APOE is a major susceptibility gene for AD. In the top genes selected by proposed method, several could be functionally linked to AD. In particular, a pathway analysis of these genes also highlights the metabolism of cholesterol, that is known to play a key role in AD pathogenesis. Interestingly, many of these top genes can be integrated in a hypothetic signalling network.Results from Genome-Wide Association Studies (GWAS) have shown that the genetic basis of complex traits often include many genetic variants with small to moderate effects whose identification remains a challenging problem. In this context multi-marker analysis at the gene and pathway level can complement traditional point-wise approaches that treat the genetic markers individually. In this paper we propose a novel statistical approach for multi-marker analysis based on the Rasch model. The method summarizes the categorical genotypes of SNPs by a generalized logistic function into a genetic score that can be used for association analysis. Through different sets of simulations, the false-positive rate and power of the proposed approach are compared to a set of existing methods, and shows good performances. The application of the Rasch model on Alzheimer’s Disease (AD) ADNI GWAS dataset also allows a coherent interpretation of the results. Our analysis supports the idea that  With the recent improvement of high-throughput genotyping technologies, the use of Genome-Wide Association Studies (GWAS) has become widespread in genetic research to identify significant associations between genetic markers such as Single Nucleotide Polymorphisms (SNPs) and complex phenotypes such as common diseases. GWAS generally yield results at the SNP-level, that are sets of SNPs associated with the disease. However, the vast majority of loci that have been identified for common diseases show modest effects and generally explain only a small part of the variance or heritability of the phenotype observed . In a reIt is increasingly recognized that analyzing the combined association of multiple markers at the gene or pathway level may provide a complementary approach to the more common single SNP association approach, with several key benefits . First ip-value of all the SNPs assigned to the set of SNPs, i.e. the p-value of the most significant SNP The following short Trait’ is a binary trait disease coded by 1 for cases and 0 for controls, the association of the multi-marker Rasch genetic score to the disease can then simply be assessed with a logistic model:>glmIf ‘Trait’ is a quantitative trait, the association of the multi-marker Rasch genetic score to the disease can then simply be assessed with a linear model:>lm(Trait ∼ score)If ‘false-positive rate and power based on simulations over three scenarios of dependence between SNPs and varying levels of association. For each scenario, we consider:–Kp = 0.05.a binary disease trait (500 cases and 500 controls) of prevalence –a set of 24 SNPs including 12 disease susceptibility loci (DSL) simulated with relative risks ranging from 1 (no association) to 2 (strong association).The performances of our Rasch-based multi-marker genetic association test are first evaluated in term of This simulation framework detailed hereafter follows principles widely used previously –22.a and A having the frequencies pa and pA = 1 − pa. p0, p1 and p2 are the frequencies of genotypes aa, aA/Aa and AA defined by the Hardy-Weinberg proportions:F is the consanguinity coefficient. This coefficient can indicate a deficit (F > 0) or conversely an excess (F < 0) of heterozygous. Here, we consider F = 0, so that the locus is under the Hardy-Weinberg equilibrium. We then want to compute the genotype frequencies of the SNP for cases and controls pDi and pHi where i = 0, 1 or 2 using the disease prevalence Kp, the penetrances f0, f1 and f2 of the genotypes and the mode of inheritance. The main modes of inheritance can be defined by considering the relative risks f0 = Kp/(p0 + RR1 × p1 + RR2 × p2), f1 = RR1 × f0, f2 = RR2 × f0 and the Bayes formulas, we can easily derive the desired frequencies:The simulation model for one SNP is based on the Wright’s model  applied The 24 SNPs are simulated independently according to this model, the 12 non-associated SNPs with a relative risk of 1 and the 12 DSLs with a relative risk ranging from 1 to 2.Scenario 1. Then the remaining SNPs are completed based on a real dataset (here the chromosome 6 of the ADNI dataset described below) in order to generate one LD blocks of moderate magnitude (0.4–0.7) around each DSL. Simulating this way leads to genetic patterns similar to those found in real data and therefore allows us to finely control the level of LD between SNPs.To account for SNPs in Linkage Disequilibrium (LD), our simulation model follows an approach based on the diplotype frequencies of real datasets. These frequencies are used as an empirical distribution of the range of possible diplotypes. First, 12 DSLs are simulated independently from the model described in Scenario 2 with the difference that we consider SNPs in strong LD (0.8–1).The simulation is the same as for B = 1000 simulations in order to provide accurate Monte-Carlo estimates of false-positive rate and power. For each simulation we obtain a p-value of association of the set of SNPs simulated by applying our Rasch-based multi-marker association test. The false-positive rate is estimated by PrH0 and the power is estimated by PrH1, with α the significance level usually set to 5%. Consequently in our simulations, by placing ourselves under the null hypothesis H0 of no association (RR2 = 1), then under the alternative hypothesis H1 of association (RR2 > 1), we can respectively estimate both false-positive rate and power of our method by considering the same quantity:p-values inferior or equal to α.For each scenario and each level of DSL relative risk, we ran –minP [p-value of the set of SNPs considered as the p-value of the set. This method is obviously biased since it does not take the multiple-testing and the dependence of tests into account. It is used here as a negative control and also because it is nevertheless the most widely used approach in practice.minP  is the s–GATES [p-values computed by a standard linear trend test of association on each SNP are combined with the control of correlation structure: significant p-values in high LD count less than significant p-values of independent SNPs.GATES  is a mul–Fisher [p-values. For m SNPs, the multi-marker test statistic is given by m degrees of freedom under the null hypothesis when the m tests are independent. An adjustment to dependent tests is also available and used here [Fisher  is the wsed here .–SKAT [SKAT  is SNP-sWe compared the performances of our Rasch-based multi-marker association test to three existing methods:APP, PSEN1 and PSEN2 genes) and Late-Onset AD  [Alzheimer’s disease (AD) is the most common neurodegenerative disorder and affects more than 35 million people worldwide. It is characterized by brain atrophy reflecting neuronal and synaptic loss and the presence of amyloid plaques and neurofibrillary tangles, leading to a progressive deterioration of cognitive functions involving memory, reason, judgment and orientation . AD pathfactors) .APOE is a major susceptibility gene for LOAD [APOE, markers within several other genes gave replicated evidence of association with LOAD [CR1 has been demonstrated to be able to produce an AD up-regulated protein [In recent years, several Genome-Wide Association Studies (GWAS) were performed to detect genetic loci associated with LOAD –31. Thesfor LOAD . In addiith LOAD . The ide protein . AlthougPlink  [By way of illustration, we applied our Rasch-based multi-marker association test to the genes of the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu) . The stu of 0.2) . Missing of 0.2) . SNPs arThe top genes identified by the Rasch analysis were integrated into a hypothetic signalling network. Protein-protein interaction data and functional findings were extracted from QIAGEN’s Ingenuity Pathway Analysis , manually analysed and supplemented by literature curation.Rasch, minP, GATES, Fisher and SKAT across the three scenarios are given minP has a strongly inflated false-positive rate, far above the expected 5% level and decreasing with the level of LD . This observation was actually expected knowing the drawbacks of the minP method, and validates our simulations. On the other hand, Rasch, GATES, Fisher and SKAT have a correct control of the false-positive rate to 5% (Scenario 3 (Rasch is the closest to the 5% level (estimated to 0.043) whereas GATES and Fisher are more conservative (estimated to 0.039 and 0.034 respectively) and SKAT is more inflated. In term of power, Rasch has the best performances on independent SNPs followed by Fisher . It is well-known that its ɛ4 allele has been associated with an increased risk of developing Alzheimer’s disease [ZNF398 (p = 9.71e−6) and AEN (p = 1.27e−5). AEN encodes an enhancing apoptosis nuclease, a process that takes part to the neuronal loss observed in AD. We unfortunately did not find any indication about the possible functional implication of ZNF398 in AD.The association of 16514 genes to the Alzheimer’s disease (AD) was analyzed with our Rasch-based multi-marker association test. Standard QQ plots is given  disease . This re−3  were identified as potent modulators of insulin signalling [Alzheimer’s disease is sometimes named ‘type 3 diabetes’ due to twice more frequent occurrence in diabetic patients , 46. Twognalling , 48. Notgnalling .–NTM encodes a neural cell adhesion molecule that modulates neurite outgrowth and adhesion via a homophilic mechanism [NTM might directly bind to amyloid beta [echanism . Some daoid beta . It has oid beta  and liesoid beta –SEMA7A belongs to the semaphorins family involved in neuronal processes. Semaphorins and their downstream signaling components regulate synaptic physiology and neuronal excitability in the mature hippocampus, and these proteins are also implicated in a number of developmental, psychiatric, and neurodegenerative disorders [SEMA7A not only enhances axon growth via beta1-integrin, but equally processes immune-modulatory activity and regulates endothelial functions [ADAMTS12) is also implicated in control of immune responses and angiogenesis, deregulated in course of Alzheimer’s disease [isorders . Remarkaunctions , 56. As  disease , 58.–LARP1 protein associates with the mTOR complex 1 (mTORC1) regulating global protein synthesis. Functional importance of mTOR signalling has been experimentally confirmed in Alzheimer’s disease, and therapeutic targeting of this signalling module is considered as a promising strategy for developing neuro-protective treatments [Finally, eatments –61.As we noticed a slight deviation from the QQ-line at 10−3 , we alsoITGB1 mediates effect of SEMA7A on axon growth [CASR gene that forms a functional complex with metabotropic glutamate receptor GRM5 [GRM5 is a co-receptor for cytotoxic Aβ oligomers bound to prion PRNP protein [We also performed a formalized network analysis based on these top genes with the Ingenuity Pathway Analysis. The resulting network is given n growth . The inttor GRM5 , 67. It  protein .With the recent improvement of high-throughput genotyping technologies, the use of Genome-Wide Association Studies has become widespread in genetic research. However, the high dimension of the genetic data, the simultaneous testing of many markers and the necessity to account for the complex genetic structure of human populations are, among others, tricky issues that have raised doubts about the relevance of these studies’ findings. The development of methods in Statistical Genetics is therefore very important to ensure that such studies are correctly conducted and to provide a proper interpretation of their findings, and this research has involved scientists from many disciplines. In this context, applying the Rasch model initially developed for psychometric data to the analysis of genetic data can be viewed as a new link between two areas of research that was not obvious before. Our novel statistical approach may be useful to complement at the gene or pathway level, the findings of significant associations made at the single SNP level.minP, GATES, Fisher and SKAT). We noticed that the benefits of Rasch in terms of power were more important when applied to independent SNPs which is coherent with one of the assumptions of the model that the response to different items are independent. As this loss of power is observed for all the methods when the level of dependence between the SNPs (Linkage Disequilibrium) increases, a Rasch model taking dependence into account could be of interest and further increase the power of the method.Based on simulations, it showed in different situations good performances in terms of false-positive rate and power compared to other popular methods  can be functionally linked to Alzheimer’s disease. A pathway analysis of these genes also highlights the metabolism of cholesterol, that is known to play a key role in AD pathogenesis. All these elements can be integrated in a hypothetic signalling network based on known protein-protein, functional and phenomenological interactions (The application of the Rasch model to the genes of the Alzheimer ADNI GWAS data allowed a coherent interpretation of the data. Our analysis supports that ractions . IntereseRMR package does not converge and consequently does not provide any results. It happened for instance to 9 genes over the 16514 genes analyzed in the ADNI GWAS data and the reasons of that problem were not clear to us. Finally applying a Rasch model necessitates accessing individual level genetic data. But often, only summary statistics are available for published GWAS. This is a real limitation for most of the existing multi-marker methods in order to correctly account for gene size and LD, although some authors have found a solution in using the genotype data from a reference panel such as the 1000 Genomes or the HapMap projects [Through this study, we encountered three limitations for the application of the Rasch model. First, it works on complete data without missing values. However missing values are a common problem in most scientific research domains as they can arise from different sources such as mishandling of samples, low signal-to-noise ratio, measurement error, non-response or deleted aberrant value. Consequently the application of the Rasch model requires preliminary imputation of missing values. This imputation is a general and separate scientific topic that has been thoroughly discussed to date –76. Secoprojects –79 whichThe application of the Rasch model also opens two opportunities that were not yet considered here. The analysis of multiple markers is not limited to the gene level, and the Rasch-based multi-marker genetic association test could also be applied to the analysis whole pathways. In addition, this genetic score could also be used as a predictor of the disease for the supervised classification of cases versus controls. The Rasch model is also suitable to the inclusion of rare variants, as most rare variants analyses focus on gene level test by collapsing the effects of all rare SNPs in a gene into a single test of association . These aFrom a broader point of view, given the urgent need to understand how the thousands of loci that have been identified in genome-wide association studies contribute to the genetic basis complex traits, the application of multi-marker methods at the gene or pathway level becomes an increasingly important approach for secondary analysis of GWAS data –82. Main


Phalaris arundinacea L., Phalaris caesia Nees), a cool-season grass species with potential as a biofuel crop. Our study involved two linkage populations and an association panel of 590 reed canarygrass genotypes. Plants were assayed for up to 5228 single nucleotide polymorphism markers and 35 traits. The genotypic markers were derived from low-depth sequencing with 78% missing data on average. To soundly infer marker-trait associations, multiple imputation (MI) was used: several imputes of the marker data were generated to reflect imputation uncertainty and association tests were performed on marker effects across imputes. A total of nine significant markers were identified, three of which showed significant homology with the Brachypodium dystachion genome. Because no physical map of the reed canarygrass genome was available, imputation was conducted using classification trees. In general, MI showed good consistency with the complete-case analysis and adequate control over imputation uncertainty. A gain in significance of marker effects was achieved through MI, but only for rare cases when missing data were <45%. In addition to providing insight into the genetic basis of important traits in reed canarygrass, this study presents one of the first applications of MI to genome-wide analyses and provides useful guidelines for conducting GWAS based on genotyping-by-sequencing data.Genotyping by sequencing allows for large-scale genetic analyses in plant species with no reference genome, but sets the challenge of sound inference in presence of uncertain genotypes. We report an imputation-based genome-wide association study (GWAS) in reed canarygrass ( In its 2005 billion-ton supply report, the US Department of Agriculture (USDA) and the US Department of Energy (USDOE) set the goal of a 30% replacement of US petroleum consumption with biofuels by 2030. This goal implies a production of approximately 1 billion dry matter tons of biomass per year from forest and agricultural lands. According to the report’s projections and assumptions, achieving that objective will require the following: (i) the conversion of active croplands, pasture land, and lands under the conservation reserve program (CRP) into perennial crop lands and (ii) achieving biomass yields ranging between 5.5 and 8 dry tons per acre from perennial crops (between 12.4 and 19.8 Mg ha-1) . Plant b-1) , resista-1) , leaf arPhalaris arundinacea L., Phalaris caesia Nees) is a promising HEC in North America. It belongs to the tribe Avenae , which includes the oat genus Avena (Brachypodium distachyon (Brachypodium), Oryza sativa (Rice), Zea mays (maize), and Sorghum bicolor (Sorghum), Brachypodium is the most closely related to reed canarygrass , which is thought to be native to Europe, Asia, and North America  (P. arundinacea found in North America are of European ancestry (P. arundinacea trace back to the early 20th century in North America (th century in Europe (Reed canarygrass (6x = 42) . Most cu America , but thin Europe . Some non Europe , and papn Europe . HoweverPanicum virgatum L.)  or sliding-window algorithms, may be used. However, other methods are available for imputing unordered marker data  provides the opportunity to call polymorphisms without prior marker development. With this technology, a genome-wide association study (GWAS) can be performed on reed canarygrass, but because no reference sequence is available, one limitation of such study is the inability to assign single nucleotide polymorphisms (SNP) to specific physical positions in the reed canarygrass genome. This has two important implications: (i) SNP-trait associations cannot be mapped to a particular region of the genome and (ii) no imputation method based on marker sequences, Phalaris aquatica  in reed canarygrass. GWAS was performed under the assumption of disomic inheritance in reed canarygrass, which is suggested by the alloploidy of a closely related species, Supporting Information, Table S1), and one association panel (AP) comprising 590 clones originating from North America and Europe (Table S1). LP1 and LP2 were derived from two distinct crosses involving genotypes of WR00, a tetraploid cultivar originating from Wisconsin. In AP, four populations  were accessions of P. caesia (hexaploid). The AP subset consisting of only the 550 P. arundinaceae clones is referred to as AP-4x.Three panels were assessed in this study: two randomly segregating populations derived from biparental crosses (LP1 and LP2) comprising 177 and 189 clones, respectively  and Ithaca, New York . Soil type was Plano silt loam  in Arlington and was Niagara silt loam  in Ithaca.http://dfrc.wisc.edu/sniper/.Traits related to disease resistance (Ds), morphology , phenology , and quality  were determined on plants grown in Ithaca in 2009–2011 and in Arlington in 2010–2011 . Qualityi, modeled as fixed . For all other terms, the corresponding effects were considered random, independent, and identically normally distributed: j; l within location j; m within location j; × indicates interactions; and r and column c, within an environment (year l within location j). Plot effects within environments were modeled asAssociation analyses were performed on best linear unbiased estimations (BLUEs) of genotypes’ performance for a given trait, inferred from the following linear mixed model:http://dfrc.wisc.edu/sniper/.The matrix of genotypes’ BLUEs, computed as described above, can be downloaded from ApeKI restriction, and sequencing were performed according to http://www.maizegenetics.net/gbs-bioinformatics; http://dfrc.wisc.edu/sniper/.Genome reduction, by e.g., PMV <0.20). However, we did not filter these markers to avoid removing potentially useful information, and also because a rigorous study of the behavior of the MI approach at different missing rates is merited. After this first filtering step, 18,818 markers were retained. Marker variables were then discarded if they met one of the following criteria: (i) being “constant”, i.e., having a variance close to 0 (only 24 markers were discarded based on that criterion) or (ii) being “collinear”, i.e., being correlated by at least 0.999  to some other marker variable with a smaller amount of missing values in the dataset. This filtering step was recommended by Table S2). Also, for markers not meeting the criteria of Association analyses) tended to show larger deviations from those obtained based on nonmissing data only ((CC); see Association analyses), no matter what the imputation uncertainty was . Finally, effects of discarded markers as estimated from multiple imputes ((MI); see Association analyses) tended to show excessively strong shrinkage in comparison with (AD) estimates, with little adjustment of estimates in response to imputation uncertainty .Markers were selected based on proportion of missing values (PMV) <0.90 in each of three panels, separately. It is typical to filter out GBS SNPs by a predetermined low missing rate  reflect low sequencing depth of the GBS. From the quality control step, the marker data matrix θ , multiple imputation (MI) aims at producing a correct Monte Carlo (MC) approximation of m imputes of i.e., the sampling model is correct. Another implication that is of particular concern when performing MI is that the distribution i.e., the imputation procedure is proper  i.e., that the total variance of Consider some incomplete data set s proper , which i≥Var(θ¯) . Conditi≥Var(θ¯)  is neces≥Var(θ¯) , i.e., te.g., linear regression) or nonparametric models . This flexibility is particularly valuable when dealing with missing values at many variables, in which case JM may simply not be practical , one must sample from the joint distribution of missing values at all variables. To achieve this, two strategies have been proposed: joint modeling (JM) and fully conditional specification (FCS). JM consists of sampling from the joint distribution directly, by ordinary MC . This meractical .In this study, we sampled missing values from The general algorithm was:For impute For For sample For Return m is the total number of imputes of q is the number of marker variables, and L is the number of MCMC iterations.where m was set to 20, based on available computational and memory resources. The number of iterations L up to which the actual sampling occurred  was set to 10, based on previous studies . For the MI algorithm to be computationally tractable, only a subset of SNPs was considered for fitting the CART models: for each SNP k, only the 500 SNPs showing the highest marginal mutual information with SNP k were considered as potential predictors.MI, based on CART, was implemented according to MI implemented as described above took 1 d per imputation (chain of 10 iterations) on a workstation consisting of 24 Intel Xeon X7460 CPU processors at 2.66 GHz with 264 Gb of RAM. The procedure was parallelized on five i.e., each element in http://dfrc.wisc.edu/sniper/.Two types of marker data were generated from the multiple imputation procedure: the MI set of matrices P. arundinacea samples) consisting of Figure S1 shows, for each subset separately, the distribution of PMV and MAF for markers selected prior to conducting GWAS. The distributions are equivalent across subsets and similar to that observed with the larger set of 6138 markers, with the exception that the average MAF increased from 0.28 to 0.32.Only tetraploid samples were considered for GWAS, and the following subsets of the data were examined separately when testing marker-trait associations: LP1, LP2, and AP-4x  approximation for computational efficiency , the linear mixed model of e rrBLUP  was usedQ and K estimated from γ; see below) was high. Estimates from (MI) were expected to show appropriate control over imputation uncertainty, i.e., shrinkage toward zero as γ increases. In (MI*), marker effects are assessed from Q and K is also accounted for, through β. Although (CC), (AD), and (MI) are convenient to assess the impact of imputation uncertainty on β estimates, (MI*) should be the method permitting the most statistically sound inferences regarding marker-trait associations.(CC) is the analysis restricted to complete cases  for a given marker tested, with the approximation of In AP-4x, population structure and relatedness were, respectively, accounted for by one principal component and by matrix β, as described in p value was  and (AD), significance of marker-trait associations was assessed by performing an F-test for the regression coefficient estimate alue was :Pr, (AD), or (MI) were transformed into adjusted false discovery rates (FDR) using the method of Q and K estimated from Discussion). The R package qvalue , panel (second and third PCs), and geographical origin—Eastern Europe vs. North America or Western Europe (fourth PC). The fifth and sixth PCs do not seem to reflect any grouping and explain a small proportion of the total genotypic variation compared with the first four PCs.To analyze population structure, principal component analysis (PCA) was performed on the average dosage matrix s on PCs . GroupinP. arundinacea samples). The numbers of selected PCs for those subsets were 0, 0, 2, and 1, respectively (data not shown). The relevant PCs in AP were consistent with race and geographical origin, respectively, in the same way as in the whole data set. The relevant PC in AP-4x was consistent with geographical origin.PCA was also performed for four other subsets of the data: LP1, LP2, AP, and AP-4x , in absolute value, averaged over imputes. Such value aims at assessing to what extent it is appropriate to use β based on the hypothetically complete data set. It follows from the MCAR assumption that imputations are unbiased if estimates of marker effects from (AD), 2 = 0.94). However, for marker-trait associations with moderate-to-high imputation uncertainty β and opportunities for false positives or false negatives  to (MI). This behavior results in higher p values (lower significance) for those associations in which imputation uncertainty is high: whether associations inferred in (AD) are true or false, their p values will tend to fall above the significance threshold in (MI) if their among-impute variability is too large, but there is good agreement between (AD) and (MI) for low values of γ  to (MI) is shown.The purpose of MI is to make sound inferences when one bases statistical analyses on imputed data. Therefore, MI should produce more conservative estimates The plot  and the The plot  of β^MI ues of γ . Only foX, but also the principal component and the genetic relationship matrices Q and K (both estimated from X) are allowed to vary over imputes. Although considering Q and K as fixed (estimated by ) was convenient for studying the behavior of inferences from (CC) to (AD) to (MI), (MI*) should permit the safest inferences by appraising the imputation uncertainty in the imputed genotypes as well as in the resulting Q and K estimates. β estimates from (MI) to (MI*), despite the relatively low consistency of Q estimates across imputes  than in (MI) (γ were quite consistent from (MI) to (MI*), with increased noise around perfect concordance for values of γ above 0.2 (In (MI*), the marker data matrix  imputes . Also, m in (MI) . Values bove 0.2 .i.e., markers with a modest missing data problem) there were generally small differences in significance γ is a good guideline for setting apart inferences with excessively high imputation uncertainty.As γ and the apparent usefulness of imputation for increasing power, it would be helpful to identify the factors that allow the analyst to effectively predict γ and determine how likely imputation is to generate gains in power. γ and proportion of missing values (PMV), MAF, or the average mutual information (AMI) between given markers and all other markers in the data set. There seems to be a strong positive relationship between γ and PMV, with values of γ that are below 0.2 for markers that have up to approximately 25% of missing values. As PMV increases, the slope for γ decreases and the variability around the conditional mean, as determined by a smoothing curve, increases. There seems to be a weak correlation between γ and MAF, with markers having high MAF showing slightly higher imputation uncertainty, especially for MAF <0.1. However, this relationship might be an artifact from markers with low PMV, which tended, by happenstance, to have lower MAF. There seems to be some (nonlinear) relationship between γ and AMI, but it is not clear whether it is due to the fact that markers with very low AMI tended to have lower PMV on average: for AMI <0.02, average PMV is 0.45 (SD 0.32); for AMI >0.02, average PMV is 0.80 (SD 0.071). A simple additive model fitted to the data with 2CV, prediction reliability in 10-fold cross-validation, was 0.21), but a rather large part of the variation could not be accounted for , marker effects estimated from (MI*) were closer to zero and significance of associations decreased, certainly as a result of the shrinkage of marker effects as well as the high among-impute variance. These marker-trait associations that lost significance from (CC) to (MI*) generally had a high PMV (above 0.69). However, associations with Significant associations, for which FDR <0.1 in (CC) or (MI*), were detected in the AP-4x subset and in the LP2 subset; no significant association was detected in LP1. These involved nine markers (thereafter “significant markers”): TP140584; TP184396; TP191264; TP217634; TP268059; TP341988; TP477925; TP521945; and TP87762. The one association involving TP341988 was detected only in LP2 , whereas the others were detected only in AP-4x. As shown in p values for (CC) suggests that potential confounders (due to population structure and relatedness in AP-4x) were well accounted for in the GWAS models  . This grBrachypodium distachyon genome (v1.0) (Brachypodium distachyon genome: (i) TP184396, shown to be negatively associated to Lignin , is in a region homologous to the last intron of an arginine-tRNA ligase gene ; (ii) TP268059, shown to be negatively and positively associated with CP and XYL_Eff, respectively, is in a region homologous to the only exon in a phenylalanine/histidine-ammonia-lyase gene ; and (iii) TP341988, shown to be negatively associated with GLC in LP2, is in a region homologous to the second exon of a translation elongation factor G gene . The moderate significance of TP184396's alignment to the B. distachyon genome may be due to the fact that the marker is located in an intron, which is usually more likely than exons to be under low selection pressure and diverge substantially across species.For each marker, a 64-bp read containing the corresponding SNP was obtained from the UNEAK pipeline. The reads corresponding to the nine significant markers were analyzed for homology with the e (v1.0) . Three oe.g., calling individuals homozygous for a SNP if >80% of their reads is of a particular allele) or probabilistic methods (returning posterior probabilities of genotypes), which rely on some estimates of error rate and population allele frequency at SNPs , should be pertinent: although computationally intensive, the proposed imputation procedure was flexible, easy to implement, and appropriate for modeling marker data, as was suggested by Here, no reference genome or genetic map was available for imputing the GBS marker data. Although it was possible to generate genetic maps on LP1 and LP2, the proportion of missing values was such that, in both populations, parental SNP genotypes could not be determined clearly  and imputation uncertainty would have to be accounted to produce reliable genetic maps, which would have been a serious statistical and computational challenge. Consequently, imputation methods based on HMM, which can be very accurate, were not used. In presence of unordered marker data with a general pattern of missingness and no reference panel, the strategy described here to impute missing values  , Table 7γ lower on average than 0.3 would correspond to PMV lower than approximately 0.45 according to the model presented in Values of This GWAS revealed associations of markers with multiple traits, which may indicate pleiotropy of one single tagged causal variant, LD between distinct causal variants affecting different traits, or genetic correlation among traits. For example, TP87762 was negatively associated with TSC and positively associated with Ds, St, ARA, and Mg. This result would probably suggest that, through lower tiller density, reed canarygrass plants carrying the nonreference allele at TP87762 were less prone to disease and lodging and had higher concentration of arabinose and magnesium.Significant markers could not be mapped to a particular region of the reed canarygrass genome, because no reference map or genome sequence is available for that species. However, the significant markers identified here may be used in further studies making use of DNA sequences in reed canarygrass. Potentially, our studies may bring some insight about the function of genes in related species such as Brachypodium or oat. However, actual causal genes were probably not directly tagged by significant markers. Markers that did not map to the Brachypodium genomes  were probably in LD with unmarked functional regions, which possibly could have been mapped to the Brachypodium genome. TP184396 and TP 341988 could be mapped to the Brachypodium genome, but within genes with very general purposes (both matched genes involved in mRNA translation). Such results could be “true hits,” but this does not seem very likely. However, TP268059, negatively associated to CP and positively associated to XYL_Eff, was mapped to the exon of a gene coding for Phenylalanine/Histidine-Ammonia-Lyase. Although it is entirely possible that TP268059 did not directly tag the true causal gene, it is plausible that TP268059 is a true hit: Phenylalanine-Ammonia-Lyase (PAL) is involved in the very first step of the monolignol biosynthetic pathway leading to the synthesis of lignin . Howeverand marker effects are large, which is consistent with the results obtained here (K matrix), and uncertainty in covariates cannot be conveniently accounted for, as was done in (MI*) here. Although in human GWAS such models may be appropriate  here , such sined here . A more is|Xobs) , featureis|Xobs) . Unfortui.e., the variability among imputed values is equivalent to what it would be if the data had been complete. Here, we made the MCAR assumption stating that no factor  influenced missingness at the marker data  and the information available for imputation (no reference panel and no genetic map of markers), imputation uncertainty was too prohibitive for detecting novel significant markers when performing imputation-based association tests. Gains of significance were nonetheless possible, mostly for low values of γ , correspγ . Pasaniuq on the order of q, memory usage can be reduced by replacing the adjacency matrix with a e.g., 500 in this study) or determining the set of candidate predictors at each iteration with no storage, hence trading computational efficiency for memory efficiency. Both of these measures would involve modifications of the mice code. Importantly, if a (completely typed) reference panel H is available for imputation, then one can base imputations of H only, i.e., k, there would be no uncertainty about values at the predictors to account for . As a result, the multiple imputed datasets in MI could be sampled by ordinary MC instead of MCMC. This would dramatically reduce the computational and memory requirements when implementing MI. Note also that when imputing at markers that are completely untyped in X (in silico genotyping), basing imputations on When dealing with marker datasets that are much larger than the one considered here (numbers of markers utations . Thus, i


In many traits, not only individual trait levels are under genetic control, but also the variation around that level. In other words, genotypes do not only differ in mean, but also in  variation around the genotypic mean. New statistical methods facilitate gaining knowledge on the genetic architecture of complex traits such as phenotypic variability. Here we study litter size  and its variation in a Large White pig population using a Double Hierarchical Generalized Linear model, and perform a genome-wide association study using a Bayesian method.Sus scrofa chromosome (SSC) 11. A possible candidate gene for TNB is ENOX1, which is involved in cell growth and survival. On SSC7, two possible candidate genes for varTNB are located. The first gene is coding a swine heat shock protein 90 (HSPCB = Hsp90), which is a well-studied gene stabilizing morphological traits in Drosophila and Arabidopsis. The second gene is VEGFA, which is activated in angiogenesis and vasculogenesis in the fetus. Furthermore, the genetic correlation between additive genetic effects on TNB and on its variation was 0.49. This indicates that the current selection to increase TNB will also increase the varTNB.In total, 10 significant single nucleotide polymorphisms (SNPs) were detected for total number born (TNB) and 9 SNPs for variability of TNB (varTNB). Those SNPs explained 0.83 % of genetic variance in TNB and 1.44 % in varTNB. The most significant SNP for TNB was detected on Sus scrofa genome involved in maintaining low variability of litter size, but further studies are needed to identify the causative loci.To the best of our knowledge, this is the first study reporting SNPs associated with variation of a trait in pigs. Detected genomic regions associated with varTNB can be used in genomic selection to decrease varTNB, which is highly desirable to avoid very small or very large litters in pigs. However, the percentage of variance explained by those regions was small. The SNPs detected in this study can be used as indication for regions in the  Conventional methods for studying the genetic architecture of complex traits focus on the level of those traits. In other words, the focus is on variation in trait level among genotypes. This implies that quantitative trait loci (QTL) can be defined as mean-controlling genes, as they affect the observed average phenotype of specific genotype. However, in many traits not only the mean is under genetic control, but also variation around the mean. Hence, genotypes not only differ in their average phenotypic level, but also in the variation around this average. In the following, we label this phenomenon as “phenotypic variability”, not to be confused with ordinary variability in average trait levels among genotypes. Therefore, development and application of statistical methods that allow studying phenotypic variability are required for a better understanding of the genetic architecture of complex traits –4.Arabidopsis thaliana [Drosophila melanogaster [Hsp90). This gene was described in Drosophila and Arabidopsis as a gene stabilizing developmental and morphological traits [The variation around the expected mean of a trait given its genotype, can be studied by analyzing the heterogeneity of residual variance across the observations . It has thaliana ) and aninogaster , rat [19nogaster  and in hnogaster , 21). Onl traits –24.Sus scrofa chromosomes (SSC) except SSC11, SSCX and SSCY  and QTL for the mean litter size of a genotype, and such QTL have been found on all igQTLdb, ). HoweveigQTLdb,  and shouigQTLdb, , 28. MorigQTLdb, –33. DecrigQTLdb, , RönnegåigQTLdb,  showed oigQTLdb, . A genomENOX1, which is involved in cell growth and survival. On SSC7, two possible candidate genes for varTNB are located. The first is a gene coding a swine heat shock protein 90 (HSPCB = Hsp90), which is a well-studied gene stabilizing morphological traits in Drosophila and Arabidopsis. The second is VEGFA, which is activated in angiogenesis and vasculogenesis in the fetus. We also found a positive genetic correlation between TNB and its variance, indicating that single-trait selection for TNB will increase the varTNB. To our knowledge, this is the first study reporting SNPs for TNB on SSC11 and SNPs associated with varTNB in pigs.The main objective of this study was to identify SNPs associated with litter size (TNB) and its variation (varTNB), through a multi-SNP GWAS applying a Bayesian method. In total, 10 SNPs were detected for TNB and 9 SNPs for varTNB. The most significant SNP for TNB was detected on SSC11 and for varTNB on SSC7. A possible candidate gene for TNB on SSC11 is The main objective of this study was to detect regions associated with litter size  and its variation (varTNB) in a Large White pig population using a genome-wide association study (GWAS). Prior to the GWAS, the phenotypes for the association study had to be obtained. Therefore, as the first step, a Double Hierarchical GLM (DHGLM) was used to estimate variance components and estimated breeding values (EBV) of TNB and varTNB. Second, the EBV obtained with DHGLM were deregressed. Finally, the deregressed EBV were used as phenotypes in the GWAS. In this section, we present and discuss all the results that lead to detecting regions associated with TNB and varTNB.Table The variance components and heritability for TNB from the DHGLM presented in Table SDe, i.e. the genetic standard deviation in residual standard deviation divided by the mean residual standard deviation of the trait (see Methods section for more details). The GCVSDe for varTNB in this study is 0.09, which is slightly lower than in previous studies focusing on litter size variation in pigs  of TNB can be reduced by 9 % in one generation.To show the potential response to selection we propose to use the Genetic Coefficient of Variation on the standard deviation level , asTable To investigate further the large difference between the permanent and genetic correlations obtained with the DHGLM, we also performed a conventional bivariate analysis. To compare methods, models need to be on the same scale. A DHGLM takes a logarithm of residual variance in exponential model. Thus, in the conventional analysis we used mean TNB and the log-transformed variance of mean TNB (log(var(TNB))) per sow. The estimated additive genetic variances for mean TNB and for log(var(TNB)) were similar to values obtained from the DHGLM Table . The conFelleki et al.  reportedSus scrofa chromosomes (SSC) as reported in previous GWAS for this trait [In total, 10 significant SNPs were detected for TNB Fig. . Associais trait . Since tOverall, the significant SNPs explained 0.83 % of the total genetic variance in TNB, and 1.44 % of the genetic variance in varTNB Tables . The SNPThe estimated genetic correlation between TNB and its variation 0.49) could suggest presence of pleiotropic effects or SNPs in linkage disequilibrium (LD). However, we did not identify overlap in regions significant for TNB and varTNB Figs  and 2. O9 could sThe two regions detected on SSC11 are the first SNPs associated with TNB on this chromosome. Thus far, no other study available in PigQTLdb (based on February 2015 search) reported significant associations for TNB on SSC11. Only one study reported QTL for a reproduction trait within the region of the most significant SNP (ASGA0050328) for TNB, which was a QTL for number of teats .ENOX1 was found at 24.16-24.48 Mb. One of the SNPs associated with TNB (with BF = 10.2) was located in this region. The ENOX1 is a protein coding gene from the ecto-CNOX family being part of electron transport pathways associated with mitochondrial membranes [ENOX1 are cellular defense and growth as well as cell survival [ENOX1 indicate that this gene might be a new region relevant for TNB in pigs.No candidate genes could be identified within the region of ±50kbp around ASGA0050328 . The nearest candidate gene named embranes . Functiosurvival . The funLGALS8 within the detected region. The LGALS8 is widely expressed in tumoral tissues and seems to be involved in integrin-like cell interactions, cell-cell adhesion, cell-matrix interaction and growth regulation [In addition, the region detected on SSC18 (58.86-58.88 Mb) shows relevance for TNB in pigs. Three QTL related to reproduction traits were previously described within this region  CUL9 is a cytoplasmic anchor protein in complex associated with p53 [CUL9 is controlling the localization and the function of p53 in the cell [CUL9 was not yet described in swine, its functions can be important in affecting litter size variability in pigs, especially since CUL9 is expressed in embryonic, placental, and uterus tissues in the human [Within the region of the most significant SNP INRA0025193) for varTNB at 43.76 Mb on SSC7, one candidate gene was located named 93 for vawith p53 . The p53the cell . Even thhe human .HSPCB (SSC7: 45.11-45.12 Mb) and VEGFA (SSC7: 44.46-44.47 Mb). The first gene belongs to the Sus scrofa heat shock protein family. This protein family is referred to as molecular chaperones since they are activated under various stress condition, such as heat [Drosophila and Arabidopsis as Hsp90, it is well described as a gene stabilizing developmental and morphological traits. The Hsp90 was describe to buffer environmental  and genetic (e.g. unfavorable mutations) factors, resulting in low variation of developmental and morphological traits [Hsp90 was shown to be one of many genes with buffering effects [VEGFA, is a vascular endothelial growth factor. A VEGFA is a protein mediator growth factor activated in angiogenesis and vasculogenesis in the fetus (and adult) [HSPCB) and provide the vascular network to the placenta (VEGFA).Two more SNPs on SSC7 associated with varTNB (with BF 10.2 and 17.5) were located within the regions of two other possible candidate genes already described in swine:  as heat , hyperth as heat , and inf as heat . Their f as heat , 59. Knol traits –24. Even effects , 61, it d adult)  as well d adult) . These tAn important aim in pig production is to obtain a high number of slaughter pigs per sow per year , 64, 65.We showed that residual variance in litter size has a genetic component, and can thus be changed by selection. The results of our study also show presence of an unfavorable positive genetic correlation between litter size and its variation. So far, main inefficiency in fecundity was the presence of too small litters. More and more, however, too large litters are becoming the prevailing problem. Therefore, simultaneous selection of litter size and it variation is necessary to achieve a higher mean litter size and at the same time a lower variance in litter size. This is important for pig production, since both small and oversized litters can be detrimental for farm economy. Although the genetic correlation was 0.49, we did not identify overlap in regions significant for TNB and varTNB Tables .Current breeding programs can use the knowledge of this study in the genomic evaluation of selection candidates. Genomic selection can greatly increase accuracy of selection also for non-phenotyped individuals. This is beneficial for traits such as litter size variability, which has low heritability and is recorded only long after the moment of selection of the candidate.ENOX1, is involved in cell growth and survival. Also on SSC18, another possible candidate gene for TNB is located, named LGALS8. Two genes located on SSC7 (HSPCB and VEGFA) are the most promising candidate genes identified for varTNB. The HSPCB is coding the heat-shock protein involved in buffering environmental and genetic factors, whereas VEGFA is activated in angiogenesis and vasculogenesis in the fetus. We also found a positive genetic correlation between TNB and its variation. This indicates that in breeding practice simultaneous selection of those traits is necessary to achieve a higher mean litter size and lower variance of this trait. The SNPs detected in this study can be used as an indication for regions in the Sus scrofa genome involved in maintaining low variability of litter size, but further studies are needed to confirm causative mutations.To our knowledge, this is the first study reporting SNPs for TNB on SSC11 and first SNPs associated with varTNB in pigs. In total, 10 SNPs were detected for TNB and 9 SNPs for varTNB. The most significant SNP for TNB was detected on SSC11. A possible candidate gene for TNB on SSC11, named Data for this study were collected between February 1998 and July 2014. In total 264,419 litter size  observations were available from 69,549 Large White sows. Litters were kept in the data if they contained at least 4 piglets in TNB , whereas litters of 27 piglets or larger were all considered “27” (43 litters). Most sows had repeated observations; only 15,379 sows had a single observation. Number of parities recorded per sow varied between 1 and 16, with an average of 3.8 per sow, and with 974 sows with 10 or more parities recorded. The parities 10 and higher were put on the same level of explanatory variable in the model . After data editing 263,088 litters from 69,238 Large White sows remained for the analysis. The average TNB in edited data was 13.5 (±3.5). The pedigree was traced back 5 generations if available and consisted of 83,571 animals.The conventional univariate analysis of TNB was performed in ASReml 2.0  using thy is a vector of observation on TNB in the litter; b is a vector of fixed effects (parity of the sow and farm_year_season of the farrowing) on y; a is a vector of random additive genetic effects on y, with a ~ N; pe is a vector of random non-genetic permanent sow effects on y, with pe ~ N; and e is a vector of residual, with e ~ N.where The variance component estimates obtained with the conventional univariate analysis were used as starting values in the Double Hierarchical Generalized Linear model (DHGLM).The conventional bivariate analysis of litter size and its variation was performed on average TNB per sow (meanTNB) and log-transformed variance of TNB per sow (log(var(TNB))). The log-transformation was necessary to obtain results on the same scale in the conventional analysis and in the DHGLM. In the conventional bivariate analysis, only sows with 3 or more parities were used to allow proper estimation of the variance. In total, observations on 43,490 sows were used. The bivariate analysis was performed in ASReml 2.0  with modymean is a vector of observations on average TNB per sow and yv is a vector of the log-transformed variance of meanTNB; bmean and bv are vectors of fixed effects of farm_year_season of the farrowing on ymean and yv; amean and av are vectors of random additive genetic effects on ymean and yv, with e and ev vectors of residuals, with W* is the weighting factor based on the number of litters the sow had. The conventional bivariate analysis has no permanent sow effect, since there is only a single observation per sow.where ei2 is the squared residual from the level part of the model for observation i, and hi is the leverage, being the diagonal element of the hat matrix of y corresponding to observation i. A log link function was used, because χ2-distributed with one degree of freedom. Felleki et al. [i and ψ is a vector with the response variable in the variance part of the model. Note that ψi is a linearized working variable for log(φi). The DHGLM is then as follows:Estimation of residual variance was performed on the full data set. A Double Hierarchical Generalized Linear model (DHGLM) as presented by Rönnegård et al.  allows ei et al.  showed ty is a vector of observations on TNB in the litter and ψ is a vector of response variables for the variance part of DHGLM; the residuals e and ev are assumed to be independent and normally distributed, but with heterogeneous variances across the observations; b and bv are vectors of fixed effects (parity of the sow and farm_year_season of the farrowing) on y and ψ; a and av are vectors of random additive genetic effects on y and ψ, with pe and pev are vectors of random non-genetic permanent sow effects on y and ψ, with e and ev vectors of residuals, with e2 and σev2 are scaling variances, which are expected to be equal to 1 [ψ in the previous iteration of the algorithm. The method iterates the bivariate model a number of times until convergence, since the residual variance part (varTNB) depends on the level part (TNB) of the model and vice versa. In this study, 22 iterations were needed.where mentclasspt{minimaSDe, a measure of ability to respond to selection) was applied to transform the estimates from the variance to the SD level. The GCVSDe is calculated as the genetic standard deviation in residual standard deviation divided by the mean residual standard deviation of the trait: av is the genetic SD in residual variance. The GCVSDe shows the proportional change in residual SD, when the residual variance would be changed by one unit σav. This allows seeing the magnitude of potential response to selection in the SD of TNB. Note that in Hill and Mulder [SDe. The literature on genetic analyses of residual variance defines also the heritability for residual variance at the level of the squared phenotype (hv2), which is equal to the genetic variance in residual variance as a proportion of the phenotypic variance of P2 and equals: hv2 = σav2/(2σP4 + 3(σav2 + σpev2)) [hv2 is a measure of reliability of EBV obtained on a single observation per animal; in contrast to classical heritability, it does not reflect the potential of the trait to respond to selection.A DHGLM uses the log-transformed variance of the trait. To help with practical interpretation of the results, two measures will be used: Genetic Coefficient of Variation at standard deviation level and the heritability for residual variance. The Genetic Coefficient of Variation on standard deviation level (GCVmentclass2pt{minim σpev2)) . The hv2Genotypes below). In addition, the boars had only observations through their daughters’, sisters’ and mothers’ performance. The use of deregressed EBV of animals instead of their phenotype is expected to give more reliable results in genome-wide association study; since accounting for offspring and parents information increases the power of the GWAS [In this study, a large number of phenotypic observations for TNB were available, but much lower number of sows and boars was genotyped . The EBV from univariate analysis were used to avoid EBV for one trait to be affected by the other trait in the bivariate analysis. The deregressed EBV obtained with the DHGLM for TNB, were also compared with those from the conventional bivariate analysis. The correlation between deregressed EBV for TNB from both methods was 0.988.Deregressed EBV were obtained based on EBV from two univariate analyses of the traits TNB and varTNB using the results from the final iteration of the DHGLM  or with unknown position on build 10.2. After quality control, 40,969 out of 64,232 genotyped SNPs remained in the data set. In addition, animals were removed from the data set if their call rate was <95 % and if pedigree or genotype led to many Mendelian inconsistencies .The deregressed EBV for litter size variability have overall low reliabilities, due to the low heritability of that trait. To maintain a sufficient number of genotyped animals for the genome-wide association study (GWAS), a threshold of 0.05 was used as an acceptable reliability of the deregressed EBV of the animal. Nonetheless, this caused a difference in number of animals used in GWAS for TNB and varTNB. Subsequently, 2,351 animals remained in the set with litter size observations and 2,067 in the set for residual variance of litter size.Multi-SNP genome-wide association was performed using a Bayesian Variable Selection method , which ey is an n-vector of deregressed EBV for the litter size or its variation on n animals; μ is an n-vector equal to the mean; Xb is a matrix with dimensions n by p, where p SNPs are coded as 0, 1, 2 copies of specific allele vector; and β is a p-vector with the markers effects fitted as random effects; e is an n-vector of weighted random residual effects assumed to be normally distributed N, where Wt is a diagonal matrix with wt1,…,wtn elements. On the marker effect, the Bernoulli distribution was applied:where 1 = 0.001, meaning that on average only 1 in 1,000 SNPs will be in the second distribution in each cycle. This allowed only ~41 SNPs per cycle to have a large effect on the traits. To secure that all the SNPs were used, 500,000 MCMC cycles were performed. Selecting a stringent prior provides a more precise distinction between SNPs with large and small effects on the trait [where the first distribution refers to the null distribution and it is assumed that the SNPs have small effect , which yielded the same results as the analysis with prior of 0.001.The Bayes Factor (BF) was calculated for each SNP to determine the significant associations:1 and π0 are the prior probabilities and where π Raftery , the SNP Raftery  by enterhttp://www.animalgenome.org/QTLdb/pig.html).The dataset used in this study is available upon request. Contact Egbert Knol by e-mail: Egbert.Knol@topigsnorsvin.com. Furthermore, the SNPs detected within this study are submitted to an open access animal QTL database – Pig QTL database (


The elusive but ubiquitous multifactor interactions represent a stumbling block that urgently needs to be removed in searching for determinants involved in human complex diseases. The dimensionality reduction approaches are a promising tool for this task. Many complex diseases exhibit composite syndromes required to be measured in a cluster of clinical traits with varying correlations and/or are inherently longitudinal in nature . A multivariate approach for detecting interactions is thus greatly needed on the purposes of handling a multifaceted phenotype and longitudinal data, as well as improving statistical power for multiple significance testing via a two-stage testing procedure that involves a multivariate analysis for grouped phenotypes followed by univariate analysis for the phenotypes in the significant group(s). In this article, we propose a multivariate extension of generalized multifactor dimensionality reduction (GMDR) based on multivariate generalized linear, multivariate quasi-likelihood and generalized estimating equations models. Simulations and real data analysis for the cohort from the Study of Addiction: Genetics and Environment are performed to investigate the properties and performance of the proposed method, as compared with the univariate method. The results suggest that the proposed multivariate GMDR substantially boosts statistical power. Many complex disorders such as asthma, diabetes, cardiovascular disease, Alzheimer’s disease, hypertension, mental disorders, and drug addictions are generally multifaceted phenotypes, measured by a set of scales and/or intermediate phenotypes that can be correlated as the outputs of a metabolic network with many interwoven pathways On the other hand, as a natural property of complex networks and the widespread intermolecular dependence in gene regulation and biochemical and metabolic systems, the presence of interactions appears to be the norm rather than the exception. Ever-growing evidence has pointed to the view that risk factors act in concert, rather than isolated, to affect complex diseases–the joint actions or interactions of multiple genetic and non-genetic factors play an important role in the etiology of complex diseases The data reduction approaches such as the multifactor dimensionality reduction method (MDR) The datasets used for the analyses described in this article were obtained from the database of Genotypes and Phenotypes (dbGaP). All the study protocol and forms/procedures were approved by the Institutional Review Board of the University of Alabama at Birmingham.m attributes either genes or discrete environmental factors are chose to span an m-dimensional contingency table. Each subject is allocated into a cell in this m-dimensional space based on the observations on these attributes and every nonempty cell can be labeled as either “high-risk” if the ratio of cases to controls in the cell is larger than a pre-specified threshold or “low-risk” otherwise. A new dichotomous attribute  is formed by pooling the high-risk cells and the low-risk cells into the high-risk group and the low-risk group, respectively, thus changing the space of the data from originally higher dimensions to one dimension. The resulting model is evaluated in its ability to classify the phenotype; accuracy, defined as the proportion of the correct classifications , is a commonly used measure. Cross-validation and/or permutation testing can be integrated into the above process for evaluation of model, and the optimal subset(s) of features can be selected in terms of the classification ability measured by accuracy or its derivatives such as p-value.In the original MDR method for a case-control study While sharing the same variable construction algorithm as in MDR, GMDR uses a general statistic, instead of affection status, to classify the two divergent groups. The statistic of an individual corresponding to a certain cell in a given contingency table can be generally expressed as the product of its membership coefficient belonging to this cell and its residual under the null hypothesis, which will be elaborated in the following subsections, respectively.Multivariate traits can be represented by an appropriate statistical model corresponding to data nature; generalized linear model (GLM) The linear predictor can be used for various scenarios. For a repeated measurement study, In application to detection of overall effects and/or pleiotropic effects on multiple traits, the components of The residuals can be computed under the null hypothesis of no target effects, i.e., A GLM is characterized by three parts: the response distribution, the linear predictor, and the link function between the linear predictor and the mean of the response variable. Specifically, the density function of a GLM has such a general form,i, The score of likelihood for a set of independent observations The GLM can be fitted by maximum likelihood (ML) estimation method. Note that estimation of The unknown scale parameter Under the null hypothesis of no target effects  of interest and the target factor(s), offering a possibility for variable construction to create plausible new attributes that maximize the residual phenotypic correlation. When the null hypothesis holds true, the association is expected to be zero and the classification model is formed purely by chance. The variable construction process is illustrated with the In Step 1 the data are randomly split into The accuracy is defined by,The statistic is by nature quantitative. However, the above accuracy, measured by the proportion of correct classifications in all classifications, only makes use of the qualitative information in the statistics, i.e., whether or not the statistic value of an individual in the high-valued  group is larger (less) than the threshold. As an alternative, the accuracy may also quantitatively be measured by the differences in the averaged statistic values between the high-valued and the low-valued groups AAbb, AaBb and aaBB were considered as high-value groups and the rest as a low-value group), 3 uppercase-letter model (3ULM) , 4 uppercase-letter model (4ULM) , in which the marginal effects of each disease locus were close to zero. For each individual, two continuous traits were simulated conditional on genotype data, interaction model, heritability of each trait and correlation between traits. Phenotypes were generated based on the following multivariate linear model,i on phenotype j, and To verify the validity and demonstrate the high statistical power of the new approach, we performed extensive simulations for continuous phenotypes. To simplify our exposition, here we considered a total of 10 unlinked loci with equi-frequent causative alleles, including two, three, and four disease-causing genes. Hardy-Weinberg equilibrium and linkage equilibrium were assumed in the simulations. Genotype data were simulated for 1,000 individuals with minor allele frequency of 0.3. Bivariate traits were simulated under the digenic espistatic , trigenic , and tetragenic  interaction models commonly used in simulation studies, for example antidiagonal model  were computed based on To illustrate use of the GEE-GMDR approach proposed here, a real data set from the Study of Addiction: Genetics and Environment (SAGE) was analyzed to identify interactions among genes. A large proportion of SAGE samples were unrelated except a few siblings. After quality control, a total of 3,897 individuals from three subsamples: the Collaborative Study on the Genetics of Alcoholism (COGA) , the Collaborative Study on the Genetics of Nicotine Dependence (COGEND)  and the Family Study of Cocaine Dependence (FSCD)  were obtained. Using Illumina Human 1M platform, 1,069,796 SNP markers were genotyped for each participant. Self-reported ethnicities indicate that about 35% of the participants are black and 65% are white. Detailed genotype information and demographic characteristics of SAGE cohort can be obtained from the database of Genotypes and Phenotypes (dbGaP) through dbGaP accession number phs000092.v1.p. Three common different measurements of ND were selected from the recorded traits: the lifetime score on FagerstrÖm Test for Nicotine Dependence (FTND), the DSM4 Nicotine Dependence (DSM4ND) and the largest number of cigarettes smoked in 24 hours (MC).p<10−7 using PLINK software α4 subunit (CHRNA4), 3 in the nAChR β2 subunit (CHRNB2), 56 in the neurotropic tyrosine kinase receptor 2 , and 18 in the brain derived neurotropic factor (BDNF) were chosen to detect gene-gene interactions among the four genes. In total, 15,120 (5×3×56×18) tetragenic interactions with one SNP from each of the four genes were examined.We excluded SNPs that had missing genotype rate >0.1, minor allele frequency <0.05 and a Hardy-Weinberg equilibrium test p-value can be approximately calculated under the null distribution by the approximated Z score, which is Owing to the fact that self-identified ethnicity often partially reflects one’s genetic ancestral origins, especially for populations that have complicated migration or admixture histories, the principal components analysis was performed using GCTA software for the SAGE data in which both unrelated samples and relatives are included to identify the population structure The Type I error rates under no residual correlation are presented in In summary, GEE-GMDR method maintains appropriate Type I error rate and thus is a valid test. The comparison of the multivariate analysis of GEE-GMDR with the univariate analysis of GMDR to identify combinations of multiple target genes demonstrates that the GEE-GMDR has higher or at least equal power in most situations. The GEE-GMDR method is able to substantially improve test accuracy, cross-validation consistency and power with inclusion of covariance structure of multiple phenotypes in GEE model.p value of 2.62e-04 and test accuracy of 0.5780, supporting our hypothesis of a possible interaction among CHRNB2, NTRK2, BDNF, and CHRNA4 underlying ND . The GEE-GMDR method detected the most significant tetragenic interaction model (rs2072660- rs1209068- rs11030134- rs6011770) with an empirical lying ND . DetaileCHRNA4NTRK2 and BDNFNTRK2 to BDNF regulates the short-term synaptic functions and long-term potentiation of brain synapses. Biological interactions of BDNF with NTRK2 and CHRNA4 with CHRNB2 have been constructed experimentally under vitro and animal models. Statistical interactions among CHRNA4, CHRNB2, BDNF and NTRK2 underlying ND had been discovered in our previous studies Recent evidence revealed genetic associations between ND and Identification of epistatic and/or gene-environment interactions underlying complex traits variation is one of the important and challenging tasks in human genetics and epidemiology studies. In the literature, joint actions of multiple factors are the norm rather than the exception in the genetic architecture of complex traits Complex diseases can be multidimensional and defined by a set of intermediate phenotypes that cluster together and are correlated. Complex traits can often be grouped into symptom groups. In some cases, data are collected from longitudinal studies. Using a univariate analysis, instead of multivariate analysis, will substantially reduce the power or likely yield a misleading result. It is desirable to to extend GMDR method for multivariate data to analyze complex multifaceted disorders and longitudinal data and/or toincrease test power. Here, we have proposed a natural extension of GMDR to multiple phenotypes based on GLM, QLM and GEE models in terms of the nature of data available, named multivariate GMDR approaches. In GLM, we need to specify all the probability distribution, the relationship between the mean and the variance, and the covariance. In QLM, we only need to specify the relationship between the mean and the variance, and the multivariate correlation structure. GEE model requires only the relationship between the mean and the variance of the marginal distribution. GEE, the extension of GLM and QLM, is more flexible, and has been widely applied to correlated multivariate data because of its robustness and no need for specifying the exact covariance structure One of the key advantages of the proposed method, as shown in both the simulation and the real data analyses of this study, is that this new multivariate approach is more powerful via borrowing information across multiple correlated phenotypes. We have compared the power of GEE-GMDR with the univariate GMDR method through Monte Carol simulations under the digenic, trigenic and tetragenic interaction models. Remarkable increase in power has been observed using our proposed multivariate method in the simulations, especially for trigenic and tetragenic models. The lower power of identifying high order interaction may mostly be attributed to smaller sample size, low allele frequency and low genetic variance The proposed approach has been applied to the SAGE sample with inclusion of multiple phenotypes related to ND. Conventional single-marker methods that separate interacting genes from their context fail to interpret the whole genetic architecture and seem to be inefficient. It is appropriate and powerful to identify high order gene-gene interaction underlying the ND, which is controlled by a large number of genes with a modest effect size, as demonstrated in our study and previous reports Very recently a multivariate GMDR method based on GEE was just proposed in the literature et al have developed a graphics processing unit (GPU)-based GMDR program (GMDR-GPU) As is the same as the original GMDR, computational limitations for high dimensional interactions still remain with our proposed method, especially for identifying high order interaction for data of the genome wide association studies (GWAS). However, Zhu GEE statistic used in our GEE-GMDR approach makes use of all dimensions of data but sometimes a part of them will dilute useful information. The possible directions for future research are to explore constructing an overall or extracting few summary statistics by multivariate statistical techniques . Principal components may be useful, in particular, when the phenotypes of interest are highly correlated and/or highly dimensional, i.e., nearly collinear or degenerated multivariate data. The principle is to construct the test statistic based on the first few principal components from a principal components analysis or canonical correlation analysis, instead of all components. Principal components analysis or canonical correlations, as a kind of statistical method of linear transformations, are effective to reduce the number of phenotypes surveyed.


The genetic disorder known as ‘crumbly’ fruit is becoming a serious problem in the European raspberry industry. The study set out to examine the crumbly phenotype in a red raspberry mapping population under two environments (field and polytunnel) across six seasons in an effort to understand variability of the syndrome and to examine whether genetic factors were important and if so, whether QTL associated with the phenotype could be identified. This highlighted that seasonal, environmental (field or polytunnel) and genetic factors all influence the condition. Two QTL that are important for the genetic control of the condition have been located on linkage groups one and three, and an association with ripening time has been identified.The online version of this article (doi:10.1186/s40064-015-1010-y) contains supplementary material, which is available to authorized users. Raspberry fruits are formed from an aggregation of multiple fertilized ovaries each of which are referred to as drupelets as they become fleshy. In the condition known as crumbly fruit, which has been linked with pollen abortion and embryo sac degeneration, drupelets are generally reduced in number but greatly enlarged or, in the case of small reductions, cohere imperfectly so fruit readily crumbles when picked  : crumbly (Stcr) : sterile (stCr or stcr) as st is epistatic to Cr. Cr were postulated to be linked to Gene H (pubescent canes) and also gene T (fruit colour)  of other years with the 2008 scores, which had the lowest level of crumbliness. The associations of other years with the 2004 scores, which were based on MP1 only, were also generally small. The associations among the scores on MP1 and MP2 in 2007, 2009, 2010, 2011 and 2012 were all highly significant , with a particularly high gamma statistic of 0.99 between the severity scores in 2011 and 2012.Table No progeny were always scored as crumbly, though some were assessed as being crumbly 75 % or more of the times scored. Some individuals never exhibited the crumbly phenotype. The crumbly phenotype was always more severe under the open field conditions than in the polytunnel and varied significantly from season to season, according to the over years and sites analysis. As well as the environmental and seasonal effect, the agreement over years measured by the gamma scores indicated a strong genetic effect for the crumbly trait. Met Office monthly weather data was examined on maximum and minimum temperature, frost, rain and sunshine but no associations could be identified between weather conditions and extent of the crumbly condition across seasons. For example in the two severe seasons 2004 and 2012 the weather conditions were very different, with 2004 being a warm dry season and 2012 cooler and wetter. There was a late spring frost in 2012 which did not occur in 2004. In 2008 where little crumbly fruit occurred, the major difference was in the amount of rainfall at the open flower stage.p = 0.05. The Kruskal-Wallis analysis indicated associations above this threshold of the crumbly phenotypes with markers on LG 1 for the field scores in 5 of the 7 seasons analysed . No significant associations with this region were detected in the field trials in 2004 (the first full fruiting year) or in 2008, when the incidence of crumbly was very low (4 %), or with any of the polytunnel trials, which had a much lower incidence of the condition. The most significant region included marker RUB256e, an SSR with four alleles  at 101 cM, although typically markers between 90 cM and 110 cM were significant. Figure p < 0.001) in the GLM, but the additive effect of the Moy parent and the dominance effect were not significant (p > 0.05). For each of these traits the direction of the effect was consistent, with the mean proportion of crumbly fruit being significantly higher in the genotypes carrying the Latham ‘b’ allele than in those with the Latham ‘a’ allele. For the field scores from 2004 and the polytunnel scores from 2007 and 2009, the additive effect of the Latham allele at RUB256e was significant (0.005 < p < 0.05) but again the additive effect of the Moy parent and the dominance effect were not significant (p > 0.05). Again, the mean proportion of crumbly fruit was higher in the genotypes carrying the Latham ‘b’ allele than in those with the Latham ‘a’ allele. The last two columns of Table Based on permutation tests, a threshold of 13.8 for the Kruskal-Wallis (KW) statistic with one degree of freedom was used, corresponding to a genome-wide significance of p = 0.003 for the polytunnel in 2007 and p < 0.001 for the other traits. ERubLR_SQ05.3_D11AOC also showed a lower significance in the GLM for traits 2004 Field B (p = 0.003), 2010 Field (p = 0.004), 2011 Field incidence (p = 0.017) and 2012 Field incidence (Field = 0.011). For all of these traits the direction of the effect was consistent, and there was no significant interaction (p > 0.05) between the two markers ERubLR_SQ05.3_D11AOC and RUB256e. Table The Kruskal-Wallis analysis also indicated associations above the genome-wide permutation threshold between the crumbly scores from the polytunnel in 2007, 2009 and 2010 and the field severity scores from 2011 and 2012 with markers segregating on LG 3, although the best marker varied slightly (region 107-133 cM). Again the significant markers segregated in the Latham parent. Figure A combined analysis over years and sites (field or polytunnel) was conducted on the binary scores, omitting the data from 2004 as this was only scored on the MP1 lines. The analysis of deviance table for the incidence of crumbliness is shown in Additional file p < 0.05) with some of the time to ripening scores recorded for this population in 2006 by Graham et al. ) were searched using BLAST  and Methyl transferase (AER13155.1) containing the Rub256e marker and those containing a polymorphism between the parents were mapped using primers in Table This study has highlighted that environmental, seasonal and genetic factors all play a role in the development of crumbly fruit in red raspberry. A region on LG 1 at the Rub256e marker, and an association with ripening time and ripening associated markers on LG 3 were identified for further analysis. No association with crumbly fruit and Gene H was determined. This work has allowed us to identify a genetic component to the condition which can be assessed for breeding lines less prone to crumbly fruit. Controlled environment studies will be carried out in an attempt to define triggers of the condition in those samples where the phenotype varies between seasons and environments.The population, as described previously  of 94 seedlings, and a further mapping population of an additional 94 seedlings (MP2). Phenotypic data on crumbly fruit, scored as crumbly or not according to Jennings http://www.fruithealth.co.uk).Raspberry Bushy Dwarf Virus (RBDV) testing was carried out as standard to ensure plants were free of the virus , the paternal additive effect (P2) and the dominance effect (D) can be derived for each offspring at each position as:pr(ac) is the probability that the offspring has genotype ac at that position, and these can be used in the linear model. For the binary traits here, the linear model was replaced by a generalised linear model (GLM) with binomial errors and a logit link function to relate the crumbly trait to the additive and dominance effects in the genetic regions identified by the KW analysis. The genetic predictors of the additive effects of each parent and the dominance effect were calculated at each marker position using the QIBDPROBABILITIES procedure of GenStat 16 for Windows  test was used initially to test each marker on the map for associations with the crumbly fruit scores for each year and environment, using the MapQTL 5 software (Van Ooijen pws.byu.edu)) were searched using BLAST  1, therefore the raspberry genome pseudomolecules (provided by Joshua Udall BYU, Genetics and Biotech Faculty (Graham et al.  studied


Medicago sativa) association panel comprised of 336 genotypes from 75 alfalfa accessions represented by four to eight genotypes for each accession. Each genotype was genotyped using 85 simple sequence repeat (SSR) markers and phenotyped for five fiber-related traits in four different environments. A model-based structure analysis was used to group all genotypes into two groups. Most of the genotypes have a low relative kinship (<0.3), suggesting population stratification not be an issue for association analysis. Generally, the Q + K model exhibited the best performance to eliminate the false associated positives. In total, 124 marker-trait associations were predicted (p < 0.005). Among these, eight associations were predicted in two environments repeatedly and 20 markers were predicted to be associated with multiple traits. These trait-associated markers will greatly help marker-assisted breeding programs to improve fiber-related quality traits in alfalfa.Association mapping is a powerful approach for exploring the molecular genetic basis of complex quantitative traits. An alfalfa ( Medicago sativa) is one of the most important forage crops in the world due to its high biomass and choice nutritional profiles and it provides reliable sources of protein and minerals to animals. The main objective, however, in many alfalfa quality breeding programs is to improve the digestibility , and joint-mapping by combining linkage and association analysis (Ed Buckler's NAM population in maize) are the three main methods for dissecting complex quantitative traits. Compared with the traditional linkage analysis based on mapping populations, association mapping, has been proposed as an alternative powerful tool to overcome limitations of pedigree based QTL mapping for it has higher mapping resolution, reduced research time, and greater allele number  were 0.9317 for ADF, 0.9634 for neutral detergent fiber (NDF), 0.6769 for ADL, 0.4950 for NDF digestibility , and 0.9001 for NDFD 48 h.Plants were tested under four environments . Plant leave tissue samples were ground and then passed through a 1-mm mesh screen . Aliquots of each sample were scanned by near-infrared reflectance spectroscopy. Measurements were obtained using a FOSS 5000 scanning monochromator  and recorded at 2-nm intervals between 1100 and 2500 nm. A subset of 100 samples was selected for the calibration of spectroscopy using chemical analyses. Coefficients of determination  was calculated as the genotypic variance divided by the total variance.Analysis of variance (ANOVA) of all phenotypic data based on the means of traits of each accession under four environments was conducted as model: Phen = genotypes + environments + e. where Phen as the phenotypic observation, genotypes as the genetic effect, environments as the effect of the four environments, and e as the residual. All analyses were performed using SAS 8.02  ranging from 2.48 to 8.13%  may be obtained.Most of the loci that were associated with the five traits could only be identified in a specific environment, indicating that the fiber-related traits in the study are variously influenced by the environment. However, some stable associations were identified in our study, such as the allele m350_342 which located in chromosome 1 were repeatedly detected in two environments and associated with ADF, NDF, NDFN30h, and NDFD48h. Three markers, m115_183, m520_134, and m520_137, located in chromosome 7 were repeatedly detected in two environments and associated with NDFD48h. Markers with significant traits associated over multiple environments may indicate that the associated genes are more stably expressed  (Ray et al., Among these associated alleles, different distributed patterns were observed among eight chromosomes in alfalfa. Eight alleles from seven markers which associated all five traits were observed in the chromosome 1, while only one allele of one marker which associated two traits was observed in the chromosome 6 Table . In the The present study is the first attempt in associating alfalfa fiber-related traits with the genotyping results derived from SSR markers using a diverse set of global collection of alfalfa genotypes. Our results demonstrated that this alfalfa panel is suitable for association mapping analysis targeting complex quantitative traits with optimal association models. The markers associated to the QTLs in the study can be effectively used in further alfalfa marker assisted breeding programmers for introgression of alleles into locally well adapted germplasm.ZW designed the experiments performed the statistical analysis and drafted the manuscript. HQ performed SSR genotyping. HZ, GL, ZZ, RX, and YZ conducted the quality analysis. XW and HG revised manuscript. All authors have read and approved the final manuscript.The reviewer HC declared a shared affiliation, though no other collaboration, with the authors RX and YZ to the handling Editor, who ensured that the process nevertheless met the standards of a fair and objective review. The other 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.