The additive genomic variance in linear models with random marker effects can be defined as a random variable that is in accordance with classical quantitative genetics theory. Common approaches to estimate the genomic variance in random-effects linear models based on genomic marker data can be regarded as estimating the unconditional (or prior) expectation of this random additive genomic variance, and result in a negligence of the contribution of linkage disequilibrium (LD). We introduce a novel best prediction (BP) approach for the additive genomic variance in both the current and the base population in the framework of genomic prediction using the genomic best linear unbiased prediction (gBLUP) method. The resulting best predictor is the conditional (or posterior) expectation of the additive genomic variance when using the additional information given by the phenotypic data, and is structurally in accordance with the genomic equivalent of the classical additive genetic variance in random-effects models. In particular, the best predictor includes the contribution of (marker) LD to the additive genomic variance and possibly fully eliminates the missing contribution of LD that is caused by the assumptions of statistical frameworks such as the random-effects model. We derive an empirical best predictor (eBP) and compare its performance with common approaches to estimate the additive genomic variance in random-effects models on commonly used genomic datasets. THE additive genetic variance is defined as the variance of the breeding value (BV) and is the most important determinant of the response of a population to selection . The addThe additive genomic variance is defined as the variance of a trait that can be explained by a linear regression on a set of markers  p is much larger than the number of observations n. It is well known that a lot of traits are influenced by many genes, and that at least some loci with tiny effects are missed when using variable selection or even single-marker regression models.Consequently, Then, p markers to the phenotypic values are assumed to be independent normally distributed random variables with equal variances. Thus, they claim that the random contribution made by each marker is not correlated with the random contributions made by any other marker, which leads to a negligence of the contribution of LD to the additive genomic variance.Recently, there has been a general discussion whether estimators for the genomic variance account for LD between markers, which is defined as the covariance between the marker genotypes . Some auArabidopsis thaliana  expectation of the random additive genomic variance. Combined with the assumptions on the unconditional distribution of the marker effects in the gBLUP-method, this leads to an insufficient adaptation to the data and a negligence of the contribution of LD.We show that the additive genomic variance in linear models with random marker effects (RME) can be defined as a random variable. Based on this premise, we propose a novel predictor of the additive genomic variance and place existing estimators in a joint framework permitting comparison with the new predictor. We contribute to the solution of many of the above mentioned controversies by reviewing common approaches to estimate the additive genomic variance, i.e., we use the conditional (or posterior) expectation of the random additive genomic variance given the additional information by the phenotypic values for an improved adaptation to the data. We decompose the best predictor into the GCTA-GREML estimator and a function for the contribution of marker LD, which determines whether GCTA-GREML is biased up- or downward. The best predictor is structurally in accordance with the genomic equivalent of the additive genetic variance from classical quantitative genetics, i.e., it explicitly includes the contribution of LD. We propose an empirical best predictor (eBP), and illustrate our theoretical results on several commonly used genomic datasets.We introduce a novel best prediction approach for the additive genomic variance in both the current and the base population, n-vector y of phenotypic values and the n-vector g of genomic values is given byn-row-vector containing 1s, n.The connection of the G is a variance-covariance matrix and g.The statistical model that is probably most popular in genomic applications is the random-effects model (REM), in which the genomic values are assumed to be normally distributedX the In the following, we assume that the genome is mapped with G introduced in j  is called linear equivalent model  to the “1n+PXβ+ε.The centeg of genomic values in g.The restriction of the column-means of the marker genotypes in model (6) to be 0 guarantees that the sample mean of the genomic values in Genomic Best Linear Unbiased Prediction.Inferences on quantities based on genomic data in model (1) and (6) are often performed with the gBLUP method . Model (n draws from the underlying data-generating process of the (mean-centered) marker genotypes Model (6) is a realization of X1,…,Xp) . This diR. Instead of the genomic values g or n draws from the underlying data-generating process When we are interested in the genomic values in the corresponding consistent base population, we should take the relationship (correlation) between the individuals into account . In whatThe additive genetic variance  with respect to a trait is defined as the theoretical variance of the genetic value based on the respective QTLs . The genHere, we give an overview of different approaches to define a genomic variance in the framework of the linear models (1) and (6).n-vector of genomic values Without further assumptions on the nature of the genome, we can define the sample variance=PXβ see  in the cIn the base population, we define the sample variance=X*β see . Here, ΣAlternatively, we can define the theoretical variance of the genomic values directly in the REM. The linear model is generated by drawing from the data-generating process of the marker genotypes , and the model assumptions in the REM dictate that marker effects are random variables. This gives rise to three different scenarios for the variance of the genomic values in the REM .The additive genomic variance of a randomly sampled (representative) individual (The variance of a randomly sampled (representative) individual with RME is given byX equalsThe variance of a randomly sampled trait averaged over individuals with fixed genotypes Theoretical Variances of the Genomic Values in the REM. These quantities refer to the genotypes in the current population. We can apply the same definitions in the base population by considering the data-generating process of the genotypes in the base population  see , or the 0.5) see . Becauseβ in model (6) and the genomic values g in model (1) are random variables. Consequently, the additive genomic variance in β are predicted by the BLUP and then estimated by the eBLUP  best predictor for the additive genomic variance variance —the genoa, i.e., The expectation of the random variable ormation . The uncV for the additive genomic variance are considered in, for example, For the additive genomic variance in the current population, For the additive genomic variance in the base population, R used for the transformation to the base population is assumed to be the GRM G defined in Often , the matG is in general singular (because P is singular), and therefore We recommend caution when using the simplification in V, V, We emphasize that only the diagonal elements of the sample variance-covariance matrix ], however, make use of the data y to a certain extent.The unconditional (prior) expectation of y, makes best use of the information provided by y even before the final estimation is performed.By contrast, the conditional (posterior) expectation, given the phenotypic values Z  of Z onto the linear space of all functions in Y  expectation of a random variable diction”  of the rons in Y .Z and i.e., we can replace the target random variable Z by the best predictor defined in The best predictor in y, because X is constant, and, therefore, independent of y. The matrix of conditional second moments of the marker effects β is usually nondiagonal , as well as in the base population V, We emphasize that the best predictor of the additive genomic variance in the current population (G for transformation to the base population), will be given in the Appendix Estimation of the Additive Genomic Variance in the REM. We compare the use of the unconditional expectation and the best predictor for the prediction of the random additive genomic variance in the REM in Explicit formulae for the eBPs of the additive genomic variance, as well as a formula for R-package BGLR  under consideration was body length (BL). The relationship of the mice is recorded in the R, and is used for the transformation to the base population.For an illustration of the theoretical results of the previous sections, we used the mice dataset that comes with the age BGLR . The datage BGLR  in a mouR-package “BGLR” . Although the GRM is not invertible, we will show in the Appendix Estimation of the Additive Genomic Variance in the REM how to use the GRM for a transformation to the base population, and to calculate the corresponding unconditional expectation We used this outcome for the estimation of the unconditional expectation Detailed information about the calculations and the programming code is provided in the Supplemental Material, File S1.R-code and its output. Supplemental material available at Figshare: https://doi.org/10.25386/genetics.8114117.The authors affirm that all data necessary for confirming the conclusions of this article are represented fully within the manuscript and the supplemental material that has been uploaded to figshare. File S1 contains a detailed description of the estimation of the genomic variances for the gBLUP-method as well as the corresponding V and the best predictor W for the additive genomic variance in the current population. In the mice and wheat datasets, Arabidopsis data, the eBP is about double the size of the unconditional expectation.In the first section of Arabidopsis data. Technically, it is possible to define the heritability in two ways, namely with respect to the phenotypic variance, and with respect to the sum of the additive genomic variance and the residual variance. The sum of the eBP The sample variance of the phenotypic values has been scaled to 1. The sum of In the second section of Arabidopsis data the eBP exceeds the estimated unconditional expectation. This conforms to the behavior of V and W in the current population.The estimation results for the unconditional expectation We have shown that commonly used estimators for the additive genomic variance in the REM with genomic marker data are based on the unconditional expectation of the random additive genomic variance. We have introduced a novel best prediction approach for the random additive genomic variance in both the current and the base population. In the following, we discuss several important implications.Common ways of estimating the additive genomic variance focus on the base population. These approaches are independent of the actual current population, and, consequently, valid even if the generations change.If one aims at the response of a population to selection, however, it might be more meaningful to estimate the additive genomic variance in the actual given population. This implies that the estimation of the genomic variance has to be conducted again when the individuals change. A formal definition of the heritability is best possible in the current population, where the phenotypic and residual variance are estimable.We have preferred to use given relationship matrices for the transformation of the genomic values to the base population. In the case that such a matrix is not available, we have shown how to use genomic relationship matrices for the transformation, although a formal inversion of GRMs is, in general, not possible.In β. In that approach, e.g., in The frequentist gBLUP-method can also be set up in the context of Bayesian regression models  of the random additive genomic variance using the standard output of the gBLUP-method, which goes hand-in-hand with a fast implementation of the empirical version of the predictors. The connection between the BLUP Definitions of the genomic variance, we saw that the (random) additive genomic variance equalsIn the section variance , which iIn the REM, however, the marker effects are random, with unconditional expectation 0 and unconditional diagonal variance-covariance matrix with equal variances Estimation of the Additive Genomic Variance in the REM that we can partition the best predictor of the random additive genomic variance We show in the Appendix i.e., it determines the direction and the magnitude of the bias of GCTA-GREML . In addition, we notice that this bias depends not only on the sign of the covariance between the marker genotypes, but both on the sign and the magnitude of the weighted covariances.The best predictor, therefore, consists of the unconditional expectation of the additive genomic variance (no contribution of LD) and a function that explicitly contains the weighted contribution of marker LD. This function determines whether estimators like GCTA-GREML  are biased upwards or downward, When best  estimators are considered, the inclusion of LD is intrinsic; in that case, discussions about whether or not LD should be included or even optimally included become immaterial.L can be decomposed into the weighted sample variances and covariances of the marker genotypes. Instead of the marker effects, the components of the matrix We emphasize that, contrary to the unconditional expectation, the best predictor maintains the structure of the additive genomic variance variance .V and W .The best prediction approach eliminates the problem of the missing contribution of LD to the additive genomic variance that is caused by mathematical modeling  that uses the additional information given by the phenotypic data in an optimal way, minimizes the mean square error of prediction, includes the contribution of LD, and maintains the structure of the genomic equivalent of the classical additive genetic variance.


Plants can produce different phenotypes when exposed to different environments. Understanding the genetic basis of these plastic responses is crucial for crop breeding efforts. We discuss two recent studies that suggest that yield plasticity in maize has been under selection but is controlled by different genes than yield. Figure 1; Phenotypic plasticity is phenotypic variation that results from the complex relationships between an individual’s genotype (G) and the environment (E), including management decisions, in which it is grown. The ability of a single genotype to produce different phenotypes in different environments is termed phenotypic plasticity. The amount of phenotypic change across environments describes the degree of plasticity. When this change is far from zero, the phenotype is plastic; when this change is near zero, the phenotype is stable. Phenotypic plasticity is a characteristic of an individual genotype; when variation for plasticity exists within a population, it is termed genotype-by-environment interaction (GxE) . Responses to this macroenvironmental variation are termed plasticity . By contThe adaptive significance of phenotypic plasticity is situational . For exaConsequently, accounting for plastic responses in applications such as plant breeding is critical because plants are sessile and must respond to the environments in which they are grown. Breeders have approached this challenge in two ways . First, post hoc manner  and QTLs that only have effects in certain environments (gene regulation model) . ControvMaize is an ideal model and crop species in which to study phenotypic plasticity. It was domesticated in the lowlands of southwestern Mexico and subsequently adapted to highland and temperate environments . FurtherSTF and low nucleotide diversity) between 30 temperate and 30 tropical inbred maize lines along with a control group of SNPs that exhibited no such evidence. These SNPs were used to test the amount of phenotypic variation among 858 temperate maize hybrids that could be explained by SNP-environment interactions within each group. They found that high STF SNPs explained an average of 8.1% of the GxE variance for yield compared to 18.7% for low STF SNPs. High STF SNPs also explained less genetic variance for yield than low STF SNPs. This suggests that selection, especially in genomic regions showing high differentiation between temperate and tropical germplasm, has increased yields in temperate environments at the expense of the amount of GxE variation associated with yield that is explained by those SNPs. This reduction in GxE variance could be the result of increased uniformity in either the yield plasticity or yield stability of modern temperate maize germplasm. Because breeders have heavily selected for yield stability in modern temperate maize, To address this question, 1, will be required to test this hypothesis.If loci for yield and yield stability are genetically linked, selection for yield would be expected to affect allele frequencies at loci for yield stability  and vice versa. Because breeders have selected for both yield and yield stability, the simultaneous reduction in genetic and GxE variances explained by putatively selected genomic regions suggests that these regions contain genetically linked genes for yield and yield stability where the favorable alleles for both traits are in coupling linkage. Simultaneous selection for yield and yield stability by breeders has been remarkably successful; for example, modern commercial hybrids have improved drought tolerance  while also increasing yields in well-watered and drought conditions . HoweverAdditionally, yield integrates plastic responses to multiple abiotic and biotic stresses; the genes that mediate each plastic response may or may not be linked to yield-related genes and/or each other, increasing the complexity of breeding decisions when selecting for adaptive loci for particular combinations of environmental conditions. This view of yield as a function-valued trait of multiple environmental factors has a rich theoretical background , but manIdentifying such loci will allow for the identification and selection of desirable recombinants in elite breeding germplasm. However, if the findings of Further experiments to identify adaptive variation in maize paired with emerging technologies present an opportunity to improve yields not only by increasing average yield but also by maintaining and manipulating genetic variation that helps individuals maximize productivity across different environments. New cultivars can then be tailored to expected future conditions and agronomic practices in their target performance environments. Such an approach could not only create higher-yielding cultivars that exploit favorable conditions but also higher-yielding, stable cultivars that provide predictable yields in challenging environments where subsistence farming is often found.All authors contributed to the writing and editing of the manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Epistasis is an important contributor to genetic variance. In inbred populations, pairwise epistasis is present as additive by additive interactions. Testing for epistasis presents a multiple testing problem as the pairwise search space for modest numbers of markers is large. Single markers do not necessarily track functional units of interacting chromatin as well as haplotype based methods do. To harness the power of multiple markers while minimizing the number of tests conducted, we present a low resolution test for epistatic interactions across whole chromosome arms. Epistasis covariance matrices were constructed from the additive covariances of individual chromosome arms. These covariances were subsequently used to estimate an epistatic variance parameter while correcting for background additive and epistatic effects. We find significant epistasis for 2% of the interactions tested for four agronomic traits in a winter wheat breeding population. Interactions across homeologous chromosome arms were identified, but were less abundant than other chromosome arm pair interactions. The homeologous chromosome arm pair 4BL/4DL showed a strong negative relationship between additive and interaction effects that may be indicative of functional redundancy. Several chromosome arms appeared to act as hubs in an interaction network, suggesting that they may contain important regulatory factors. The differential patterns of epistasis across different traits demonstrate that detection of epistatic interactions is robust when correcting for background additive and epistatic effects in the population. The low resolution epistasis mapping method presented here identifies important epistatic interactions with a limited number of statistical tests at the cost of low precision. Arabidopsis , the added power from accurately tracking relevant LD blocks make these methods attractive.The multiple test correction problem is not the only challenge to identifying epistatic interactions. Allele frequency, linkage disequilibrium and the number of alleles at a given locus can all reduce the efficacy of pairwise marker epistasis detection. Low allele frequencies at either locus reduce the epistatic effect, partitioning it to the additive instead  markers can be modeled by first calculating an additive covariance for each marker set, We propose a method to greatly reduce the number of statistical tests while taking advantage of multiple markers to determine importance of epistatic interactions across chromosome arms of an allohexaploid wheat population. This method is similar to the “divide and conquer” method of Common wheat is an important allohexaploid crop with three subgenomes, A, B and D, resulting from hybridization events approximately 500 thousand and 10 thousand years ago. Due to the allopolyploid nature of wheat, we were interested in identifying interactions across homeologous loci. Interactions at homeologous loci are analogous to dominance effects in diploid hybrids, and could be used to fix favorable homeoallelic interactions in inbred lines , 2013. OEach chromosome arm of the wheat genome was sequenced independently using flow cytometry to assist in the assembly of the large complex genome . The lonWe demonstrate the low resolution epistasis mapping methodology using the CNLM winter wheat dataset from Details of the CNLM population used in this study can be found in Chromosome centromere positions were provided by the International Wheat Genome Sequencing Consortium (IWGSC) for all chromosomes except 3B . The positions provided by the IWGSC were used for all chromosomes other than our estimate for the 3B centromere position for all additional analyses.i and i.e., S) and long (L) arms of each chromosome, where μ is the population mean, i and The low resolution epistasis mapping approach employed here uses markers from two defined regions, vs. model 1) and interaction (model 3 vs. model 2) variance estimates of the chromosome arms were greater than zero. From the Neyman-Pearson lemma  of each region were subsequently used to look for patterns between additive and interaction effects for the chromosome arm pair. The pairwise product of the additive chromosome arm BLUPs was then compared to the chromosome arm interaction BLUPs, in a manner analogous to the Additive For the 14 three-way homeologous arm sets, a three-way interaction was included and tested against a model with only the three two-way interaction terms. We did not attempt to run all three-way chromosome arm combinations, as this would have been computationally infeasible, with In summary, this method estimates a random additive effect for each region and a random interaction effect between these regions for all individuals while correcting for the background genetic effects of the remainder of the genome. The likelihood ratio test is used to determine if the variability of the interaction effects is greater than zero. If the interaction effects do significantly deviate from 0, then we reject the null hypothesis that there are no interacting loci between the two regions.Variance component estimation was accomplished using restricted maximum likelihood (REML) implemented in ‘ASReml-R’ . Other chttps://doi.org/10.25387/g3.7311797.All data used for this study can be found in Most of the GBS marker density estimates of centromere locations agreed well with the positions provided by the IWGSC . Chromosomes 1D and 4A were exceptions. We estimated the 3B centromere to be positioned between 344.4 Mbp and 345.0 Mbp .Homeologous chromosome arm pair models each had five random genetic effects and therefore five covariance structures for the two-way interaction models. All models converged, but some variance parameter estimates were often close to the parameter boundary and were considered to be zero. Variance component estimates on the boundary did not occur for the background additive or epistatic effects, but often occurred for one or both of the additive chromosome arm effects or the interaction effect. This resulted in a relatively large number of additive and interaction variance component tests with a p-value of 1. As a result, p-value distributions were heavily skewed toward 0 and 1 . Most chromosome arms had low additive effect p-values, whereas most interaction p-values were high, indicating that the majority of chromosome arm pairs do not have effect interactions large enough to detect.i.e., 0), and when it was positive, the p-value tended to be low. Larger sample sizes may be necessary to obtain uniform p-value distributions when the null hypothesis is true. We therefore considered all homeologous arm pairs with an interaction variance p-value less than 0.05 that also had positive additive variance component estimates to determine the relationship between additive chromosome arm effects and their interaction.The U-shaped distribution of the p-values suggested that when the true variance was very small or zero, the average information algorithm estimated the parameter on the boundary .Relationships between chromosome arm additive and interaction effects were only considered for the ten chromosome arm pair trait combinations that had all chromosome arm additive and interaction effects with significant non-zero variance components. Of these ten, six had significant correlations between the additive product and the interaction with an absolute value ≥0.1 . Four ofρ=−0.65,, indicatFor all There were several chromosome arms that appeared to be interacting with multiple loci . Of these, several clearly stand out . Chromosi.e., Most correlations between the additive products and the epistatic effect were low in magnitude . The centromere position estimate reported here should be sufficient to assign most of the 3B markers to the correct chromosome arm for the subsequent analyses.i.e., zero), resulting in the U-shaped distribution. When variance parameters are estimated on the parameter boundary, the p-value becomes 1 simply due to the fact that the variance component is zero. This is likely due to a lack of sufficient population size to distinguish and resolve multiple small variance components. Perhaps another explanation may be provided by the use of the the average information algorithm to fit the mixed model, which may lose a small portion of information by avoiding the calculation of the second derivative of the likelihood function. While other algorithms exist for solving REML problems, the computational burden of resolving multiple variance components with dense covariance structures may be restrictive. Further investigation is necessary to determine how large a population need be to resolve multiple genetic variance parameters with magnitudes of 1% or less of the total variance.The distribution of p-values from the likelihood ratio test should be uniform if no true interactions exist. If interactions are important, then we would expect to see a skewed distribution with many small p-values. However, the p-values were often calculated to be 1 because the variance components were estimated on the parameter boundary , and result in a semi-dwarf plant stature which greatly reduces lodging in high nitrogen environments. Plants with both mutant alleles are far shorter than expected based on the additive effect of the two mutant alleles. From the perspective of functionality, one functional homeolog is able to recover most of the gene pathway function, with relatively little gained from the addition of a second functional homeolog. Therefore, the loss of one gene does not result in complete loss of the gene pathway, but merely a non-additive reduction in its total activity.A negative correlation for PH was not observed for the 4BS/4DS chromosome arm pair, as would be expected from previous results for the The lack of a negative correlation between the additive and interaction effects for the 4BS/4DS pair cast some doubt on the usefulness of these correlations to infer the direction of the epistatic effect. The relationship between the product of the additive effects and the interaction was thought to mirror the For inbred allopolyploids, multi-subunit protein complexes can be comprised of genes from a single subgenome, or from multiple subgenomes. If functional copies of subunits exist on both genomes, the formation of subgenome hetero-complexes may occur. Protein complexes comprised of evolutionarily divergent subunits may have increased or, more likely, decreased functionality. If hetero-complexes display decreased functionality, then we would expect the relationship between the additive and epistatic effects to be negative.It is unlikely that all homeologous interactions are so large in effect that they are quickly fixed after the hybridization event. The distribution of epistatic effects is likely similar in shape to the distribution of additive effects. These distributions will change based on the complexity of the trait. If a trait is governed by relatively few loci, the relatively few epistatic interactions may have larger effects, and may be easier to detect. In contrast, a large number of small effect additive loci may also result in a large number of small effect interactions that are too small to detect in populations of moderate size.PH appears to exhibit a higher degree of epistasis than either TW or HD. However, the number of interacting loci or chromosome arms detected was not directly related to the observed increase in genomic prediction accuracy by inclusion of epistatic predictors. GY showed no evidence of important epistatic interactions in this study, as has been previously shown , it is possible that residual structure is causing the observed additive and epistasis relationships. The drastically different patterns in the arm pair test results for each trait suggests otherwise. If these interactions were due to population structure, we would expect to see similar patterns of significance across all traits. When we omitted the background epistatic effect, most of the 861 interactions were significant (results not shown). We deemed this to be due to chromosome arm epistatic relationship matrices modeling close relationships in the population regardless of which unit of chromatin was used to determine those relationships. However, it is possible that these interactions are far more prevalent than suggested here, and that correction for background epistatic effects is diluting true genetic signal.While we corrected for population structure on both the additive and epistatic levels (The prevalence of a few chromosome arms interacting with many other arms is of particular interest, due to the potential for one site to influence the expression of so many other sites. These sites appear to act as hubs in interaction networks, and have been shown to be prevalent in yeast (The detection of chromosome arm interactions not identified in the homeologous marker sets of It should be noted that epistatic relationships formed from the Hadamard product of covariance matrices have the property of shrinking distant relationships while emphasizing close ones. For example, two lines with an additive covariance of 0.1 will have an epistatic covariance of 0.01, whereas two lines with an additive covariance of 0.9 will have an epistatic covariance of 0.81. It may be that there are several levels of relatedness that must be considered to properly account for genetic relatedness. The pedigree is an example of a covariance estimation procedure that emphasizes close relationships and deemphasizes more distant ones. Considering both pedigree and marker based covariance matrices has been shown to be more predictive than using either alone (The utility of the low resolution approach will depend on the magnitude and direction of effects, as well as the number and distribution of causal interacting loci. This is further complicated by the relative LD between these loci, the markers tagging them, and the loci with which they interact. While single markers may not be in high LD with causal loci, the combination of some markers can form haplotypes that are in high LD with one or more causal loci. Haplotype combinations across regions may then flag true interacting loci that would be otherwise undetectable using single marker interactions. While our method does not explicitly define these haplotypes, it does capture these relationships through the additive genetic covariance of the region. We suspect that this method will be the most appropriate for traits with moderate genetic complexity, as demonstrated by the lack of interacting regions detected for perhaps the most complex trait, grain yield.The interacting pairs presented here do not have the precision to make claims of interacting genes. Nor are these interactions necessarily targets for selection. They do, however, demonstrate that there appears to be global patterns of epistasis across the genome. Seemingly additive only traits have often been shown to be under a high degree of epistasis when careful investigation is used to elucidate the trait (This is evident when we consider the the complexity of the cell, where no genes truly work independently of one another. In order to create the complex structure of the cell, proteins may interact with other proteins, both alike and dislike to them, to form multi-subunit complexes. Therefore allelic variation alone should be sufficient to produce epistatic variation. It is merely our inability to separate this variation from “additive” variation under classic parameterizations that leads many to conclude that epistasis is not important (Further research into this methodology might be used to identify meaningful haplotypes. Once interacting segments are identified, they can each be split into multiple pieces for further refinement of the method, while nominally increasing the number of tests performed. The low resolution epistasis mapping approach presented here emphasizes the power of using multiple genetic markers to test for interacting genomic regions, albeit at the cost of low precision.


E.coli dataset on ampicillin resistance validates the approach.Genome wide association studies (GWAS) rely on microarrays, or more recently mapping of sequencing reads, to genotype individuals. The reliance on prior sequencing of a reference genome limits the scope of association studies, and also precludes mapping associations outside of the reference. We present an alignment free method for association studies of categorical phenotypes based on counting  Association mapping refers to the linking of genotypes to phenotypes. Most often this is done using a genome-wide association study (GWAS) with single nucleotide polymorphisms (SNPs). Individuals are genotyped at a set of known SNP locations using a SNP array. Then each SNP is tested for statistically significant association with the phenotype. In recent years thousands of genome-wide association studies have been performed and regions associated with traits and diseases have been located.However, this approach has a number of limitations. First, designing SNP arrays requires knowledge about the genome of the organism and where the SNPs are located in the genome. This makes it hard to apply to study organisms other than human. Even the human reference genome was shown to be incomplete  and assoSome of these limitations can be overcome by utilizing high throughput sequencing data. As sequencing gets cheaper association mapping using next generation sequencing is becoming feasible. The current approach to doing this is to map all the reads to a reference genome followed by variant calling. Then these variants can be tested for association. But this again requires a reference genome and it may induce biases in variant calling and regions not in the reference genome will not be included in the study. Moreover, sequencing errors make genotype calling difficult when sequencing depth is low  and in rIn the past, alignment free methods have been developed for a number of problems including transcript abundance estimation , sequenc k-mers.  presenteecently,  proposedecently,  showed tHowever, these methods for association mapping in bacterial genomes use only the presence and absence of possible  and 3. HIn contrast to the approach in We have implemented our method in a software called ‘hitting associations with In a pairwise comparison of the Toscani in Italia (TSI) and the Yoruba in Ibadan, Nigeria (YRI) populations we find that sequences identified by our method largely agree with results obtained using standard GWAS based on variant calling from mapped reads . Agreement with sites found using read alignment and genotype calling indicate that Apolipoprotein B (ApoB) gene.We also analyze data from the Bengali from Bangladesh (BEB) population to explore possible genetic basis of high rate of mortality due to cardiovascular diseases (CVD) among South Asians and find significant differences in frequencies of a number of non-synonymous variants in genes linked to CVDs between BEB and TSI samples, including the site rs1042034, which has been associated with higher risk of CVDs previously, and the nearby rs676210 in the E. coli data set on ampicillin resistance and find hits to the -lactamase TEM (blaTEM) gene, the presence of which is known to confer ampicillin resistance, validating our overall approach.We then demonstrate that population structure can be inferred from We present a method for finding regions associated with a categorical trait using sequencing reads without mapping reads to reference genomes. The workflow is illustrated in Jellyfish .Our method is implemented in a tool called ‘hitting associations with software . The logThe sequences obtained by merging overlapping ng BLAST  to checkng BLAST  to be cong BLAST .Escherichia coli strain. We introduced different types mutations - single nucleotide changes, short indels (less than 10 bp) and long indels (between 100 bp and 1000 bp) into the genome. Then wgsim of SAM tools   since thHawk is an alignment free method for mapping associations, it is able to find associations in regions that are not in the human reference genome, hg38. The analysis resulted in 94,795 and 66,051 sequences of lengths up to 2,666 bp and 12,467 bp associated with YRI and TSI samples respectively that did not map to the human reference genome. Similarly BEB-TSI comparison yielded 19,584 and 18,508 sequences with maximum lengths of 1761 bp and 2149 bp associated with BEB and TSI respectively.As We found that few of the sequences enriched for in TSI samples, with lengths up to 12kbp and 2kbp in comparisons with YRI and BEB respectively, mapped to the Epstein–Barr virus (EBV) genome, strain B95-8 [GenBank: V01555.2]. EBV strain B95-8 was used to transform B cells into lymphoblastoid cell lines (LCLs) in the 1000 Genomes Project and was shown to be a contaminant in the data .Interheart Study, the mean age of MI among the poeple from Bangladesh is considerably lower than non-South Asians and the lowest among South Asians (We noted that cardiovascular diseases (CVD) are a leading cause of mortality in Bangladesh and age standardized death rates from CVDs in Bangladesh is higher compared to Italy see . Moreoveh Asians . This moThe sequences of significant association with the BEB sample were aligned to RefSeq mRNAs and the ones mapping to genes linked to CVDs  were anaApolipoprotein B (ApoB) has been associated with increased levels of HDL cholesterol and decreased levels of triglycerides  of differential allele frequencies in BEB and TSI samples  gene which is known to confer resistance to ampicillin  and jus-lactamase TEM-1 (blaTEM-1) gene or are in linkage disequilibrium with it, the null is not true which results in deviations from the diagonal line. However, majority of the points are close to the origin which can be seen from the cumulative distribution of p-values in QQ plot of the p-values obtained is shown in -lactamase TEM-1 (blaTEM-1) as the top hit while the best -lactamase TEM-1 (blaTEM-1) gene. Freebayes  merits particular mention as the CC genotype at that site has been associated with higher risk of CVDs.We explored variants in genes linked to cardiovascular diseases in the BEB, TSI comparison as South Asians are known to have a higher rate of mortality from heart diseases compared to many other populations. We find a number of non-synonymous mutations in those genes are more common in the BEB samples in comparison to the TSI ones underscoring the importance of association studies in diverse populations. The SNP rs1042034 in the gene E. coli lead to hits to a gene, the presence of which is known to provide the resistance.We also outlined an approach to uncover population stratification, a known confounder in association studies, from E. coli datasets provide a proof of principle of this approach and motivate extension of this method to quantitative phenotypes and modeling of randomness of counts in population stratification detection and correction of confounder steps and then application to association studies of disease phenotypes in humans and other organisms.The results on simulated data, real data from the 1000 genomes project and  In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.k-mers" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Detlef Weigel as the Senior Editor. The following individual involved in review of your submission has agreed to reveal his identity: Casey S Greene (Reviewer #3).Thank you for submitting your article "Association Mapping from Sequencing Reads using The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.Summary:k-mers are differentially represented in the reads from cases vs. controls, and concatenates overlapping k-mers to find out longer sequences are associated with the trait of interest. The method is particularly well suited to detecting associations with structural variants, since it avoids the problem of SV calling, which is currently challenging with current methods.The paper describes a method and associated software (HAWK) that performs association mapping on sequencing data without the use of read alignment and variant calling. The method tests if Overall, there was much enthusiasm for the work, but there were serious concerns regarding how the authors would account for population structure. This will need to be thoroughly addressed in a revision.Essential revisions:1) Demonstrate that the method is robust to problems of population structure.k-mers are differentially represented in cases vs. controls, the authors do a simple two-sample Poisson comparison using likelihood ratio test. This makes the implicit assumption that the read counts are Poisson distributed and the Poisson rates are constant in cases and in controls. The latter assumption is generally problematic for read count data, as biological differences can create additional variations of read counts. Please show empirical evidence that the counts are Poisson, and indicate what could go wrong if they are not.2) Statistical problem: to test if It is known in RNA-seq data, for example, that read counts from multiple samples cannot be adequately modeled by a Poisson distribution, and a negative binomial distribution is used to model "overdispersion"  and WASP (PMID: 26366987)).k-mers, there is no association.It is not entirely clear that using negative binomial (NB) would solve the problem in this case, as the underlying Poisson rates may be discrete. For instance, suppose the association signal is driven by a deletion (CNV). Each individual either has the deletion or not. NB makes the assumption that the rate follows a gamma distribution. The authors should show the signal is not driven by overdispersion or confounders. At the minimum, they should have a QQ plot, which shows that for most k-mers somehow depends on coverage , one could imagine that this may create some false associations.3) A related problem is the possible confounders. For example, if cases and controls have different coverage, and representation of 4) The validation of the method is inadequate. The experiment of classifying population ancestry is a quite artificial – few people will do this kind of analysis. What is much more relevant is application of the method on real sequencing data of some complex phenotype. The authors did do it on a CVD dataset, but only superficially. They applied it only on genes known to be linked to CVD, for example, and it's unclear how the results compare with the alternative approach. To really demonstrate the benefits of HAWK, they should apply it to the full data, and evaluate the results more deeply. How do the results compare with standard GWAS? Any new associations? Can these new associations by validated? Do we learn any biology?5) The authors state that Cortex  is not suited to a large number of individuals . Please explain why this is the case. A comparison with HAWK would help to motivate their approach.6) Iqbal et al. say that "Current assembly methods also typically ignore pre-existing information, such as a reference sequence or known variants. Although variant discovery should not be biased by such information, neither should this information be discarded." The authors should comment on this with respect to their approach.7) For the comparison between HAWK and traditional genotype GWAS (subsection “Verification with 1000 genomes data”), restrict the analysis to only SNPs, i.e. to variants that a normal GWAS is meant to capture. Then we can see the power difference afforded by Poisson and access to counts vs genotype calls. Also, include the use of dosages, a softer version of a hard variant calls. If the false positive rate is well-controlled here , and you are well-powered, then the much-harder-to-verify claims about finding structural variants are better supported.q-values may be more appropriate (Bonferroni is overly conservative).8) The authors should discuss the error rate they are interested in and why, and then explain what an appropriate error measure would be. For example, it seems that Storey et al.'s 9) In the comparison between HAWK and more conventional approaches, were the SNP calls used in the conventional approach based on the same data that HAWK had access to? If not, it's not really a fair comparison. Also, add in what happens if you use dosage calls.10) Include explanation of all the ways HAWK might lead to false positives, how likely they are to occur, and what can be done to find out if they are happening.11) HAWK appears only applicable to case-control, and not quantitative traits. This should be mentioned in the Abstract, and also more directly in the Introduction. The authors should explain how HAWK could be applied to quantitative traits .12) Define the scope where HAWK is to be applied more carefully than "sequencing data". Please provide empirical support using real data for any claims . The current results only appear to support whole genome sequencing use cases.k-mers" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Detlef Weigel as the Senior Editor. The following individual involved in review of your submission has agreed to reveal his identity: Casey S Greene (Reviewer #3).Thank you for submitting your article "Association Mapping from Sequencing Reads using The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.Essential revisions:k-mers would mostly reflect ancestry-specific markers. Indeed, in Table 3, for several variants, the difference between the frequencies of BEB and TSI groups are so large that they unlikely result from their disease effects. The authors show only results from genes already linked to CVD. It's not clear at all if these genes are ranked high in their findings, had they analyzed all genes. As such, it is almost impossible to interpret the results here – whether it's due to population difference or disease association.1) The CVD study seems very odd. The authors use BEB or TSI as response variables, and do association, instead of using CVD status (i.e. cases or controls). Since BEB and TSI are obviously different in population ancestry, the resulting E. coli. analysis: this is more like a true association study. Application to E. coli instead of human is a bit odd though. Other than that, if one uses traditional procedure of variant calling and association testing, will the same gene(s) be found? The authors need to demonstrate that their method adds something new.2) In the k-mers into larger sequences, providing an approach to discarding false positives".3) "We agree that deviations from the Poisson assumption are possible, and this motivated us to perform the conservative Bonferroni correction and then assemble To appease the concern about the distributional assumption, it is not really sufficient to state that you are using a conservative type 1 error control. First, because that's not really a sensible way to fix the potential problem . Second, because who's to say if Boneferroni (or any method X) is conservative enough to outweigh potential false positive of a wrongly specified model? One would at least have to specifically demonstrate that.Also, it would be good, as a baseline, to use an expensive negative binomial test to see how much better things would be if you could afford to do that. Otherwise we don't have a good sense of the cost of assuming Poisson (if any). Additionally, if you did negative binomial with a less conservative measure than Bonferroni, we'd have still an even better sense of how bad the assumptions of your approach are. . At the very least, it would be good to see some kind of further justification that Poisson is reasonable (maybe it is in there and I missed it).4) "We then perform principal components analysis (PCA) on the matrix".How does this compare to doing a "standard" genetics PCA on the same samples? Which would you expect to be better, and for what reason? It would be nice to see some sort of head-to-head to get a sense of how different they are.k-mers counts".5) "We ﬁt a logistic regression model involving potential confounders and k-mer counts instead of on SNPs? If so, this should be expanded on and explained, and the cost in power discussed; if there is no cost in power, then why not use just that method instead of HAWK? In fact, it would be good to see a head-to-head comparison of this logistic regression approach with HAWK, when there is no confounding. This section needs more explicit description of what you are doing, why you are doing it, and what the pros and cons are."involving"? What precisely does this mean? Not clear what you did from this sentence. More importantly, it looks like you are here abandoning the fundamental part of HAWK, the Poisson testing, in this (population confounding) setting, and instead using standard GWAS techniques on the k-mers in each sample and gender of individuals along with k-mer counts and obtain ANOVA p-values of the k-mer counts using χ2-tests"6) "We then ﬁt logistic regression models to predict population identities using ﬁrst two principal components, total number of k-mer counts? Why not just the PCA? What are you trying to demonstrate exactly? Do you want to show that the PCs by themselves correct for confounding? Are you saying that gender is also a confounder here? If so, why? And if so, how would you account for it or other necessary covariates in your Poisson model? Please expand the thought process here.You need to expand conceptually what is going on here. Why do you need the genders here? And why the total 7) "We observed that the ﬁrst two principal components separate the two populations and the ﬁrst principal component correlates with sequencing depth indicated by size of the circles in the ﬁgure".only PC2 is separating the populations; PC1 is most certainly not. Did you remove the mean from your data? PCA is a low rank approximation to the covariance matrix, and thus if you don't remove the mean, you don't really have a covariance matrix, and the first PCA will then just capture the mean, which may be what you are seeing here. When you say with "sequencing depth", specifically what sequencing depth do you mean here? Maybe this is related to the issue of the mean. It seems that 8) "We then performed simulation experiments to test whether associations can be detected after correcting for confounding factors. […] The fraction of runs association was detected are shown in Appendix 1 Figure 9"You do this experiment, you have a plot in the appendix, but you fail to tell the reader what the results/take-home message are for it in the main text. Please add this.10 are close to zero which is expected since the ﬁrst two principal components separate the populations"9) "It shows that all of the −logPlease draw a QQ plot for the adjusted p-values so we can see just how calibrated thing are. I'd also like to see QQ plots for the distribution of p-values in your simulations in general and on the real data analyses (both for population trait and otherwise). Essential revisions:1) Demonstrate that the method is robust to problems of population structure.k-mer data and that population stratification and other confounders can be used to adjust p-values. We have added subsections titled “Detecting population structure” and “Correcting for population stratification and other confounders” in the Materials and methods section and “Detection and correction for confounding factors” in the Results section. These describe our approach. We have also provided some validation results.We have shown that population stratification can be inferred from 2) Statistical problem: to test if k-mers are differentially represented in cases vs. controls, the authors do a simple two-sample Poisson comparison using likelihood ratio test. This makes the implicit assumption that the read counts are Poisson distributed and the Poisson rates are constant in cases and in controls. The latter assumption is generally problematic for read count data, as biological differences can create additional variations of read counts. Please show empirical evidence that the counts are Poisson, and indicate what could go wrong if they are not.It is known in RNA-seq data, for example, that read counts from multiple samples cannot be adequately modeled by a Poisson distribution, and a negative binomial distribution is used to model "overdispersion"  and WASP (PMID: 26366987)).It is not entirely clear that using negative binomial (NB) would solve the problem in this case, as the underlying Poisson rates may be discrete. For instance, suppose the association signal is driven by a deletion (CNV). Each individual either has the deletion or not. NB makes the assumption that the rate follows a gamma distribution. The authors should show the signal is not driven by overdispersion or confounders. At the minimum, they should have a QQ plot, which shows that for most k-mers, there is no association.k-mers”. This assumption also has a computational advantage, allowing us to compute p-values quickly compared to negative binomial distribution based models which would require an iterative process. We agree that deviations from the Poisson assumption are possible, and this motivated us to perform the conservative Bonferroni correction and then assemble k-mers into larger sequences, providing an approach to discarding false positives. We also report fractions of individuals in groups with corresponding k-mers present for specific variants highlighted in the paper. Furthermore, the top hits eventually go through another step of adjustment for confounders. We have added the CDF of p-values in the YRI-TSI comparison which shows that for most k-mers p-values are not highly significant.Our rationale behind the Poisson assumption has been explained in the subsection “Finding significant 3) A related problem is the possible confounders. For example, if cases and controls have different coverage, and representation of k-mers somehow depends on coverage , one could imagine that this may create some false associations.k-mers per sample in our confounder correction step explained in “Correcting for population stratification and other confounders” in Materials and methods and “Detection and correction for confounding factors” in Results.To address the problem of potential false associations due to different coverage, we propose to include total number of 4) The validation of the method is inadequate. The experiment of classifying population ancestry is a quite artificial – few people will do this kind of analysis. What is much more relevant is application of the method on real sequencing data of some complex phenotype. The authors did do it on a CVD dataset, but only superficially. They applied it only on genes known to be linked to CVD, for example, and it's unclear how the results compare with the alternative approach. To really demonstrate the benefits of HAWK, they should apply it to the full data, and evaluate the results more deeply. How do the results compare with standard GWAS? Any new associations? Can these new associations by validated? Do we learn any biology?k-mers in a setting not confounded by population structure. We have subsequently proposed an approach to correct for population stratification and other potential confounders and applied it to data to map ampicillin resistance in E. coli and updated the manuscript. We find hits to a gene known to confer the said resistance.The objective of the association mapping with population identity as the phenotype was to check whether associations of diverse types of variants can be captured using 5) The authors state that Cortex  is not suited to a large number of individuals . Please explain why this is the case. A comparison with HAWK would help to motivate their approach.We have provided an explanation in the third paragraph of the Introduction.6) Iqbal et al. say that "Current assembly methods also typically ignore pre-existing information, such as a reference sequence or known variants. Although variant discovery should not be biased by such information, neither should this information be discarded." The authors should comment on this with respect to their approach.k-mers. We have commented on this in the sixth paragraph of the Introduction.A major motivation of our approach is to allow association mapping in organisms with no or incomplete reference genome and not be biased by reference genome in variant calling. An intriguing question is at what stage we can better utilize reference genomes? One answer may be when assembling 7) For the comparison between HAWK and traditional genotype GWAS (subsection “Verification with 1000 genomes data”), restrict the analysis to only SNPs, i.e. to variants that a normal GWAS is meant to capture. Then we can see the power difference afforded by Poisson and access to counts vs genotype calls. Also, include the use of dosages, a softer version of a hard variant calls. If the false positive rate is well-controlled here , and you are well-powered, then the much-harder-to-verify claims about finding structural variants are better supported.k-mers due to structural variants and contaminants make a straightforward comparison between the approaches hard. Hence we resorted to simulation. We stress that the major motivation of our approach is to allow association mapping without a reference genome. The power issue was discussed to partly explain the larger number hits found by k-mer based approach.A limitation of our approach is it doesn’t allow us to consider only one type of variant. While a single SNP can lead to sequences of maximum length of 61, there are often multiple SNPs nearby making restricting analysis to SNPs difficult. Presence of 8) The authors should discuss the error rate they are interested in and why, and then explain what an appropriate error measure would be. For example, it seems that Storey et al.'s q-values may be more appropriate (Bonferroni is overly conservative).k-mers”.We have used the conservative Bonferroni correction specifically because it is conservative and thus can ameliorate deviation from the Poisson assumption. We have added an explanation in “Finding significant 9) In the comparison between HAWK and more conventional approaches, were the SNP calls used in the conventional approach based on the same data that HAWK had access to? If not, it's not really a fair comparison. Also, add in what happens if you use dosage calls.Yes, we used the same set of reads used by the 1000 genomes project to call variants.10) Include explanation of all the ways HAWK might lead to false positives, how likely they are to occur, and what can be done to find out if they are happening.We have listed possible reasons behind false positives in the subsection “Correcting for population stratification and other confounders” and discussed an approach to deal with them.11) HAWK appears only applicable to case-control, and not quantitative traits. This should be mentioned in the Abstract, and also more directly in the Introduction. The authors should explain how HAWK could be applied to quantitative traits .k-mers” and “Correcting for population stratification and other confounders”.We have mentioned it in the Abstract and the Introduction. We have mentioned how we may extend this to quantitative phenotypes at the ends of the subsections “Finding significant 12) Define the scope where HAWK is to be applied more carefully than "sequencing data". Please provide empirical support using real data for any claims . The current results only appear to support whole genome sequencing use cases.We have applied our method to whole-genome sequencing data so far. The manuscript has been updated to reflect that.Essential revisions:1) The CVD study seems very odd. The authors use BEB or TSI as response variables, and do association, instead of using CVD status (i.e. cases or controls). Since BEB and TSI are obviously different in population ancestry, the resulting k-mers would mostly reflect ancestry-specific markers. Indeed, in Table 3, for several variants, the difference between the frequencies of BEB and TSI groups are so large that they unlikely result from their disease effects. The authors show only results from genes already linked to CVD. It's not clear at all if these genes are ranked high in their findings, had they analyzed all genes. As such, it is almost impossible to interpret the results here – whether it's due to population difference or disease association.Since the sites were obtained using population identity and not CVD status, they are due to population difference. We merely wanted to explore whether any of those could potentially contribute to higher mortality from CVDs in BEB. The listed sites are not the highest ranked ones and are included as they are in genes known to be linked to CVDs. We have provided clarification in the second paragraph of the subsection “Differential prevalence of variants in genes linked to CVDs in BEB-TSI comparison”. We feel that insights gained from such unconventional methods may help researchers to narrow down on specific variants and study their effects.2) In the E. coli. analysis: this is more like a true association study. Application to E. coli instead of human is a bit odd though. Other than that, if one uses traditional procedure of variant calling and association testing, will the same gene(s) be found? The authors need to demonstrate that their method adds something new.E. coli”.We have performed a conventional association study, presented the results in Figure 14 in the appendix and discussed our findings and limitations of such approaches in the last paragraph of the subsection “Association mapping ampicillin resistance in 3) "We agree that deviations from the Poisson assumption are possible, and this motivated us to perform the conservative Bonferroni correction and then assemble k-mers into larger sequences, providing an approach to discarding false positives".To appease the concern about the distributional assumption, it is not really sufficient to state that you are using a conservative type 1 error control. First, because that's not really a sensible way to fix the potential problem . Second, because who's to say if Boneferroni (or any method X) is conservative enough to outweigh potential false positive of a wrongly specified model? One would at least have to specifically demonstrate that.Also, it would be good, as a baseline, to use an expensive negative binomial test to see how much better things would be if you could afford to do that. Otherwise we don't have a good sense of the cost of assuming Poisson (if any). Additionally, if you did negative binomial with a less conservative measure than Bonferroni, we'd have still an even better sense of how bad the assumptions of your approach are. . At the very least, it would be good to see some kind of further justification that Poisson is reasonable (maybe it is in there and I missed it).k-mers”. Earlier we had meant to say, albeit not quantitatively, that when there is deviation from the Poisson assumption, a series of measures is likely to reduce the number of false positives.We have presented our reasoning behind the Poisson assumption in the first paragraph of the subsection “Finding significant We have performed a simulation where data is simulated from negative binomial distributions and p-values were then calculated using likelihood ratio tests for both Poisson and negative binomial distributions. The results are in Appendix 1 Figure 1 along with p-values calculated in the same way on some real data. The results do not indicate that p-values calculated assuming Poisson distribution is notably lower than those found using negative binomial distributions.4) "We then perform principal components analysis (PCA) on the matrix".How does this compare to doing a "standard" genetics PCA on the same samples? Which would you expect to be better, and for what reason? Would be nice to see some sort of head-to-head to get a sense of how different they are.k-mer counts in the first paragraph of the subsection “Detection and correction for confounding factors”.We have added a PCA plot obtained from variant calls in Appendix 1 Figure 11A and explained why those should be cleaner than PCA done with 5) "we ﬁt a logistic regression model involving potential confounders and k-mers counts"."involving"? What precisely does this mean? Not clear what you did from this sentence. More importantly, it looks like you are here abandoning the fundamental part of HAWK, the Poisson testing, in this (population confounding) setting, and instead using standard GWAS techniques on the k-mer counts instead of on SNPs? If so, this should be expanded on and explained, and the cost in power discussed; if there is no cost in power, then why not use just that method instead of HAWK? In fact, it would be good to see a head-to-head comparison of this logistic regression approach with HAWK, when there is no confounding. This section needs more explicit description of what you are doing, why you are doing it, and what the pros and cons are.k-mers directly. The LRT based on Poisson lets us effectively and quickly test k-mers when there are no confounders and when there are confounders, lets us quickly filter k-mers out so that more computationally expensive tests can be performed on the rest. We have conducted simulations to assess loss in power and shown results in Appendix 1 Figure 4. We observe that there is cost in power. We leave designing tests taking into account stochasticity in k-mer counts which will hopefully improve power as future work. In any case, since we are fitting logistic regressions, it will be difficult to scale up to all the k-mers. The section has been expanded (subsection “Correcting for population stratification and other confounders”).We have rewritten this part to clarify things. We actually think the fundamental part of HAWK is the testing of 6) "We then ﬁt logistic regression models to predict population identities using ﬁrst two principal components, total number of k-mers in each sample and gender of individuals along with k-mer counts and obtain ANOVA p-values of the k-mer counts using χ2-tests".You need to expand conceptually what is going on here. Why do you need the genders here? And why the total k-mer counts? Why not just the PCA? What are you trying to demonstrate exactly? Do you want to show that the PCs by themselves correct for confounding? Are you saying that gender is also a confounder here? If so, why? And if so, how would you account for it or other necessary covariates in your Poisson model? Please expand the thought process here.k-mers from X and Y chromosomes in case of sex imbalance in cases and controls. Total number of k-mers was included in the model as sequencing biases such as the GC content bias are known which may lead to false positives if the cases and controls are sequenced at different sequencing depths. We have provided a detailed description . These were included to illustrate that they could be confounders in some datasets. However, for this dataset first two PCs adequately adjust for confounders . So, gender is not a confounder here. The Poisson model is used when there is no significant confounding. When there are we perform a second phase of adjusting p-values using logistic regression.The genders of individuals are included to prevent false positives from 7) "We observed that the ﬁrst two principal components separate the two populations and the ﬁrst principal component correlates with sequencing depth indicated by size of the circles in the ﬁgure".It seems that only PC2 is separating the populations; PC1 is most certainly not. Did you remove the mean from your data? PCA is a low rank approximation to the covariance matrix, and thus if you don't remove the mean, you don't really have a covariance matrix, and the first PCA will then just capture the mean, which may be what you are seeing here. When you say with "sequencing depth", specifically what sequencing depth do you mean here? Maybe this is related to the issue of the mean. We meant PC1 and PC2 together separates the populations. The mean was indeed removed from the data. The correlation between PC1 and sequencing depth i.e. average number of reads covering a base is possibly due to not getting reads from hard to sequence regions when the depth is low and may be due to other batch effects (sequencing depth is linked here to the sequencing platform used).8) "We then performed simulation experiments to test whether associations can be detected after correcting for confounding factors. […] The fraction of runs association was detected are shown in Appendix 1 Figure 9".You do this experiment, you have a plot in the appendix, but you fail to tell the reader what the results/take-home message are for it in the main text. Please add this.We have added a comment to summarize the plot in the last paragraph of the subsection “Detection and correction for confounding factors”.10 are close to zero which is expected since the ﬁrst two principal components separate the populations".9) "It shows that all of the −logPlease draw a QQ plot for the adjusted p-values so we can see just how calibrated thing are. I'd also like to see QQ plots for the distribution of p-values in your simulations in general and on the real data analyses (both for population trait and otherwise).E. coli ampicillin resistance association mapping  among others.We have added QQ plots for adjusted p-values for 1000 genomes , QQ plots for Poisson LRT p-values for YRI-TSI comparison , QQ plots for adjusted p-values


Female reproductive behaviours have important implications for evolutionary fitness and health of offspring. Here we used the second release of UK Biobank data  to evaluate the association between five female reproductive traits and polygenic risk scores (PRS) projected from genome-wide association study summary statistics of six psychiatric disorders . We found that the PRS of attention-deficit/hyperactivity disorder (ADHD) were strongly associated with age at first birth (AFB) (genetic correlation of −0.68 ± 0.03), age at first sexual intercourse (AFS) (−0.56 ± 0.03), number of live births (NLB) (0.36 ± 0.04) and age at menopause (−0.27 ± 0.04). There were also robustly significant associations between the PRS of eating disorder (ED) and AFB (0.35 ± 0.06), ED and AFS (0.19 ± 0.06), major depressive disorder (MDD) and AFB (−0.27 ± 0.07), MDD and AFS (−0.27 ± 0.03) and schizophrenia and AFS (−0.10 ± 0.03). These associations were mostly explained by pleiotropic effects and there was little evidence of causal relationships. Our findings can potentially help improve reproductive health in women, hence better child outcomes. Our findings also lend partial support to the evolutionary hypothesis that causal mutations underlying psychiatric disorders have positive effects on reproductive success. Some of these traits have been shown to associate with the physical and mental health of offspring3, and there has been growing evidence that maternal AFB is associated with increased risk of psychiatric disorder5 and behavioural problems in their children7. For instance, both early and late maternal AFB are associated with an increased risk to schizophrenia (SCZ)4, bipolar disorder (BIP)8, attention-deficit/hyperactivity disorder (ADHD)7, autism spectrum disorder (ASD)9 and depression10 in offspring. Age at menopause and menarche also tend to correlate with the risk of adverse mental health outcomes in offspring14.Female reproductive behaviours, including age at first birth (AFB), age at first sexual intercourse (AFS), age at menarche (AMC), age at menopause (AMP) and number of live births (NLB) have important implications in reproductive health and evolutionary fitness20.The relationship between reproductive behavior and susceptibility to psychiatric disorders is likely complex and bidirectional. Individuals with psychiatric illnesses and their relatives may be more prone to risk taking and impulsive behaviors, which cause early pregnancy and childbirth in women, or they may exhibit poor social interactions that result in delays in key reproductive transitions such as marriage, pregnancy and childbirth26. Genome wide association studies (GWAS) that examined reproductive behaviors and psychiatry disorders separately have identified several genome-wide significant genetic loci influencing AFB21, NLB21 and AFS3, and ones affecting susceptibility to ADHD27, ASD23, eating disorders (ED)28, BIP29, major depressive disorder (MDD)30, and SCZ31. When reproductive traits and psychiatric disorders are examined together, a complex genetic relationship between the two clusters of traits begins to emerge32. For instance, recent studies26 showed a U-shaped relationship between AFB and the polygenic risk score of SCZ24, which echoes with the phenotypic association between maternal age and SCZ risk in offspring found in epidemiological studies . However, for most other psychiatric disorders, the latent genetic architecture shared with reproductive traits remains unknown.In addition to the epidemiological evidence, the phenotypical relationship between female reproductive traits and psychiatric disorder risk in offspring may have a genetic basisIn this study, we investigate the genetic association of five specific reproductive traits in women  with six common psychiatric disorders  using polygenic score and LDSC approach. Causal relationship between the risk of psychiatric disorders and female reproductive traits is further tested using Mendelian randomization. This will shed light on the complex bio-psychosocial risk factors associated with shared genetic effects between female reproductive and psychiatric traits. Furthermore, this study provides insight into whether complex phenotypes of reproductive behaviour can be caused by mental health or rather by pleiotropic genes associated with both traits.UK Biobank’s scientific protocol and operational procedures were reviewed and approved by the North West Multi-centre Research Ethics Committee (MREC), National Information Governance Board for Health & Social Care (NIGB), and Community Health Index Advisory Group (CHIAG). UK Biobank also obtained informed consent from the study participants. Research Ethics approval was obtained from University of South Australia Human Research Ethics Committee (HREC). All methods were performed in accordance with the relevant guidelines and regulations.35. The quality control (QC) criteria for the imputed genotype data included an imputation reliability (INFO score) >0.638, minor allele frequency >0.01, p-value for Hardy-Weinberg equilibrium test >10E-7, and SNP missingness <0.05. Women with non-white British ancestry, individual missingness >0.05, individuals with putative sex chromosome aneuploidy or one individual per pair of relatives was excluded. After the QC above, we compared discordance between the first and second release of QCed UK Biobank data for each SNP and individual that were excluded if the discordance rate was above 0.05. In addition, we excluded genome-wide significant SNPs in a GWAS analysis where individuals in the first and second release UK Biobank data were treated as cases and controls. Subsequently, 220,685 individuals and 7,253,311 SNPs remained, from which 1,133,064 Hapmap3 SNPs were selected for the analyses. Traits of interest were AFB, AFS, AMC, AMP, and NLB. Number of observations for each trait are provided in Table We used the second release of the UK Biobank dataset that initially included 264,859 women with imputed genotypes for 35.6 million SNPs. The imputation was based on the Haplotype Reference Consortium reference panelhttp://www.med.unc.edu/pgc), including ADHD27, ASD23, ED28, BIP29, MDD30, and SCZ31. The number of cases and controls in these studies are provided in Table The GWAS summary results of six psychiatric disorders were obtained from the PGC (http://www.med.unc.edu/pgc/) and not likely to include UK Biobank sample. To compute PRS for the UK Biobank sample we used approximately 1,133,064 HapMap 3 SNPs from each study. The projection of the SNP effects onto the UK Biobank data was conducted using MTG239.For each psychiatric disorder, PRS were computed for the UK Biobank sample as the sum of the risk alleles weighted by the estimated SNP effects from the PGC GWAS that are in public domain  provided by the UK Biobank. For sensitivity analyses, we repeated the prediction using additional variables including educational attainment, income level, and smoking and alcohol consumption status.41. GWAS was carried out and SNP effects were estimated for each of the reproductive traits using PLINK 1.942. In the GWAS, phenotypes were adjusted for age at interview, year of birth, assessment center at which participant consented, genotype batch and the first 15 PCs. We also used additional covariates such as educational attainment, income level, smoking and alcohol consumption status to further adjust the phenotypes in sensitivity analyses.Genetic correlation is a classic population parameter used to infer the geometric mean of trait variance . Pairwise genetic correlations were estimated between the reproductive traits and psychiatric disorders using linkage disequilibrium score regression (LDSC) based on GWAS summary statisticsNs overlapping individuals in the two studies with sample sizes of N1 and N2, respectively40. To check if there were any overlapping samples between UK Biobank and PGC, we explicitly estimated the intercept when using LDSC to estimate genetic correlations between the reproductive traits and psychiatric disordersThe intercept of LDSC is estimated as 42. Prior to the association test, we adjusted the outcomes, the phenotypes of the female reproductive traits, for age at interview, year of birth, assessment centre at which participant consented, genotype batch and the first 15 PCs. We used an inverse-variance weighted (IVW) regression43 as the primary analysis to determine a causal relationship and used MR-Egger44, weighted median45 and weighted mode46, which rely on different assumptions regarding the validity of instruments, for sensitivity analyses. Details of those assumptions are described elsewhere46. MR analyses were performed using a R package, TwoSampleMR47, and P-values were obtained using a two-tailed t-test with the respective degree of freedom. The instruments were available for ADHD, BIP and SCZ only, therefore ASD, ED, MDD were not tested for the MR analyses.We performed two-sample Mendelian randomization (MR) analyses to test whether there were causal relationships between the risk of psychiatric disorders and female reproductive traits. We used psychiatric disorder PRS as exposure data and female reproductive traits as outcomes. Genome-wide significant SNPs (p < 5E-08) from the PGC GWAS summary statistic results were used as instruments in the MR analyses. Associations between the instruments and the outcomes were testedIn this study, we analyzed genetic data from 220,685 women who were part of the UK Biobank. The average age of the study participants at recruitment was 57, with a range from 40 to 71. The number of observations for each reproductive trait is provided in Table First, we computed PRS of the six psychiatric disorders for the UK Biobank sample and estimated the mean of the PRS for each age category of each reproductive trait Figs –5. Then,24. The mean difference in PRS was statistically significant for most pairwise comparisons of age classes for ADHD, ED and MDD .We regressed each of pre-adjusted reproductive traits  on the PRS of each of the psychiatric disorders. Using the linear regression prediction modeling, we showed that the PRS of each six psychiatric disorder were significantly associated with at least one of the five reproductive traits, confirming some of the robust associations from the earlier analyses of mean difference of PRS across the five age categories Tables –S6. Of aWe conducted a sensitivity analysis in which dependent variables were further adjusted for educational attainment, income level, smoking and alcohol consumption status Fig. . Most ofWhen using GWAS p-value thresholds to filter SNPs, the significance of linear prediction decreased for most of the association analyses between the reproductive traits and six psychiatric disorders Table . This inWe used LDSC to estimate genetic correlations between the reproductive behavior traits and six psychiatric disorders (see Methods). We estimated genetic correlations between the five reproductive traits to reveal the shared genetic architecture of the traits Fig. . As expeFigure In the analyses where dependent variables were further adjusted for education, income levels, smoking and alcohol consumption status, the estimated genetic correlations between reproductive traits and psychiatric disorders were not substantially changed Fig. , compare41. Most of the intercepts from the cross-trait LDSC analyses of the five reproductive traits were significantly different from zero, as expected  for a causal relationship between ADHD and AFB , and between ADHD and AFS . None of the sensitivity analyses confirmed the significance except the weighted median approach, i.e. p-value = 0.042 for the causal relationship between ADHD and AFB and p-value = 0.016 for that between ADHD and AFS  (Table 40. The strong genetic associations between ADHD and most reproductive traits (except AMC) were supported by three different approaches, the analyses of mean difference of PRS across five different (age) categories, linear prediction and genetic correlation using LDSC. Genetic associations between two of the reproductive traits (AFB and AFS) and both MDD and ED were consistently significant across the various analyses. The association between SCZ and AFB (and NLB) was significant in the analyses of mean difference of PRS and linear prediction although the estimated genetic correlation from LDSC analysis was not significant.We revealed the complex psychosocial genetic risk architecture underpinning the association between reproductive traits and six psychiatric disorders using the HapMap3 SNPs that are known to be reliable50. Our findings of significant associations between female reproductive traits and between these traits and psychiatric disorders may be useful in improving female reproductive health, hence better child outcomes. For example, the high-risk group for ADHD could be informed about features of ADHD, such as impulsive behaviour and possible consequences of impulsivity. This intervention may lead to prevent them giving birth at an immature age, which can improve their reproductive health and the maternal environment for their baby. Furthermore, this information of genetic predisposition, combined with information on psychosocial factors, can be recorded as a form of family medical history, and used to monitor health of offspring.It is known that modelling genetic correlation between traits can increase accuracy significantly in predicting individual genetic risk for the traits51. Third, causal mutations underlying psychiatric disorders have positive effects on reproductive success. The last hypothesis can be supported by our observation that NLB and ADHD have a strong positive genetic correlation  while they persist in the population in a stable prevalence rate. It is questioned why natural selection has not excluded the causal mutations underlying psychiatric disorders. Three plausible hypotheses have been proposed. First, the impact of natural selection on the removal of existing causal mutations may be slower than the addition of new genetic mutations causing, for example, ADHD or SCZ. Second, pre-existing neutral mutations may interact with environments that have been new exposures for the population, and cause the disordersion Figs  and 8. Fion Figs  and 8, w53 reporting that there is no evidence of a causal effect of genetic liability for schizophrenia on NLB or AFB. It was noted that we used a small number of SNPs as instruments in the MR analyses especially for ADHD and BIP. Therefore, a further study with a large number of instruments should be warranted when more genome-wide SNPs become available.ADHD is shown to be genetically associated with a set of reproductive traits such as AFB or AFS. However, we found no clear evidence that higher genetic risk of ADHD decreased AFB or AFS, from the MR analyses. The negative association between ADHD PRS and AFB or between ADHD PRS and AFS found in linear prediction and genetic correlation might be mostly because of pleiotropic effects. MR-Egger result also showed that pleiotropic effects explained the relationship between ADHD PRS and AFS. We did not find any evidence to support a causal relationship for any pair of psychiatric disorder PRS and reproduction traits, which agree with a recent study54.We report a novel shared genetic architecture between female reproductive traits, i.e. including strong positive genetic correlation between AFB and AFS, high negative genetic correlation between AFB and NLB, and AFS and NLB, moderate positive genetic correlation between AMP and AFB (AFS), moderate negative genetic correlation between AMP and NLB, and negligible genetic correlation between AMC and other reproductive traits. These reported relationships among the reproductive traits may help shed light on how causal genetic variants are involved and shared in the regulation of reproductive mechanisms at the genome-wide level. The strong positive and negative associations between AFB and AFS, and AFB and NLB are intuitive and well expected. However, other relationships (e.g. moderate positive association between AFB and AMP and negligible association between AMC and AMP) should be further investigated in a future study to identify the exact mechanism of menarche and menopause in relation to reproductive success55. Secondly, we tested causal relationship between the PRS of ADHD, BIP and SCZ and five reproductive traits in women and found no evidence of causality. A relatively small number of instruments might result in lack of power to detect such causal relationship if there is any. Thirdly, we only focused on reproductive traits in women because data on related male traits  are not fully available in the UKB. The collection of data on reproductive traits such as AFB in men, that would enable the study of genetic associations between psychiatric disorders and these reproductive traits should be a priority for the field. Fourthly, for the female reproductive traits, we used self-reported data and ascertained sample that is less representative of the general population. A fifth limitation is the possible confounding from factors such as a history of early childhood adversity and socioeconomic status (SES). A history of childhood adversity has been associated with a range of adverse mental health outcomes, including psychosis56, BIP57, and MDD58. With regard to SES, children living in low-income households have been found to have an increased risk for ADHD59. Low access to economic resources have been associated both with an earlier age of menarche, and in both females and males, a larger number of offspring at an earlier age60. In particular, a strong link has been observed between lower SES during childhood and parenthood during adolescence61. Therefore, in sensitivity analyses, we repeated the prediction using income level (as a measure of SES) as an additional variable.There are a number of limitations in this study. Firstly, we used the PRS or LDSC approach using GWAS summary statistics instead of individual-level genotype data for psychiatric disorders, for which we do not have access permissions. The power and accuracy of detection and estimation in the analyses could be increased when using individual-level genotype dataIn conclusion, we revealed the latent genetic architecture between the five reproductive traits in women and six psychiatric disorders. In particular, ADHD PRS and a set of reproductive traits were substantially associated through common genetic factors. There were also a number of robust associations between reproductive traits and ED, SCZ and MDD PRS. There was no evidence that the risk of any psychiatric disorder modulated the phenotypes of the female reproductive traits. Therefore, the associations found in this study were mostly due to genetic pleiotropic effects. Our findings can have potential to help improve reproductive health in women and their child outcomes. These findings also can help address an evolutionary hypothesis that causal mutations underlying ADHD, MDD and SCZ have positive effects on reproductive success.http://www.med.unc.edu/pgc/.PGC: https://sites.google.com/site/honglee0707/mtg2.MTG2: https://github.com/bulik/ldsc.LDSC: Supplementary file