see G component of fitness variation behaves very differently in the G × Gmaternal model than it does in the G × E model. Using the Price Equation, I showed how, in both models, the genetic variance affects the rate of increase in W, mean fitness, as expected from Fisher’s Fundamental Theorem. However, there is a second component to the change in W that appears in the G × G model that is absent in the G × E model. Here, it proves to be as important as the familiar first component, the additive genetic variance in fitness, and, in other cases, it may be more or less important, depending upon the nature of the fitness inter.There are many other types of fitness relationships between maternal and direct genetic effects that could be modeled. Often, in natural systems, a negative genetic correlation is observed between maternal and direct effects, possibly owing to natural selection favoring an intermediate phenotype although there are other causes of negative correlations (cf. review by ’s time” . However’s time”  “…about nges see . By part


N = 373 K as training data). LDpred-funct attained a +4.6% relative improvement in average prediction accuracy  compared to SBayesR . For height, meta-analyzing training data from UK Biobank and 23andMe cohorts (N = 1107 K) increased prediction R2 to 0.431. Our results show that incorporating functional priors improves polygenic prediction accuracy, consistent with the functional architecture of complex traits.Polygenic risk prediction is a widely investigated topic because of its promising clinical applications. Genetic variants in functional regions of the genome are enriched for complex trait heritability. Here, we introduce a method for polygenic prediction, LDpred-funct, that leverages trait-specific functional priors to increase prediction accuracy. We fit priors using the recently developed baseline-LD model, including coding, conserved, regulatory, and LD-related annotations. We analytically estimate posterior mean causal effect sizes and then use cross-validation to regularize these estimates, improving prediction accuracy for sparse architectures. We applied LDpred-funct to predict 21 highly heritable traits in the UK Biobank (avg  Incorporating functional information has shown promise for improving polygenic risk prediction of complex traits. Here, the authors describe polygenic prediction method LDpred-funct, and demonstrate its utility across 21 heritable traits in the UK Biobank. In this study, we aim to leverage functional priors to improve polygenic prediction8. Several studies have shown that incorporating prior distributions on causal effect sizes can improve prediction accuracy16, compared to standard Best Linear Unbiased Prediction (BLUP) or Pruning + Thresholding methods22. Recent efforts to incorporate functional information have produced promising results24 , but maybe limited by dichotomizing between functional and non-functional variants23 or restricting their analyses to genotyped variants24.Genetic variants in functional regions of the genome are enriched for complex trait heritability25, which includes coding, conserved, regulatory, and LD-related annotations. LDpred-funct first analytically estimates posterior mean causal effect sizes, accounting for functional priors and LD between variants. LDpred-funct then uses cross-validation within validation samples to regularize causal effect size estimates in bins of different magnitude, improving prediction accuracy for sparse architectures. We show that LDpred-funct attains higher polygenic prediction accuracy than other methods in simulations with real genotypes, analyses of 21 highly heritable UK Biobank traits, and meta-analyses of height using training data from UK Biobank and 23andMe cohorts.Here, we introduce a method, LDpred-funct, for leveraging trait-specific functional priors to increase polygenic prediction accuracy. We fit functional priors using our recently developed baseline-LD model25, fit using stratified LD score regression (S-LDSC)25 applied to height summary statistics from British-ancestry samples from the UK Biobank interim release. We randomly selected 10,000, 20,000, or 50,000 unrelated British-ancestry samples as training samples, and we used 7585 unrelated samples of non-British European ancestry as validation samples. By restricting simulations to chromosome 1 (≈1/10 of SNPs), we can extrapolate results to larger sample sizes , analogous to the previous work16.We performed simulations using real genotypes from the UK Biobank interim release and simulated phenotypes (see Methods). We simulated quantitative phenotypes with SNP-heritability R2) for seven main methods: P + T19, LDpred16, SBayesR9, P + T-funct-LASSO23, AnnoPred24, LDpred-funct-inf and LDpred-funct (see Methods). Results are reported in Fig. 16 (see Supplementary Tables N = 10 K training samples (+19% relative improvement for sparse architecture and +8.6% relative improvement for polygenic architecture) but attained prediction R2 close to 0 at larger sample sizes (N = 20 K and N = 50 K), perhaps because the algorithm failed to converge . These results are consistent with the fact that LDpred is known to be sensitive to model assumptions at large sample sizes16. Accounting for sparsity using LDpred-funct further improved prediction accuracy, particularly for the sparse architecture . LDpred-funct attained substantially higher prediction accuracy than P + T-funct-LASSO in most settings . LDpred-funct also attained higher prediction accuracy than AnnoPred at large sample sizes  . The difference in prediction accuracy between LDpred and each other method, as well as the difference in prediction accuracy between LDpred-funct and each other method, was statistically significant in most cases (see Supplementary Table P < 10−125 for sparse architecture and P < 10−75 for polygenic architecture at N = 50 K training samples). Simulations with 1000 or 10,000 causal variants generally recapitulated these findings, although SBayesR, P + T-funct-LASSO and AnnoPred performed better than LDpred-funct for the very sparse architecture at N = 10 K  produced improvements that varied with sample size , the average running time was 71 min for LDpred-funct vs. 5249 min for AnnoPred, not including the time to compute LD matrices.The average running time for all 7 methods is reported in Supplementary Table K = 40 posterior mean causal effect size bins to regularize effect sizes in our main simulations. We determined that results were not sensitive to this parameter . LDpred-funct-cheat performed only slightly better than LDpred-funct, indicating that LDpred-funct is not sensitive to the imperfect estimation of functional enrichment parameters .We performed four secondary analyses. First, we assessed the calibration of each method by checking whether regression of true vs. predicted phenotype yielded a slope of 1. We determined that LDpred-funct was well-calibrated (regression slope 0.98–0.99), LDpred and AnnoPred were fairly well-calibrated (regression slope 0.85–1.00), and other methods were not well-calibrated (Supplementary Table N = 373 K) and validation samples of non-British European ancestry (avg N = 22 K). We included 6,334,603 imputed SNPs in our analyses (see Methods). We computed summary statistics and 26 , consistent with simulations under a polygenic architecture  performed much worse (see Supplementary Table R2), with no convergence issues in the full UK Biobank analysis ; we note that expanding the set of SNPs analyzed worsened the performance of SBayesR (see below).Among methods that do not use functional information, LDpred outperformed P + T (+18% relative improvement in average prediction R2), but greatly outperformed LDpred-inf , consistent with simulations. The relative improvement in avg prediction R2 for LDpred-funct vs. LDpred was +9.7% for quantitative traits (higher prediction R2 for 14/14 traits), and +11% for binary traits (higher prediction R2 for 5/7 traits). We observed a positive but non-significant correlation across traits between R2), consistent with simulations under a polygenic architecture. AnnoPred performed slightly but non-significantly worse than LDpred-funct  performed only slightly better than LDpred  in each cross-validation fold. It is possible that incorporating data from an additional 20 K samples could confer an unfair advantage for LDpred-funct compared to other methods. To assess this, we performed three additional experiments. First, we repeated the LDpred-funct analyses using smaller validation sample sizes (as low as 1 K), again using 10-fold cross-validation. We determined that results were little changed , instead of the trait-specific functional enrichments used in our primary analyses, slightly reduced the prediction accuracy of LDpred-funct-inf (Supplementary Table R2 vs. LDpred-funct (random), which performed similarly to LDpred (3.1% decrease in average prediction R2 vs. LDpred)  , LDpred and AnnoPred were fairly well-calibrated , and other methods were not well-calibrated . Training sample sizes were equal to 408,092 for UK Biobank and 698,430 for 23andMe, for a total of 1,106,522 training samples. For comparison purposes, we also computed predictions using the UK Biobank and 23andMe training data sets individually, as well as a training data set consisting of 113,660 British-ancestry samples from the UK Biobank interim release. , and were lower for the 23andMe training data set (R2 = 0.344 for LDpred-funct), consistent with the ≈30% higher heritability in UK Biobank as compared to 23andMe and other large cohorts28; the higher heritability in UK Biobank could potentially be explained by lower environmental heterogeneity. We note that in the meta-analysis, we optimized the meta-analysis weights using validation data (similar to ref. 29), instead of performing a fixed-effect meta-analysis. This approach accounts for differences in heritability as well as sample size, and attained a + 3.3% relative improvement in prediction R2 compared to fixed-effects meta-analysis (see Supplementary Table K training samples (−10% to +0.2% change in average prediction R2) but attained prediction R2 close to 0 in the height analysis with 113 K training samples, perhaps because the algorithm failed to converge . LDpred-funct attained +10% (P < 2 × 10−4) and +4.6% (P = 0.04) relative improvements compared to LDpred16 and SBayesR9, two state-of-the-art methods that do not model functional enrichment. Thus incorporating functional annotations improves polygenic prediction accuracy. We note that our main analyses used baseline-LD model v1.1, but using the updated baseline-LD model v2.1 yields slightly higher prediction R2 for LDpred-funct-inf and LDpred-funct  that corrects marginal effect sizes for winner’s curse using LASSO and incorporates functional data to define high-prior and low-prior SNP sets. LDpred-funct attained a +19% average relative improvement vs. P + T-funct-LASSO across 21 UK Biobank traits. Second, ref. 24 introduced AnnoPred, which uses a Bayesian framework to incorporate functional annotations. AnnoPred models sparsity differently than LDpred-funct, as it uses a point-normal prior to estimating posterior mean effect sizes via Markov Chain Monte Carlo (MCMC), whereas LDpred-funct performs a regularization step to account for sparsity. We note that ref. 24 considered only genotyped variants and binary annotations. As noted above, functional enrichment information is far less useful when restricting to genotyped variants  across 21 UK Biobank traits, consistent with slightly worse results for AnnoPred in simulations at large sample sizes. We emphasize that our work combines binary and continuous-valued functional annotations to improve polygenic risk prediction using imputed variants.Two previous studies have highlighted the potential advantages of leveraging functional enrichment to improve prediction accuracy30), but methods that use raw genotypes/phenotypes as training data have the potential to attain higher accuracy26; incorporating functional enrichment information into prediction methods that use raw genotypes/phenotypes as training data remains a direction for future research. Second, the regularization step employed by LDpred-funct to account for sparsity relies on heuristic cross-validation instead of inferring posterior mean causal effect sizes under a prior sparse functional model; we made this choice because the appropriate choice of sparse functional model is unclear, and because inference of posterior means via MCMC may be subject to convergence issues. As a consequence, the improvement of LDpred-funct over LDpred-funct-inf may be contingent on the number of validation samples available for cross-validation; in particular, for very small validation samples, the number of cross-validation bins is equal to 1 , as our simulations focused exclusively on prediction in Europeans. However, we evaluated the performance of LDpred-funct in predicting 21 UK Biobank traits in diverse populations using European training data (as in recent studies35). The results were promising, particularly in Africans (+23% vs. LDpred (P <  10−5), +18% vs. SBayesR (P = 0.001); see Supplementary Table 39. A more thorough investigation, e.g. incorporating non-European training data29, is an important direction for future research. Sixth, we have not performed a comprehensive assessment of how much different functional annotation models contribute to improvements in prediction accuracy, which remains an important future direction, particularly as functional annotation models will improve as increasingly rich functional data is generated. Specifically, the improvements in prediction accuracy that we reported are a function of the baseline-LD model25, but there are many possible ways to improve this model, e.g. by incorporating tissue-specific enrichments43, modeling MAF-dependent architectures46, and/or employing alternative approaches to modeling LD-dependent effects47; we anticipate that future improvements to the baseline-LD model will yield even larger improvements in prediction accuracy. As an initial step to explore alternative approaches to modeling LD-dependent effects, we repeated our analyses using the baseline-LD + LDAK model (introduced in ref. 48), which consists of the baseline-LD model plus one additional continuous annotation constructed using LDAK weights47.  We determined that results were virtually unchanged , LDpred16, SBayesR9, P + T with functionally informed LASSO shrinkage23 (P + T-funct-LASSO), AnnoPred24, our LDpred-funct-inf method, and our LDpred-funct method; we also included LDpred-inf16, which is known to attain lower prediction accuracy than LDpred16, in some of our secondary analyses. P + T, LDpred-inf, LDpred, and SBayesR are polygenic prediction methods that do not use functional annotations; we did not include RSS12 and SBLUP11 methods in our comparisons, because ref. 9 reported that SBayesR performed as well or better than both RSS and SBLUP and was more computationally efficient . P + T-funct-LASSO is a modification of P + T that corrects marginal effect sizes for winner’s curse, accounting for functional annotations. AnnoPred is which uses a Bayesian framework to incorporate functional annotations. LDpred-funct-inf is an improvement of LDpred-inf that incorporates functionally informed priors on causal effect sizes. LDpred-funct is an improvement of LDpred-funct-inf that uses cross-validation to regularize posterior mean causal effect size estimates, improving prediction accuracy for sparse architectures. Each method is described in greater detail below. In both simulations and analyses of real traits, we used squared correlation (R2) between predicted phenotype and true phenotype in a held-out set of samples as our primary measure of prediction accuracy.We compared 7 main prediction methods: Pruning + Thresholding19  followed by P-value thresholding18. Specifically, the method has two parameters, PT, and proceeds as follows. First, the method prunes SNPs based on a pairwise threshold PT. Letting M be the number of SNPs remaining after LD-clumping, polygenic risk scores (PRS) are computed asgi is a vector of normalized genotypes for SNP i. The parameters PT are commonly tuned using validation data to optimize prediction accuracy19. While in theory, this procedure is susceptible to overfitting, in practice, validation sample sizes are typically large, and PT are selected from a small discrete set of parameter choices, so that overfitting is considered to have a negligible effect29. Accordingly, in this work, we consider PT ∈ {1, 0.3, 0.1, 0.03, 0.01, 0.003, 0.001, 3 ∗ 10−4, 10−4, 3 ∗ 10−5, 10−5, 10−6, 10−7, 10−8}, and we always report results corresponding to the best choices of these parameters. The P + T method is implemented in the PLINK software (see Code availability).The P + T method builds a polygenic risk score (PRS) using a subset of independent SNPs obtained via informed LD-pruning16. The infinitesimal model assumes that normalized causal effect sizes have prior distribution βi ~ N, where M is the number of SNPs. The posterior mean causal effect sizes areD is the LD matrix between markers, I is the identity matrix, N is the training sample size, k SNPs in the region of LD; following ref. 16 we use the approximation M > > k. D is typically estimated using validation data, restricting to non-overlapping LD windows. We used the default LD window size, which is M/3000. 28 , or can be estimated from summary statistics using the aggregate estimator as described in ref. 16. To approximate the normalized marginal effect size ref. 16 uses the p-values to obtain absolute Z scores and then multiplies absolute Z scores by the sign of the estimated effect size. When sample sizes are very large, p-values may be rounded to zero, in which case we approximate normalized marginal effect sizes pi is the minor allele frequency of SNP i, and 50 the system of linear equations The LDpred-inf method estimates posterior mean causal effect sizes under an infinitesimal model, accounting for LDi ~ N0, σ, where \16. It assumes a Gaussian mixture prior: p, and βi ~ 0 with probability 1 − p, where p is the proportion of causal SNPs. The method is optimized by considering different values of p ; in the special case where 100% of SNPs are assumed to be causal, LDpred is roughly equivalent to LDpred-inf. We excluded SNPs from long-range LD regions (reported in ref. 51), as our secondary analyses showed that including these regions were suboptimal, consistent with ref. 9.The LDpred method is an extension of LDpred-inf that uses a point-normal prior to estimating posterior mean effect sizes via Markov Chain Monte Carlo (MCMC)9. It assumes a finite mixture of normal distributions to account for sparsity, defined as: πc, where c ranges from 1 to C, the total number of components in the mixture model. We used as input the recommended parameters from ref. 9, with C = 4 mixtures with parameters γc = . The method requires a shrunk LD matrix12. The authors of ref. 9 made available shrunk LD matrices estimated from 50,000 randomly selected white British individuals from the UK Biobank51 for two different SNPs sets. The 1.1M SNP set consists of 1,094,841 variants, constructed by restricting 1,365,446 SNPs from HapMap352 to MAF > 0.01 and removing strand ambiguous SNPs and long-range LD regions (as reported in ref. 51). The 2.9M SNP set consists of 2,865,810 variants, constructed by applying LD-pruning (R2 > 0.99) to a larger set of 8 million variants from the UK Biobank51 with MAF > 0.01, overlapped with a previous large GWAS53 and present in 1000 Genomes54. We note that we could not scale the SBayesR analysis to the full set of 6,334,603 variants used in other analyses due to computational constraints. We used the 1.1M SNP set in our primary analyses as it achieved the highest average prediction R2 in our real traits analyses (see Results section), but we also considered the 2.9 M SNP set in secondary analyses. For analyses that use BOLT-LMM summary statistics we used Neffective as reported in ref. 26.The SBayesR method infers posterior mean causal effect sizes from GWAS summary statistics and an LD matrix23 proposed an extension of P + T that corrects the marginal effect sizes of SNPs for winner’s curse and incorporates external functional annotation data (P + T-funct-LASSO). The winner’s curse correction is performed by applying a LASSO shrinkage to the marginal association statistics of the PRS:−1 is the inverse standard normal CDF. Functional annotations are incorporated via two disjoint SNPs sets, representing "high-prior” SNPs (HP) and "low-prior” SNPs (LP), respectively. We define the HP SNP set for P + T-funct-LASSO as the set of SNPs in the top 10% of expected per-SNP heritability under the baseline-LD model25, which includes coding, conserved, regulatory, and LD-related annotations, whose enrichments are jointly estimated using stratified LD score regression25 (see Baseline-LD model annotations section). We also performed secondary analyses using the top 5% (P + T-funct-LASSO-top5%). We define PRSLASSO,HP(PHP) to be the PRS restricted to the HP SNP set, and PRSLASSO, LP(PLP) to be the PRS restricted to the LP SNP set, where PHP and PLP are the optimal significance thresholds for the HP and LP SNP sets, respectively. We define PRSLASSO = PRSLASSO,HP(PHP) + PRSLASSO,LP(PLP). We also performed secondary analyses where we allow an additional regularization to the two PRS: PRSLASSO = α1PRSLASSO,HP(PHP) + α2PRSLASSO, LP(PLP); we refer to this method as P + T-funct-LASSO-weighted.Reference24 uses a Bayesian framework to incorporate functional priors while accounting for LD, optimizing prediction R2 over different assumed values of the proportion of causal SNPs. Reference24 proposed two different priors for use with AnnoPred. The first prior assumes the same proportion of causal SNPs but different causal effect size variance across functional annotations, and uses a point-normal prior to estimating posterior mean effect sizes via Markov Chain Monte Carlo (MCMC). In the special case where 100% of SNPs are assumed to be causal, AnnoPred is roughly equivalent to LDpred-funct-inf (see below). The second prior assumes different proportions of causal SNPs but the same causal effect size variance across functional annotations. We only consider the first prior, since the second prior cannot be extended to incorporate continuous-valued annotations from the baseline-LD model. We excluded SNPs from long-range LD regions (as reported in ref. 51) when running AnnoPred. We used the default LD window size, which is M/3000.AnnoPred25, which includes coding, conserved, regulatory, and LD-related annotations, whose enrichments are jointly estimated using stratified LD score regression25. Specifically, we assume that normalized causal effect sizes have prior distribution c is a normalizing constant such that M+ is the number of SNPs with D is estimated using validation data, restricting to windows of size 0.15%M+. In principle, it is possible to use banding to define the LD matrices, where LD between distant pairs of SNPs (10 Mb or more) is rounded to zero55, but we elected to use the simpler window-based approach (as in ref. 16).We modify LDpred-inf to incorporate functionally informed priors on causal effect sizes using the baseline-LD modelK bins  where each bin has roughly the same sum of squared posterior mean effect sizes, and determine the relative weights of each bin based on the predictive value in the validation data. Intuitively if a bin is dominated by non-causal SNPs, the inferred relative weight will be lower than for a bin with a high proportion of causal SNPs. This non-parametric shrinkage approach can optimize prediction accuracy regardless of the genetic architecture. In detail, let b1 as the smallest set of top SNPs with bk as the smallest set of additional top SNPs with αk are optimized using validation data via 10-fold cross-validation. For each held-out fold in turn, we split the data so we estimate the weights αk using the samples from the other nine folds  and compute PRS on the held-out fold using these weights ; thus, in each cross-validation fold, the validation samples used to estimate regularization weights are disjoint from the validation samples used to compute predictions. We then compute the average prediction R2 across the 10 held-out folds. To avoid overfitting when K is very close to N, we set the number of bins (K) to be between 1 and 100, such that it is proportional to K weights in each fold is at least 100 times larger than K:N is the number of validation samples. For highly heritable traits , approximately concordant with Eq.  posterior mean causal effect sizes, partition the SNPs into R2 of each method and the difference in prediction R2 between two methods were computed via block-jackknife using 200 genomic jackknife blocks5; this is more conservative than computing standard errors based on the number of validation samples, which does not account for variation across a finite number of SNPs. For each method, we first optimized any relevant tuning parameters using the entire genome and then analyzed each jackknife block using those tuning parameters.Standard errors. Standard errors for the prediction 57), while preserving a large number of training samples. We restricted our simulations to 459,284 imputed SNPs on chromosome 1 (see below), fixed the number of causal SNPs at 2,000 or 5,000 , and fixed the SNP-heritability βi for causal SNPs from a normal distribution with variance equal to p is the proportion of causal SNPs and 25, fit using stratified LD score regression (S-LDSC)25 applied to height summary statistics computed from unrelated British-ancestry samples from the UK Biobank interim release . We computed per-allele effect sizes bi as pi is the minor allele frequency for SNP i estimated using the validation genotypes. We simulated phenotypes as 16 according to the ratio N/M, where N is the training sample size. We compared each of the seven methods described above. For LDpred-funct-inf and LDpred-funct, for each simulated trait we used S-LDSC (applied to training data only) to estimate baseline-LD model parameters. For LDpred-funct, we report R2 as the average prediction R2 across the 10 held-out folds.We simulated quantitative phenotypes using real genotypes from the UK Biobank interim release (see below). We used up to 50,000 unrelated British-ancestry samples as training samples, and 7,585 samples of other European ancestries as validation samples (see below). We made these choices to minimize confounding due to shared population stratification or cryptic relatedness between training and validation samples . We selected 21 UK Biobank traits (14 quantitative traits and 7 binary traits) with phenotyping rate >80% , SNP-heritability 26). We restricted training samples to 409,728 British-ancestry samples51, including related individuals . We restricted validation samples to 24,436 samples of non-British European ancestry, after removing validation samples that were related ( >0.05) to training samples and/or other validation samples , while preserving a large number of training samples. We analyzed 6,334,603 genome-wide imputed SNPs, after removing SNPs with minor allele frequency <1%, removing SNPs with imputation accuracy <0.9, and removing A/T and C/G SNPs to eliminate potential strand ambiguity. We used 26 as input to LDpred, AnnoPred, LDpred-funct-inf, and LDpred-funct.The full UK Biobank data set includes 459,327 European-ancestry samples and ~ 20 million imputed SNPsmentclasspt{minimamentclasspt{minima58. We used the UK Biobank interim release both in simulations using real genotypes, and in a subset of analyses of height phenotypes (to investigate how prediction accuracy varies with training sample size).The UK Biobank interim release includes 145,416 European-ancestry samples59, estimated using FastPCA60 (see Code availability). For consistency, we used the same set of 24,351 validation samples of non-British European ancestry with height phenotypes as defined above. We analyzed 5,957,957 genome-wide SNPs, after removing SNPs with minor allele frequency <1%, removing SNPs with imputation accuracy <0.9, removing SNPs that were not present in the 23andMe height data set (see below), and removing A/T and C/G SNPs to eliminate potential strand ambiguity.In our analyses of height phenotypes, we restricted training samples to 113,660 unrelated (≤0.05) British-ancestry samples for which height phenotypes were available. We computed association statistics by adjusting for 10 PCsIn our simulations, we restricted training samples to up to 50,000 of the 113,660 unrelated British-ancestry samples, and restricted validation samples to 8441 samples of non-British European ancestry, after removing validation samples that were related ( >0.05) to training samples and/or other validation samples. We restricted the 5,957,957 genome-wide SNPs (see above) to chromosome 1, yielding 459,284 SNPs after QC.61. Summary association statistics were computed using linear regression adjusting for age, gender, genotyping platform, and the top five principal components to account for residual population structure. The summary association statistics will be made available to qualified researchers (see Data availability).The 23andMe data set consists of summary statistics computed from 698,430 European-ancestry samples (23andMe customers who consented to participate in research) at 9,898,287 imputed SNPs, after removing SNPs with minor allele frequency <1% and that passed QC filters . Analyses were restricted to the set of individuals with >97% European ancestry, as determined via an analysis of local ancestryWe analyzed 5,808,258 genome-wide SNPs, after removing SNPs with minor allele frequency <1%, removing SNPs with imputation accuracy <0.9, removing SNPs that were not present in the full UK Biobank data set (see above), and removing A/T and C/G SNPs to eliminate potential strand ambiguity.i is the PRS obtained using training data from cohort i. The PRS can be obtained using P + T, P + T-funct-LASSO, LDpred-inf, or LDpred-funct. The meta-analysis weights γi can either be specified via fixed-effect meta-analysis . In our primary analyses, we fit the weights γi in-sample and report prediction accuracy using adjusted R2 to account for in-sample fitting29. We also report results using 10-fold cross-validation: for each held-out fold in turn, we estimate the weights γi using the other nine folds and compute PRS on the held-out fold using these weights. We then compute the average prediction R2 across the 10 held-out folds.We meta-analyzed height summary statistics from the full UK Biobank and 23andMe data sets. We defineγi. Then, we use γi to compute  weighted posterior mean causal effect sizes (PMCES) via PMCES = γ1PMCES1 + γ2PMCES2, which are binned into k bins. Then, we estimate bin-specific weights αk  weighted posterior mean causal effect sizes When using LDpred-funct as the prediction method, we perform the meta-analysis as follows. First, we use LDpred-funct-inf to fit meta-analysis weights 25. For each trait, we used the τc values estimated for that trait to compute i under the baseline-LD model, asac(i) is the value of annotation c at SNP i.The baseline-LD model (v1.1) contains a broad set of 75 functional annotations , whose enrichments are jointly estimated using stratified LD score regressionτc for each annotation c are estimated vial is the LD score of SNP i with respect to annotation ac and i. We note that τc quantifies effects that are unique to annotation c. In all analyses of real phenotypes, τc and Joint effect sizes 54 as the reference data set to compute LD scores, as in ref. 25.In our primary analyses, we used 489 unrelated European samples from phase 3 of the 1000 Genomes Project62 as the reference data set (as in ref. 48), ensuring a closer ancestry match with British-ancestry UK Biobank samples. We also repeated our LDpred-funct analyses using S-LDSC with the baseline-LD + LDAK model (instead of the baseline-LD model), with UK10K as the reference data set. The baseline-LD + LDAK model (introduced in ref. 48) consists of the baseline-LD model plus one additional continuous annotation constructed using LDAK weights47, which has values α = −0.25, pj is the allele frequency of SNP j, and wj is the LDAK weight of SNP j computed using UK10K data.To verify that our 1000 Genomes reference data set produces reliable LD estimates, we repeated our LDpred-funct analyses using S-LDSC with 3,567 unrelated individuals from UK10KFurther information on research design is available in the Supplementary InformationPeer Review FileReporting Summary


PRICKLE1 and FOCAD affecting cranial base width. We replicated this G × G interaction signal at the locus level in an additional 5,128 Korean individuals. We used the hypomorphic Prickle1Beetlejuice (Prickle1Bj) mouse line to directly test the function of Prickle1 on the cranial base and observed wider cranial bases in Bj/Bj.Prickle1 Importantly, we observed that the Prickle1 and Focadhesin proteins co-localize in murine cranial base chondrocytes, and this co-localization is abnormal in the Prickle1Bj/Bj mutants. Taken together, our findings uncovered a novel G × G interaction effect in humans with strong support from both epidemiological and molecular studies. These results highlight the potential of studying measures of phenotypic variability in gene mapping studies of facial morphology.The human face is a highly complex and variable structure resulting from the intricate coordination of numerous genetic and non-genetic factors. Hundreds of genomic loci impacting quantitative facial features have been identified. While these associations have been shown to influence morphology by altering the mean size and shape of facial measures, their effect on trait variance remains unclear. We conducted a genome-wide association analysis for the variance of 20 quantitative facial measurements in 2,447 European individuals and identified several suggestive variance quantitative trait loci (vQTLs). These vQTLs guided us to conduct an efficient search for gene-by-gene (G × G) interactions, which uncovered an interaction between  Human genetic studies have been remarkably successful at detecting variants with an impact on mean trait values. On the other hand, the genetic basis of phenotypic variability remains largely uncharacterized. Variance quantitative trait loci (vQTLs) are genetic variants exhibiting inter-individual intra-genotypic variability, where one of the alleles is associated with a larger phenotypic variance compared to the other . vQTLs aThe human face is a highly complex structure resulting from the intricate coordination of multiple genetic and epigenetic factors. Beyond the individual effect of common single nucleotide polymorphisms (SNPs) identified through GWAS , gene-byOne sensible strategy is variance prioritization. It has been shown, both theoretically and empirically, that a heterogeneous phenotypic variance across genotype groups can arise from genetic interaction effects . AlthougPRICKLE1, a gene known to contribute to the control of craniofacial development in mouse mutants , and as we expect some false associations, to prioritize SNPs for the G × G interaction stage, we further considered the biological relevance of the loci where the vQTLs are located rather than relying on their p-value alone.Detecting differences in group variances demands larger sample sizes than when detecting differences in group means. Given that our cohort is smaller than the typical sample size of previous vQTL studies, we increased the minimum minor allele frequency (MAF) cutoff for the sake of maintaining statistical power. SNPs with MAF <0.2 were excluded, giving a total of 3,104,639 qualified autosomal SNPs (genotyped and imputed). Because SNPs with MAF >0.2 represent much less of the genome-wide variation than typically interrogated in a GWAS, to reduce burden of multiple comparisons, we used a relaxed threshold of 5 × 101 to identify SNPs associated with the variance of 20 quantitative facial measurements. The Levene’s test is equivalent to a test of differences in mean deviation across subgroups, with the deviation calculated as the absolute difference between an individual’s phenotypic value and their group-specific mean/median. We used the median as the central measure because it provides more robustness to non-normality and outliers than does the mean. To control for covariates, age, age2, sex, height, weight, facial size, and four genetic ancestry PCs were first regressed on facial traits in a linear regression model, and the residuals were then used as the phenotype in statistical tests.We used the Levene’s test with median implemented in the OSCA software packagep-values, and when there was evidence of inflation , we used the robust standard error to obtain unbiased standard errors of the coefficients under variance heteroscedasticity (R package sandwich). This technique was able to correct the inflation and generate well-behaved Q–Q plots. In the discovery analysis, all G × G interaction analyses were performed between a single SNP and all other SNPs across the genome, and therefore a p-value below the conventional threshold of 5 × 10–8 was considered significant. Co-localization plots for the vQTL analysis and the G × G interaction summary statistics were generated by R package LocusCompareR (The G × G interaction test was done in PLINK  with theCompareR .df) test for the interaction effect only and the 2 df test for the main and the interaction effects jointly were used. Only a limited set of environmental factors were collected in our cohort, and we chose to examine the interactions of SNPs with age and sex.Although the current study had a primary focus on G × G interactions as the underlying genetic process for variance heterogeneity, we also considered possible gene–environment (G × E) interactions by using a linear regression model in PLINK. Both the 1 degree-of-freedom (Meff), which was computed according to the eigenvalue-based procedure by p-value below 0.05 divided by the product of the Meff of the two regions under test was considered significant.The replication cohort consisted of 5,128 unrelated individuals of Asian ancestry collected for the Korean Genome and Epidemiology Study (KoGES) in South Korea from 2009 to 2012 (Ansan–Ansung cohort), for which standardized two-dimensional (2D) frontal photographs were available. The cohort, phenotyping approach, and genotype data have been described in detail previously . The maxPrickle1Bj mouse line is maintained as heterozygotes outbred to wild-type J6C57/Bl individuals. All animal work is approved by the University of Pittsburgh Institutional Animal Care and Use Committee. To collect embryos, we performed timed mating, where the presence of the plug was designated as embryonic day 0.5 (E0.5). On the appropriate day of pregnancy, we euthanized the pregnant dam and collected the embryos via caesarian section. We confirmed the embryonic development stage based on morphology. Genotyping was performed using custom SNP Taqman assays .The Embryos were fixed overnight in 4% paraformaldehyde, followed by dehydration and storage in 70% ethanol until further processing. For the tissue sections, the embryos were embedded in paraffin wax, sectioned at 8-μm thickness, and placed on (3-aminopropyl) triethoxysiloxane (TESPA)-treated slides that were stored at 4°C until use.Prickle1+/+, Bj/10 Prickle1+, and Bj/Bj13 Prickle1 heads at E14.5 were captured on a Leica M165FC dissecting microscope with a DFC 450 camera at 1.6× magnification . Care was taken to avoid excess canting of the heads during imaging. Six landmarks were placed around the cranium using 3D Slicer  and the anti-Prickle1  antibodies and performed citrate buffer antigen retrieval. After antigen retrieval, the slides were briefly blocked and then incubated overnight in the fridge with the primary antibodies (diluted 1:100). The next day, the slides were washed followed by incubation with secondary antibodies . The sections were mounted with Prolong Gold with DAPI (Invitrogen). Fluorescent images were captured on a Nikon TE2000 inverted fluorescent microscope  and are represented as a single slice of an image stack. Institutional ethics (IRB) approval was obtained at each recruitment site . All adult subjects gave written informed consent prior to participation; for children, written consent was obtained from a parent or legal guardian. All procedures performed in this study were conducted in accordance with the guidelines of the Declaration of Helsinki. All experimental protocols using mice were approved by the University of Pittsburgh Institutional Animal Care and Use Committee (#17050839) and carried out in accordance with institutional animal care protocols.p-values below 5 × 10–7 were observed , in animal models and in humans   . This mut allele . Pricklenomalies .Prickle1Bj allele affected the cranial base width in a dosage-dependent fashion using data from 30 mice. At E14.5, Prickle1Bj/Bj mice exhibited significantly wider cranial bases (p = 0.01) compared to Prickle1+/+ and Prickle1Bj/+ littermates (p = 0.72), which could be due to a lack of power given the limited number of mice.We explored to what extent the termates . The varp < 5 × 10–7 after randomly shuffling the cranial base width among our samples  to ensure well-behaved statistics for the genome-wide search of SNPs interacting with the lead PRICKLE1 SNP rs1796391   for cranial base width residuals in the unstratified and the stratified data. The minor allele of rs10511683 (FOCAD) was associated with a wider cranial base in people with the AA genotype at rs1796391 (PRICKLE1), whereas it showed an opposite effect of similar magnitude in the PRICKLE1 GG group. FOCAD SNP rs10511683 had small p-values for its association with the phenotypic mean in the stratified analyses of the AA and GG groups of rs1796391 (PRICKLE1), but not in the combined sample, where the opposite effects offset each other (p = 0.13)  . Figure 10511683 . The rigalso see .FOCAD) overlaps with weak signals of histone modifications, signifying enhancer activity in chondrocyte cells and osteoblast primary cells. SNP rs1796391 (PRICKLE1) also showed a potential enhancer signature in osteoblasts. These cell types are important players during craniofacial morphogenesis, and studying cell type-specific roles of PRICKLE1 and FOCAD in future work could potentially provide more insights.We queried the human cell line experimental data from the Roadmap Epigenomics Project and the ENCODE Project for potential regulatory roles of the identified non-coding SNPs . The genFOCAD locus (PRICKLE1 SNPs in each of the three genotype groups defined by rs10511683 (FOCAD) separately. If the variance heterogeneity at rs1796391 (PRICKLE1) was induced by its interaction with rs10511683 (FOCAD), we would expect to see weakened signals in each group. In accordance with this expectation, the group-specific vQTL signals were either absent or much attenuated compared to the unstratified analysis , but demonstrating low LD (r2 = 0.11) with the original signal at rs1796391 (PRICKLE1), had a significant interaction p-value (1.88 × 10–10) smaller than that of the original one. SNP rs10880322 had neither a strong mean effect (GWAS p = 0.025) nor a variance effect (Levene’s test p = 0.428) on cranial base width when tested alone; however, when taking into account its interaction with rs10511683 (FOCAD), both the main and interaction effects of rs10880322 became highly significant  for the original rs1796391 (PRICKLE1) × rs10511683 (FOCAD) interaction than that reported in the section G × G interaction of PRICKLE1 for cranial base width (5.82 × 10–7) above (where the robust standard error technique was used).We then turned our focus to s1796391 . The leanificant . Note thnificant , we needPRICKLE1). The outlier is visible in PRICKLE1) yielded a p = 9.03 × 10–7 in the Levene’s test, and its interaction with rs10511683 (FOCAD) had a p = 1.07 × 10–6 (using robust standard error). Interaction is only one of the possible explanations for variance heterogeneity; other genetic processes can also give rise to genotype-specific variances. One group of such processes reflects genuine genetic effects, such as the presence of linked causal variants with a mean effect, nearby rare variants with a mean effect, and G × E interactions . AnotherPRICKLE1) did not show a mean effect on cranial base width (p = 0.18), nor did any common SNP (MAF > 1%) located within ±500 kb have a small p-value (PRICKLE1 coding variants using the sequence kernel association test (SKAT) , although variants that are rarer or undetected could still potentially have an effect. Thirdly, the results from the G × E tests did not support an interaction effect of rs1796391 with either sex or age, which were the only two non-genetic factors available in this cohort   and the t (SKAT) . There ws cohort . These rPRICKLE1 vQTL signal and the interaction between PRICKLE1 and FOCAD. The SNP-level replication considered the exact same SNPs detected in the discovery stage or, when unavailable, proxy SNPs based on LD. The locus-level replication expanded the genetic markers under consideration using a window size of 500 kb on either side of the genes of interest. We used a minimum MAF of 0.1 for SNP inclusion, which is lower than that of the discovery analysis as the replication cohort is roughly twice as large. For simplicity and clarity, below, we follow a SNP rs identifier with either “PRICKLE1” or “FOCAD” in parentheses to indicate its locus. Note that this notation does not imply that the SNP falls inside the gene itself, but reflects our locus-level strategy considering a surrounding region (approximately 1 Mb) encompassing the gene. A total of 906 SNPs in and nearby PRICKLE1 and 796 SNPs in and nearby FOCAD were available for replication analysis. The numbers of independent SNPs were estimated to be 138 and 162, respectively, according to the eigenvalue-based procedure by p-value thresholds were therefore set to be 0.05/138 = 3.62 × 10–4 for replicating the PRICKLE1 vQTLs and 0.05/138/162 = 2.24 × 10–6 for replicating the PRCKLE1 × FOCAD effect.The replication cohort consisted of 5,128 unrelated individuals of Asian ancestry collected in South Korea for which a suitable measure of maximum upper facial width was available. We conducted both a SNP-level and a locus-level replication of the PRICKLE1 locus SNPs yielded a significant replication p-value in the Levene’s test (r2 = 1)  (No e’s test . The dis(r2 = 1) , yet nonfor all) .FOCAD) interacting with PRICKLE1 in the discovery cohort was not available in the replication cohort. The best proxy SNPs had only moderate LD with rs10511683 (r2 = 0.63–0.77)  . Supplemp-values from the G × G interaction test was consistent with the null expectation, except for those at the tail (–6 (FOCAD locus SNPs (rs10964862 and rs10123324) and the different PRICKLE1 locus SNPs (FOCAD) showed significant interaction with 10 PRICKLE1 SNPs, and the top two interacting SNP pairs surpassed the conventional genome-wide threshold of 5 × 10–8: rs10964862 (FOCAD) × rs11181736 (PRICKLE1) with p = 4.53 × 10–9 and rs10964862 (FOCAD) × rs11181735 (PRICKLE1) with p = 9.33 × 10–9. These interaction signals were several hundred kilobase pairs away from the original SNP pair identified in the discovery samples (rs1796391 × rs10511683), and the signal at the PRICKLE1 locus did not overlap with the known coding sequence (FOCAD) is 9.4 kb toward the 5′-UTR of the IFNW1 gene and overlaps with a sequence showing enhancer signatures in mesenchymal stem cell-derived chondrocyte cells  were also a few hundred kilobase pairs distant to the target SNP pair , failed the quality control procedure in the United States cohort. Since most of the SNPs that the other G × G replication, rs10964862 (FOCAD), interacted with  with p = 6.0 × 10–4, which did not pass a Bonferroni threshold correcting for the effective number of tests [120; by the method of p-values can be found in We tried to replicate the interacting SNP pairs identified in the South Korean cohort in our United States sample. One of the G × G replication SNPs, rs10123324  in the cranial base, we performed immunofluorescence staining with antibodies to Prickle1 and Focadhesin at E12.5 and E15.5 cranial base. Focad was observed ubiquitously throughout the at E12.5 . In contial base . In the drocytes . Consiste furrow . These dPRICKLE1 and FOCAD in shaping the cranial base/upper facial width and provided strong biological support for this statistical interaction by showing the co-expression of the two genes in relevant craniofacial tissues during early development. Our findings suggest that future genetic studies of human facial morphology should expand their perspective from a sole focus on phenotypic mean to include measures of phenotypic variance and to move beyond the marginal effects of genetic variants in order to investigate their joint and interaction effects.By focusing on the variance of human facial features, this study uncovered a novel G × G interaction effect between PRICKLE1 serves as a signaling factor in the non-canonical Wnt pathway, the disruption of which is known to cause cleft palate and to stunt limb growth   . There iadhesin) . Intriguof FOCAD . The assof FOCAD . In addit palate . Our finPrickle1Bj/Bj mice had significantly wider heads and abnormal localization of the Focadhesin protein. These data, together, strongly support the relevance of Prickle1 and Focad and their potential interaction in the development of the cranial base. Proximal distal growth of the cranial base is accomplished by the expansion of the paired synchondroses. Growth plate development involves a process where chondrocytes in a quiescent resting zone mature into proliferating chondrocytes that line up in columns that are oriented in the proximal distal axis of the cranial base . It is a limitation that we followed up on a strictly speaking non-significant signal (PRICKLE1), and in the next step of the analysis, its interaction with FOCAD again did not reach the genome-wide significance threshold. However, three things together motivated us to pursue a replication and mouse experiments. Firstly, the suggestive p-values of a true positive might be a mere result of insufficient power. Secondly, the disruption of PRICKLE1 was known to associate with craniofacial phenotypes, but nearby SNPs were never significant in the GWAS, implying a role of a rare mutation or a complicated mode of action, such as interaction. Thirdly, the genome-wide search of loci interacting with FOCAD did identify PRICKLE1 as the most and only significant locus (p = 1.88 × 10–10).Although the statistical results of our human data that initially led to the experimental interrogation of ant loci , we did PRICKLE1 and FOCAD were not known to relate in any way before this study, few other pre-selection strategies would have been able to sort them out and make a discovery. Note that the significant rs10880322 (PRICKLE1) × rs10511683 (FOCAD) interaction exemplified how interaction effects need not cause variance heterogeneity at either locus. Likewise, variance heterogeneity can arise from genetic processes other than interaction, as well as statistical artifacts such as outliers. We carefully scrutinized alternative explanations for the PRICKLE1 vQTL (see section “Results”), ensuring that, although not impossible, they were not likely to explain our finding. Despite the lack of definite relationship between heterogeneous variance and interaction, a detected vQTL does signify a possible presence of unmodeled statistical interaction, which can be further examined relatively easily. Our results highlighted the potential of vQTL analysis in revealing G × G interactions and underscored the need to explore how genetic factors crosstalk in facial genetic studies.The strategy of variance prioritization showed a clear benefit in studying G × G interactions in the present study. With a sample size of 2,447 individuals, an exhaustive search for pairwise interactions between genome-wide SNPs would not have sufficient power given the huge burden of multiple comparisons, even with the scope restricted to SNPs with a MAF >0.2. As https://dbgap.ncbi.nlm.nih.gov/) at accession phs000949.v1.p1. The raw source data for the phenotypes – the 3D facial surfaces – are available for the 3D Facial Norms dataset through the FaceBase Consortium (www.facebase.org). The genotypic and phenotypic data of the Korean sample is available upon request from the Korean Genome and Epidemiology Study (KoGES) of the Center for Disease Control and from the Korea Medicine Data Center of the Korea Institute of Oriental Medicine, respectively. The mouse data is available upon request from HSR.Publicly available datasets were generated for this study. These data can be found here: the genetic data and the 20 facial distances evaluated in the United States cohort are available to the research community through the dbGaP controlled-access repository .DL, SW, and JS conceived and designed the study. EF, MM, SW, and JS provided funding for data collection. JH, GW, and LM provided the discovery human data. H-JB and SC provided the replication human data and involved in the replication with Korean subjects. DL spearheaded the statistical analysis and wrote the manuscript. H-JB and ML contributed to the statistical analysis. AE and HS-R conducted the mouse experiments. All authors reviewed and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.