Social genetic effects (SGE) are an important genetic component for growth, group productivity, and welfare in pigs. The present study was conducted to evaluate i) the feasibility of the single-step genomic best linear unbiased prediction (ssGBLUP) approach with the inclusion of SGE in the model in pigs, and ii) the changes in the contribution of heritable SGE to the phenotypic variance with different scaling ω constants for genomic relationships.xx, 0.1 to 1.0; with a 0.1 interval) were scaled with the genomic relationship for best model fit with Akaike information criterion (AIC).The dataset included performance tested growth rate records (average daily gain) from 13,166 and 21,762 pigs Landrace (LR) and Yorkshire (YS), respectively. A total of 1,041 (LR) and 964 (YS) pigs were genotyped using the Illumina PorcineSNP60 v2 BeadChip panel. With the BLUPF90 software package, genetic parameters were estimated using a modified animal model for competitive traits. Giving a fixed weight to pedigree relationships (τ: 1), several weights (ωT2) were mostly higher with ssGBLUP than in the pedigree-based analysis. The model AIC value increased with any level of ω other than 0.6 and 0.5 in LR and YS, respectively, indicating the worse fit of those models. The theoretical accuracies of direct and social breeding value were increased by decreasing ω in both breeds, indicating the better accuracy of ω0.1 models. Therefore, the optimal values of ω to minimize AIC and to increase theoretical accuracy were 0.6 in LR and 0.5 in YS.The genetic variances and total heritability estimates (In conclusion, single-step ssGBLUP model fitting SGE showed significant improvement in accuracy compared with the pedigree-based analysis method; therefore, it could be implemented in a pig population for genomic selection based on SGE, especially in South Korean populations, with appropriate further adjustment of tuning parameters for relationship matrices. The genetic effect of an individual on the phenotypes of its social partners  is often termed the social genetic effect (SGE) or the indirect genetic effect . The groIn recent years, a single-step genomic BLUP (ssGBLUP) which provides better predictions for all animals in a pedigree , is a meThe phenotypic dataset on the growth rate of animals was obtained through performance tests of Yorkshire  and Landrace  pigs between 2005 and 2015. These pigs were born and raised in a closed nucleus (breeding) farm in South Korea. Animals were reared in pens where each pen housed 4 to 10 YS or 3 to 8 LR pigs of the same sex. Each group of animals comprised 1 to 7 YS or 1 to 6 LR full-sibs. The performance evaluations on the ADG of pigs started soon after each animal reached a live body weight of 30 kg, and were finished when a target weight of 90 kg was attained. On average, fewer than 160 days were required to attain this target weight. The animal phenotypes selected to be studied were based on the size of the group (frequency >10%). The average ADG was recorded to be 787.9±88.3 g/d in LR and 792.5±92.1 g/d in YS.Complete pedigrees of the studied animals were obtained from the Korean Animal Improvement Association database. The total numbers of animals in the pedigree of the two breeds were 14,013 (LR) and 22,536 (YS). The numbers of individuals for which both parents were known were 13,916 (LR) and 22,324 (YS). In the whole pedigree, about 96% (LR) and 62% (YS) of the animals were inbred. The average inbreeding coefficients were 0.035 and 0.015 in LR and YS, respectively. The ranges of inbreeding coefficients were 0.0001 to 0.263 (YS) and 0.001 to 0.274 (LR). The observed average family sizes were 3.97 and 4.11 in LR and YS, with ranges of 2 to 15 and 2 to 17, respectively. The population structures of these breeds were determined using the CFC v1.0 software package [2/pig) in size with solid concrete flooring. Pigs were fed ad libitum and water was constantly accessible through nipple drinkers. The feeding program was applied in accordance with pig testing standards of the Korean Animal Improvement Association (http://www.aiak.or.kr/eng/index.jsp).The experimental protocols describing the management and care of the animals were reviewed and approved according to the guide for the care and use of laboratory animals  on 7 March 2014 . The management practices on the studied population stated that each pen was 2.5×3.6 m  and YS 964) breeds were genotyped using the Illumina PorcineSNP60 v2 BeadChip panel, which comprised 61,565 SNP (single nucleotide polymorphism) markers . The qua64 breedsclassic) and a social model pedigree relationships only modified for competitive traits  [The variances and covariances of the studied traits were estimated by an animal model applying the REML approach. The effects of batch, sex, and group size were fitted as fixed effects. In the model, age at target weight was fitted as a covariate effect. The effect of batch was formed by combining year, month, and week based on each performance test. The models also included two nongenetic random effects, namely, birth litter and group effects ,11. To tze-1)0.5 , was addze-1)0.5 ,11,13. Aze-1)0.5  was usedy is the vector of observations (ADG), b is the vector of fixed effects, aD is the vector of random direct additive genetic effects, aS is the vector of random SGEs, l is the vector for random birth litter, g is the vector of random group, and e is the vector of residuals. X, ZD, ZS, W, and V are the corresponding incidence matrices. Assumptions for the probability distributions were I is an identity matrix of appropriate dimensions; and A is the numerator relationship matrix and C is defined by the matrix σDaSa is the covariance between direct and indirect genetic effects, and C⊗A denotes the Kronecker product of two matrices.where ith animal is defined as follows:According to Bijma et al  for traiaD,i) on its own phenotype and its  on the phenotypes of its n – 1 group mates. Bijma et al [The TBV is the heritable effect of an individual on trait values in the population, which is the sum of its direct genetic effect −1 – ω A22−1, given to the genotyped animals [1.0, ω0.9, ω0.8, ω0.7, ω0.6, ω0.5, ω0.4, ω0.3, ω0.2, and ω0.1, in accordance with their values of the constant ω, whereas a model with pedigree information only is denoted as PED in later sections.where VanRaden . Both A2 animals ,18. The ith individual with the mth model was calculated as follows:Accuracy of breeding value was calculated in two different ways  for the validation pigs. Accuracy was defined aswhere yc is corrected ADG for fixed effects.where CBV is the sum of pig’s own direct breeding value and SBVs of pen mates, 0.6 and ω0.5 in LR  were mostly higher with ssGBLUP than in the pedigree-based analysis . The best single-step models (ΔAIC = 0) showed larger estimates of direct and social variances than pedigree-based methods, and thus also larger covariance estimates, resulting in higher total heritability estimates with those models. The T2 estimates with the best fitting models were 0.54 and 0.80 in LR and YS, respectively. They were also greater than those of the pedigree-based analysis method by 0.13 and 0.22 in these two breeds, respectively. However, our T2 estimates for LR with the ω10 model coincided strongly with those of Bergsma et al [T2 estimates also exhibited the same trend. Note that even when the social variance is markedly smaller than direct genetic variance, its contribution to 2, especially when group sizes are large, as was the case with YS. The lower T2 estimates in LR could also be due to the larger nongenetic litter effects and negative covariances between direct and social effects. According to Bijma et al [2 pen) was lower than that of YR (4 to 10 pigs/9 m2 pen).The genetic variances and total heritability estimates . The ω1.0 models also performed poorly in DBV prediction . Among the ssGBLUP models, the theoretical accuracy of DBV was increased by decreasing ⌐ω in both breeds . The best fit models based on AIC exhibited an increase of accuracy by 5% to 8% compared with the ω1.0 models in both breeds. The ranges of SBV accuracies with the PEDsocial in LR and in YS were 0.16 and 0.31, respectively. Similar to DBV, both PEDsocial and ω1.0 models performed poorly in SBV prediction. However, unlike the DBVs from the single-step methods, the best fitting models exhibited notable increases in SBV accuracies by 39% (LR) and 19% (YS) with ω0.6 and ω0.5, respectively, compared with each of the breed’s worst fit (ω1.0) model. In cross validation, the correlations between CBV and corrected phenotype were also mostly higher with ssGBLUP than in the pedigree-based analysis. However, there were little differences among the ssGBLUP models. The ranges of correlations between CBV and corrected phenotype in LR and YS were 0.31 to 0.33 and 0.21 to 0.22, respectively. The correlative prediction methods showed more variability in terms of ranking of models across traits and replicates so care should be taken interpreting these results with small sample sizes [H matrix.le sizes . Putz etle sizes  also sugle sizes . Therefole sizes  reportedThe phenotypic variability of some traits that are expressed in the social environment could be significantly influenced by SGEs. Earlier reports on such traits, for instance, social dominance or aggressiveness, also suggested that SGEs can substantially influence total phenotypic variability –32. The A22 matrix also differed between the best fitting models for the LR and YS breeds. But it was clear that the models with ω of 1.0 in the H matrix were the worst fitting. Our study also indicated the possibility of further adjustment of other model parameters  in the H matrix to reduce inflation of the estimated breeding values. Our results also indicated the value of further analysis with a greater sample size to obtain a more robust estimation of breeding values. We believe that our results provide useful insights for future modeling of SGE in the genomic selection of pig breeds, especially in South Korea.For SGEs, our study showed greater improvement in parameter estimates through ssGBLUP over the traditional pedigree-based method. Both breeds differed to some extent for their estimated parameters. The value of ω used for adjusting


Selection of an appropriate statistical significance threshold in genome-wide association studies is critical to differentiate true positives from false positives and false negatives. Different multiple testing comparison methods have been developed to determine the significance threshold; however, these methods may be overly conservative and may lead to an increase in false negatives. Here, we developed an empirical formula to determine the statistical significance threshold that is based on the marker-based heritability of the trait. To develop a formula for a significance threshold, we used 45 simulated traits in soybean, maize, and rice that varied in both broad sense heritability and the number of QTLs.10 (P)-values. For all species, the threshold –log10 (P)-values increased as both marker-based and broad-sense heritability increased. Higher broad sense heritability in these crops resulted in higher significant threshold values. Among crop species, maize, with a lower linkage disequilibrium pattern, had higher significant threshold values as compared to soybean and rice.A formula to determine a significance threshold was developed based on a regression equation that used one independent variable, marker-based heritability, and one response variable, − logOur formula was less conservative and identified more true positive associations than the false discovery rate and Bonferroni correction methods.The online version of this article (10.1186/s12864-019-5992-7) contains supplementary material, which is available to authorized users. Linkage mapping (LM) and genome-wide association studies (GWAS) are the two most popular methods to decipher genetic architectures of complex traits in crops . With adA major problem in GWAS is false positives that arise from population structure and family relatedness. Several statistical models have been developed to control false positives in GWAS. Mixed linear model (MLM) has become the most popular approach with the ability to consider population structure and family relatedness , 4. SincP-value threshold for statistical significance is critical to differentiate true positives from false positives and false negatives. To determine the statistical significance threshold in GWAS, different statistical procedures accounting for multiple testing have been proposed, including the Bonferroni correction, Sidak correction, False Discovery Rate (FDR), permutation test, and Bayesian approaches. Bonferroni correction and FDR [Determining the correct  and FDR –15 are t and FDR .P-value due to the assumption that every genetic variant tested is independent of the rest. The False Discovery Rate controls the expected proportion of false positives among the rejected null hypotheses and is a popular, less conservative approach compared to the Bonferroni correction [Zea mays L.), should be more stringent than a population with higher LD pattern, such as soybean (Glycine max L.) or rice (Oryza sativa L.), as the number of independent markers tends to be greater in maize than soybean. The LD decay rate (r2 = 0.25 level) was much greater in maize (1 kb) [The Bonferroni correction method is considered the most conservative method for selecting a threshold rrection . Howeverrrection . An imbarrection . Additioe (1 kb)  than soye (1 kb) –21. or re (1 kb) . Therefo10 (P)-values, and vice versa for a complex trait with low heritability.As trait complexity increases, the number of loci affecting the trait increases along with environmental interactions with an expected decrease in heritability. Conversely, for less complex traits, fewer loci affect the trait, there is less interaction with the environment, and there is an expected increase in heritability. For a trait with a high heritability, the threshold value for significance of associating loci with a trait would have high – logHere, we develop an empirical formula to determine the statistical significance thresholds that is based on the marker-based heritability of the trait. The objective of this study was to develop an empirical formula that can determine the statistical significance thresholds for GWAS using a large number of simulated phenotypes that varied in heritability and the number of QTLs for soybean, maize, and rice. These crops were selected because of differences in LD pattern with maize having a lower LD pattern compared with soybean and rice. The phenotypes were simulated and associated with freely-available SNP marker datasets for all these crops.In this study, we developed a method to determine the significant threshold value for GWAS using the 45 simulated phenotypic traits that varied in both the broad sense heritability and the number of QTLs in three crop species that differed in their LD patterns. We repeated the simulation of these traits 10 times so that simulated QTLs were randomly assigned to different parts of the genome in order to obtain unbiased results.10 (P)-values  -values for all three crops (Table 10 (P)) also increased among the crop species for these simulated traits as the LD decreased. Specifically, maize had higher significant threshold (−log10 (P)) values as compared to soybean and rice for simulated traits when they had more than 50% broad sense heritability ) value as the response variable in the multiple regression analysis, we obtained an equation for determining significant threshold values in GWAS for each crop. We observed that marker-based heritability showed a significant effect on the response variable (P < 0.05) , a was the intercept, and b was the slope of the regression coefficient for the marker-based heritability (X) in maize, soybean, and rice. Table 2 = 0.14) and rice (R2 = 0.16), and was moderate for soybean (R2 = 0.35), these regressions were highly significant (P < 0.0001) and indicate that the predictor variables still provide information about the response even though data points fall further from the regression line.Using both broad-sense heritability and marker-based heritability as independent variables and the selected significant threshold (−log5) Table , but the10 (P)) values as shown in Figs. For datasets based on previously reported results, estimated marker-based heritability was 66.8% for DP and 84.9% for ED in maize, 28.6% for C13 and 77.8% for CW in soybean, and 42.8% for SD and 68.8% for PH in rice. These marker-based heritability values were used to determine significant threshold . Higher broad sense heritability traits in these crops had higher significant threshold values. Among crop species, maize, with a lower LD pattern, had higher significant threshold values as compared to soybean and rice , 43.2% (R7), and 39.1% (R9). Using this marker-based heritability, significant threshold P-values were determined for the simulated trait in all three repetitions. Results indicated that our formula-based threshold values identified 10 QTLs for this simulated trait in these three repetitions across different parts of the genome  to determine if our formula accurately estimated threshold P-values with other multiple testing comparison methods using the GWAS results from the previously-published phenotypic datasets in maize [P ≤ 0.05) with our formula, Bonferroni correction, and FDR. Because Bonferroni, Šidák, Hommel, and Hochberg corrections had similar results, and False Discovery Rate and Positive False Discovery Rate had similar results, only Bonferroni correction and FDR are shown in Table 10P ≥ 3.5), which was the arbitrary selection. Our formula identified a greater number of markers than the uncorrected method for the C13 trait in soybean, which might be due to the generation of false negatives in the uncorrected method.Using the equation developed from marker-based heritability, we evaluated our threshold in maize , soybeanin maize , 20, andin maize . The resThese results indicate that selection of significant threshold values vary in different populations and crop species, which depend on the heritability of the trait in a particular environment. The GWAS results for these comparisons were obtained from the FarmCPU model because this multi-locus model effectively controlled false positives that arise from population structure and family relatedness as compared to all MLM models , which are single-locus models.P-value for GWAS based upon the marker-based heritability of a trait in a specific environment. This method is simple and robust across a wide range of heritabilities and species with different LD. This method is less conservative and captures more true positives as compared to more conservative methods such as FDR and Bonferroni corrections.We developed a simple method for determining the threshold www.soybase.org) for 346 accessions that were previously reported by Kaler et al. [OryzaSNP project, an oligomer array-based re-sequencing effort using Perlegen Sciences technology, and (2) BAC clone Sanger sequencing of wild species from the OMAP project [To develop a formula for a significance threshold, we used 45 simulated traits in soybean, maize, and rice that varied in broad sense heritability and the number of QTLs (Q). We used an R code script for simulation, where real genotypic data of each crop was used and different number of QTLs and heritability were assigned to create a simulated phenotype. In soybean, genotypic data consisted of 42,509 SNP markers -values. Broad-sense heritability was the heritability that was used to simulate the trait, and marker-based heritability was estimated using genetic variance determined from a simulated trait and genotypic marker data [Y = Xβ + Zu + e, where where Y is the vector of observed phenotypes; β is an unknown vector containing fixed effects, including the genetic marker, population structure (Q), and the intercept; u is an unknown vector of random additive genetic effects from multiple background QTL for individuals/lines; X and Z are the known design matrices; and e is the unobserved vector of residuals. The u and e vectors are assumed to be normally distributed with a null mean and a variance of: 2aK with σ2a as the additive genetic variance and K as the kinship matrix. Homogeneous variance is assumed for the residual effect; i.e., R = σ2eI, where σ2e is the residual variance. The proportion of the total variance explained by the genetic variance is defined as marker-based heritability.A formula to determine a significance threshold was developed based on a multiple regression equation that used two independent variables, broad-sense heritability and marker-based heritability, and one response variable, − logker data  that werker data . In the 10 (P)-value determined from the association analysis of a simulated trait that identified the number of QTLs for that simulated trait. For example, if a simulated trait had 10 QTLs, then the significant – log10 (P)-value was selected that identified these 10 QTLs after performing association analysis using the FarmCPU model [The response variable was the – logPU model .We validated this formula using the GWAS results from previously-published phenotypic datasets in soybean, maize, and rice. The GWAS results were obtained after performing association analysis on the datasets including carbon isotope ratio (C13)  and canoAdditional file 1:Figure S1. Scatter plots between significant threshold and marker-based heritability in maize, soybean, and rice. Table S1. Phenotypic data of canopy wilting (CW) and carbon isotope ratio (C13) from 346 soybean accessions previously reported by Kaler et al. . Table S2. The R code script used for trait simulation for rice data. Similar programming can be used for other crops by changing the genotypic data. (DOCX 176 kb)


The timing of fruit maturity is an important trait in sweet cherry production and breeding. Phenotypic variation for phenology of fruit maturity in sweet cherry appears to be under strong genetic control, but that control might be complicated by phenotypic instability across environments. Although such genotype-by-environment interaction (G × E) is a common phenomenon in crop plants, knowledge about it is lacking for fruit maturity timing and other sweet cherry traits. In this study, 1673 genome-wide SNP markers were used to estimate genomic relationships among 597 weakly pedigree-connected individuals evaluated over two seasons at three locations in Europe and one location in the USA, thus sampling eight ‘environments’. The combined dataset enabled a single meta-analysis to investigate the environmental stability of genomic predictions. Linkage disequilibrium among marker loci declined rapidly with physical distance, and ordination of the relationship matrix suggested no strong structure among germplasm. The most parsimonious G × E model allowed heterogeneous genetic variance and pairwise covariances among environments. Narrow-sense genomic heritability was very high (0.60–0.83), as was accuracy of predicted breeding values (>0.62). Average correlation of additive effects among environments was high (0.96) and breeding values were highly correlated across locations. Results indicated that genomic models can be used in cherry to accurately predict date of fruit maturity for untested individuals in new environments. Limited G × E for this trait indicated that phenotypes of individuals will be stable across similar environments. Equivalent analyses for other sweet cherry traits, for which multiple years of data are commonly available among breeders and cultivar testers, would be informative for predicting performance of elite selections and cultivars in new environments. The fruiting season of a cherry is short and sweet, but at least it’s relatively consistent across different environments, a new genetic analysis suggests. The development of additional early- and late-maturing sweet cherry cultivars is a major objective for cherry-breeders, but the influence of the environment on the timing of fruit maturity—which could reduce the accuracy of selective breeding efforts - was unclear. So, Craig Hardner at the University of Queensland in St Lucia and colleagues used DNA markers to  model relationships among individuals and examined the dates at which their fruit ripened at four different locations in Europe and the USA across two seasons. The timing of fruit maturity was relatively stable between related individuals across similar environments, suggesting that new cherry cultivars could be developed without having to test them at multiple sites. Prunus avium L.) production, particularly given the brief shelf life of the fruit. Fruit of an individual tree tends to ripen over a narrow window of 3–7 days and firmness and taste decline rapidly over 4–7 days after harvest, even under cold storage4. Similarly, the harvest window for cherry is relatively short7. Fruit produced in the early or later part of the production season attracts higher prices than in the mid-season8.The timing of fruit maturity is important for sweet cherry 11 and large-effect Quantiatative Trait Loci (QTLs) on linkage groups 1, 4, 5, and 6 have been detected within a full-sib family (‘Regina’ × ‘Lapins’), each explaining up to 20% of phenotypic variation12.There is evidence that phenotypic variation in phenology of fruit maturity in sweet cherry is under strong genetic control, suggesting breeding opportunities for this trait. Traditionally, sweet cherry cultivars are divided according to their maturity date into early , mid-  and late season 14. From the early 20th century, modern breeding involving selection of elite parents was undertaken in the USA and Canada, followed by European countries such as the UK, Russia, Ukraine, and others by the 1950s. In the last 30 years, breeding has progressed rapidly, fuelled in part by the adoption of dwarfing rootstocks allowing intensive orchard production15. Sansavini and Lugli16 noted that during the period 1991–2004, 230 new cultivars were released, this number being second in stone fruit only to peach. Nevertheless, some very old selections, such as ‘Burlat’ or ‘Bing’, and cultivars with unclear origin such as ‘Ambrunés’, ‘Emperor Francis’, ‘Germersdorfer’, ‘Hedelfingen’, ‘Napoleon’ or ‘0900 Ziraat’, are still widely planted. Cherry breeding has relied on a narrow genetic base for the main founders17, and modern cultivars are only a few generations removed from early ancestors18.While cherry has been cultivated for more than 2000 years, breeding commenced around the early 1800s13. Replacing ‘Burlat’ has been an important target for decades, although most new cultivars of the same early maturity period have undesirable fruit quality in terms of firmness, tolerance to rain-induced fruit cracking, or post-harvest shelf life. The breeding program conducted at Summerland, Canada, has released several commercially successful late-maturing cultivars, such as ‘Sumtare’ (Sweetheart™), ‘13S2009’ (Staccato™) and ‘13S2101’ (Sovereign™).The development of early- and late-maturing cultivars is a major objective in sweet cherry breeding across the globe20. Confirming the degree of G × E is an important initial step in developing strategies for commercial deployment of improved germplasm in target production environments21. If the degree of G × E is small, germplasm can be selected for overall mean performance across the target region. Alternatively, if the magnitude of G × E is large and repeatable factors can be identified that explain some of the pattern in G × E, higher gains can be achieved by using those factors to subdivide production regions into ‘mega-environments’ and select germplasm targeted for each. Otherwise, the presence of G × E represents experimental noise, reducing selection accuracy.Genotype-by-environment interaction (G × E) is a common phenomenon in crop plants that can complicate selection25, or are within related families connected through a pedigree structure that is used to estimate an expected relationship matrix26. Thereby, the performance of an individual in one environment can be genetically correlated with its performance in other environments28 and that correlation can be used to predict the performance of genetic treatments  in environments in which they have not been directly tested29. However, conducting conventional METs in horticultural tree crops is expensive, particularly due to the large size of experimental units and long juvenility periods. As a consequence, tree fruit breeding and elite selection or cultivar evaluation programs are often localised and have limited replication of germplasm among programs. These features limit the opportunity to evaluate G × E for horticultural tree crops.Detection of G × E in plants requires replication of genetic effects across multi-environment trials (METs). Commonly, METs in horticulture are established with individuals that are clonally replicated across environments, e.g., refs. 30. Such models support estimation of variance and covariances to describe G × E, incorporation of complex models that might include different experimental designs among trials or spatial effects, and prediction of unbiased genetic values from unbalanced data. A common model used to describe G × E from METs is a simple main effect and homogeneous interaction (G + G × E) linear mixed model. However, this model is restrictive as it unrealistically assumes a common genetic variance across environments and a common covariance (and hence correlation) among all pairs of environments30. More general models allow greater flexibility, although a highly unstructured covariance matrix might be difficult to estimate, particularly when the number of dimensions is large and the data for each dimension are limited33. The interaction term in the simple G + G × E model is an average of the individual pairwise covariances in the general model34 A parsimonious alternative is the FA parameterisation of the genetic-by-environment covariance matrix, which estimates environmental loadings for a reduced number of hypothetical orthogonal dimensions that maximise differences among environments (λ), and estimates specific variances for each environment for genetic effects not accounted for by the loadings (ψ)31.Linear mixed models are the preferred approach for analysing data from METs35, to approximate identity-by-descent as replication of chromosome segments among individuals deployed across METs. An advantage of the GRM is that it can describe heterogeneity of realised relationships within families that occur as a consequence of Mendelian sampling36. A realised relationship matrix is expected to be more accurate than a pedigree-derived relationship matrix, because the latter assumes a mean expected relationship among relatives. In addition, realised relationships might capture cryptic or undocumented relationships38. The GRM could therefore be used to replace the pedigree-derived relationship matrix to study G × E. Considerations in using the GRM with a factor analytic (FA) parameterisation of the genetic-by-environment covariance matrix to model G × E have been described previously39.Genetic relationships among individuals can be described using a DNA marker-based genomic relationship matrix (GRM)The aim of this study was to use a G × E model involving a GRM to combine phenotypic data on fruit maturity timing collected in multiple locations for cherry germplasm having little clonal replication among locations, to thereby estimate patterns in G × E and predict performance of individuals in environments in which they have not been tested. By leveraging the information from existing breeding program datasets in this way, we hypothesised achieving improved accuracy of genetic predictions within and across locations compared to predictions arising from datasets of single breeding programs.Germplasm used consisted of 597 sweet cherry cultivars, selections and unselected offspring, subsets of which were grown at four locations: Prosser, WA, USA ; Bourran  and Balandran , France; and Forlí , Italy , belonged to sweet cherry trials of the ‘Chart of evaluation of fruit varieties’ program coordinated by Centre Technique Interprofessionnel des Fruits et Légumes (CTIFL) and involving Institut National de la Recherche Agronomique (INRA), regional experimental stations, representatives from growers associations, and nurserymen and private breeders from France and beyond. Balandran is one of the three locations in France where new cultivars are initially evaluated and is 44 m above sea level. The climate is classified as Mediterranean and the trial was planted on Costiéeres soil. Each cultivar was grafted onto rootstock ‘Colt’ and planted as two clonal replicates, adjacent wherever possible. Trees were trained to a vase with four to five permanent branches. The trial was irrigated and managed using standard approaches. Fruit maturity timing was measured as the date (in Julian days) on which 50% of the fruit on both accessions were mature.Prunus Genetic Resources Center of INRA at Bourran (Lot & Garonne), near Bordeaux40. The site is 50 m above sea level. The climate is classified as oceanic and the soil type is clay alluvium. Two replicate trees for each individual on the rootstock ‘Maxma 14′ were planted adjacently in a random design. Approximately half were of French breeding origin and the remainder were from 15 other countries in North America, Asia and Europe. Accessions were historic landrace cultivars (n = 85) and recently bred cultivars currently planted in Europe (n = 106). Two accessions of each of the cultivars Noir d’Ecully and Giorgia were also included, each corresponding to a different period of introduction into the INRA collection. Trees were trained to an open vase and were irrigated and managed using a standard commercial regime. Fruit maturity timing was measured as the date (in Julian days) on which at least 70% of the fruit on both replicate trees were mature, using firmness, taste and skin colour indicators.The 193 individuals assessed at Bourran, France  were accessions of a germplasm collection maintained by the 41 using firmness, taste and skin colour indicators.The pool of 56 individuals assessed at Forlí, Italy , was composed of traditional germplasm from the Emilia-Romagna territory, modern cultivars from various breeding programs, breeding selections, and a few clones of known cultivars selected in the 1970s after irradiation of bud sticks and are part of the sweet cherry collection at CREA —Forlì. The site is 34 m above sea level, with Mediterranean climate and silty-clay soil with medium-low organic matter soil. Each individual was represented by two to three adjacent replicate trees grafted onto the rootstock ‘Colt’. Trees were managed according to integrated production practices, irrigated and trained to a vase. Fruit maturity timing was measured as the date (in Julian days) that the first 10% of fruit on all replicate trees were ripe for consumptionn = 268)42 and the Pacific Northwest Sweet Cherry Breeding Program at Washington State University. The site is ~200 m above sea level. The climate is classified as semi-arid and the soil is well-drained non-saline from the Aridisols order. The RosBREED Crop Reference Set consists of cultivars, founders, ancestors, advanced selections and unselected offspring representing germplasm used in US breeding programs43. A total of 53 Prosser.US individuals were cultivars or breeding selections, with the remainder consisting of unselected offspring from 98 families (53 families with two or more offspring). The germplasm was planted between 2006 and 2008 as single trees of each individual, with offspring often arranged in families. Trees were managed using conventional orchard management practices. All individuals were grown on ‘Gisela 6’ rootstock except for own-rooted offspring. Fruit maturity timing was measured in 2011 and 2012 within the RosBREED project44 as the date (in Julian days) at which 50% of fruit on a tree were mature using firmness, taste and skin colour indicators43.The 384 individuals evaluated at Prosser, USA  were within the RosBREED sweet cherry Crop Reference Set , GenomeStudio results were exported and used as input to the software program ASSIsT47 to obtain a subset of robust SNPs with reliable genotype calls. A multi-step pipeline using FlexQTL, VisualFlexQTL48 and Pedihaplotyper49 to detect genotyping errors was applied to obtain a high-quality genotypic dataset of 1617 SNPs. Data for 1215 high-quality SNPs loci were available from INRA, and for 1636 SNPs from CREA.SNP data generated using the RosBREED cherry 6K SNP array v1ove Fig. . BrieflyA data-curation pipeline was developed to manage germplasm, SNP and phenotypic data contributed from the various sources. The first step in the process was standardisation of names of individuals common across datasets. Initially, the format of an individual’s name was simplified by removing special characters, changing all upper case letters to lower case, and inserting an underscore between characters separated by a space. Formatted names were then compared to an existing dictionary of original names, formatted names and final names  created from previous data sources. Full and partial matching were used to identify potential matches in the existing dictionary. New names that could not be reconciled were added to the dictionary.51 in the R package synbreed52 using only flanking markers, i.e., without considering family information.Curation of SNP data involved standardisation of SNP names across the datasets against NCBI standard identities. SNP data from individuals common to multiple locations were compared to ensure consistency and fill in missing genotypic calls. The very few inconsistencies in genotypes among common individuals were set to missing, except in one case: ‘Noire de Meched’ genotyped by INRA and found to be substantially different to an individual with the same name in the CREA collection . In this case, cultivar names were modified to distinguish the difference. SNP loci with missing data of >30% or a minor allele frequency <0.05 were removed, leaving 1273 loci for 550 individuals with unique genome-wide DNA profiles . The missing 33,704  individual SNP locus genotypes were imputed with a hidden Markov Model implemented in Beagle53. Fields not in the list of standard design and sampling effects were retained. Phenotypic data was reshaped to long format .Phenotypic data curation began with incorporation of standardised individual names. Next, field names in original data files that identified experimental and sampling design factors were identified and renamed using standardised terms Table . A fieldnG individuals, evaluated at nL locations and nSj seasons within the jth location, was:y was a vector of observations, b was vector of unknown fixed effects , X was the design matrix for fixed effects, u was a vector of unknown random effects , Z was the design matrix for random effects and r was a vector of unknown residual effects, with variation of observations assumed to follow a multivariate normal distribution:G was the variance–covariance of additive genomic effects for the nG individuals among the nE location-by-season environments and R was the variance–covariance among residual effects.A multivariate genomic prediction linear mixed model was implemented to estimate model parameters and predict fixed and random effects for fruit maturity data collected from individuals planted across the four locations and assessed over Table nG × nE) × (nG × nE) matrix G was separated into the nG × nG additive genomic relationship matrix (GA), estimated following VanRaden35 method 1 using the complete imputed individual-by-SNP matrix and the nE × nE additive genetic-by-environment covariance matrix (GA×E), i.e.:nE × nE additive genomic-by-environment covariance matrix, GA×E, can be modelled using various structures30. Assuming homogeneous genomic variances across all environments and perfect genomic correlations  among all pairwise environmental combinations leads to 1nE was a nE × nE matrix of 1 s. A compound structure, i.e., homogeneous variances on the diagonal and homogenous covariances on the off-diagonals, is equivalent to a main effect interaction model 30. Alternatively, a FA parameterisation models GA×E as a reduced set of parameters that are environmental loadings (ΨA×E). In this study, only a first-order FA model was examined. The variance–covariance matrix of residual effects (R) was modelled as independent blocks with each block representing the variance–covariance among observations across the seasons at the jth location (Rj), which was separated into RSj  and Inj .The  of fixed effects and best linear unbiased predictions (E-BLUPs) of random effects  were produced following Henderson’s55 mixed model equations using estimated G and R matrices.Parameters of the random and residual models were estimated using average information approaches implemented in the statistical software ASRemlρ2) estimated using the software PLINK56 against pairwise physical distances. Average linkage disequilibrium (LD) decay in physical distance was converted to genetic distance using the map presented in Guajardo et al.57The pattern of linkage disequilibrium across the experimental population was examined by plotting individual pairwise squared correlations of SNP locus genotypes among individuals (C) was undertaken to examine the germplasm genetic structure across the four locations. The matrix C was estimated as:M was an m × nG incidence matrix  that mapped the incidence of an individual onto specific individual-by-environment realisations. A pedigree-derived additive genetic relationship matrix was estimated from known pedigree records to examine the similarity between expected (pedigree-derived) and estimated realised GRMs.An eigen-value decomposition of the individual-by-location GRM . Firstly, a within-location validation population was created by subsampling individuals within locations, as the main interest of this study was performance predictions of individuals in environments in which they have not been tested, but phenotypes of other genotyped individuals have been used to characterise the environment of interest. Five-fold validation within locations was used. Adjusted phenotypes were obtained by removing from the phenotypic observations the effects of location, season, location-by-season, and year of planting at the Balandran location estimated from the fit of the FA model to the full dataset. Individuals at each location were then randomly assigned to one of the five sets to define location-by-set validation populations. For each validation analysis, adjusted phenotypes for the individuals in the particular location-by-set validation population were set to missing and parameters for the FA A × E model were re-estimated using the training population and then used to predict additive genomic values for the validation population. Predictive ability (PA) was estimated as the correlation between the genomic predicted breeding values for the particular location-by-set validation population and the original adjusted phenotypes. Prediction accuracy was estimated for the particular location-by-set validation population by dividing PA by the square-root of the individual narrow-sense genomic estimated heritability of genomic values from the fit of the FA A × E model to the full dataset, as independent heritability estimates were unavailable. Standard error of PACC was estimated as A fourfold cross-validation was undertaken for a second scenario of dropping out the complete phenotypic data from one entire location for each validation iteration. Predictive ability was estimated as the correlation among the adjusted phenotypic performance of individuals in the removed location and the mean of the predicted breeding values for these individuals across the three preserved locations. Prediction accuracy was again estimated using genomic narrow-sense heritability of the removed environments estimated from the fit to the full dataset.ρ2-value <0.2 between 140 and 240 kb, which corresponds to ~0.30 and 0.52 cM. At 1 cM/465 kb, LD declined to <0.14.The average LD between SNPs within a 100-kb window was 0.36 with the value for 20% of pairs being >0.80  better fit to the data than a model that did not account for A × E (logl = −2560.7). Estimated average p < 0.001) than the compound symmetry (A + A × E) model.The compound symmetry A + A × E model (log-likelihood = −2554.2) was a significantly . Estimates of the correlations between the Prosser and European environments were between 0.87 and 0.92.Estimated additive genomic correlations among the European locations and between years within these locations from a FA A × E model with the GRM were close to 1 Table . In contr = 0.80, n = 8) with estimated narrow-sense heritability.The overall mean genomic predictive ability of using genome-wide SNP information from phenotyped individuals at a particular location to predict performance of individuals untested at that location was 0.81, with mean PACC 0.91 ± 0.24 . The stronger LD in our study might be a consequence of the inclusion of full-sib families from the Prosser location. The degree of decay in LD with increasing distance observed in this study and that of Campoy et al.40 is similar to that recently reported in a Belgian apple collection59, but higher than that reported for peach  is able to capture unknown pedigree relationships, and heterogeneity in relationships due to Mendelian sampling, that is not possible using only pedigree information to estimate relationship coefficients 36. Differences between the pedigree-derived and SNP-derived relationship matrices might also be a consequence of incorrect or incomplete pedigree information or improper scaling. The high frequency of pedigree-derived pairwise relationship coefficients  with a value of 0, compared to the more balanced distribution of realised relationships estimated from genotypic data, suggests that pedigree records are incomplete. Germplasm individuals are generally assumed to be unrelated in the base generation of the pedigree-derived relationship matrix, but they might instead be related and those relationships can be captured in the GRM. Estimation of the GRM is also dependent on the estimated allele frequency for each SNP35. Ideally, allele frequencies for the base population are needed, but they are difficult to obtain. Allele frequencies in our study were computed from available genotyped accessions and not from a base population, which could have resulted in improper scaling. Nonetheless, Wang et al.64 suggests with strong quality control , differences between the elements of matrices can be reduced.The contrast in distribution of diagonals and off-diagonals between the SNP and pedigree-derived relationship matrices observed here has been discussed by othersA, suggests predictions using a realised relationship matrix will be more accurate36. In this study in particular, the GRM might also more accurately estimate additive effects compared to a relationship matrix based on expected relationship coefficients derived from pedigree information, as there were only a small number of offspring per family and so there might have been insufficient replication of Mendelian sampling events within each family to accurately obtain a family average, which is an important source of information for prediction of genetic effects65.The higher variance of the GRM diagonal and off-diagonal elements, compared to those for 66 This conclusion agrees with general knowledge of sweet cherry cultivar pedigrees, which involved a strong bottleneck, repeated use of a small number of parents, and crossing of individuals that shared recent ancestors62. For example, breeding programs have continuously used the cultivar ‘Stella’ and its descendants in breeding for self-compatibility, presumably dramatically reducing the genetic diversity of released cultivars14.Estimates >1 for the expected relationship coefficients of an individual with itself dervied  from pedigree information, and large realised relationship coefficients estimated from genotypic data, suggest that some individuals included in this study are inbred, as was also observed in western larch by, e.g., Klápste et al.67 approaches, AMMI is a fixed effects approach, whereas FA mixed models offer more flexibility such as fitting genotypes and genotype-by-environment interaction as random effects30. A mixed model approach more easily deals with unbalanced datasets and supports incorporation of heterogeneous variances and covariances among environments—all of which are essential features of our dataset. Superiority of FA models over AMMI models have been demonstrated using empirical MET eucalypt and teak datasets68.This study employed a FA parameterisation to estimate parameters of the genetic-by-environment covariance matrix. While FA models have similarities with additive main effects multiplicative interactions (AMMI)Despite the high average additive genomic correlation among environments  where individuals can be vegetatively propagated74. In contrast to the single-step approach taken here, other studies have applied a two-stage analysis which  adjusts phenotypes for fixed effects prior to fitting of the genetic model, e.g., refs. 72. A single-step analysis as reported here was possible due to the small size of the population. Single-step analyses allow incorporation of complex models in the analysis75. While Welham et al.76 suggest that PACC from a single-step analysis is similar to that from a two-stage analysis, de los Campos et al.77 have reported that two-stage approaches for genomic prediction might produce biased genetic marker effects and correlation among residuals.The FA G × E model supported the combination of adjustment of phenotypes for fixed effects and estimation of genetic parameters into a multivariate single-step analysis. The realised relationship matrix was used to describe correlated genetic effects across environments so that data from various sources could be combined. Prediction accuracy from multivariate approaches is generally higher than that from univariate approaches and supports the prediction of genetic effects for individuals in environments in which they have not been tested by leveraging of correlated information79. The within-location validation strategy adopted here is similar to the second validation strategy adopted by Kumar et al.72 Here, training populations were constructed by excluding a subset of individuals within a location to simulate the strategy of predicting the performance of an individual within a target environment for which phenotypic data was available from other genotyped individuals. The high accuracy of this strategy (>0.8) might be explained in part by the high apparent heritability of fruit maturity timing (>0.60). The high prediction accuracies of the model for individuals at the Balandran and Bourran locations (>0.85) are also likely to be a consequence of the high genomic correlation (0.98), and large number of individuals replicated (68), between these locations. The high PACC for individuals at the Prosser location (>0.95) is likely to be a consequence of the random sampling strategy used to form the validation population as we did not account for the known complex family  structure. It is well known that close relationships between the training and validation populations increases PACC80. The lower accuracy of the across-location validation strategy of predicting across environments (0.50–0.95) is probably a consequence of the confounding of G × E with within-environment PACC.The estimates of PACC in all environments reported here are considerably higher than those reported previously for other traits in other horticultural tree cropsThis study appears to be the first to quantify differences among locations in timing of fruit maturity in sweet cherry, particularly on a global scale. The advantage of undertaking a combined multivariate analysis like presented here is that location means are unbiased because they are adjusted for differences in germplasm composition among environments. In this way, comparisons are not confounded by the use of different germplasm in each environment. Differences among seasons within a location were not as great as differences between the European locations and Prosser, suggesting that fruit maturity across the European locations is generally earlier than at the Prosser location, at least for the seasons sampled here. More locations and seasons evaluated could be used to develop a more general predictive equation. A joint analysis of flowering timing and fruit maturity timing could help determine the extent of this phenological phenomenon.10, suggesting that an individual’s breeding value for timing of fruit maturity is reliably predicted from the phenotype of that individual. However, our model only included additive genetic effects, and others81 have demonstrated that heritability estimates can be inflated if large non-additive effects are present but omitted from the training model. Deflated estimates of additive genetic variances in complex models might also be a consequence of correlation among additive and non-additive genetic effects82.High heritability estimates for timing of fruit maturity in sweet cherry have been reported previously83. Differences in ranking between seasons at Prosser was the major source of G × E detected in this study. The limited number of locations in this study prohibits the development and testing of a hypothesis to explain this pattern. The generally high correlation among all locations suggests that the timing of fruit maturity of an individual relative to other individuals is expected to be stable across production environments similar to these test locations. This conclusion is supported by the high correlation of predicted genetic values at Prosser with the average of predicted values across the three European locations  suggests that individuals can be selected for elite average performance across environments similar to those examined here, and elite germplasm in one environment for fruit maturity is expected to be elite in other environments. However, it is difficult to generalise this recommendation to new environments. In addition, other traits might exhibit different G × E patterns. However, the genetic potential of genotyped but untested individuals can be predicted using the methods outlined here if genomic and phenotypic data are available for other individuals and are used to estimate a prediction model.12 have suggested the presence of large-effect QTLs contributing to genetic variation of fruit maturity timing, the model of genetic architecture used in this study might not be accurate. Incorporation of known QTL effects in the linear model should improve accuracy, as it is expected that such a model would more closely approximate the true distribution of QTL effects77. One of the challenges of building these models is that characterisation of the variation around QTL regions as individual biallelic SNPs might not capture the complexity of variation in the region. This variation might be better modelled using the same genotypic data by describing SNP haplotypes that contain QTL regions. The level of LD in cherry is sufficiently high that this information could be used to phase the SNP genotypes . Otherwise, the model could use separately obtained genotypic data that describes the allelic composition of each individual at large-effect QTLs with multiple effective alleles.This study is based on a model of individual SNP loci in linkage disequilibrium with chromosome segments containing numerous small-effect QTLs such that replication of SNP alleles across environments models the replication of causal alleles at the small-effect QTLs. However, as previous studies85.Incorporation of data from additional environments would be useful to develop a more general description of G × E patterns for the trait, enable development and testing of hypotheses to explain the G × E patterns, and increase PACC. Only a minor degree of G × E was detected in this study and thus there is insufficient variation to develop and test hypotheses of environmental drivers of G × E for fruit maturity timing. Increasing the environmental range might induce greater G × E. In this case, it might be that repeatable prediction of G × E could be achieved through incorporation of environmental variables or through use of crop growth models that associate variability in underlying physiological process with variation in environmental driversThis study has demonstrated that genome-wide SNP-derived relationship matrices can be used to quantitatively describe cryptic genetic relationships. In addition, this study has demonstrated that such realised relationships can be used to combine phenotypic data collected on individuals evaluated across disparate locations into a single analysis without the need for clonal replication or knowledge of pedigree relationships. These approaches now allow the latent value of existing, otherwise unconnected, historical phenotypic datasets collected by numerous local breeding and cultivar testing programs to be exploited. Such analyses can increase knowledge of the genetic architecture of important traits to support ongoing genetic improvement, as well as having the immediate practical value of predicting genetic effects of individuals  in untested environments. This type of analysis would be informative for other important horticultural traits of sweet cherry, such as bloom timing, productivity, fruit size, fruit firmness, fruit sweetness and fruit acidity, for which phenotypic data are largely available among breeders and cultivar testers.


Specific combining abilities for yield had significant (p < 0.001) positive effect from the cross GU-154-L x UF-273 Type 2 (0.703) and strong negative interaction between ICS-43 and LF-1 (−0.975). A weighted index was used to select the top performers while providing the corresponding genetic gains. At an 1% selection intensity, yield component gains ranged from 17.8 to 331.9%. Agronomic traits such as branch angle, trunk diameter and jorquette height had lower genetic gains and lower heritabilities. In addition, the parents in this study were genotyped with a 96-SNP marker off-typing set and a significant positive correlation of 0.39 (p = 0.019) was found between genetic distance and specific combining abilities for yield. Preliminary comparison of clonal parents vs. seedlings yield in the family with the highest SCA suggest for the first time presence of heterobeltiosis in cacao.A comprehensive understanding of the genetic basis of target traits in any crop is critical to design breeding strategies for the development and release of new improved varieties. In this study, 34 cacao families were evaluated for vigor and yield related traits over the course of 6 years in Costa Rica. Linear mixed models provided the variance components for the partitioning of additive and non-additive effects. Heritabilities of yield over time ranged from 0.085 to 0.576, from 0.127 to 0.399 for vigor, and 0.141 to 0.146 for disease resistance traits. Significant ( Theobroma cacao L is a diploid species with 10 pairs of chromosomes (2n = 2x = 20). It is an understory fruit-tree species from the Malvaceae family native to South America  of a given trait, the magnitude of partition of the genetic variance into additive and non-additive genetic effects, the type and amount of genetic correlation that exists between traits, and, in cases where there are multiple site experiments, the magnitude and dynamics of genotype-by-environment interactions. All of the above elements are required to understand the genetic basis of the traits of interest and they are essential to evaluate the potential outcomes in terms of future genetic gains of an operational breeding program.Cacao breeding programs, with the implementation of recurrent selection from a diverse population began in the 1930s Dias, . SeveralThe main objective of this study is to contribute to a better understanding of the phenotypic variation of target breeding traits in cacao through the evaluation of families from a long-term field experiment established in Costa Rica and to provide insight into the genetic basis of target traits through the evaluation of the parental population planted in a clonal trial. At the same time, these trails are part of a breeding program aiming at ultimately delivering new clones to farmers. Indeed, farmers have limited access to improved cacao cultivars and breeding is necessary to develop the planting material farmers need. To improve yields, farmers require cultivars that have low vigor, high yield, and disease resistance so that land and pesticide use becomes more efficient while increasing farmers' income. The trial was the result of a factorial crossing scheme measured for 6 years in mature trees. The specific objectives of this work include: (1) to calculate narrow-sense heritability and dominance ratio for various key traits, (2) to estimate general and specific combining ability effects and their relationships with genetic distance, (3) to determine additive and dominant correlation between contrasting traits, (4) to estimate genetic correlations for yield between different measurement years, (5) to explore response to early or late selection on yield, and (6) to calculate genetic gains for different selection intensities. Special emphasis was given to the understanding of the dynamics of cocoa yield in regard to time and tree age.Musa spp.), planted at a spacing of 6 × 6 m, and immortelle trees (Erythrina poeppigiana), planted at 12 × 12 m, were added to the experiment as temporary and permanent shade, respectively. Cacao trees were given structural pruning at the beginning of the trial and periodic maintenance pruning thereafter. Every 3 months, 150 g of granular fertilizer (formula 18-5-15-6-0.3-7) were applied. Manual weed control was carried out every 2 months, complemented by two annual directed applications of paraquat herbicide (0.2 kg/ha). No disease control was carried out in the trial other than cutting diseased fruits at the time of the monthly evaluations, which were then left on the soil without applying any treatment. The L7 trial consists of 34 full-sib families that originated from a full factorial crossing scheme based on 12 parents   in mid-2003 at the CATIE's “La Lola” experimental farm under a randomized complete-block-design (RCBD) spanning an area of 2.6 ha at a density of 3 × 3 m. The layout considered five replications, each with four plots of each family, and plots were formed by four trees. Thus, there is a total of 80 plants per family and 2,720 trees for the experiment, plus additional border trees, with Banana plants , four replications and 8 plants per replication and clone was used. Cacao trees were established at 3 × 3 m. Banana plants and immortelle trees were planted at 9 × 9 m as temporary and permanent shade species, respectively.Moniliophthora roreri), a clonal fungal pathogen /1,000 for each age , also an average yield (Ym) and its standard deviation (Ysd) were calculated across these years. Additionally, precocious yield was defined as the average yield for the ages 2 and 3 (Yinit), and full production yield was defined as the average yield between ages 4 and 7 (Ylate). Pod index (PI), defined as the number of healthy pods needed to obtain 1 kg of healthy dry bean weight, was calculated as PI = HP*1000/FWB*0.4, where 0.4 is the FBW to dry bean weight conversion factor , number of total pods (TP), number of healthy pods (HP), number of pods infected with Monila , which are weakly informative for assessing incidence of disease. In addition, %PHY was not considered as the level of Phytophthora in this trial was very low (~0.5%). Note also that with low levels of PHY, then %HP ~ 100%–%MO; hence, there is a high negative correlation between %HP and %MO. Natural logarithm and logit transformations were implemented in some traits to approximate normality of residuals ; f is the random vector of family effects, with f~MVN; p is the random vector of plot effects, with p~MVN; and e is the random vector of errors, with e~MVN. The matrix A is the numerator relationship matrix obtained from pedigree, the I matrix is an identity matrix of its corresponding size, 1 is a vector of ones, and the incidence matrices for the respective effects are represented by X, Z1, Z2, and Z3.where h2) and dominance ratio (d2) (their standard errors were obtained by approximation using the Delta method). These expressions corresponded to:After fitting the above model, estimated general combining ability ; ft is the random vector of family effects within trait, with ft~MVN; pt is the random vector of plot effects within trait, with pt~MVN; and e is the random vector of errors, with e~MVN. The 2 × 2 matrices Ga and Gf are additive and dominant variance-covariance matrices of structure CORUH between traits, respectively; D are 2 × 2 diagonal matrices, and the other terms were previously defined. The above bivariate model provides directly with an estimate of the additive and dominant trait-to-trait genetic correlations.where All linear mixed models were fitted with the statistical software R  was done on a representative clone from each replication using the 96 SNP off-typing marker described in DuVal et al. . A totaln = 27) and 5% (n = 136) performing hybrids were calculated. Genetic gains were expressed as: (i) the percent increase of the 1 or 5% selection intensities over the overall mean in L7 (Table 3), (ii) the gain (based on phenotypic data) of the entire L7 progeny population relative to the parents planted as clones in the L10 clonal trial  values from both hybrids and parents were obtained from the single-trait animal model  and definitions of yield  were explored. Heritabilites and dominance ratios were obtained from the single-site model for all traits, together with genetic additive and dominant correlations between traits based on the above definitions from the bivariate models. Of interest is the genetic association between precocious yield (Yinit) and full production yield (Ylater) in order to assist with response to early selection. Additionally, predictive performance of parents (GCA) was compared across these traits.The centered and scaled predicted values (z-scores) of the individuals and parents from the single trait analyses were ranked and used in a linear combination . The rankings of the z-scores were first obtained for each individual by trait, where the low ranks are favorable and the high ranks are least favorable. Percent monilia infected pods (%MO), branch angle (BA2), pod index (PI), jorquette height (JH2), and trunk diameter (TD5) were ranked in ascending order . Average yield across years (Ym) and Yield efficiency at age 5 (YE5) were ranked in descending order . All rankings were adjusted for ties by averaging those equally ranked. The weights chosen to construct the index were based on heritability and the known economic importance of each trait. The best individuals correspond to those whose index value (based on the rank for each trait) is the lowest. The expression for the index is denoted as:Z is a matrix of the ranked z-scores for individual i for each of the k traits, w is a vector of length k for the weights (Table 5) assigned to each trait.where, Phenotypic measurements for the L7 population , in contrast to early years (0.352).d2) is highest for Ym, Ysd, Ylate, PI, and TP (0.426–0.573). Moderate d2 levels are found for TD5, Y3 to Y7 and YE5 with d2-values ranging from 0.237 to 0.314. Low dominance ratios were observed for %MO, %HP, BA2, JH2, and Y2 (0.057–0.115). It is important to note that PI, TD5, Ysd, TP, and Ylate showed a larger dominance ratio than heritability .In general, the L7 population has overall high dominance ratio, likely due to the high genetic diversity of the parents and heteroric effects. Dominance ratio , Ylate (Y4–Y7), and Yinit (Y2–Y3) are all positive and relatively high with values ranging from 0.778 to 0.999. As expected, higher correlations were found between close ages. However, Y2 presents the lowest heritability and also the lowest dominance correlation with other traits; for example, the dominance correlation between Y2 and Y3 is 0.683, and close to zero for Ym, as well as for the later years Y6 and Y7. The additive correlation between Ylate and Yinit was 0.933, indicating that, regardless of the differences in heritabilities, these two traits provide with similar ranking of genotypes. The genetic additive correlations between Y2 with Ym is 0.835, and Y3 to Ym is 0.935; hence, there is better genetic agreement from Y3 for indirect selection of high overall yielding genotypes.The estimates of dominance genetic correlation showed, generally, weaker associations, in absolute value, than the majority of the additive correlations, but they follow similar trends Figures , S4. Hown = 27), and 5% (n = 136). As expected, given that cacao is in its early stages of domestication , followed closely by Ylate (0.542). The dominance ratio for the yield traits are close to their heritability estimates, indicating high levels of non-additive genetic variance. The dominance correlation between Yinit against Ylate is 0.345, indicating different behavior for ages 2–3 (juvenility period) compared to the later years. However, the additive correlation between Yinit and Ylate is 0.933, and at Y4 at age 4, there is very strong correlations with ages 5, 6, and 7. This suggests that Y4 can be used for early selection, which also has a strong correlation of 0.946 with Ym.The variability of yield production, measured by the standard deviation of yield over the measured ages (Ysd) provided with a high heritability of 0.459 Table . AdditioFigure The top 1% best ranked individuals for the weighted index come from only four families: IMC-60 × PA-150, IMC-60 × IMC-67, UF-273 Type1 × IMC-67, and and UF-273 Type 2 × IMC-67 . This is likely due to the small number of parents and families evaluated in this study, in comparison with larger breeding trials, together with the potential presence of off-types in some families. Some traits showed low heritability values, such as %MO, a result that might be due to large environmental and micro-environmental confounding effects in the field. Indeed, the narrow-sense heritability for %MO was low with a value of 0.141 and broad-sense heritability 0.190 (SD 0.075). In our study, we aimed at pyramiding potential different sources of Moniliosis resistant genes by crossing parents with low Moniliosis incidence  from various cacao genetic groups .In most cases, the levels of dominance were similar to the additive component, reflecting large levels of heterosis. Nevertheless, the dominance ratio inflation is likely a result of the diverse and divergent parents in the population. For example, the cross UF-273(T2) × GU-154-L has the largest positive SCA of 0.703 for Ym, where GU-154-L comes from the Guiana structural group and the UF-273(T2) is of Trinitario background, comprised of Amelonado, Criollo, and Nacional genetic groups . Furthermore, a significant positive correlation of 0.39 (p = 0.019) was observed between the genetic distance (GD) of the parents and SCA for Ym, which corroborates that heterosis and heterobeltisios occur in cacao perhaps due to allele complementarity between specific diverse backgrounds.Similarly, the SCA for Ym between the clone UF-273(T2) and LcTeen-37, which belongs to the Curaray group, have a negative SCA of (−0.34). As a possible result of diverse backgrounds, the traits PI, TD5, Ysd, TP, and Ylate have a larger dominance ratio than heritability  and Guiana backgrounds (0.991), respectively.p = 0.00834) but the ancestral groups showed no significance at α = 0.05 in this study.As described in Cornejo et al. (in review), Criollo ancestry has a negative effect on yield and the Criollo genetic group has a higher level of cumulated deleterious mutations than other genetic groups. However, in this population, the genetic distance is the main predictor of yield. To test the association between Criollo ancestry and SCA for Ym, a generalized linear model was run on SCA as the response variable and the average proportion of the 10 genetic groups between the parents, plus the genetic distance (GD) as covariates. GD had the most significant association to SCA, with an estimate of 0.018 .One of the objectives of this study was to evaluate how yield heritability and dominance effects differ for several evaluations based on measurements that range from 2–7 years of age. The lowest heritability and genetic correlations with any other years was found for Y2. We observed that at age 2, 80% of the trees were unproductive, thereby driving the estimate of heritability to be close to zero. However, there is yield consistency among years, particularly after year three. Although the additive correlations between Yinit (years 2 and 3) and the Ylate (years 4–7) were high (0.933) and it favors early selection of high yielding genotypes; however, Yinit has a lower heritability of 0.352, in contrast to Ylate of 0.542. Hence, even that early selection is possible, by using Yinit, higher precision and certainty will be achieved by concentrating selection on later years by focusing on Ylate. The dominance correlations between these traits was low (0.345). In this case, attention is toward the dominance correlation because the As mentioned, the first years of measurement (Yinit) do not correlate well with average yield production, Ym (0.411), but at age 4 (Y4), dominance correlations with Ym are high (0.946). If an accelerated breeding approach were to be used by rapidly cloning the best individual hybrid trees, we would then recommend using Y4 as the appropriate age to perform the early selection for yield. In this case, if feasible, we would clone at least the top half at year 4 since the top 10% for Ym is spread within the 62% at Y4 in this population Figure . This cah2 = 0.459) and also has a strong additive correlation with Ym (0.981), which indicates first, that important improvements can be done by reducing the yield variability on some selected genotypes. However, as we reduce this variability we also reduce the genetic gains for Ym. Here, it is important to select some genotypes that are correlation breakers, but this should be done carefully, as it requires further investigation and understanding of the underlying physiological basis of season to season and year to year productivity variation in cacao as it is known in other perennial crops . In regard to increasing genetic gains for disease, improvements can be made by focusing on best parents or families. For example, the strongest GCA effect for %MO comes from PA-169 (−0.338), which belongs to the Marañon structural group. Other parents that are favorable for lowering the effect of proportion of Monilia infected pods in the progeny are GU-154-L and IMC-67. In addition, the family ICS-43 × LF-1 has a favorable interaction (SCA = −0.271). Therefore, selection of the top progenies from the top crosses can improve the genetic gains in disease resistance terms for the next breeding round. However, we observed a low correlation between Ym and %MO despite that no fungicide treatment was applied, therefore, we believe when there's low heritability for disease resistance, that it should have a relative low weight regarding yield when constructing a selection index.Vigor traits are of importance since they can have an impact in income (less pruning leads to lower labor costs) and total yield per hectare if lower vigor is selected and higher planting densities are used. In this study we found a correlation with high yield and high vigor as has been previously reported (Glendinning, In this study, the L7 population has large dominance effects, and for some traits such as pod index, trunk diameter and yield, it is larger than the additive effects. As previously mentioned, the inflated dominance is likely in part due to the diverse and divergent genetic backgrounds of the parents Dias, . Often, GM and SG analyzed the data and wrote the manuscript. AA-L collected and analyzed data. AM-Q collected and analyzed data. GM, SG, and JM designed the analysis. WP-M managed collection of field data. JM conceived the experiment, analyzed data and wrote 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. The reviewer NBMA declared a shared affiliation, with no collaboration,with several of the authors, WP-M, AM-Q, AA-L, to the handling Editor.


Identifying variants, both discrete and continuous, that are associated with quantitative traits, or QTs, is the primary focus of quantitative genetics. Most current methods are limited to identifying mean effects, or associations between genotype or covariates and the mean value of a quantitative trait. It is possible, however, that a variant may affect the variance of the quantitative trait in lieu of, or in addition to, affecting the trait mean. Here, we develop a general methodology to identify covariates with variance effects on a quantitative trait using a Bayesian heteroskedastic linear regression model (BTH). We compare BTH with existing methods to detect variance effects across a large range of simulations drawn from scenarios common to the analysis of quantitative traits.We find that BTH and a double generalized linear model (dglm) outperform classical tests used for detecting variance effects in recent genomic studies. We show BTH and dglm are less likely to generate spurious discoveries through simulations and application to identifying methylation variance QTs and expression variance QTs. We identify four variance effects of sex in the Cardiovascular and Pharmacogenetics study. Our work is the first to offer a comprehensive view of variance identifying methodology. We identify shortcomings in previously used methodology and provide a more conservative and robust alternative. We extend variance effect analysis to a wide array of covariates that enables a new statistical dimension in the study of sex and age specific quantitative trait effects.https://github.com/b2du/bth.Bioinformatics online. Drosophila melanogaster  is central to the study of statistical genetics , α is the heteroskedastic effect with a prior α = 1, the variance of the response is not a function of the covariate, whereas when α ≠ 1, the variance term is associated with the covariate. We put priors on each of these parameters in order to incorporate biologically appropriate and computationally tractable forms of uncertainty in the test (The Bayesian test for heteroskedasticity (BTH) models a continuous trait across the test .H0, α = 1) with the likelihood of the data under the alternative hypothesis . In particular, for each application of the model , the BF has the formβ and evaluating the resulting multivariate integral using a multivariate Laplace approximation similar to the integrated nested Laplace approximation (INLA) method  to) method .permBF corresponding to data in which the variance of the phenotype is independent of the covariate. Thus, we compute FDR by considering, for any BF threshold t,t. We used the FDR-calibrated BF thresholds to discover heteroskedastic associations in our data, and we compared our discoveries to the discoveries from existing tests for heteroskedasticity.The BFs provide a measure of the heteroskedasticity of the association between a covariate and a phenotype of interest under certain assumptions, which we examine carefully in the simulations. To quantify the global false discovery rate (FDR) of the quantified BFs, we designed and performed permutations of the covariate-trait pair such that any mean effects are maintained but variance effects are removed . Furthermore, we generated a distribution of dglm) test.We compared results from our BTH against four tests for heteroskedasticity: i) the Brown-Forsythe test ; ii) theEach of these statistical tests makes assumptions about the underlying data by design. The Levene test, which has been used in a number of biological studies . The CLSix. This assumption becomes particularly meaningful in the case where the variants considered are non-binary, such as age. The dglm approach makes the same assumptions as BTH. In contrast to the above methods, including dglm, our test incorporates estimates of uncertainty, integrating over all possible mean effects in both the null and the alternative hypothesis.Our model for BTH makes the following assumptions: i) the noise has a Gaussian distribution; ii) the covariate is a continuous or ordered value; and iii) the functional form of the heteroskedasticity is dosage or variant dependent, with monotone effects on the variance. We make these assumptions to gain statistical power in identifying heteroskedastic effects in genomic studies, and to avoid spurious results. Assumption iii) is illustrated through modelling the variance as an exponential function We show the value of BTH with respect to these related approaches in extensive simulations and in three genomic data applications. In the simulations, for data that violate the model assumptions, we provide prescriptive tests and transformations to enable a well-powered application of BTH. We then apply BTH to methylation QTLs, gene expression QTLs and gene expression data versus biological covariates to illustrate the promise of BTH for identifying variance effects in diverse genomic data.To compare results from BTH with state-of-the-art tests for variance QTLs, we simulated data across a range of possible scenarios in genomic studies. We account for discrete and continuous covariates, different parameter settings and a number of distributions of the quantitative trait.ix ∈ {0, 1, 2} from individual mafπ): iz ∈ {0, 1, 2} are sampled from a Bernoulli distribution: For discrete covariates, each simulated biallelic, diploid variant β0, effect size β, and variance parameters σ2, α, we simulated the quantitative trait iy for individual i from a Gaussian distribution, using a linear model: Then, given intercept x and corresponding traits y. For each simulation with heteroskedasticity, we performed a single permutation of the quantitative trait sample labels and included for comparison this null simulation with identical MAF and trait distribution (see Online Methods). Thus, each simulation result contains 2000 tests, half of which are from a null distribution constructed using permutations, and the other of which are simulated to have variance effects.For each parameter configuration, we generated 1000 simulated datasets of covariates dglm. We compared performance using precision-recall curves, which quantify the proportion of true associations discovered  versus the proportion of discoveries that are truly associated . When the curves are close to precision = 0.5 across most values of recall, this means that the method cannot differentiate between non-associations and true associations in this scenario with equal numbers of true and null associations. The closer the curves are to precision = 1 across values of recall, the greater the area under the curve (AUC) is (with a maximum of one), and the better the performance of that method.For discrete genotypes, we compared results from BTH against results from the Brown-Forsythe test , the Levdglm are illustrated: when variance effect log (α) = –0.2, we see at high levels of recall as much as a 10% improvement in precision , and we discourage this rounding approach with real data .Table 1dglm show uniform improvement in AUC across all of the simulations, considering increasing mean effects  = 0;Weak null: the simulated mean effects β ≠ 0 and the simulated variance effects log (α) = 0;Weak alternative: the simulated mean effects β = 0 and the simulated variance effects log (α) ≠ 0;Strong alternative: the simulated mean effects β ≠ 0 and the simulated variance effects log (α) ≠ 0.We quantified the relative performance of the tests using precision-recall curves as above; however, caution must be used here in interpreting the relative AUC. We consider four possibilities for the simulated mean and variance effects with respect to the statistical test we perform perform :Strong β ≠ 0, this will present as variance effects in these tests. The BTH model integrates over the mean effect size, β, testing the union of the weak and strong alternative hypotheses against the union of the weak and strong null hypotheses. Moreover, in the permutations, we specifically remove variance effects while maintaining mean effects. These design decisions lead to different behavior of the test on these simulations from non-ideal scenarios.These definitions become important when discussing the log Gaussian and gamma simulations: for both, the variance is a function of the mean, inducing an explicit relationship between the two. In other words, when there are mean effects, β ≠ 0, log (α) ≠ 0; β ≠ 0, log (α) = 0; α. This suggests that the performance in the strong alternative simulations is due to mean effects. We verified this by considering simulations from the weak alternative , finding that all of the tests fail to detect signs of heteroskedasticity in the gamma simulations  and returns the probability of each of seven distributions within and across groups for discrete covariates and across values for continuous covariates  ≠ 0; dglm uniformly had the largest AUC across the five methods. We also compared results on the transformed weak null simulations  = 0; To address the problem of distributional misspecification of the model, we developed a statistical classifier that takes as input the variates . Given avariates ; when thvariates . We compWe applied BTH and the alternative tests for variance effects to a genome scale differential DNA methylation study  at the GEO accession numbers GSE24260  and GSE24274 . We removed eight individuals that did not have methylation data, and combined the genotypes from 280 individuals with 170 063 SNPs common to both genotype platforms and without missing data.Following previous work, we removed CpG probes of poor quality or with common mutations. We used the For each CpG site, we tested for association with cis-variants, defined as variants within 10KB of the CpG site. We evaluated the global FDR of our association results using a single permutation of the methylation data. Significance was assessed using a global FDR and FDR stratified by MAF. Using our distribution classifier, we found that most of the methylation level traits were gamma distributed .BTH does not find any significant variant-mediated associations between genetic variants and methylation levels at CpG sites at a global FDR of 0.05 and a MAF-stratified FDR of 0.05 .dglm identified three significant associations, CLS identified 549 significant associations, and the Levene test identified 878 significant associations  study.Gene expression values for 10 195 genes in lymphoblastoid cell lines (LCLs) from 480 Caucasian individuals were assayed on human microarray platforms. Genotypes were assayed using genotyping arrays and subsequently imputed using IMPUTE2 to yield 33 386 856 total markers across the 22 autosomal chromosomes. We removed variants with MAF below 0.05.The preprocessing of gene expression data for testing of variance eQTLs is somewhat different than the preprocessing for mean effect eQTLs. To test for variance eQTLs, we log transform the microarray gene expression data so they do not have a log normal distribution, we control for outliers, and we control for known (directly measured) and unknown (inferred) confounders (Methods). After preprocessing the genotype and gene expression data, we performed association mapping between each gene and the cis-SNPs local to that gene; here, cis-SNPs are defined to be ≤ 200 Kb from the gene transcription start or end site. There were 9862 genes with at least one cis-SNP in these data, and, on average, each gene had 847 cis-SNPs. We computed the test statistic for the putative association between each cis-SNP gene pair with these processed gene expression data.dglm and CLS tests report P-values for each association , we chose to base the significance of the dglm and CLS tests on our previously described FDR method using permutations. We calibrate the FDR for each test according to this same permutation strategy for consistency. When subsetting the statistical tests by MAF range, BTH uncovers six associations, including a locus regulating variance of the gene DIS3L, recently identified as a possible risk factor for myocardial infarction , we normalized the values—dividing each covariate by its maximum such that each covariate had a maximum value of one—for stability of parameter estimation; this does not change the interpretation of our results. Both sex and smoking status are binary covariates, so the application of BTH is equivalent to testing for differential variance across the binary covariate. Overall, these data contain 46% females and 87% non-smokers.dglm and 17 associations using CLS , C14orf166 and FRG1. Among these, of interest is the gene C14orf166 which is involved in viral replication in the case of infection with influenza virus A  = 0;AH, the alternative hypothesis where α ≠ 1 or, equivalently, log (α) ≠ 0.BFs are computed by integrating over uncertainty in each of the model parameters β0, σ2 and α after computing a closed-form integral over the effect size β using multivariate Laplace approximations  of the log BFs was quantified using permutations. To do this, we developed a permutation that preserved the mean effects but removed any variance effects. In particular, for trait d, true positives (TP) and false positives (FP) are estimated using these BFs as d is computed as FDRd is computed from the data, and the pairs  with Global FDR calibration was performed after computing the unpermuted and permuted BFs, dglm and CLS tests were implemented and applied for comparison with BTH. The Brown-Forsythe and Levene tests both belong to the general Levene family of tests for equality of variance across k subgroups. For n samples corresponding to categorical covariate H0 corresponds to equal variances across subgroups AH corresponds to unequal variances across subgroups tn be the number of samples in x. The corresponding vectors y. For each t, let tω(s) be entry s of the t is the mean of the entries tsz, and y. When each tω(s) values instead of their mean, this becomes the Brown-Forsythe test. Significance and global FDR were computed based on permutations as with BTH, replacing BFs with P-values.The Levene, Brown-Forsythe, iy conditional on the genotype ix for each individual i using linear regression. The Spearman rank correlation test between the squared residuals, ix, correspond to a test for variance QTLs. This implementation of CLS followed the description in prior work . While dglm performs similarly to BTH in simulation, its reliance on glm results in convergence problems of the iteratively reweighted least squares procedure, thus making it unreliable when scaling to large applications.Finally, r models , where tage dglm . Signifiα  ≠  0 and β  ≠  0, α  ≠  0 and β  =  0, α  =  0 and β  ≠  0, and α  =  0 and β  =  0), we generated 50 samples of observed data from those six models. We computed one-sided Kolmogorov-Smirnov (KS) statistics between each of these samples and 100 samples, with matched sample sizes, generated from 79 distinct probability density functions with matched mean and variance. Thus, every sample is represented as a point in a 79-dimensional feature space. Performance of the RF classifier was evaluated using five-fold cross validation. The performance of the classifier was quantified using a precision recall curve on cross-validated simulation data (We trained a random forest classifier [RandomForest in scikit-learn (ion data .Genotype information for the HapMap phase 2 methylation study data are available from HumanHap550k and HumanHap650k genotype arrays (Illumina) at the GEO accession numbers GSE24260 . Using oHumanMethylation450 BeadChip platform were downloaded from the Gene Expression Omnibus (GEO), accession number GSE36369 .Control for known covariates; mean center: For each vector jy in matrix Y, corresponding to single gene j across all n samples, a linear model jr vectors gives us the normalized expression matrix.Control for unknown covariates: We computed the first two PCs of the normalized expression matrix using SVD. We used linear regression as in the previous step to control for the linear effects of these two PCs in the normalized gene expression matrix.The resulting matrix is the processed gene expression matrix. After empirical quantile normalization (Gene expression levels from 10 195 genes in lymphoblastoid cell lines (LCLs) created from 480 genotyped individuals were downloaded from the Gene Expression Omnibus (GSE36868). Genotypes for 387 514 variants and eight other covariates were available through dbGaP (Study Accession phs000481.v1.p1)  that allows for continuous covariates and incorporates uncertainty in estimates of mean and variance effects of covariates to robustly test for variance QTLs and QTCs. We evaluated our approach and compared it to state-of-the-art methods on extensive simulated datasets conforming to, and in violation of, the assumptions in our model. We described a prescriptive procedure to ensure a well-powered application of our model to diverse genomic and epigenetic study data. Although we are mainly focused on variance effects of genotype on quantitative traits, this approach may be used broadly in testing for heteroskedastic associations, and we show this application by discovering meaningful associations between non-genetic covariates and gene expression data.dglm are more conservative and less sensitive to multi-modal distributions than both CLS and the Levene test, as we showed in the multi-modal simulation studies and through spurious results from CLS and the Levene test in the methylation data. In scenarios where the data are close in distributional form to the modeling assumptions, as in the gene expression data, BTH finds similar numbers of associations as CLS. While our findings show that dglm outperforms the Levene, Brown-Forsythe and CLS tests in multiple simulation settings, its iterative approach to fitting often fails to converge because of sensitivity to step size. This shortcoming makes dglm cumbersome for genome-wide variance eQTL analysis, and we recommend BTH as a more reliable alternative. While BTH is three times slower than dglm, which often converges in under 1s on a machine with Intel Core running at 2.2 GHz, BTH is easily parallelized for genome-wide associations.In the Results, we note that BTH and The lack of results from BTH in the methylation data raise an important discussion point. In particular, the signature of gene × gene or gene × environment epistatic interactions may show up as a bimodal distribution of the trait: consider the distribution of a trait that has an eQTL with mean effect in women but not in men. We note that our statistical test was robust to deviations from unimodality, but CLS and Levene were not, making the purpose of these tests somewhat orthogonal. Thus, to identify candidate epistatic associations, CLS and Levene are the appropriate methods to use; on the other hand, to identify variance effects, our method is superior in terms of statistical power. We also hypothesize that the permutations that are used for these tests, while appropriate, lead to conservative estimates of FDR, which impacted all of the statistical tests calibrated using permutations.n samples are well powered to detect mean effects of a certain size, to detect comparable variance effects at the same precision, a sample of O(n2) is needed; thus additional samples will facilitate finding these effects. We also propose that these measurements of cellular traits are inappropriate candidates for variance QTLs because the variance effects will not be across individuals but instead across cells within an individual, as shown in previous studies (The lack of power in the variance QTL studies was clear: unlike mean effects, we found no variance effects of genetic variants on methylation, and six significant variance effects on gene expression levels. We note that, if  studies .Our BTH framework improves on existing methods with a flexible modeling framework, integrating over uncertainty, and fast robust statistical inference, leading to improved power to detect heteroskedastic associations. Identifying heteroskedastic associations in quantitative traits will augment our catalog of quantitative trait regulation and lead to an improved understanding of the mechanisms of genetic control over phenotypes.Supplementary FiguresClick here for additional data file.Supplementary TablesClick here for additional data file.Supplementary TextClick here for additional data file.