AprioriGWAS, which brings two innovations. (1) Based on Apriori, a successful method in field of Frequent Itemset Mining (FIM) in which a pattern growth strategy is leveraged to effectively and accurately reduce search space, AprioriGWAS can efficiently identify genetically associated genotype patterns. (2) To test the hypotheses of epistasis, we adopt a new conditional permutation procedure to obtain reliable statistical inference of Pearson's chi-square test for the AprioriGWAS to age-related macular degeneration (AMD) data, we found that: (1) angiopoietin 1 (ANGPT1) and four retinal genes interact with Complement Factor H (CFH). (2) GO term “glycosaminoglycan biosynthetic process” was enriched in AMD interacting genes. The epistatic interactions newly found by AprioriGWAS on AMD data are likely true interactions, since genes interacting with CFH are retinal genes, and GO term enrichment also verified that interaction between glycosaminoglycans (GAGs) and CFH plays an important role in disease pathology of AMD. By applying AprioriGWAS on Bipolar disorder in WTCCC data, we found variants without marginal effect show significant interactions. For example, multiple-SNP genotype patterns inside gene GABRB2 and GRIA1 (AMPA subunit 1 receptor gene). AMPARs are found in many parts of the brain and are the most commonly found receptor in the nervous system. The GABRB2 mediates the fastest inhibitory synaptic transmission in the central nervous system. GRIA1 and GABRB2 are relevant to mental disorders supported by multiple evidences.Identifying gene-gene interaction is a hot topic in genome wide association studies. Two fundamental challenges are: (1) how to smartly identify combinations of variants that may be associated with the trait from astronomical number of all possible combinations; and (2) how to test epistatic interaction when all potential combinations are available. We developed  AprioriGWAS that employees Apriori, an algorithm that pioneers the branch of “Frequent Itemset Mining” in computer science to cope with daunting numbers of combinations, and conditional permutation, to enable real signals standing out. By applying AprioriGWAS to age-related macular degeneration (AMD) data and bipolar disorder (BD) in WTCCC data, we found interesting interactions between sensible genes in terms of disease. Consequently, AprioriGWAS could be a good tool to find epistasis interaction from GWA data.Genes do not operate in vacuum. They interact with each other in many ways. Therefore, to figure out genetic causes of disease by case-control association studies, it is important to take interactions into account. There are two fundamental challenges in interaction-focused analysis. The first is the number of possible combinations of genetic variants easily goes to astronomic which is beyond current computational facility, which is referred as “the curse of dimensionality” in field of computer science. The other is, even if all potential combinations could be exhaustively checked, genuine signals are likely to be buried by false positives that are composed of single variant with large main effect and some other irrelevant variant. In this work, we propose  Gene-gene interactions have been proposed as one potential explanation of the well-known problem of missing heritability AprioriGWAS, a tool to address this problem. This tool is based on a successful algorithm in the field of computer science, AprioriThe main obstacle of interaction analysis is that the large number of multi-locus genotype combinations generated from large numbers of genetic variants is too high for current computational resources. This is in fact a well-known computational problem, known in the field of computer science as the ‘curse of dimensionality’ Apriori was originally designed for supermarket data mining to assist shop owners in designing the layout of displayed products. Given customers' transactions, the algorithm can identify sets of items that frequently co-exist in transactions. For example, by knowing that customers usually buy milk and bread together, the shop owner can put them near each other in the store.item is defined as an individual product, for example, bought by a customer; itemset stands for a set of items purchased together; length of itemset is defined as the number of items in the itemset. The process of growing a short itemset to a longer itemset is referred to as pattern growth. Generally, the key insights of Apriori are that: (1) frequent itemset with many items can be gained by growing itemset of short length; and (2) since subsets of any frequent itemset should also be frequent during pattern growth, itemsets predicted not to have any effect can be dropped during pattern growth, thereby significantly reducing the search space. In the case of GWAS, the number of individual genotypes is analogous to the number of transactions in supermarket data. The genotype of a variant is an item, and genotype combinations of different variants are an itemset, here also called a genotype pattern. Instead of just finding frequent genotype patterns, we want to find genotype patterns with different frequencies in cases and controls. We call them differential genotype patterns. While Apriori originally works on one database to find the most frequent itemsets, we are interested in patterns with different frequencies in two databases (cases and controls). To assess whether a pattern should be retained during pattern growth, we make use of the proportion testMethods).Before describing the algorithm, we briefly give definitions of a few key terms: Methods). In this step, we try to distinguish whether a differential pattern is caused by variants with marginal effects or by interaction effect. The process of pattern growth helps to narrow down the number of variant combinations to be tested for interaction effect.Interaction among variants is carried out after obtaining all differential genotype patterns. We test the possibility of interaction among variants involved in a differential genotype pattern by conducting Pearson's Chi-square test for the contingency tables composed of all genotype patterns found for variants and phenotypes . We demonstrate that our new approach has a magnitude lower false discovery rate (FDR) compared with regular permutation, while maintaining comparable power.Determining all potential combinations that are statistically associated with disease does not automatically identify genuinely interacting genes. The daunting number of all combinations of variants heavily increases the load of multiple tests and mixes genuine signals with noise. As summarized by Anderson AprioriGWAS to age-related macular degeneration (AMD [MIM 153800]), which has been deemed a good example of a small number of common variants explaining a large proportion of heritability AprioriGWAS to bipolar disorder; we found potential interactions inside individual gene  and interactions across gene or chromosomes. Further results will be presented in full later.We applied AprioriGWAS algorithm for mining possible interaction variants, as well as the conditional permutation approach for testing interactions. We then evaluate the performance of AprioriGWAS with simulated data and compare it with logistic regression implement in Epistasis function of PLINK. Lastly we demonstrate applications of AprioriGWAS to AMD and WTCCC bipolar data and exciting findings from both datasets.The remainder of this paper is organized as follows. In the next section we introduce the Apriori algorithm can be traced back to the seminal paper published by IBM Research in 1993 n. Apriori will usually scan dataset in n rounds . In the first round, it will initiate the 1-itemsets that are frequent. In each subsequent round, it will take the frequent itemset generated in the last round as starting point and grow any itemset by adding one more item. Retention of the new itemset will be decided by firstly predicting how likely it will be and then, given a positive prediction, by checking the actually supporting transactions. Finally, the collection of all frequent itemset in all rounds will be reported.Historically, the AT_609GG_1099CC, denoting a pattern composed of a variant with id 46 and genotype AT combined, a variant with id 609 and genotype GG, and a variant with id 1099 and genotype CC. The key goal is to find genotype patterns that have a significant frequency difference in cases and controls .In this paper, genotype patterns are defined as genotype combinations of different variants. We use integer numbers as ids of variants; then we can have, for instance, a pattern like 46AprioriGWAS is divided into two steps. First, detecting differential genotype patterns by an Apriori-like strategy. Obviously, the same set of variants can lead to several differential genotype patterns. Second, testing interaction among a set of variants by testing association of all possible combinations of genotype patterns against case/control status. The first step helps to narrow down the combinations of variants need to be tested. Due to multiple test problems and potential association of single variants involved in the differential genotype pattern, we adopt a new conditional permutation in the second step to control the marginal effect of single variants for testing of variant interactions.The algorithm of AprioriGWAS generally follows the flow of the original Apriori described above. Genotype patterns start from one single genotype, then, in each subsequent round, genotype pattern growth occurs by adding one more genotype of a new variant. Explicitly, for a given pattern length, we scan each pattern in the candidate set against all candidate genotypes of the remaining variants to see whether a variant should be included. The main difference is that the criteria of predicting whether the growth of a pattern should be retained is replaced by a proportion test The first step of vsπ. In the following equation, Essentially, the proportion test is to test whether a genotype pattern has the same frequency between cases and controls , one may be interested in comparing the null hypothesis of Y within individual strata of E, is not valid for specifically testing no interaction , there is no extra interaction among the variants that contribute to the association level. Precisely speaking, for any significance level, v in single marker test, and v has the highest marginal effect compared with other involved variants in the pattern.The null hypothesis Hv, To test whether A modified conditional permutation. Formally, the procedure is as follows: assuming variant v has the strongest marginal effect among the variants involved in a given differential pattern, we retain the association of v with the phenotype outcome v. This permutation thereby yields the null distribution of the p-value of Pearson's Chi-Squared test of the contingency table when a main effect of v is present.N to denote the vectors composed of n is the sample size, and use m to denote the number of variants. A permutation is denoted by a mapping More precisely: we use As explained above, performing parametric bootstrap tests for each pair of variants would be computationally expensive; now the same problem applies to our initial strategy. To make the test of interaction feasible for GWAS data, some computational tricks have to be employed. Since different variants with the same significance level of marginal effect will share the same null distribution, it is feasible to calculate the null distributions in advance and use that for each variant. We therefore choose to group variants with a similar level of marginal effects and use the same threshold for each group. For example, variants with p-value between (0.001∼0.0001) in the single marker test will use the same threshold calculated in advance from the null distribution. We thereby obtain the table of thresholds for patterns composed of variants with different categories of marginal effect and make use of them as a lookup table during the analysis.n≥3). For all other variants with p-value larger than 0.001 , we treat them as one extra group. For each group of variants we choose the lower limit of the p-value to do conditional permutation. For example, in the analysis of AMD data, among 103,611variants, 62 variants have p-values within , 8 variants have p-values in , one variant has a p-value in , one variant has a p-value in , and all others have p-values exceeding 103−. We choose the most significant one in each class, and do a conditional permutation for that variant, thereby obtaining the critical value for the contingency table composed of variants no more significant than the lower limit of each class. We then compare results from the non-permuted dataset with the pre-calculated lookup table to obtain the significant combinations of variants.Formally, we calculate the table of thresholds as follows. For variants with p-values in the single marker test of less than 0.001, we set one threshold for each order of magnitude  have population allele frequencies 0.05, 0.1, 0.2, and 0.5.For power simulations, we adopt all the parameters   (sample size 180). We simulate 1000 datasets for each model and calculate powers as described above.In classical textbooks on genetics, a technique to detect epistasis  is to check whether the proportions in an F2 population fit theoretical predictions of hypothetical interaction type Table 4.et alAprioriGWAS with the default parameter setting on further quality-controlled data.The AMD dataset analyzed in this paper was published by Klein Bipolar disorder data used is available from WTCCC AprioriGWAS, we compared significant differential patterns  detected by the default setting of AprioriGWAS with an exhaustive search in 3200 simulated datasets. Coverage is defined as the percent of differential patterns detected in each simulated dataset.To quantitatively estimate how many genuine differential patterns could be detected by AprioriGWAS is done by conditional permutation, as stated above.To assess the power of different methods, we simulated 1000 cases and 1000 controls genotyped at 1,000 variants with a single pair of causative interacting loci. For each model and combination of parameters, we simulated 200 datasets. The power for each model and parameters is thus defined as the number of datasets from which we find the two simulated interaction variants divided by the total number of simulated datasets (here 200). To make a fair comparison of power, we control family-wise type I error by conducting permutation for both methods. Controlling the FDR  of et alMethods), in which three types of interactions are generated. We then applied regular permutation and conditional permutation to control family-wise type I error.We simulated data by two-locus interaction models proposed by Marchini Methods) are demonstrated in The performances of regular permutation and conditional permutation test .As mentioned, AprioriGWAS) and exhaustive search. We found that 97% of all differential genotype patterns found by exhaustive search were covered by the results from AprioriGWAS. With such high coverage, the chance of losing possible interaction variants is minimized. There are a few points below 85%, reflecting that there is variation of power to cover all potential combinations. It is true that the overall coverage is subject to lots of parameters, like sample size and allele frequency. To minimize this variation, larger sample size is always desirable.AprioriGWAS to find interacting variants with traditional single locus genotypic test and exhaustive search in PLINK epistasis function). The epistasis function in PLINK for case control data is basically stepwise logistic regression. We chose to use the all combinations option. The power comparison is based on two levels: finding at least one casual variant, or finding both interacting variants  in stepwise logistic regression the genuine interactions are buried by the noise of a too large number of combinations, whereas with the conditional permutation test used in AprioriGWAS, genuine interactions are able to stand out.For Level 2, detecting both interacting variants, it is evident that  2 and 3 Figure 4AprioriGWAS and PLINK epitasis, maintained similar power for both levels. This was not unexpected since the interaction-based strategies should be better able to find an epistasis effect.When comparing corresponding panels in Figure S1)We also simulated data that have more SNPs  and find that the relative power between three methods and interaction models remain similar although the absolute powers are all decreased.  while AprioriGWAS has around 50% power.epistasis function in PLINK. AprioriGWAS was approximately a magnitude faster. Although retaining candidate genotype patterns in memory can help speed up the algorithm, its affordability is subject to the particular computational resources.We compared the speed of our method with the AprioriGWAS using HDF5-based data format We took the strategy of writing candidate patterns on hard disk for each round of pattern extension. The genotype data used to be relatively small comparing with the patterns however is getting larger and larger empowered by new sequencing platforms. To solve this problem, we implemented AprioriGWAS on published AMD data Table S1. By checking published functional literals and gene annotations, as well as GO enrichment of the genotype patterns, we learned that the findings are well validated by existing functional studies and clinical applications.We applied We found that ANGPT1, BBS9, PP7, MED27, CHRM2 interact with a well-known AMD gene, Complement Factor H (CFH). The most exciting finding here is ANGPT1, a protein with important roles in vascular development and angiogenesis. In fact, drugs for anti-angiogenic activity have been approved by the FDA in the United States for the treatment of cancer and AMD http://llama.mshri.on.ca/funcassociate/). We performed GO term enrichment analysis for significant pairs of genes with an online tool developed by Roth lab , presented in Go terms “synaptic membrane” (GO:0097060), “synaptic transmission” (GO:0007268), “transmission of nerve impulse” (GO:0019226) and “multicellular organismal signaling” (GO:0035637) are barely significantly enriched in found SNPs pairs Table 6.Table S2 are related with mental disorder). Good examples are GABRB2 and GRIA1 (α-amino-3-hydroxy-5-methyl-4-isoxazole proprionic acid (AMPA) subunit 1 receptor gene). AMPARs are found in many parts of the brain and are the most commonly found receptor in the nervous system. The GABRB2 mediates the fastest inhibitory synaptic transmission in the central nervous system. Multiple evidences showed that GRIA1 and GABRB2 are relevant to Bipolar Disorder and Schizophrenia http://www.genecards.org), we found that diseases associated with GRIK3, SULT4A1 and LRFN2 are schizophrenia, schizotypal personality disorder and neuronitis respectively, and SORBS1 is associated with insulin resistance.We found multiple-SNP genotype patterns inside individual genes , as a tool for detecting main and interaction effects of genetic variants in case-control association studies. One of its outstanding properties is that it can find variants whose disease association lives solely from their interaction without having (appreciable) main effects. We applied our approach to a published dataset on AMD and documented that AprioriGWAS furnishes sensible results. In fact, it found an AMD-associated variant (ANGPT1) not previously reported to be associated with AMD. We also identified interesting genes from WTCCC bipolar disorder data. One good point is that GO term enrichment analyses of all the genes identified, always show sensible terms for relevant disease. Our description of these findings is primarily intended to show the efficacy of our approach rather than to provide research findings about AMD and bipolar disorder.We have introduced AprioriGWAS can also use this to filter results before doing experimental validations. As an example, we use another independent dataset for AMD study Regardless the goal being interaction or single gene, statistical tests all suffer from the problem of false positives. Since the numbers of variants (and their combinations) are usually a few magnitudes larger than the sample size for most association studies, it will be common to see false positives. The current practice in the community is that researchers who would like to claim association or carry out experimental validations usually have to check whether the results are replicable in other independent dataset(s) Researchers who use The most commonly used multiple variants analysis is stepwise regression, in which variants are added to the regression equation one after another by some suitable criteria. But statistical analysis shows that the usual stepwise model selection methods are path dependent and therefore suboptimal Apriori corresponds to sample size in GWAS; the number of items is equivalent to the number of variants and the itemsets. In contrast to supermarket data, GWAS data have a limited number of “transactions”, but a large number of “items” in two datasets, cases and controls. Both conditions make the problem more difficult. The time spent reading the data in each round of pattern growth is constant. In addition, the computational resources cost depends on how many combinations of genetic variants will be generated and tested. The more combinations are tested, the less likely it is that genuine patterns are missed, though of course more resources will be used. In AprioriGWAS, there are several parameters for the user to specify according to their computer resources and understanding of the disease model. The threshold for the proportion test and minimal support of concerned itemsets are parameters that affect candidate search space, algorithm speed, and power of detecting all distinct genotype patterns. When these parameters are set to zero, AprioriGWAS will exhaustively search all possible combinations. Computation time and spatial complexities of the tool may be interesting to the reader. The number of transactions for original Apriori may suggest that, given Apriori's ability to also mine association rules, one could also treat the case control label as items and directly adopt Apriori for case/control data. The result will then be a subset of variants that can imply the case/control labels. But searching frequent itemsets and then mining the association between genotype pattern and disease status is inefficient, since frequent genotype patterns are not necessarily associated with phenotype; on the other hand, genotype patterns strongly associated with phenotype may not necessarily be in high frequency, and such an association could be distributed in different patterns than the same variants combinations.Those familiar with n variants with search length of m, the total number of combinations is huge. Given the natural correlation of the combinations, it is clearly far more stringent than necessary. However, only correcting on the number of differential pattern tested produces a bias in the other direction, since the nominal value of the significance level of the chi-square test for the Instead of the conditional permutation proposed here, one could also consider Bonferroni correction. For However, regular permutation suffers from an inflated significance level for contingency tables containing variants with marginal effects. This is due to the fact that when a contingency table is composed of at least one variant with strong marginal effect, the p-value for that contingency table becomes extremely small compared with regular permutation results. The FDR is therefore very high, even close to 1.v, with strong marginal effect, we developed a conditional permutation procedure (Methods), which helps get the null distribution of the p-value of a contingency table composed of the variant and other variants. Simulation results show that, when we control the family-wise type I error by conditional permutation, we also keep FDR well controlled. Compared with INTERSNP AprioriGWAS keeps FDR well controlled in a systematic way.To solve the problem of an inflated significance level by a contingency table composed of at least one variant, Another concern might be whether these differential genotype patterns are artifacts caused by linkage disequilibrium (LD). We believe this is not the case, since the LD should impact both cases and controls, and therefore the pattern created by LD will not be differential unless the LD structure is significantly different in cases and controls for particular genetic variants. If that is the case, then there must be some reason of selection to explain the deviation in the genotype pattern, and it is difficult to judge whether this is an artifact or something of interest. In addition, our conditional permutation also breaks LD between interacting variants.AprioriGWAS toward that direction. There may be non-trivial statistical challenges since the low-frequency or rare variants are usually less shared by the individuals therefore their combinations that form genotype patterns will be even less shared by individuals. For a given set of variants, we will have many patterns with little supports.Low-frequency or rare variation might impact the performance of the method, even when explicitly only testing for interactions among common variants. What matters is the extent of LD between causal rare variants and testing common variants. We haven't addressed this problem in the current method. It would be interesting to extend Figure S1Power comparison using 1,000,000 genetic variants. a. Power of finding both interacting variants for model 1; b. Power of finding both interacting variants for model 2; c. Power of finding both interacting variants for model 3. Power of finding both interacting variants for model 1, 2, and 3 . AprioriGWAS has much better power for Models 2 and 3, which do not show explicit marginal effect. The X-axis is the same as (TIFF)Click here for additional data file.Table S1Results of Age related Macular Degeneration (AMD). 168 pairs of variants show significant genotype pattern difference between case and control samples.(PDF)Click here for additional data file.Table S2Results of Bipolar Disorder from WTCCC. 200 pairs of variants show significant genotype pattern difference between case and control samples.(PDF)Click here for additional data file.


Genomic selection (GS) has become a commonly used technology in animal breeding. In crops, it is expected to significantly improve the genetic gains per unit of time. So far, its implementation in plant breeding has been mainly investigated in species farmed as homogeneous varieties. Concerning crops farmed in family pools, only a few theoretical studies are currently available. Here, we test the opportunity to implement GS in breeding of perennial ryegrass, using real data from a forage breeding program. Heading date was chosen as a model trait, due to its high heritability and ease of assessment. Genome Wide Association analysis was performed to uncover the genetic architecture of the trait. Then, Genomic Prediction (GP) models were tested and prediction accuracy was compared to the one obtained in traditional Marker Assisted Selection (MAS) methods.Several markers were significantly associated with heading date, some locating within or proximal to genes with a well-established role in floral regulation. GP models gave very high accuracies, which were significantly better than those obtained through traditional MAS. Accuracies were higher when predictions were made from related families and from larger training populations, whereas predicting from unrelated families caused the variance of the estimated breeding values to be biased downwards.We have demonstrated that there are good perspectives for GS implementation in perennial ryegrass breeding, and that problems resulting from low linkage disequilibrium (LD) can be reduced by the presence of structure and related families in the breeding population. While comprehensive Genome Wide Association analysis is difficult in species with extremely low LD, we did identify variants proximal to genes with a known role in flowering time (e.g. CONSTANS and Phytochrome C).The online version of this article (doi:10.1186/s12864-015-2163-3) contains supplementary material, which is available to authorized users. Lolium perenne L.) is one of the most cultivated forage species in temperate grasslands, mainly farmed for its re-growth capacity after defoliation, and for the high value as feed for ruminants, due to palatability, digestibility, and nutritive contents [Perennial ryegrass  is an important trait for forage species, often used as a model trait , 6 due tDue to its significant effects on other traits, breeding has always aimed to exploit the natural variation in HD, in order to create mixtures of varieties that could give high performances throughout the whole year. Since International listing of new varieties requires fulfilment of the three criteria; distinctiveness, uniformity, and stability (DUS) there is also a strong breeding focus on HD in order to create uniform varieties. While the uniformity of inbred varieties is rather easy to control it can be more challenging in outbreeding grass varieties that are breed as families.In order to ensure more stability in forage quality over the season, cultivars have been divided into different earliness groups. The number- and extend of each HD group differs between countries, with some countries defining up to nine HD groups. However, HD appears to behave as a continuous character, and the distinction between early and late material is not always clear, with new candidates that may be classified in different neighboring HD groups, depending on the definitions used in the different countries. The trait always showed medium to high heritability , 14–16. Arabidopsis thaliana L., as well as in cereals like wheat and rice heading or the control of flowering has been the subject for numerous studies and publications (reviewed in [Vrn), which regulates heading after low temperature periods; (ii) photoperiod response genes (Ppd), which is active/inactive with a certain day length; (iii) ‘earliness per se’ factors, which seems to be independent of light and cold requirements [In the latter decade, the genetic control of HD was better understood thanks to the use of molecular markers and comparative genome analyzes. In model species, such as iewed in –19). Espirements .Triticeae’s maps [Vrn1 gene [LpCO, homologous to the CONSTANS of Arabidopsis and the Hd1 of rice, involved in the photoperiodic regulation of flowering time [Hd3 region of rice [FLOWERING-LOCUS-T (FT) orthologue of Arabidopsis. FT gene is involved in induction to reproductive growth at the meristem [L. perenne [Lolium multiforum Lam. and Festuca pratensis L. [In perennial ryegrass a number of flowering genes were previously identified by sequence homology with flowering genes found in Arabidopsis, rice, and maize , 21–23. e’s maps . QTLs wee’s maps , 25–29. e’s maps , and lowrn1 gene , and itsrn1 gene . A relating time , 31, 32.meristem , 34 and  perenne . Other hWhile these studies identified some of the key genes in floral control in ryegrass, the biology of the trait is still far from being understood. Furthermore, the use of QTL analyses was shown to be not effective in capturing small effect genes  and to oGS is practically implemented trough different steps: (i) model development on a set of individuals/families that are both genotyped and phenotyped (training set); (ii) estimation of GEBVs for a set of individuals/families that are only genotyped , based on their relationship with the training set; (iii) selection of the best breeding material. In this paper we will refer to the first two steps as Genomic Prediction (GP). GS is now widely used in animal breeding , but it 2 families (F2s), phenotyped for HD and genotyped with high marker coverage, were used to dissect the genetic and genomic structure of the trait. First, a Genome Wide Association Analysis (GWAS), to check for the presence of major QTL was conducted. Second, significant markers were used to calculate the GEBVs in a set of synthetic (SYN) families, a part of which was related with the training set. Third, GP models were tested within the F2 set, using different cross-validation (CV) schemes and different population sizes, and then used to predict breeding values of SYN families. Predictive ability of GP was compared with predictions based on GWAS results.This paper represents our first attempt to introduce GP in a breeding program of forage perennial ryegrass, using HD as model trait. 1757 FResults from the Principal Component Analysis (PCA) showed the presence of some degree of population structure Fig. . The ‘elResults from the LD analysis are displayed in Fig. The total amount of phenotypic variance, together with different definition of heritability, is shown in Fig. 10(P) is clear from the QQ-plots reported in Fig. P < 0.05) was 10 using the t-test with Bonferroni correction, and 19 using False Discovery Rate (FDR) .Using the Bayesian Information Criterion, the optimal number of PCs for population structure correction was determined to be four, confirming the visual identification of the ‘elbow’ point. The effect of the correction on the significance levels expressed as –logHd1 homolog of the LpCO gene was also mapped. The marker 2801|42855 locates in a gene encoding for Phytocrome C (PHYC). The scaffold 1379 harbored two significant SNPs: 1379|60655, situated in a gene encoding for a Pectate Lyase 4, and 1379|64623, located outside the gene space, but only 3500 bp away from the Pectate Lyase 4 coding region.The SNP 5059|6359 is situated in a scaffold where the Prediction of SYNs based on SNP markers is shown in Fig. 2s, predictive abilities were extremely high, reaching the value of 0.84 in the absence of related families in the two sets (pp-fold scheme), and 0.90 when training and validation set contained related families, in the so called k-fold scheme . For the k-fold scheme, the accuracy computed using the approximation in (14) gave an estimate that was higher than one (1.04). However, the standard error (SE) of this estimate was 0.07, indicating that the actual accuracy could range between 0.91 and 1.00. Analyses on reduced training sets  was 1.39 using 175 families, and 1.10 using the full dataset. Bias for population sizes below 175 families is not shown, because affected by very large SE and not indicative of any trend. Accuracies within the set (UK) were lower than the ones found on an equal number of randomly chosen F2s, especially in the pp-fold scheme. Bias was also slightly higher. The CV within the other cluster gave more or less the same results as the CV within all F2s.For each Cross-validation (CV) scheme, predictive abilities, accuracies, and bias are displayed in Table eme Fig. . The Hot2 families was slightly lower . Predictions for GEBVs were better when the set (UK) was used as training set. In this case, accuracies were comparable to the ones obtained within all F2s, with a pp-fold scheme, and using a similar population size. The bias level indicates that the GEBVs variance was underestimated when the set (UK) was used as training population, and slightly overestimated when the prediction was performed in the opposite direction. Predictive ability for SYNs . The accuracy was 0.93, 14 % higher than the in prediction based on the significant markers. In this case, the linear regression of mean corrected phenotypes on GEBVs indicated no bias in the GEBVs variance.Predictions across sets also worked well. Accuracy of predicting UK set from the other FYNs Fig.  was simi2 families were grouped in one big cluster. This may lead to the hypothesis of a common European genetic pool. This pool is likely to originate from the continuous and (more or less) free exchange of breeding material among the different breeders. The parents of the set (UK) may be an exception to that pool, and their genetic origin need to be further investigated. The relation between PC3 and HD indicates the need to correct for population structure while performing GWAS, in order to avoid false positives. Further variance analyses were performed by adding fixed regressions for the first 1, 2, 3, and 4 PCs to the equation shown in formula (2). Result indicates a correlation of HD with the PC3, but not with the other three main PCs. When accounting for the first two PCs, the additive genomic variance across location was equal to the 98 % of the ones of the model without any PC. When accounting for the first three PCs, the additive genetic variance left was the 89 %. Adding a regression for PC4 had a negligible effect.The population structure was mainly defined by the origin of the PPs, which was correlated with the first principal component. The majority of the F2 > 0.25) to extend at best 1 kb. However, in the present breeding material, the presence of relatedness and population structure generally increased the LD, bringing to an increase by the order of three in the average distance between those markers. This fact suggests that population structure, which is known to be mostly responsible for the long range LD, also plays an important role in increasing the level of LD within scaffolds. The correction also led to a decrease in the average distance between markers in LD, which was more pronounced for higher LD levels.The LD within scaffolds showed to decay rapidly, confirming the concerns expressed by Hayes et al. , who repσ2pply may indicate the presence of dominance acting between families and within single environments. In the literature, the presence of non-additive effect is reported also across location [2 families.Estimation of variance components confirm results obtained by Fè et al.  on a sublocation . HoweverThe GWAS analysis revealed a rather complex genetic architecture of HD in ryegrass. Several markers with significant effect were identified. In the most significant SNPs, a shift from one homozygous form to the other can cause changes of up to 2.78 days in the date of heading. That is a remarkable difference, if compared with the level of variation in the phenotypes: average phenotypes corrected for fixed effect had a SD of 4.92. However, due to low Minor Allele Frequency (MAF), these markers were only able to explain a small proportion of the additive variance, which may indicate the presence of a large number of genes also affecting the trait, but with effects lower than the detection limit. There were a high number of significant SNPs found outside the gene space. This is not too surprising considering a recent study in maize found that the majority of trait associated variants were located outside annotated genes, but within 5 Kb of transcriptional start and stop sites .CONSTANS (CO) plays a crucial role in promoting flowering in response to long days [CO homolog. It has already been established that a ryegrass homolog to CO exhibits expression patterns consistent with its function in Arabidopsis, and can complement co mutants [CO homolog co-located on linkage group seven with a large effect QTL for HD [CO was found to be significantly associated with HD in a collection of 96 perennial ryegrass genotypes (originating from nine populations) [CO contributes to variation in HD in perennial ryegrass, specifically within a large collection of breeding families.Some of the significant SNPs were clearly linked to genes that may have a direct or indirect influence on HD. It is well established that ong days . We iden mutants . FurtherL for HD , 31. Alllations) . The ~29PHYC. Phytochromes are red/far-red photoreceptors that play a role in how a plant responds to light, and adapts its growth and development. It was recently demonstrated in wheat that PHYC plays a major role in accelerating flowering under long-days [PHYC represses flowering under non-inductive conditions. The fact that loss-of-function mutations in PHYC resulted on average in a 108 day delay in flowering of wheat under long days emphasizes the potential for allelic variation at this gene to greatly alter flowering times. A similar role for PHYC has also been recently reported in Brachypodium distachyon [Brachypodium, and a similar role for PHYC in floral induction of perennial ryegrass is possible. A homolog to PHYC has been mapped to linkage group four of perennial ryegrass [FT. That may be due to different reasons: (i) absence of causative polymorphisms in the breeding material; (ii) no or low LD between markers and the causative polymorphisms ; (iii) low MAF at the causative polymorphisms (about 45 % of the markers had MAF lower than 0.05); (iv) polymorphisms not detected because they are correlated with the family structure and shrunken by the correction with G-matrix and PCs (the third PC was clearly correlated with HD).We also identified a significant SNP within the coding region of a homolog to ong-days , in contstachyon . Perenniryegrass , althoug2 population. However, the presence of a certain level of population structure , even tA very high accuracy may also seem in contrast with what reported in the literature for traits affected by major effect SNPs . Theoretk-fold scheme) were significantly higher than the ones obtained in the absence of related families (pp-fold scheme), for any population size. Regarding the relationship between predictions and the size of training population, increasing the training size led to pronounced gains in predictive abilities for population sizes lower than 500 families, and to smaller gains in the case of larger populations. Despite that, due to the higher predictive abilities, accurate predictions could be obtained even using a relatively small training set. Problems of underestimations of the GEBVs variance may arise in absence of closely related families. Regarding predictions across sets, the relative difficulty in predicting the set (UK) may be due to families in this set having fewer relatives in the chosen training population. The level of bias may arise due to the differences in genetic variance or to genetic correlations that are less than unity between the two sets. This problem could compromise a correct comparisons between GEBVs from the training set (which have known bias) and the validation set (which have unknown bias). This should be taken into consideration during the design of the training population, allowing the presence a wide variety of genotypes in as many environments as possible.Accuracies reached by predicting from related families . Despite this, the technique used for GWAS has limitations, mainly due to the marker density given the rapid decay of LD, and due to the strong structure in the population.The study has also revealed important details about the genetic architecture of HD in Both Phenotypic and genomic data were available for a total of 1846 families of forage diploid perennial ryegrass. All breeding material was part of a standard forage breeding program run by DLF A/S . Unlike cereal breeding, population based forage breeding usually does not advance further than the second generation. Each year, the best breeding material is selected and added to the company’s gene bank, which also includes European varieties, commercialized both by DLF and other companies.2s: 1757 F2 families produced across 13 years (between 2000 and 2012) from a seed bank of 198 PPs. Development of F2 families was detailed in Fè et al. [1 seeds; (iv) isolated multiplication of F1 populations in isolated plots for random mating; (v) harvesting of F2 seeds; (vi) field trials of F2 families (assumed to be in Handy-Weinberg equilibrium).Set 1. F2 families, by visual merits and according to the synchronous heading time. After crossing, SYNs production followed the same protocol described for F2 families, involving pooling, multiplication of the seed in isolated plots, and testing in field trials.Set 2. SYNs: 89 families obtained by random mating between 5–11 single plants. Single plants were selected from the highest biomass yielding FThe plant material consisted of two different sets:2 families, and sequenced on multiple lanes of on an Illumina HiSeq2000 (single-end). After basic data filtering, the average number of reads per family was about 20 million. Data for each family was then aligned against a draft sequence assembly. 1,879,139 SNPs were identified, distributed across 30,285 scaffolds. Sequencing depth at a SNP ranged from 1 to 250 (upper limit) reads per family. SNP positions having more than 60 reads were discarded, as suspected to be originated from plastid genomes or from highly repetitive regions not captured in the draft assembly. No threshold was set in relation to the minimum number of reads. That could lead to a poor estimation of the allele frequencies and, consequently, to underestimation of allele substitution effect. However, it is possible to take account of this problems by using specific corrections, as showed by Ashraf et al. [Sequence data was produced by Genotyping-By-Sequencing (GBS) . GBS usef et al. . Markersst’. Data were available for a period of 11 years (between 2003 and 2013), and for two locations: Store Heddinge (South-Eastern Denmark) and Didbrook (Southern England). Fields were divided in trials, each consisting of randomized 24 sward plots, arranged in 2 sub-trials. Plot size was 1.5*10 m in Denmark and 0.5*4 m in England. Randomization was ensured within trials, but not always across trials. In some cases, especially in the oldest experiments, families were sorted according to the flowering time, or to the year of origin. That resulted in a certain degree of unbalance, within locations, between trials and PPs. A summary of the phenotypic data is displayed in Table Phenotypic data were collected, within the standard breeding procedures of DLF. Families were sown during spring and scored during the following season. HD was assessed on family means by visual scoring, and defined as the day in which, two-thirds of the spike is visible on at least one plant in the plot or one third of the spike is visible in three plants in the plot. The character was expressed as ‘days after May 1Xi×j) for all families was calculated from all SNP markers, after filtering for SNP depth and allele frequency (MAF > 0.02). Firstly, allele frequencies were arranged in a matrix K is a scaling parameter, corresponding to the sum of expected SNP variances as computed by Ashraf et al. [2 families on a set of 100 scaffolds larger than 20 kbp, randomly sampled across the whole genome. The LD was expressed as squared correlation between markers (r2). Corrections for both relatedness and for population structure were performed according to the method described by Mangin et al. [where f et al. , being 0NbClust’ . The proNbClust’ . LD withn et al. .2 set and compared by F-test (for the fixed part) and Akaike test (for the random part). The models that showed the best fit to the data is reported below:Data were analyzed by linear mixed models, using the software DMU , 56. They is the vector of phenotypes; X is the design matrix for the fixed factor; t is the vector of trial effects nested within location and year; Zi are design matrices for random factors; i is a vector of breeding values ~ N, where G is the G-matrix; ily is a vector of genotype × environment interactions ~ N; p is a vector of the originating PPs ~ N, with P being a genomic relationship matrix among PPs (P-matrix) built as described in the following paragraph; pply is the vector of interaction between PPs (which would mainly arise from dominance effects) nested within environments ~ N; e is a vector of random residuals ~ N. Additional factors for breeding values and PPs, with identity matrices as variance-covariance structure were tested to check for presence of genetic effects not explained by G- and P- matrix. However, such effects turned out to be not significantly different from zero and were left out from the model. The same was for the interactions among PPs and between PPs and environments, and for the spatial effect within trials. Breeding values were calculated by summing the corresponding solutions for i and p:where j indicates family and j1 and j2 indicates the parents population for family j.where Z3 was built to account for the presence of multiple PPs, as shown in Additional file P was computed based on the estimated frequencies of the PPs, following the same procedure that used to compute G-matrix. PPs frequencies were estimated for each SNP marker, using the following model:Matrix fi is the vector of frequencies for marker i; μi is the mean frequency for marker i; Z is a matrix of random effect, accounting for the presence of multiple PPs, built as explained in Additional file pi is a vector of originating PPs ~ N. The estimated PPs frequency for a marker i was equal to:where 1 and F2 families; (ii) PPs in Hardy-Weinberg equilibrium; (iii) large number of families originated by each parent combination; (iv) parent plants chosen at random from the PPs; (v) absence of self-pollination; (vi) no intercross among F1 families; (vii) absence of selection between F1s and F2s; (viii) uniform variances across different factors. Here, the only difference in respect to the original model is represented by the relationship among PPs. That would cause inbreeding between the F1’s, [2s (described by P-matrix and by G-matrix respectively), and within F2s. The latter component can be ignored, as analyses are based on family means. The G-matrix also accounts for the increase in inbreeding within the F2 families. Variance components were estimated by restricted maximum likelihood method (REML), and can be interpreted as follows: σ2i is the additive genetic variance among families, across environments; σ2ily is the additive G × E variance; σ2p is the variance among PPs across environments; σ2pply is the variance of the G × E for dominance; σ2e is the variance of residuals, which includes environmental effects within plots and measurement errors.The model was based on the additive biallelic infinitesimal model described by Ashraf et al. , which whe F1’s,  and leadAcross PPs, it was possible to compute three kinds of heritabilities for a single observation: (6) narrow sense heritability across environments; (7) narrow sense environment-specific heritability; (8) broad sense environment-specific heritability:σ2p was added twice, as each F2 family was originated from two PPs, and where σ2P is the phenotypic variance, calculated as:where the component GWAS analysis was performed by using the software GAPIT . CorrectXi and Z are design matrices for fixed and random effects respectively; α0i is the allele substitution effect for locus i; pc is the vector for PCs effects; i is the vector of breeding values with G-matrix as variance-covariance structure distributed as N; e is a vector of random residuals, distributed as N. Missing genotypes, for each marker, were imputed with the average allele frequencies across families, as was done for computing G-matrix. The significance of each marker effect was evaluated using t-test after Bonferroni correction and FDR [2 > 0.10), only the marker with the lowest P-values was regarded as significant. Allele substitution effects were corrected for low sequencing depth [where ĝ is a vector of breeding values, calculated from the model shown in equation (3), but assuming all variance covariance matrices to be identity matrices;  and FDR , using ang depth , using t0 is the allele substitution effect as estimated from GWAS, α is the corrected allele substitution effect, D is the average sequencing depth across families, and i refers to a given locus.where αAs the allele frequencies are expressed on family means, the genetic variance would be half of the variance between individuals , and thei refers to a given allele. For all the SNPs that were declared significant in at least one of the tests, σ2gi was calculated both within the F2 families, and using the MAF of the SYN families. Then, a single marker regression was performed in the SYN families, in order to check the association between marker effects in the two sets. Later, all SNPs were ordered based on their P-values, and then used to estimate the GEBVs of the SYN families:where p is the MAF, α is equal to the allele substitution effect, and ĝ is the vector of GEBVs, M is equal to the number of significant markers, αi is the allele substitution effect for marker i, and pi is the MAF at marker i. The calculation was performed multiple times, assuming different values for M (from 1 to 50 markers).in this equation, 2 sets: (i) all F2 families; (ii) different clusters of F2 families, previously identified during the PCA; (iii) reduced sets of randomly chosen F2s, differing for size of the training populations. Within each set, CV was performed according to two different schemes testing different hypothesis: (a) k-fold (k = 100) tests predictions in case of presence of related individuals in the training and in the validation set, leaving out families in random order and estimates their breeding values; (b) pp-fold tests predictions in case of absence of related individuals in the training and in the validation set, estimating all the families originated by a certain PPs combination, after having left out everything that had at least a PP in common. As pp-fold implied a greater reduction in terms of training population compared to k-fold, a pp-like strategy was also tested, in order to ensure the same training population size is used in both schemes. This strategy exactly replicated the cross-validation scheme used in pp-fold, but leaving out random families instead. Analyses on set (iii) were repeated ten times, each time using a different set of randomly chosen F2s, and the average predictive abilities and bias were calculated. Then, CVs were performed across clusters and, finally, all the F2s were used to predict the breeding values of the SYN families.GP studies were carried out by GBLUP , 61, usiρg,ĝ). In this case, its value is not known, but can easily be computed by using the following equation [Accuracy is defined as the correlation between true breeding values and GEBVs (equation :14\documyf), defined as predictive ability, and the denominator represents the expected correlation between GEBVs and g and ĝ, which is not guaranteed to fall within the theoretically defined range of the parameters. The expected correlation between GEBVs and where the nominator is the true correlation between GEBVs with the average phenotypes corrected for the fixed effect , and represents the upper limit for the prediction accuracy. However, this formula refers to a very simplified model with only genomic and residual variances. In the present paper, the equation needs to account for the other random components:where n is the average number of replicates across all fields (nplots/nfamilies); nily is the average number of environments per each family (nily/nfamilies) and npply is the average number of environments per each PP (nPPly/nPP). When predictions were performed on the same dataset, comparison between predictive abilities from different models was performed with Hotelling-Williams test [As different families were replicated a different number of times, ams test , using tams test . The biaĝ is the vector of GEBVs from the CV procedure. Absence of bias will result in a regression coefficient (b) of 1. A significant deviation from 1 indicates bias in the estimation of the GEBVs’ variance.where


B. vulgaris) is an example of binomial trait of agronomic importance. In this paper, a panel of 192 SNPs (single nucleotide polymorphisms) was used to genotype 124 sugar beet individual plants from 18 lines, and to classify them as showing “high” or “low” root vigor.Genomic information can be used to predict not only continuous but also categorical  traits. Several traits of interest in human medicine and agriculture present a discrete distribution of phenotypes (e.g. disease status). Root vigor in sugar beet  approach. From a 5-fold cross-validation scheme, 500 testing subsets were generated. The estimated average cross-validation error rate was 0.000731 (0.073%). Only 9 out of 12326 test observations (500 replicates for an average test set size of 24.65) were misclassified.The estimated prediction accuracy was quite high. Such accurate predictions may be related to the high estimated heritability for root vigor (0.783) and to the few genes with large effect underlying the trait. Despite the sparse SNP panel, there was sufficient within-scaffold LD where SNPs with large effect on root vigor were located to allow for genome-enabled predictions to work. Most of current research and applications in genetics are driven by the large quantity of data on individual genomic polymorphisms produced by modern high-throughput genotyping and sequencing technologies . A thrivGenomic data are used to predict future or unobserved events (e.g. disease risk ), or theB. vulgaris) is an example of binomial trait of agronomic importance.Genomic information can in principle be used to predict continuous or categorical (ordered or unordered) polygenic traits. Most works so far focussed on continuous traits, while fewer studies dealt with genomic predictions for categorical traits -11. HoweIn this paper, SNP (single nucleotide polymorphisms) genotypes were used for the classification problem of predicting “high” or “low” root vigor inidividual plants in sugar beet.Pioneering works on genomic predictions for continuous traits in sugar beet already exist ,14. HoweB. vulgaris) plants from 18 high- and low-root-vigor lines were available. These lines were characterised by different productivity and were provided by Lion Seeds Ltd. (UK). Root vigor is related to nutrient uptake from the soil and plant productivity |X) of having either high or low root vigor given the predictors was modeled assuming a continuous underlying latent variable l (“liability”). A threshold model [In a GBLUP (Genomic Best Linear Unbiased Predictions) framework , the prold model  of the fl the vector of continuous gaussian liabilities, g the vector of additive genetic values of individual plants, and e the vector of logistically distributed residuals; X is a design matrix that allocates records to genetic values. The genetic and residual variances were s=1;I is the identity matrix). The narrow sense heritability for root vigor was then estimated as: with i into high- or low-root-vigor plant was obtained from the cumulative distribution function of the logistic distribution (i.e. the logistic function: pi >/≤ 0.5.Low and high root vigor (coded as 0 and 1 respectively) phenotypes were the input of model 2, which returned a probability  of belonging to either class. The probability of classifying each observation In order to obtain a valid estimate of the classification error from model (2), a 5-fold cross validation procedure was adopted . The 123n is the number of observations in the test set and I (·) an indicator function which returns a value of 1 if the predicted and observed phenotypes are different, 0 otherwise. The cross-validation (CV) error rate was then estimated averaging the test error rate over all replicates: where Beagle was used to impute missing genotypes [Asreml was used to fit the threshold model in (2) and estimate variance components [Python code. Data preparation and figures were produced with the open source statistical environment R[The programme enotypes . The commponents . Genomicronment R.Figure h2 = 0.783, with a standard error of 0.086.Model fit was evaluated comparing the full (model (2)) and the reduced (null model: the intercept only) models through a likelihood ratio test. Deviance dropped significantly , showing good fit of the model. The estimated genetic variance was 11.856, on the liability scale. From Eq. 3 heritability was then estimated as From cross-validation, 500 testing subsets were generated. In each of these, observations were classified according to the model fitted to the corresponding training subset, and the classification error calculated as in (4), then averaged over all subsets. The estimated cross-validation error rate from (5) was 0.000731 (0.073%). Only 9 out of 12326 test observations (500 replicates for an average test set size of 24.65) were wrongly classified. All 9 missclassified observations belonged to a single low-root vigor line (line “LOW1”).P (Y = 0|X )= 0.5 dotted threshold line.In the training sets observations were always correctly classified (training error rate = 0). The average estimated probability of having low or high root vigor was calculated for each plant across all 500 replicates of the training and testing sets. The box plots in Figure B. vulgaris) has been addressed. A very low cross-validation test error rate (<1%) was estimated for the genome-based classification of root vigor in sugar beet lines. Genomic predictions with different accuracies have been reported in literature: high  content [In this paper, the problem of classifying binomial phenotypes using SNP markers in sugar beet , moderale trees ) and lowle trees ) accuracle trees ) reporte content ,14.%).The few prediction errors were all observed in line “LOW1”. This line had the strongest off-diagonal genomic relationship with line “HIGH1”. “LOW1” and “HIGH1” differ for root vigor (and related genes) but share most of their genetic basis. The close relationship of “LOW1” to “HIGH1” (a high root vigor line) may well explain why all 9 misclassifications were observed in this line, considering that SNP genotypes -through the genomic relationship matrix- were used as predictors. The error rate in line “LOW1” was nonetheless very low  and to Some relevant aspects are discussed below.The high predictive ability for root vigor estimated in this study (>99% of correct classifications) may be related to the heritability of the trait and to the number of segregating QTL underlying its expression: this is sometimes referred to as the “genetic architecture” of the trait .%). For a highly heritable trait, using genetic markers to correctly predicting phenotypes is expected to yield good results. The effect of heritability on error rate was checked by fixing the heritability in model (2) instead of estimating it from the data. This has the effect of altering the covariance structure in the GBLUP mixed model equations from model (2) by loosening the relationship between genotypes and phenotypes. Artificially lower heritabilities of 0.5, 0.33 and 0.20 were tested by rerunning the analysis with the same cross-validation scheme and number of replicates. Figure h2 = 0.5, 0.33 and 0.2 was 0.058 (5.8%), 0.128 (12.8%) and 0.181 (18.1%), respectively.Root vigor has been estimated to have high heritability (≈80logit (pi) = μ + SNPm (pi = P (Y = 1|μ + SNPm); SNPm: individual genotype at SNP m) was fit to the data. The magnitude of estimated SNP effects is reported in the barplot in Figure Not only the total genetic variance of a trait plays a role in determining the accuracy of genomic predictions, but also how this variance is spread along the genome (i.e. the distribution of QTL/genes underlying the trait). For a polygenic trait (i.e. determined by a large number of genes) a large number of markers would be needed to ensure that the genetic variance is fully captured. Contrariwise, for an oligogenic trait (i.e. determined only by few genes), even a small number of markers -as in the present study- may be sufficient to capture the genetic variance of the trait, provided that these markers are close to the relevant QTLs. A single-SNP genome-wide association study was performed in order to estimate marker effects for root vigor. A logistic regression model of the form r2[B. vulgaris genome was not assembled in chromosomes, but organised in 82305 scaffolds (and contigs). The sugar beet genome sequence comprises 567 Mbps of which 85% could be assigned to chromosomes [The extent of linkage disequilibrium (LD) in the experimental population is a parameter relevant to the success of genomic predictions. The basic assumption underlying genome-wide predictions is indeed that observed genetic markers and unobserved QTLs are in LD . LD betwr2- is deemr2. Dense mr2, was 0.061. This values is below what is needed for genomic predictions to work, but refers to all markers, not only adjacent markers. Adjacent markers could be determined only within-scaffold; the average within-scaffold pairwise LD was r2 = 0.404, which seems to be largely sufficient for reliable genomic predictions. Also the LD between markers with large effect on root vigor  randomly distributed along the genome. Such missing genotypes are imputed using pedigree-based or pedigree-free methods. Imputation accuracy is typically very high; for instance, >95% correctly imputed genotypes were reported in maize [Imputing missing genotypes is usually a preliminary step to the analysis of genomic data. After markers and individuals with low call-rate are edited out, there is usually still a small proportion of uncalled genotypes .In order to estimate the accuracy with which genotypes were imputed in the present study, a subset with no missing genotypes was extracted from the total dataset. Increasing proportions of missing data were then artificially introduced in the data: 1%, 2%, 3%, 5%, 10% and 20%. For each proportion of missing genotypes, 5 random replicates were generated. The average proportion of correctly imputed genotypes over 5 replicates for each proportion of missing data was then used to estimate an empirical curve of the imputation accuracy. Results are summarised in Table The threshold model used in this study for genome-enabled prediction of the binary trait root vigor in sugar beet was compared with Support Vector Machine (SVM), another widely adopted method for classification of categorical observations ,33.C to be used in SVM were chosen so to minimize the classification error through 5-fold cross-validation. A linear kernel (i and p parameters) and C=0.01 were chosen and used to classify sugar beet individual plants with SVM in the same cross-validation procedure adopted for the threshold model . The estimated error rate was close to zero (0.025%), in line with what was obtained with the threshold model (0.073%). The two classifiers were compared also by looking at the ROC curves [The kernel function and tuning parameter C curves : the twoRoot vigor, expressed as high root elongation rate, is essential for the efficient acquisition of mobile soil nutrients ; this isRoot traits are difficult to be measured accurately and this is an obstacle to reliable and effective selection. Genomic data can be used for early and accurate prediction of root vigor in sugar beet seeds, thereby enhancing the efficiency of breeding for rhizospheric stress tolerance and yield in sugar beet. Improvements are likely to come from shortened breeding cycles and more accurate and less expensive phenotypic evaluation.In this paper, the use of genomic information to predict a binomially distributed phenotype (root vigor) in sugar beet populations was presented. Prediction accuracy proved to be quite high, with an estimated cross-validation error rate close to zero (0.073%). Such excellent prediction performance may be related to properties of the analysed trait and available population. Root vigor was estimated to have high heritability (0.783) and to be determined by few genes with large effect. Despite the sparse SNP panel, there was sufficient within-scaffold LD where SNPs with large effect on root vigor were located. For an oligogenic highly heritable trait with a favorable distribution of markers on the genome, even with relatively few SNPs very accurate predictions can be achieved. The results described in this paper constitute an interesting application of genomic predictions to binomial  traits, and may lead to promising applications of genomic selection in sugar beet breeding programmes.The authors declare that they have no competing interests.FB carried out all statistical analyses and drafted most of the manuscript. PS, CB and MS selected the experimental population and generated all molecular and phenotypic data. AS contributed ideas to the work and drafted parts of the manuscript. All authors read and approved the final manuscript.


A penalized likelihood based method called Firth logistic regression method may provide a simple yet effective solution. It is easier to implement and less computational intensive than alternative approaches such as permutation or bootstrapping, and worthy of more attention in association studies of sequencing data.In association tests of sites with low minor allele frequency or count, it is known that single-variant tests are impractical to use because the results from which will be either underpowered or unreliable. Joint analyses by pooling or “collapsing” multiple variants based on annotated gene group information are thus more preferred in rare variant association tests. However, the issue remains in a genome-wide association scan because there is always a portion of regions containing less number of variant sites. Moreover, most current exome or genome sequencing association studies are still limited to small sample sizes. Standard testing methods that rely on the asymptotic theories will also not preserve the type I error rate. These factors together will distort the final genome-wide quantile–quantile plot of the testing p-values —sometimes even without a warning message. Although approaches like Fisher's exact test and exact logistic regression can be used to handle the separation problem, their use become problematic when there are continuous covariates need to be considered. The implementation of firth logistic regression is fairly easy as it is now available in many standard packages (such as R package “logistf”). In a recent work, Ma et al.  (Heinze and Puhr, The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Genome researches have revealed that a large portion (over 50%) of genetic variants on human chromosomes are rare variants (RVs) with extremely low allele frequency  in populations. In recent years, advances of DNA genotyping and sequencing technologies have been facilitating the discovery of RVs, and the association between RVs and human diseases is of rapidly growing interest in understanding genetic and molecular mechanism of both common and rare diseases , RVs have been identified to be associated with some human diseases or disease-related complex traits, such as cholesterol level  of a set of RVs have effects on a trait. However, the power can be compromised by neutral or “noise” variants  mingled in analysis. To deal with this issue, many bioinformatics databases and tools , how to select a RV set is critical and will significantly affect the power. Although typically RVs are grouped and selected by gene, there are many other ways to determine the RV sets, for example, by chromosomal region, by exon, by functional pathway, etc. The allele frequency cutoff can be fixed  or a variable threshold method . Analysis of data from such designs can be performed in either a simple way that tests the enrichment of RVs in one group (vs. another group), or a more sophisticate way that models the group assignment as a binary outcome variable dependent on individual and/or summarized RV variables as well as other confounding covariates  if necessary. Although most designs and statistical methods were originally introduced for binary outcomes from unrelated subjects, many of them have been extended to quantitative traits and family data (Fang et al., Since the potential of power improvement via statistical methods is limited, the ultimate and most effective way of increasing power yet relies on larger sample sizes, which can be realized by either increasing the sample size in a single experiment, or combining the results from multiple experiments into a meta-analysis using recently developed methods (Hu et al., 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.


Dominance effects may contribute to genetic variation of complex traits in dairy cattle, especially for traits closely related to fitness such as fertility. However, traditional genetic evaluations generally ignore dominance effects and consider additive genetic effects only. Availability of dense single nucleotide polymorphisms (SNPs) panels provides the opportunity to investigate the role of dominance in quantitative variation of complex traits at both the SNP and animal levels. Including dominance effects in the genomic evaluation of animals could also help to increase the accuracy of prediction of future phenotypes. In this study, we estimated additive and dominance variance components for fertility and milk production traits of genotyped Holstein and Jersey cows in Australia. The predictive abilities of a model that accounts for additive effects only (additive), and a model that accounts for both additive and dominance effects (additive + dominance) were compared in a fivefold cross-validation.Estimates of the proportion of dominance variation relative to phenotypic variation that is captured by SNPs, for production traits, were up to 3.8 and 7.1 % in Holstein and Jersey cows, respectively, whereas, for fertility, they were equal to 1.2 % in Holstein and very close to zero in Jersey cows. We found that including dominance in the model was not consistently advantageous. Based on maximum likelihood ratio tests, the additive + dominance model fitted the data better than the additive model, for milk, fat and protein yields in both breeds. However, regarding the prediction of phenotypes assessed with fivefold cross-validation, including dominance effects in the model improved accuracy only for fat yield in Holstein cows. Regression coefficients of phenotypes on genetic values and mean squared errors of predictions showed that the predictive ability of the additive + dominance model was superior to that of the additive model for some of the traits.P < 0.01) for all milk production traits but not for fertility. Accuracy of prediction of phenotypes was slightly increased by including dominance effects in the genomic evaluation model. Thus, it can help to better identify highly performing individuals and be useful for culling decisions.In both breeds, dominance effects were significant ( There is limited knowledge on how much genetic variation is explained by non-additive effects in dairy cattle, especially for fertility traits. Dominance is a non-additive genetic effect that arises because of the interactions between alleles at the same locus . The totHowever, dominance effects are usually ignored in the genetic evaluation of livestock. In livestock, pedigree information is rarely informative enough to accurately estimate dominance effects since it is difficult to find large populations with considerable dominance relationships such as full-sib families and these effects are often confounded with other non-genetic effects. The prediction of dominance effects using pedigree information is computationally very demanding since it requires large datasets and complex algorithms to compute and invert dominance covariance matrices between individuals , 3. EstiInformation from genome-wide single nucleotide polymorphism (SNP) panels have been commonly used in dairy cattle to detect additive effects of SNPs in association studies (e.g. ), or to The objectives of this study were: (1) to estimate additive and dominance variance components for fertility and milk production traits in Holstein and Jersey cows using covariance matrices between individuals for genomic effects obtained from high-density SNP genotypes; and (2) to investigate whether future phenotypes could be predicted with a higher accuracy by including both additive and dominance effects compared to a model that accounts for additive effects only. We also investigated the relationship between estimated dominance effects and level of inbreeding.2) between imputed genotypes and real genotypes were greater than 0.9 for both breeds. After imputation, 632,003 SNPs distributed over 29 Bos taurus autosomes (BTA) and the two sex chromosomes remained. Genotypes of 7902 and 7501 Holstein and 4014 and 3685 Jersey cows with calving interval (CI) and milk production records, respectively, were used in this study.Genotypes were obtained from the Australian national genomic reference population. Genotyping was done using the Illumina BovineSNP50 v2 BeadChip  and these genotypes were imputed to the 800 k high-density SNP panel using 1785 animals (including 1534 Holstein and 251 Jersey key ancestor bulls and heifers) with genotypes from the Illumina BovineHD BeadChip . Imputation was done using BEAGLE 3.1  on qualiPhenotypic records were 305-day milk, fat and protein yields and a measure of fertility, i.e. calving interval (CI). Records for all traits were extracted from the Australian Dairy Herd Improvement Scheme  database. To accurately remove contemporary group effects from the cow phenotypes, a larger dataset comprising 11,294,808 multiple parity records of respectively 3,425,911 and 482,325 Australian Holstein and Jersey lactating cows between 1985 and 2011 were used to estimate fixed effects of age at calving, month of calving, parity and herd-year-season (HYS) management group in a single-trait model. All cows in the full dataset were required to have at least four grandparents of the same breed (Holstein or Jersey). For production traits, milk, fat and protein yields less than 2000 L, 50 and 45 kg and more than 15,000 L, 800 and 600 kg, respectively, were discarded and a minimum and maximum of 270 and 370 days, respectively, for days in milk were set to minimize the effect of extreme lactations on estimates of means and variances. For the fertility dataset, CI had to be within 290 and 550 days to be valid and CI between 551 and 762 days were set to a maximum of 551 days in accordance with . FurtherEstimates of variance components for each trait were obtained by using the genomic best linear unbiased prediction (GBLUP) method in two univariate models that either included only additive (A) or both additive and dominance (A + D) genetic effects as follows.For the model with additive effects only (A):y is a vector of pre-corrected phenotypes for each trait; 1n is a vector of ones and µ is the population mean term; u1 and u2 are vectors of breeding values (BV) of animals and are assumed to be distributed as G being the additive relationship matrix; d is the vector of dominance deviations (DV) for animals and is assumed to be distributed as D is the dominance relationship matrix; pe1 and pe2 are vectors of random permanent environmental effects with e1 and e2 are vectors of random residual terms distributed as  W is the incidence matrix relating observations to animals and the variances are as follows: For the model with both additive and dominance effects (A + D):G) and dominance (D) relationship matrices were constructed based on the information from genome-wide SNPs as described in [The additive  and 2qi for aa, Aa and AA genotypes, respectively, with pi and qi being the allele frequency of A and a alleles at marker i in the population. For the dominance relationship matrix D, aa, Aa and AA genotypes in M were respectively coded as −2pi2, 2piqi and −2qi2. This way of coding the genotypes guarantees the absence of confounding between additive and dominance variances and allows a direct comparison of genomic-based estimates of variance components with the pedigree-based counterparts [where terparts . Variancterparts . Model AN is the total number of SNPs, pi is the frequency of the most common allele at SNP i, and xi is coded as 0, 1 and 2 for aa, Aa and AA genotypes, respectively. We also calculated the proportion of the SNPs that were homozygous (i.e. the SNP genotypes that were either aa or AA), which was used as another measure of inbreeding.To investigate the relationship between dominance effects and inbreeding, the correlations between estimated dominance deviations and inbreeding coefficients of animals were calculated. Two different measures of inbreeding based on genomic data were used. First, the diagonal elements of the genomic relationship matrix based on , which aχ(1)2 at P = 0.01 is actually the critical value for the observed test statistic to be in the top 0.005 of the null distribution. To compare the accuracy of prediction of phenotypes obtained from each model, a fivefold cross-validation was performed. Five partitions of the data with approximately equal sizes were generated at random for each trait, except that paternal half-sibs were only present in onefold, so that animals across folds were less related. Such a cross-validation scheme was also used in Khansefid et al. [t test.The goodness of fit of models A and A + D were compared based on the maximum likelihood (ML) ratio test. Two times the difference between maximum log of likelihood (LogL) of models A and A + D for each trait were compared to their expected values under a null model with mixed Chi squared distributions with 1 and 0 degree of freedom. In such a comparison, the d et al.  and Bolod et al. . Furtherd et al. ). For reTo evaluate the effect of accounting for dominance variation on genetic evaluation, animals were ranked based on their estimated breeding values or total genetic values and additional comparisons were made based on Spearman׳s rank correlations across the two models under different hypothetical percentages of selection.Estimates of additive variance and additive heritabilities for CI and production traits were similar regardless of whether dominance effects were included in the model Table . AdditivCorrelations between dominance deviations obtained from the dominance model and two measures of inbreeding level  are in Table P values of their conservative likelihood ratio tests are in Table P < 0.01) for all production traits but not for CI, which shows less dominance variation than for production traits in the Holstein breed and no dominance variance in the Jersey breed.The maximum LogL values from models A and A + D and the We assessed the predictive ability of the models in three ways: (1) Pearson’s correlations between estimated genetic values and phenotypes adjusted for fixed effects; (2) regression coefficients of adjusted phenotypes on estimated genetic values; and (3) mean squared errors (MSE) of prediction of adjusted phenotypes from the estimated genetic values. The results are presented as averages for five validation datasets.P < 0.01) only for fat yield in Holstein cows and for the total genetic values compared to predictions based on additive effects only.Average correlations between estimated breeding values or total genetic values and adjusted phenotypes in the validation data for fivefold cross-validation showed a slightly better predictive ability with model A + D than with model A for most of the traits Table . Standarbadjusted phenotype, BV) and A + D  in the fivefold cross-validation analysis are in Table Average regression coefficients of adjusted phenotypes on estimated genetic values obtained with models A (P < 0.01) for fat and protein yields in Holstein cows . The average rank correlations of animals in the five validation sets according to their estimated breeding values from model A and total genetic values from model A + D are in Table 2 (squared coefficient of correlation between SNP and causative mutation), whereas for the dominance variance, it depends on (r2)2, i.e. r4 [The proportion of dominance variance relative to phenotypic variance was small for all traits studied here. Dominance variance accounted for 3.3, 3.8 and 3.2 % of the phenotypic variance for milk, fat and protein yields, respectively in Holstein cows. Larger proportions of dominance variance relative to phenotypic variance were estimated for milk (5.1 %), fat (6.6 %) and protein (7.1 %) yields in Jersey cows. For CI, estimated dominance variance was about 1.2 % in Holstein but very close to zero in Jersey cows. The additive and dominance effects estimated for a SNP depend on the extent of linkage disequilibrium (LD) between the SNP and the causative mutation. Specifically, the proportion of additive genetic variance observed at the SNP depends on the r i.e. r4 . Thus, mEstimation of additive and non-additive variances for economically important traits in dairy cattle is usually based on pedigree relationship matrices. Tempelman and Burnside  reportedhd2 = 0.23) and casein contents (hd2 = 0.21) showed a significant (P < 0.05) contribution of dominance variance to the total phenotypic variance.The use of genomic data instead of pedigree information in genetic evaluations has led to a renewed interest in the prediction of non-additive genetic effects. Sun et al.  reportedThe range of the estimated contributions of non-additive genetic variance to total genetic variance varies considerably across studies. The reasons that may explain these inconsistencies could be differences in the definition of traits, the size of the dataset used in the analyses, inconsistencies in the estimation of dominance relationships between individuals (pedigree vs. genomic data), the models used to estimate yield deviations, pre-selection of genotyped animals, breed- and population-specific differences and also confounding between additive and dominance variances as argued by . For exaVariation in the level of inbreeding within populations or breeds could be another important factor to explain the inconsistencies reported for the level of contribution of dominance gene actions to the total phenotypic variation of the trait of interest. Dominance variance increases with inbreeding up to a point (inbreeding level of about 0.25 ) and popInbreeding influences complex traits by increasing the number of homozygous loci and corresponding reductions in heterozygosity. It has been hypothesised that the action of (semi) lethal recessive mutations is the major reason for this , but it P < 0.01) in Holstein cows. We found similar results to those reported in [Using additive plus dominance effects, rather than additive effects only, led to slightly better predictions of phenotypes Table . The accorted in , which sorted in . RegressOne of the important consequences of ignoring non-additive effects could be the biased evaluation of individuals and subsequently incorrect rankings based on estimated breeding values. It has been argued in other studies that including dominance effects in genetic evaluation models could result in more accurate estimates of additive genetic variance and breeding values, thereby increasing the accuracy of predictions , 33. ThiAnother use of dominance information would be to allocate matings by considering specific combining ability of the parents and predicting the outcome of every possible mating to choose the matings that can maximize the productive performance of the offspring. Ertl et al.  comparedThe small dominance variances estimated for CI could also be related to the measure of fertility used in this study. Calving interval has a low heritability, and the very large environmental variance makes it difficult to estimate dominance effects (and additive effects for that matter) accurately. Other fertility measures with a higher heritability including detailed traits may help to better understand the role of non-additive gene actions for traits that are closely related with fitness. For example, Bolormaa et al.  estimateDominance variance contributed up to 3.8 and 7.1 % of the phenotypic variation of milk production traits in Australian Holstein and Jersey cows, respectively, whereas this contribution was much smaller in the case of CI. However, when compared to the additive component, dominance represented a significant proportion of the total genetic variance for all milk production traits. Dominance correlated unfavourably with inbreeding which suggests that more inbred animals benefit less from dominance in the direction of improved performance. The better performance (better goodness of fit and prediction accuracy) of model A + D for some traits compared to model A suggests that dominance should be included in models for the genetic evaluation of animals to improve the accuracy of prediction of future phenotypes. This could also become a useful tool for culling decisions on farms, and use of the total genetic of potential progeny in mating plans may improve progeny performance.