Polygenic risk scores (PRSs) are a method to summarize the additive trait variance captured by a set of SNPs, and can increase the power of set‐based analyses by leveraging public genome‐wide association study (GWAS) datasets. PRS aims to assess the genetic liability to some phenotype on the basis of polygenic risk for the same or different phenotype estimated from independent data. We propose the application of PRSs as a set‐based method with an additional component of adjustment for linkage disequilibrium (LD), with potential extension of the PRS approach to analyze biologically meaningful SNP sets. We call this method POLARIS: POlygenic Ld‐Adjusted RIsk Score. POLARIS identifies the LD structure of SNPs using spectral decomposition of the SNP correlation matrix and replaces the individuals' SNP allele counts with LD‐adjusted dosages. Using a raw genotype dataset together with SNP effect sizes from a second independent dataset, POLARIS can be used for set‐based analysis. MAGMA is an alternative set‐based approach employing principal component analysis to account for LD between markers in a raw genotype dataset. We used simulations, both with simple constructed and real LD‐structure, to compare the power of these methods. POLARIS shows more power than MAGMA applied to the raw genotype dataset only, but less or comparable power to combined analysis of both datasets. POLARIS has the advantages that it produces a risk score per person per set using all available SNPs, and aims to increase power by leveraging the effect sizes from the discovery set in a self‐contained test of association in the test dataset. These include sample stratification, risk prediction, and the detection of relationships between different subphenotypes  may be found to be associated with a disease in different samples. Gene‐based analyses also directly provide information for functional analysis  method and also on summary statistics using Brown's method  method aims to improve upon the standard PRS method by correcting the inflated Type I error observed both in standard PRS in the presence of LD  is present in the regression model, rather than the number of predictors being equal to the number of markers, or the number of chosen PCs. Like the standard PRS approach, an advantage of our method is that it performs a self‐contained test of association in the test dataset, leveraging the discovery set to increase the power of this test. A significant test statistic implies significant association specifically in the test sample, unlike a significant meta‐analysis result, where the association evidence could result from other samples. This might be important if the test sample is of specific interest, for example, a different ethnicity, or a different, but related, phenotype.In the present paper, POLARIS is evaluated by comparing the set‐based results calculated using LD‐adjusted PRS with those found using MAGMA  into a single regression predictor using single‐SNP effect sizes (log(ORi) = βi) taken from a previous study as coefficients,g is obtained from the test set. The underlying assumption is that individual genotypes are available for the test set, but only summary data (effect sizes β) for the discovery set are available.For The standard PRS method does not adjust for LD between markers and thus requires LD pruning  eigenvectors xk. The correlation matrix is the covariance matrix of the joint distribution of individual genotypes after standardization of each SNP. Its eigenvectors indicate the directions of the principal axes of this standardized distribution, and the corresponding eigenvalues give the variances of the distribution in the corresponding directions. In the absence of LD, these variances will be equal to 1, and the distribution will be isotropic. However, if there is LD, then these variances will in general be different, and the standardized distribution will be more elongated in some principal directions and flattened in others.We correct for this imbalance due to LD by replacing the vector This anisotropy can be removed by scaling the standardized joint distribution in the direction of each principal axis with the inverse square root of the eigenvalue in this direction. However, adjusting the standardized distribution in this way will not only remove LD, but also equalize the single marker variances, thus discarding information such as the minor allele frequencies. As our aim is to adjust for LD only, but not for single‐SNP variances, we therefore have chosen to apply the same scaling transformation to the original, unstandardized joint distribution instead.xkxkTg, which is the part of g along the kth principal axis, has correlation matrix eigenvalue λk and therefore contributes a disproportionate amount of variance to PRS unless λk≈1. For an uncorrelated marker, one spectral component will be concentrated on this marker, and the corresponding eigenvalue λk≈1.More specifically, due to the orthonormality of the eigenvectors, the PRS can be expressed in a spectral decompositiong in the direction of the kth principal axis, xkTg, with the inverse square root of the correlation eigenvalue, giving an adjusted coordinate 1λkxkTg, and hence the rescaled spectral component 1λkxkxkTg.For our adjustment, we rescale the coordinate of x, xTS−1x=∥S−12x∥2, where S is the covariance matrix −12g=∑k=1M1+λ0λk+λ0xkxkTg, and I is the M×M unit matrix. Note that if all markers are uncorrelated, then λk≈1 for all k, which makes g∼≈g, and consequently POLARIS≈PRS.Using the adjusted dosages g∼=C−12g (or the extension with a ridge parameter) directly to the vector of genotypes. More precisely, an adjustment of the variance only will be achieved by removing the sample mean vector g^ before the adjustment, givingβT(I−C−12)g^, which is irrelevant in the subsequent regression analysis.We remark further that we applied the adjustment 2.2To understand detailed differences and similarities between MAGMA  r2=0.8 between consecutive SNPs. The “causal” SNP is associated with disease with OR=1.1 and the remaining nine SNPs have an OR closer to the null value of 1. An additional 90 independent unassociated SNPs are also present in the set, see supplementary Scenario A (“One LD Block”): 10 SNPs in an LD Block with (i) Scenario B : 115 SNPs from real genetic and environmental risk in Alzheimer's disease (GERAD) data  is a large study that used genotyped and imputed data on 7,055,881 single nucleotide polymorphisms (SNPs) to meta‐analyze four previously published GWAS datasets consisting of 17,008 AD cases and 37,154 controls .For this study, we used only directly genotyped SNPs from the GERAD data  gene models  data using POLARIS were compared to those from the MAGMA‐PCA approach in GERAD genotype data and also the MAGMA‐SUMMARY approach in IGAP data . For the latter, we only consider SNPs present in both IGAP and GERAD. Prior to the gene‐set analysis, SNP summary statistics for the whole IGAP data were adjusted for the genomic control parameter, λ=1.087, as reported in Escott‐Price et al. .33.13.1.1We investigated the Type I error rates in simulations where none of the SNPs have an association to disease  in either the discovery or test sets, for Scenarios A (one LD block) and B , termed A(null) and B(null), respectively. Type I error is deemed acceptable if the nominal value is included in the 95% CI for estimated Type I error rate. The expected Type I error is displayed on the Type I error plots (gray dashed line).Figure P‐value threshold of 0.001 when the test and discovery set have N = 10000 and r2=0.2.The Type I error rate is reasonable in the majority of cases; the nominal value is included in the 95% CI. MAGMA in the combined data has slightly inflated Type I error at a The Type I error for scenario B(null), with the case‐control status randomly permuted in order to remove the effect size of any SNPs, is shown in Figure 3.1.2The power of the POLARIS method , MAGMA in the test set only  and MAGMA in the combined discovery and test sets  are displayed for each simulated scenario. The power graphs for Scenario A are shown in Figure N = 10,000 and discovery N = 50,000, but here MAGMA is applied to the individual genotypes of the discovery and test sets combined , so the sample used to estimate LD and perform the statistical test is very large, whereas POLARIS uses the discovery set N = 50,000 for effect size estimation and only N = 10,000 for LD estimation, and importantly, for statistical testing. In all cases, POLARIS has higher power than MAGMA in the test set only, as is expected, because POLARIS increases power by incorporating information from the discovery set. The power for POLARIS increases when the size of the test set increases, as this improves the estimate of LD between markers.POLARIS has equivalent power compared with MAGMA in the combined discovery and test sets in all cases. In the most likely realistic situation, the discovery set is larger than the test set, but only summary statistics are available for the discovery set. MAGMA on the combined dataset has higher power where the test Figure These power results were also compared with the power of the summary statistics based approach implemented in MAGMA, see supplementary 3.2P‐value threshold for POLARIS, MAGMA‐PCA in GERAD genotype data and MAGMA‐SUMMARY in IGAP summary statistic data. Statistically independent associations in some instances implicate overlapping regions. To define genes as physically independent, we have annealed associated genes that were not separated by at least 250 kb in each analysis separately. In the APOE region, significant genes on chromosome 19 between 44.4 and 46.5 Mb were counted as one. We also present the results for all genes in supplementary Table P‐value thresholds is higher or equal for POLARIS compared to the MAGMA‐PCA approach in GERAD data. This is expected as POLARIS uses both GERAD and IGAP‐noGERAD data, while MAGMA‐PCA uses GERAD genotypes only. The results for the summary statistic approach show higher numbers of significant genes for higher significance thresholds. The five gene‐wide significant genes found by the summary statistics approach are: TOMM40, CLU, BIN1, MS4A4E, and CR1, which have all been previously reported as being associated with AD from single SNP analyses .The number of independent significant genes for all P values caused by gene size is insufficiently tackled by most available methods  was computed. The correlation between the number of SNPs per gene and the empirical P value of each gene in AD data was then determined. We found no evidence  of a correlation between the number of SNPs in a gene and the gene P‐value for the POLARIS method. Therefore, associations with disease observed in larger genes is not simply due to a greater number of SNPs in the gene. Similarly, we observed no evidence  of an inflation in P‐value for increasing gene size using MAGMA‐PCA on GERAD data. When considering the correlation between the IGAP gene‐based P‐value and set size, we observe a statistically significant negative correlation  when MAGMA‐SUMMARY is used on summary data, indicating that the higher the number of SNPs, the lower the set‐based P‐value.The issue of bias in the estimation of set‐based 4In this paper, we present a method for accounting for LD in the calculation of a PRS. The resulting individual LD‐adjusted PRS can also be used for analyzing whether a set of SNPs is associated with disease. This method combines the advantages of PRS and spectral analysis of the genetic data. The latter suggests a mathematically sound adjustment for LD and includes a stabilization parameter (similar to ridge regression) to cope with cases of extreme LD. It adjusts for LD between SNPs and informs the analysis with previously reported effect sizes of a SNP's association with disease. In the present study, we have chosen to do this adjustment using the SNP‐SNP correlation matrix; however, one could alternatively use the SNP‐SNP covariance matrix. For all examples above, this gives very similar results. Partitioning the overall polygenic risk based on meaningful SNP sets, the method allows both to test for significance of association of these sets  and to provide individual set‐specific risk scores for subjects, which can further be used for risk prediction of subphenotypes with respect to the SNP sets. To assess the quality of the proposed approach, we compare its use for set‐based analysis with the widely used MAGMA software. We show that POLARIS gives the correct Type I error and its power lies between that of MAGMA applied to the test dataset only and MAGMA applied to the combined test and discovery datasets. In practice, researchers would use all the available genotype data, and would use PRS‐based methods only if effect sizes only are known for an additional dataset.P‐value for the strength of association between the set and disease. This set risk score can be used to stratify individuals for follow up studies  and also prioritize genes for further functional studies , supporting the development of precision medicines. (2) POLARIS can increase power by leveraging the discovery set to perform a self‐contained test of association in the test dataset. Another way to incorporate the discovery set would be to use meta‐analysis, however, this detects an association in the combined set rather than the test set only. This may be important when the test data differs in some way from the discovery data, for example different ethnicity, or different phenotype. A good example might be where the test sample uses different diagnostic criteria to measure the same phenotype  and one wishes to validate these criteria by showing that they show association to the same genes as those implicated by the standard diagnosis. (3) POLARIS is not inflated by set size. This is an issue previously reported in summary statistic based approaches  It produces a risk score per person per set, unlike other set‐based methods which only provide a For set‐based analyses, in situations where only summary statistics are available for part of the data, we suggest to use the POLARIS method which unites advantages of the PRS approach and the PCA‐based set‐based method, while taking into account the LD structure for the genetic region of interest.POLARIS can be used for any set of SNPs, for example, the whole genome, genes, or pathways. Therefore, it has the potential of data‐driven discovery of pathways.POLARIS can also be utilized in a number of cross‐disorder analyses to determine commonality between disorders at a gene‐based or pathway‐based level. There are a number of common disorders for which the GWAS summary data are publically available . The GWAS data for one disorder can be used to generate scores per person per gene in another disorder or subphenotype of interest, and thus test for overlap between disorders at a gene‐based level.The POLARIS method can be extended to add additional information into the score, such as rare variants from exome sequencing studies. The POLARIS set‐based method is implemented into a freely accessible platform independent software. Large sets have a high computational burden due to the spectral decomposition of large correlation matrices; recommendations on maximum set size and the corresponding required computational resource are included with the software.In this study, POLARIS was applied to test binary traits. However, the POLARIS score can also be used as a variable  in regression models for quantitative traits.A limitation of the POLARIS implementation is that currently it is only available as a self‐contained set‐based method. However, POLARIS can in principle be used as a competitive set analysis, adjusting for the baseline level of association in the data either by including a general PRS in the analysis or comparing the set‐based PRS to those generated from random sets of genes (matched for number of genes/gene size/numbers of SNPs) or random sets of SNPs .Another limitation of PRS‐type approaches is the imperfect tagging of the underlying causal variants by SNPs and imperfect effect size estimates. The challenge of selecting the true set of susceptibility SNPs for PRS modeling to capture heritability has been pointed out ‐ Simulation of 10 SNPs in LD and 90 independent SNPs. POLARIS (purple), MAGMA in the test set only (blue), MAGMA in combined test and discovery sets (green) and MAGMA using the combined test and discovery set summary statistics (red). Expected Type I Error is shown by the grey dashed line.Click here for additional data file.Figure S4: Type I Error Comparison of Set‐Based Methods; Scenario B(null) ‐ Simulation of 115 SNPs from Real Data, with Permuted Phenotypes to Remove Effect Sizes. POLARIS (purple), MAGMA in the test set only (blue), MAGMA in combined test and discovery sets (green) and MAGMA using the combined test and discovery set summary statistics (red) are compared. Expected Type I Error is shown by the grey dashed line.Click here for additional data file.Figure S5: Power Comparison of Set‐Based Methods at Different P‐value Thresholds; Scenario ASimulation of 10 SNPs in LD with OR=1.1 and 90 independent, unassociated SNPs. POLARIS (purple), MAGMA in the test set only (blue), MAGMA in combined test and discovery sets (green) and MAGMA using the combined test and discovery set summary statistics (red) are compared.Click here for additional data file.Figure S6: Power Comparison of Set‐Based Methods; Scenario B – Simulation of 115 SNPs, with a Proportion of Phenotypes Permuted to Maintain Effect Sizes. POLARIS (purple), MAGMA in the test set only (blue), MAGMA in combined test and discovery sets (green) and MAGMA using the combined test and discovery set summary statistics (red) are compared.Click here for additional data file.Table SI: Comparison of the Number and Proportion of Independent Genes Below a P‐value Threshold for POLARIS, MAGMA‐PCA in GERAD data and MAGMA‐SUMMARY in IGAP data.Click here for additional data file.


The family social relations model (SRM) is applied to identify the sources of variance in interpersonal dispositions in families, but the antecedents or consequences of those sources are rarely investigated. Simultaneous modeling of the SRM with antecedents or consequences using structural equation modeling (SEM) allows to do so, but may become computationally prohibitive in small samples. We therefore consider two factor score regression (FSR) methods: regression and Bartlett FSR. Based on full information maximum likelihood (FIML), we derive closed-form expressions for the regression and Bartlett factor scores in the presence of missingness. A simulation study in both a complete- and incomplete-case setting compares the performance of these FSR methods with SEM and an ANOVA-based approach. In both settings, the regression FIML factor scores as explanatory variable produces unbiased estimators with precision comparable to the SEM-estimators. When SRM-effects are used as dependent variables, none of the FSR methods are a suitable alternative for SEM. The latter result deviates from previous studies on FSR in more simple settings. As an example, we explore whether gender and past victimhood of relational and physical aggression are antecedents for family dynamics of perceived support, and whether those dynamics predict physical and relational aggression. Families can be seen as dynamic systems where the family members form interdependent subsystems  method, where the measurement and structural part are subsequently estimated  that participated in the RADAR-Y (Research on Adolescent Development and Relationships - Younger Cohort) study, from which we consider the first three waves. Previous studies of perceived support mainly focused on differences in perceived relational support between families, but it is also interesting to examine the relationship between the SRM-effects of perceived support and some external variables , mother (M), target adolescent (T) or sibling (S). In our illustration XTF, for example, represents the support the target adolescent perceives from the father. Self-ratings  are not considered here.The SRM decomposes dyadic measurements into four different effects at three different levels: the individual, the dyadic and the family level. Consider a dyadic measurement, Xij can be expressed as a linear function of a family effect (Fam), an actor effect (Act), a partner effect (Par), a relation-specific effect Rel, and measurement error (ϵ):According to the SRM, each Relij cannot be disentangled from the measurement error ϵij. To separate the relationship effect from the measurement error at least two observations or indicators of each dyadic relation are needed  that is typically used for the SRM , while only 12 means of the 12 dyadic measurements are observed. Therefore, restrictions are typically applied to the means such that the mean actor effects sum up to zero, the mean partner effects sum up to zero, and the mean SRM relationship effects sum up to zero for a given actor or a given partner. The family mean is simply defined as the average of the 12 dyadic measurements. These constraints are needed for model identification. By doing so the SRM-effects are unambiguously defined, a prerequisite for estimating the factor scores.Fam) or grand mean of each family is defined as the average of the 12 dyadic measurements. For the calculation of the actor and partner effects, it is important to point out that the table contains empty cells. These empty cells are present since the round-robin design that we consider does not include self-ratings. If self-ratings were to be allowed, the row mean could be considered as an estimate of the actor effect and the column mean as an estimate of the partner effect  i.e.,Alternatively, a weighted sum score can be obtained for each SRM-effect using the raw data. These scores are obtained by organizing the raw dyadic values for each of the relation-specific measures from each family in a two-way table in which the rows are the actors and the columns the partners (see Table A detailed derivation of these formulas for the SRM effects can be found in the SD = 0.5), their siblings were, on average, 15 years old (SD = 3.1), their mothers were, on average, 44 years old (SD = 4.5), and their fathers were, on average, 47 years old (SD = 5.1). The majority of the target adolescents were Dutch. At the time of the first wave of data collection, most adolescents were living with both parents, and most of their families were classified as having medium to high socioeconomic status. 466 families have data at the first three waves that we will consider.The RADAR study has been approved by the Medical Ethical Committee of Utrecht University Medical Centre (Netherlands). Participants of the RADAR-Y study were four-member families of 497 adolescents, who completed questionnaires during six annual home visits. Before the start of the study, adolescents and their parents received written information about the study and they provided their written informed consent in accordance with the Declaration of Helsinki. Parents gave written informed consent for the participation of their children. The sample of target adolescents consists of 283 boys and 214 girls. At the first wave, these adolescents were, on average 13 years old  and relational aggression  are taken from wave one, the perceived support measurements from wave two, physical  and relational aggression  as reported by the target adolescent from the third wave.The proactive/reactive aggression  was used to measure the target adolescent's perception of victimhood of physical aggression, victimhood of relational aggression, physical aggression and relational aggression. The measures of victimhood of physical  were all significant .First an SRM with structured means, shown in Figure Rosseel, , hereby The family effect explained 7–19% of the total variance and the actor variance accounted for 19–56% of the total variance in the dyadic measurements of support. The explained variance in dyadic support explained by the partner effects ranged from 2 to 21%. We can thus conclude that perceived support depends considerably on the characteristics of the actor . But what are possible antecedents and consequences of these SRM-effects?The ANOVA-based approach is used to estimate the SRM-effects in each family. Table Second, possible antecedents  of the SRM-effects of perceived support at wave 2 are investigated Figure . The firThe ANOVA-based approach presented here is often used but limited. First, it entails a limited information analysis instead of a full information analysis, that is only families with complete data are included. In our data, SRM-effects are missing in 22% of the families. Second, the unbiasedness of the estimator of the path coefficients is questionable  based extension of the regression factor score  and their estimated means  can be specified asAlternatively, the FSC of the regression factor score estimator  estimator.Conditional unbiasedness is an appropriate criterion when the factor scores are used as dependent variables. Such conditional unbiased estimator was introduced by Bartlett , the Barand Yuan  considerz, the estimated regression coefficients γ as follows:FSCB) and the factor loading matrix should equal the identity matrix. Since the factor loading matrix of the SRM model is not full rank, we are forced to use the generalized inverse as in (14) for FSCB. As we will see later, the product of the FSCB of the SRM-effects and its factor loading matrix will not necessarily result in an identity matrix.The Bartlett estimator yields unbiased regression coefficients when it is applied as a dependent variable  is used as predictor, or the Bartlett factor score (12) is used as outcome, has been shown analytically and in simulation studies in simple settings  and ANOVA result in biased estimates in both the complete case-setting and in the presence of missingness;When the underlying true regression coefficient is 0, the type I error rate of all the methods is correct;SEM shows the best coverage.When the SRM-effects are used as dependent variable, we hypothesize:In the complete-case setting SEM results in unbiased regression coefficients for the effect of a predictor on SRM-components;SEM FIML result in unbiased regression coefficients if missingness depends on other observed dyadic measurements;SEM FIML results in unbiased regression coefficients if missingness also depends on the observed predictor;In the presence of missing dyadic measurements:Regression (FIML), Bartlett and ANOVA result in biased estimates in both the complete case-setting and in the presence of missingness;When the underlying true regression coefficient is 0, the type I error rate of all the methods is correct;SEM shows the best coverage.R  and Bartlett (FIML) with SEM and the ANOVA scores. All analyses were performed using the statistical software Rosseel, , while oRosseel, . Since nyi is role-specific. Therefore, only the following SRM-effects are considered as predictors of yi: the family effect, the actor effect of role i and the partner effect of role i. In this data-generating model the variance of the family effect is set to 1, the variances of all the perceiver effects are set to 1, the variances of all the partner variances are set to 0.5 and the variances of all the relationship-effects are set to 1.5. Additionally, the generalized reciprocities are set to 0.05 or −0.05 and the dyadic reciprocities are all set tot 0.02. The data were randomly generated from the model-implied multivariate normal distribution using the R-package lavaan. A complete-case scenario and two missing data scenarios are considered. In the first missing data scenario, missingness in some of the dyadic measurements is related to other dyadic measurements. In detail, missing values in the dyadic measurements XTM and XTF were randomly generated with a missing rate of 25% when XTS had a value that was smaller than its first quartile. In the second scenario missingness is related to the outcome. Missing values were then randomly generated in XTM, XTF, and XTS with a missing rate of 25%, when yT had a value that was smaller then its first quartile. For each scenario 1,000 simulations were performed. We considered sample sizes of either n = 50 or n = 500 four-person families.In this condition the SRM-components of four-member families are used as independent variables. We assume a SEM model with four regressions as depicted in Figure An overview of the performance of the methods in the complete-case scenario is presented in the upper part of Table The middle part of Table An overview of the methods, in the presence of missing values related to the outcome can be found in the lower part of Table In sum, all our hypotheses are confirmed. We conclude that there is no method that outperforms SEM (FIML) when the SRM-components are used as predictor. But if SEM fails to converge regression FIML is an acceptable alternative (provided that missingness in the dyadic scores only depend on other measured dyadic scores), since no convergency issues were encountered when fitting the SRM that was used as base model for the computation of the factor scores. Alternatively, one could use the estimates from regression FIML as starting values in SEM FIML to overcome the convergency issues.z1, z2, and z3. Zero-effects are assumed between the predictors and the partner effects. This SEM can be viewed as a multiple indicators multiple causes (MIMIC) model, and is equivalent to a multiple group CFA with equal latent variances, but different latent means across groups  result in the same conclusions (results not shown). Again, 1,000 simulations were performed for each scenario. Sample sizes of either n = 50 or n = 500 four-person families were considered.In this second condition, the SRM-components are used as dependent variables. Data were generated from the SEM model shown in Figure In the upper part of Table The middle part of Table In the lower part of Table In sum, SEM (FIML) clearly outperforms all the other methods in this condition. However, it is important to stress that in the SEM-approach one needs to constrain the regression coefficients such that conditional on the predictor the actor effects  sum to zero. Furthermore, almost all of our hypotheses are met, concluding that in this condition there is no suitable alternative for SEM. As suggested before, the observed bias in the Bartlett FSR may be explained by the relation between the estimated regression coefficients  and Bartlett (FIML) approaches with SEM and the ANOVA-based approach.The simulation study showed that when the SRM-components are used as predictor, regression (FIML) FSR is a valuable alternative for SEM. There are, however, some limitations with regression (FIML) FSR. First, the standard errors of the FSR are underestimated even in large samples, but bootstrap may help overcome this limitation. Second, regression FIML has stronger missingness assumptions than SEM. Indeed, for regression FIML missingness may only depend on observed dyadic scores used in the SRM. The upside of regression FIML FSR is that it has less convergency issues than SEM in small samples.In the second condition, where the SRM-components are used as outcome, SEM clearly outperforms all the other methods concerning bias, coverage and precision. These results are not in line with previous research that suggested the Bartlett factor scores as a valuable alternative. An explanation for this deviation was found in the use of the Moore-Penrose generalized pseudo-inverse that rendered the unbiasedness characteristic of the Bartlett factor scores void. Factor loading matrices that are not full rank may also appear in other settings, such as multitrait-multimethods models  study were used. RADAR-Y has been financially supported by main grants from the Netherlands Organization for Scientific Research , and Stichting Achmea Slachtoffer en Samenleving (SASS), a grant from the Netherlands Organization for Scientific Research to the Consortium Individual Development , and various other grants from the Netherlands Organization for Scientific Research (NWO), the VU University of Amsterdam and Utrecht University. JL, AB, and TL would like to thank the Research Foundation Flanders (FWO) for financial support (Grant G020115N).JL, TL, and AB contributed to the conception and design of the manuscript. Data collection was carried out by SB and WM. Data analysis and interpretation for this study was performed by JL, TL, and VE. Next, JL developed the initial draft of the manuscript. Finally, all authors approve the version to be published.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.


Potato is an important food crop due to its increasing consumption, and as a result, there is demand for varieties with improved production. However, the current status of breeding for improved varieties is a long process which relies heavily on phenotypic evaluation and dated molecular techniques and has little emphasis on modern genotyping approaches. Evaluation and selection before a cultivar is commercialized typically takes 10–15 years. Molecular markers have been developed for disease and pest resistance, resulting in initial marker-assisted selection in breeding. This study has evaluated and implemented a high-throughput transcriptome sequencing method for dense marker discovery in potato for the application of genomic selection. An Australian relevant collection of commercial cultivars was selected, and identification and distribution of high quality SNPs were examined using standard bioinformatic pipelines and a custom approach for the prediction of allelic dosage. As a result, a large number of SNP markers were identified and filtered to generate a high-quality subset that was then combined with historic phenotypic data to assess the approach for genomic selection. Genomic selection potential was predicted for highly heritable traits and the approach demonstrated advantages over the previously used technologies in terms of markers identified as well as costs incurred. The high-quality SNP list also provided acceptable genome coverage which demonstrates its applicability for much larger future studies. This SNP list was also annotated to provide an indication of function and will serve as a resource for the community in future studies. Genome wide marker tools will provide significant benefits for potato breeding efforts and the application of genomic selection will greatly enhance genetic progress. Most breeding methods rely heavily on phenotypic selection for germplasm improvement with little importance on genotypic selection at the molecular level. Conventional breeding aims to combine desirable traits together from elite individuals in a step wise manner that is both laborious and time consuming. The resulting time frame can be 10–15 years before the commercial release of a new cultivar . Marker-Genomic selection (GS) predicts the performance of individuals that have been genotyped based on a prediction equation that is derived from an extensively phenotyped and genotyped reference population . Recent The composition and size of the reference population is critical as only the genetic variation that is present will be predicted, and any that is missing will not have their performance predicted in populations that contain the genetic variants . Stich aAmong the various types of markers used, SNP markers are the most abundant in the genome. The generation of the potato genome sequence  revealedEcoR1 and MspI to reduce chloroplast and rDNA sequences in the libraries, they could call 5,000 variants in 12 potato genotypes. Despite differences in methodology, both GBS and RAD-seq approaches operate by sequencing a set of restriction fragments, typically between 150 and 400 bp . GBS-t can enhance sequence alignment by removing complex repeat regions in the genome to focus on the transcriptome, where the size is relatively conserved across species. Other benefits include the low read depth required per genotype for accurate imputation (c. 3 million reads for diploid organisms), even sampling distribution of genes across the genome and that all SNPs identified are found in genic regions . PreviouRecent developments of large genomic, genetic, and phenotypic data sets have assisted with breeding approaches  yet moreThe germplasm collection used in this study comprised a total of 181 unique tetraploid potato cultivars. Cultivar names and market class are listed in Phenotypic data were collected from trials conducted between 2007 and 2012 at Toolangi, 169 Victoria, Australia. Specific trait analyses for quality traits were conducted at Knoxfield, 170 Victoria, Australia. A minimum of two replicates per trial were used for all cultivars, up to a 171 maximum of 10. All field based and quality phenotypic data were the result of the average of a 172 minimum of 3 years of trialling. An initial analysis of the raw phenotypic data was performed and only phenotypic traits that had the necessary variance and where more than 50% of cultivars had phenotypes were included in the analysis. Flesh color, color when boiled (base flesh color remaining post boiling), skin texture , eye depth, and maturity were scored visually, and crisp score was assessed visually after frying , while dry matter was calculated by weighing tubers in air and in water. All the historic phenotypic data, where necessary, was converted from a descriptive to a numerical scale. Conversion of phenotypic data to numerical scales, representation of the variance, and description of numerical phenotypes are provided in Tubers from each of the 181 potato cultivars were grown in a pine bark-based potting mix with appropriate nutrients in 6-inch pots under glasshouse conditions. In all cases, the sixth leaf was sampled for an automated 96 well format mRNA extraction. A Dynabead extraction and library preparation method was performed as per Solanum tuberosum Group Phureja DM pseudomolecule (v4.03) assembly (http://broadinstitute.github.io/picard) the resulting sam files were cleaned using cleansam for soft-clipping beyond-end-of-reference alignments and setting MAPQ scores to 0 for reads that were unmapped. Files were then converted to bam files with Picard v2.1.0All data were processed with a single bioinformatic pipeline for SNP discovery. Initial sequence data fastq files were processed through a custom perl script for read quality trimming  and adaptor removal using cutadapt v1.9 . Alignmeassembly  was perfassembly . Using PBam files were initially processed with Picard v2.1.0 to mark and remove duplicate reads. The Genome Analysis Toolkit GATK;  was usedSolanum tubersosum annotation file v4.03 due to overlapping entries.The data were further filtered, and the extraction of all genotypes was processed through R . Variantk-nearest neighbor imputation method as described in k) and 17 closest loci, and these values were previously found to be highly accurate for more diverse outcrossing populations  and observed phenotypic values. A mean prediction accuracy and standard error were then calculated across the five-fold accuracies. Bias in genomic prediction was also calculated by regressing the observed phenotypic value on the all predicted GEBVs and calculating the slope. A slope coefficient of one was taken as representing no bias, while coefficients greater than one represent underprediction.where To further evaluate marker-trait linkages, the same traits in the above section were evaluated for marker effects. Effects were squared for better discrepancy between markers. Data for estimating marker effects were processed using BGLR, and Manhattan plots were generated in R for eye depth and maturity . Final eAn average of 3,108,151 reads per sample was generated from the 181 cultivars, with >79% of all cultivars having over 1M reads and over 89% of the high-quality reads aligned to the reference genome. A total of 3,971,538 SNPs were initially discovered, then following the application of stringent read depth, missing data and minor allele frequency filters, 183,848 high confidence SNPs remained . The higTo obtain genomic variant annotations (defined as assignment of variant function) and ascertain potential functional effects for the complete SNP set identified, all 183,848 high confidence SNPs were annotated using SnpEff to provide the most extensive resource possible . The larThe potato cultivars that were sequenced more than once and other highly related cultivars showed close relatedness . Small clades formed for some usage groups such as French fry, fresh and crisping, for example, the russet samples have a slightly reduced distance (0.26) when compared to the average distance for the complete data. Overall, with the exception of the russet varieties that group together and have the same processing purpose, a lack of population structure is apparent between the 181 potato cultivars, demonstrating broad genetic diversity in the samples chosen. It was also apparent that usage is not a good demarcation of population structure .From the complete sample data set, a total of 169 individuals had phenotypic information generated from the historic breeding program activities  and werePhenotypic data were available for a range of traits, from highly complex polygenic to known single gene genetic control. Consequently, BayesA and BayesB genomic prediction selection models were explored. BayesA accuracies of 0.81, 0.77, 0.46/0.45, 0.49/0.55, 0.42/0.42, 0.37/0.37, and 0.23 were achieved for flesh color, color when boiled, skin texture, dry matter, eye depth, crisp score, and maturity, respectively . For theVarying magnitudes of marker effects were observed across chromosomes. For the trait eye depth , the locThe difficulties of breeding with autotetraploids are well known but not impossible to overcome. Phenotypic selection has long been the chosen method of breeding for potato when improvement to agronomic and quality traits has been targeted. With the release of the potato genome sequence , reducinThis study confirmed the high sequence diversity of potato previously reported . This apThe samples used in this study were selected for its diversity and applicability to the potato industries within Australia and globally. There was an apparent lack of population structure within our germplasm, especially when compared to other studies . HoweverThe GBS-t approach identifies large numbers of variants that do require strict filtering in order to identify those of high quality. This results in a large amount of unusable data. However, the SNP density in potato is sufficiently high so that less than 5% of SNPS can pass these filters and there are still c. 184 K remaining. This volume of variants is more than sufficient for GS as this study has investigated. An additional issue of the GBS-t method in autotetraploid samples would be accurate allele dosage detection and allele specific gene expression (ASE) causing incorrect genotypes which could have implications on genotype accuracy. In this study, cost effectiveness was a consideration, which limited the sequencing depth performed. As a result, absolute accuracy in genotypes, particularly in the heterozygous class, cannot be evaluated. For this, tetraploids and other polyploids require more sequencing depth which in turn, increases the cost per sample. However, the GBS-t approach was being validated with GS to asses its applicability in routine genotyping in breeding programs. Previous studies have identified that the modest error in genotyping c. 10% has limited effect on GS accuracy . While tvia different methods; however, the method of GS does not typically incorporate such values, and it is proposed that all breeding values be genomically predicted . This trend was expected given a higher proportion of the observed phenotype is explained by genetics, rather than environment in those traits with high heritability, such as flesh color. While GS is optimal for highly complex traits, studies have shown that it is still highly effective for traits under simpler control . Predictredicted .The small population size of 169 cultivars in this study likely contributed to the lower prediction accuracy of those highly complex polygenic traits with lower heritability, such as yield. Traits such as these would require a larger reference population to accurately estimate the individual effects of the large number of loci controlling the trait. This is also supported by the underprediction bias observed for these traits, indicating the genetic effect was not fully estimated, likely due to shrinkage, given the small population size. It is recommended that further studies aimed at developing prediction equations for these traits utilize a much larger reference population. Alternative strategies would be to either include a more diverse selection of cultivars in the population to make a more broadly applicable prediction equation or to focus on restricted specific genetics within specific breeding programs for specific targets. It has been shown that populations with a higher degree of genetic variance have displayed better GS accuracies . It shouFrom the evaluation of marker effects, some loci were identified as having larger effects and were concordant with the identified loci previously published. This was shown for traits including maturity and eye depth, where the major locus controlling eye depth has been located on chromosome 10  and the In conclusion, this study validates the application of GBS-t as a method for genome wide genotyping in potato with some advantages over other commonly used genotyping techniques. The GBS-t method has also delivered downstream benefits establishing a comprehensive genomic resource of annotated variants for the community. Functional genomic studies on any trait of interest would also benefit from this approach as the resulting data set enables a more complete characterization of the regions of effect, as well as providing gene expression information. It also enables regions of small effect to be identified in a reduced time frame compared to other classical methods. The results will assist in the application of GS in breeding programs for tetraploid potato which enables greater genetic gain per unit time that has been previously described in BC prepared plant materials, performed RNA extraction, and prepared sequencing libraries. BC, NC, and LP performed data analysis. AS provided the phenotypic data, and BC, FC, BR, AS, and NC conceived the study. All authors read and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Triticum turgidum subsp. durum) farmer varieties (FV) with high adaptability and breeding potential. To harness their unique allelic diversity, we produced a large nested association mapping (NAM) population intercrossing fifty Ethiopian FVs with an international elite durum wheat variety (Asassa). The Ethiopian NAM population (EtNAM) is composed of fifty interconnected bi‐parental families, totalling 6280 recombinant inbred lines (RILs) that represent both a powerful quantitative trait loci (QTL) mapping tool, and a large pre‐breeding panel. Here, we discuss the molecular and phenotypic diversity of the EtNAM founder lines, then we use an array featuring 13 000 single nucleotide polymorphisms (SNPs) to characterize a subset of 1200 EtNAM RILs from 12 families. Finally, we test the usefulness of the population by mapping phenology traits and plant height using a genome wide association (GWA) approach. EtNAM RILs showed high allelic variation and a genetic makeup combining genetic diversity from Ethiopian FVs with the international durum wheat allele pool. EtNAM SNP data were projected on the fully sequenced AB genome of wild emmer wheat, and were used to estimate pairwise linkage disequilibrium (LD) measures that reported an LD decay distance of 7.4 Mb on average, and balanced founder contributions across EtNAM families. GWA analyses identified 11 genomic loci individually affecting up to 3 days in flowering time and more than 1.6 cm in height. We argue that the EtNAM is a powerful tool to support the production of new durum wheat varieties targeting local and global agriculture.The Ethiopian plateau hosts thousands of durum wheat ( Triticum turgidum subsp. durum, genome AB, 2n = 4× = 28) is cultivated worldwide for traditional and high‐value food preparations thanks to its capacity to produce hard grains with remarkable nutritional value. World durum wheat production in 2016–2017 was an estimated 39.9 million tonnes, which is approximately 5% of global wheat production . Nevertheless, durum wheat is a cultural crop of major importance in the Mediterranean basin, Central Asia, Central America, and Sub‐Saharan Africa. For this reason, durum wheat breeding efforts have been aimed at both quality and productivity traits. In the 1910s, the work of Nazareno Strampelli incorporated landrace germplasm with early varieties producing wheat lines that are still grown today . In the second half of the 20th century, Norman Borlaug's work introduced dwarfing genes into wheat breeding germplasm, increasing grain yield in durum elite lines that are still widely employed  gene pool. With its intermediate tetraploid karyotype, durum wheat may be crossed with wild wheats to incorporate their haplotypes in a cultivated background  may also be used to accelerate the introduction of novel diversity into wheat breeding through durum wheat  families that share the recurrent founder haplotype. The NAM design was developed in maize , is essential to achieve predictive wheat improvement. However, limitations in allelic diversity and genetic mapping resolution of currently available durum wheat mapping populations pose a limit to the capacity to pinpoint genes responsible for traits of agronomic interest. While diversity panels employed in genome‐wide association (GWA) studies increase the diversity available to search for causative alleles of interest, their historical genetic recombination history is unknown and does not allow the elevated statistical power of classical pedigree‐based haplotype mapping  released for cultivation in the country as demonstrated by their genotypic  population built by intercrossing of a MV with international background with 49 Ethiopian FVs and one Italian MV, which were chosen on the basis of their phenotypic and molecular diversity. A subset of 12 EtNAM families, 100 RILs each, was genotyped with 13 000 single nucleotide polymorphisms (SNPs), allowing the characterization of the genetic diversity and structure of the population. We conclude by conducting a GWA study for phenology and plant height, reporting a number of loci accounting for a relevant amount of the phenotypic variation in the EtNAM. We make the population available to the wheat community, and argue that the EtNAM may act as a breeding tool and a research tool that supports the identification of QTL of agronomic relevance and their incorporation into pre‐breeding materials.9 RILs divided into 50 families , 12 114 markers were retained. The EtNAM families in the subset have a number of polymorphic SNPs ranging from 1665 (EtNAM 51) to 4110 (EtNAM 3), with a mean of 3479 Table . Family The molecular diversity in the EtNAM subset overlaps the breeding scheme of the population Figure . The MVsFollowing the filtering parameters listed in the Methods, 9383 SNP markers were physically mapped to the A and B genome of wild emmer wheat Table . DiversiP‐values, computed with a Bonferroni multiple test correction, is 13. Family EtNAM 5 presents the highest number and extension of such regions on Chr 2A, 2B, 3A, 4B, and extensively on Chr 5B and Chr 6A. Some regions of biased inheritance of parental allele are present in more than one EtNAM family, including regions at the beginning of Chr 2B , on Chr 3A , on Chr 3B , and on Chr 4A . In some of these genomic bins, the founder unbalance is contrasting among families, with some showing an overrepresentation of the RF and others showing overrepresentation of the alternative founder line  to 18.7 Mb (Chr 3B)  decay is different across EtNAM families, and Chr 3B, 4B, and 6B show the slowest decay in several of them Figure . Overall) Figure a. Perice) Figure . The est) Figure . Areas o) Figure .2) was 90.1% for DB, 89.5% for DH, 60.8% for DM and 59.5% for PH. Best linear unbiased prediction (BLUP) values were derived from trait values and were used to conduct a GWA study to identify quantitate trait nucleotides (QTNs), i.e. SNPs in linkage with causal variants underlying the trait variation in the EtNAM  and plant height (PH) Figure . The broAM Table . The EtNld Table . QTNs fold Table . PH QTNset al., et al., The founder lines contribute with broad phenotypic and molecular diversity to the EtNAM population. Having been chosen from the most divergent genotypes of Ethiopian FVs Figure , they coet al., exotic alleles available in Ethiopian FVs . The accessions deriving from the EBI (5 g each) were purified during 2011, selecting one representative spike each , F2 progenies were produced by subsequent selfing of F1 plants under controlled conditions during 2013. Four additional cycles of controlled selfing were carried out starting with 252 single plants per F2 family and eventually producing RIL‐F6 in 2015 by growing two cycles per year when possible. During the inbreeding phase we followed an SSD approach by harvesting only the main tiller and planting only three seeds per line starting from F3. EtNAM RILs were coded with an increasing number from 1 to 252 attached to the NAM family number . There are currently gaps in the numeric series corresponding to genotypes lost during the SSD procedure due to infertility or stochastic reasons. The data reported in this manuscript are based on the F6 generation. When this manuscript was written, the EtNAM was at the RIL‐F9 stage. All the crosses were conducted at the Sirinka Agricultural Research Center, Sirinka, Ethiopia. The EtNAM lines are stored at the Amhara Regional Agricultural Research Institute (Ethiopia) and are available to researchers upon signing a material transfer agreement, free of charge for public institutions.Fet al., The EtNAM comprises 6280 RILs. A subset of EtNAM families was selected to provide an initial characterization of the population. Twelve EtNAM families were selected because of genetic diversity, traits of interest, and RILs availability. For each of the 12 EtNAM families in the subset, 100 RILs were germinated in an open field, five seedlings each, at the Sirinka Agricultural Research Center, Sirinka, Ethiopia. When they reached sufficient biomass, seedlings for each RIL were harvested and pooled in equal amounts. Green tissues were oven‐dried at 45°C for 72 h, and later maintained in sealed bags with silica gel beads . Samples were shipped to Italy and genomic DNA was extracted with the GenElute Plant Genomic DNA Miniprep Kit  following the manufacturer's instructions. DNA was checked for quantity and quality by electrophoresis on a 1% agarose gel and NanoDrop 2000 . Genotyping was performed on the Infinium 15K Ultra HD chips, derived by subsampling a set of markers optimized for durum wheat from the wheat 90K SNP chip , DH (days to 50% heading), and DM (days to full maturity). PH was measured on three random plants per plot, in cm. BLUPs were calculated for all traits with restricted maximum likelihood (REML) analysis using SAS ® . A mixed model describing the variate Yijk from replication i, column j, and row k, can be expressed by the following equation (Eq. 1): Ga, the genotype main effect; Lb, the locations main effect; GLab, the interaction effect between genotypes and locations. The random model terms are: bi, the effect of replication b; cj, the effect of columns within replication b; rk, the effect of rows within replication b and ∈ijk, the overall error terms. The overall error term was estimated from the random terms of replications, rows and columns where rows and column were nested in replication. h2 across the three locations was calculated from variance estimated for each component by REML analysis in SAS ®  according to Vargas et al.  in 2016, Geregera (11°40′N/38°52′E) in 2016, and Kulumsa (8°01′N/39°09′E) in 2017. Seed rate of 100 kg/ha and fertilizers rate of 92 N + 46 Pgithub.com/XiaoleiLiuBio/MVP) with custom scripts. The estimation of variance components was performed using the EMMA method  decay as a function of physical distance in each of the EtNAM families.Click here for additional data file.Figure S7 Pairwise linkage disequilibrium (LD) in the EtNAM subset.Click here for additional data file.Figure S8 Linkage disequilibrium (LD) evolution in the EtNAM subset.Click here for additional data file.Figure S9 Phenotypic distribution of days to booting (DB), heading (DH), maturity (DM), and plant height (PH) in the EtNAM population subset.Click here for additional data file.Table S1 Founders of the EtNAM population.Click here for additional data file.Table S2 Genotyping statistics for the EtNAM subset.Click here for additional data file.Table S3 Physical position of markers on wild emmer genome.Click here for additional data file.Table S4 Distortion in founder contributions to EtNAM RIL families.Click here for additional data file.Table S5 Outcome of the GWA study conducted on phenology traits and plant height.Click here for additional data file.Table S6 Gene models within ± 500 Kb of QTNs.Click here for additional data file. Click here for additional data file.


With the increasing amount of high-throughput genomic sequencing data, there is a growing demand for a robust and flexible tool to perform interaction analysis. The identification of SNP-SNP, SNP-CpG, and higher order interactions helps explain the genetic etiology of human diseases, yet genome-wide analysis for interactions has been very challenging, due to the computational burden and a lack of statistical power in most datasets.The wtest R package performs association testing for main effects, pairwise and high order interactions in genome-wide association study data, and cis-regulation of SNP and CpG sites in genome-wide and epigenome-wide data. The software includes a number of post-test diagnostic and analysis functions and offers an integrated toolset for genetic epistasis testing.https://CRAN.R-project.org/package=wtest.The wtest is an efficient and powerful statistical tool for integrated genetic epistasis testing. The package is available in CRAN:  The etiology of complex disorder involves an interplay of polygenic biomarkers, lifestyle and environmental factors . Robust p-value bias caused by complicated genetic architectures. Flexible implementation options are provided. The package can calculate SNP-CpG epistasis for biomarkers located in physical proximity of the input genome and epigenome. A number of post-test diagnostic, visualization and statistical genetic analysis functions are provided for model diagnosis. This is the first statistical software providing functions for direct gene-methylation interaction and high-order interaction evaluations in genome and epigenome dataset.Here we introduce a software that provides estimations for different types of genetic associations, including the main effect, second or higher order interaction, and gene-methylation interaction. This package is built upon the W-test  to perfok by 2 contingency table can be formed, where k is the number of non-empty category combination formed by the SNP-set, and 2 is the binary phenotype. The statistic tests for the existence of distributional difference of a subset in the case group from a comparison control group, and it takes the following form,The wtest package is based on the W-test  to measuni1 and ni0 are the number of cases and controls in the ith cell of the contingency table; N1 and N0 are the total cell counts of cases and controls; ith cell of the contingency table; and SEi is the standard error of the ith log odds ratio. The W-test follows a chi-squared distribution of f degrees of freedom. The scalar h and degree of freedom f take forms of covariance matrices of the log odds ratios and are estimated from bootstrapped samples under the null hypothesis by the large sample theory. The W-test inherits a data-set adaptive degree of freedom that absorbs the genetic variation not attribute to phenotypes, therefore robust to complicated genetic architectures. In this software, we further extend it to evaluate high-order interaction effect and gene-methylation interaction effect. For gene-methylation interaction, methylation data are clustered into two categories according to high and low methylation levels by two-mean clustering algorithm. We also use a novel triangular network diagram to display interaction effects up to the third order. Extensive simulation studies testing the power and type I error of the W-test can be found in Wang, Sun et al. (2016) [where entclass1pt{minima. (2016) .h and f; (2) Testing by the W-test. Step 1. Estimation of parameters h and f. In genotype data, the hf function is called, and in genotype and methylation data, the function hf.snps.meth is called. Parameter h is the scaler in Eq. (f is the degrees of freedom of a chi-squared distribution of the W-test. The two parameters are estimated using bootstrap samples with permutated phenotypes (null hypothesis) for B times. Simulations suggest that the estimation converges at B>400 when the number of variables is 1000 and the number of subjects is 1000  and f=k−1. In this case, k is the integer categorical combinations formed by the marker set. When k=2, the W-test is equivalent to the odds ratio test for a 2-by-2 table.Figure wtest evaluates main and second order interaction and wtest.high evaluates third or higher order interaction in genotype data. The wtest.snps.meth calculates SNP-CpG interactions for genome and epigenome data. Oftentimes users are interested to explore the interactions among biomarkers with a certain level of main effect signals. The input.pval option in the function can be used to screen candidate SNPs according to their p-values to form interaction sets. While the output.pval option allows the convenient output of interaction sets reaching a p-value threshold. In function wtest.snps.meth, positions of the biomarkers are input alongside the genome and epigenome data sets, and the window size to calculate cis-regulation relationship can be specified. The methylation.recode function transforms the methylation data into high and low methylated levels. For high order interaction calculation, a simple check for sample size can be done by estimating the average number of cell counts formed by a set, and a high order is feasible if the number is at least two. A reference table could be found in Additional file Step 2. Testing by the W-test. The w.diagnosis, which plots the W-test statistics histograms from the observed data and the curve of the chi-squared distribution using estimated parameters, indexed by the number of categorical combinations k. Close overlaying of the densities indicates the goodness of fit of estimation. An example is shown in the real data application section. The w.qqplot function assists the diagnostic of probability distribution and degree of population stratification.Diagnostic checking for test statistic distribution can be performed by The software is applied to a number of real data analysis with novel biomarker findings and interesting implications –9. Here h and f estimation can be found in Additional file order=2), setting input.pval=0.001 and output.pval=0.001, the function would output second order epistasis marker pairs with p-value <0.001. Figure w.diagnosis function. The estimated red color chi-square curve follows closely with the histogram of the test statistics calculated from the observed data, showing a good estimation of the parameters.This data contains 653 bipolar disorder patients and 1767 healthy controls, and 46,181 SNPs of chromosome 6 . The resGRM4, p-value =2.06×10−7. GRM4 is a major excitatory neurotransmitter in central nervous system and it is a susceptible gene for bipolar disorder and schizophrenia [SYNE1, BTBD9, RPL12P2) in the middle of the plot, in which BTBD9 plays a key role and extends to form significant combinations with FGD2 and CDKAL1. The BTBD9 is reported to be associated with neuropsychiatric disorders such as restless legs syndrome in Schizophrenia and the Tourette Syndrome [BTB/POZ domain-containing protein that involved in protein-protein interactions [Data analysis identified one SNP with significant main effect: rs2495982 near ophrenia , 12. ForSyndrome , 14. Theractions , and is ractions . It is vet al 2018 [window.size=10,000). The phenotype is whether or not a subject responded to the treatment, calculated by comparing the before and after treatment triglyceride levels [h and f are estimated by hf.snps.meth, and the gene-methylation interactions are calculated by wtest.snps.meth. Table MPPED2 [GUCY2E [This application was originally reported in Sun  al 2018 . The date levels . The h ae MPPED2  and GUCYB=200, n.sample=1000, the time elapsed for estimating h and f is 40.5s. After h and f calculation or assuming default values, the time used to evaluate main effects is 0.04s, and took 1.69s for second order interaction. In the same environment, the running time for existing tests for interaction yields 36.41s by chi-squared test and 130.56s by logistic regression. In the real data set, the genome-wide main effect calculation on 5000 subjects and 500,000 SNPs took around 5 min; and second order interaction calculation on 8000 SNPs used around 3.5 h.The speed of the wtest package is evaluated on a laptop computer of 1.6GHz Intel Core i5 processor and 4GB RAM. Simulation data are used to compare the speed of different methods. On a data set consists of 5000 subjects and 100 SNPs, when https://CRAN.R-project.org/package=wtest under the GPL-2.0 license.Genetic epistasis testing is important to fathom the massive genomic data, and it also provides a way to explore the relationship between diseases and various types of biomarkers. This package offers an integrated toolset to analyse the association of genetic signals at all levels: from main effects, high order interactions, to gene-methylation interactions. The software is available in CRAN from Project name: wtestProject home page: https://CRAN.R-project.org/package=wtestOperation systems: Platform independentProgramming language: R (>= 3.1), C++License: GPL (>= 2)Restrictions to use by non-academics: NoneAdditional file 1 Convergency simulation study. The coefficient of variance of h at different B for pairwise interactions. Simulated dataset contains 1000 subjects and 1000 SNPs. A convergent h and f estimation is reached at B>400.Additional file 2 Reference table of sample size estimation. When the averaged MAF is 0.3 and the sample size is greater than the estimated sample size, no more than 25% cells have averaged cell count less than 2 in the contingency tables.Additional file 3h and f estimation for main effects, second order interaction, and third order interaction analysis.Additional file 4 Top three identified sNPs at different levels of interaction orders. Note: Bonferroni corrected significant thresholds: main effect p-value <1.1×10−6, second order interaction p-value <4.69×10−11, and third order interaction p-value <3.05×10−15. Gene: the gene located within 35kb of the identified SNPs.


A) and intrinsic water use efficiency (WUE) are physiological traits directly influencing biomass production, conversion efficiency, and grain yield. Though the influence of physiological process on yield is widely known, studies assessing improvement strategies are rare due to laborious phenotyping and specialized equipment needs. This is one of the first studies to assess the genetic architecture underlying A and intrinsic WUE, as well as to evaluate the feasibility of implementing genomic prediction. A panel of 383 soybean recombinant inbred lines were evaluated in a multi-environment yield trial that included measurements of A and intrinsic WUE, using an infrared gas analyzer during R4–R5 growth stages. Genetic variability was found to support the possibility of genetic improvement through breeding. High genetic correlation between grain yield (GY) and A (0.80) was observed, suggesting increases in GY can be achieved through the improvement of A. Genome-wide association analysis revealed quantitative trait loci (QTLs) for these physiological traits. Cross-validation studies indicated high predictive ability (>0.65) for the implementation of genomic prediction as a viable strategy to improve physiological efficiency while reducing field phenotyping. This work provides core knowledge to develop new soybean cultivars with enhanced photosynthesis and water use efficiency through conventional breeding and genomic techniques.Photosynthesis ( Although a positive relationship between photosynthesis and yield is not always observed  from 32 families (12 RILs per family) coming from a subset of the Soybean Nested Association Mapping (SoyNAM) panel  were ass-1 . The pan-1 . The NAM-2. In 2017, 66 RILs were discarded as consequence of non-uniform emergence. In 2018, the same experimental design was planted in two locations: ACRE (ACRE_2018) and Romney, IN , with data collected from 382 and 368 RILs, respectively. Soil types included Chalmers silty clay loam (Typic Endoaquolls) and Raub-Brenton complex (Aquic Argiudolls) for ACRE and Drummer soils (Typic Endoaquolls) for Romney (Kc) were obtained from Experiments were grown in one location in 2017 and two locations in 2018. An alpha lattice incomplete block design with 383 RILs, two complete replications and 32 incomplete blocks per replication was planted at the Purdue University Agronomy Center for Research and Education – ACRE (40°28′20.5″N 86°59′32.3″W) in 2017 (ACRE_2017). Experimental units corresponded to six rows plots (0.76 m × 3.35 m) with a seeding rate of 35 seeds mr Romney . High nar Romney , along wr Romney . Absencer Romney –S6. FAO r Romney , while t2 concentration, water and nitrogen status, and temperature (-2 s-1) was confirmed in different random selected cultivars and subsequently set as constant for the measurements. The LED light source within the 6 cm2 chamber was used. To control other variables affecting the gas exchange, CO2 concentration and temperature were also set as constant at 400 μmol mol-1 and 25°C, respectively. The relative humidity was restricted to 75 ± 10%. To avoid non-adapted reading as consequence of significant differences between the external environment and the chamber, each leaf was previously adapted for at least three minutes or until getting stable readings. Outlier determination following the criterion 1.5 × interquantile range (IQR) were carried out and atypical values caused by non-adapted leaves were eliminated. Normality in the data set was confirmed through histograms , COperature . To accoperature . The constograms .gs) in field under comparable conditions. Finally, the rows four and five of each plot were mechanically harvested and weighted for yield calculation. Moisture for grain yield (GY) was standardized to 13% and extrapolated to hectare.The gas exchange parameters were measured before the seed filling phenological period, from late R4 and early R5 , in the finhap f90 (pb) were used. Quality control for minor allele frequency (MAF < 0.15)  markers . These mnhap f90 , the seg < 0.15)  was perf < 0.15) .Data collected were consolidated and analyzed using the mixed model approach in the software R through the package “lme4” . SourcesY is the vector of phenotypes measured with the ith equipment in the jth environment into the kth block, μ is the intercept, α accounts for the effect of the covariate equipment, β corresponds to the effects of environment, δ accounts for the block effect, βγ corresponds to the interaction environment × block, δ accounts for the genetic effect, and e controls the error. The covariate was treated as a fixed effect while the other sources of variation were considered as random. Given the limitations in humidity control offered by the LI-COR 6400XT and the range set in this research for relative humidity, leaf vapor pressure deficit (VPDL) reported by the equipment was also used as covariate in the model fitted for WUE.where fam in the function gwas2. The base model for the genome scanning is described by:A genome wide association analysis under the empirical Bayesian framework was performed using the R package “NAM” . In eachy corresponds to the BLUPs values, Zu is the incidence matrix of haplotypes generated from marker data, μ is the vector of regression coefficients of within-family marker effects, g corresponds to the polygenic coefficients accounting for population structure, and e is the vector of residuals. Statistical significance of each marker was calculated using the likelihood ratio test (LRT) between a full model including the marker, Zu, and a reduced model without marker. The association between markers and traits was evaluated using a corrected p-value threshold of 0.0002. This threshold corresponds to the p-value of 0.01 divided by the number of unique segments (58), estimated as the number of significant eigenvalues computed from the spectral decomposition of the genetic relationship matrix. The R package CM-plot was used to create the Manhattan plots (where an plots . To discan plots . The expan plots .H) on an entry mean basis and plot basis was calculated from model (1) using the variance components from the mixed model in the Eq. 1. Heritability on an entry mean basis was calculated through the equation below  was also calculated from different whole-genome regressions using the expectation-maximization restricted maximum likelihood method from the “NAM” package .To determine the genetic correlations, a multivariate mixed model using the restricted maximum likelihood (reml) approach was solved using as response variable a matrix with the BLUPs values for  package  was usedp-values from 0 to 2 from the genomic information already described, a set of whole-genome regressions were computed using the Bayesian framework , Bayes ridge regression (BayesRR), BayesCpi and BayesDpi (emBA), BayesB (emBB), BayesC (emBC), BLASSO (emBL), BLASSO2 (emDE), maximum likelihood (emML), and Bayesian ridge regression (emRR) were also fitted. Five-fold cross-validation was implemented splitting the data set randomly in proportions 80 (training): 20  each time. Correlation coefficients between measured and predicted breeding values in the validation set were calculated each time. Function mcmcCV and emCV from the R package bWGR were used to perform the cross-validations.Using the SNPs above the significance threshold of -log ramework . Seven wramework : BayesA,BayesDpi . LikewisA and WUE  among environments were detected, with means of 26.9, 27.3, and 26.6 μmol CO2 m-2 s-1 for ACRE_2017, ACRE_2018, and RMN_2018, respectively  among environments were detected, with means of 20.2, 17.1, and 22.7 μmol CO2 mol-1 H2O for ACRE_2017, ACRE_2018, and RMN_2018, respectively  between predicted and observed values. Maximum correlation for A , the variable selection methods BayesB and BayesC displayed the lowest correlation through the different SNPs data sets. BayesB reported 22–27% less predictive ability compared with the best methods, while BayesC exhibited 24–30% lower correlation. Similar results were also found when the same set of models was fitted through EM regressions  might be possible. Intrinsic WUE here documented is similar to previous reports of 2 mol-1 H2O) with the maximum attainable (54.4 μmol CO2 mol-1 H2O), it is hypothesized until 34 μmol CO2 (1.5 mg CO2) extras can be fixed with the same rate of H2O efflux through the stomata. Better understanding in genetic basis of stomatal control seems to play a key role to reduce the gap between actual and potential WUE in soybean H H . In plan NAD(P)H . Though  NAD(P)H , there in CYP1A1 . Inductin CYP1A1 . Likewisn CYP1A1  and CYP7n CYP1A1  were carn CYP1A1 .The other association found on the chromosome 15 was linked to a gene encoding for NADH oxidoreductase-related. This type of protein catalyzes the oxidation of NADH and the reduction of other compound . The mosWUE was associated with regions on the chromosome 4, 7, 15, and 16  and Glyma04g41150   . A carbo cytosol . In C3 ppiration .WUE that we observed on chromosome 7 at the position 29855554 bp is associated with a trehalose-6-phosphate synthase. Trehalose is a disaccharide present in bacteria, fungi, and invertebrates linked with abiotic stress tolerance given its role as energy source, compatible osmolyte, and protein/membrane stabilizer . The additive-by-additive matrix improved the heritability in the range of 11–34% when the number of SNPs was the lowest  but it did not show substantial improvement when -log p-value > 1.5 matrix was considered. Introducing epistatic effects might help to explain genetic effects when the number of markers is reduced. The fact heritability is relatively constant in the interval -log p-value > 0.0 to -log p-value > 1.0 but decreases when data set becomes smaller aligns with previous reports for flowering date, height and nodule number in alfalfa  that agrees with previous reports summarized by A may positively affect grain yield.We observed a lack of phenotypic correlation between p-value is higher than 1.5 and 2.0, the reduction in predictive ability for none of the traits was higher than 20%. The use of selected sets of SNPs for genomic prediction was previously reported in crops and animals. In eucalyptus, for instance, similar predictive ability between large  and reduced  SNPs data sets was reported by p-value ≥ 1.0) is an appropriate and consistency approach to predict photosynthesis and intrinsic WUE across most of the methods considered. Although the implementation of different methods for genome regression in soybean yield and protein did not show significant improvement (BayesDPi showed a better performance for the traits evaluated here. The fact Bayes B and BayesC had shown lower predictive ability suggests that variable selection did not favor the predictive ability for these physiological traits. Contrarily, a quality control based on the significance of the genome-wide association results was beneficial.The complexity of field measurements, the limited variance explained by significant SNPs, and the moderate to low heritability for these traits, suggest genomic prediction-selection as suitable methodology to approach their breeding . This worovement , the metWUE indicate these traits can be improved through breeding strategies. New technologies like genomic selection-prediction arises as valuable approach to overcome the phenotyping bottleneck in gas exchange. Pre-selecting SNPs for genomic prediction modeling based on the significance of associations can benefit the predictive ability. Improving photosynthesis through breeding techniques is a viable strategy to increase yield in soybean, considering their high positive genetic correlation.The existence of natural diversity and a preliminary genetic architecture for photosynthesis and intrinsic SoyBase.org, https://www.soybase.org/SoyNAM/.The datasets generated for this study can be found in ML and KR conceived and designed the experiments. ML and AX conducted the data analysis and interpretation. ML wrote and edited the manuscript. KR coordinated and supervised the research, and acquired funding.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.